Testing MathJax-LaTeX

At first, we sample $$f(x)$$ in the \(N\) ($N$ is odd) equidistant points around \(x^*\):

\[
f_k = f(x_k),\: x_k = x^*+kh,\: k=-\frac{N-1}{2},\dots,\frac{N-1}{2}
\]

where \(h\) is some step.

Then we interpolate points \((x_k,f_k)\) by polynomial

\begin{equation}
\label{eq:poly} \tag{1}
P_{N-1}(x)=\sum_{j=0}^{N-1}{a_jx^j}
\end{equation}

Its coefficients \(a_j\) are found as a solution of system of linear equations:
\begin{equation} \label{eq:sys} \tag{2}
\left\{ P_{N-1}(x_k) = f_k\right\},\quad k=-\frac{N-1}{2},\dots,\frac{N-1}{2}
\end{equation}
Here are references to existing equations: \ref{eq:poly}, \eqref{eq:sys}.
Here is reference to non-existing equation \eqref{eq:unknown}.

The Curiously Inscrutable Principles of Trade Mechanics of Europa Universalis 4 (aka another EU4 trade guide)

Prologue

 

My goal here is to explain from first principles the fundamentals of trade mechanics in EU4.  This will not attempt to be a comprehensive guide on all aspects of trade, but it does attempt to be a comprehensive guide and tutorial on the mechanics of trade nodes and merchants.  I will try to explain the system from the ground up as well as communicate an intuitive understanding and interpretation.  I will attempt to be rigorous with the mechanics but will be more brush-strokey with tactics, strategy, and modifiers (since modifiers just complicate things and in application are a part of strategy).  My hope is to explain the basics well enough that the reader can contemplate tactics and strategy on their own.  I will also attempt my explanations to be very beginner-friendly and clear enough so that not only newbies but theoretically even people who haven’t ever seen or touched EU4 will understand the mechanics by the end of this (probably not possible).  (The exception is my brush-strokey section at the end on tactics and strategy, which assumes that you know the mechanics by then.)  This guide will start out very slowly with a leisurely pace with a lot of hand-holding in a tutorial style, so for people who already have a basic feeling for trade, skip to section “II.b The Trade Power of a Trade Node” to get to the meat of it or section “III Trade Value Flow in a Trade Node” to jump straight to the details.

Note: The screenshots use version 1.15 of the game.

 

 

0 Tooltips

 

In general, never blindly trust the tooltips and their numbers.  The descriptions are often more frustrating and confusing than informative.  However, I actually will use screenshots of tooltips where they actually are helpful or where I have been able to decipher them.  I will highlight those tooltips in screenshots as the key to playing this game (or any computer activity that shows you a bunch of numbers on the screen) is to stare at the numbers that are actually useful to the human, since often the screen in EU4 is just an onslaught of numbers that sometimes make sense and sometimes don’t.  Gradually, though, I will attempt to explain all the numbers on the tooltips that appear in the trade node screens and map mode.

 

 

I Trade Value and Trade Power.  And Trade Value =/= Trade Power.

 

Trade value =/= trade power.  These two terms are completely different things in EU4.  Imagine a town: the trade value of a town is the value of the goods it produces.  Let’s say the town produces salt.  A pile of salt sitting on the dock of the town is the town’s *trade value*.  A boat comes along to the dock.  That boat is going to carry that salt to be sold in the global salt market at the nearest big marketplace, which is located in a big city some distance away.  Let’s call that city the main trading city of the country that the town is in.  So the boat comes to the town, picks up the salt, and makes its way to the main trading city.  But who owns or operates that boat?  The answer to that question is *trade power*.  Whoever controls the means to sell that salt to someone else in the country or the world and the means to continue selling that salt as it gets sold and bought repeatedly across the world in different countries is who controls trade power.  More concretely, if you’re the ruler of an empire, if you control the points of sale of salt as it moves further away – dock by dock, market by market – from the town that the salt was produced in, you should be able to somehow earn money from that control over trade flow (think sales taxes at different market towns that salt is sold in).  That control – owning those boats and those market towns – is called having trade power.  Trade value is the value of goods produced by a town.  Trade power is the amount of control that a ruler and his or her country has over the flow of trade in and even beyond the country.

 

I.a Trade Value Produced by a Province

 

01f-tvtpFigure I.a.1: Town A produces salt.  That is town A’s trade value.  Boats take the salt from town A and move them to a market in town B.  Other boats then move them to town C, etc.  Who has control or jurisdiction over the activities of those boats (e.g. through taxing or regulating the merchants who operate the boats)?  Who controls the laws and taxes in these market towns of B and C?  That’s trade power.  Trade value is produced by a town.  Trade power is controlled by the ruler of the country.

 

01s-tv-granadaScreenshot I.a.1: Granada produces salt.  (Every province produces one and only one type of good.)  The game tells us that 1 unit of salt is worth 3.00 ducats (the bottom-right of the province info screen, under the picture of a pile of salt).  In the bottom-left, it says that the trade value of Granada’s production is 3.60.  Hover your mouse over that, and the tool tip tells us that this comes from 3.60 = 3.00 (value of 1 unit of salt) × 1.20 (units produced by the province).  Where does this 1.20 come from?

 

01-a-2s-tv-granadaScreenshot I.a.2: Granada’s base production.  Hover over where I’ve put a red rectangle and circle on that “6,” and the tooltip tells us that Granada has a base production of 6 and that each unit of base production gives 0.20 units of local production.  This is how Granada produces 1.20 units of goods.  But it doesn’t tell us if this is a monthly or yearly amount…

 

01-a-3s-tv-granadaScreenshot I.a.3: Granada’s production income.  Hover over the “Production   0.37” on the left (of course the number will be different on your screen).  How does it get 0.37?  The “Trade Value 0.30” is showing us the monthly trade value produced by Granada.  So 3.60 = 0.30 (Granada’s monthly trade value) × 12 is Granada’s yearly trade value produced.  The modifiers here are +25%.  Note that the total Production Efficiency here is +25%, of which there is just one type of modifier supplying that 25%, which is From Technology (yeah, the line saying From Technology is indented.  I.e. there are different modifiers that will add up to give you your total Production Efficiency.  Here, we only have From Technology giving us a Production Efficiency modifier).  0.30 × (1 + 25%) = 0.37, so 0.37 is the monthly production income from Granada’s salt.  And 3.60 × (1 + 25%) = 4.50 is Granada’s yearly production income from salt.

 

01-a-4-prod-incScreenshot I.a.4: In the Economy tab, the red box I highlighted shows the country’s total Production Income from all the country’s provinces.

 

Summary: A province’s trade value is measured in ducats and comes from the value of the trade good it produces multiplied by its production amount (which comes from its base production, which Granada has 6).  Modify by modifiers such as technology, production buildings (workshops), etc. and you arrive at the province’s production income.  The country earns this as revenue from this province every month (the 0.37 from above).

 

I.b Provincial Trade Power

 

While a province produces *trade value* via [trade good × base production] (× modifiers) as we have seen above, a province produces *trade power* by [province development + geographical bonuses + buildings + …] (× modifiers) where we’ll just call the first part flat bonuses, i.e. a province’s trade power = [flat bonuses] (× modifiers).

 

01-b-1-stp-sevprovScreenshot I.b.1: Trade power of the province of Sevilla.  Hover over the “Trade Power 120.4” in the bottom-left.  A province’s trade power comes from its flat bonuses to trade power, and then adjusted by modifiers.

Flat bonuses in Sevilla:

4.60 from “Development” comes from its total provincial development of 23 (boxed in red in the screenshot) and then divided by 5 (apparently, the game just assumes you like doing this kind of mental arithmetic);

10.00 from having an “estuary” comes from Sevilla’s geographical position – some provinces, like Sevilla here, are hardcoded by the game to have a boost in trade power;

another 10.00 from being a “coastal center of trade” is also hardcoded by the game for Sevilla here;

and 20.00 from Port to the New World is from an event (an exception here, so don’t worry about it).

Modifiers in Sevilla:

50% from giving the burghers estate control of the province (not important here);

50% from having a trade building (market place) built in Sevilla;

25% for Sevilla being a coastal province;

and some other modifiers that aren’t important to know what they are at this point (treasure fleet, mercantilism).

 

Summary: A province’s trade power comes from flat bonuses (development/5 + hardcoded “geographic” bonuses for certain provinces, usually called estuaries or coastal/inland centers of trade + etc.), which are then modified by trade buildings (e.g. market place) and other modifiers.  This gives us the provincial trade power created by a province.  (Note that there is no monthly or yearly rate of creation for trade power.  It’s just a number that represents the amount of “control” over the means of trade that that province provides for its country and ruler.)

 

I.c Trade Value =/= Trade Power

 

Note how there is almost no overlap here between a province’s trade value and its trade power.  (The only overlap is where base production determines both a province’s trade value (via amount of goods produced) and total province development (province development = base tax + base production + base manpower) and total province development divided by 5 is a flat bonus for trade power.  But really, this connection isn’t worth keeping in your mind).  So you can imagine that there can be provinces rich in trade value (e.g. produces tons of salt) but poor in trade power (e.g. doesn’t operate the boats that come to its docks to take the salt to the global trade network) and there can be provinces poor in trade value (e.g. produces a bucket of grain that no one will pay much for) but has a lot of trade power (e.g. located in a river delta, has a big market, i.e. all the boats in that region that collect salt in nearby provinces come to this province to offload their salt because foreign boats from faraway lands come here to buy that salt in bulk).

 

 

II Trade Node

 

There are two inherent parts to a trade node: its trade value and its trade power.  Note that in section I we started with a province’s trade value and trade power.  Now we’re going to trade nodes.

A trade node is basically a geographically partitioned region of the world.  Every province in the world is assigned to one and only one trade node.  So each trade node in the world is made up of a collection of neighboring provinces.  For example, the provinces of Granada and Sevilla and a bunch of other provinces on the Iberian Peninsula are part of the Sevilla trade node.  (The *province of Sevilla* is NOT the “capital” or “center” or anything like that of the *Sevilla trade node*.  The province of Sevilla is simply one province of many provinces that make up the trade node that’s located at the Iberian region.  That trade node just happens to also be called the “Sevilla trade node.”)

 

II.a The Trade Value of a Trade Node

 

So a trade node is made up of a collection of provinces.  The sum of the trade values produced in all the provinces in a trade node is the trade value of that trade node.  Another name for this is the “Local” trade value of the trade node.

 

02-a-1f-tntvFigure II.a.1: Trade Value (“Local”) of a Trade Node.  Let’s say our super-simplified Sevilla trade node has only 3 provinces in it: the Portuguese province of Algarve on the left (green), and the Spanish provinces of Sevilla in the middle and Granada on the right.  Add their (monthly) provincial trade values (0.24, 0.40, and 0.30), and you get the trade value of the Sevilla trade node, 0.94, aka the “Local” trade value of the trade node.  Note how it doesn’t matter which country owns those provinces.  What matters is only whether the province is a geographical member of the trade node.

 

02-a-1s-sevila-tn-screenScreenshot II.a.1: Trade Value of a Trade Node.  Go to the trade map mode and click on the Sevilla trade node.  Hover over where it says “Local    +7.29” on the left.

 

02-a-2s-sevilla-tn-tvScreenshot II.a.2: Trade Value of a Trade Node.  By hovering over the value of “Local,” we get a list of all the trade values of all the provinces in the Sevilla trade node.  We can find Granada’s trade value in the list, 0.30 monthly and 3.60 yearly.  The total monthly trade value of the Sevilla node contributed by the trade values of provinces in the node is 7.29.

 

Note that in a way, the trade value produced by a province has been double-counted in the game.  First, it turns into production income that the owner of the province earns directly (back to Screenshot I.a.3.):

01-a-3s-tv-granada

 

But it also enters the trade node that the province is a part of and becomes this entity called Trade Value or “Local” Trade Value of a Trade Node (back to Screenshot II.a.2.):

02-a-2s-sevilla-tn-tv

 

The simple way to think about this is to pretend that each unit of salt in the game actually represents two portions of salt.  One portion is sold locally for cash and the other portion is shipped to the nearest major trading area, i.e. the trade node that the salt-producing province is a part of, where it’s sold to traders who are part of the global trading market.  Note that the first portion of salt has been immediately and fully converted to cash while the second portion of salt has entered the global market so that it’s likely that it will be sold and bought and sold and so on by different people anywhere in the world that trade flows to.  Who knows where that portion of salt will end up?  Note that we – rulers of this salt-producing province (e.g. Spain in the screenshots) – have not seen the cash revenue for this second portion of salt yet.  That is to be earned at a later stage of the trade game.

(A more precise way to think about this double-counting in my opinion is to pretend that production income from producing a good (the part that’s immediately converted to cash) is actually a “production tax” that you charge producers in your provinces.  For every unit of salt produced in a province, you charge the salt-producing factory 3.00 ducats as a production tax.  The salt-producer then sells that salt to some traders on boats, who move it around the world (the salt becomes trade value in a trade node).  At some point along this trip around the world, a ruler can charge a sales tax on the transaction of salt between merchants, also at 3.00 ducats per 1 unit of salt.  We haven’t gotten to the part yet about where and who can charge this sales tax on salt transactions.)

 

II.b The Trade Power of a Trade Node

 

The trade power of a trade node is made up of different countries’ trade power in that node.  Here, it matters whose trade power a unit of trade power is (i.e. it matters now whether it’s Portugal’s trade power or Spain’s trade power in the Sevilla node).  Furthermore, trade power comes from different sources.  The different sources are:

1) provincial trade power (as mentioned above in I.b),

2) “trade power transferring/propagating upstream,”

3) light ships protecting trade in the trade node (if it’s a sea node), and

4) the presence or not of a merchant in the trade node (very small flat +2.0 trade power).

5) Also, if the trade node is your home trading node, you get +5 trade power there (your home trading node is the trade node that your province with your main trading city is in.  Your main trading city is your capital city unless you move your main trading city to another city later in the game.  The terms “city” and “province” are completely interchangeable in the scope that we’re discussing here).

6) Also, there are events that just seemingly randomly give you bonuses out of thin air.  So we’ll just ignore these in this guide.  (These are likely hardcoded by the game for certain countries, like Spain getting trade bonuses for exploring the New World).

 

II.b.1) Provincial Trade Power

 

Note that provincial trade power in a trade node can only come from your owning a province in that trade node.  Everything that was used to explain I.b gives us this number for each province.  So, just for now, pretend that there is only provincial trade power and ignore all the other sources of 2) ~ 6) from above.  The state of trade power in the simplified Sevilla trade node would be like this:

 

Figure II.b.1: Provincial Trade Power in the Sevilla node.  Portugal holds 13.6/139.1 = 9.8% and Spain holds 125.5/139.1 = 90.2% of the trade power in the Sevilla node.

 

II.b.2) “Trade Power Transferring/Propagating Upstream”

 

What the hell is “trade power transferring/propagating upstream?”  Different trade nodes are connected by links and each link has a direction of flow to it.  For example, the Tunis trade node is linked to the Sevilla trade node and has an arrow pointing from Tunis to Sevilla.  I.e., the Tunis trade node is upstream of the Sevilla trade node.

 

02-b-2-1s-tunis-sevillaScreenshot II.b.2.1: The Tunis node upstream of the Sevilla node.

 

This is the game hardcoding that “trade wealth flows from Tunis to Sevilla.”  To put it another way, “trade in Sevilla is more powerful than in Tunis,” and one way this is manifested in the game is by this “trade power transferring/propagating upstream” mechanic.  What happens is that countries who have (at least 10) provincial trade power in Sevilla will have 20% of that provincial trade power in Sevilla “propagated” to upstream nodes like Tunis.  So out of nowhere, Spain gets 125.5 × 20% = 25.1 trade power in Tunis and Portugal gets 13.6 × 20% = 2.72 trade power in Tunis just because the arrow on the trade map mode points from Tunis to Sevilla.

 

02-b-2-1f-upstream-propFigure II.b.2.1: Provincial trade power of each country in a downstream node (Sevilla) is “propagated upstream” (to Tunis).  Note that Portugal’s trade power of 13.6 > 10, so it just barely made the cutoff for this upstream propagation.  If Portugal’s trade power were 9, then it would get 0 trade power propagated upstream to Tunis.

 

Summary: Tunis is upstream of Sevilla.  “Trade wealth” flows downstream from Sevilla to Tunis, i.e. trade in Sevilla is “more powerful” than trade in Tunis.  So having provincial trade power in Sevilla automatically grants you 20% of that amount of trade power up in Tunis.

 

This amount of upstream propagation from the Sevilla trade node is the same for all trade nodes that happen to be upstream of the Sevilla node.  The Safi, Ivory Coast, and Caribbean trade nodes are also upstream of the Sevilla node, so Spain and Portugal would also respectively get 25.1 and 2.72 trade power in every one of these nodes in addition to the Tunis node.  There is no propagation effect from Sevilla to any nodes that are downstream of it, like the Genoa node.  The upstream propagation only goes upstream by one link.  Thus, Spain’s provincial trade power in the Sevilla node that propagates up to the Tunis node won’t propagate further up to the Katsina node even though Katsina is upstream of Tunis.  (If Spain has provincial trade power in Tunis, then that amount of provincial trade power would propagate upstream to the Katsina node.)

 

02-b-2-2f-sevilla-propFigure II.b.2.2: Upstream propagation is the same amount in all upstream nodes.

 

You can see how a country dominating (via having a lot of provincial trade power in) a trade node that is downstream to a lot of other nodes is powerful because that country will get bonus trade power in all those neighboring upstream trade nodes.  Note that it’s only *provincial* trade power in a trade node that gets this upstream propagation.

 

02-b-2-2s-sevilla-upstreamScreenshot II.b.2.2: Spain has 207.5 provincial trade power in the Sevilla node.  (That column tells us the amount of provincial trade power that each country has in the node.  I think the icon for the column is supposed to represent the shapes of provinces next to each other on a map.)

 

Screenshot II.b.2.3: Now, let’s look at the Safi node, which is upstream of the Sevilla node.  Mousing over the 166.5 in the right-most column (this column tells us the total trade power in the node of each country), we get a tooltip.  One of the items is “Transfers from traders downstream: +41.5”.  This comes from Spain’s provincial trade power downstream in the Sevilla node, where 207.5 × 20% = 41.5.  Spain having provincial trade power of 207.5 in the Sevilla node has given it 41.5 trade power in the Safi node.

 

02-b-2-4s-genoaScreenshot II.b.2.4: Looking at the Genoa node, here, Spain has 185.7 provincial trade power.  If you mouse over the blue arrow pointing left, it says, “Trade propagation reduction: -80%.”  I believe this is the game’s obscure, inscrutable way of telling us that there is 20% provincial trade power upstream propagation originating from here.

 

02-b-2-5s-tunisScreenshot II.b.2.5: Looking at the Tunis node.  The Tunis node is directly upstream of both the Sevilla node and the Genoa node.  Thus, Spain has 207.5 × 20% (from Sevilla downstream) + 185.7 × 20% (from Genoa downstream) = 78.6 trade power in the Tunis node from “upstream propagation.”

 

02-b-2-6s-sevilScreenshot II.b.2.6: Going back to the Sevilla node, if you mouse over the yellow plus sign, the tooltip tells us nothing about the upstream propagation that is happening from here.  Both Genoa in the screenshot above and Sevilla here are nodes that are causing upstream propagation.  We’re mousing over the same location in the table, Spain’s second column from the left, yet we’re not getting info about the upstream propagation here in the Sevilla node screen.  This is why you can’t trust the tooltips.  (We’ll get to what the plus sign and the green arrows mean later.)

 

II.b.3,4,5) Other Sources of Trade Power

 

If a country uses light ships to do a “protect trade” action in a trade node (possible only in sea nodes), that country will gain trade power in that trade node.  If a country places a merchant at a trade node (whether to “collect” or “transfer trade power”), that country gets a small (usually negligible +2.0) additional amount of trade power in that trade node.  (Which trade nodes you can do these are limited by, according to the wiki http://www.eu4wiki.com/Trade, things like existing trade power in that node and supply range (including fleet basing rights in other countries) for light ships and your country’s trade range for merchants.)  Finally, you get a bonus +5.0 trade power in your home trading node.  Every country has 1 and only 1 home trading node, which is the trading node your main trading city is located in, which is the same as your capital city at the beginning of the game (you can later change the location of your main trading city to be different from where your capital city is).

 

02-b-3s-sevillaScreenshot II.b.3: In the Sevilla trade node screen, in the “right half” of the table, you have, from the left column, each country’s trade power coming from light ships and merchants, each country’s provincial trade power, modifiers, and finally total trade power in the Sevilla node.  Hover over the total trade power of Spain and we get the breakdown, giving us the provincial trade power of 207.5, trade power from light ships of 154.5, (no merchant trade power here since I don’t have a merchant here right now), trade power of 37.1 that has propagated upstream from Genoa since Spain in this game has provincial trade power in the Genoa node (which is downstream of the Sevilla node), and trade power of +5.0 since Sevilla is the home trading node of Spain, plus modifiers (pretty much aka “global trade power,” a term the game uses elsewhere).

 

We also note that somehow, France, the Papal State, and Savoy have some trade power in this node, even though they have 1.) no provincial trade power and thus also 5.) can’t have their main trading city here to make it a home node, 3.) light ships, or 4.) merchants here (0s in those two columns).  Knowing all the above, we know that the only way they could have trade power here is by 2.) having provincial trade power in the Genoa node that’s propagating upstream to Sevilla or 6.) some other random bonuses, maybe from special events, who knows.

 

III Trade Value Flow in a Trade Node

 

We’re now going back to trade value (not trade power) in a trade node.  Let’s look at the Basra trade node.

 

03-1s-basraScreenshot III.1: The Basra node.  Basra has a link incoming from the Hormuz node (Basra is downstream of Hormuz) and outgoing links to the Aleppo and Persia nodes (Basra is upstream of Aleppo and Persia).

 

Note on the left the little table that lists Incoming, Local, Total, and Outgoing.  Whatever that all means…  But this is actually a crucial indicator that helps us understand what’s going on in a trade node.  These tell us the flow of *trade value* within a trade node.  “Incoming” is trade value that has somehow by some mechanic (to be explained soon) flown from upstream nodes (i.e. Hormuz) down into our node in question (Basra).  “Local” is what we figured out before in II.a: it’s the total trade value (aka “Local” trade value in a trade node) produced by provinces that are geographically located in this trade node.  Thus, “Incoming” + “Local” is the actual total trade value that exists in the trade node.  For Basra here, it’s 1.89 + 3.13 = 5.02 ducats.  (Ignore the “Total” label that appears in the table for now, as that will be explained soon.)

We’ve established that in each trade node, the total *trade power* in the trade node is the sum of the trade powers of different countries in that node.  A country’s proportional trade power in a node (country’s trade power in a node divided by the total trade power of all countries in that node) is like that country’s proportional “claim” on the total *trade value* (“Incoming” + “Local”) that exists in that trade node.  So if Spain has 40% trade power in a trade node that has a total trade value of 10 ducats, Spain’s “claim” in that trade node is 4 ducats out of that 10 ducats.  You can use your trade power to “claim” your portion of the trade value in that node and do something with that amount of trade value.

 

You can take the trade value you have claimed with your trade power and either:

A) “Collect” it for income (either because that’s your home node where you automatically collect or because it’s not your home node but you place a merchant there to collect. The game calls both of these actions “collecting” or “retaining” trade value in the trade node.) (My intuitive interpretation of this action is that a Spanish merchant collecting in a node is a Spanish government representative placed there to collect sales tax on the trade occurring there between Spanish traders.)

B) “Transfer it downstream or steer it as it goes downstream” (or as the game calls it, “transfer trade power,” “transferring,” “transporting forward,” or “steering.” I prefer to call it steering.  The terms “transferring” and “steering” will be used interchangeably in this guide to mean the same thing.). (My intuitive interpretation of this action is that a Spanish merchant steering trade in a node is a Spanish government representative placed there to enforce a mercantilist policy of only allowing Spanish traders in this node to interact with other Spanish traders that do business in a downstream node.)

C) Not use your existing trade power in a node to claim any of the trade value there. (This occurs in a special situation, described in IV.b.1.) If this happens, other countries will still use their trade power in the node to proportionally claim that portion of trade value that you aren’t claiming.

 

The amount of trade value that has been steered/transferred to downstream nodes is the “Outgoing” amount in that table on the left side of the trade node screen and the amount of trade value that has been “retained” (i.e. collected) in a trade node is the “Total” amount in that table.  So “Incoming” + “Local” – “Outgoing” = “Total”.  This is the trade flow in a trade node.  I think of “Local” as “locally produced trade value” and “Total” as “remaining or collected trade value in the node.”  For Basra above, it’s “Incoming” (1.89) + “Local” (3.13) – “Outgoing” (2.66) = “Total” (2.35).

 

03-2s-basraScreenshot III.2: Spain’s Trade Power in Basra.

Here, Spain has 38.2 trade power, seen in the right-most column of that table.  If I hover over the yellow portion in the pie chart on the right, it says “Spain has 22% of the Trade Power among countries Collecting from Trade.  Their trade power is 38.29.”  The first sentence is confusing.  Spain is not collecting from trade here.  We also see that in the map, in the top row of the sign/icon for the Basra node (around the center of the screen, a bit to the right), it says “38.2   (21%)”.  So some rounding of numbers is going on.  So ignore the confusion and just take the info that Spain has either 38.2 trade power here, which is 21% of the total trade power, or 38.29 trade power here, which is 22% of the total trade power.  Close enough.  Let’s go with the former.

 

From this info, what we know is that we can use our 21% of trade power in the Basra node to “claim” (and do something with) 21% of the total trade value in the Basra node.  Again, the total trade value that exists in the Basra node is “Incoming”(1.89) + “Local”(3.13) = 5.02 ducats.  So we have a claim to 5.02 × 21% ducats if all other countries are also claiming their share of the trade value here to either collect/“retain” or transfer downstream.  But if some countries aren’t claiming their share of trade value with their trade power, then there’s more trade value up for grabs for all the other countries that are claiming trade value.  (This situation will be described below in IV.b.1.)  So if Spain is collecting or transferring (in the screenshot, Spain is transferring in the node) and at least one other country is not claiming their share of trade value here, then Spain will be able to claim more than 5.02 × 21% ducats here.  Let’s look at all the possible actions for a country in a trade node.

 

 

IV Merchant Actions in a Trade Node

 

At a trade node, you can either have no merchant there, a merchant collecting there, or a merchant steering trade (“transferring trade power”) there.  Any country who has a trade node within its trade range can put a merchant there to collect or transfer, so it’s very common for a given node to have many merchants there from different countries.  However, a country can at most place one of their merchants at a node (so you can’t both collect and transfer in a single node or put two merchants to simultaneously collect or simultaneously transfer there).  In end nodes, you can only place a merchant there to collect.  For your home node:

 

IV.a Home Trading Node

 

Let’s momentarily go back to the home node at Sevilla.  In your home node, you automatically collect from there even without a merchant.  If you place a merchant there to collect (and you can only choose to collect at your home node.  No transferring/steering), you get a 10% bonus to the amount of ducats you collect there.

 

04-a-1s-sevillaScreenshot IV.a.1: Sevilla node.  In the first column of the table, you might see an icon of a house (meaning that a merchant is here collecting), an icon of a ship and wagon (a merchant is here transferring/steering), or a dash.  In the second column of the table, you might see either a yellow plus sign (collecting), a green arrow pointing right (transferring downstream), or a blue arrow pointing left (propagating upstream).  Here, we see that Spain has a dash and a yellow plus.  This is Spain’s home trading node, so Spain is collecting (the yellow plus) but has chosen not to place a merchant here for the 10% bonus income.  This is also Portugal’s home node, and we see that they are collecting (the yellow plus) and have placed a merchant here (the house icon) for the 10% bonus income.

Note however that Spain and Portugal also have provincial trade power here, so they are also propagating 20% of that trade power upstream, but the blue arrow pointing left isn’t shown anywhere.  Thus, don’t trust the tooltips.  Lack of a blue arrow doesn’t mean that countries here aren’t propagating trade power upstream.  But the two columns here do tell us the countries’ actions: Spain’s dash and plus sign mean that this is Spain’s home node and Spain has not placed a merchant here, while Portugal’s house icon and plus sign mean that this is Portugal’s home node and Portugal has placed a merchant here for the +10% bonus to income.

 

If you have no merchants collecting in any of your non-home trading nodes in the entire world, for every merchant that is transferring in any non-home node in the entire world, you get an extra +10% bonus trade power modifier in your home node.  Don’t ask me why.  Example:

04-a-2s-sevillaScreenshot IV.a.2: We have no merchants collecting in any non-home nodes and have merchants transferring in Alexandria, Constantinople, and Safi.  The tooltip for Spain’s trade power modifier (mousing over where it says 111.9%) tells us that we get a +10% bonus to trade power in Spain’s home node of Sevilla for each of these three merchants transferring.  (The merchant in Alexandria is steering trade value to Genoa, which is downstream of Sevilla, so none of that trade value gets to Sevilla, yet we get this bonus in Sevilla.  Go figure.)

 

If you have even one merchant collecting in any non-home nodes in the entire world, no matter how many merchants you have transferring trade in non-home nodes, all of the abovementioned 10% bonus trade power modifiers in your home node disappear.  Don’t ask me why.

 

Example:

04-a-3s-sevillaScreenshot IV.a.3: We have merchants in Tunis and Venice collecting.  The tooltip for Spain’s trade power modifier (where it says 81.9%) tells us that there is “No transfer bonus due to merchant collecting in Tunis.”  For whatever reason, we aren’t told that we also have a merchant collecting in Venice…  In any case, if there is even one merchant collecting anywhere in the world that is not your home node, this is what you’ll see in the home node trade power modifiers.

 

If you have a merchant collecting at your home node (i.e. providing you with a 10% bonus to trade income at your home node), this will not cause these +10% trade power bonuses at your home node from merchants transferring in other nodes to go away.  It is only if you have a merchant collecting in a non-home node that the +10% trade power bonuses at home go away.

 

04-a-4s-sevillaScreenshot IV.a.4: We have a merchant collecting in Sevilla for the 10% bonus income at our home node, and merchants transferring in Alexandria, Constantinople, and Safi.  We still get the +10% bonus to trade power from each merchant transferring at those three nodes.

 

This means that there may be a moment in the game when you’re wondering tactically whether to collect from a non-home node for extra income or maintain all merchants to be transferring trade in different nodes so that you get this boost in trade power in your home node (and thus a boost to your income from your home node).

 

04-a-5s-trade-tabScreenshot IV.a.5: In the trade tab, the big number in the top-right tells you your total trade income.  Trade income comes from your automatic collection at your home node and other merchants collecting in non-home nodes.  The numbers on this screen (and many other numbers in the game) update on the first of every month.  So try different configurations for your merchants and see how the big number changes.  This trial and error is often the best way to tell what’s best for your country at the moment.

 

IV.b Non-Home Trading Node with No Merchant Collecting or Transferring There

 

I am going to call both your home node (where you automatically collect) and non-home nodes where you’ve placed a merchant to collect: “collection points” or “nodes that you collect in.”

 

IV.b.1 The Node is Not Upstream of Any of Your Collection Points, and You Don’t Have a Merchant There

 

If the node is not upstream of any of your collection points (even via multiple links) and you don’t have a merchant placed at that node, the game just assumes that you aren’t going to bother claiming trade value in that node.  This is the special situation mentioned in III C).

 

04-b-1-1f-not-upstreamFigure IV.b.1.1: We collect in nodes A and B.  Node C goes to D, which goes to E, A, and J.  Node J is an end node.  Node E goes to B, which goes to G and H.  Node H also goes to A.  Node A goes to F, which then goes to K and L, which are end nodes.  Nodes C, D, E, and H are upstream of nodes we are collecting in (to be precise, C, D, E, and H are all upstream of at least one node we are collecting in).  Nodes F, G, L, K, and J are not upstream of any nodes we are collecting in.  This section is relevant to nodes like F, G, L, K, and J.  (Note that node H is downstream of a node we are collecting in, but as long as it is also upstream of a node we are collecting in, node A here, then it isn’t relevant to this section.  Also, I don’t say “nodes that are downstream of all nodes that you are collecting in,” since then:

 

04-b-1-2f-not-upstreamFigure IV.b.1.2: assuming node A is an end node here, node G isn’t “downstream of all nodes you collect in” since it isn’t downstream of node A, but it’s still relevant here.  The technical point is that node G is not upstream of any of your collection points.

 

In a way, it makes sense that you might forfeit the use of your trade power in nodes F, G, L, K, and J.  If you don’t have a merchant collecting in F, G, L, K, or J, you aren’t trying to directly gain income there.  And if you don’t have a merchant transferring at F (you can’t place a merchant to transfer at end nodes K, L, G, or J), the game interprets that as your saying that you don’t care how the trade value flows after F since none of that trade value downstream of F can possibly get to any of your collection points anyway.  So for nodes F, G, L, K, and J, if you don’t have a merchant there, the game mechanic will automatically have you not use your trade power to claim your share of trade value there.

 

04-b-1-1s-champScreenshot IV.b.1.1: The Champagne node.  Spain has a dash and a blue arrow pointing left.  The blue arrow pointing left technically is supposed to mean that “Spain is transferring trade power upstream,” but this isn’t important nor helpful here, since Spain has 0 provincial trade power in Champagne here and is thus propagating 0 × 20% = 0 trade power to upstream nodes Rheinland and Bordeaux.  When you see a dash and a blue arrow in the same row, it means that the country has no merchant here and the node is not upstream of any of the country’s collection points.  I’m mousing over the yellow slice of the “Trade Power” pie chart.  Spain has 10% of the trade power in the Champagne node, but Spain is not using that 10% trade power to claim their share of the trade value here.

 

04-b-1-2s-champScreenshot IV.b.1.2: I’m mousing over the green pie of the “Retained Trade Value” pie chart.  It shows France with 54%.

 

04-b-1-3s-champagScreenshot IV.b.1.3: I’m mousing over the red portion and it shows the Netherlands, Great Britain, the Papal State, and Savoy.

 

The “Retained Trade Value” pie chart is badly labeled.  I would call it “Claimed Trade Value.”  As you can see, Spain is missing from this pie chart as it wasn’t mentioned in the above 5 countries.  That’s because Spain hasn’t claimed its trade value here.  Green + red in that pie chart is the amount of trade value that has been claimed in this node by countries that are claiming here.  Green is the share of trade value that has been claimed and collected in this node (via automatic home node collection or non-home node merchant collecting) and Red is the share of trade value that is claimed and is being transferred downstream.  Spain is not claiming any trade value here, so it isn’t part of this pie chart.

 

04-b-1-4s-champScreenshot IV.b.1.4: Mousing over the blue portion of the pie chart on the right, we see that France has 48% of the trade power here.  Note how that is less than the 54% for France in the pie chart on the left.  The pie chart on the right is every country’s portion of trade power in the node.  Spain has trade power in this node, so it’s included in this pie chart (the yellow slice in the 6 o’clock to 7:30 portion).  The pie chart on the left is every country’s claimed trade value in the node, and thus has excluded countries like Spain that aren’t using their trade power to claim trade value here.  This is why France’s share is higher in the left pie.  In the right pie, France has 48% of the trade power and Spain has 10% of the trade power in Champagne.  In the left pie, France has claimed 54% of the trade value in Champagne due to some countries like Spain not using their trade power here to claim any trade value.  In that same left pie, Spain has not claimed any trade value with its 10% trade power and so is left out.

 

IV.b.2 The Node is Upstream of Any One of Your Collection Points, and You Don’t Have a Merchant There (and No One Else Has a Merchant Transferring There)

 

Back to Figure IV.b.1.1:

04-b-1-1f-not-upstreamSo now we’re talking about nodes C, D, E, and H.

 

If it’s not your home node and you’re not doing anything there (no merchant is placed there) and there are no other merchants from other countries there transferring trade (other countries can have a merchant collecting there, that’s fine), then the trade value that you are able to claim with your trade power there “transfers downstream” “equally”.   You may have trade power in a node like this, but you haven’t placed a merchant there to collect nor to transfer/steer trade downstream.  But since trade flowing downstream from this node has a chance to eventually arrive at one of your collection points, the game basically assumes that even though you have no merchant here, you still want to use your trade power here and claim trade value and transfer it downstream.  Take node D.  Trade value in this node may all flow to J and never reach your collection points of A and E, who knows… but by virtue of being upstream of A or E, there is still a chance that trade value from D will reach A or E and then B, who knows.  No one knows, really, at this point.  So even if you have no merchant in D, the game assumes that you’ll want your claimed trade value to flow downstream rather than not claim your trade value here at all.  What the game will do is it will have your claimed trade value flow downstream “equally” without preference among the downstream nodes connected to it.  Thus, your claimed trade value in node D will be divided equally and flow downstream to nodes A, E, and J.

 

04-b-2-1s-basraScreenshot IV.b.2.1: Back to the Basra node.

 

Hormuz flows to Basra, which flows to Aleppo and Persia.  Basra is not my home node and I have no merchant there.  I have 22% of the trade power here.  Note for now how it says “We transfer 1.18 ducats to Aleppo,” whatever that means.

 

04-b-2-2s-basraScreenshot IV.b.2.2: Note that the flags under the trade node’s “main sign”/icon or whatever it’s called (where it says “37.9 (21%)” on the first line and “2.3” on the second line) show the countries that either have a home node here or don’t have a home node here but have a merchant here collecting.  There are no flags under the two smaller signs each saying “1.32,” which means that there are no countries that have placed a merchant here to transfer/steer to Aleppo or Persia.

I’m currently mousing over the red part of the pie chart on the left.  Of the trade value in this node that has been claimed by countries, Spain has claimed 24% of that.  The total trade value in this node up for grabs is “Incoming” 1.86 + “Local” 3.13 = 4.99 ducats, so 4.99 × 24% = 1.20 ducats.  This is slightly off from 1.18 ducats but is probably from rounding that the game does.  So Spain has claimed 1.18 ducats of trade value in this node.  Other countries in this node that aren’t collecting but have claimed their trade values are Tabarestan and the Ottomans.  The red part of the pie chart is 53% of 4.99, so that is 4.99 × 53% = 2.64 ducats claimed by these three countries.  There are no merchants transferring/steering here, so Tabarestan and the Ottomans also simply don’t have a merchant here.  Note that Spain, Tabarestan, and the Ottomans all have a dash and then a green arrow pointing right in the first two columns.  This means that these countries don’t have a merchant here but are “transferring trade downstream.”  What this means is that the combined claims of Spain, Tabarestan, and Ottomans cause 2.64 ducats of trade value in this trade node to be transferred downstream (which is why it says “Outgoing: -2.64” on the left) without preference, i.e. equally between Aleppo and Persia.  That is why we see that 2.64/2 = 1.32 ducats are flowing each to Aleppo and Persia.  Of that 2.64 ducats that are “Outgoing,” Spain is causing 1.18 ducats of that.  That also means that Spain is causing 1.18/2 = 0.59 ducats to flow to each of Aleppo and Persia.  (This is why the “We transfer 1.18 ducats to Aleppo” message is confusing.  It should be something like “We cause 1.18 ducats to flow out of Basra downstream.  And in the tooltip, say: Specifically, 0.59 of that is going to Aleppo and 0.59 of that is going to Persia.”)

 

IV.c Non-Home Trading Node with a Merchant Transferring There

 

Now, let’s place a merchant in a node to “Transfer Trade Power.”

 

04-c-as-basraScreenshot IV.c.a: I’m at the Basra node and click “Transfer Trade Power” and select a merchant.

 

04-c-bs-basraScreenshot IV.c.b: Then I see this while I wait for the merchant to arrive in Basra.  I can’t choose whether to put him on the link to Aleppo or Persia while the merchant is still traveling.  (And if you try to send another merchant to Basra when the screen looks like this, the game will try, but in the end only the last merchant you send will matter.  The earlier merchants will magically teleport back to “Free” status.)  Once the merchant arrives, he’ll appear on one of the downstream links, and you can then instantaneously move him to steer on any one of the downstream links or to collect in the node, but you have to wait until the first of the next month to see the change in the numbers.

 

IV.c.1 Only One Merchant Transferring at a Node

 

04-c-1-1s-bScreenshot IV.c.1.1: Here, we have a merchant in Basra steering to Aleppo.  The “sign”/icon that says “3.03” is the link between Basra and Aleppo.  The red check mark means I have a merchant steering on this link.  Under that there is a small Spanish flag, showing that there is a Spanish merchant here.  Under the “sign”/icon that says “0.00” for the link between Basra and Persia, we see no flag underneath it, so there is no merchant steering here.  Note that in the table at the bottom-left of the screen, Spain has a “ship and wagon” icon and a green arrow pointing right.  This means that Spain has a merchant in this node transferring/steering trade.  We are the only one here with a merchant steering.

 

Note that the “Incoming”, “Local,” “Outgoing”, and “Total” numbers on the left have changed slightly from before due to some months passing.  Spain currently has 23% of the total trade power here and has claimed 25% of the trade value here.  So Spain has claimed [2.13 (“Incoming”) + 3.13 (“Local”)] × 25% = 1.31 ducats of trade value here.  Note how it says, “We transfer 1.31 ducats to Aleppo,” which I suppose can be interpreted to be technically correct.  Now we look at the sign that says “3.03.”  What has happened is that by having the only merchant here transferring, Spain actually now gets to steer all 2.84 Outgoing ducats in the direction that Spain wants to direct them.  Even though Spain only has the trade power to claim 1.31 ducats here, Spain having the only merchant here transferring overrules that in terms of steering trade, allowing Spain to decide where all the downstream trade steers to.  Other countries that have no merchants collecting or steering here but have claimed trade value with their trade power (Tabarestan and the Ottomans) cannot influence where their claimed trade value steers downstream to.  We have all 2.84 ducats going to Aleppo instead of divided equally between Aleppo and Persia due to the Spanish merchant steering here.  The growth of 2.84 to 3.03 is because:

 

04-c-1-2s-bScreenshot IV.c.1.2: I’m currently mousing over the “3.03.”  The first line of the tooltip is saying that 2.84 is the original amount of trade value that is “Outgoing” out of Basra.  The second line means that 100.00% of that is going to Aleppo due to the one Spanish merchant steering here.  The third line means that 53.90% of trade power among countries that are claiming trade value in this node (Spain, Tabarestan, and the Ottomans) are using it to transfer their claimed trade value downstream.  In other words, this just tells us the amount of the red portion of the pie chart on the left.  The fourth line tells us that the Spanish merchant somehow is able to apply a +6.90% bonus to the trade value that it steers from Basra to Aleppo.  This comes from:

 

04-c-1-3s-bScreenshot IV.c.1.3: In the trade tab, there’s an item called Trade Steering, which here shows 38.1% (coming from Spain’s Naval Tradition).  One merchant steering trade on a route applies a bonus of 5% to the trade value modified by Trade Steering.  So 5% × (1 + 38.1%) = 6.90%.  (Although the Trade Steering bonus comes from Naval Tradition, it applies to any link between nodes that your merchant is steering, even when steering from an inland node to another inland node.  Shrug.)

 

IV.c.2 Merchants Working in Different Directions

 

If one merchant is steering trade in one direction and another merchant in another direction, the two countries of those merchants will fight over the “Outgoing” trade value using their country’s trade power in the node to determine how much of the trade value flows to where.  Countries who don’t have merchants collecting or transferring/steering in the node but have their claimed trade value flowing downstream – as described in the previous section – can’t decide where their outgoing trade value flows to.  Let’s look at it in action.

 

04-c-2-1s-bScreenshot IV.c.2.1: Let’s look at the Tunis node.  There are a bunch of countries collecting here (either because it’s their home node or it isn’t their home node but they have placed a merchant here to collect).  Portugal is the only country transferring, and it’s steering trade to the Sevilla node.  Because it’s the only one steering here, 4.30 ducats are going to Sevilla and 0.00 are going to Genoa.  Let’s place a merchant here to steer towards Genoa.

 

04-c-2-2s-bScreenshot IV.c.2.2: Our merchant is at Tunis steering towards Genoa.  I’m mousing over the “0.83” that’s highlighted with a red oval and box, the Tunis to Sevilla route.  What the tooltip tells us is that of all of the 4.09 “Outgoing” trade value, 19.50% of that is going to Sevilla (0.83/(3.52+0.83) ≈ 19.50%, probably to do a rounding error).  Of all the trade value that initially exists at Tunis, which is 0.00 “Incoming” + 4.93 “Local” = 4.93, 83.00% of that has been claimed by countries that are letting it transfer downstream.  And the Portuguese merchant has a 5.20% bonus to trade value that he steers.

 

04-c-2-3s-bScreenshot IV.c.2.3: I’m now mousing over the “3.52” that’s highlighted with a red oval and box, the Tunis to Genoa route.  What the tooltip tells us is that of all of the 4.09 “Outgoing” trade value, 80.40% of that is going to Genoa (3.52/(3.52+0.83) ≈ 80.40%).  Of all the trade value that initially exists at Tunis, which again is 0.00 “Incoming” + 4.93 “Local” = 4.93, 83.00% of that has been claimed by countries that are letting it transfer downstream (this is the same as the above.  Hasn’t changed).  And the Spanish merchant has a 6.90% bonus to trade value that he steers.

 

Thus, Spain and Portugal are fighting over the “Outgoing” 4.09 trade value at this node.  Spain has 213.6 trade power here and Portugal has 68.0.  Thus, Spain gets to decide that 213.6/(213.6 + 68.0) = 76% of that 4.09 trade value will go to Genoa (plus a +6.90% bonus to that trade value), while Portugal gets 24% of 4.09 to go to Sevilla (plus a +5.20% bonus to that trade value).  From the table, we see that France and The Papal State don’t have merchants here but are having their claimed trade value flow downstream.  Thus, while they are contributing to trade value to flow out of Tunis, they aren’t deciding on where it goes to, Sevilla or Genoa.  Only Spain and Portugal are deciding that because they are the only ones that have placed merchants to steer here.

 

IV.c.3 Merchants Working in the Same Direction

 

What if Spain and Portugal both steer trade from Tunis to Sevilla?

 

04-c-3s-bScreenshot IV.c.3: Now, both the Spanish and the Portuguese merchant are steering towards Sevilla.  I’m mousing over the 4.45.  Below that, we see both the Spanish and Portuguese flags, showing that these two merchants are steering on this link.  Of the Outgoing 4.10 ducats, 100.00% of that is going on this route.  Now what happens is that the bonuses that each merchant applies to the trade value on this link changes.  The first merchant still starts at 5% modified by their country’s Trade Steering bonus (Portugal), resulting in 5.20%.  The second merchant starts at 2.5% modified by their country’s Trade Steering bonus, so for Spain we have 2.5% × (1 + 38.1%) = 3.4%.  This goes on until the fifth merchant steering on the same link.  For more: http://www.eu4wiki.com/Trade#Multiple_merchant_bonus.

 

IV.d Non-Home Trading Node with a Merchant Collecting There

 

When placing a merchant to collect in a non-home node, your trade power in the node is halved, and then that amount of trade power is used to claim trade value for direct income, plus modifiers (called Trade Efficiency in the game).

 

04-d-1s-tScreenshot IV.d.1: Now, let’s have our merchant in Tunis collect there.  I’m mousing over the 106.6.  Note that our trade power here has halved compared to the 213.3 trade power from the previous screen when we were transferring trade.  Somehow our modifier here is -11.3%, shown in the column to the left.  We’ll mouse over that in the next screenshot.  Now that we are collecting here (with our halved trade power), we are now one of the countries that is “retaining” trade value here, so note how the green in the pie chart on the left has increased compared to before.  Also, using our trade power, we have shifted trade value from “Outgoing” to “Total.”  Spain’s icons down in the table, a house icon and a green arrow pointing right, means that there is a merchant here collecting in a non-home node.  It doesn’t make sense why it’s still the green arrow (which is supposed to mean transferring trade downstream).

 

04-d-2s-tScreenshot IV.d.2: Mousing over the -11.3% now.  Our total positive (green) modifiers here are +77.4%.  So when we didn’t have a merchant collecting here, we had: 120.2 trade power from trade power sources × 177.4% modifiers = 213.3 trade power.  Our halving modifier from collecting here is applied to this final 213.3 and thus takes away 213.3/2 = 106.6 trade power.  So, effectively, if we pretended that this halving modifier was an additive modifier like the other green modifiers that get applied to the original 120.2 trade power, it is doing -106.6/120.2 = -88.7%.  Thus, our total additive modifiers is +77.4% – 88.7% = -11.3%.  Thus, the halving modifier is (confusingly) expressed as -88.7% in the tooltip.

 

Summary: So first, as noted in IV.a, a merchant collecting in any non-home node voids all possible +10% bonuses to trading power at your home node that come from other merchants transferring in other nodes.  Second, as noted here, a merchant collecting in a non-home node also first reduces your trade power there by half before you get to collect trade value as income.

 

Note that in this whole, complex, convoluted-tooltipped trade game, the ONLY instances where you get any trade income is only in your “collection points” – either your home trading node or in non-home nodes where you’ve place a merchant to collect.  All your provincial trade power, your light ships placement, your merchants steering trade – all of that is there solely to determine the amount you collect at your collection points.  My interpretation is that your collection points are where you are charging sales tax on trade that you control.  In your non-home nodes, I suppose your sales tax is causing traders to avoid your flag and your control (they’d rather trade with other countries not collecting sales tax there), and thus your trade power halves there.  In your home node, I suppose you have enough control in that market so that your trade power isn’t affected by collecting sales tax there.

 

 

V Tactics and Strategy

 

V.a Basic Trade Power Increase

 

The simplest ways to increase trade power in a node is to increase provincial trade power by conquering other provinces in the node, increasing trade power in the provinces (buildings, modifiers like mercantilism), and building light ships to protect trade in the node if it’s a sea node.  You should mainly target provinces in a node that have those hardcoded bonuses, like estuaries and “centers of trade,” as those flat bonuses pretty much always make them the key provinces to control in the node for the sake of trade power.  Just owning those few provinces in the node can often get you majority trade power in the node.

 

V.b Basic Steering

 

04-b-1-1f-not-upstreamBack to Figure IV.b.1.1.  We collect in nodes A and B.  Assume that there are no other merchants from other countries steering at any nodes in the figure.  So D’s Outgoing is divided equally between J, A, and E.  If we were to place a merchant at node D to steer trade, in which direction would we want to steer it to?

Clearly not J, since then the trade value isn’t going anywhere that we’re collecting.  If J wasn’t an end node and instead led to other downstream nodes but none of them led to our collection nodes, we still know that we wouldn’t want to steer to J.  We want to steer trade value towards our collection nodes – this is basic steering.

How to choose between steering to A or E?

 

See Appendix below for unnecessary math used to show that:

 

**The answer is not J and then trial and error between A and E by looking at the trade tab on the first of the month, lol.**

 

V.c Steering Towards Your Home Node or Collecting at a Non-Home Node.  A Trade Power Threshold for Your Home Node?

 

Early on, you might have a home node and one other non-home node and you might have decent trade power in both.  Should you use a merchant in the non-home node to transfer towards your home node or should you collect at that non-home node?  This is a question I had from the get-go and is also a common question I see by others, and for good reason, since it’s basically what you encounter at the beginning of the game.

 

See Appendix below for unnecessary math used to show that:

 

**The answer is use trial and error by looking at the trade tab on the first of the month, lol.  The best answer to this question is “just try both out.”**

 

V.d What To Do About Nodes that Aren’t Upstream of Any of Your Collection Points

 

04-b-1-1f-not-upstreamBack to Figure IV.b.1.1: I’m talking about those F, G, L, K, and J nodes.  I’ll also call these nodes “strictly downstream” nodes just because it’s less unwieldy.

 

If you aren’t worried about the erasure of the +10% trade power bonuses at home from merchants transferring at other nodes, you can collect at these strictly downstream nodes.  It’s a comparison of whether you have enough trade power at the strictly downstream nodes to warrant a merchant collecting there, or if your home node is rich enough in trade value and you have enough pre-modified trade power at your home node such that those +10% bonuses in trade power at home get you more income.

Let’s say your home node is A, which is upstream of node F.  In terms of long-term conquest-related strategy, you want to increase your (usually provincial and light ships) trade power in A so that you’re simply claiming more trade value there for your own collection instead of allowing other countries to transfer it downstream to F, and you also want to increase your trade power in F so that your option of collecting that “run-off” trade value in F with a merchant is better (“run-off” as in the trade value that runs off downstream out of A to F).  You would also want to increase your provincial trade power in F so that you get upstream propagation in A (depends on how large the provincial trade power at F is compared to the total pre-modified trade power at A).

 

If F connects to further downstream nodes and there are rivals you are competing against that are collecting either at F or further downstream:

if there are rivals downstream of F, you might place a merchant to steer trade at F if you can steer it away from their collection nodes or at least in a way that it helps your rivals the least;

if there are rivals collecting at F, you might place a merchant at F simply to steer it in any direction downstream so that at least you’re claiming your trade value there and letting it flow out of F (remember that if you don’t place a merchant at a node that is strictly downstream of your collection points, the game will cause your country to not claim any of the trade value there).

 

Note that as long as you dominate your home node of A (say you basically have 100% trade power there), then you’ll simply collect all the trade value that’s flowing to A and none of it will flow downstream to F.  Increasing your trade power % in A is the best, but increasing in both A and F are useful.

 

V.e Web/Net vs. Trail/Route 

 

The TL;DR to all of this muck is that in the long run if you’re conquering lands and you’re increasing the number of your merchants through technology/ideas or colonization and trade companies, you’ll want a web of merchants steering trade towards your home node and/or a trail of merchants steering trade in one long trail along trade nodes starting from some far away trade node and ending at your home node.  A web synergizes with blobbing around you.  A trail synergizes with deliberately picking off high trade power provinces in trade nodes upstream of you and utilizing the 5% merchant steering bonus to trade value, as that means that you’re multiplying the trade value by 105% at every one of the links between trade nodes on the trail.  In the former, you’re like the ancient Roman Empire drawing trade from all around you towards Rome.  In the latter, you’re a colonial power with dominance in key places that connect back to your country, all connected together.  The shape of the nodes and the placement of Europe usually mean that in the mid to late game, a successful trail strategy from Asia and the Americas to Europe is king.

 

 

VI Other Stuff

 

Skipped stuff: Light ships, Privateering, Embargoes, Modifiers: Trade Efficiency (modifier to trade income at your collection points), Global Trade Power (modifier to trade power) plus domestic (nodes where you have provincial trade power) and foreign (nodes where you don’t have provincial trade power) trade power modifiers, mercantilism (boost to provincial trade power), Colonial Nations (only in the Americas and Australia), Trade Companies (only the coasts of Africa and Asia.  Built for the trail strategy), Burghers Estate, Buildings, Trade Goods, Caravan Power/Inland Bonus (merchants that are collecting at or steering towards or away from inland trade nodes give you a trade power boost, probably to simulate them being relatively more powerful in inland nodes than in sea nodes or sea node to sea node links because there are no light ships in inland nodes).

 

Other Recommended Guides:

 

/u/Conditionable’s Trade: Deconstructed

https://www.reddit.com/r/eu4/comments/4njv9d/trade_deconstructed/?st=iu29ccph&sh=5b59ad7f

A comprehensive guide with screenshots.  Also, check out /u/Llama-Guy’s comment where he makes some points.

 

/u/issoweilsosoll’s Trade – a case study

https://www.reddit.com/r/eu4/comments/37b98o/trade_a_case_study/?st=iu9ww22f&sh=22d29f71

The final form of the trade “trail” strategy.  The real TL;DR of trade in EU4 is to try to copy this, basically.  Even if you don’t understand or don’t want to understand any of the details of trade mechanics, this guide and demonstration alone will help a lot for the general feeling.

 

/u/Llama-Guy’s So I made a flowchart of trade flow

https://www.reddit.com/r/eu4/comments/3xuufh/so_i_made_a_flowchart_of_trade_flow/?st=iu3tp5hj&sh=f1ac7a12

Don’t use the version at the top of that page.  Use this corrected version in the comments here:

https://www.reddit.com/r/eu4/comments/3xuufh/so_i_made_a_flowchart_of_trade_flow/cy84q5d

A streamlined flow chart of the game’s trade nodes and links.  It’s probably only useful after you’re very comfortable with how trade works in the game, but it’s a great way to show what makes which nodes important in the game.  The comments are interesting, too, pointing out how Aden is a key chokepoint for Asian trade flowing towards Europe, how Genoa is “better” than Venice despite both being end nodes since there are multiple opportunities upstream to steer trade away from Venice towards Genoa and Venice doesn’t have access to the New World, how the English Channel tends to grab loads of trade from the New World, and how if you’re a non-European country thinking about colonizing the New World by crossing the Pacific, the only places from which you can hope to get trade value to flow to you is Mexico and upstream of that – all other trade nodes in the New World flow east to Europe.

 

 

VII Appendix

 

The following isn’t recommended if you aren’t interested in pointless math.

 

V.b Basic Steering

 

04-b-1-1f-not-upstreamBack to Figure IV.b.1.1.  We collect in nodes A and B.  Assume that there are no other merchants from other countries steering at any nodes in the figure.  So D’s Outgoing is divided equally between J, A, and E.  If we were to place a merchant at node D to steer trade, in which direction would we want to steer it to?

Clearly not J, since then the trade value isn’t going anywhere that we’re collecting.  If J wasn’t an end node and instead led to other downstream nodes but none of them led to our collection nodes, we still know that we wouldn’t want to steer to J.  We want to steer trade value towards our collection nodes – this is basic steering.

 

How to choose between steering to A or E?

 

If we steer it to A, the additional amount of trade value that’s steered to A × the trade power % you have in A gives your additional income (let’s ignore all modifiers and small bonuses in these examples).  The original (before placing a merchant to steer) amount of trade value flowing from D to A is: (1/3 × D’s Outgoing), and the post-steering amount of trade value flowing from D to A is: (1 × D’s Outgoing).  So the additional amount of trade value that’s steered to A from having a merchant steer from D to A is 2/3 × D’s Outgoing.  So the additional income we get is (2/3 × D’s Outgoing) × (our trade power % in A).

If we steer to E, the amount of trade value steered from D to E becomes part of E’s “Incoming.”  We have increased the flow of trade value from D to E from 1/3 × D’s Outgoing to 1 × D’s Outgoing.  So we have increased E’s Incoming by (2/3 × D’s Outgoing).  This amount is then diluted by the % of countries’ trade power that is not being used to let trade value flow downstream, i.e. the % of countries’ trade power that is being used to collect at E.  Finally, we have our trade power in B itself to worry about (which has been halved by the fact that it is a non-home node we’re collecting in).  So the additional income we get is (2/3 × D’s Outgoing) × (trade power % of countries in E that are transferring trade value downstream) × (our trade power % in B).

 

Which is larger,

(2/3 × D’s Outgoing) × (our trade power % in A)

vs.

(2/3 × D’s Outgoing) × (trade power % of countries in E that are transferring trade value downstream) × (our trade power % in B)?

Rearranging, we get:

(Our trade power % in A) vs. (trade power % of countries in E that are transferring trade value downstream) × (our trade power % in B),

where if the first is larger, we transfer to A, and if the second is larger, we transfer to E.  Where do we steer?

 

**The answer is not J and then trial and error between A and E by looking at the trade tab on the first of the month, lol.**

 

You can do the math in the last equation above, but in-game, modifiers and other nations steering and whatnot are gonna screw with the final result.  But the basic conclusion is as expected (and a bit obvious), which is that in a chain of trade nodes, trade value originating from a far-away node gets multiplied by (trade power % in in-between nodes that are transferring trade value downstream) (which will always be ≤ 1, of course) at each in-between node.  So to maintain a chain of trade value flowing back to your home node, you want to try to dominate every one of those trade nodes.

 

V.c Steering Towards Your Home Node or Collecting at a Non-Home Node.  A Trade Power Threshold for Your Home Node?

 

Early on, you might have a home node and one other non-home node and you might have decent trade power in both.  Should you use a merchant in the non-home node to transfer towards your home node or should you collect at that non-home node?  This is a question I had from the get-go and is also a common question I see by others, and for good reason, since it’s basically what you encounter at the beginning of the game.

 

First, let’s imagine an extremely simple example.  Node B is upstream of node A and node A is your home node.  Node A has only one upstream link, which is the link it has to node B.  And node B only has one downstream link, which is the link it has to node A.  So the only benefit to placing a merchant at B to transfer/steer (to A) is for the 5% steering bonus, so let’s just ignore that bonus for now.  What I want to ask is, do we place a merchant at B to collect or not?  Let’s say you have x trade power % at A and y trade power % at B, with y = Y/(Y + U + V) where Y is your trade power in B, U is the combined trade powers of other countries collecting there, and V is the combined trade powers of other countries transferring trade downstream there.  Without collecting at B (we are letting our claimed trade value flow from B to A), this is the income we earn at A due to trade inflow from B:

 

A’s “Incoming” (trade value) × Our trade power % at A =

A’s “Incoming” × x =

“B’s “Outgoing” × x =

(B’s “Incoming” + “Local”) × (Trade power % at B of all countries transferring downstream) × x =

(B’s “Incoming” + “Local”) × (Our trade power % at B + trade power % at B of other countries transferring downstream) × x =

(B’s “Incoming” + “Local”) × (Y/(Y + U + V) + V/(Y + U + V)) × x

 

If we collect at B, y goes down by some amount to a new value, which we’ll call y’ (aka “y prime”).  We have y’ = (Y/2)/(Y/2 + U + V).

 

(The change from y to y’ is an amount that is hard to predict.  We have:

y = Y/(Y + U + V), so U + V = Y/y – Y

y’ = (Y/2)/(Y/2 + U + V) = (Y/2)/(Y/2 + U + V) = (Y/2)/(Y/2 + Y/y – Y) = (1/2)/(1/2 + 1/y – 1) = (1/2)/(1/y – 1/2) = 1/(2/y – 1)

Plotting this gives:

https://www.wolframalpha.com/input/?i=plot+1%2F(2%2Fx+-+1)+from+x+%3D+0+to+1

where the horizontal axis is y before collecting and the vertical axis is y’.  Generally, y’ is about 0 to 0.17 below y.  Here is another way to look at it:

https://www.wolframalpha.com/input/?i=plot+(1%2F(2%2Fx+-+1))%2Fx+from+x+%3D+0+to+1

where the horizontal axis is y before collecting and the vertical axis is y’/y.  So when y starts out large (near 1), y’ is also not that much smaller.  But when y is small, y’ can be almost half of y.)

 

The amount of trade value flowing from B to A coming from our trade power at B disappears (since we are now collecting at B).  Our trade power of Y/2 at B is now used to collect.  A trade power of U is still collecting at B (the other countries collecting there), and a trade power of V is still transferring trade downstream at B (the other countries transferring downstream there).  So now, V/(Y/2 + U + V) is the share of trade power used to transfer trade value from B down to A.  So the amount of income we earn at A due to trade flow from B to A is:

 

(B’s “Incoming” + “Local”) × V/(Y/2 + U + V) × x

 

The amount of income we earn at B due to collecting at B is:

 

(B’s “Incoming” + “Local”) × (Y/2)/(Y/2 + U + V) =

(B’s “Incoming” + “Local”) × y’

 

Thus, (Income due to trade flow from B to A while collecting at B) vs. (Income due to trade flow from B to A without a merchant at B) is:

 

(Income from collecting at B) + (Income at A due to trade flow from B to A) vs. (Income at A due to trade flow from B to A without collecting at B)

 

After dividing (B’s “Incoming” + “Local”) from both sides of the “vs.”:

 

(Y/2)/(Y/2 + U + V) + V/(Y/2 + U + V) × x vs. (Y/(Y + U + V) + V/(Y + U + V)) × x

 

(If you want to change the right-hand side to (Income due to trade flow from B to A with a merchant steering from B to A), you can add a 1.05 multiplier to that side, but I won’t do that here.  Continued:)

 

(Y/2)/(Y/2 + U + V) + V/(Y/2 + U + V) × x vs. (1 – U/(Y + U + V)) × x

(Y/2)/(Y/2 + U + V) vs. (1 – U/(Y + U + V)) – V/(Y/2 + U + V)) × x

 

If we let u = U/(Y + U + V), u’ = U/(Y/2 + U + V), and v’ = V/(Y/2 + U + V), we have:

 

y’ vs. (1 – u – v’) × x

y’ vs. (1 – (u + v’)) × x

 

Note that while u’ > u and v’ > v, it’s always true that 1 ≥ u’ + v’.  So 1 ≥ u + v’ and thus (1 – (u + v’)) ≥ 0.

 

Let’s say we dominate trade power in B.  That means both u and v’ are small so that (1 – (u + v’)) is close to 1, so we basically have something close to

 

y’ vs. x

 

If y’ is larger than x, then we collect at B.  Otherwise, we don’t collect at B.  y’ is our post-collection trade power % at B, so it’s simply saying that if we dominate trade at B so much that our post-collection trade power % at B is larger than our trade power % at A, then collect at B.  (This is probably unlikely since usually, you have more trade power % at your home node A when you start the game.)  If we dominate trade power at B even post-collection but we also dominate at A, then it’s a straight comparison between these two values.

 

Back to y’ vs. (1 – (u + v’)) × x

 

If we don’t dominate trade power in B, u + v’ is larger, (1 – (u + v’)) becomes smaller, and so the left-hand side becomes relatively larger, meaning that collection at B becomes more likely to be better compared to when we did dominate trade power in B.  However, of course, this all depends on x, our trade power % at our home node A, which is likely to be larger than y or y’ in most early game periods.  What the above is saying is that if we don’t dominate at B (and thus (1 – (u + v’)) is some value between 0 and 1), but we are strong enough at A (x is large), it’s still better not to collect at B.  But if we aren’t strong enough at A, then it’s better to collect at B by just locking in an income at B instead of letting it flow to A and fighting over it again with our trade power % of x at A.

 

A simple example with numbers: say that at node B, we have 10 trade power. Other countries transferring trade downstream at B have 10 trade power, and other countries collecting at B have 20 trade power.  So we have 10/40 trade power at B, or 25%.  As long as we are not collecting at B, (10 + 10)/40 = 50% of trade power at B is being used to transfer downstream and 20/40 = 50% to collect at B.  Our current income at A is (B’s Incoming + Local) × 50% × x.  Once we’re collecting at B, we have 5 trade power at B, giving us 5/35 = 14% trade power share at B.  There is 10/35 = 29% trade power share of other countries still transferring trade downstream at B now.  So our income at A now is (B’s Incoming + Local) × 29% × x and our income at B is (B’s Incoming + Local) × 14%.

 

So income during collection at B is 0.14 + 0.29 × x and income before collection was 0.50 × x.  When x < 0.66, it’s better to collect at B, and when x > 0.66, it’s better to not collect at B.  This shows us that even if you have 25% trade power at B, to make it worth steering from B to A in the purest circumstances (instead of collecting at B), you need more than 66% trade power at A.  Otherwise, just collect at B.

 

For a more realistic/general situation, let’s say node B is upstream of node A and but has b downstream links, one of which is to A.  Like before, we have y trade power % at B, coming from Y trade power that we have, U trade power of other countries collecting, and V trade power of other countries transferring trade downstream so y = Y/(Y + U + V).  Again, assume that there are no merchants steering at B.  So initially, trade value of 1/b × B’s Outgoing is flowing to each of B’s downstream nodes, including A.  Including the 5% bonus that a merchant steering gives to trade value on its link:

 

Placing a merchant to collect at B vs. Placing a merchant at B to steer from B to A:

 

(Income from collecting at B) + (Income at A due to trade flow from B to A) vs. (Income at A due to trade flow from B to A with merchant steering from B to A):

(Y/2)/(Y/2 + U + V) + (1/b) × V/(Y/2 + U + V) × x vs. 1 × (Y/(Y + U + V) + V/(Y + U + V)) × x × (1 + 5%)

y’ + (1/b) × v’ × x vs. 1 × (1 – u) × x × (1 + 5%)

 

Now even more generally, allow merchants from other countries to steer at B, let f = the percent of (B’s Incoming + Local) that flows to A when we have a merchant collecting at B, let g = the percent of (B’s Incoming + Local) that flows to A when we have a merchant steering from B to A, let h = the sum of additional percent bonuses that other countries’ merchants steering on the B to A route provide to trade value on that link, and j = the additional bonus percent we provide to the trade value in the B to A link by placing our merchant to steer on that link.  Then, we have:

 

y’ + f × (1 + h) × x vs. g × (1 – u) × (1 + h + j) × x

 

So how do we choose what to do?

 

**The answer is use trial and error by looking at the trade tab on the first of the month, lol.  The best answer to this question is “just try both out.”**

 

The intuition provided by the above simple example (the one with x > 0.66) probably still holds though, which is that domination of your home node is the most important thing to have when you want to start thinking about a global steering strategy.  Next most important would be domination at your nodes closest to your home node in the connections, and so on.  If A is your home node and A <- B <- C <- D, dominating trade power in C doesn’t help as much if you aren’t also dominating trade power in B, and of course most of all at A.  Until then, mindlessly collecting at all non-homes nodes could be worth it.

 

Here is a messy in-game example:

 

spain-trade-tableTable 1

 

Spain’s home node is Sevilla, where it has no merchant, and then it has 9 merchants in non-home nodes that I generally felt were where Spain had the best presence.  Other than Sevilla, Spain is collecting in Genoa and Constantinople since it has decent trade power in those high trade value nodes.  The other merchants are generally steering towards one of these collection points.  Ignoring modifiers and bonuses, total trade income from the three collection points is 17.68 + 8.68 + 9.86 = 36.22.  In-game, with all the modifiers and bonuses obviously, the total trade income was actually 63.43.

 

If we were steering at Constantinople and Genoa, we get the +10% bonus to trade power at Sevilla per merchant steering elsewhere (for a total of +90% trade power at Sevilla).  Our trade power at Sevilla becomes 1423 and the total trade power at Sevilla becomes 2114, giving us 67% trade power there.  The total trade value at Sevilla in the above table is 34.  Genoa is downstream of Sevilla so steering there won’t change the trade value at Sevilla.  Constantinople is upstream of Sevilla and has 17 trade value.  If all 17 of that could be steered straight to Sevilla, that means we obtain (34 + 17) * 0.67 = 34.3 trade income (before modifiers) at at our only collection point, Sevilla.

 

spain-trade-table-collectTable 2

 

I then switched things so that all 9 merchants are collecting in non-home nodes.  Ignoring modifiers and bonuses, total trade income from the ten collection points is 39.23.  If we choose to take the merchant in Ragusa (who is collecting the least) and put him in Sevilla for the 10% income bonus at home there, the total trade income becomes 40.88.  In-game, with all the modifiers obviously, the total trade income was actually around 61.8 with a collector at Ragusa and 62.3 after moving him to Sevilla.

 

(Also, for a moment, let’s take away Constantinople and Genoa and pretend that those two nodes are completely off-limits to us (since including them makes collecting from them obviously lucrative).  In such a scenario, only our home node of Sevilla is a node where we have a lot of trade power and that has a lot of trade value.  Then, having all merchants steering (to get the bonus trade power at the home node of Sevilla) should be better than having all merchants collecting in the other less valuable nodes.  If we have a very significant presence (but not totally overwhelming dominance that is close to 100%) at our home node and at all other trade nodes we have either little trade power or the nodes are poor in trade value, every bit of additional trade power at home increases our income a lot there.  Thus, we’d basically be placing merchants to steer at other nodes more for the bonus to income at home than the additional trade value that gets steered to our home node.)

 

What I’m trying to demonstrate is not the differences in total income between the scenarios, but how similar the incomes are despite the first strategy attempting an optimal balance between collecting and funneling, the second briefly-described and approximated strategy of steering everywhere and only collecting at home, and the third strategy of blindly “collect everywhere, forget about steering.”  But it’s another example of how real domination of your home trading node (or collection points in general) is needed before worldwide trade-steering strategy becomes much more important and powerful than ignoring everything and simply collecting with all your merchants even though it may feel like the least strategic method.  The real answer to all of this is still to just do trial and error, of course.

 

Cyberpunk Images: Japan and China

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Shibuya Center Street, https://www.flickr.com/photos/isaacpacheco/10579038705

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Nail House in Chongqing, China. http://www.theatlantic.com/photo/2015/04/and-then-there-was-one/390501/

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Jiutian International Plaza, Zhuzhou, Hunan, China

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Caiyuanba Bridge on-ramp, Chongqing. Elevated Roads Encroaching Farmhouses Chongqing, Mark Homs, Getty Images

IMHO, Japan is “fantasized cyberpunk.”  People (especially in the west) look at the night lights of Japanese cities and it makes them fantasize about cyberpunk, conjuring images from Neuromancer, Blade Runner, various anime, and the Matrix (the green Japanese-looking code on the screens).  China is real “high tech-low life” cyberpunk, because there’s actual low life in the midst of development and high tech, and it’s an actual dystopia.  Like, that last part isn’t an exaggeration – it actually is a dystopia.  It is utopian in its amazing, unprecedented growth out of poverty since 1979, but dystopian in its government, economic inequality, and environmental issues.  Japan on the other hand is a clean, environmentally-minded developed country (EPI ranks Japan in the 20s, ranked 39th in 2016) (first link archived) with economic equality that’s the same (archived) as developed Western Europe.  There are some who don’t join the rat race and feel alienated (e.g. NEETs, hikikomoris, or just plain otaku geeks, I guess, and they are probably overrepresented in anime-related media) and that can certainly be cyberpunk, but in terms of inequality, it doesn’t compare to China’s.

Brainteaser: The Monty Hall Problem

You are on a game show and presented with 3 doors.  Behind one is a car and behind the other 2 are goats.  You want to choose the door with a car behind it, as if you do so, you win the car.  You choose one door.  Then, the host opens one of the other doors, which reveals a goat behind it.  The host gives you a choice to either switch your door to the other one that’s still closed or keep your original choice.  Should you switch doors?

 


 

Answer:

If your strategy is to stick to your original choice, your probability of choosing the door with the car behind it is 1/3.  Let’s see what happens if you switch.  So you choose a door, the host reveals one of the other doors with a goat behind it, and asks if you want to switch.  What has happened up to this point?  There’s a 1/3 chance that you picked the door with the car behind it, which means that if you switch, you are switching to a door with a goat behind it.  There’s a 2/3 chance that you picked a door with a goat behind it, which means that if you switch, you are switching to a car behind it.  So if your strategy is to always switch, there’s a 1/3 chance you get a goat in the end (because you happened to choose a car on your first choice, which has a probability of 1/3) and a 2/3 chance you get a car in the end (because you happened to choose a goat on your first choice, which has a probability of 2/3).  So the best strategy is to switch.

The host revealing one of the doors gives you additional information.  Switching lets you use that information, assuming that it was unlikely that you got a car on your original choice.

Perhaps a more intuitive answer is if there are 100 doors.  One has a car behind it and 99 of them have goats behind them.  Choose one door, the hosts reveals another door with a goat behind it, and asks if you want to switch.  If you don’t switch, there’s a 1/100 chance that you chose the door with a car behind it.  But if you switch, assuming that you probably didn’t choose the right door on your first try (because 1/100 is small), now, you have a 1/98 chance of choosing the right door (because the host as revealed one door with a goat behind it and you’re giving up your original door).  Of course 1/98 is better than 1/100.  The exact probability of getting the right door with the switching strategy is 99/100 × 1/98 (probability that you chose the wrong door on the first try × probability of choosing the right door after accepting the offer to switch).  99/100 × 1/98 = 1/100 × (99/98) > 1/100 where 1/100 is the probability of getting the car with not switching, and so switching is better than not switching.

Brainteaser: 100 Prisoners in a Line and Extension

There are 100 prisoners.  An executioner tells them that tomorrow morning, he will line them up so that each of them is facing the back of the head of another prisoner, except for one prisoner at the end of the line.  In other words, prisoner 1 sees the back of the head of prisoner 2 as well as the backs of the heads of prisoners 3-100, prisoner 2 sees the back of the heads of prisoners 3-100, …, prisoner 99 only sees the back of the head of prisoner 100, and prisoner 100 doesn’t see any prisoners in front of him.  The executioner tells them that he will put either a red or blue hat on each prisoner, then starting with prisoner 1 (the one who can see 99 other prisoners in front of him), will ask him what color hat he is wearing.  The prisoner says a color and if he is wrong, he will silently kill that prisoner (prisoner 1 would be killed in a way that prisoners 2-100 won’t know if he was killed or not).  If he is right, he will keep him alive.  Then, the executioner will move to prisoner 2, ask the same question, and kill if he’s wrong, keep him alive if he is right.  The executioner keeps doing this for every prisoner to prisoner 100.  The prisoners are allowed to discuss together in the night what to do for the next day.  What should their plan be in order to maximize the number of survivors?  For clarity, what should their plan be in order maximize the number of survivors in the worst case scenario (any random guess by a prisoner ends up being wrong)?

 


 

Not the answer:

A sort of baseline answer is that prisoner 1 says the color of the hat worn by the prisoner right in front of him or her, thus sacrificing his or her life with a guess.  Prisoner 2 is guaranteed to live.  Repeat this for every pair of prisoners, giving us at least 50 prisoners alive at the end.  With 2 colors of hats, it makes intuitive sense that this would be an answer.  Unintuitively, this is far from the answer :-/

 


 

Answer:

One key, or hint, that may remain unemphasized when this brainteaser is presented to people, is that when a prisoner makes and says his guess for his own color, that guess is heard by all the other prisoners.  If each guess is correct, that provides valuable information to the later prisoners.

Let’s say there are only 3 prisoners and we are the middle prisoner.  We see only one prisoner in front of us and say he is wearing a red hat.  From the perspective of the prisoner behind us, either 1 or 2 red hats are seen.  So it’s possible for the prisoner behind us to announce through some code (e.g. “Red” = there is 1 red hat in front of me, “Blue” = there are 2 red hats in front of me) to tell us this.  This allows us to answer our own hat correctly.  Additionally, the prisoner in front of us will have gained two pieces of information: how many red hats there are with the 2 last prisoners and what hat the middle prisoner was wearing.  In other words, initially, there were either 1 or 2 hats worn by the last two prisoners.  The middle prisoner has the ability to answer correctly after the first prisoner sacrifices himself or herself by announcing the code.  If the first prisoners announces that there are 2 red hats in front of him, the middle prisoner will definitely say that he himself is wearing a red hat, leaving 1 red hat for the last prisoner.  If the first prisoner announces that there is 1 red hat in front of him, and then the middle prisoner says “Red,” the last prisoner knows that they are Blue, while if the middle prisoners says “Blue,” the last prisoner knows that they are Red.

Let’s say there are 4 prisoners in a line.  The first prisoner sees 1, 2, or 3 red hats in front of him or her.  But as long as the second prisoner announces his or her own hat color correctly, that will provide information for the later prisoners.  So how can the first prisoner announce information so that at least just the second prisoner will get his or her own hat color correctly?  The second prisoner sees 1 or 2 hats in front of him or her.  The answer is that the first prisoner announces the oddness or evenness of the number of red hats he or she sees.  From the second prisoner’s perspective, whatever he sees in front of him and whatever the last prisoner sees in front of him can only differ by 0 red hats or 1 red hat (whatever hat the second prisoner is wearing).  Thus, the key is, when there is only a difference of one change at each increment, oddness and evenness conveys enough information to tell us what has changed.  So the first prisoner sacrifices himself by announcing, say “Red” for an even number of red hats and “Blue” for an odd number of red hats that he sees in front of him.  This allows the second person to say his hat color correctly.  The third person has information that among the last 3 people, the number of red hats was either odd or even, plus what exact hat color the second person has, plus, of course, what exact hat color the first person, the person in front of him, has.  Effectively, the second person knows the hat colors of all 3 people at the end of the line except his own color plus the information that the first person provides, what the oddness or evenness of the number of red hats was for those 3 people.  This is enough information for the second person to figure out what color hat he has.  It’s the same with the last person. 

So with 100 people, the first person sacrifices himself by announcing the oddness or evenness of one of the colors that exist by code.  The second person has exact knowledge of the colors of the 98 people in front of him plus the oddness or evenness of one of the colors for all 99 people excluding the first person (i.e. the 98 people in front of him plus himself), giving him correct knowledge of his own color.  The third person know has exact knowledge of the color of the person behind him and the colors of the 97 people in front of him, plus the oddness or evenness of one of the colors for the 99 people that includes him, giving him enough information to figure out his own color.  This continues until the whole line is finished.  Thus, at least 99 out of 100 people can be saved with this strategy.

 


 

Extension:

What if the executioner uses more colors?

In our above case, we had 2 colors, and we sacrificed 1 prisoner at the beginning of the line to announce the oddness or evenness of one of the colors for the 99 people he sees in front of him.  Since all prisoners know the number of prisoners that the first prisoner sees (99), everyone only needs to keep track of one of the colors, say red.  The first prisoner announces the oddness or evenness of red, and each subsequent prisoner counts how many reds there are in the 99 to see if they also have a red hat or not.

If we have 3 colors, the first prisoner that can be saved would see x prisoners in front with 3 different colors and needs to figure out what color hat he has on.  Extending the strategy from above, if we sacrifice two prisoners before him, they can announce the oddness or evenness of two of the colors.  This is enough information for the first prisoner we save what color hat he has.  All subsequent prisoners will then have exact knowledge of the hat colors of all prisoners that can be saved except for their own, which they deduce by the oddness or evenness of the 2 colors that that first two prisoner we sacrifice first announce.  So in this case, we sacrifice 2 prisoners at the start and the 98 subsequent prisoners can be saved.

Let us apply the same logic to more colors.  If the executioner uses y different colors where 1 ≤ y ≤ 100, the first y – 1 prisoners sacrifice themselves by announcing the oddness or evenness of y – 1 colors.  The remaining 100 – (y – 1) prisoners will have enough information to correctly state their hat color.  If the executioner uses more colors than there are prisoners, we don’t have enough prisoners we can sacrifice to convey accurate information about the oddness or evenness of the colors we have to prisoners at the end.  In addition, we can always default back to the “baseline” solution, where each pair works together by sacrificing one prisoner (who simply announces the color of the hat in front of him) and saving the other one (who simply says the color that was announced by the prisoner before him), and guarantee at least 50 prisoners saved.  Thus, for 1 ≤ y ≤ 49, the “sacrifice for odd or even” strategy saves 99 to 51 people.  For y = 50, the strategy saves 50 people, which is the same as the result for the “default pair sacrifice” strategy.  For y > 50 (and even if y > 100), the “default pair sacrifice” strategy can always save 50 people and becomes better than the “sacrifice for odd or even” strategy.

Brainteaser: Blue Foreheads

100 people are in a room.

 

  1. All 100 of them are perfect logicians.
  2. They are told that at least one person in the room has blue paint on their forehead.
  3. They are told that once you deduce that you have blue paint on your forehead, the next time that the lights are turned off, leave the room.

 

All 100 people have actually had their foreheads painted blue (but of course, each of them don’t know this at this point – they can only see the other people’s foreheads).  The light is turned off, then on, then off, on, etc.  What happens?

 


 

Answer:

So each person sees 99 other people with blue paint on their heads.  While this is the situation we begin with, it doesn’t seem to help with solving the problem at all.  The key for this problem is to start as small as possible and then expand.

 

Start with 1 person.  1 person in a room sees 0 other people.  Thus, if there is at least 1 person in the room with blue paint, he or she must be it.  The light goes off, and then on, and we see 0 people in the room, as the person has left.

 

Let’s say we have 2 people.  Put ourselves in to the shoes of one of them.  They see 1 person in the room with blue paint on their forehead, and don’t now if there is blue paint on their own forehead.  But if there was no blue paint on their forehead, then the other person should deduce that they must be the one with blue paint on their forehead, and will be gone by the next light.  The light is turned off, then on.  Since both people see the other person with blue paint, both remain.  Now, each person knows that the other person looked at their forehead and saw blue paint, and so each person knows that they have blue paint on their own forehead.  The lights turns off and on, and there are 0 people in the room.

 

I think you know where this is going (although I find the logic the most difficult from here).  3 people in the room.  Each person sees 2 other people with blue paint on their foreheads.  The additional key here is, each person needs to think, “What if I don’t have blue paint?  If what happens then is a contradiction, then I must have blue paint.”  Choosing one person’s perspective – our “first” person – we first posit that we don’t have blue paint.  In that case, each of the other 2 people sees 1 person without blue paint and 1 person with blue paint.  Our existence as someone without blue paint doesn’t matter in their calculations.  Each of them thinks, “There is one other person in this room with blue paint.  If they see me without blue paint as well, then they should disappear by the next light.  The light turns off, then on.  All 3 people are still there.  So each of the other 2 people think, “Since that other person didn’t leave, I must have blue paint.  So I will leave by the next light.  The light turns off and on.  But since the truth is that all 3 people have blue paint, the other 2 people won’t disappear.  Instead, each of them are thinking the same thing about the other 2 people in the room that they see have blue paint on their foreheads.  Everyone waited two turns to see if the other people would make a move.  Since they didn’t, everyone has found a contradiction to “If I had blue paint,” and thus everyone deduces that they have blue paint on their own forehead.  Thus, the third time that the light goes off and on, the 3 people have left the room.

 

4 people in the room.  Assume you don’t have blue paint, so your being there doesn’t affect the others’ logic.  There are 3 people wondering if they have blue paint and the each see 2 other people with blue paint.  After 3 turns of the light going off and on, they should all leave.  If they don’t, we have a contradiction, so we have blue paint.  So on the 4th light, all 4 people leave.

 

5 people in the room.  Described another way: Let’s say we don’t have blue paint.  There are 4 other people with blue paint.  Let’s label them A, B, C, and D.  D is wondering if he or she has blue paint, looking at A, B, and C.  D first assumes he has no paint and is thinking, “C is thinking if he doesn’t have blue paint, then after 2 turns, A and B will disappear.”  After 2 turns, A and B remain.  D is thinking, “So now, C will conclude that he has blue paint.  So on the 3rd turn, A, B, and C should leave.”  After the 3rd turn, A, B, and C remain.  D is thinking, “OK, so there’s a contradiction to the assumption that I don’t have blue paint.  Thus, I have blue paint, and will disappear on the 4th turn.”  On the 4th turn, we see that A, B, C, and D still remain.  Thus, we have a contradiction to our first assumption that we have no blue paint.  We have blue paint, so on the 5th turn, we leave.  Everyone else also has the same logic process, so on the 5th turn, everyone leaves.

 

If there are 100 people in the room, all with blue paint on their foreheads, first assume that you don’t have blue paint on your forehead.  So then, your existence shouldn’t matter to the other 99 people’s logic.  Let’s label us A.  There are 100 people in the room: A, B, C, …, X, Y, Z, AA, AB, …, CV.  Person A first assumes they have no paint, and thinks, “B must be thinking, if I don’t have paint, then, C would think, if I don’t have blue paint… etc.”  Basically, we are testing the assumption that everyone first assumes that they themselves don’t have blue paint on their forehead.  It doesn’t make intuitive sense since anyone can see that there are at least 99 other people with paint, but it’s the key step.  Assume, what if everyone from A to CV thought that they didn’t have blue paint?  Or rather that A assumes they don’t have blue paint and that B assumes that B doesn’t have blue paint and B assumes that … CU assumes CU doesn’t have blue paint and that CV assumes that they don’t have blue paint?  Well, this is a contradiction, because at least 1 person must have blue paint.  Now, let’s assume A to CU thinks that they don’t have blue paint and CU sees CV has blue paint and must assume that CV sees everyone else with no paint.  After 1 turn, CV doesn’t leave (because it’s not true that the other 99 people don’t have blue paint), and thus we have a contradiction and CU must believe that they have blue paint on their forehead as well.  After turn 2, CU doesn’t leave though (because it’s not true that the 98 other people other than CV and CU don’t have blue paint), so we have a contradiction and CT must believe that they have blue paint.  Keep going until turn 99, where B doesn’t leave because it’s not true that A doesn’t have blue paint (if B saw that A doesn’t have blue paint, B should have left on turn 99).  We have a contradiction, so A concludes that they have blue paint, and so on turn 100, everyone leaves.

 

It’s a lot easier to rely on the formula we built from the smaller examples that “With a room of x people, they all leave at once after x turns.”  But I find the intuition disappears with large numbers.  The above paragraph is an attempt to describe the intuition, the key being that we assume that all x people assume that they don’t have blue paint, and then one by one contradict that (because in reality, everyone has blue paint), until we’ve contradicted all cases down to 1 person assuming they have no paint.  Once that is contradicted on the xth turn, after that, everyone leaves at once, since everyone has the same logic process.

Brainteaser: Forehead Numbers

There are 3 people placed in a room.  They all have perfect logic.  The 3 people are told by a host that a number has been written on each of their foreheads.  Each of the 3 numbers are unique, they are all positive, and they relate to each other such that A + B = C (i.e. one is the sum of the other two).  In the room, each person can only see the other two people’s numbers, as they cannot see their own foreheads.

 

Suppose you are one of the 3 and you see one person with “20” on their forehead and the other person with “30.”  The host asks you, then the person with “20,” and then the person with “30” what number is on their heads and all 3 say that they don’t know.  The host then asks, again, you, then the person with “20,” and then the person with “30” what number is on their head, and all 3 answer correctly.  How does this happen?

 


 

Answer:

The key to this brainteaser is to calculate the logic of each person’s point of view, i.e. put yourself in each of their shoes.  The annoying part of solving this brainteaser, then, is having to keep track of 3 different points of view.

“First” person: If you see “20” and “30,” that means you are either 10 or 50.  So you don’t know what’s on your forehead among these two numbers.

The “20” person: You see either 1.) “30” and “10” or 2.) “30” and “50.”  In case 1.) you are either 20 or 40.  In case 2.) you are either 20 or 80.  So you don’t know.

 

The “30” person: You see either 1.) “20” and “10” or 2.) “20” and “50.”  In case 1.) you must be 30 because you cannot be 10 as well the “First” person.  So the key here is that if you see one person has number “x” and another has number “2x,” you know you cannot also have “x” on your forehead.  You must be “3x.”  So in this case, the “30” person would know the answer that he or she has 30 on his head.  In case 2.) the “30” person has either 30 or 70, and so he or she wouldn’t know.

 

Since after the first round, everyone answered that he or she did not know, that means that we cannot have the “30” person’s case 1.), which is that he sees “20” and “10.”  In other words, our “First” person cannot have 10.  He has 50 on his forehead.  So when the host asks the “First” person the second time, he or she will answer 50.

 

The most illuminating and clean part of the problem is just up to here, but in an attempt for completeness, I kept going.

 

From the “20” person’s point of view, we assume that he or she is able to figure out the above sort of logic on his or her own.  What the “20” person sees is “30” and “50,” which means that he or she is either 20 or 80.  Somehow, the “50” person figured out on his or her own on the second round of questioning that they have 50 on their head.  The logic is that in order to find out what your number is on the second round, you are using someone’s “I don’t know” answer in the first round of questioning.  So if the “20” person indeed has 20 on his or her head, they can deduce that the “50” person is able to figure out all the above and that his or her number is 50 on the second round.  If the “20” person has 80 instead, the “First” person sees “80” and “30” and is thus wondering if his or her number is 50 or 110 and the “30” person sees “80” and 50 and wondering if they’re 30 or 130.  In none of these cases is a person announcing that they are not seeing an “x” and “2x” situation (which is what the “First” person experiences: seeing a “2x” and “3x” situation, and then seeing that the “3x” person doesn’t immediately say that he or she knows that his or her number is “3x.”).  If the “20” person has 20, then, again, the “First” person sees that the “30” person is announcing that they aren’t seeing an “x” and “2x” situation, which means that the “First” person can’t have 10 and must have 50.  This causes the “20” person to know that his or her number is 20.

 

Similarly, the “30” person sees “50” and “20” initially doesn’t know if he or she is 30 or 70.  If it’s 70, the other people either see “70” and “50” or “70” and “20,” which doesn’t allow the situation described above of someone announcing that he or she doesn’t see an “x” and “2x” situation.  If it’s 30, then everything that’s been discussed happens, and so it must be 30.

 

The key basically is that if someone sees “x” and “2x,” they should know immediately that they are 3x.  If someone sees “2x” and “3x,” they are immediately on high alert to see if the “3x” person immediately knows that he or she is 3x.  If the “3x” person doesn’t know, that is an announcement that the “3x” person did not see an “x” and “2x” situation, which means that the person we started with must be “5x.”  So, in an “x” and “2x” situation, you know immediately that you are 3x.  In a “2x” and “3x” situation, if everyone says that he or she doesn’t know in the first round, that announces that no one saw (and the “3x” person in particular did not see) an “x” and “2x” situation, which means that you must be 5x.

Arendt, Action, and Psychological Stuff

I know next to nothing about Hannah Arendt except what I’ve read on Wikipedia and Reddit (archived).  Nevertheless, it sounds really cool.

She defined the three human activities as labor, work and action, with two mutually exclusive spheres: the political and everything else.

Arendt introduces the term “vita activa” (active life) by distinguishing it from “vita contemplativa” (contemplative life), which represents her understanding of Western society. There are only three human activities: labor, work and action. They correspond to the three basic conditions under which humans live. Action corresponds to the political actions of anyone…

What’s striking to me (and I agree with it) is the stark division between the active life and the contemplative life.  There’s action and there’s talk (so I’m already disregarding Arendt’s actual philosophical definition of “action,” but oh well.)

According to Arendt, modern life is divided between two realms: that of the public in which “action” is performed, and that of the private, site of family life where the father ruled. It is in the public realm where one distinguishes oneself through “great words and great deeds” in the same way as personal glory is attained on the battlefield.

Again, emphasizing how what you do matters.  Your private life is just for yourself.  Your public life affects you and can bring you things (reputation, resources, i.e. money, connections) that don’t really exist or have meaning with just yourself in your private, home life.

Arendt claims that her distinction is unusual and new as it has not been attempted previously by the thinkers who concerned themselves with the subject of ‘human activity’, e.g. Karl Marx. She goes on to explain that “labor” is one of the only three fundamental forms of activity that are the human condition. It is repetitive and only includes the activities that are necessary to mere living, such as the production of food and shelter as well as any material production, with nothing beyond that. The condition to which ‘labor’ corresponds is sheer biological life.

So “labor” is stuff like eating, bathing, cleaning and hygiene, and perhaps some other health and maintenance-related activities.

“Work”, on the other hand, has a clearly defined beginning and end. It leaves behind a durable object, such as a tool.

I don’t grasp the meaning of this.  I’d like to think it means the same as “work” in our normal sense, although that’s probably wrong.  But anyway, surely there’s something other than the “life maintenance” things we do (like eat, sleep, and bathe in private at home) and the political life, which I’d guess, is our normal definition of “work.”  And the point of “work” is basically accumulating resources that allow us to enrich our lives further than what bare life maintenance provides us.

 

On the other hand, exercise improves mood (archived), and not in a new age-y way but physiologically:

Looking deeper, Lehmann and his colleagues examined the mice’s brains. In the stimulated mice, they found evidence of increased activity in a region called the infralimbic cortex, part of the brain’s emotional processing circuit. Bullied mice that had been housed in spartan conditions had much less activity in that region. The infralimbic cortex appears to be a crucial component of the exercise effect. When Lehmann surgically cut off the region from the rest of the brain, the protective effects of exercise disappeared. Without a functioning infralimbic cortex, the environmentally enriched mice showed brain patterns and behavior similar to those of the mice who had been living in barebones cages.

Humans don’t have an infralimbic cortex, but we do have a homologous region, known as cingulate area 25 or Brodmann area 25. And in fact, this region has been previously implicated in depression. Helen Mayberg, MD, a neurologist at Emory University, and colleagues successfully alleviated depression in several treatment-resistant patients by using deep-brain stimulation to send steady, low-voltage current into their area 25 regions (Neuron, 2005). Lehmann’s studies hint that exercise may ease depression by acting on this same bit of brain.

I couldn’t find anything on the following from a bit of low-effort Googling, but I think there are some who say that action helps to alleviate depression or improve moods as well.  And when I say action, I don’t mean Arendt’s political action nor physical exercise, but just general action, like doing things, instead of sitting and contemplating about doing.

Walking helps with thinking (archived), especially creative thinking.  Dipping into fanciful evolutionary psychology headcanon (or I might’ve also read this somewhere else), if we were all persistence hunters once (or simply had to walk a lot to gather berries and water for our hunter-gatherer tribe), we walked a lot daily, and the body uses this time and monotonous activity to let the brain think.  Many people talk about their subconscious helping them figure out problems – problems that they couldn’t when they were face-to-face with it at a desk – while they were doing something completely unrelated, even really smart people (archived):

Poincaré deliberately cultivated a work habit that has been compared to a bee flying from flower to flower. He observed a strict work regime of 2 hours of work in the morning and two hours in the early evening, with the intervening time left for his subconscious to carry on working on the problem in the hope of a flash of inspiration. He was a great believer in intuition, and claimed that “it is by logic that we prove, but by intuition that we discover”.

Of course, I don’t know what exactly Poincare did during his off-hours and it may not have been persistence hunting.  But even if it wasn’t, as long as it’s something habitual (archived):

To illustrate the differing thoughts and emotions involved in guiding habitual and nonhabitual behavior, 2 diary studies were conducted in which participants provided hourly reports of their ongoing experiences. When  participants  were  engaged  in  habitual  behavior,  defined  as  behavior  that  had  been  performed almost  daily  in  stable  contexts,  they  were  likely  to  think  about  issues  unrelated  to  their  behavior, presumably because they did not have to consciously guide their actions. When engaged in nonhabitual behavior,  or  actions  performed  less  often  or  in  shifting  contexts,  participants’  thoughts  tended  to correspond to their behavior, suggesting that thought was necessary to guide action. Furthermore, the self-regulatory benefits of habits were apparent in the lesser feelings of stress associated with habitual than nonhabitual behavior.

 

It seems that action improves mood, and I assume an improved mood improves action.  But then if the opposite might be true, why does inaction worsen mood and a worsened mood worsen action?  Again into fanciful evolutionary psychology, perhaps it’s a survival mechanism.  When circumstances are bad, you want to conserve energy and action and stay away from possibly taxing or dangerous situations.  Basically, sit tight and wait out the night/rain/drought.  Of course, when this occurs not because of a lack of resources but some internal mental reason (which is much more likely in modern life), it’s much harder or at least more mentally complex to get out of that spiral.  A modern person’s way out of inaction or depression is not the same as a hunter-gatherer seeing food for the first time in a while (some external thing happening to him) that might quickly improve his mood, spur him into action, and so on and so forth.  The tricky thing is that while action may improve mood, if an internal mental reason is what caused mood to worsen to begin with, the action isn’t targeting the source of depression.  Action in this instance is a solution that’s unrelated to the source of depression/the bad mood.  Action and its positive effects on mood might still be good enough to overcome internally-caused depression.  But I imagine that the disconnect between action and source of depression is why modern depression isn’t easily solved by action and exercise even if it undoubtedly helps physiologically.  It’s interesting and crazy that we’re such contemplative beings with such big brains but we’re still meat and water bags that are heavily influenced by physical, biological, neurochemical existence.

The Theory of Interstellar Trade

The Theory of Interstellar Trade, by Paul Krugman (1978)

Archived

 

Assume we have two planets, Earth and Trantor, separated by a large distance, the traversal of which necessitates travel at velocities comparable to the speed of light.  Assume that Earth and Trantor are in the same inertial reference frame, i.e. they are not accelerating with respect to each other.

Assume that a spaceship traveling between the two planets travels at a constant v\ .

Let’s say that from the perspective of an observer on one of the planets, the time it takes for a spaceship to make the trip is n\ .

Then, the time it takes for a spaceship to make the trip from the perspective of someone on the spaceship is

\overline{n}=n\sqrt{1-\frac{v^2}{c^2}}

(which is shorter than n\ ).  The factor on the right is a well-known result from relativity, derived mathematically from the Lorentz transformation, that gives us “time dilation” from traveling at relativistic speeds.  Krugman demonstrates the above relation by representing the voyage in Minkowski space-time in a figure using imaginary axes.

 

Let

p_E\textup{, } p_T be the price of Earth goods and the price of Trantorian goods on Earth, respectively.

p_E^*\textup{, } p_T^* be the price of Earth goods and the price of Trantorian goods on Trantor, respectively.

r\textup{, }r^* be the interest rates on Earth and Trantor, respectively.

 

The first question Krugman asks is what is the correct interest rate on a planet with regard to interstellar trade, since depending on whether you’re on a planet or on a spaceship engaging in trade, the passage of time is different.  So putting ourselves on Trantor, we compare what happens to a Trantorian trader who can engage in investing in a Trantorian bond or engage in interstellar trade with Earth.

Interstellar trade with Earth would involve buying goods on Trantor, traveling to Earth, selling the Trantorian goods there and then buying Earth goods with those proceeds, and then traveling back to Trantor and selling the Earth goods there.  Let c\ be the cost of outfitting the spaceship.  The cost of buying goods on Trantor is q_T^* p_T^* where we define q_T^* as the quantity of Trantorian goods to be traded.  Thus, the initial expenditure of the Trantorian trader while still on Trantor is

c + q_T^* p_T^* .

The trader then travels to Earth and then sells its (not his or her) q_T^* goods at price p_T .  The trader now has money q_T^*p_T and will buy Earth goods with this money to bring back to Trantor.  Earth goods cost p_E so the quantity of Earth goods that can be bought is q_T^*p_T/p_E .  Let us call this quantity of Earth goods to be brought over to Trantor q_E^* .  So we have

q_E^* = \frac{q_T^*p_T}{p_E}

When the trader arrives back on Trantor, it will sell these Earth goods at price is p_E^* , resulting in a revenue of

\frac{q_T^* p_T p_E^*}{p_E}

But the Trantorian trader also had the choice of investing its money in a Trantorian bond before leaving for Earth.  The expenditure needed for the trading venture of c + q_T^* p_T^* invested into a Trantorian bond would:

after time 2n , which is the time it takes for a round trip from the point of view of a trader who decided to stay on Trantor for that duration of time instead of go to Earth and back, become

(c + q_T^* p_T^*)(1+r^*)^{2n}

or

after time 2\overline{n}=2n\sqrt{1-\frac{v^2}{c^2}}, which is the time it takes for a round trip from the point of view of a trader who decided to actually make the round trip and thus be on a spaceship for that duration of time (maybe he puts some money into a Trantorian bond and some money into the trading venture that he personally goes on), become

(c + q_T^* p_T^*)(1+r^*)^{2n\sqrt{1-\frac{v^2}{c^2}}}

So which perspective is right?  The perspective from Trantor or from the spaceship?  Krugman answers this by reminding us of the reasoning behind present value calculations, which is that of opportunity cost – any money that you choose not to possess today (by delay possessing it until the future for a larger amount) is money that you could have invested in a riskless bond today that would have also grown in value in the future.  And so the way I’ve framed it here sort of answers the question in advance, which is that the correct value of the bond is from investing it risklessly with some riskless bond issuer, like the Trantorian government.  The opportunity cost of spending money to outfit a trading venture to Earth is the lost opportunity of buying a Trantorian bond with that money and then receiving the proceeds by the time the trader would return from the venture, which according to the bond issuer, the Trantorian government, is 2n time.  Thus, we have Krugman’s First Fundamental Theorem of Interstellar Trade:

When trade takes place between two planets in a common inertial frame, the interest costs on goods in transit should be calculated using time measured by clocks in the common frame, and not in the frames of trading spacecraft.

Another way to think about it is that a bond earns interest because the bond issuer gets to possess cash now, invest it during some time, and afterwards will have more cash as a result.  The location that this is occurring is at the bond issuer’s location, which is Trantor.  The trader can always come back to Trantor and earn his bond interest and principal (assuming it’s a bond that automatically renews after maturing every time).  All this occurs on Trantor, and so the bond’s time progression is according to Trantor’s time progression.

 

Now, for simplification Krugman assumes that perfect competition reduces the profits of Earth-trading to 0.  In other words, we force the revenue earned from a trading venture to Earth that started with expenditure (c + q_T^* p_T^*) to equal the revenue earned from if we had invested that same expenditure into a Trantorian bond and waited the time that it takes to make a round trip to Earth and back:

\frac{q_T^* p_T p_E^*}{p_E} = (c + q_T^* p_T^*)(1+r^*)^{2n}

Also for simplification, Krugman assumes away c\ , the cost of outfitting a spaceship:

\frac{q_T^* p_T p_E^*}{p_E} = ( q_T^* p_T^*)(1+r^*)^{2n}

\frac{  p_E^*}{p_E} =  \frac{p_T^*}{p_T}(1+r^*)^{2n} \hfill \textup{(Relative goods prices)}

Krugman asks (or rather, asks what if someone asks) about a Trantorian trader who instead of going back to Trantor, decides to settle on Earth, thus making a one-way trading trip.  Krugman writes that the Trantorian trader has two options: either buy Trantorian goods, bring them to Earth, and sell them there, or buy Trantorian bonds, bring them to Earth, and sell them there.  While in a “realistic” situation, we would just need to assume that there is a liquid market for Trantorian bonds on Earth to make the second option possible, let us restrict our picture to just imagining one Trantorian in the whole universe who wants to migrate to Earth and then as many other traders as we want to imagine who all only intend to make round trips and live on their home planet and thus have no interest in the bonds of the other planet (thus, no Trantorian bond market on Earth).  In this case, Krugman assumes that we just have at least one Earthling who wants to migrate to Trantor and thus may be interested in buying a Trantorian bond before making his or her one-way trip.

So pursuing option 2, what price could this Trantorian bond fetch on Earth?  The Earthling has a choice of either buying and then bringing Earth goods to Trantor for sale or buying this Trantorian bond instead that can be redeemed when he or she arrives on Trantor.  The value of a one “T$ (Trantorian Dollar) bond on arrival at Trantor will be (1+r^*)^{2n} in Trantorian Dollars.  Thus, we want to know what value of Earth goods, when sold on Trantor, would net the same amount of Trantorian Dollars, for that is the Earth price of the Trantorian bond.

Suppose the Earthling buys q amount of goods on Earth to bring to Trantor.  The cost of that purchase is p_E q in Earth Dollars.  Once on Trantor, those q goods are sold for a total of p_E^* q Trantorian Dollars.   Let this equal the amount that would have been earned if a Trantorian bond was redeemed instead.  Thus, we have

p_E^* q = (1+r^*)^{2n}

q = \frac{1}{p_E^*}(1+r^*)^{2n}

The original Earth price for this transaction was

p_E q = \frac{p_E}{p_E^*}(1+r^*)^{2n}

Thus, \frac{p_E}{p_E^*}(1+r^*)^{2n} is the fair price in Earth Dollars for a Trantorian bond that can be redeemed in Trantor for (1+r^*)^{2n} .

If the Trantorian trader went with option 1, that means investing that one Trantorian Dollar in buying Trantorian goods to sell on Earth instead of buying a Trantorian bond and bringing it over.  One Trantorian Dollar buys \frac{1}{p_T^*} quantity of Trantorian goods, which on Earth will sell for a total of \frac{p_T}{p_T^*} Earth Dollars.  But from the “Relative goods prices” equation, we have that

\frac{p_T}{p_T^*} = \frac{p_E}{p_E^*}(1+r^*)^{2n}

Thus, one Trantorian Dollar invested in Trantorian goods and brought over to Earth can be sold for an amount of Earth Dollars that is the same as the fair price in Earth Dollars that a Trantorian bond will fetch on Earth.  So as long as there is at least one Earthling who also wants to make a one-way trip to Trantor and is open to buying a Trantorian bond instead of buying and bringing over Earth goods to sell, the Trantorian trader is indifferent between bringing over Trantorian goods to sell or bringing over Trantorian bonds to sell – both will earn the same profit.  This shows that the First Fundamental Theorem of Interstellar Trade holds as long as for one migrant going one way, we have another migrant going the other way.  What we seem to have is that as long as there is an “effective” round trip (either made by one Trantorian trader, or one Trantorian and one Earthling migrant) and the assumption of no arbitrage (so on each leg of the trip, the traveler is indifferent between carrying goods or bonds), we have a relation between the Trantorian interest rate and the prices of goods, and that none of this challenges the statement that the Trantorian bond’s interest gains run according to Trantorian time (the First Fundamental Theorem).  For Earth, we can construct the same scenarios except with the trip originating from Earth.  Krugman writes that he proves the theorem in the presence of transportation costs in a paper from the future.

Krugman then asks whether interest rates on Earth and Trantor will be the same or not.  The “Relative goods prices” equation (and its associated assumptions) are kept.  Even though transportation costs, as in c , the cost of outfitting a spaceship, was assumed away earlier, the “Relative goods prices” equation shows that there is an effective cost to transporting goods across stars that comes from the transportation time needed.  (If you are on Earth and you desire Trantorian goods, even if it costs no money for the Trantorian trader to outfit its spaceship to make the journey to Earth and sell its goods to you, the profit that the Trantorian trader makes by the time it returns to Trantor after its round trip needs to match the profit it would have made from just investing in Trantorian bonds and sitting at home for the same duration of time.  Thus, that gets built into the cost of Trantorian goods in planets that are interstellar distances away from Trantor.)  Could interstellar distances cause differences in the planets’ interest rates?

We imagine a scenario where a Trantorian trader buys Trantorian goods, travels to Earth, sells the goods there, invests the proceeds into Earth bonds, spends k time on Earth, redeems the bonds and buys Earth goods with that money, travels back to Trantor, and sells the goods there.  This scenario involves the prices of goods and Earth’s interest rate; and there is always another option: invest in Trantorian bonds at the beginning, sit at home for 2n+k time, and then redeem after that.  Then by forcing a no arbitrage condition, we are then able to obtain a relation between the prices of goods, Earth’s interest rate, and Trantor’s interest rate.

In the first option, the trader buys p_T^* q_T^* (where q_T^* is the quantity of Trantorian goods bought) worth of Trantorian goods, travels to Earth, sells these goods for p_T q_T^* , invests that in Earth bonds for k time, earns p_T q_T^* (1+r)^k $ after that, buys \frac{p_T q_T^* (1+r)^k}{ p_E} quantity of Earth goods, travels back to Trantor, and then sells these goods for a total value of \frac{p_E^* p_T q_T^*}{p_E}(1+r)^k Trantorian Dollars.  Forcing a no arbitrage condition, we require that this revenue equal the revenue from investing the initial expenditure in Trantorian bonds for the same amount of total time, giving us:

p_T^* q_T^* (1+r^*)^{2n+k} = \frac{p_E^* p_T q_T^*}{p_E}(1+r)^k

(1+r^*)^{2n+k} = \frac{p_E^*}{p_T^*}\frac{p_T}{p_E}(1+r)^k

From earlier, we had

\frac{p_E^*}{p_E} = \frac{p_T^*}{p_T}(1+r^*)^{2n} \hfill \textup{(Relative goods prices)}

Putting the two equations together gives us:

(1+r^*)^{2n+k} = (\frac{p_E}{p_T}(1+r^*)^{2n})\frac{p_T}{p_E}(1+r)^k = (1+r^*)^{2n}(1+r)^k

(1+r^*)^{k} = (1+r)^k

r = r^*

Thus, we have the Second Fundamental Theorem of Interstellar Trade:

If sentient beings may hold assets on two planets in the same inertial frame, competition will equalize the interest rates on the two planets.

In his conclusion, Krugman writes

I have not even touched on the fascinating possibilities of interstellar finance, where spot and forward exchange markets will have to be supplemented by conditional present markets.

I was thinking this was a light-hearted paper on a light-hearted topic but looks like there’s a lot more out there.  Thirty-five years later, Krugman gives a mention (archived) on the topic.

Hong Kong’s Infamous Kowloon Walled City Rebuilt as Amusement Park (Gizmodo)

Hong Kong’s Infamous Kowloon Walled City Rebuilt as Amusement Park (Gizmodo)

 

“The juxtaposition of a high-tech Japanese toilet in an authentically grimy bathroom has to be seen to be believed,” he writes.

I do have an objection to that, which is that you can find plenty of “authentic,” though perhaps not often grimy, Japanese toilets in modern Japan today.