Help your rivals when they are numerous

by lsusr1 min read22nd Jan 20213 comments

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World OptimizationPractical
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Suppose you are playing a game of chess. Chess is a zero-sum bipolar game. Anything your opponent wins is something you lose, and vice versa. If a trade becomes available then you should take if the benefit to you is greater than the benefit to your opponent. If the benefit to your opponent is greater than the benefit to you then you should not take the trade.

Suppose you're playing a 4-player game of Catan. You care about your rank. That is, you care about whether you get 1st place, 2nd place, 3rd place or 4th place. Suppose an opponent offers you a trade that generates 3 non-zero-sum utilons between you and her. She offers to keep 2 utilons for herself and give you 1 utilon. You cannot negotiate. You have no reputation to uphold. Should you accept the trade?

Generally-speaking, yes.

Explanation

If you had one opponent then you should not accept her trade. If you had two opponents then the trade would neither harm nor hurt you on average. In this scenario, you have three opponents. The average benefit to a single opponent is only which is less that the gain to yourself. You gain a edge against your average opponent.

To put this another way, if you repeated the trade with each of your three opponents then each of them would gain 2 utilons while you gained a total of 3 utilons. You would land 1 utilon ahead.

Let's call the benefit to you , the benefit to a single opponent and the benefit to the average opponent . You should make a trade if .

The more opponents you have, the more advantageous is for your to accept win-win trades with a single opponent. In the limit case , you can ignore the benefits to individual opponents entirely and accept every trade that benefits you at all. Free trade between a large number of independent actors turns globally zero-sum games into locally non-zero-sum games.

When everyone is a rival, nobody is.

Second-Order Approximation

The above analysis treats opponents as all equal. Opponents are not all equal. If an opponent is far behind with little chance of catching up then you can accept higher for equivalent . The same goes for an opponent far ahead of you.

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Opponents are not all equal.

Nor is their inequality linear.  You should also reject it if you're near-even with this person, and there is a larger gap between you and the other players (in either direction).  But really, it's all down to expected value - take your distribution of possible paths through the game, conditional on each side of this decision.  Take the one with the highest mean.

Aside: you are misusing "utility" in this example.  What is offered is not utilons, but resources that (may or may not) impact the utility you get from the final position of the game.

Part of what this post says is that you shouldn't use timeless decision theory or use any game theory analysis  when playing a 4-player game of Catan.

I don't think you do a good job at supporting that claim. Interactions between players during a game of Catan are an itereated game.

I have a number of issues with this post.

 

First, as others have mentioned, opponents are very much not equal. Further, timing is important: certain trades you should be much more or less likely to take near the end of the game, for example.

Second, I don't think it's valid to look at expected values when all you care about is rank.  Expectation is very much a concept for when you care about absolute amounts.

Third, which perhaps sums everything up: I don't see a valid notion of utility / utility maximization for board games, other than perhaps "probability of winning," which makes this circular ("if you're trying to win, you should make moves that increase your probability of winning"). Utility is meant to put a linear scale on satisfaction with a given state of the world. When discussing what to do in a board game, one usually presumes the objective is to win, and satisfaction derives ultimately from winning. The closest thing you usually see to a "utility" number on an intermediate state is a heuristic, as used in e.g.: chess AIs, where you might give yourself 5 points for having a pawn in a center square. If I'm remembering my undergrad correctly, these heuristics are intended to approximate log-likelihoods of victory, but they certainly lack the soundness required to think about expected utility.

 

Let's switch out of Catan, and to a game that hopefully people here know but is more directly combative: Diplomacy. Pray tell me how you propose to assign a utility score to putting a navy in the Black Sea.