Help your rivals when they are numerous

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.


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.