Wednesday, October 09, 2013

What are the equilibria of this game?

In the course of writing a paper, we often construct examples of the sort of mathematical objects under analysis, as a means to figure out what’s true in general. Then the paper gets written, and the examples are stripped away, like the scaffolding from a completed building. Before it gets forgotten, I now provide an online description of the following game (that came up while writing this paper), since it sets you thinking about alternative solution concepts (exact/approximate Nash/correlated equilibria (well-supported or otherwise)), and which one(s) we are supposed to like best. Or maybe it could be used as part of a game theory exercise. (It may be helpful to point out that a well-supported ε-approximate equilibrium is one where no player ever uses a strategy that pays more than ε less than his best response. An approximate equilibrium that is NOT well-supported could allow some player to occasionally play a very poor strategy, provided that it’s with low probability.)

Suppose there are n players, numbered 1 through n. Each player has 2 pure strategies, call them a and b. Payoffs will be either 0 or 1, according to the following rule. Player 1 gets paid 1 to play a and 0 to play b, UNLESS all other players play a, in which case it’s the other way around. Meanwhile, for i>1, player i gets paid 1 to play the same way as player i−1 (otherwise is paid 0).

(If instead player 1 was paid to differ from player n, we would have a generalized Jordan game, with a unique Nash equilibrium where all players flip fair coins to choose a or b.)

So — spoiler alert — I reckon the answers are as follows. For exact Nash equilibrium, it’s not hard to see that players 1 and 2 can’t play pure strategies, and in fact players 1 and 2 should flip fair coins, meanwhile the other players all play 1. For exact correlated equilibrium, one solution (different from the NE) is for player 1 to flip a fair coin, while all the other players flip a single shared fair coin, ie they play the all-a’s vector or the all-b’s vector with equal probability. It looks like there is some freedom for any player to deviate slightly from this, while keeping it well-supported. In the neighbourhood of the original NE, only players 1 and 2 can deviate slightly, if we want the approximate equilibrium to be well-supported. (In fact, I think we can say more strongly, that all the well-supported Nash equilibria are in the vicinity of the exact NE.)

Interestingly, when we give up on the “well-supported” requirement for correlated equilibria, lots of other things become possible. Notably, all the players can play a with high probability 1−O(1/n), and this is what would start to happen if natural learning dynamics were applied to learn an equilibrium. Eventually, the dynamics should do something different; maybe converge to something in the region of the exact NE.

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