# On chimpanzees and Nash equilibria.

This is a pretty cool study, I think.  Amongst other things, the authors compared the performance of pairs of chimpanzees in a zero-sum game to the performance of human pairs.  And they noticed that pairs of chimpanzees typically perform much closer to the Nash equilibria than do human pairs.

I do wish the authors had done slightly more, though.  With their set-up, they were testing pairs who could not communicate with one another, and after each game the winner received some reward in sight of the loser.  So it seems as though the set-up could be extended to testing human / chimpanzee pairs.  The same prizes, food pellets or whatever, could be given to each, with the understanding that the human can exchange their reward for money after the entire series of games.

Because the authors stress that chimpanzees performed better than humans – “better” here meaning “closer to Nash equilibrium.”  As in, the pair performed close to what you’d expect for perfect rational actors.

But there is another clear sense of “better,” for individuals.  If you were playing, and you noticed your partner deviating from Nash equilibrium, you ought to deviate even farther.  A Nash equilibrium is the probabilistic set of strategies that eliminates any incentive for your partner to change their plan, while maximizing your expected rewards.  But the Nash equilibrium only maximizes your rewards against a perfect opponent: if your opponent is making mistakes, then you could win more often by using a strategy other than the one you’d pick at Nash equilibrium.

As an example, you could consider the “symmetric matching” game from the paper.  Nash equilibrium is for both players to have P=0.5 for each choice.  But if one player, the matcher, say, is choosing “left” with 60% probability, and “right” with 40%, then the opponent has an incentive to deviate from P=0.5 as well.  In this case, the highest possible payoff would be to start choosing “right” all the time, but for a repeated game that might be counterproductive: the matcher would notice and compensate for his or her original error.  But by choosing “right” with any probability greater than 50%, the mismatcher is still  capitalizing on the opponent’s error and will win a greater than expected share of the prizes.

This can be visualized by modifying graphs like those shown in Figure 2 of the paper.  Specifically, horizontal and vertical lines passing through the Nash equilibrium could be added, dividing the graph into four quadrants.  Points on the lines would represent  cases where one player had deviated from Nash equilibrium but the other did not take advantage of that deviation.  In top right and bottom left quadrants, the deviations of each player aligns to benefit the “matcher,” and the other quadrants represent outcomes benefiting the “mismatcher.”

Which is a lot of pre-amble to get to my point.  Sorry about that.  But my question is, are chimpanzees actually better at these sorts of games?  As in, if you paired a human with a chimpanzee, given that humans routinely deviate from Nash equilibrium in the asymetric type of game, would the final outcome typically favor the chimpanzee?  That’s something that seems to be implied by the paper, but they didn’t test it.  And it seems like they could.