## “What to Maximize if you Must,” A. Heifetz, C. Shannon & Y. Spiegel (2007)

Here’s the other interesting evolution of preferences paper that I mentioned in the previous post. Fair warning: the mathematical level of the paper is very high unless, perhaps, you are used to using the Whitney C(k) topology and relative prevalence on infinite dimensional manifolds in your everyday work! The basic idea is fairly straightforward, though.

We know from many examples that being “irrational” can help you in games. For example, if you are the “crazy” type who always plays “In” in the Centipede Game, your fully rational opponent will wait until the very last stage to end the game, giving both of you much higher payoffs than in the Nash equilibria. Equate payoffs with evolutionary fitness in some sense – perhaps higher payoffs in the game means more influence, or more frequent transactions in the future market, or whatever. The centipede game example, among many others, suggests that deviations and biases may not be weeded out by evolutionary selection (a technical point, but for the curious, here we allow more robust forms of selection than just the replicator dynamic). But how general is this argument? Heifetz et al prove that is quite general indeed.

Let each player get a payoff equal to their standard utility from the outcome of the game plus a disposition which is a function of the player’s own strategy, other players’ strategies and a parameter lying somewhere close to zero. For instance, if u(x,y) represents player 1’s utility from his own strategy x and opponent strategy y, while v(x,y) represents player 2’s utility, a payoff with disposition for player one might be p(x,y,e)=u(x,y)+ev(x,y). If e is greater than zero, then player 1 is altruistic. If e is less than zero, he is spiteful. If e equals zero, he has standard preferences. The conception of disposition here is broad enough to account for a variety of psychological tendencies, among other interpretations.

Let players both choose strategies from open subsets of finite-dimensional Euclidean space. Now consider a “generic” game, where a game is a set of payoffs (continuous in both player’s strategies) and a set of dispositions for each player, and where genericity has a standard measure-theoretic definition. For almost every set of payoffs and dispositions, can we find a parameter e which gives the player higher payoff than he would get if e=0? If so, then there is a disposition which will not disappear by evolutionary selection in that particular game. And it turns out that, for almost every such manifold, we can find an appropriate e. Why are these dispositions useful? We assumed payoff functions are in C^3, so in general, (pure) Nash equilibria will be locally unique. Since e is required to be close to 0, a minor disposition in one’s own strategy will directly have very little effect of payoffs. But it may have a large indirect effect by causing opponent’s to change their behavior as a result of the disposition. Think of the centipede game: a unilateral deviation to “In” in the first period is not very useful for increasing payoffs since rational opponents will play “Out” the next period anyway. But a disposition which causes you to play “Out” often is useful when known to the opponent, because she will then also play “In” until the last period, even if she a standard homo economicus.

3 final notes: First, the centipede game is an imperfect example because it has a finite strategy space. Technically, the “almost every” result applies only to games with open set of R^n strategy spaces (marginal analysis is used in the proof), but the intuition on why dispositions are useful remains the same. Second, there is a sense in which the results still hold if the strategy space is infinite dimensional (an infinitely repeated game, for instance). The basic problem there is that Nash equilibria are not usually locally unique because of various kinds of folk theorems. See the paper for details on this point. Third, the main proof has an interesting implication for delegation. If you can delegate game-playing to someone who is almost but not exactly like you, in almost every game there exists a delegate who would make it worth your while. Rarely, it turns out, is it best to play your own games yourself!

http://elsa.berkeley.edu/users/cshannon/wp/what.pdf (July 2004 Working Paper – final version in JET 2007)