“Unbeatable Imitation,” P. Duersch, J. Oechssler & B. C. Schipper (2012)

People, particularly in relatively unimportant situations, rely on heuristics rather than completely rational foresight. But using heuristics in modeling seems to me undesirable, because players using heuristics can easily be abused by more strategic players. For instance, consider the game of fighter pilot Chicken as in the movie Top Gun. Both players prefer going straight while the opponent swerves to swerving when the opponent swerves (hence showing lack of nerve) to swerving when the opponent goes straight (hence showing a unique lack of nerve) to going straight when the opponent goes straight (hence crashing). Consider playing Chicken over and over against an heuristic-based opponent. Perhaps the opponent simply best responds to whatever you did in the previous period. In this case, if I go straight in period 1, the opponent swerves in the next period, and if I swerve, the opponent goes straight. Therefore, I’ll go straight in periods 1 through infinity, knowing my opponent will swerve in every period except possibly the first. The sophisticated player will earn a much higher payoff than the unsophisticated one.

Duersch et al show that, in every 2×2 symmetric game and in a large class of N-by-N symmetric two-player games, a simple heuristic called “imitation” has an undiscounted average payoff identical to that which can be achieved by an opponent playing any strategy at all. In imitation, I retain my strategy each period unless the opponent earned strictly more than I did in the previous period, in which case I copy him. Consider Chicken again. If I go straight and the opponent swerves, then I know he will go straight in the following period. In the next period, then, I can either crash into him (causing him to swerve two periods on) or swerve myself (causing him to go straight two periods on). In any case, I can at best get my opponent to swerve while I go straight once every two periods. By symmetry, in the periods where this doesn’t happen, I can at best get the payoff from swerving when my opponent goes straight, meaning my average payoff is no better than my heuristic-based opponent! This is true no matter what strategy is used against the imitating opponent.

Now imitation will fail in many games, of course. Consider Rock-Paper-Scissors. If I play Rock when you play Scissors, then since you imitate, you will switch to Rock in the next period. Knowing this, I will play Paper, and so on, winning every period. Games that have this type of cycling possibility allow me to extract arbitrarily larger higher payoff than the imitating opponent. What’s interesting is that, in finite symmetric two-player games between an imitator and an agent with perfect rationality, games with a possibility of cycling in some subgame are the only ones in which the imitator does not earn the same average payoff per period as the rational player! Checking this condition is difficult, but games with no pure equilibrium in the relative payoff game (i.e., the game where payoffs for each player are equal to the difference in payoffs between players in the original game, hence making the original game zero-sum) always have a cycle, and games which are quasiconcave never do. Many common games (oligopoly competition, Nash bargaining, etc.) can be written as quasiconcave games.

Imitation is really pretty unique. The authors give the example of a 3×3 symmetric oligopoly game, where strategy 1 is “produce Cournot quantity”, strategy 2 is “produce Stackelberg follower quantity” and strategy 3 is “produce Stackelberg leader quantity.” The game has no subgames with cycles as defined above, and hence imitators and the rational player earn the same average payoff (if rational player plays Stackelberg leader and I play something else, then he earns more than me, hence I imitate him next period, hence he best responds by playing Stackelberg follower). Other heuristics do much worse than imitation. A heuristic where you simply best reply just plays Stackelberg follower forever, for example.

This result is quite interesting, and the paper is short; the “useful insight on the worst page” test of a quality paper is easily satisfied. I like this work too because it is related to some ideas I have about the benefits of going first. Consider shifting a symmetric simultaneous game to a symmetric sequential game. Going first has no benefit except that it allows me to commit to my action (and many negatives, of course, including the inability to mix strategies). Likewise a heuristic rule allows the heuristic player to commit to actions without assuming perfection of the equilibrium. So there is a link between “optimal” heuristics and the desire of a rational player to commit to his action in advance if he could so choose.

November 2011 Working Paper (IDEAS version). Final version published in the September 2012 issue of Games and Economic Behavior.