Playing Games without Observing Payoffs

Optimization under uncertainty is central to a number of fields, and it is by now well-known that one can learn to play a repeated game without a priori knowledge of the game, as long as one observes the payoffs (even if one does not see the opponent’s play). We consider the complementary scenario in which one does not observe the payoffs but does observe the opponent’s play. Curiously, we show that for an interesting class of games one can still learn to play well. In particular, for any symmetric two-person game, one can nearly match the payoff of the opponent (who may have full knowledge of the game), without ever observing a single payoff. The approach employed is a familiar one: attempt to mimic the opponent’s play. However, one has to be careful about how one mimics an opponent who may know that they are being mimicked. This paper has two contributions: it (a) extends our understanding of optimization under uncertainty by modeling a new setting in which one can play games optimally; and (b) introduces a new algorithm for being a copycat, one which is strategically sound even against an opponent with a superior informational advantage.