Best Response - Smoothed Best Response

Smoothed Best Response

Instead of best response correspondences, some models use smoothed best response functions. These functions are similar to the best response correspondence, except that the function does not "jump" from one pure strategy to another. The difference is illustrated in Figure 8, where black represents the best response correspondence and the other colors each represent different smoothed best response functions. In standard best response correspondences, even the slightest benefit to one action will result in the individual playing that action with probability 1. In smoothed best response as the difference between two actions decreases the individual's play approaches 50:50.

There are many functions that represent smoothed best response functions. The functions illustrated here are several variations on the following function:

where represents the expected payoff of action, and is a parameter that determines the degree to which the function deviates from the true best response (a larger implies that the player is more likely to make 'mistakes').

There are several advantages to using smoothed best response, both theoretical and empirical. First, it is consistent with psychological experiments; when individuals are roughly indifferent between two actions they appear to choose more or less at random. Second, the play of individuals is uniquely determined in all cases, since it is a correspondence that is also a function. Finally, using smoothed best response with some learning rules (as in Fictitious play) can result in players learning to play mixed strategy Nash equilibria (Fudenberg & Levine 1998).