Quantal Response Equilibrium

An alternative game theoretic equilibrium concept, with bounded rationality, from economists Richard D. McKelvey and Thomas R. Palfrey. The QRE concept applies discrete choice models from econometrics to finding mixed strategy equilibria of games. These are not (usually) Nash equilibria.

The most commonly used model is the "logit equilibrium", based on the multinomial logit:

                      exp( λ(EU_i) )
P_i =         ----------------------------
                  Σ_{j=1 to N}exp( λ(EU_j) )

P_i is the probability of a particular player choosing strategy i, out of the N possible strategies for that player.

EU_i us the expected utility from choosing strategy i. But, since this is a game, EU_i depends on what the other player(s) are doing:

EU_i = Σ_{k=1 to M}( Q_kU_ik )

where Q_k is the probability of the other player choosing their strategy k out of M possible strategies, and U_ik is the (certain) utility if strategies i and k are chosen.

So now, you've got a gnarly nonlinear system of equations in your P's and Q's. Solve it (I'd suggest numerically) and you've got a Quantal Response Equilibrium.

The randomness works nicely in this, and λ's really cool. It's a rationality parameter. When λ gets close to zero, the resulting probability distributions approach the uniform distribution, implying complete irrationality. When λ gets large, the differences in utilities are magnified, and the play approaches rational, in fact approaching a Nash equilibrium.

With an intermediate λ value, the result is often intuitively appealing. It's not merely adding noise around the Nash equilibrium. Since all players are noisily responding to each others' noisy behavior, the results can be strikingly different.