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_{k}U_{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.