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(EUi) )
Σj=1 to Nexp( λ(EUj) )
Pi is the probability of a particular player choosing strategy i, out of the N possible strategies for that player.
EUi us the expected utility from choosing strategy i. But, since this is a game, EUi depends on what the other player(s) are doing:
EUi = Σk=1 to M( QkUik )
where Qk is the probability of the other player choosing their strategy k out of M possible strategies, and Uik 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.