Sampling equilibrium through descriptive simulations

A definition of sampling equilibrium was introduced in (Osborne and Rubinstein 1998). A dynamic version of the model was introduced in (Sethi 2000). However, a descriptive simulation based on the above idea of procedural rationality (i.e. using the same algorithm of behavior) gave different results, than those achieved in (Osborne and Rubinstein 1998) and (Sethi 2000). The simulation was a starting point for new definitions of both sampling dynamics and sampling equilibrium

Naive fictitious play in an evolutionary environment

A fictitious play algorithm with a unit memory length within an evolutionary environment is considered. The aggregate average behavior model is proposed and analyzed. The existence, uniqueness and global asymptotic stability of equilibrium is proved for games with a cycling property. Also, a noisy version of the algorithm is considered, which gives rise to a system with a unique, globally asymptotically stable equilibrium for any game

Fictitious play in an evolutionary environment

We consider continuous time versions of the fictitious play updating algorithm in an evolutionary environment. We derive two forms of continuous-time limit, both defining approximations to this algorithm. The first has the form of a first-order partial differential equation, which we solve explicitly. The dynamic for a distribution of strategies is also derived, which we show can be written in a form similar to a positive definite dynamic. The asymptotic solution (in the ultra long run) is discussed for 2-player, 2-strategy co-ordination and anti-coordination games, and we show convergence to Nash equilibrium in both cases. The second, and better, approximation is in the form of a diffusion equation. This is considerably more difficult to analyze. However, we derive a formal solution and show that it leads to the same asymptotic limit for the distribution of strategies as the 1st-order approximation for 2-player, 2-strategy anti-coordination games.