Sampling dynamics: an alternative to payoff-monotone selection dynamics

Osborne and Rubinstein introduced sampling equilibria which are based on the concept of ``procedural rationality''. Sethi extended their idea to a dynamic framework which leads to the so called sampling dynamics. Unlike e.g the replicator dynamics this selection dynamics turns out to be neither payoff-monotone nor payoff-positive which has interesting consequences. This can be demonstrated by application to the travelers dilemma, a deliberately constructed social dilemma. The game has just one symmetric Nash equilibrium which is Pareto inefficient. Especially when the travelers have many options this result is rather counter intuitive and indeed there is experimental evidence which indicates that deviation will be likely even though the Nash equilibrium is strict. One can use the fact that strict Nash equilbria must be also sampling equilibria to test for the ``plausibility'' of the standard game theory result. Both, analytical tools and agent based simulation are used to investigate the dynamic stability of sampling equilibria in a generalized travelers dilemma. Two parameters are of interest: the number of strategy options (m) available to each traveler and an experience parameter (k), which indicates the number of samples an agent would evaluate before fixing his decision. The special case (k=1) can be treated analytically. The stationary points of the dynamics must be sampling equilibria and one can calculate that for m>3 there will be an interior solution in addition to the pure Nash equilibrium. Furthermore one can prove that this interior solution is asymptotically stable under the sampling dynamics while the strict Nash equilibrium is unstable (for m>3). This could not happen with any payoff-positive selection dynamic. Even more interesting is the dynamical behavior for k>1. For sufficiently large experience parameters one can observe limit cycles by means of agent based simulation. On the other hand, if k grows too large these limit cycles will be destroyed and all trajectories approach the Nash equilibrium. For any number of options one can can analytically derive a threshold k(m) such that above k(m) the Jacobean of the dynamical system, evaluated for the Nash equilibrium, can only have eigenvalues with negative real parts. One might well argue that for biological systems payoff-monotonicity of selection dynamics should be better preserved. For social systems, on the other hand, the sampling dynamics offers an interesting alternative which may help to explain deviations from Nash behavior in experiments. [1] Osborne, M. J. and Rubinstein, A.: Games with Procedurally Rational Players, in: American Economic Review, 88(4), p.834-847, 1998 [2] Sethi, R.: Stability of Equilibria in Games with Procedurally Rational Players, in: Games and Economic Behavior, 32(1), p.85-104, 2000