This can be demonstrated by application of the dynamics to the travelers’ dilemma which is a deliberately constructed social dilemma. The game has only a 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.

Both, analytical tools and agent based simulation are used to investigate the strategy evolution 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. There is an interior solution in addition to the pure Nash equilibrium and for m>3 one can prove the stability of the interior solution while the strict Nash equilibrium is unstable. Even more interesting are the results for k>1. By means of agent based simulation one can observe for sufficiently large experience parameters limit cycles where a few of the more cooperative strategies alternate. On the other hand, if k grows too large these limit cycles will be destroyed and all trajectories approach the uncooperative Nash equilibrium. One can even prove analytically the asymptotic stability if the experience parameter exceeds the threshold k(m) = (m+1)/2.

So far, only a homogeneous setting was discussed. Next, a homogeneous population (agents have experience parameter k=10) is injected with more sophisticated agents which will use 100 samples before they fix their decisions (k=100). One might suggest that the limit cycles get easily destroyed, since the parameters k=10 and m=20 are already close to the threshold where the inefficient Nash equilibrium should result. But it turns out that the heterogeneous population can cope with a remarkably high share of sophisticated agents.

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