Game theory is widely used as a behavioral model for strategic interactions in biology and social science. It is common practice to assume that players quickly converge to an equilibrium, e.g. a Nash equilibrium, but in some situations, convergence fails.

Extending the work of Pangallo/Heinrich/Farmer (SciAdv: eaat1328) to multiplayer games on networks, we generate payoff matrices at random to calculate the likelihood of convergence over ensembles of normal-form games. We study two possible formalisms based on best reply dynamics, in which each player myopically uses the best response to the opponents’ last actions. We show that the presence of best reply cycles predicts nonconvergence of six well-known learning algorithms and show that cycles become more dominant as games get more complicated and more competitive. This suggests a problem, since many situations in economics are complicated and competitive, and if the key behavioral assumption is wrong, then the predictions of the model are likely wrong too.

We give analytical results and numerical evidence for a scaling property, which allows us to relate games of many players and few actions to 2-player games with more actions and conjecture a similar property on ensembles of games where the underlying network of players is not fully connected.


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