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Probabilistic Interpretations of Integrability for Game Dynamics

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Abstract

In models of evolution and learning in games, a variety of proofs of convergence rely on the assumption that the players’ choice functions are integrable. This assumption does not have an obvious game-theoretic interpretation. We address this question by introducing probability models defined in terms of piecewise-smooth closed curves through \(\mathbb{R}^{n}\); these curves describe cycles in the performances of the available actions. We establish that a choice function is integrable if and only if in the probability model induced by each such curve, the rate at which players switch to a randomly drawn action is uncorrelated with a certain binary signal. The binary signal specifies whether the performance of the randomly drawn action is improving or worsening, and can also be interpreted as a signal about the performances of actions other than the one randomly drawn.

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Notes

  1. For overviews of the heuristic learning literature, see Young [24] and Hart [10].

  2. This requirement on switching rates is not unreasonable in the models described above, in which agents are assumed to act in a completely myopic way. It would be much less reasonable in alternative models in which agents make simple forecasts about the likely directions of change in the performances of their actions before deciding which action to play. The use of such forecasts can generate dynamics with excellent convergence properties—see Shamma and Arslan [22] and Arslan and Shamma [1].

  3. For background on population games and evolutionary dynamics, see Sandholm [21].

  4. For formal analyses linking the stochastic decisions of individual agents to a deterministic description of aggregate behavior via a differential equation, see Benaïm and Weibull [2] and Roth and Sandholm [17].

  5. The restriction of the excess payoff vector π to the complement of \(\mathbb{R}^{n}_{-}\) in condition (1) reflects two facts. First, the excess payoff vector cannot lie in \(\mathrm{int}(\mathbb{R}^{n}_{-})\), since this would mean that all actions generate a lower than average payoff. Second, the excess payoff vector \(\hat{F}(x)\) lies on \(\mathrm{bd}(\mathbb{R}^{n}_{-})\) if and only if x is a Nash equilibrium of F: see Proposition 3.4 of Sandholm [19].

  6. See Sandholm [18, 20].

  7. Here and in what follows, we call π a “performance vector” to emphasize that it is an element of \(\mathbb{R}^{n}\), but despite this terminology, π should be viewed as a point in space rather than a velocity through space. Also, the interpretation of \(\rho(\pi) \in\mathbb{R}^{n}\) suggests that its components should be nonnegative, but this property is not needed in our analysis.

  8. Our results depend on defining our probability model using the parameterization of C that moves at a constant ℓ 1 rate. In particular, if C is a closed curve of performance vectors induced by a closed trajectory through the set of population states, it can be endowed with this trajectory’s parameterization, but we do not use this parameterization to define our probability model.

  9. Without the independence of Z and I, this implication would not hold. The situation is analogous to a basic one in Bayesian statistics. There, two observations that are independent conditional on an unknown (i.e., random) parameter are not unconditionally independent, because the value of the first observation provides information about the parameter, which in turn provides information about the second observation.

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Acknowledgements

I thank George Mailath for asking the question that motivated this paper, and Larry Samuelson and two referees for helpful comments. Financial support from the National Science Foundation under Grants SES-0092145 and SES-1155135 is gratefully acknowledged.

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  1. Department of Economics, University of Wisconsin, 1180 Observatory Drive, Madison, WI, 53706, USA

    William H. Sandholm

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Sandholm, W.H. Probabilistic Interpretations of Integrability for Game Dynamics. Dyn Games Appl 4, 95–106 (2014). https://doi.org/10.1007/s13235-013-0082-y

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