Probabilistic Interpretations of Integrability for Game Dynamics

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 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.

Population Games and Deterministic Evolutionary Dynamics

“…36 For a game-theoretic interpretation of this integrability condition, see Sandholm (2014b). 37 We can replace actual payoffs with excess payoffs as the arguments of M andM, since maximizers of actual payoffs are also maximizers of excess payoffs.…”

Riemannian game dynamics

“…18 See Hart and Mas-Colell (2001), Hofbauer and Sandholm (2009), and Sandholm (2010a, 2014. 19 Weibull (1995) and Björnerstedt and Weibull (1996) introduce revision protocols for imitative dynamics.…”

Passivity and Evolutionary Game Dynamics

“…where the function ξ ji : X → TX assigns a tangent vector to each x in X, and ji : R n → R n is called a revision protocol which depends on a directional derivative of the total payoff. We refer to (17) as perturbed if the perturbation v satisfies the conditions (16), and as unperturbed if v = 0. 7 The idea of imposing perturbations on the average payoff appeared in game theory and economics to investigate the effect of random perturbations or disutility on choice models [15], [20], [21], to model human choice behavior [22], and to analyze the effect of social norms in economic problems [23].…”

“…The condition (P1) ensures that the integral P V(p, x)•dp does not depend on the choice of the path P . This condition is an important requirement in establishing stability [16]. The condition (P2) implies that with the payoff vector fixed, the population state x(t) evolves along a trajectory for which the function S decreases.…”

“…Since the EDM (17) is δ-passive if v = 0, by Theorem III.3, we can find a storage function S : R n × X → R + for which the conditions (P1) and (P2) hold for η = 0. In what follows, we show that when the perturbation v satisfies (16), the resulting perturbed EDM is strictly output δ-passive with a storage function S(p, x)…”