NoteThe logit dynamic for games with continuous strategy sets
Introduction
Evolutionary game theory seeks to provide dynamic foundations to equilibrium behavior. The two most well known evolutionary dynamics are the replicator dynamic (Taylor and Jonker, 1978) and the logit dynamic (Fudenberg and Levine, 1998; Hofbauer and Sandholm, 2002, Hofbauer and Sandholm, 2007). The usual approach in this field of research is to consider games with a finite number of strategies and assess whether dynamics converge to Nash equilibria in such games. Alongside this approach, however, there has also emerged a significant literature that seeks to extend such evolutionary dynamics to games with continuous strategy sets. For example, the continuous strategy version of the replicator dynamic has been studied by Bomze, 1990, Bomze, 1991, Oechssler and Riedel, 2001, Oechssler and Riedel, 2002, Cressman (2005), Cressman and Hofbauer (2005), and Cressman et al. (2006). Infinite dimensional extensions of the Brown–von Neumann–Nash (BNN) dynamic and the pairwise comparison dynamic have been developed by Hofbauer et al. (2009) and Cheung (2014) respectively. The continuous strategy version of the logit dynamic, however, has remained relatively unexplored. This paper seeks to address this lacuna and provide foundations to the logit dynamic for games with continuous strategy sets.
The interest in the logit dynamic in the evolutionary literature emerges from the fact that it preserves a close approximation of the canonical game theoretic best response behavioral model while still being amenable to analysis using standard ODE techniques. Since the finite dimensional logit dynamic has been extensively studied, we believe it is a worthwhile exercise to establish its continuous strategy version. Our more general interest in infinite dimensional evolutionary dynamics is due to two reasons. First, these dynamics are not straightforward extensions of their finite dimensional counterparts. Instead, their analysis requires us to address certain subtle mathematical and technical issues whose resolution offers further insight into how these dynamics behave. For example, since the state space of continuous strategy evolutionary dynamics is the space of probability measures, defining these dynamics requires us to make a non-trivial choice of topology with which to define neighborhoods in this space. As Oechssler and Riedel (2002) show, this has implications on important issues like the choice of stability criterion to assess convergence of solution trajectories to equilibria. Second, economic problems are usually modeled as games with continuous strategy sets. It is to be hoped, therefore, that the development of a theory of infinite dimensional evolutionary dynamics will make evolutionary game theory more amenable for application to economic problems.
We consider a population game with a strategy set that is continuous. In such a game, a population state is a probability measure that specifies the distribution of the use of the various strategies in the population. The logit dynamic for such games is based on the logit choice measure which is a probability measure that maximizes a perturbed version of the players' payoffs, where the perturbation arises from the entropy of the strategy the agent uses (Mattsson and Weibull, 2002). The logit dynamic moves the current population state towards the logit choice measure. Rest points of this dynamic are fixed points of the logit choice measure. We call such a fixed point a logit equilibrium. We note that the basic definition of the continuous logit dynamic has already been provided in Lahkar (2007). That work was, however, of a preliminary nature since it did not contain a very rigorous analysis of the properties of the dynamic. This paper rectifies that problem and establishes a set of technical results that provides precise foundations to the dynamic; namely, the existence of a logit equilibrium, its convergence to a Nash equilibrium as the perturbation factor becomes small, and the existence, uniqueness and continuity of solution trajectories of the logit dynamic. Given the abstract nature of the state space, showing these properties requires us to make appropriate choice of topologies in the space of probability measures. Following earlier work such as Oechssler and Riedel, 2001, Oechssler and Riedel, 2002 and Hofbauer et al. (2009), we choose between the strong and weak topologies for establishing these results. We use the weak topology to show existence of a logit equilibrium and the strong topology to establish results about the solution trajectories of the logit dynamic.
We apply the dynamic to two classes of games—potential games and negative semidefinite games. A potential game is characterized by a real valued function which summarizes information about payoffs in the game (Sandholm, 2001). Negative semidefinite games have the property of “self-defeating externalities” (Hofbauer and Sandholm, 2009) due to which small groups of agents who change strategies find themselves at a payoff disadvantage. It is well known that a wide range of finite dimensional evolutionary dynamics converge to equilibria in both these classes of games (Sandholm, 2010). For games with continuous strategy spaces, it is known that the BNN dynamic (Hofbauer et al., 2009) and pairwise comparison dynamics (Cheung, 2014) converge to Nash equilibria in these games.1 We establish similar convergence results under the logit dynamic. Subject to the technical condition that the initial population state can be characterized with a bounded density function, we show that solution trajectories of the logit dynamic converge to a logit equilibrium under the strong topology in both potential games and negative semidefinite games. In proving these results, we extend the Lyapunov functions that Hofbauer and Sandholm (2007) use to establish convergence of the logit dynamic in finite strategy potential games and negative semidefinite games. For potential games, the Lyapunov function is the entropy-adjusted potential function which increases in value along every solution trajectory of the logit dynamic.
Apart from being of theoretical interest, these results may also be applied to economic problems that can be modeled as potential or negative semidefinite games. For example, Hu (2011) applies the logit dynamic to a continuous strategy model of congestion. His model satisfies negative semidefiniteness and uses the same Lyapunov function that we use. It concludes that the logit equilibrium in the model is asymptotically stable, but does so without clarifying the technical issues that arise in defining the dynamic or in proving convergence results. By establishing the technical foundations of the dynamic, this paper puts Hu's (2011) application on a sounder footing.
Finally, we note that this paper is most closely related to Perkins and Leslie (2014). The authors, in work done simultaneously but independently from our paper, consider a model of stochastic fictitious play in a finite player game with a continuous strategy set. They show that the evolution of mixed strategies of the players in this framework is determined by the logit dynamic. While the two papers have different motivations, they are complementary in certain aspects. First, Perkins and Leslie (2014) combine our dynamical systems results with stochastic approximation theory to develop the learning variant of the logit dynamic. Hence, their approach provides alternative microfoundations to the dynamic. Second, they extend our single population logit dynamic to a multipopulation one. Third, in showing convergence of the logit dynamic to equilibria in zero-sum games, they use the multi-player version of the Lyapunov function that we use for negative semidefinite games. In doing so, they establish certain crucial technical results which we have, in turn, been able to use to prove our results on convergence.
The rest of the paper is structured as follows. In Section 2, we introduce the logit dynamic for population games with continuous strategy spaces. Section 3 establishes the fundamental properties of existence of a logit equilibrium, its convergence to a Nash equilibrium as the perturbation factor becomes small, and the existence, uniqueness and continuity of solution trajectories of the logit dynamic. In Section 4, we discuss the microfoundations of this dynamic using the idea of maximization of the perturbed payoff. Section 5 establishes the convergence of the logit dynamic to logit equilibria in potential games and negative semidefinite games. Section 6 concludes. Some proofs are in Appendix A.
Section snippets
The logit dynamic
We consider a strategic situation in which a continuum of agents, called a population, play a game with a strategy set S which is a compact and convex subset of R. We assume that the mass of agents in the population is 1. Let be the Borel σ-algebra on S. We identify a population state with a probability measure over the measurable space . We denote the set of probability measures on by Δ. A population state describes the distribution of strategies over S. For example, if is a
Logit equilibrium
In order to address fundamental issues like the existence of a logit equilibrium and the existence of a solution trajectory to the logit dynamic, we need to make an appropriate choice of topology. In the finite dimensional case, the choice of topology is not of much consequence since all norms are equivalent. However, in the infinite dimensional case, this choice is important since the structure of the neighborhood of a probability measure can depend upon the topology we choose.
As is standard
Microfoundations of the logit dynamic
Cheung (2014) provides a general derivation of evolutionary dynamics in population games with continuous strategy space by using the notion of revision protocols. Recall that is the space of bounded measurable functions from S to R and that the payoff function . We now define the conditional switch rate ρ as the bounded measurable function such that for a given population state P, is the rate at which an agent who is currently playing strategy
Potential games and negative semidefinite games
We apply the logit dynamic to two classes of games—potential games and negative semidefinite games. Sandholm (2001) and Hofbauer and Sandholm, 2007, Hofbauer and Sandholm, 2009 establish convergence results in such games with finite strategy sets under different evolutionary dynamics.6
Conclusion
In this paper, we have introduced the logit dynamic for population games with continuous strategy sets. We have established the fundamental properties of existence of a logit equilibrium, its convergence to a Nash equilibrium as the perturbation factor becomes small, and the existence, uniqueness and continuity of solution trajectories of the logit dynamic. We have then applied the dynamic to potential games and negative semidefinite games and established convergence to logit equilibria in
Acknowledgements
The authors thank an outstanding referee of this journal for careful reading and many excellent suggestions that improved the paper. They also thank David Leslie, Steven Perkins and Bill Sandholm for comments and suggestions. This work was partially done when the first author was employed in IFMR, Chennai. The author acknowledges the generous support and contribution of IFMR in completing this paper.
References (33)
Pairwise comparison dynamics for games with continuous strategy space
J. Econ. Theory
(2014)Stability of the replicator equation with continuous strategy space
Math. Soc. Sci.
(2005)- et al.
Measure dynamics on a one-dimensional continuous trait space: theoretical foundations for adaptive dynamics
Theor. Popul. Biol.
(2005) - et al.
Stability of the replicator equation for a single-species with a multi-dimensional continuous trait space
J. Theor. Biol.
(2006) - et al.
Evolution in games with randomly disturbed payoffs
J. Econ. Theory
(2007) - et al.
Brown–von Neumann–Nash dynamics: the continuous strategy case
Games Econ. Behav.
(2009) A note on best response dynamics
Games Econ. Behav.
(1999)- et al.
Probabilistic choice and procedurally bounded rationality
Games Econ. Behav.
(2002) - et al.
On the dynamics foundations of evolutionary stability in continuous models
J. Econ. Theory
(2002) - et al.
Stochastic fictitious play with continuous action sets
J. Econ. Theory
(2014)
Potential games with continuous player sets
J. Econ. Theory
Evolutionarily stable strategies and game dynamics
Math. Biosci.
Infinite Dimensional Analysis
Noisy directional learning and the logit equilibrium
Scand. J. Econ.
Stability Theory of Dynamical Systems
Convergence of Probability Measures
Cited by (34)
The logit dynamic in supermodular games with a continuum of strategies: A deterministic approximation approach
2023, Games and Economic BehaviorGeneralized perturbed best response dynamics with a continuum of strategies
2022, Journal of Economic TheoryCitation Excerpt :For example, an important question to resolve is the choice of topology in the space of population states, which takes a measure theoretic form. Following much of the literature (for example, Oechssler and Riedel (2001, 2002), Lahkar and Riedel (2015)) we apply the strong and weak topologies in our analysis. The weak topology is helpful in resolving questions like the existence of a perturbed equilibrium and convergence of solution trajectories of the perturbed best response dynamics to perturbed equilibria in potential and negative definite games.
Monotonicity in the trip scheduling problem
2021, Transportation Research Part B: MethodologicalEvolutionary implementation in aggregative games
2021, Mathematical Social SciencesEvolutionary implementation in a public goods game
2019, Journal of Economic TheoryNonatomic potential games: the continuous strategy case
2018, Games and Economic BehaviorCitation Excerpt :These papers, as well as Hofbauer et al. (2009), also examine a specific class of games called doubly symmetric games, which are examples of potential games. Cheung (2014, 2016) and Lahkar and Riedel (2015) provide a more general definition of potential games with continuous strategy sets. Analogous to the definition of a potential game in Sandholm (2001), they define a potential game as a population game in which the payoff function is equal to the gradient of a potential function.