Evolutionary Games on Infinite Strategy Sets: Convergence to Nash Equilibria via Dissipativity

TL;DR

This paper develops a comprehensive dissipativity-based framework for evolutionary games with infinite, compact strategy sets, recasting population states as probability measures and payoffs as continuous functions on a strategy space. By extending delta-dissipativity and related stability concepts to infinite-dimensional Banach spaces, it establishes conditions under which Nash equilibria attract and rest points converge under a broad class of dynamics, including monotone games and dynamic payoff models. The theory recovers classical finite-strategy results (e.g., Brown-von Neumann-Nash and impartial pairwise comparison dynamics) as special cases and demonstrates stability via modular verification of the dynamics map and payoff structure. Case studies, notably the continuous war of attrition and smoothing payoffs, illustrate both the power and limitations of finite-dimensional approximations and underscore the need for infinite-dimensional analysis. The framework broadens applicability to dynamic payoffs, ensures robustness to model variations, and provides a foundation for analyzing complex, continuous-action strategic interactions in engineering and economics.

Abstract

We consider evolutionary dynamics for population games in which players have a continuum of strategies at their disposal. Models in this setting amount to infinite-dimensional differential equations evolving on the manifold of probability measures. We generalize dissipativity theory for evolutionary games from finite to infinite strategy sets that are compact metric spaces, and derive sufficient conditions for the stability of Nash equilibria under the infinite-dimensional dynamics. The resulting analysis is applicable to a broad class of evolutionary games, and is modular in the sense that the pertinent conditions on the dynamics and the game's payoff structure can be verified independently. By specializing our theory to the class of monotone games, we recover as special cases existing stability results for the Brown-von Neumann-Nash and impartial pairwise comparison dynamics. We also extend our theory to models with dynamic payoffs, further broadening the applicability of our framework. Throughout our analyses, we identify and elaborate on new technical conditions that are key in extending dissipativity theory from finite to infinite strategy sets, such as compactness of the set of Nash equilibria and evolution of dynamic payoffs within a compact positively invariant set. We illustrate our theory using a variety of case studies, including a novel, continuous variant of the war of attrition game.

Figures (9)

• Figure 1: We study the evolutionary dynamics model \ref{['eq: edm']} from a control-theoretic lens by viewing the evolution as "open-loop" dynamics $v$ controlled by the game's feedback payoffs $\rho(t) = F(\mu(t))$.
• Figure 2: The infinite-strategy BNN dynamics do not converge to the Nash equilibrium since $\int_S g d\mu(t) \not\to \int_S g d\mu^\star$ as $t\to\infty$ for the bounded continuous function $g(s) = \max\left\{0, \frac{1}{T-s^\star}(s-s^\star)\right\}$.
• Figure 3: Evolution of the distribution function $s \mapsto \mu(t)([0,s])$ for continuous war of attrition on $S = [0,2]$ under BNN dynamics with uniform initial distribution $\mu_0$.
• Figure 4: Evolution of the distribution function $s \mapsto \mu(t)([0,s])$ for continuous war of attrition on $S = [0,2]$ under BNN dynamics with Gaussian initial distribution $\mu_0$ (mean $1$, variance $0.1$).
• Figure 5: Evolution of the distribution function $s \mapsto \mu(t)([0,s])$ for continuous war of attrition game under BNN dynamics with smoothing, together with Gaussian initial distribution $\mu_0$ (mean $1$, variance $0.1$) and initial payoff $\rho_0 = F(\mu_0)$.

Theorems & Definitions (55)

• Lemma 1: parthasarathy1967probability
• Definition 1
• Example 1
• Definition 2
• Example 2
• Example 3
• Definition 3
• Definition 4
• Definition 5
• Definition 6