Game Dynamics and Equilibrium Computation in the Population Protocol Model

TL;DR

The paper studies how local game dynamics in a population protocol setting give rise to global distributions over strategies. It introduces a distributional equilibrium (DE) concept for pairwise, locally updated RD games, and analyzes a concrete k-IGT family of dynamics where GTFT agents adjust their generosity across a discretized set G. By connecting these updates to high-dimensional weighted Ehrenfest processes, the authors derive exact stationary distributions (multinomial with weights), mixing-time bounds, and conditions under which the population converges to an ε-approximate DE, with ε = O(1/k) and convergence time O(k n log n). The results illuminate fundamental tradeoffs between local state space size, convergence rate, and equilibrium quality, and open avenues for broader equilibrium computation in population protocols and multi-agent dynamics.

Abstract

We initiate the study of game dynamics in the population protocol model: $n$ agents each maintain a current local strategy and interact in pairs uniformly at random. Upon each interaction, the agents play a two-person game and receive a payoff from an underlying utility function, and they can subsequently update their strategies according to a fixed local algorithm. In this setting, we ask how the distribution over agent strategies evolves over a sequence of interactions, and we introduce a new distributional equilibrium concept to quantify the quality of such distributions. As an initial example, we study a class of repeated prisoner's dilemma games, and we consider a family of simple local update algorithms that yield non-trivial dynamics over the distribution of agent strategies. We show that these dynamics are related to a new class of high-dimensional Ehrenfest random walks, and we derive exact characterizations of their stationary distributions, bounds on their mixing times, and prove their convergence to approximate distributional equilibria. Our results highlight trade-offs between the local state space of each agent, and the convergence rate and approximation factor of the underlying dynamics. Our approach opens the door towards the further characterization of equilibrium computation for other classes of games and dynamics in the population setting.

Figures (2)

• Figure 1: When $k=6$, three examples showing how the parameter value of a $\mathsf{GTFT}$ agent is probabilistically updated under the $k$-IGT dynamics. Note that conditioned on a $\mathsf{GTFT}$ agent $u$ being sampled as the first agent, $u$ increments its parameter value with probability $(1-\beta)$ and decrements its parameter value with probability $\beta$ (where in both cases, the values are truncated to the range $[0, {\widehat{g}}]$).
• Figure 2: Example when $k=3$ and $m=3$ of the space $\mathcal{X}$ and the set of non-zero transitions specified by $P$ (denoted by a directed edge between two nodes). Transitions colored blue have coefficient $a$, and transitions colored red have coefficient $b$. Considering the outward transition edges from a node $(\genfrac{}{}{0pt}{1}{i}{j}) \in \mathcal{X}$, downward vertical transitions are weighted by $j/n$, upward vertical transitions are weighted by $(1-(i+j)/n)$, downward diagonal transitions are weighted by $j/n$, and upward diagonal transitions are weighted by $i/n$.

Theorems & Definitions (45)

• Definition 1.1
• Definition 1.2
• Definition 2.0: Incremental-Generosity-Tuning (IGT) Dynamics
• Proposition 2.0
• Definition 2.0: $(k, a, b ,m)$-Ehrenfest Process
• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Theorem 2.4
• Proposition 2.4