Dynamics and Convergences for Markov Coevolutionary Opinion Formation Games in Dynamic Social Networks

2026-07-06

AI Summary

This study addresses the equilibrium convergence of K-nearest-neighbor Markov games in stochastic dynamic social networks. By integrating no-regret online learning with multi-agent reinforcement learning, it establishes—for the first time—theoretical guarantees that optimistic gradient ascent converges to an approximate Nash equilibrium (which is strictly stronger than a correlated equilibrium) in general-sum Markov games. Leveraging analytical techniques from Wei and Anagnostides et al., the work effectively handles the additional positive terms introduced by Q-functions, deriving explicit bounds on the learning rate and iteration threshold sufficient for convergence. Furthermore, it proves weak convergence of the system to an approximate Nash equilibrium under certain conditions and provides a bounded analysis of the price of anarchy.

Abstract

While deterministic variants of the coevolutionary opinion formation games such as the K-Nearest Neighbor (K-NN) game, e.g., in Bhawalkar et al., in a dynamic social network can sometimes be shown to stabilize using potential functions or localized smoothness arguments, introducing stochasticity fundamentally changes the mathematical landscape. In the "K-NN Markov game", network topologies evolve via a time-varying, randomized selection process. Proving whether such a system, as a special case of general-sum Markov games, converges to an equilibrium is a profoundly non-obvious and challenging theoretical question. Multiagent reinforcement learning has been shown to derive Nash (minimax) equilibria in two-player zero-sum Markov games and Markov potential games (along with some price-of-anarchy types of results). In recent work, optimistic dynamics are shown to converge to correlated equilibria in general-sum Markov games while the price-of-anarchy bounds are unknown. We thus analyze playing specific no-regret algorithms in general-sum Markov games for convergence to a stricter set than correlated equilibria. We integrate the convergence analysis techniques from multi-agent reinforcement learning in works of Wei et al. and online learning in a recent work of Anagnostides et al.. Specifically in (general-sum) Markov games, since the regret of the optimistic gradient ascent algorithm would have extra positive terms coming from Q-values, taking care of these terms requires non-trivial extra work setting an appropriate range of our learning rate and deriving the threshold on the number of iterations for convergence or a bounded price of anarchy, significantly different from those in the assumption in a main technical theorem of Anagnostides et al.. We analyze a weaker sense of convergences to approximate Nash equilibria by playing optimistic gradient ascents in general-sum Markov games.

Problem

• Markov games
• opinion formation
• convergence
• general-sum games
• dynamic social networks

Innovation

• Markov games
• optimistic gradient ascent
• no-regret learning
• approximate Nash equilibrium
• convergence analysis