[Paper Review] Pure Nash Equilibria and Best-Response Dynamics in Random Games

Ben Amiet, Andrea Collevecchio+2 more|arXiv (Cornell University)|May 26, 2019

Game Theory and ApplicationsDecision Sciences48 references3 citations

TL;DR

This paper studies pure Nash equilibria (PNE) and best-response dynamics (BRD) in large random games with two strategies per player and i.i.d. payoffs allowing ties. By linking random games to percolation theory, it shows that the number of PNE grows geometrically with the number of players when ties have positive probability, and that BRD almost surely converges to a PNE when the tie probability is small—establishing a phase transition governed solely by the tie probability.

ABSTRACT

In finite games mixed Nash equilibria always exist, but pure equilibria may fail to exist. To assess the relevance of this nonexistence, we consider games where the payoffs are drawn at random. In particular, we focus on games where a large number of players can each choose one of two possible strategies, and the payoffs are i.i.d. with the possibility of ties. We provide asymptotic results about the random number of pure Nash equilibria, such as fast growth and a central limit theorem, with bounds for the approximation error. Moreover, by using a new link between percolation models and game theory, we describe in detail the geometry of Nash equilibria and show that, when the probability of ties is small, a best-response dynamics reaches a Nash equilibrium with a probability that quickly approaches one as the number of players grows. We show that a multitude of phase transitions depend only on a single parameter of the model, that is, the probability of having ties.

Motivation & Objective

• To understand the existence and distribution of pure Nash equilibria (PNE) in large random games where payoffs are i.i.d. and ties are allowed.
• To analyze the convergence behavior of best-response dynamics (BRD) in such games, particularly whether PNE are reachable via iterative strategy updates.
• To establish a formal connection between random games and percolation models to characterize the geometric structure of PNE sets.
• To identify the critical role of the tie probability α in determining phase transitions in PNE count and BRD convergence.
• To provide asymptotic results, including a central limit theorem and error bounds, for the number of PNE under general i.i.d. payoff distributions with ties.

Proposed method

• Model a random game with N players, each choosing one of two strategies, and i.i.d. payoff distributions that allow ties with probability α.
• Construct a random oriented graph representing best-response relationships between strategy profiles, where edges indicate profitable deviations.
• Establish a correspondence between this game graph and a percolation model on the hypercube, using the set of PNE as a key structural component.
• Use tools from percolation theory to analyze the connectivity and size of components in the graph, particularly those containing PNE.
• Apply probabilistic bounds and summability arguments (via Borel-Cantelli and Markov's inequality) to prove almost sure convergence of BRD to a PNE when α is small.
• Derive asymptotic results on the number of PNE, including geometric growth and a central limit theorem with explicit error bounds.

Experimental results

Research questions

• RQ1 How does the number of pure Nash equilibria grow asymptotically as the number of players increases, when ties in payoffs are allowed?
• RQ2 What is the probability that best-response dynamics converges to a pure Nash equilibrium in large random games with positive tie probability?
• RQ3 How does the geometry of the set of pure Nash equilibria—its connectivity and component structure—depend on the tie probability α?
• RQ4 Can the random game structure be mapped to a percolation model, and what insights does this mapping provide about equilibrium existence and dynamics?
• RQ5 What phase transitions occur in the behavior of PNE and BRD as the tie probability α varies?

Key findings

• The number of pure Nash equilibria grows geometrically with the number of players N when the probability of ties α > 0.
• A central limit theorem holds for the number of PNE, with explicit bounds on the approximation error.
• When the tie probability α is sufficiently small, best-response dynamics converges to a pure Nash equilibrium with probability approaching one as N increases.
• The set of PNE exhibits a phase transition: for larger α, some equilibria become inaccessible via BRD due to disconnected components in the best-response graph.
• The geometry of PNE is characterized via a percolation model on the hypercube, where the existence of large connected components in the preimage of best-response maps determines reachability.
• The entire asymptotic behavior—number of PNE, convergence of BRD, and geometric structure—depends on a single parameter: the probability of ties α.