Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Finding pure Nash equilibria in large random games

Andrea Collevecchio, Tuan-Minh Nguyen, Ziwen Zhong

TL;DR

The paper analyzes BRD-like dynamics on random two-action, N-player games with i.i.d. payoffs that may contain ties. By coupling the oriented hypercube of BRD with bond percolation on the cube, it proves that BRD and a broad class of nearest-neighbor processes reach a Pure Nash Equilibrium before encountering any trap with high probability as $N\to\infty$, for all $\alpha\in(0,1)$ (i.e., $\beta=(1-\alpha)/2\in(0,1/2]$). The proof uses a strong construction of the process and a cluster-based absorbing-state analysis to show PNE-containing clusters outnumber traps and that boundary visits efficiently funnel the walk to a PNE. This establishes a robust convergence guarantee for BRD in large random games and demonstrates that percolation geometry provides a powerful framework for understanding stochastic dynamics on high-dimensional strategic spaces.

Abstract

Best Response Dynamics (BRD) is a class of strategy updating rules to find Pure Nash Equilibria (PNE) in a game. At each step, a player is randomly picked, and the player switches to a "best response" strategy based on the strategies chosen by others, so that the new strategy profile maximises their payoff. If no such strategy exists, a different player will be chosen randomly. When no player wants to change their strategy anymore, the process reaches a PNE and will not deviate from it. On the other hand, either PNE may not exist, or BRD could be "trapped" within a subgame that has no PNE. We consider a random game with $N$ players, each with two actions available, and i.i.d. payoffs, in which the payoff distribution may have an atom, i.e. ties are allowed. We study a class of random walks in a random medium on the $N$-dimensional hypercube induced by the random game. The medium contains two types of obstacles corresponding to PNE and traps. The class of processes we analyze includes BRD, simple random walks on the hypercube, and many other nearest neighbour processes. We prove that, with high probability, these processes reach a PNE before hitting any trap.

Finding pure Nash equilibria in large random games

TL;DR

The paper analyzes BRD-like dynamics on random two-action, N-player games with i.i.d. payoffs that may contain ties. By coupling the oriented hypercube of BRD with bond percolation on the cube, it proves that BRD and a broad class of nearest-neighbor processes reach a Pure Nash Equilibrium before encountering any trap with high probability as , for all (i.e., ). The proof uses a strong construction of the process and a cluster-based absorbing-state analysis to show PNE-containing clusters outnumber traps and that boundary visits efficiently funnel the walk to a PNE. This establishes a robust convergence guarantee for BRD in large random games and demonstrates that percolation geometry provides a powerful framework for understanding stochastic dynamics on high-dimensional strategic spaces.

Abstract

Best Response Dynamics (BRD) is a class of strategy updating rules to find Pure Nash Equilibria (PNE) in a game. At each step, a player is randomly picked, and the player switches to a "best response" strategy based on the strategies chosen by others, so that the new strategy profile maximises their payoff. If no such strategy exists, a different player will be chosen randomly. When no player wants to change their strategy anymore, the process reaches a PNE and will not deviate from it. On the other hand, either PNE may not exist, or BRD could be "trapped" within a subgame that has no PNE. We consider a random game with players, each with two actions available, and i.i.d. payoffs, in which the payoff distribution may have an atom, i.e. ties are allowed. We study a class of random walks in a random medium on the -dimensional hypercube induced by the random game. The medium contains two types of obstacles corresponding to PNE and traps. The class of processes we analyze includes BRD, simple random walks on the hypercube, and many other nearest neighbour processes. We prove that, with high probability, these processes reach a PNE before hitting any trap.
Paper Structure (11 sections, 9 theorems, 62 equations, 4 figures, 1 table)

This paper contains 11 sections, 9 theorems, 62 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction.
  2. Motivation
  3. Further background
  4. Basic concepts from Game Theory
  5. Random games
  6. Main result
  7. Strategies of the proof
  8. Randomly oriented hypercube and bond percolation
  9. Proof of Theorem 2.3
  10. Strong construction of X
  11. Absorbing states clustering together

Key Result

Theorem 1.3

Assume that $\alpha\in (0,1)$ i.e. $\beta\in (0, 1/2)$. Let $(\mathbf{X}^{(N)})_N$ be a sequence of processes such that for each $N$, the distribution of $\mathbf{X}^{(N)}$ is given by annealed and its quenched law satisfies assump. Then the sequence of events $(\{\tau^{(N)}<\xi^{(N)}\})_{N\ge 2}$ h

Figures (4)

  • Figure 1: Iterations needed for Best Response Dynamics (red) and Simple Random Walk (blue) to reach a PNE conditional on not hitting any trap for $N=15$ and $\alpha\in \{0.5, 0.6, 0.7, 0.8, 0.9 \}$, with 500 trials. The boxes and the whiskers are corresponding to (0.25, 0.75) and (0.05, 0.95) quantile intervals respectively.
  • Figure 2: Representation of $\Gamma_{3}$ on $\{0,1\}^{3}$. Red vertices are PNE and greens lines represent possible best-response paths.
  • Figure 3: Representation of $\Gamma_{4}$ on $\{0,1\}^{4}$. Blue vertices form a trap, red vertices represent PNE and green lines correspond to possible best-response paths.
  • Figure 4: Every pair of designated vertices (in red) is connected with a dash line if the Hamming distance between them is less than or equal to $2m_{\beta}-2$. The union of discrete balls with radius $m_{\beta}-1$ centered at connected designated vertices forms a cluster.

Theorems & Definitions (23)

  • Definition 1.1
  • Example 1.1
  • Example 1.2
  • Theorem 1.3
  • Remark 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • ...and 13 more