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Zero-sum Random Games on Directed Graphs

Luc Attia, Lyuben Lichev, Dieter Mitsche, Raimundo Saona, Bruno Ziliotto

TL;DR

This paper considers a class of two-player zero-sum games on directed graphs whose vertices are equipped with random payoffs of bounded support known by both players and shows convergence at a double-exponential rate in terms of the expansion.

Abstract

This paper considers a class of two-player zero-sum games on directed graphs whose vertices are equipped with random payoffs of bounded support known by both players. Starting from a fixed vertex, players take turns to move a token along the edges of the graph. On the one hand, for acyclic directed graphs of bounded degree and sub-exponential expansion, we show that the value of the game converges almost surely to a constant at an exponential rate dominated in terms of the expansion. On the other hand, for the infinite $d$-ary tree that does not fall into the previous class of graphs, we show convergence at a double-exponential rate in terms of the expansion.

Zero-sum Random Games on Directed Graphs

TL;DR

This paper considers a class of two-player zero-sum games on directed graphs whose vertices are equipped with random payoffs of bounded support known by both players and shows convergence at a double-exponential rate in terms of the expansion.

Abstract

This paper considers a class of two-player zero-sum games on directed graphs whose vertices are equipped with random payoffs of bounded support known by both players. Starting from a fixed vertex, players take turns to move a token along the edges of the graph. On the one hand, for acyclic directed graphs of bounded degree and sub-exponential expansion, we show that the value of the game converges almost surely to a constant at an exponential rate dominated in terms of the expansion. On the other hand, for the infinite -ary tree that does not fall into the previous class of graphs, we show convergence at a double-exponential rate in terms of the expansion.
Paper Structure (10 sections, 11 theorems, 50 equations, 1 figure)

This paper contains 10 sections, 11 theorems, 50 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weakly transitive games with sub-exponential expansion
  4. Oriented directed games
  5. Weakly transitive games with controlled expansion
  6. Main results
  7. Outline of the proofs.
  8. Perspectives
  9. Plan of the paper.
  10. Classical results

Key Result

Proposition 1

Every oriented directed game is $\delta$-transient for all $\delta\in (0,1/2)$.

Figures (1)

  • Figure 1: The figure depicts part of a tiling with two types of square tiles. The vertices and the edges of the planar graph originating from the tiling are depicted in blue and red, respectively. Each horizontal edge is oriented from left to right and every vertical edge is oriented from bottom to top. One may choose $z^*$ to be the bottom left vertex of a small square and $M=6$.

Theorems & Definitions (26)

  • Definition 1
  • Definition 2: $\delta$-transient games
  • Remark 1
  • Remark 2
  • Proposition 1
  • Example 1: Games on tilings
  • Example 2: Games on directed chains of graphs
  • Theorem 1
  • Theorem 2
  • Lemma 1: Corollary 2.27 in JLR00
  • ...and 16 more