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Non-alternating mean payoff games

Tom Meyerovitch, Aidan Young

TL;DR

The paper studies a non-alternating variant of mean payoff games where Alice commits to an infinite sequence before Bob responds with full knowledge of that sequence. It frames the problem on two graphs $G$ and $H$ with a score function $P$ on edge pairs and connects to the covering radius in constrained coding theory and to adversarial ergodic optimization. The main results show that for irreducible $G$ and $H$, there exists a Nash equilibrium for the infinite-round non-alternating game, and the equilibrium payoff equals the limit of the normalized finite-round payoffs, with a subadditivity argument guaranteeing the limit and constructive strategies illustrating optimality. The paper also demonstrates that irreducibility is necessary for the convergence guarantee, provides counterexamples in reducible cases, and raises open questions about computation, periodicity of equilibria, and comparisons with the alternating game.

Abstract

We present and study a variant of the mean payoff games introduced by A. Ehrenfeucht and J. Mycielski. In this version, the second player makes an infinite sequence of moves only after the first player's sequence of moves has been decided and revealed. Such games occur in the computation of the covering radius of constrained systems, a quantity of interest in coding theory.

Non-alternating mean payoff games

TL;DR

The paper studies a non-alternating variant of mean payoff games where Alice commits to an infinite sequence before Bob responds with full knowledge of that sequence. It frames the problem on two graphs and with a score function on edge pairs and connects to the covering radius in constrained coding theory and to adversarial ergodic optimization. The main results show that for irreducible and , there exists a Nash equilibrium for the infinite-round non-alternating game, and the equilibrium payoff equals the limit of the normalized finite-round payoffs, with a subadditivity argument guaranteeing the limit and constructive strategies illustrating optimality. The paper also demonstrates that irreducibility is necessary for the convergence guarantee, provides counterexamples in reducible cases, and raises open questions about computation, periodicity of equilibria, and comparisons with the alternating game.

Abstract

We present and study a variant of the mean payoff games introduced by A. Ehrenfeucht and J. Mycielski. In this version, the second player makes an infinite sequence of moves only after the first player's sequence of moves has been decided and revealed. Such games occur in the computation of the covering radius of constrained systems, a quantity of interest in coding theory.
Paper Structure (4 sections, 4 theorems, 47 equations, 4 figures)

This paper contains 4 sections, 4 theorems, 47 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Defining the non-alternating game
  3. Nash equilibrium in the non-alternating game
  4. Open questions about the non-alternating game

Key Result

Theorem 2.1

For any initial edges $(e_0,f_0) \in \mathcal{E}_{G} \times \mathcal{E}_{H}$, the normalized Nash-equilibrium payoff of the $n$-round alternating game converges to the Nash equilibrium of the infinite-round alternating game. That is: Moreover, there exist positional strategies $\psi_A : \mathcal{E}_{G} \times \mathcal{E}_{H} \to \mathcal{E}_{G}, \psi_B : \mathcal{E}_{G} \times \mathcal{E}_{H} \to

Figures (4)

  • Figure 1: Pictured here are the graphs referenced in \ref{['ex:non_alt_reducible_graphs_infinite_limit_counterexample']}, where the top row depicts the graph $G$ with vertices $\mathcal{V}_{G} = \{P, M\}$, while the bottom row depicts the graph $H$ with vertices $\mathcal{V}_{H} = \{X,Y, Z\}$.
  • Figure 2: The graphs $G, H$ referenced in \ref{['ex:alt_and_non_alt_games_different_limits']}. Solid arrows denote edges present in both $G$ and $H$. Dashed arrows denote edges of $G$ that are not edges of $H$.
  • Figure 3: Pictured here are the graphs $G, H$ referenced in \ref{['ex:irrational_example']}, where the top row depicts a graph $G$ with vertices $\mathcal{V}_{G} = \{O\}$, and the bottom row depicts a graph $H$ with vertices $\mathcal{V}_{H} = \{X, Y, Z\}$.
  • Figure 4: Pictured here are the graphs $G, H$ referenced in \ref{['ex:irrational_example']}, where the top row depicts the graph $G$ with vertices $\mathcal{V}_{G} = \{P, M\}$, and the bottom row depicts a graph $H$ with vertice $\mathcal{V}_{H} = \{X, Y, Z\}$.

Theorems & Definitions (11)

  • Theorem 2.1
  • Theorem 3.1
  • Remark 3.2
  • Lemma 3.3
  • proof
  • proof : Proof of \ref{['thm:non_alt_nash_lim']}
  • Example 3.4
  • Proposition 4.2
  • Example 4.3
  • Example 4.7
  • ...and 1 more