The Attractor of the Replicator Dynamic in Zero-Sum Games

TL;DR

This work addresses the long-standing question of the day-to-day behavior of the replicator dynamic in two-player zero-sum games. It introduces the preference graph to capture discrete player priorities and proves a unique global attractor exists, equal to the content of the sink component of this graph, independent of payoff magnitudes. The symmetric case yields a tournament structure, while the non-symmetric case uses von Neumann symmetrisation to embed the flow into a symmetric framework, establishing global asymptotic stability via a Lyapunov-type argument on the sink mass $x_H$. The results clarify the connection between chain recurrence and Nash equilibria through the graph structure, and offer a robust, payoff-free prediction tool for learning dynamics in zero-sum settings.

Abstract

In this paper we characterise the long-run behaviour of the replicator dynamic in zero-sum games (symmetric or non-symmetric). Specifically, we prove that every zero-sum game possesses a unique global replicator attractor, which we then characterise. Most surprisingly, this attractor depends only on each player's preference order over their own strategies and not on the cardinal payoff values, defined by a finite directed graph we call the game's preference graph. When the game is symmetric, this graph is a tournament whose nodes are strategies; when the game is not symmetric, this graph is the game's response graph. We discuss the consequences of our results on chain recurrence and Nash equilibria.

Figures (2)

• Figure 1: The preference graphs (Definition \ref{['def: fundamental graph']}) of two zero-sum games: (left) Rock-Paper-Scissors (symmetric), and (right) Matching Pennies (non-symmetric).
• Figure 2: A zero-sum game (left) and its associated preference graph (right). The sink component of the graph consists of all profiles other than $(a,a)$. By Theorem \ref{['attractor characterisation']}, this game's unique replicator attractor is the content (Definition \ref{['def: content']}) of the sink component, which is the union of the strategy spaces of the subgames $\{a,b,c\}\times \{b,c\}$ and $\{b,c\}\times\{a,b,c\}$, represented by the shaded region on the graph. Note that the strategy space of the game is 4-dimensional, with the attractor a 3-dimensional region on the boundary.

Theorems & Definitions (36)

• Definition 3.1
• Definition 3.2
• Definition 3.3: Non-Symmetric Zero-Sum Replicator Equation
• Definition 3.4: Symmetric Zero-Sum Replicator Equation
• Definition 3.5
• Definition 4.1: biggar2023replicator
• Lemma 4.2: Uniqueness
• Theorem 4.3: The Attractor of the Replicator
• proof
• Lemma 4.4: Invariance and Minimality