Preference graphs: a combinatorial tool for game theory

TL;DR

This paper investigates the preference graph, a combinatorial representation of normal-form games where edges encode unilateral improvements and sinks correspond to pure Nash equilibria. It shows the preference graph is a foundational tool for understanding game structure, dominated strategies, and key classes (ordinal potential, supermodular, weakly acyclic), and it connects this graph to core dynamics such as fictitious play and the replicator dynamic. The authors articulate basic graph-theoretic properties, contrast the preference graph with the best-response graph, and demonstrate how sink equilibria provide a computable and predictive lens for a wide range of dynamics, including Markov-based approximations and the Price of Sinking. Overall, the work advocates a graph-centric framework for analyzing and applying game-theoretic concepts to new problems, with potential for broader modeling via partially ordered preferences and dynamics.

Abstract

The preference graph is a combinatorial representation of the structure of a normal-form game. Its nodes are the strategy profiles, with an arc between profiles if they differ in the strategy of a single player, where the orientation indicates the preferred choice for that player. We show that the preference graph is a surprisingly fundamental tool for studying normal-form games, which arises from natural axioms and which underlies many key game-theoretic concepts, including dominated strategies and strict Nash equilibria, as well as classes of games like potential games, supermodular games and weakly acyclic games. The preference graph is especially related to game dynamics, playing a significant role in the behaviour of fictitious play and the replicator dynamic. Overall, we aim to equip game theorists with the tools and understanding to apply the preference graph to new problems in game theory.

Figures (15)

• Figure 1: Examples of preference graphs: the $2\times 2$ games Matching Pennies (\ref{['fig:MP 1']}) and Coordination (\ref{['fig:CO 1']}), and the symmetric zero-sum game Rock-Paper-Scissors (\ref{['fig: RPS 1']}). In symmetric zero-sum games the preference graph has a special form, where nodes are strategies and arcs represent the winning strategy in any pair. See Section \ref{['sec: symmetric zero-sum']}.
• Figure 2: The preference graphs of $2\times 2$ games biggar_graph_2023: Matching Pennies (MP, \ref{['fig:MP']}), Coordination (CO, \ref{['fig:CO']}), Single-Dominance (SD, \ref{['fig:SD']}), Double-Dominance (DD, \ref{['fig:DD']}).
• Figure 3: The relationship between affine, ordinal, strategic and preference equivalence, from biggarthesis. The arrows denote that one is a sub-relation of the other.
• Figure 4: Two drawings of Shapley's graph shapley_topics_1964, and a typical payoff matrix representation krishna_convergence_1998. Figure \ref{['fig:shapley-square']} reflects the structure of the payoff matrix in Figure \ref{['fig:shapley payoff']}, while Figure \ref{['fig:shapley-cycle']} more clearly shows the connectivity. The graph has three sources which all have arcs into a single sink equilibrium, which is a 6-cycle. This graph appears as the 6-cycle-source graph in biggar_graph_2023.
• Figure 5: The Inner Diamond graph biggar_graph_2023, its best-response subgraph and a zero-sum payoff matrix which generates it. The preference graph has a single sink equilibrium, which is a pure Nash equilibrium. The best-response subgraph has an additional sink equilibrium, which is a 4-cycle.

Theorems & Definitions (32)

• Definition 3.1: Preference graph
• Definition 3.2: Weighted preference graph
• Definition 3.3
• Definition 3.4
• Remark 1
• Theorem 3.5: biggar_graph_2023candogan_flows_2011
• Theorem 3.6: biggar_graph_2023
• Definition 3.7
• Lemma 3.8
• Lemma 3.9