Edge-dominance games on graphs

TL;DR

Edge-dominance games on graphs study a symmetric zero-sum dynamic where players move between adjacent states with a geometric payoff tied to parent-child relations. The authors develop a topology- and geometry-based framework, leveraging the block-cut tree and thinned block-cut tree to characterize equilibria under girth constraints and absence of certain small unbalanced cycles. They identify three on-path pure equilibria (k-chase, walking together, static) and provide necessary/sufficient conditions for their existence, with explicit results for trees, graphs with girth at least six, and outerplanar graphs. They further show that a data structure for all pure equilibria can be computed efficiently and that mixed equilibria follow from standard Bellman-based value iteration, connecting to pursuit-evasion and discrete Hotelling models while highlighting unique dynamic and symmetric aspects. The work offers both deep structural insights and practical algorithms for analyzing equilibrium structure in edge-dominance games on broad graph classes, with implications for conflict modeling and competitive feature-location dynamics.

Abstract

We consider zero-sum games in which players move between adjacent states, where in each pair of adjacent states one state dominates the other. The states in our game can represent positional advantages in physical conflict such as high ground or camouflage, or product characteristics that lend an advantage over competing sellers in a duopoly. We study the equilibria of the game as a function of the topological and geometric properties of the underlying graph. Our main result characterizes the expected payoff of both players starting from any initial position, under the assumption that the graph does not contain certain types of small cycles. This characterization leverages the block-cut tree of the graph, a construction that describes the topology of the biconnected components of the graph. We identify three natural types of (on-path) pure equilibria, and characterize when these equilibria exist under the above assumptions. On the geometric side, we show that strongly connected outerplanar graphs with undirected girth at least 4 always support some of these types of on-path pure equilibria. Finally, we show that a data structure describing all pure equilibria can be efficiently computed for these games.

Figures (6)

• Figure 1: Example graphs.
• Figure 2: The two winning configurations for player $y$ in a tree.
• Figure 3: The four unbalanced small cycles, with their minimal vertex highlighted.
• Figure 4: Constructions without cycle-based or static equilibria.
• Figure 5: Illustrations for Theorem \ref{['thm:outerplanar']}.

Theorems & Definitions (52)

• Theorem 1.1: Informal version of Theorem \ref{['thm:char4']}
• Definition 2.1
• Definition 2.2
• Remark 1
• Definition 2.3
• Definition 2.4
• Remark 2
• Lemma 2.5
• proof
• Remark 3