Colonel Blotto Game: An Analysis and Extension to Networks

TL;DR

This paper extends the classical Colonel Blotto game to network settings, analyzing how network topology influences equilibrium existence, uniqueness, and players' payoffs, with implications for resource allocation contests.

Contribution

It introduces a network-based framework for the Colonel Blotto game, revealing how topology affects equilibrium properties and strategic outcomes.

Findings

Unique equilibrium exists in 2-regular topologies under certain conditions.

Regular topologies lead to better player payoffs and higher field entry rates than random topologies.

Dispersion of field weights reduces average player payoff.

Abstract

The Colonel Blotto game, introduced by Borel in the 1920s, is often used for modeling various real-life settings, such as elections, lobbying, etc. The game is based on the allocation of limited resources by players to a set of fields. Each field is ``won'' and a corresponding field-specific value is obtained by the player who sends the most resources. In this paper, we formulate a discrete Blotto game played on a general \textit{accessibility network} (i.e., the bipartite graph made of players and the fields they can allocate resources to). The primary goal is to find how the topology of the accessibility network controls the existence and uniqueness of equilibrium allocations, and how it affects the fraction of fields that are entered and the average payoff of players at equilibrium. We establish that, in a 2-regular topology, when the values of fields are close enough and the number of players is not a multiple of 4, then there is a unique equilbrium. We also prove that players are better off and fields are more likely to be entered in a regular topology than a random topology. We find numerically that dispersion of field weights negatively affects average player payoff. The main contribution is a framework for analyzing contests where players are permitted access to some (but not necessarily all) venues of competition.