[Paper Review] Optimal Strategies of Blotto Games: Beyond Convexity

Soheil Behnezhad, Avrim Blum+4 more|arXiv (Cornell University)|Jan 14, 2019

Merger and Competition AnalysisEconomics, Econometrics and Finance3 citations

TL;DR

This paper presents polynomial-time approximation schemes for Colonel Blotto games under two objectives: maximizing guaranteed expected payoff and maximizing the probability of achieving a minimum utility threshold. By partitioning the non-convex solution space into polynomially many convex polytopes, the authors overcome traditional convex programming limitations and establish the first complexity result showing General Colonel Blotto is exponential time-complete.

ABSTRACT

The Colonel Blotto game, first introduced by Borel in 1921, is a well-studied game theory classic. Two colonels each have a pool of troops that they divide simultaneously among a set of battlefields. The winner of each battlefield is the colonel who puts more troops in it and the overall utility of each colonel is the sum of weights of the battlefields that s/he wins. Over the past century, the Colonel Blotto game has found applications in many different forms of competition from advertisements to politics to sports. Two main objectives have been proposed for this game in the literature: (i) maximizing the guaranteed expected payoff, and (ii) maximizing the probability of obtaining a minimum payoff $u$. The former corresponds to the conventional utility maximization and the latter concerns scenarios such as elections where the candidates' goal is to maximize the probability of getting at least half of the votes (rather than the expected number of votes). In this paper, we consider both of these objectives and show how it is possible to obtain (almost) optimal solutions that have few strategies in their support. One of the main technical challenges in obtaining bounded support strategies for the Colonel Blotto game is that the solution space becomes non-convex. This prevents us from using convex programming techniques in finding optimal strategies which are essentially the main tools that are used in the literature. However, we show through a set of structural results that the solution space can, interestingly, be partitioned into polynomially many disjoint convex polytopes that can be considered independently. Coupled with a number of other combinatorial observations, this leads to polynomial time approximation schemes for both of the aforementioned objectives.

Motivation & Objective

• To develop efficient algorithms for computing near-optimal strategies in Colonel Blotto games with bounded support, addressing the challenge of non-convex solution spaces.
• To provide polynomial time approximation schemes (PTAS) for both maximin expected payoff and (u,p)-maximin strategies, where p is the probability of achieving a minimum utility u.
• To resolve the computational complexity of a generalized Colonel Blotto game, proving it is exponential time-complete.
• To enable practical deployment of equilibrium strategies by reducing support size while preserving near-optimality.

Proposed method

• Partitioning the non-convex solution space of Colonel Blotto into polynomially many disjoint convex polytopes to enable convex optimization techniques.
• Using combinatorial structure and strategic constraints to limit support size in optimal strategies.
• Reducing the Succinct Linear Inequalities problem to General Colonel Blotto via a modified version of Adler’s zero-sum game reduction.
• Encoding feasible troop allocations as subsets of battlefields with complementary assignments to ensure correspondence with payoff matrix entries.
• Defining a utility function that penalizes infeasible allocations and maps feasible ones to entries in a skew-symmetric payoff matrix.
• Leveraging the equivalence between the maxmin of the General Colonel Blotto game and the maxmin of the constructed payoff matrix to prove completeness.

Experimental results

Research questions

• RQ1 Can optimal strategies in Colonel Blotto games be computed with bounded support despite the non-convexity of the solution space?
• RQ2 Is it possible to design polynomial-time approximation schemes for both maximin expected payoff and (u,p)-maximin objectives?
• RQ3 What is the computational complexity of finding the maximin in a generalized version of the Colonel Blotto game?
• RQ4 Can the solution space of Colonel Blotto be decomposed into manageable convex components to enable efficient optimization?
• RQ5 Does a reduction from succinct linear inequalities to General Colonel Blotto preserve the maxmin outcome and establish completeness?

Key findings

• The solution space of Colonel Blotto games can be partitioned into polynomially many disjoint convex polytopes, enabling convex optimization techniques despite overall non-convexity.
• The paper presents the first polynomial-time approximation schemes (PTAS) for both the maximin expected payoff and (u,p)-maximin objectives in Colonel Blotto games.
• General Colonel Blotto is proven to be exponential time-complete via a reduction from Succinct Circuit Value through Succinct Linear Inequalities and Adler’s zero-sum game construction.
• Feasible allocations in the game are encoded as subsets of battlefields with complementary troop assignments, ensuring one-to-one correspondence with payoff matrix rows and columns.
• Infeasible allocations are shown to be dominated and can be safely eliminated from consideration without affecting the game’s maxmin outcome.
• The maxmin of the General Colonel Blotto game is equivalent to the maxmin of the constructed skew-symmetric payoff matrix, with the last component of the maxmin strategy being nonzero iff the original system has a solution.