Optimal Strategies of Blotto Games: Beyond Convexity

Soheil Behnezhad; Avrim Blum; Mahsa Derakhshan; MohammadTaghi Hajiaghayi; Christos H. Papadimitriou; Saeed Seddighin

Home/Archive/arXiv

arXiv · 1901.04153

Optimal Strategies of Blotto Games: Beyond Convexity

Soheil Behnezhad·Avrim Blum·Mahsa Derakhshan·MohammadTaghi Hajiaghayi·Christos H. Papadimitriou·Saeed Seddighin

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

cs.GT

Keep this discovery

BibTeX RIS

Soheil Behnezhad, Avrim Blum, Mahsa Derakhshan, MohammadTaghi Hajiaghayi, Christos H. Papadimitriou, Saeed Seddighin. 2019-01-14. Optimal Strategies of Blotto Games: Beyond Convexity. https://arxiv.org/abs/1901.04153

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

MMS Allocation for Chores with Online Agent Arrivals

We study the fair allocation of $m$ indivisible chores to $n$ agents with subadditive cost functions arriving online in an arbitrary order. Upon an agent's arrival, we are informed of her cost function and must irrevocably assign her a set of chores. We focus on the Maximin Share (MMS) fairness notion and aim to compute an allocation in which all items are assigned, and no agent incurs a cost more than $\alpha$ times her MMS. Without any prior information about the instance (other than $n$ and $m$), we design an algorithm with a competitive ratio of $O(\min\{n, k\log^{1+\epsilon}k, \log m\})$ for any constant $\epsilon > 0$, where $k$ denotes the number of cost function types. Our bound matches the best known offline approximation guarantees for MMS under subadditive costs and is nearly optimal with respect to all three parameters: we show that even for binary additive cost functions, no online algorithm can achieve a competitive ratio of $o(\min\{n, k\log k, \log m\})$. We then consider the setting in which the $k$ cost function types are known in advance (though the realized types of arriving agents are not). For additive cost functions, we provide an algorithm with a competitive ratio of $O(\min\{\log k, \log(kn)/\log\log(kn)\})$, and show that constant-competitive algorithms do not exist for general $k$, even for the binary additive setting. For binary additive functions when $k \le n$, we propose a $3$-competitive algorithm and establish a lower bound of $2$.

cs.GT↗

arXiv2026

Truncated Noisy Best-Response Algorithms: Toward Game Theoretic Learning with Safety Guarantees

We consider a game theoretic approach to solve multi-agent coordination problems with submodular maximization objectives. It is known for such problems that the Nash equilibria for the corresponding game are always within 50% of the optimal, but that the equilibria which achieve this worst-case bound are not stable. To exploit this instability, we propose a family of algorithms which we call Truncated Noisy Best-Response (TNBR) Algorithms. These algorithms are flexibly characterized by agents asynchronously and stochastically selecting actions from a neighbourhood of their best response payoffs. We compute bounds on the recurrent classes of TNBR algorithms' associated Markov chains. Our bounds fall into two categories: first, "Performance" bounds ensure that TNBR algorithms always have a high-value recurrent state; second, "Safety" bounds ensure that TNBR algorithms never have arbitrarily-bad recurrent states. Furthermore, these two types of bounds are linked by a waterbed-like effect: every game with a poor Safety guarantee necessarily has a favorable Performance guarantee.

cs.GT↗

arXiv2026

Existence of the Core in Approval-Based Committee Elections

We settle the main open question in the theory of approval-based multi-winner elections: we show that there always exists a committee in the core. The core is a stability and group fairness concept. The proof introduces a new voting rule that optimizes an entropy-like objective function over committees and payment systems. All local optima of this objective function lie in the core, which implies that a core committee can be found in polynomial time.

cs.GT↗