We initiate the study of the natural multiplayer generalization of the classic continuous Colonel Blotto game. The two-player Blotto game, introduced by Borel as a model of resource competition across $n$ simultaneous fronts, has been studied extensively for a century and seen numerous applications throughout the social sciences. Our work defines the multiplayer Colonel Blotto game and derives Nash equilibria for various settings of $k$ (number of players) and $n$. We also introduce a "Boolean" version of Blotto that becomes interesting in the multiplayer setting. The main technical difficulty of our work, as in the two-player theoretical literature, is the challenge of coupling various marginal distributions into a joint distribution satisfying a strict sum constraint. In contrast to previous works in the continuous setting, we derive our couplings algorithmically in the form of efficient sampling algorithms.
In the Colonel Blotto game, two players with a fixed budget simultaneously allocate their resources across n battlefields to maximize the aggregate value gained from the battlefields where they have the higher allocation. Despite its long-standing history and important applications, the Colonel Blotto game still lacks a complete Nash equilibrium characterization in its most general form where players are asymmetric and battlefields' values are heterogeneous across battlefields and different between the two players---this is called the Generalized Colonel Blotto game. In this work, we propose a…
The Colonel Blotto Problem proposed by Borel in 1921 has served as a widely applicable model of budget-constrained simultaneous winner-take-all competitions in the social sciences. Applications include elections, advertising, R&D and more. However, the classic Blotto problem and variants limit the study to competitions over a finite set of discrete battlefields. In this paper, we extend the classical theory to study multiplayer Blotto games over arbitrary measurable battlegrounds, provide an algorithm to efficiently sample equilibria of symmetric "equipartionable" Generalized Blotto games, and…
We introduce the Colonel Blotto game with favoritism, an extension of the famous Colonel Blotto game where the winner-determination rule is generalized to include pre-allocations and asymmetry of the players' resources effectiveness on each battlefield. Such favoritism is found in many classical applications of the Colonel Blotto game. We focus on the Nash equilibrium. First, we consider the closely related model of all-pay auctions with favoritism and completely characterize its equilibrium. Based on this result, we prove the existence of a set of optimal univariate distributions -- which ser…
This paper explores the Nash equilibria of a variant of the Colonel Blotto game, which we call the Asymmetric Colonel Blotto game. In the Colonel Blotto game, two players simultaneously distribute forces across $n$ battlefields. Within each battlefield, the player that allocates the higher level of force wins. The payoff of the game is the proportion of wins on the individual battlefields. In the asymmetric version, the levels of force distributed to the battlefields must be nondecreasing. In this paper, we find a family of Nash equilibria for the case with three battlefields and equal levels …
In the Colonel Blotto game, which was initially introduced by Borel in 1921, two colonels simultaneously distribute their troops across different battlefields. The winner of each battlefield is determined independently by a winner-take-all rule. The ultimate payoff of each colonel is the number of battlefields he wins. This game is commonly used for analyzing a wide range of applications such as the U.S presidential election, innovative technology competitions, advertisements, etc. There have been persistent efforts for finding the optimal strategies for the Colonel Blotto game. After almost a…
We consider a stochastic version of the well-known Blotto game, called the gladiator game. In this zero-sum allocation game two teams of gladiators engage in a sequence of one-to-one fights in which the probability of winning is a function of the gladiators' strengths. Each team's strategy consists of the allocation of its total strength among its gladiators. We find the Nash equilibria and the value of this class of games and show how they depend on the total strength of teams and the number of gladiators in each team. To do this, we study interesting majorization-type probability inequalitie…
The Complexity of Computing Nash Equilibria in Colonel Blotto Games
National and Kapodistrian University of Athens（雅典国立卡波蒂斯坦大学）；Archimedes, Athena Research Center（阿基米德雅典研究中心）；National Technical University of Athens（雅典国立技术大学）
The Colonel Blotto game is a renowned resource allocation problem with a long-standing literature in game theory (almost 100 years). However, its scope of application is still restricted by the lack of studies on the incomplete-information situations where a learning model is needed. In this work, we propose and study a regret-minimization model where a learner repeatedly plays the Colonel Blotto game against several adversaries. At each stage, the learner distributes her budget of resources on a fixed number of battlefields to maximize the aggregate value of battlefields she wins; each battle…
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) m…
We study the problem of computing Nash equilibria of zero-sum games. Many natural zero-sum games have exponentially many strategies, but highly structured payoffs. For example, in the well-studied Colonel Blotto game (introduced by Borel in 1921), players must divide a pool of troops among a set of battlefields with the goal of winning (i.e., having more troops in) a majority. The Colonel Blotto game is commonly used for analyzing a wide range of applications from the U.S presidential election, to innovative technology competitions, to advertisement, to sports. However, because of the size of …
In competitive adversarial environments, it is often advantageous to obfuscate one's strategies or capabilities. However, revealing one's strategic intentions may shift the dynamics of the competition in complex ways. Can it ever be advantageous to reveal strategic intentions to an opponent? In this paper, we consider three-stage Colonel Blotto games in which one player can choose whether or not to pre-commit resources to a single battlefield before play begins. This pre-commitment is public knowledge. In response, the opponent can either secure the battlefield by matching the pre-commitment w…
Multi-agent systems are designed to concurrently accomplish a diverse set of tasks at unprecedented scale. Here, the central problems faced by a system operator are to decide (i) how to divide available resources amongst the agents assigned to tasks and (ii) how to coordinate the behavior of the agents to optimize the efficiency of the resulting collective behavior. The focus of this paper is on problem (i), where we seek to characterize the impact of the division of resources on the best-case efficiency of the resulting collective behavior. Specifically, we focus on a team Colonel Blotto game…
Not Yet: Humans Outperform LLMs in a Colonel Blotto Tournament
HSE University（俄罗斯高等经济学院）；New Economic School（新经济学校）；Central Economic Mathematical Institute, Russian Academy of Sciences（俄罗斯科学院中央经济数学研究所）；Adyghe State University（阿迪格国立大学）；Moscow Institute of Physics and Technology（莫斯科物理技术学院）；Innopolis University（因诺波利斯大学）
Resource allocation games such as the famous Colonel Blotto (CB) and Hide-and-Seek (HS) games are often used to model a large variety of practical problems, but only in their one-shot versions. Indeed, due to their extremely large strategy space, it remains an open question how one can efficiently learn in these games. In this work, we show that the online CB and HS games can be cast as path planning problems with side-observations (SOPPP): at each stage, a learner chooses a path on a directed acyclic graph and suffers the sum of losses that are adversarially assigned to the corresponding edge…
Successful algorithms have been developed for computing Nash equilibrium in a variety of finite game classes. However, solving continuous games -- in which the pure strategy space is (potentially uncountably) infinite -- is far more challenging. Nonetheless, many real-world domains have continuous action spaces, e.g., where actions refer to an amount of time, money, or other resource that is naturally modeled as being real-valued as opposed to integral. We present a new algorithm for {approximating} Nash equilibrium strategies in continuous games. In addition to two-player zero-sum games, our …
Learning and equilibrium computation in games are fundamental problems across computer science and economics, with applications ranging from politics to machine learning. Much of the work in this area revolves around a simple algorithm termed \emph{randomized weighted majority} (RWM), also known as "Hedge" or "Multiplicative Weights Update," which is well known to achieve statistically optimal rates in adversarial settings (Littlestone and Warmuth '94, Freund and Schapire '99). Unfortunately, RWM comes with an inherent computational barrier: it requires maintaining and sampling from a distribu…
Nokia Bell Labs（诺基亚贝尔实验室）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔国立理工学院）；MPI-SWS（马克斯·普朗克软件系统研究所）；Safran Tech（赛峰科技）；Univ. of Southampton（南安普顿大学）
Resource allocation games such as the famous Colonel Blotto (CB) and Hide-and-Seek (HS) games are often used to model a large variety of practical problems, but only in their one-shot versions. Indeed, due to their extremely large strategy space, it remains an open question how one can efficiently learn in these games. In this work, we show that the online CB and HS games can be cast as path planning problems with side-observations (SOPPP): at each stage, a learner chooses a path on a directed acyclic graph and suffers the sum of losses that are adversarially assigned to the corresponding edge…
The General Lotto game is a popular variant of the famous Colonel Blotto game, in which two opposing players allocate limited resources over many battlefields. In this paper, we consider incomplete and asymmetric information formulations regarding the resource budgets of the players. In particular, one of the player's resource budget is common knowledge while the other player's is private. We provide complete equilibrium characterizations in the scenario where the private resource budget is drawn from an arbitrary Bernoulli distribution. We then show that these characterizations can be used to…
Many efficient algorithms have been designed to recover Nash equilibria of various classes of finite games. Special classes of continuous games with infinite strategy spaces, such as polynomial games, can be solved by semidefinite programming. In general, however, continuous games are not directly amenable to computational procedures. In this contribution, we develop an iterative strategy generation technique for finding a Nash equilibrium in a whole class of continuous two-person zero-sum games with compact strategy sets. The procedure, which is called the double oracle algorithm, has been su…
