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Generalized Colonel Blotto Game Generalized Colonel Blotto Game Aidin Ferdowsi1,∗, Anibal Sanjab1,∗, Walid Saad1, and Tamer Bas¸ar2 Abstract— Competitive resource allocation between adver- many variants of the CBG have been studied including those sarial decision makers arises in a wide spectrum of real- with homogeneous valuations [8], [10], symmetric resources, world applications such as in communication systems, cyber- multiple players [4], and heterogeneous resources [11]–[13]. physical systems security, as well as financial and political competition. Hence, developing analytical tools to model and This existing body of literature has primarily focused analyze competitive resource allocation is crucial for devising on analyzing the existence of the equilibrium in a CBG. optimal allocation strategies and anticipating the potential For instance, the seminal work in [9] has proven that an outcomes of the competition. To this end, the Colonel Blotto equilibrium in deterministic strategies (i.e. pure-strategies) game is one of the most popular game-theoretic frameworks for of the Blotto game does not exist when the number of modeling and analyzing such competitive resource allocation problems. However, in many practical competitive situations, considered battlefields is greater than two. As a result, most the Colonel Blotto game does not admit solutions in determin- of the literature has henceforth focused on analyzing the istic strategies and, hence, one must rely on analytically complex mixed-strategy equilibrium of the CBG, which essentially mixed-strategies with their associated tractability, applicability, corresponds to a multi-variate probability distribution over and practicality challenges. In this regard, in this paper, the resources to be allocated over each of the battlefields. In a generalization of the Colonel Blotto game which enables the derivation of deterministic, practical, and implementable this regard, the works in [9], [12], [13] have studied uniform equilibrium strategies is proposed while accounting for scenar- marginal distributions of resources on each battlefield. ios with heterogeneous battlefields. In addition, the proposed However, relying on mixed strategies, as has been the case generalized game factors in the consumed/destroyed resources in all of the CBG literature [8]–[13], presents serious chal- in each battlefield, a feature that is not considered in the lenges to the tractability, applicability, and implementability classical Blotto game. For this generalized game, the existence of a Nash equilibrium in pure strategies is shown. Then, closed- of the derived solutions in real-world environments. In fact, form analytical expressions of the equilibrium strategies are deriving such mixed strategies is often overly complex re- derived and the outcome of the game is characterized, based quiring various mathematical simplifications along the way on the number of each player’s resources and each battlefield’s for tractability [1]–[4], [14], [15]. Moreover, in terms of valuation. The generated results provide invaluable insights on applicability, the optimality of such mixed strategies assumes the outcome of the competition. For example, the results show that, when both players are fully rational, the more resourceful that the game is actually played for an infinitely large number player can achieve a better total payoff at the Nash equilibrium, of times [16]. In addition, previous CBG models assume a result that is not mimicked in the classical Blotto game. win-or-lose settings over each battlefield. In other words, I. INTRODUCTION the player that allocates more resources over a battlefield wins all of its valuation and the other player receives Game-theoretic resource allocation models have become nothing. However, given that resources are consumed over prevalent across a variety of domains such as wireless com- a certain battlefield, even when this battlefield is won or munications [1] and [2], cyber-physical security [3]–[5], fi- lost, these consumed resources must be accounted for in the nancial markets, and political/electoral competitions [6]–[8]. game model. For instance, even when losing a battlefield, One of the mostly adopted game-theoretic frameworks for destroying a portion of the resources of the opponent can be modeling and analyzing such competitive resource allocation considered a gain for the non-winner. arXiv:1710.01381v2 [cs.GT] 2 Mar 2018 problems is the so-called Colonel Blotto game (CBG) [9]. The main contribution of this paper is therefore to develop The CBG captures the competitive interactions between two a fundamentally novel generalization of the CBG dubbed players that seek to allocate resources across a set of battle- as the Generalized Colonel Blotto Game (GCBG) which fields. The player who allocates more resources to a certain captures the rich and broad settings of the classical CBG, battlefield wins it and receives a corresponding valuation. and, yet enables the derivation of tractable, deterministic, The fundamental question that each player seeks to answer and practical solutions to the two-player competitive resource in a CBG is how to allocate its resources to maximize the allocation problem while not being limited to studying win- valuations acquired from the won battlefields. In recent years, or-lose settings over each battlefield. To this end, we first introduce the basics of the proposed GCBG and, then, we This research was supported by the U.S. National Science Foundation under Grants OAC-1541105 and CNS-1446621, and by the Army Research compare its features to the classical CBG. Subsequently, we Laboratory under IoBT Cooperative Agreement Number W911NF-17-2- prove the existence of an equilibrium to the GCBG – a pure- 0196. strategy Nash equilibrium (NE) – and provide a detailed 1Aidin Ferdowsi∗, Anibal Sanjab∗ and Walid Saad are with Wireless@VT, Bradley Department of Electrical and derivation as well as closed-form analytical expressions of Computer Engineering, Virginia Tech, Blacksburg, VA, USA, the pure NE strategies of the GCBG. In addition, using the {aidin,anibals,walids}@vt.edu derived NE strategies, we prove that when both players are 2Tamer Bas¸ar is with the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, IL, USA, [email protected] fully rational, the more resourceful player is able to achieve ∗The first two authors have equally contributed to this paper. an overall payoff that is greater than that of its opponent, and hence, win the game. In contrast with the classical CBG, in For the CBG, it has been proven [9] that for Rb > Ra and our proposed GCBG: nRa > Rb, there exists no pure-strategy NE. As a result, • We derive deterministic equilibrium strategies which for solving the CBG, a common methodology is to explore are practical, in the sense that the players can derive mixed strategies based on which each player j chooses a ∈P exact optimal strategies and do not need to randomize probability distribution (or a multi-variate probability density between possible strategies, function) over j to maximize its potential expected payoff. Q • We characterize low complexity solutions even for a However, the reliance on such mixed strategies introduces large number of battlefields with heterogeneous valua- high analytical complexity and presents challenges in terms tions, of the tractability of the derivation of the solutions as well • We model realistic applications of competitive resource as to the practicality and applicability of these solutions. allocation problems by taking into consideration partial Hence, to enable the derivation of tractable, deterministic, winning and losing over a battlefield. and practical solutions to the competitive resource allocation Finally, we provide a numerical example which showcases problem, we next propose a generalization of the CBG the effect of the number of resources of each player on the dubbed generalized Colonel Blotto game. NE strategies and outcomes of the GCBG. B. Proposed Generalized Colonel Blotto Game II. PROPOSED GENERALIZED COLONEL BLOTTO GAME j j n ˜ j A GCBG , j∈P , R j∈P , , vi i=1, U j∈P , A. Classical Colonel Blotto Game P {Q } { } N { } { } n j j n k is defined by seven components that include the players, A standard CBG [9] , ∈P , R ∈P , , v , P {Q }j { }j N { i}i=1 strategy spaces, available resources, battlefields, and valua- j o U j∈P is defined byn six components: a) the players in the tion of battlefields as is the case in the CBG. However, we { } ˜ j set , , b) the strategy spaces j for , c) the define a new utility function U j∈P as an approximation to oa,b j { } availableP { resource} Rj for j , d) aQ set of n ∈Pbattlefields, the utility function of the CBG. This approximation is based ∈ P , e) the normalized value of each battlefield, vi for i , on a continuous differentiable function and an approximation andN f) the utility function, U j, for each player. The set∈ N of parameter, k. When k increases, the GCBG will very closely possible strategies for each player, j , corresponds to the resemble the CBG. At the limit, when k goes to infinity, the different possible distributions of its QRj resources across the GCBG converges to the CBG. However, in contrast with the battlefields. Hence, j is defined as CBG, the differentiability of the approximate utility function Q n enables the derivation of deterministic equilibrium strategies. j = rj rj Rj , rj 0 , (1) In this respect, as defined in (2), the payoff from each Q i ≤ i ≥ ( i=1 ) battlefield resembles a step function. Hence, we propose an X − j j j j where r is the number of allocated resource units by player approximation of this function, u˜i (ri , ri , k), referred to i j to battlefield i, and rj j is the vector of these allocated hereinafter as k approximation, defined as follows: ∈Q − resources (an allocation strategy of player j) . To this end, − vi − vi u˜j (rj , r j , k) , arctan k(rj r j ) + , (5) the payoff of player j, for a battlefield i, is defined as: i i i π i − i 2 − j j j vi, if ri > ri , and the utility function of each player U˜ is the summation − − uj(rj , r j )= vi , if rj = r j , (2) of the k approximation payoffs from each battlefield.
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