From Battlefields to Elections: Winning Strategies of Blotto and Auditing Games

Behnezhad, Soheil; Blum, Avrim; Derakhshan, Mahsa; Hajiaghayi, MohammadTaghi; Mahdian, Mohammad; Papadimitriou, Christos H.; Rivest, Ronald L.; Seddighin, Saeed; Stark, Philip B.

doi:10.1137/1.9781611975031.148

Cited by 12 publications

(25 citation statements)

References 22 publications

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“…where with some abuse of notation, we use ω i to refer to the battlefield valuation set in state i. The ex-interim utility for player I, given type t i is the expected payoff (8) to the informed player in the Blotto game CB(XI , XU , V, 1 2 ), wherev = 1 and v = α ∈ (0, 1). It is negative when γ ∈ ( 1 2 , 1) and α ∈ (0, 1).…”

Section: Informationmentioning

confidence: 99%

“…(c) The value of information, π * I − π GW I , is non-negative. Information offers the most improvement for low budget (γ near 1 2 ) and when there is higher priority in the diagonal battlefields (α low).…”

Section: Informationmentioning

confidence: 99%

“…In adversarial interactions, informational asymmetries may provide an advantage to one competitor over the other. Adversarial contests appear in a wide range of applications, such as political campaigns [1], [2], security of cyberphysical systems [3], and competitive advertising [4]. Rigorous studies on the interplay between asymmetric information and strategic decision-making in adversarial settings has received attention in the recent literature [5], [6].…”

Section: Introductionmentioning

confidence: 99%

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Characterizing the interplay between information and strength in Blotto games

Paarporn

Chandan

Alizadeh

et al. 2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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In this paper, we investigate informational asymmetries in the Colonel Blotto game, a game-theoretic model of competitive resource allocation between two players over a set of battlefields. The battlefield valuations are subject to randomness. One of the two players knows the valuations with certainty. The other knows only a distribution on the battlefield realizations. However, the informed player has fewer resources to allocate. We characterize unique equilibrium payoffs in a two battlefield setup of the Colonel Blotto game. We then focus on a three battlefield setup in the General Lotto game, a popular variant of the Colonel Blotto game. We characterize the unique equilibrium payoffs and mixed equilibrium strategies. We quantify the value of information -the difference in equilibrium payoff between the asymmetric information game and complete information game. We find information strictly improves the informed player's performance guarantee. However, the magnitude of improvement varies with the informed player's strength as well as the game parameters. Our analysis highlights the interplay between strength and information in adversarial environments.

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Section: Informationmentioning

confidence: 99%

Section: Informationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Characterizing the interplay between information and strength in Blotto games

Paarporn

Chandan

Alizadeh

et al. 2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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show abstract

“…Recent studies have made significant progress in understanding the optimal response problem in Colonel Blotto when the support size is unbounded [2,3,4,13,14,15,24,25,27]. Note that the unbounded case is also challenging to solve due to the exponential number of pure strategies of the players in the Colonel Blotto game.…”

Section: Introductionmentioning

confidence: 99%

“…The case of (u, p)-maximin strategies is generally harder to solve. However, Behnezhad et al [3] show that it is possible to obtain a logarithmic factor approximation in polynomial time.…”

Section: Introductionmentioning

confidence: 99%

Optimal Strategies of Blotto Games

Behnezhad

Blum

Derakhshan

et al. 2019

Proceedings of the 2019 ACM Conference on Economics and Computation

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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.We also provide the first complexity result for finding the maximin of Blotto-like games: we show that computing the maximin of a generalization of the Colonel Blotto game that we call G C Bis exponential time-complete. * A portion of this work was completed while some of the authors were visiting Simons Institute for Theory of Computing.

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Computing Stackelberg Equilibria of Large General-Sum Games

Blum

Haghtalab

Hajiaghayi

et al. 2019

Algorithmic Game Theory

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We study the computational complexity of finding Stackelberg Equilibria in general-sum games, where the set of pure strategies of the leader and the followers are exponentially large in a natrual representation of the problem.In zero-sum games, the notion of a Stackelberg equilibrium coincides with the notion of a Nash Equilibrium Korzhyk et al. (2011b). Finding these equilibrium concepts in zero-sum games can be efficiently done when the players have polynomially many pure strategies or when (in additional to some structural properties) a best-response oracle is available Ahmadinejad et al. (2016); Dudík et al. (2017); Kalai and Vempala (2005). Despite such advancements in the case of zero-sum games, little is known for general-sum games.In light of the above, we examine the computational complexity of computing a Stackelberg equilibrium in large general-sum games. We show that while there are natural large general-sum games where the Stackelberg Equilibria can be computed efficiently if the Nash equilibrium in its zero-sum form could be computed efficiently, in general, structural properties that allow for efficient computation of Nash equilibrium in zero-sum games are not sufficient for computing Stackelberg equilibria in general-sum games.

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From Battlefields to Elections: Winning Strategies of Blotto and Auditing Games

Cited by 12 publications

References 22 publications

Characterizing the interplay between information and strength in Blotto games

Characterizing the interplay between information and strength in Blotto games

Optimal Strategies of Blotto Games

Computing Stackelberg Equilibria of Large General-Sum Games

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