We study Nash equilibria in the setting of network creation games introduced recently by Fabrikant, Luthra, Maneva, Papadimitriou, and Shenker. In this game we have a set of selfish node players, each creating some incident links, and the goal is to minimize α times the cost of the created links plus sum of the distances to all other players. Fabrikant et al. proved an upper bound O (√α) on the price of anarchy: the relative cost of the lack of coordination. Albers, Eilts, Even-Dar, Mansour, and Roditty show that the price of anarchy is constant for α = O (√ n ) and for α ≥ 12 n ⌈ lg n ⌉, and that the price of anarchy is 15(1+(min{α 2 / n , n 2 /α}) 1/3 ) for any α. The latter bound shows the first sublinear worst-case bound, O ( n 1/3 ), for all α. But no better bound is known for α between ω(√ n ) and o ( n lg n ). Yet α ≈ n is perhaps the most interesting range, for it corresponds to considering the average distance (instead of the sum of distances) to other nodes to be roughly on par with link creation (effectively dividing α by n ). In this article, we prove the first o ( n ε ) upper bound for general α, namely 2 O (√ lg n ) . We also prove a constant upper bound for α = O ( n 1-ε ) for any fixed ε > 0, substantially reducing the range of α for which constant bounds have not been obtained. Along the way, we also improve the constant upper bound by Albers et al. (with the lead constant of 15 ) to 6 for α < ( n /2) 1/2 and to 4 for α < ( n /2) 1/3 . Next we consider the bilateral network variant of Corbo and Parkes, in which links can be created only with the consent of both endpoints and the link price is shared equally by the two. Corbo and Parkes show an upper bound of O (√α) and a lower bound of Ω(lgα) for α ≤ n . In this article, we show that in fact the upper bound O (√α) is tight for α ≤ n , by proving a matching lower bound of Ω(√α). For α > n , we prove that the price of anarchy is Θ( n /√ α). Finally we introduce a variant of both network creation games, in which each player desires to minimize α times the cost of its created links plus the maximum distance (instead of the sum of distances) to the other players. This variant of the problem is naturally motivated by considering the worst case instead of the average case. Interestingly, for the original (unilateral) game, we show that the price of anarchy is at most 2 for α ≥ n , O (min {4 √lg n , ( n /α) 1/3 }) for 2√ lg n ≤ α ≤ n , and O ( n 2/α ) for α < 2√ lg n . For the bilateral game, we prove matching upper and lower bounds of Θ( n /α + 1) for α ≤ n , and an upper bound of 2 for α > n .
We give approximation algorithms and inapproximability results for a class of movement problems. In general, these problems involve planning the coordinated motion of a large collection of objects (representing anything from a robot swarm or firefighter team to map labels or network messages) to achieve a global property of the network while minimizing the maximum or average movement. In particular, we consider the goals of achieving connectivity (undirected and directed), achieving connectivity between a given pair of vertices, achieving independence (a dispersion problem), and achieving a perfect matching (with applications to multicasting). This general family of movement problems encompasses an intriguing range of graph and geometric algorithms, with several real-world applications and a surprising range of approximability. In some cases, we obtain tight approximation and inapproximability results using direct techniques (without use of PCP), assuming just that P = NP.A preliminary version of this article appeared in
We study the problem of computing Nash equilibria of zero-sum games. Many natural zerosum 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 the strategy space, standard methods for computing equilibria of zero-sum games fail to be computationally feasible. Indeed, despite its importance, only a few solutions for special variants of the problem are known.In this paper we show how to compute equilibria of Colonel Blotto games. Moreover, our approach takes the form of a general reduction: to find a Nash equilibrium of a zero-sum game, it suffices to design a separation oracle for the strategy polytope of any bilinear game that is payoff-equivalent. We then apply this technique to obtain the first polytime algorithms for a variety of games. In addition to Colonel Blotto, we also show how to compute equilibria in an infinite-strategy variant called the General Lotto game; this involves showing how to prune the strategy space to a finite subset before applying our reduction. We also consider the class of dueling games, first introduced by Immorlica et al. (2011). We show that our approach provably extends the class of dueling games for which equilibria can be computed: we introduce a new dueling game, the matching duel, on which prior methods fail to be computationally feasible but upon which our reduction can be applied. 1 difficulty of solving for equilibrium explicitly, it is natural to revisit the equilibrium computation problem for Colonel Blotto games.An Approach: Bilinear Games How should one approach equilibrium computation in such a game? The exponential size of the strategy set is not an impassable barrier; in certain cases, games with exponentially many strategies have an underlying structure that can be used to approach the equilibrium computation problem. For example, Koller, Megiddo and von Stengel [29] show how to compute equilibria for zero-sum extensive-form games with perfect recall. Immorlica et al. [27] give an approach for solving algorithmically-motivated "dueling games" with uncertainty. Letchford and Conitzer [35] compute equilibria for a variety of graphical security games. Each of these cases involve games with exponentially many strategies. In each case, a similar approach is employed: reformulating the original game as a payoff-equivalent bilinear game. In a bilinear game, the space of strategies forms a polytope in R n , and payoffs are specified by a matrix M : if the players play strategies x and y respectively, then the payoff to player 1 is x T M y. It has been observed that such bilinear games can be solved efficiently when the stra...
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