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...

We study the problem of selling n items to a single buyer with an additive valuation function. We consider the valuation of the items to be correlated, i.e., desirabilities of the buyer for the items are not drawn independently. Ideally, the goal is to design a mechanism to maximize the revenue. However, it has been shown that a revenue optimal mechanism might be very complicated and as a result inapplicable to real-world auctions. Therefore, our focus is on designing a simple mechanism that achieves a constant fraction of the optimal revenue. Babaioff et al. [3] propose a simple mechanism that achieves a constant fraction of the optimal revenue for independent setting with a single additive buyer. However, they leave the following problem as an open question: "Is there a simple, approximately optimal mechanism for a single additive buyer whose value for n items is sampled from a common base-value distribution?" Babaioff et al. show a constant approximation factor of the optimal revenue can be achieved by either selling the items separately or as a whole bundle in the independent setting. We show a similar result for the correlated setting when the desirabilities of the buyer are drawn from a common base-value distribution. It is worth mentioning that the core decomposition lemma which is mainly the heart of the proofs for efficiency of the mechanisms does not hold for correlated settings. Therefore we propose a modified version of this lemma which is applicable to the correlated settings as well. Although we apply this technique to show the proposed mechanism can guarantee a constant fraction of the optimal revenue in a very weak correlation, this method alone can not directly show the efficiency of the mechanism in stronger correlations. Therefore, via a combinatorial approach we reduce the problem to an auction with a weak correlation to which the core decomposition technique is applicable. In addition, we introduce a generalized model of correlation for items and show the proposed mechanism achieves an O(log k) approximation factor of the optimal revenue in that setting.

Chore division, introduced by Gardner in 1970s [10], is the problem of fairly dividing a chore among n different agents. In particular, in an envy-free chore division, we would like to divide a negatively valued heterogeneous object among a number of agents who have different valuations for different parts of the object, such that no agent envies another agent. It is the dual variant of the celebrated cake cutting problem, in which we would like to divide a desirable object among agents. There has been an extensive amount of study and effort to design bounded and envy-free protocols/algorithms for fair division of chores and goods, such that envy-free cake cutting became one of the most important open problems in 20-th century mathematics according to Garfunkel [11]. However, despite persistent efforts, due to delicate nature of the problem, there was no bounded protocol known for cake cutting even among four agents, until the breakthrough of Aziz and Mackenzie [2], which provided the first discrete and bounded envy-free protocol for cake cutting for four agents. Afterward, Aziz and Mackenzie [3], generalized their work and provided an envy-free cake cutting protocol for any number of agents to settle a significant and longstanding open problem. However, there is much less known for chore division. Unfortunately, there is no general method known to apply cake cutting techniques to chore division. Thus, it remained an open problem to find a discrete and bounded envy-free chore division protocol even for four agents.In this paper, we provide the first discrete and bounded envy-free protocol for chore division for an arbitrary number of agents. We produce major and powerful tools for designing protocols for the fair division of negatively valued objects. These tools are based on structural results and important observations. In gen- * The omitted proofs can be found in the full version of this paper.

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People make decisions and express their opinions according to their communities. An appropriate idea for controlling the diffusion of an opinion is to find influential people, and employ them to spread the desired opinion. We investigate an influencing problem when individuals' opinions are affected by their friends due to the model of Friedkin and Johnsen [4]. Our goal is to design efficient algorithms for finding opinion leaders such that changing their opinions has great impact on the overall opinion of the society. We define a set of problems like maximizing the sum of individual opinions or maximizing the number of individuals whose opinions are above a threshold. We discuss the complexity of the defined problems and design optimum algorithms for the non NP-hard variants of the problems. Furthermore, we run simulations on real-world social network data and show our proposed algorithm outperforms the classical algorithms such as degree-based, closeness-based, and pagerank-based algorithms.