Complexity of Coalition Structure Generation

We revisit the coalition structure generation problem in which the goal is to partition the players into exhaustive and disjoint coalitions so as to maximize the social welfare. One of our key results is a general polynomial-time algorithm to solve the problem for all coalitional games provided that player types are known and the number of player types is bounded by a constant. As a corollary, we obtain a polynomial-time algorithm to compute an optimal partition for weighted voting games with a constant number of weight values and for coalitional skill games with a constant number of skills. We also consider well-studied and well-motivated coalitional games defined compactly on combinatorial domains. For these games, we characterize the complexity of computing an optimal coalition structure by presenting polynomial-time algorithms, approximation algorithms, or NP-hardness and inapproximability lower bounds

On strong equilibria and improvement dynamics in network creation games

We study strong equilibria in network creation games. These form a classical and well-studied class of games where a set of players form a network by buying edges to their neighbors at a cost of a fixed parameter \xce\xb1. The cost of a player is defined to be the cost of the bought edges plus the sum of distances to all the players in the resulting graph. We identify and characterize various structural properties of strong equilibria, which lead to a characterization of the set of strong equilibria for all \xce\xb1 in the range (0, 2). For \xce\xb1> 2, Andelman et al. [4] prove that a star graph in which every leaf buys one edge to the center node is a strong equilibrium, and conjecture that in fact any star is a strong equilibrium. We resolve this conjecture in the affirmative. Additionally, we show that when \xce\xb1 is large enough (\xe2\x89\xa5 2 n) there exist non-star trees that are strong equilibria. For the strong price of anarchy, we provide precise expressions when \xce\xb1 is in the range (0, 2), and we prove a lower bound of 3/2 when \xce\xb1\xe2\x89\xa5 2. Lastly, we aim to characterize under which conditions (coalitional) improvement dynamics may converge to a strong equilibrium. To this end, we study the (coalitional) finite improvement property and (coalitional) weak acyclicity property. We prove various conditions under which these properties do and do not hold. Some of these results also hold for the class of pure Nash equilibria

Welfare guarantees for proportional allocations

According to the proportional allocation mechanism from the network optimization literature, users compete for a divisible resource -- such as bandwidth -- by submitting bids. The mechanism allocates to each user a fraction of the resource that is proportional to her bid and collects an amount equal to her bid as payment. Since users act as utility-maximizers, this naturally defines a proportional allocation game. Recently, Syrgkanis and Tardos (STOC 2013) quantified the inefficiency of equilibria in this game with respect to the social welfare and presented a lower bound of 26.8% on the price of anarchy over coarse-correlated and Bayes-Nash equilibria in the full and incomplete information settings, respectively. In this paper, we improve this bound to 50% over both equilibrium concepts. Our analysis is simpler and, furthermore, we argue that it cannot be improved by arguments that do not take the equilibrium structure into account. We also extend it to settings with budget constraints where we show the first constant bound (between 36% and 50%) on the price of anarchy of the corresponding game with respect to an effective welfare benchmark that takes budgets into account.Comment: 15 page

Solving Weighted Voting Game Design Problems Optimally: Representations, Synthesis, and Enumeration

We study the inverse power index problem for weighted voting games: the problem of finding a weighted voting game in which the power of the players is as close as possible to a certain target distribution. Our goal is to find algorithms that solve this problem exactly. Thereto, we study various subclasses of simple games, and their associated representation methods. We survey algorithms and impossibility results for the synthesis problem, i.e., converting a representation of a simple game into another representation. We contribute to the synthesis problem by showing that it is impossible to compute in polynomial time the list of ceiling coalitions of a game from its list of roof coalitions, and vice versa. Then, we proceed by studying the problem of enumerating the set of weighted voting games. We present first a naive algorithm for this, running in doubly exponential time. Using our knowledge of the

On the inefficiency of equilibria in linear bottleneck congestion games

We study the inefficiency of equilibrium outcomes in bottleneck congestion games. These games model situations in which strategic players compete for a limited number of facilities. Each player allocates his weight to a (feasible) subset of the facilities with the goal to minimize the maximum (weight-dependent) latency that he experiences on any of these facilities. We derive upper and (asymptotically) matching lower bounds on the (strong) price of anarchy of linear bottleneck congestion games for a natural load balancing social cost objective (i.e., minimize the maximum latency of a facility). We restrict our studies to linear latency functions. Linear bottleneck congestion games still constitute a rich class of games and generalize, for example, load balancing games with identical or uniformly related machines with or without restricted assignments

Efficient Equilibria in Polymatrix Coordination Games

We consider polymatrix coordination games with individual preferences where every player corresponds to a node in a graph who plays with each neighbor a separate bimatrix game with non-negative symmetric payoffs. In this paper, we study $\alpha$-approximate $k$-equilibria of these games, i.e., outcomes where no group of at most $k$ players can deviate such that each member increases his payoff by at least a factor $\alpha$. We prove that for $\alpha \ge 2$ these games have the finite coalitional improvement property (and thus $\alpha$-approximate $k$-equilibria exist), while for $\alpha < 2$ this property does not hold. Further, we derive an almost tight bound of $2\alpha(n-1)/(k-1)$ on the price of anarchy, where $n$ is the number of players; in particular, it scales from unbounded for pure Nash equilibria ($k = 1)$ to $2\alpha$ for strong equilibria ($k = n$). We also settle the complexity of several problems related to the verification and existence of these equilibria. Finally, we investigate natural means to reduce the inefficiency of Nash equilibria. Most promisingly, we show that by fixing the strategies of $k$ players the price of anarchy can be reduced to $n/k$ (and this bound is tight)

Ischemic nucleotide breakdown increases during cardiac development due to drop in adenosine anabolism/catabolism ratio

Abstract Our earlier work on reperfusion showed that adult rat hearts released almost twice as much purine nucleosides and oxypurines as newborn hearts did [Am J Physiol 254 (1988) H1091]. A change in the ratio anabolism/catabolism of adenosine could be responsible for this effect. We therefore measured the activity of adenosine kinase, adenosine deaminase, nucleoside phosphorylase and xanthine oxidoreductase in homogenates of hearts and myocytes from neonatal and adult rats. In hearts the activity of adenosine deaminase and nucleoside phosphorylase (10–20 U/g protein) changed relatively little. However, adenosine kinase activity decreased from 1.3 to 0.6 U/g (P < 0.025), and xanthine oxidoreductase activity increased from 0.02 to 0.85 U/g (P < 0.005). Thus the ratio in activity of these rate-limiting enzymes for anabolism and catabolism dropped from 68 to 0.68 during cardiac development. In contrast, the ratio in myocytes remained unchanged (about 23). The large difference in adenosine anabolism/catabolism ratio, observed in heart homogenates, could explain why ATP breakdown due to hypoxia is lower in neonatal than in adult heart. Because this change is absent in myocytes, we speculate that mainly endothelial activities of adenosine kinase and xanthine oxidoreductase are responsible for this shift in purine metabolism during development