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

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

Robust Price of Anarchy for Atomic Games with Altruistic Players

We study the inefficiency of equilibria for various classes of games when players are (partially) altruistic. We model altruistic behavior by assuming that player i's perceived cost is a convex combination of 1-\beta_i times his direct cost and \beta_i times the social cost. Tuning the parameters \beta_i allows smooth interpolation between purely selfish and purely altruistic behavior. Within this framework, we study altruistic extensions of linear congestion games, fair cost-sharing games and valid utility games. We derive (tight) bounds on the price of anarchy of these games for several solution concepts. Thereto, we suitably adapt the smoothness notion introduced by Roughgarden and show that it captures the essential properties to determine the robust price of anarchy of these games. Our bounds reveal that for congestion games and cost-sharing games the worst-case robust price of anarchy increases with increasing altruism, while for valid utility games it remains constant and is not affected by altruism. We also show that the increase in price of anarchy is not a universal phenomenon: for symmetric singleton linear congestion games we derive a bound on the price of anarchy for pure Nash equilibria that decreases as the level of altruism increases. Since the bound is also strictly lower than the robust price of anarchy, it exhibits a natural example in which Nash equilibria are more efficient than more permissive notions of equilibrium