Rahul Savani

Computing stable outcomes in hedonic games

We study the computational complexity of finding stable outcomes in symmetric additively-separable hedonic games. These coalition formation games are specified by an undirected edge-weighted graph: nodes are players, an outcome of the game is a partition of the nodes into coalitions, and the utility of a node is the sum of incident edge weights in the same coalition. We consider several natural stability requirements defined in the economics literature. For all of them the existence of a stable outcome is guaranteed by a potential function argument, so local improvements will converge to a stable outcome and all these problems are in PLS. The different stability requirements correspond to different local search neighbourhoods. For different neighbourhood structures, our findings comprise positive results in the form of polynomial-time algorithms for finding stable outcomes, and negative (PLS-completeness) results.

Power Indices in Spanning Connectivity Games

The Banzhaf index, Shapley-Shubik index and other voting power indices measure the importance of a player in a coalitional game. We consider a simple coalitional game called the spanning connectivity game (SCG) based on an undirected, unweighted multigraph, where edges are players. We examine the computational complexity of computing the voting power indices of edges in the SCG. It is shown that computing Banzhaf values is #P-complete and computing Shapley-Shubik indices or values is NP-hard for SCGs. Interestingly, Holler indices and Deegan-Packel indices can be computed in polynomial time. Among other results, it is proved that Banzhaf indices can be computed in polynomial time for graphs with bounded treewidth. It is also shown that for any reasonable representation of a simple game, a polynomial time algorithm to compute the Shapley-Shubik indices implies a polynomial time algorithm to compute the Banzhaf indices. This answers (positively) an open question of whether computing Shapley-Shubik indices for a simple game represented by the set of minimal winning coalitions is NP-hard.

Wiretapping a Hidden Network

We consider the problem of maximizing the probability of hitting a strategically chosen hidden virtual network by placing a wiretap on a single link of a communication network. This can be seen as a two-player win-lose (zero-sum) game that we call the wiretap game. The value of this game is the greatest probability that the wiretapper can secure for hitting the virtual network. The value is shown to be equal the reciprocal of the strength of the underlying graph. We provide a polynomial-time algorithm that finds a linear-sized description of the maxmin-polytope, and a characterization of its extreme points. It also provides a succint representation of all equilibrium strategies of the wiretapper that minimize the number of pure best responses of the hider. Among these strategies, we efficiently compute the unique strategy that maximizes the least punishment that the hider incurs for playing a pure strategy that is not a best response. Finally, we show that this unique strategy is the nucleolus of the recently studied simple cooperative spanning connectivity game.

Approximate well-supported nash equilibria below two-thirds

In an e-Nash equilibrium, a player can gain at most e by changing his behaviour. Recent work has addressed the question of how best to compute e-Nash equilibria, and for what values of e a polynomial-time algorithm exists. An e-well-supported Nash equilibrium (e-WSNE) has the additional requirement that any strategy that is used with non-zero probability by a player must have payoff at most e less than a best response. A recent algorithm of Kontogiannis and Spirakis shows how to compute a 2/3-WSNE in polynomial time, for bimatrix games. Here we introduce a new technique that leads to an improvement to the worst-case approximation guarantee.

Approximate Well-supported Nash Equilibria Below Two-thirds

In an

The Complexity of the Simplex Method

\epsilon

Polylogarithmic Supports Are Required for Approximate Well-Supported Nash Equilibria below 2/3

ϵ-Nash equilibrium, a player can gain at most