In this dissertation, we analyze the computational properties of game-theoretic centrality measures. The key idea behind game-theoretic approach to network analysis is to treat nodes as players in a cooperative game, where the value of each coalition of nodes is determined by certain graph properties. Next, the centrality of any individual node is determined by a chosen game-theoretic solution concept (notably, the Shapley value) in the same way as the payoff of a player in a cooperative game. On one hand, the advantage of game-theoretic centrality measures is that nodes are ranked not only according to their individual roles but also according to how they contribute to the role played by all possible subsets of nodes. On the other hand, the disadvantage is that the game-theoretic solution concepts are typically computationally challenging. The main contribution of this dissertation is that we show that a wide variety of game-theoretic solution concepts on networks can be computed in polynomial time. Our focus is on centralities based on the Shapley value and its various extensions, such as the Semivalues and Coalitional Semivalues. Furthermore, we prove #P-hardness of computing the Shapley value in connectivity games and propose an algorithm to compute it. Finally, we analyse computational properties of generalized version of cooperative games in which order of player matters. We propose a new representation for such games, called generalized marginal contribution networks, that allows for polynomial computation in the size of the representation of two dedicated extensions of the Shapley value to this class of games.
The hardness result presented in this chapter is consistent with other studies of the complexity of the Shapley value in various settings. For instance, computing the Shapley value was shown to be #P-Complete for weighted majority games and in minimum spanning tree games . @cite_31 obtained negative results for a related problem of computing the Shapley-Shubik power index for the spanning connectivity games that are based on undirected, unweighted multigraphs. Also, @cite_45 showed that the computation of the Banzhaf index for connectivity games, in which agents own vertices and control adjacent edges and aim to become connected to the certain set of primary edges, is #P-Complete. A comprehensive review of these issues, including some positive results for certain settings, can be found in .
@cite_31 @cite_45
1570872913 2005690414 2122807012 2038127015
We study the complexity of the following problem: Given two weighted voting games G? and G?? that each contain a player p, in which of these games is p's power index value higher? We study this problem with respect to both the Shapley-Shubik power index [16] and the Banzhaf power index [3,6]. Our main result is that for both of these power indices the problem is complete for probabilistic polynomial time (i.e., is PP-complete). We apply our results to partially resolve some recently proposed problems regarding the complexity of weighted voting games. We also show that, unlike the Banzhaf power index, the Shapley-Shubik power index is not #P-parsimonious-complete. This finding sets a hard limit on the possible strengthenings of a result of Deng and Papadimitriou [5], who showed that the Shapley-Shubik power index is #P-metric-complete. Preference aggregation is used in a variety of multiagent applications, and as a result, voting theory has become an important topic in multiagent system research. However, power indices (which reflect how much "real power" a voter has in a weighted voting system) have received relatively little attention, although they have long been studied in political science and economics. We consider a particular multiagent domain, a threshold network flow game. Agents control the edges of a graph; a coalition wins if it can send a flow that exceeds a given threshold from a source vertex to a target vertex. The relative power of each edge agent reflects its significance in enabling such a flow, and in real-world networks could be used, for example, to allocate resources for maintaining parts of the network. We examine the computational complexity of calculating two prominent power indices, the Banzhaf index and the Shapley-Shubik index, in this network flow domain. We also consider the complexity of calculating the core in this domain. The core can be used to allocate, in a stable manner, the gains of the coalition that is established. We show that calculating the Shapley-Shubik index in this network flow domain is NP-hard, and that calculating the Banzhaf index is #P-complete. Despite these negative results, we show that for some restricted network flow domains there exists a polynomial algorithm for calculating agents' Banzhaf power indices. We also show that computing the core in this game can be performed in polynomial time. This paper is a twofold contribution. First, it contributes to the problem of enumerating some classes of simple games and in particular provides the number of weighted games with minimum and the number of weighted games for the dual class as well. Second, we focus on the special case of bipartite complete games with minimum, and we compare and rank these games according to the behavior of some efficient power indices of players of type 1 (or of type 2). The main result of this second part establishes all allowable rankings of these games when the Shapley-Shubik power index is used on players of type 1. In this paper, we prove that both problems for calculating the Banzhaf power index and the Shapley-Shubik power index for weighted majority games are NP-complete.