Shapley Meets Shapley

This paper concerns the analysis of the Shapley value in matching games. Matching games constitute a fundamental class of cooperative games which help understand and model auctions and assignments. In a matching game, the value of a coalition of vertices is the weight of the maximum size matching in the subgraph induced by the coalition. The Shapley value is one of the most important solution concepts in cooperative game theory. After establishing some general insights, we show that the Shapley value of matching games can be computed in polynomial time for some special cases: graphs with maximum degree two, and graphs that have a small modular decomposition into cliques or cocliques (complete k-partite graphs are a notable special case of this). The latter result extends to various other well-known classes of graph-based cooperative games. We continue by showing that computing the Shapley value of unweighted matching games is #P-complete in general. Finally, a fully polynomial-time randomized approximation scheme (FPRAS) is presented. This FPRAS can be considered the best positive result conceivable, in view of the #P-completeness result.Comment: 17 page

On Demand Responsiveness in Additive Cost Sharing

We propose two new axioms of demand responsiveness for additive cost sharing with variable demands. Group Monotonicity requires that if a group of agents increase their demands, not all of them pay less. Solidarity says that if agent i demands more, j should not pay more if k pays less. Both axioms are compatible in the partial responsibility theory postulating Strong Ranking, i.e., the ranking of cost shares should never contradict that of demands. The combination of Strong Ranking , Solidarity and Monotonicity characterizes the quasi-proportional methods, under which cost shares are proportional to 'rescaled' demands. The alternative full responsibility theory is based on Separability, ruling out cross-subsidization when costs are additively separable. Neither the Aumann-Shapley nor the Shapley-Shubik method is group monotonic. On the otherhand, convex combinations of "nearby" fixed-path methods are group-monotonic: the subsidy-free serial method is the main example. No separable method meets Solidarity, yet restricting the axiom to submodular (or supermodular) cost functions leads to a characterization of the fixed-flow methods, containing the Shapley-Shubik and serial methods.

A Logic-Based Representation for Coalitional Games with Externalities

We consider the issue of representing coalitional games in multiagent systems that exhibit externalities from coalition formation, i.e., systems in which the gain from forming a coalition may be affected by the formation of other co-existing coalitions. Although externalities play a key role in many real-life situations, very little attention has been given to this issue in the multi-agent system literature, especially with regard to the computational aspects involved. To this end, we propose a new representation which, in the spirit of Ieong and Shoham [9], is based on Boolean expressions. The idea behind our representation is to construct much richer expressions that allow for capturing externalities induced upon coalitions. We show that the new representation is fully expressive, at least as concise as the conventional partition function game representation and, for many games, exponentially more concise. We evaluate the efficiency of our new representation by considering the problem of computing the Extended and Generalized Shapley value, a powerful extension of the conventional Shapley value to games with externalities. We show that by using our new representation, the Extended and Generalized Shapley value, which has not been studied in the computer science literature to date, can be computed in time linear in the size of the input

The Shapley value for airport and irrigation games

In this paper cost sharing problems are considered. We focus on problems given by rooted trees, we call these problems cost-tree problems, and on the induced transferable utility cooperative games, called irrigation games. A formal notion of irrigation games is introduced, and the characterization of the class of these games is provided. The well-known class of airport games Littlechild and Thompson (1977) is a subclass of irrigation games. The Shapley value Shapley (1953) is probably the most popular solution concept for transferable utility cooperative games. Dubey (1982) and Moulin and Shenker (1992) show respectively, that Shapley's Shapley (1953) and Young (1985)'s axiomatizations of the Shapley value are valid on the class of airport games. In this paper we show that Dubey (1982)'s and Moulin and Shenker (1992)'s results can be proved by applying Shapley (1953)'s and Young (1985)'s proofs, that is those results are direct consequences of Shapley (1953)'s and Young (1985)'s results. Furthermore, we extend Dubey (1982)'s and Moulin and Shenker (1992)'s results to the class of irrigation games, that is we provide two characterizations of the Shapley value for cost sharing problems given by rooted trees. We also note that for irrigation games the Shapley value is always stable, that is it is always in the core Gillies (1959)

Computational Aspects of Extending the Shapley Value to Coalitional Games with Externalities

Until recently, computational aspects of the Shapley value were only studied under the assumption that there are no externalities from coalition formation, i.e., that the value of any coalition is independent of other coalitions in the system. However, externalities play a key role in many real-life situations and have been extensively studied in the game-theoretic and economic literature. In this paper, we consider the issue of computing extensions of the Shapley value to coalitional games with externalities proposed by Myerson [21], Pham Do and Norde [23], and McQuillin [17]. To facilitate efficient computation of these extensions, we propose a new representation for coalitional games with externalities, which is based on weighted logical expressions. We demonstrate that this representation is fully expressive and, sometimes, exponentially more concise than the conventional partition function game model. Furthermore, it allows us to compute the aforementioned extensions of the Shapley value in time linear in the size of the input

Efficient computation of the Shapley value for game-theoretic network centrality

The Shapley value—probably the most important normative payoff division scheme in coalitional games—has recently been advocated as a useful measure of centrality in networks. However, although this approach has a variety of real-world applications (including social and organisational networks, biological networks and communication networks), its computational properties have not been widely studied. To date, the only practicable approach to compute Shapley value-based centrality has been via Monte Carlo simulations which are computationally expensive and not guaranteed to give an exact answer. Against this background, this paper presents the first study of the computational aspects of the Shapley value for network centralities. Specifically, we develop exact analytical formulae for Shapley value-based centrality in both weighted and unweighted networks and develop efficient (polynomial time) and exact algorithms based on them. We empirically evaluate these algorithms on two real-life examples (an infrastructure network representing the topology of the Western States Power Grid and a collaboration network from the field of astrophysics) and demonstrate that they deliver significant speedups over the Monte Carlo approach. Fo

The Shapley value for airport and irrigation games

In this paper cost sharing problems are considered. We focus on problems given by rooted trees, we call these problems cost-tree problems, and on the induced transferable utility cooperative games, called irrigation games. A formal notion of irrigation games is introduced, and the characterization of the class of these games is provided. The well-known class of airport games Littlechild and Thompson (1977) is a subclass of irrigation games. The Shapley value Shapley (1953) is probably the most popular solution concept for transferable utility cooperative games. Dubey (1982) and Moulin and Shenker (1992) show respectively, that Shapley's Shapley (1953) and Young (1985)'s axiomatizations of the Shapley value are valid on the class of airport games. In this paper we show that Dubey (1982)'s and Moulin and Shenker (1992)'s results can be proved by applying Shapley (1953)'s and Young (1985)'s proofs, that is those results are direct consequences of Shapley (1953)'s and Young (1985)'s results. Furthermore, we extend Dubey (1982)'s and Moulin and Shenker (1992)'s results to the class of irrigation games, that is we provide two characterizations of the Shapley value for cost sharing problems given by rooted trees. We also note that for irrigation games the Shapley value is always stable, that is it is always in the core Gillies (1959).TU games, Shapley value, Axiomatization, Cost Sharing

On the impossibility of fair risk allocation

Measuring and allocating risk properly are crucial for performance evaluation and internal capital allocation of portfolios held by banks, insurance companies, investment funds and other entities subject to financial risk. We show that by using a coherent measure of risk it is impossible to allocate risk satisfying the natural requirements of (Solution) Core Compatibility, Equal Treatment Property and Strong Monotonicity. To obtain the result we characterize the Shapley value on the class of totally balanced games and also on the class of exact games

Steady Marginality: A Uniform Approach to Shapley Value for Games with Externalities

The Shapley value is one of the most important solution concepts in cooperative game theory. In coalitional games without externalities, it allows to compute a unique payoff division that meets certain desirable fairness axioms. However, in many realistic applications where externalities are present, Shapley's axioms fail to indicate such a unique division. Consequently, there are many extensions of Shapley value to the environment with externalities proposed in the literature built upon additional axioms. Two important such extensions are "externality-free" value by Pham Do and Norde and value that "absorbed all externalities" by McQuillin. They are good reference points in a space of potential payoff divisions for coalitional games with externalities as they limit the space at two opposite extremes. In a recent, important publication, De Clippel and Serrano presented a marginality-based axiomatization of the value by Pham Do Norde. In this paper, we propose a dual approach to marginality which allows us to derive the value of McQuillin. Thus, we close the picture outlined by De Clippel and Serrano