Games on concept lattices: Shapley value and core (original) (raw)

Games on lattices, multichoice games and the Shapley value: a new approach

Mathematical Methods of Operations Research, 2007

Multichoice games, as well as many other recent attempts to generalize the notion of classical cooperative game, can be casted into the framework of lattices. We propose a general definition for games on lattices, together with an interpretation. Several definitions of the Shapley value of a multichoice games have already been given, among them the original one due to Hsiao and Raghavan, and the one given by Faigle and Kern. We propose a new approach together with its axiomatization, more in the spirit of the original axiomatization of Shapley, and avoiding a high computational complexity.

Algorithms for computing the Shapley value of cooperative games on lattices

Discrete Applied Mathematics

We study algorithms to compute the Shapley value for a cooperative game on a lattice L Σ = (F Σ , ⊆) where F Σ is the family of closed sets given by an implicational system Σ on a set N of players. The first algorithm is based on the generation of the maximal chains of the lattice L Σ and computes the Shapley value in O(|N | 3 .|Σ|.|Ch|) time complexity using polynomial space, where Ch is the set of maximal chains of L Σ. The second algorithm proceeds by building the lattice L Σ and computes the Shapley value in O(|N | 3 .|Σ|.|F Σ |) time and space complexity. Our main contribution is to show that the Shapley value of weighted graph games on a product of chains with the same fixed length is computable in polynomial time. We do this by partitioning the set of feasible coalitions relevant to the computation of the Shapley value into equivalence classes in such a way that we need to consider only one element of each class in the computation.

Games on distributive lattices and the Shapley interaction transform

Proceedings of IPMU

The paper proposes a general approach of interaction between players or attributes. It generalizes the notion of interaction defined for players modeled by games, by considering functions defined on distributive lattices. A general definition of the interaction index is provided, as well as the construction of operators establishing transforms between games, their Möbius transforms and their interaction indices.

Efficiently computing the Shapley value of connectivity games in low-treewidth graphs

Operational Research

The Shapley value is the solution concept in cooperative game theory that is most used in both theoretical and practical settings. Unfortunately, in general, computing the Shapley value is computationally intractable. This paper focuses on computing the Shapley value of (weighted) connectivity games. For these connectivity games, which are defined on an underlying (weighted) graph, computing the Shapley value is \#\textsf {P}$$ # P -hard, and thus (likely) intractable even for graphs with a moderate number of vertices. We present an algorithm that can efficiently compute the Shapley value if the underlying graph has bounded treewidth. Next, we apply our algorithm to several real-world (covert) networks. We show that our algorithm can quickly compute exact Shapley values for these networks, whereas in prior work these values could only be approximated using a heuristic method. Finally, it is demonstrated that our algorithm can also efficiently compute the Shapley value time for sev...

The Shapley value for bicooperative games

2004

and centrA: RESUMEN El objetivo de este trabajo es analizar un concepto de solución que asigna a cada juego bicooperativo un único vector. En el contexto de los juegos bicooperativos introducidos por Bilbao (2000), definimos una solución denominada valor de Shapley porque este valor puede interpretarse de una ma nera semejante al clásico valor de Shapley para juegos cooperativos. El resultado más importante del trabajo es una caracterización axiomática de este valor.

The Shapley value for capacities and games on set systems

2006

We propose a generalization of capacities which encompass in a large extent the class of Choquet’s capacities. Then, we define the class of probabilistic values over these capacities, which are values satisfying classical axioms, the well-known Shapley value being one. Lastly, we propose a value on these capacities by borrowing ideas from electric networks theory.

Values for two-stage games: Another view of the Shapley axioms

International Journal of Game Theory, 1990

This short study reports an application of the Shapley value axioms to a new concept of "two-stage games." In these games, the formation of a coalition in the first stage entities its members to play a prespecified cooperative game at the second stage. The original Shapley axioms have natural equivalents in the new framework, and we show the existence of (non-unique) values and semivalues for two stage games, analogous to those defined by the corresponding axioms for the conventional (one-stage) games. However, we also prove that all semivalues (hence, perforce, all values) must give patently unacceptable solutions for some "two-stage majority games" (where the members of a majority coalition play a conventional majority game). Our reservations about these prescribed values are related to Roth's (1980) criticism of Shapley's "),-transfer value" for non-transferable utility (NTU) games. But our analysis has wider scope than Roth's example, and the argument that it offers appears to be more conclusive. The study also indicates how the values and semivalues for two-stage games can be naturally generalized to apply for "multi-stage games."