# An axiomatic approach to the concept of interaction among players in cooperative games

@article{Grabisch1999AnAA, title={An axiomatic approach to the concept of interaction among players in cooperative games}, author={Michel Grabisch and Marc Roubens}, journal={International Journal of Game Theory}, year={1999}, volume={28}, pages={547-565} }

Abstract. An axiomatization of the interaction between the players of any coalition is given. It is based on three axioms: linearity, dummy and symmetry. These interaction indices extend the Banzhaf and Shapley values when using in addition two equivalent recursive axioms. Lastly, we give an expression of the Banzhaf and Shapley interaction indices in terms of pseudo-Boolean functions.

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#### References

SHOWING 1-10 OF 20 REFERENCES

On the axiomatization of the τ-value

- Mathematics
- 1987

SummaryThe τ-value is a solution concept for a subclass of games with transferable utility introduced and axiomatized by Tijs (1981, 1987). In this note an alternative characterization of the τ-value… Expand

Equivalent Representations of a Set Function with Applications to Game Theory and Multicriteria Decision Making

- Mathematics
- 1998

This paper introduces four alternative representations of a set function: the Mobius transformation, the co-Mobius transformation, and the interactions between elements of any subset of a given set… Expand

The selectope for cooperative games

- Economics, Computer Science
- Int. J. Game Theory
- 2000

Abstract. The selectope of a cooperative transferable utility game is the convex hull of the payoff vectors obtained by assigning the Harsanyi dividends of the coalitions to members determined by… Expand

Multilinear Extensions of Games

- Mathematics
- 1972

The multilinear extension of an n-person game v is a function defined on the n-cube I N which is linear in each variable and which coincides with v at the conrners of the cube, satisfying f(x) = v({i… Expand

On weighted Shapley values

- Mathematics, Economics
- 1983

Nonsymmetric Shapley values for coalitional form games with transferable utility are studied. The nonsymmetries are modeled through nonsymmetric weight systems defined on the players of the games. It… Expand

A Simplified Bargaining Model for the n-Person Cooperative Game

- Computer Science
- 1963

The bargaining model I propose to discuss in this paper2 is a simplified and, as I hope, improved version of my bargaining model published in the Princeton Contributions (see [3]). Like my earlier… Expand

Probabilistic Values for Games

- Economics
- 1977

Abstract : Much attention has been given to methods for measuring the value of playing a particular role in a n-person game. The study of various values is motivated by several considerations. One is… Expand

A mathematical theory of evidence

- Computer Science
- 1976

This book develops an alternative to the additive set functions and the rule of conditioning of the Bayesian theory: set functions that need only be what Choquet called "monotone of order of infinity." and Dempster's rule for combining such set functions. Expand

K-order Additive Discrete Fuzzy Measures and Their Representation

- Mathematics, Computer Science
- Fuzzy Sets Syst.
- 1997

A related topic of the paper is to introduce an alternative representation of fuzzy measures, called the interaction representation, which sets and extends in a common framework the Shapley value and the interaction index proposed by Murofushi and Soneda. Expand

Alternative Representations of Discrete Fuzzy Measures for Decision Making

- Mathematics, Computer Science
- Int. J. Uncertain. Fuzziness Knowl. Based Syst.
- 1997

This paper introduces three different representations of fuzzy measures, through the Mobius transformation, and the expression of importance and interaction. This leads naturally to the concept of… Expand