Computing power indices: Multilinear extensions and new characterizations

Alonso-Meijide, J. M.; Casas-Méndez, Balbina; Holler, Manfred J.; Lorenzo-Freire, Silvia

doi:10.1016/j.ejor.2007.04.019

Cited by 34 publications

(21 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Alonso-Meijide et al (2008) introduce procedures to calculate the Deegan-Packel index, the Public Good Index and the Johnston index by means of the multilinear extensions. defined the multilinear extension of a game.…”

Section: Other Results In Related Researchmentioning

confidence: 99%

Power, Voting, and Voting Power: 30 Years After

Holler¹,

Nurmi²

2013

View full text Add to dashboard Cite

Section: Other Results In Related Researchmentioning

confidence: 99%

Power, Voting, and Voting Power: 30 Years After

Holler¹,

Nurmi²

2013

View full text Add to dashboard Cite

“…And so is sex-ual reproduction for any kind of species that is using it. Another striking example of complementary cooperation that is observed frequently in biology is symbiosis 2 .…”

Section: Additive Cooperation Vs Complementary Cooperationmentioning

confidence: 99%

“…al. [2]. The generating function method is another important, but relatively new, way to computing power indices, see Algaba et.…”

Section: Literature Reviewmentioning

confidence: 99%

Complementary cooperation, minimal winning coalitions, and power indices

Cao¹,

Yang²

2013

Theoretical Computer Science

View full text Add to dashboard Cite

We introduce a new simple game, which is referred to as the complementary weighted multiple majority game (C-WMMG for short). C-WMMG models a basic cooperation rule, the complementary cooperation rule, and can be taken as a sister model of the famous weighted majority game (WMG for short). In this paper, we concentrate on the two dimensional C-WMMG. An interesting property of this case is that there are at most $n+1$ minimal winning coalitions (MWC for short), and they can be enumerated in time $O(n\log n)$, where $n$ is the number of players. This property guarantees that the two dimensional C-WMMG is more handleable than WMG. In particular, we prove that the main power indices, i.e. the Shapley-Shubik index, the Penrose-Banzhaf index, the Holler-Packel index, and the Deegan-Packel index, are all polynomially computable. To make a comparison with WMG, we know that it may have exponentially many MWCs, and none of the four power indices is polynomially computable (unless P=NP). Still for the two dimensional case, we show that local monotonicity holds for all of the four power indices. In WMG, this property is possessed by the Shapley-Shubik index and the Penrose-Banzhaf index, but not by the Holler-Packel index or the Deegan-Packel index. Since our model fits very well the cooperation and competition in team sports, we hope that it can be potentially applied in measuring the values of players in team sports, say help people give more objective ranking of NBA players and select MVPs, and consequently bring new insights into contest theory and the more general field of sports economics. It may also provide some interesting enlightenments into the design of non-additive voting mechanisms. Last but not least, the threshold version of C-WMMG is a generalization of WMG, and natural variants of it are closely related with the famous airport game and the stable marriage/roommates problem.Comment: 60 page

show abstract

“…. , F n in Proposition 12 are not uniquely determined by p. For instance, we could choose the power function 1] . We could as well consider the beta distribution with parameters p i and 1 − p i .…”

Section: Properties and Interpretationsmentioning

confidence: 99%

Weighted Banzhaf power and interaction indexes through weighted approximations of games

Mathonet

2011

European Journal of Operational Research

View full text Add to dashboard Cite

The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside coalitions of players. It was shown that the power and interaction indexes can be obtained as solutions of a standard least squares approximation problem for pseudo-Boolean functions. Considering certain weighted versions of this approximation problem, we define a class of weighted interaction indexes that generalize the Banzhaf interaction index. We show that these indexes define a subclass of the family of probabilistic interaction indexes and study their most important properties. Finally, we give an interpretation of the Banzhaf and Shapley interaction indexes as centers of mass of this subclass of interaction indexes.

show abstract

Computing power indices: Multilinear extensions and new characterizations

Cited by 34 publications

References 18 publications

Power, Voting, and Voting Power: 30 Years After

Power, Voting, and Voting Power: 30 Years After

Complementary cooperation, minimal winning coalitions, and power indices

Weighted Banzhaf power and interaction indexes through weighted approximations of games

Contact Info

Product

Resources

About