scholarly journals Power Indices and Minimal Winning Coalitions in Simple Games with Externalities

2015 ◽  
Author(s):  
Joss Marra Alonso-Meijide ◽  
Mikel Alvarez-Mozos ◽  
M. Gloria Fiestras-Janeiro
Keyword(s):  
Power Indices ◽  
Simple Games ◽  
Games With Externalities ◽  
Minimal Winning Coalitions

scholarly journals Power Indices and Minimal Winning Coalitions for Simple Games in Partition Function Form

2017 ◽  
Vol 26 (6) ◽  
pp. 1231-1245 ◽  
Author(s):  
J. M. Alonso-Meijide ◽  
M. Álvarez-Mozos ◽  
M. G. Fiestras-Janeiro
Keyword(s):  
Partition Function ◽  
Power Indices ◽  
Simple Games ◽  
Function Form ◽  
Partition Function Form ◽  
Minimal Winning Coalitions

scholarly journals Bounds for Owen's Multilinear Extension

2007 ◽  
Vol 44 (4) ◽  
pp. 852-864 ◽  
Author(s):  
Josep Freixas
Keyword(s):  
Game Theory ◽  
Simple Game ◽  
Practical Interest ◽  
Simple Games ◽  
Theoretical Interest ◽  
Complex Component ◽  
Multilinear Extension ◽  
Voting System ◽  
Minimal Winning Coalitions ◽  
Minimal Blocking

Owen's multilinear extension (MLE) of a game is a very important tool in game theory and particularly in the field of simple games. Among other applications it serves to efficiently compute several solution concepts. In this paper we provide bounds for the MLE. Apart from its self-contained theoretical interest, the bounds offer the means in voting system studies of approximating the probability that a proposal is approved in a particular simple game having a complex component arrangement. The practical interest of the bounds is that they can be useful for simple games having a tedious MLE to evaluate exactly, but whose minimal winning coalitions and minimal blocking coalitions can be determined by inspection. Such simple games are quite numerous.

scholarly journals Computing power indices for weighted voting games via dynamic programming

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Jochen Staudacher ◽  
László Á. Kóczy ◽  
Izabella Stach ◽  
Jan Filipp ◽  
Marcus Kramer ◽  
...  
Keyword(s):  
Dynamic Programming ◽  
Power Indices ◽  
Efficient Computation ◽  
Unified Framework ◽  
Weighted Voting ◽  
Weighted Voting Games ◽  
The Public ◽  
Voting Games ◽  
Carry Over ◽  
Minimal Winning Coalitions

We study the efficient computation of power indices for weighted voting games using the paradigm of dynamic programming. We survey the state-of-the-art algorithms for computing the Banzhaf and Shapley-Shubik indices and point out how these approaches carry over to related power indices. Within a unified framework, we present new efficient algorithms for the Public Good index and a recently proposed power index based on minimal winning coalitions of smallest size, as well as a very first method for computing Johnston indices for weighted voting games efficiently. We introduce a software package providing fast C++ implementations of all the power indices mentioned in this article, discuss computing times, as well as storage requirements.

On Ordinal Equivalence of Power Indices in Simple Games

1987 ◽  
Vol 14 (22) ◽  
pp. 49-60
Author(s):  
Yoshinori Tomiyama
Keyword(s):  
Power Indices ◽  
Simple Games ◽  
Ordinal Equivalence

SOME OPEN PROBLEMS IN SIMPLE GAMES

2013 ◽  
Vol 15 (02) ◽  
pp. 1340005 ◽  
Author(s):  
CESARINO BERTINI ◽  
JOSEP FREIXAS ◽  
GIANFRANCO GAMBARELLI ◽  
IZABELLA STACH
Keyword(s):  
Power Indices ◽  
Review Of Literature ◽  
Simple Games ◽  
Open Problems ◽  
Weighted Games

This paper presents a review of literature on simple games and highlights various open problems concerning such games; in particular, weighted games and power indices.

scholarly journals Minimal winning coalitions and orders of criticality

2021 ◽  
Author(s):  
Michele Aleandri ◽  
Marco Dall’Aglio ◽  
Vito Fragnelli ◽  
Stefano Moretti
Keyword(s):  
Simple Game ◽  
Simple Games ◽  
Hitting Sets ◽  
The Family ◽  
Minimal Winning Coalitions ◽  
Minimal Blocking

AbstractIn this paper, we analyze the order of criticality in simple games, under the light of minimal winning coalitions. The order of criticality of a player in a simple game is based on the minimal number of other players that have to leave so that the player in question becomes pivotal. We show that this definition can be formulated referring to the cardinality of the minimal blocking coalitions or minimal hitting sets for the family of minimal winning coalitions; moreover, the blocking coalitions are related to the winning coalitions of the dual game. Finally, we propose to rank all the players lexicographically accounting the number of coalitions for which they are critical of each order, and we characterize this ranking using four independent axioms.

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