On the complexity of problems on simple games

Josep Freixas; Xavier Molinero; Martin Olsen; Maria Serna

RAIRO - Operations Research (2012)

  • Volume: 45, Issue: 4, page 295-314
  • ISSN: 0399-0559

Abstract

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Simple games cover voting systems in which a single alternative, such as a bill or an amendment, is pitted against the status quo. A simple game or a yes-no voting system is a set of rules that specifies exactly which collections of “yea” votes yield passage of the issue at hand. Each of these collections of “yea” voters forms a winning coalition. We are interested in performing a complexity analysis on problems defined on such families of games. This analysis as usual depends on the game representation used as input. We consider four natural explicit representations: winning, losing, minimal winning, and maximal losing. We first analyze the complexity of testing whether a game is simple and testing whether a game is weighted. We show that, for the four types of representations, both problems can be solved in polynomial time. Finally, we provide results on the complexity of testing whether a simple game or a weighted game is of a special type. We analyze strongness, properness, weightedness, homogeneousness, decisiveness and majorityness, which are desirable properties to be fulfilled for a simple game. Finally, we consider the possibility of representing a game in a more succinct and natural way and show that the corresponding recognition problem is hard.

How to cite

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Freixas, Josep, et al. "On the complexity of problems on simple games." RAIRO - Operations Research 45.4 (2012): 295-314. <http://eudml.org/doc/222510>.

@article{Freixas2012,
abstract = {Simple games cover voting systems in which a single alternative, such as a bill or an amendment, is pitted against the status quo. A simple game or a yes-no voting system is a set of rules that specifies exactly which collections of “yea” votes yield passage of the issue at hand. Each of these collections of “yea” voters forms a winning coalition. We are interested in performing a complexity analysis on problems defined on such families of games. This analysis as usual depends on the game representation used as input. We consider four natural explicit representations: winning, losing, minimal winning, and maximal losing. We first analyze the complexity of testing whether a game is simple and testing whether a game is weighted. We show that, for the four types of representations, both problems can be solved in polynomial time. Finally, we provide results on the complexity of testing whether a simple game or a weighted game is of a special type. We analyze strongness, properness, weightedness, homogeneousness, decisiveness and majorityness, which are desirable properties to be fulfilled for a simple game. Finally, we consider the possibility of representing a game in a more succinct and natural way and show that the corresponding recognition problem is hard.},
author = {Freixas, Josep, Molinero, Xavier, Olsen, Martin, Serna, Maria},
journal = {RAIRO - Operations Research},
keywords = {Simple; weighted; majority games; NP-completeness; simple},
language = {eng},
month = {1},
number = {4},
pages = {295-314},
publisher = {EDP Sciences},
title = {On the complexity of problems on simple games},
url = {http://eudml.org/doc/222510},
volume = {45},
year = {2012},
}

TY - JOUR
AU - Freixas, Josep
AU - Molinero, Xavier
AU - Olsen, Martin
AU - Serna, Maria
TI - On the complexity of problems on simple games
JO - RAIRO - Operations Research
DA - 2012/1//
PB - EDP Sciences
VL - 45
IS - 4
SP - 295
EP - 314
AB - Simple games cover voting systems in which a single alternative, such as a bill or an amendment, is pitted against the status quo. A simple game or a yes-no voting system is a set of rules that specifies exactly which collections of “yea” votes yield passage of the issue at hand. Each of these collections of “yea” voters forms a winning coalition. We are interested in performing a complexity analysis on problems defined on such families of games. This analysis as usual depends on the game representation used as input. We consider four natural explicit representations: winning, losing, minimal winning, and maximal losing. We first analyze the complexity of testing whether a game is simple and testing whether a game is weighted. We show that, for the four types of representations, both problems can be solved in polynomial time. Finally, we provide results on the complexity of testing whether a simple game or a weighted game is of a special type. We analyze strongness, properness, weightedness, homogeneousness, decisiveness and majorityness, which are desirable properties to be fulfilled for a simple game. Finally, we consider the possibility of representing a game in a more succinct and natural way and show that the corresponding recognition problem is hard.
LA - eng
KW - Simple; weighted; majority games; NP-completeness; simple
UR - http://eudml.org/doc/222510
ER -

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