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.

Keywords: simple, weighted, majority games, NP-completeness

@article{RO_2011__45_4_295_0, author = {Freixas, Josep and Molinero, Xavier and Olsen, Martin and Serna, Maria}, title = {On the complexity of problems on simple games}, journal = {RAIRO - Operations Research - Recherche Op\'erationnelle}, pages = {295--314}, publisher = {EDP-Sciences}, volume = {45}, number = {4}, year = {2011}, doi = {10.1051/ro/2011115}, mrnumber = {2881357}, zbl = {1235.68082}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ro/2011115/} }

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 - Recherche Opérationnelle PY - 2011 SP - 295 EP - 314 VL - 45 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ro/2011115/ DO - 10.1051/ro/2011115 LA - en ID - RO_2011__45_4_295_0 ER -

%0 Journal Article %A Freixas, Josep %A Molinero, Xavier %A Olsen, Martin %A Serna, Maria %T On the complexity of problems on simple games %J RAIRO - Operations Research - Recherche Opérationnelle %D 2011 %P 295-314 %V 45 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ro/2011115/ %R 10.1051/ro/2011115 %G en %F RO_2011__45_4_295_0

Freixas, Josep; Molinero, Xavier; Olsen, Martin; Serna, Maria. On the complexity of problems on simple games. RAIRO - Operations Research - Recherche Opérationnelle, Volume 45 (2011) no. 4, pp. 295-314. doi : 10.1051/ro/2011115. http://archive.numdam.org/articles/10.1051/ro/2011115/

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