A characterization of weighted simple games based on pseudoweightings
- PDF / 305,262 Bytes
- 13 Pages / 439.37 x 666.142 pts Page_size
- 25 Downloads / 180 Views
A characterization of weighted simple games based on pseudoweightings Josep Freixas1 Received: 2 May 2020 / Accepted: 9 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The paper provides a new characterization of weighted games within the class of simple games. It is based on a stronger form of the point-set-additive pseudoweighting property of simple games. The characterization obtained is of interest in various research fields such as game theory, coherent structures, logic gates, operations research and Boolean algebra. A (monotonic) simple game corresponds to an inequivalent (monotonic) function in Boolean algebra and a weighted game corresponds to a threshold function. The characterization obtained provides a better understanding of these mathematical structures while opening new prospects for solving numerous open problems in these areas. Keywords Boolean functions · Threshold functions · Simple games · Weighted games · Pseudoweightings · Optimization
1 Introduction Simple games are at the core of voting systems, in them a single alternative, such as a bill or an amendment, is pitted against the status quo, and voters can vote for or against the bill. Simple games are very versatile structures equivalent to monotonic hypergraphs that can also be thought as coherent structures, logic gates or Boolean functions. Two subclasses of simple games stand out for their mathematical properties and applications to real voting systems. We refer to the class of complete games that contains, in turn, the class of weighted games. Both subclasses have been extensively studied in several different fields and under different denominations. The search for elegant properties to characterize these kinds of games, within the class of simple games, has been and is an important milestone. Each characterization sheds new light
B 1
Josep Freixas [email protected] Universitat Politècnica de Catalunya, Av. Bases de Manresa, 61-73, 08242 Manresa, Spain
123
J. Freixas
on the understanding of these structures and makes it easier to solve related problems and obtain new enumerations of subclasses of games. Before proceeding let’s introduce a little bit informally some of the referred notions that constitute the basis of this paper: simple games and weighted games. See Sect. 2 for the formal definitions. In the former USSR the three top state officials, the President, the Prime minister, and the Minister of Defence (Ustinov, Brezhnev, Kosygin), all had “nuclear suitcases”. Any two of them could authorize a launch of a nuclear warhead. No one could do it alone. This situation can be modeled as a simple game with grand coalition N = {1, 2, 3} and where the set of winning coalitions is given by W = {{1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}. Thus, the set of minimal winning coalitions with respect to the inclusion is given by W m = {{1, 2}, {1, 3}, {2, 3}}. This simple game is weighted since a quota can be chosen together with weights assigned to players in such a way that the sum of the weights of the
Data Loading...
