✦ LIBER ✦

The Fundamental Theorem of Voting Schemes

✍ Scribed by D.E. Loeb

Publisher: Elsevier Science
Year: 1996
Tongue: English
Weight: 298 KB
Volume: 73
Category: Article
ISSN: 0097-3165
DOI: 10.1006/jcta.1996.0006

No coin nor oath required. For personal study only.

✦ Synopsis

Introduction

Let V be a set, and call the elements of V voters or players. A subset A V is called a coalition. The compliment V&A of a coalition A is denoted A . A set of coalitions S is a game if all supersets of winning coalitions are winning as well.

A coalition A V is said to be blocking (in S) if A Â S. The set of all blocking coalitions is denoted S*, and is called the dual of S, since (S*)*=S. A game S is simple if A # S (A wins) implies A Â S (A blocks); i.e. S S*. Conversely, a game S is strong if A Â S (A blocks) implies A # S (A wins); i.e. S* S.

Given a weight w i # N for each voter i # V=[1, 2, ..., n] and a quota q # N, we can define the quota game (w 1 , w 2 , ...,

Democracy over an odd number of voters is a strong simple game. Dem 2n+1 =(11... 1 2n+1 ) n+1 =[A [1, ..., 2n+1] : |A| >n].

Similarly, dictatorships are also strong simple games.

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