An axiomatization of success

In this paper we give an axiomatic characterization of three families of measures of success de…ned by Laruelle and Valenciano (2005) for voting rules. Key words: collective decision-making, voting rules, axiomatization 1. Introduction The aim of this paper is to provide an axiomatization of the measures of success in voting rules. We look for a set of axioms, that is, assumptions that, whatever their plausibility, have a clear meaning and make sense one by one, independently of the others. What we obtain in this paper are the three families of measures of success for voting rules de…ned by Laruelle and Valenciano (2005). These measures are associated with probability distributions p over the set of all possible vote con…gurations. Measure , which is formalized in the following section, gives the probability for a voter of having the result he voted for. Measure p+ gives the probability for a voter of having the result he voted for conditioned on voting yes. And measure p gives the probability for a voter of having the result he voted for conditioned on voting no. In this paper we give three axiomatic characterizations. One for the family of measures f gp2P , where P denotes the set of all the possible probability distributions. Other for the family of measures f gp2P . And the last one for f p gp2P . The axioms we employ are some common ones together with others which are speci…c for each family. In the following section we present the measures of success de…ned by Laruelle and Valenciano (2005), and in Section 3, 4 and 5 we give the axiomatic characterizations of the three families. 2. Background We consider voting rules to make dichotomous choices (acceptance and rejection) by a voting body. Let N = f1; 2; ::; ng denote the set of seats. If any vote di¤erent from ’yes’is assimilated into ’no’, there are 2 possible vote con…gurations. Each vote con…guration can be represented by the set S N of ’yes’voters. An N -voting rule is fully speci…ed by the set WN of winning vote con…gurations, that is, those *Department of Applied Economics IV, Faculty of Business and Economics, University of the Basque Country, Lehendakari Agirre, 83, 48015 Bilbao, Spain, e-mail: [email protected] |Ikerbasque and Department of Foundations of Economic Analysis I, Faculty of Business and Economics, University of the Basque Country, Lehendakari Agirre, 83, 48015 Bilbao, Spain, email: [email protected]. 1 2 M. JOSUNE ALBIZURI*, ANNICK LARUELLE| which lead to the acceptance of a proposal (the others would lead to the rejection of the proposal): WN = fS : S leads to a …nal ’yes’g. When N is obvious from the context, we omit the subscript’N’and writeW instead of WN . In order to exclude unconsistent voting rules, we assume that the set W satis…es the following conditions: (i): The unanimous ’yes’leads to the acceptance of the proposal: N 2 W ; (ii): The unanimous ’no’ leads to the rejection of the proposal: ; = 2W ; (iii): If a vote con…guration is winning, then any other con…guration containing it is also winning: If S 2 W , then T 2 W for any T containing S; (iv): If one vote con…guration leads to the acceptance of a proposal, the opposite con…guration will not: If S 2 W , then NnS = 2 W . Let V RN denote the set of voting rules with set of seats N . A voting rule can also be described by its set of minimal winning con…gurations. A con…guration S is minimal winning if S 2 W and for any i 2 S, S n i = 2W . The set of minimal winning con…gurations of rule W is denoted M(W ). A seat i is said to be a dictator seat if for all S we have S 2W if and only if i 2 S. The T -unanimity rule, denoted W , is the voting rule W = fS N : S Tg The extreme cases are when T = N (unanimity) and T = fig (seat i is a dictator seat). For any voting rule W 2 V RN such that W 6= U , and any T 2M(W ), the modi…ed voting rule W T is the voting rule such that W T = W n fTg. Let GN denote the set of transferable utility games with player set N . That is, GN is formed by the mappings w from 2 into R such that w (;) = 0. And SGN denote the subset of GN formed by simple superadditive games such that the worth of N is 1. That is, by the mappings w 2 GN such that w (S) 2 f0; 1g for any S N , w (N) = 1 and w (S [ T ) w (S) +w (T ) whenever S \ T = ;. Notice that superadditivity implies monotonicity, that is, w (T ) w (S) whenever S T . Then we can obviously identify V RN with SGN , by associatingW 2 V RN with the game w 2 SGN that satis…es w(S) = 1 if and only if S 2 W . We distinguish the game and the procedure by using the small letter in the …rst case and the capital letter in the second case. Laruelle and Valenciano (2005) de…ne some measures of success. They consider a probability distribution over the set of all possible vote con…gurations, which can be interpreted as a ”common prior”about the voters voting behavior. Let p denote a probability distribution over the set of vote con…gurations, and let p(S) denote, for each S N , the probability of S being the vote con…guration. Let P denote the set of all probability distributions. For a given p let i(p) := Prob (i votes ’yes’) = X