Social welfare functions

Transcription

1 Social welfare functions We have defined a social choice function as a procedure that determines for each possible profile (set of preference ballots) of the voters the winner or set of winners for the corresponding election that uses those ballots. A more general situation is provided by a social welfare function. Such function also has as domain the collection of all possible voting profiles for the voters, but instead of determining a set of winners, a social welfare function determines a social preference list, a single list that ranks the alternatives from first to last for the benefit of the entire public. Of course, the alternative at the top of this list can be considered the winner of the election (more generally, we may want to consider the first two or three, or more, of the alternatives at the top of the social preference list to be the set of winners), so any social welfare function will provide the necessary elements of a social choice function. However, we can also use a social choice function to build a social welfare function.

2 If the social choice function determines, for instance, that the set of winners for the election is { a,c,d}, where the list of candidates is a,b,c,d,e, then we can construct a social preference list by placing a, c, d in some order (say alphabetical) at the top of the list. Then, after eliminating these winning alternatives from each of the voters ballots, we can apply the same social choice function to determine what alternative(s) should be ranked next on the social preference list (resolving ties by some method, like deferring to an alphabetization of the alternatives). Iterating (i.e., repeating) this method as often as necessary, we eventually rank all the alternatives on the social preference list. The above argument proves the Theorem. Every social choice function gives rise to a social welfare function, and conversely. Notice that whenever there are only two alternatives involved, it is particularly simple to see the correspondence between a social choice function and the associated social welfare function: the winner of the election is the alternative at the top of the social preference list, and the loser is the one at the bottom (and if there is a tie, both alternatives are winners).

3 May s Theorem In this simple (and very common) situation in which there are only two alternatives to choose from, it is easy to see that the plurality, Borda, Hare, and sequential pairwise with fixed agenda methods all reduce to majority rule: the one of the two alternatives that receives the majority of the first place votes wins the election. (The only case in which a tie can arise is if there are an even number of voters and both alternatives receive exactly half of the first place votes.) This suggests that in the two-alternative case, there may be very few different social welfare functions. This is not quite true. Observe that the dictator method does not in general reduce to majority rule. Moreover, there are other welfare functions that are not equivalent with majority rule: Tyrant methods Pick one of the alternatives. Declare this alternative the winner regardless of how the voters cast their ballots. Parity method An alternative is a social choice precisely when it receives an even number of votes.

4 Still, neither of these last two methods has much to offer as a rational social welfare function. As we have done in the past, let us describe some desirable properties of social welfare functions: 1. Anonymity A social welfare function is anonymous if the outcome of the election is unaffected by a shuffling of the ballots amongst the voters. Majority rule is an anonymous method, while any dictatorship is not. 2. Neutrality A social welfare function is neutral if, when the alternatives are permuted in the same manner on each voter s ballot, then the winners are permuted amongst the alternatives in exactly the same way. Majority rule is a neutral method, but any tyrant method is not. 3. Monotonicity As we defined earlier, a method is monotone if whenever a voter changes his ballot by moving a winner of the election higher in his preference list, then a redetermination of the outcome must name this alternative as a winner. Majority rule is monotone, but parity is not.

5 A quota system is a social choice method that declares that all alternatives are winners if none of them receives at least a quota of q first-place votes, where n 2 < q n +1, n being the number of voters. If one alternative does have at least q votes, then it is declared the sole winner. (Observe that it is impossible for more than one alternative to receive more than n/2 firstplace votes, else there is a total of more than n votes!) In the case of a two-alternative election, when the quota is the maximal value q = n +1, it is impossible for an alternative to achieve the quota, and both alternatives will always tie. On the other hand, if n is odd and q = n +1, then the quota system is 2 equivalent to majority rule. It is not too hard to see that any quota system is anonymous, neutral, and monotone. In fact, the converse it also true. Theorem. The only social welfare functions for two alternatives that are anonymous, neutral, and monotone are quota systems.

6 Proof. Suppose we have a social welfare function for two alternatives a and b and n voters that is anonymous, neutral, and monotone. Because of anonymity, the social choice must be determined only by the number of voters who prefer a to b. So let K be the set of these numbers k lying between 0 and n for which alternative a is the sole winner when it receives exactly k votes. If K is an empty set, a is never a sole winner. But then by neutrality, b is never a sole winner either. So the alternatives must always tie for winner (as there must be at least one winner). This is the situation that occurs under a quota method when q = n +1. If K is not empty, let q be the smallest number contained in it. Then a is the sole winner when exactly q voters vote for a. By monotonicity, every number greater than q (and less than or equal to n) must also lie in K. Next, we claim that n -q is not in K, for if a wins with exactly q votes and we switch alternatives a and b, then neutrality requires that b be the sole winner and that a lose with n -q votes. So, since n -q is not in K, then n -q < q, which implies that n < q. We have thus shown that the method is a 2 quota system, and so the proof is complete. //

7 An immediate consequence of this result is May s Theorem (1952). With an odd number of voters and no ties, the only social welfare function for two alternatives that is anonymous, neutral, and monotone is majority rule. Proof. We know that the welfare function must be a quota system by the previous theorem. Since there are an odd number of voters, the smallest q can be is the smallest whole number greater than n n +1, namely. But if q is any larger than this 2 2 number, it is possible that no sole winner could be declared (as when a receives exactly n +1 2 votes). As there are no ties allowed, it follows that q = n +1 2 and the social welfare function is equivalent to majority rule. //