May's Theorem

As promised yesterday, the rest of this week's regular voting analysis type bit...

a) Proof of May’s Theorem[1]

May gives four necessary and sufficient conditions for simple majority rule[2]:

Decisiveness: “the method must be decisive and universally applicable, or more briefly always decisive, since it must specify a unique decision (even if this decision is to be indifferent) for any individual preferences.” (p.681)
Anonymity: “The group decision function is a symmetric function of its arguments… D is determined only by the values of the Di that appear, regardless of how they are assigned to individuals as indicated by subscripts (names). A more usual name is equality.” (p.681)
Neutrality: “the method of group decision does not favor either alternative. A precise way of stating this is that if the names of x and y are reversed, the result is not changed.” (p.681)
Positive responsiveness: “if the group decision is indifference or favourable to x, and if the individual preferences remain the same except that a single individual changes in a way favourable to x, then the group decision becomes favourable to x.” (p.682)

May states these conditions are necessary for majority-rule – nothing is majority rule unless it satisfies them. Secondly he states they are sufficient, anything satisfying them is majority-rule.

That majority-rule decision function (D) can be represented as sum of N(1), N(0) and N(-1); where N(1) is the number of votes for x and N(-1) as votes for y. Those who are indifferent, N(0), don’t matter here. If D=0 the social decision is indifference. If D>0 the social decision =+1, and if D<0 it is –1.

That majority rule therefore satisfies the four conditions is fairly obvious. It always produces an outcome of +1, 0 or –1. This is true even when individual inputs (votes) change. Swapping a +1 and a –1 ballot doesn’t affect the result, the sum is unchanged. The same voting patterns on another issue (e.g. a against b) would produce the same sum, and thus the same result – there is no bias to the status quo, as there would be if, e.g. a two-thirds majority were required. Finally an increase in any Di will increase the sum D, and thus ensure positive responsiveness, e.g. if D was non-negative it would become positive.

Next we have to show any procedure satisfying the four conditions is majority-rule, i.e. that they are sufficient for majority-rule. If N(-1) = N(1), i.e. votes are tied, it follows from the first three conditions that the social decision is indecisive, i.e. D=0. If x were to win, i.e. D=1, then swapping all +1s and –1s would either produce the reverse result, so anonymity would be violated, or the same result in which case neutrality would be violated. Further, if N(1) = N+1(-1), i.e. one y-voter becomes indifferent and abstains, so there is one more x-voter than y-voter, then the social decision must prefer x, from positive responsiveness. Thus the four conditions imply simple majority-rule.

b) Evaluation of May’s Theorem

May’s theorem merely defines necessary and sufficient conditions for simple majority-rule. It is not clear that his conditions are always desirable, as is the case with Arrow’s.

Decisiveness I take to be relatively uncontroversial. We want a procedure that always yields outcomes, even if those outcomes are to be indifferent. After all, if politics is supposed to be war by other means[3], we need a political solution – otherwise we may end up resorting again to violence.

Anonymity, as May points out, is basically equality between voters. This seems obviously desirable. Even if we wanted some voters to count for more[4], in order to correctly weight their number of votes, we would want each vote to count equally, and thus be anonymous.

Neutrality is the condition that no option is favoured. This rules out requiring super-majorities, or a bias in favour of the status quo. This may be generally desirable, but it is not always so – often constitutions are ‘entrenched’ so that they require, e.g. a two-thirds majority to amend. May has nothing to say about this, other than that such is obviously not simple majority-rule. For ordinary political decisions, however, it seems neutrality is preferable. A bias in favour of the status quo would effectively mean those voting for it (though unidentified a priori) would count for more than those voting against it.

It is the final one of May’s conditions that I find most problematic. Positive responsiveness, as he defines it, is quite stringent – much more so than Arrow’s positive monotonicity or non-negative responsiveness[5]. May requires that where there is indifference, a single voter can become decisive. I am not sure I share this intuition. Suppose there are millions of voters involved – many of whom are indifferent, but those for x exactly balance those for y. What May’s condition means is that if just one of the indifferent voters decides to vote for x, then x is now an outright winner. It’s not clear to me that a conflict between 1,000 and 1,001 should be decided simply by reference to the numbers[6]. Just because the other members of society ‘balance out’ it is not obvious that one other should be able to determine the social decision[7].

Lottery-voting is decisive (the vote picked determines the outcome), anonymous (all votes have an equal chance of being picked) and neutral (no outcome is particularly favoured – the chances of each outcome depend only on the number of votes it gets). It is May’s positive responsiveness that lottery-voting ‘fails’, but this simply tells us that lottery-voting is not simple majority-rule (because satisfying these four conditions is sufficient for being such, anything that isn’t majority-rule must fail one). As I have said, however, this demand does not seem normatively compelling to me. Lottery-voting satisfies Arrow’s weaker axiom – a vote for x always increases x’s chances of victory[8].

Another advantage of lottery-voting is that it is easily applicable to cases with more than two options. May is working in a context where voters choose simply between x and y (as in a two party election). It is this context that necessarily means indifferent voters are ignored. For May, if there are two votes for x and three for y, then y is ‘socially preferred’ even if a million people are indifferent. Voters who want the social outcome to be indifference can only attempt to bring this about by voting for whichever option they think is in the minority, and even then it is a fine balancing acting, as only an exactly equal balance results in social indifference – indifference can’t be registered as a ‘positive preference’, as this would be to introduce a third option.

That May is only working in a two-option framework isn’t itself a criticism, but it shows the limits of his theorem, as politics almost always involves more than two options – and while a larger number may be artificially reduced to two (e.g. by AV or STV) there is no fair way of doing this[9], and the voting system is no longer simple majority-rule as in any case it has to eliminate the other options first. Lottery-voting can be straightforwardly applied to cases with more than two options, and still respects decisiveness, anonymity, neutrality and non-negative responsiveness.

[1] Following D. C. Mueller (2003) Public Choice III pp.135-6.
[2] K. O. May (1952) ‘A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision’ Econometrica 20:4 680-684 (references in parentheses refer to this article).
[3] I reverse Clausewitz’ aphorism. War I assume came first, before politics emerged as a peaceful way of resolving conflicts. C.f. A. Przeworski ‘Minimalist Conception of Democracy: A Defense’ in I. Shapiro and C. Hacker-Cordan (eds.) (1999) Democracy’s Value p.48. It’s possible that majority rule developed from counting the numbers on each side and giving victory to the greater number on the basis they were most likely to win a violent contest. This would also explain why the franchise was originally linked to fighting ability – i.e. restricted to men, and particularly the rich (hoplite class).
[4] See, e.g., Aristotle, Fleurbaey. C.f. Banzhaf on weighted voting.
[5] See May p.682, fn.8.
[6] In another context, I seem to be joined by Frances Kamm – who describes an extra person in this case as an ‘irrelevant’ utility’ – John Broome and Iwao Hirose.
[7] Note this is not the same as the median voter theorem, in which the one in the middle gets their way because others are balanced on either side – because (as will be highlighted later) the median is a placeholder for whoever happens to be in the middle – there is not one decisive voter who can throw the decision either way.
[8] A condition that rules out STV.
[9] Riker (1982) Liberalism Against Populism: A confrontation between the theory of democracy and the theory of social choice pp.59-60.