Kenneth May on Social Choice

Kenneth O. May and the Theory of Social Choice

Thomas L. Bartlow
Department of Mathematical Sciences
Villanova University
Villanova, PA 19085

Introduction

Kenneth O. May (1915--1977) distinguished himself as a mathematician, educator, historian of mathematics, and political activist. The story of May's early life is well told by Charles V. Jones, Philip C. Enros and Henry S. Tropp in ``Kenneth O. May, 1915-1977. His Early Life to 1946,''. This biography brings us up to the time when May earned a Ph.D. in mathematics at the University of California with a thesis, ``On The Mathematical Theory of Employment." Following the completion of his degree he joined the faculty of Carleton College in Northfield, Minnesota, served as a consultant to the Cowles Commission for Research in Economics and began investigating questions in the theory of social choice. Later his interests turned to mathematics education, history of mathematics and information retrieval. A brief summary of this later work was given in an obituary notice by Trevor H. Levere. Between 1947 and 1953 May made three contributions to the theory of social choice, related to

Election Inversions
Majority Decision and
Intransitivity.

We summarize his work in each of these area and follow with a Conclusion and a Bibliography.

Election Inversions

Let an election take place in districts. An inversion occurs when one party gains a majority of the total vote while another party gains a majority in a majority of the districts. On May 10, 1947 May presented a report on the probability of an election inversion to the Minnesota section of the Mathematical Association of America and on May 21 the American Mathematical Society received an abstract of his results; a second abstract arrived in November. A paper in the American Mathematical Monthly in April 1948.

Let there be m voters in each of n districts in which voting is uniformly distributed and independent of the other districts. Assuming m and n both odd, May developed a formula for the probability P(m,n) of an election inversion. He evaluates P(m,n) for various values of m and n and finds that its limit as m and n become large is 1/6. He conjectures that P(m,n) is monotonically increasing in both m and n, increasing from P(2,2) = 3/32 to the limit 1/6. The later abstract announces an extension of these results to the case where voting is not uniformly distributed. No full exposition of this extension ever appeared, nor has one been found among the materials in the possession of the University of Toronto Archives.

Following this theoretical work May undertook an empirical study of the frequency of election inversions. On April 16, 1949 he wrote the American Philosophical Society to inquire whether funds might be available to cover the expenses of such a study. On May 3 he submitted a formal application for 200 dollars, the grant was approved on June 3, and funds were disbursed on June 23. The librarian at the American Philosophical Society has kindly made available to me the correspondence relating to this grant, but has denied access to the referee reports. In October of 1950, 1951, and 1953, May made progress reports to the Society, in the form of a one page letter to the Executive Officer, commenting on the difficulty of obtaining information, and reporting on the expenditure of funds. Finally on April 24, 1958 he wrote that the work was complete and a final report was ready. The report appeared in the 1958 Yearbook of the Society. May was able to examine the results of 278 legislative elections and found 23 inversions, only about half the number that might be expected from the theoretical model. May remarks that the number is ``surprisingly low and suggests the need for a more realistic model.''

In his correspondence with the American Philosophical Society May says that the question of the frequency of election inversions was brought to his attention G. Lowell Field. In 1946 Field was an assistant professor of political science at the University of Texas but in 1947 he moved to Wayne University (now Wayne State) in Detroit, and later he became chair of political science at the University of Connecticut. It is not known when or under what circumstances Field posed the question to May. In the exploratory letter to the American Philosophical Society on April 16, 1949 May says that Field "was particularly interested in the electoral college system." In 1951 Field published a book on comparative government which references May's papers in connection with a discussion of the difficulties of a legislature that lacks popular support. Although May's father, Samuel Chester May (1887-1955) was a political scientist, it does not seen likely that he provided a connection between his son and Field. Samuel May was twenty-three years older than Field and their interests in political science were quite different. Samuel May was long established on the West Coast while Field earned his Ph.D. at Columbia and, after brief sojourns in Texas and Michigan, spent most of his career in Connecticut. Moreover Kenneth and Samuel May had become estranged in 1940 over political questions and, although there was a partial reconciliation in 1944, there may still have been some strain. At the University of California May had studied statistics with Jerzy Neyman and had devoted much time and energy to political activism. No doubt he saw Field's question as an opportunity to combine two important interests. (On May's work at the University of California and his relationship with his father see the biography by Jones, Enros and Tropp.)

Majority Decision

In 1952 May took up the study of two questions that seem to have been inspired by the 1951 book of Kenneth Arrow. Let there be a set of individuals confronted with the problem of making a group decision among a set of several alternatives. Let each individual rank the alternatives in order of preference. Imagine a function that accepts as input a vector of individual orderings and produces as output a group ordering. In the summer of 1948 Arrow discovered that, if there are more than two alternatives, no function can satisfy all of several conditions which Arrow put forth as desirable. The results were publicly announced at a meeting of the Econometric Society in December 1948; later Arrow recalled that May attended this session. This discovery formed the basis of Arrow's Ph.D. thesis at Columbia and became widely known with the publication of his book in 1951. May's review of the book was one of the first and he finds occasion frequently to quote the book in his own papers of 1952--54.

Arrow's discussion of his theorem begins by considering the rule which produces a group ordering with alternative a preferred to alternative b if and only if a majority of individuals prefer a to b. He shows that such a rule satisfies all but one of his conditions, failing the other one because, when there are more than two alternatives, the resulting relation may fail to be transitive, even if all the individual orders are transitive. When there are two alternatives the rule, called the method of majority decision, satisfies all of Arrow's conditions. May restricts his attention to this latter case and seeks necessary and sufficient conditions for a function to be this method, obtaining results for two papers.

In the first paper he explains that, with just two alternatives a and b, a preference order can be coded by one of three numbers, 1 if a is preferred to b, 0 if indifference obtains, -1 if b is preferred to a. Then a vector of individual preferences is an n-tuple in D^n where D={-1,0,1} and May introduces the term "group decision function" for a function f:D^n -> D. Then he proved that the following four conditions are necessary and sufficient for f to be majority rule.

The function f is defined and single valued on D^n.
f is "symmetric in its arguments." If vector u is a permutation of vector v, the f(u) =f(v).
If all voters change their minds, so does the function, f(-v) = -f(v).
f is monotonic. If f(v) is 0 or 1 and if u is identical with v except that one individual has increased his opinion of a, then f(u)=1.

The second paper, an abstract of which was received by the American Mathematical Society in November 1952, gives examples to show that these conditions are completely independent. For any subset of them there is a (non-majority) group decision function that satisfies the conditions in that subset and violates all others.

Intransitivity

Arrow assumed that both individual and group preference orders would be transitive. May's third line of investigation, also undertaken in 1952, argues against this assumption. On April 9, 1952 he initiated a correspondence with Warren McCullogh soliciting evidence of individual intransitivity of preference. He presented a paper to the Econometric society in December 1952 and shortly thereafter submitted it to Econmetrica. In June of 1953 Econometrica editor Robert H. Strotz wrote May about referees' comments, suggesting that the referees had misunderstood the paper, and offering to send a revised paper to a fourth referee. May replied rather testily defending his paper and making suggestions for a new referee, including Arrow. The paper appeared in January 1954. The correspondence mentioned in this paragraph and an early draft of the paper can be found in Box 015 of collection B83-0023 at the University of Toronto Archives.

In the first section of this paper May defines a binary preference pattern as a collection of binary choices, preference or indifference, among pairs of alternatives in any set of alternatives such that

for every pair of alternatives one is preferred to the other or indifference holds (trichotomy),
indifference is reflexive,
indifference is symmetric,
preference is anti-symmetric.

Transitivity does not follow from these properties, either for indifference or for preference and May chooses not to assert transitivity.

In the second section May proves that the existence of a cardinal or ordinal utility function on the alternatives implies transitivity of indifference and preference, and, for finite sets of alternatives, the converse is true. He further argues that the existence of an infinite set of alternatives on which a transitive preference pattern could be imposed but no utility function could be found would have cardinality larger than the continuum and would have no practical importance. Thus, from a practical standpoint May considers the existence of a utility function as equivalent to the existence of a transitive preference pattern.

The third section cites a variety of evidence that individual preferences may not be transitive, taken from May's reading and from his correspondence with McCulloch. May also reports on an experiment he had conducted on the binary preferences of college students for characteristics of potential mates (intelligence, wealth, and beauty). Students were presented with descriptions of pairs of potential mates possessing these qualities in various degrees. More than one-quarter of the students registered intransitive preferences.

The final section of the paper contains May's version of Arrow's theorem. Let individuals be presented with the problem of choice from a set of alternatives. Let each individual establish a binary preference pattern on the set of alternatives. For each ordered pair of alternatives (x,y), each individual's preference can be encoded by -1, 0, or 1 as in the majority decision paper of 1952 and we wish to have a group decision function f which acts on vectors of preference codes to establish an group preference on the pair. Let D_i be -1, 0, or 1, according to the i-th individual's opinion. The function should produce an aggregate evaluation

D=f(D_1,D_2,...,D_n).

Moreover f should have the following properties.

It acts on all vectors of individual preferences.
It respects unanimity. If all individuals have the same opinion on two alternatives, f adopts this as the group opinion.
It is "not negatively responsive." If D'_i is greater than or equal to D_i for every i, then D' is greater than or equal to D.
There is no dictator. For each individual i there is a vector in which D_i = -1 but D is not -1.

A set of such functions, one for each pair of alternatives, establishes a function which assigns to each vector of individual binary preference patterns an aggregate binary preference pattern. May shows that this function must satisfy Arrow's conditions. Since Arrow assumed transitive preferences on three or more alternatives and found a contradiction, May concludes that, "among binary preference patterns arising from the aggregation of three or more binary patterns there are non-transitive patterns arising from transitive components unless the method of aggregation fails to be non-negatively responsive or unless one component dominates.".

Conclusion

May's papers on majority decision and intransitivity are part of the mainstream of the theory of social choice in the last half of the twentieth century. A review of the Science Citation Index and the Social Science Citation Index shows that the 1952 paper on majority rule is cited by about half a dozen authors per year until 1985 and has been cited at least twice a year every year since then. (The 1953 supplement is rarely cited.) The 1954 paper on intransitivity has been cited at least once every year with two or three citations in most years and a high of eight citations in 1991. Downing and Stafford identify twenty classics in the theory of social choice based on citations in 1978 and 1979. The eleven 'major classics' had between 14 and 50 citations. The nine 'minor classics' were cited between 8 and 13 times. Although May's papers are not included in either category, his 1952 paper on majority rule had 10 citations during this two year period and his 1954 paper on intransitivity was cited 5 times. Downing and Stafford separated their major classics into a verbal and a mathematical tradition. Five of the seven in the mathematical tradition cited May, as did one in the verbal tradition.

On the other hand the work on election inversions might as well have been a private communication between May and Field. It appears that the only publications on this topic are May's paper of 1948 and his report to the American Philosophical Society in 1958. The only reference I have found to either of these is in Field's 1951 textbook.

May continued to be interested in the mathematical economics of his doctoral work and in the theory of social choice for the rest of his career. During the 1950's some of his students at Carleton College did undergraduate research projects on social choice theory. As he was making the transition to history of mathematics in the early 1960's he wrote reviews of works on social choice. In 1968 he received an invitation to present a paper at a conference on the theory of social choice and although he had to decline he offered to prepare a paper on the history of the subject for a later occasion. A curriculum vita prepared in 1975 lists mathematical economics among his research interests (University of Toronto Archives, collection B83-0023, boxes 002, 015).