Markets and Majorities

The backdrop for Professor Arrow’s influential early work was the centuries-long recognition that majority voting can produce arbitrary outcomes.

Consider a legislature choosing its leader from among three candidates: Alice, Betty and Harry. If the legislature were to vote first on Alice versus Betty, with the winner running against Harry, it could come to a different decision than had it first started by voting on Alice versus Harry. Because the order with which the legislature takes votes is arbitrary, the ultimate winner of this system of majority voting becomes arbitrary. That puts politics in an awkward corner.

In search of nonarbitrary outcomes, social scientists proffered different ways to conduct votes. For example, the legislature could run all three candidates in the initial round and structure some type of runoff. Or the legislature could give each member multiple votes to be assigned to the three candidates in proportion to the intensity of the member’s preferences.

But no voting system, however cleverly designed, resolved the problems associated with majority voting. In a theorem of stunning generality, Professor Arrow proved that no system of majority voting worked satisfactorily according to a carefully articulated definition of “satisfactory” (which social scientists generally accept).

What Professor Arrow proved in his book “Social Choice and Individual Values” (1951) was far more sweeping. Not only would majority-voting rules prove unsatisfactory; so, too, would nonvoting systems of making social choices if, as was fundamental to his way of thinking, those choices were based on the preferences of the individuals making up the society. (Professor Arrow’s rules did not allow for dictators.)