There are more strategy-proof procedures than you think

With as few as eight individuals and five alternatives, there are 561, 304, 372, 286, 875, 579, 077, 983 strategy-proof social choice rules. © 2012 Elsevier B.V. All rights reserved. The Gibbard–Satterthwaite theorem, now almost 30 years old, has become a staple of microeconomic theory, social choice theory, and mechanism design (Gibbard, 1973; Satterthwaite, 1975). Everyone understands that there are very few strategyproof procedures: Asmany dictatorial rules as there are individuals and a handful of rules that have small range, that select less than all alternatives. But there are more strategy-proof rules than you think. Suppose that there are m alternatives, making up the set X , and n individuals, making up the set N . A coalition is a subset of N . For simplicity, assume that preferences are strong, i.e., antisymmetric as well as transitive. There are then m! possible preference orderings. A profile u is an assignment of one preference ordering, ≻i, to each individual i : u = (≻1, ≻2, . . . ,≻n) and so there are (m!)n profiles. A social choice rule selects one of the alternatives at each profile. There are, therefore, m(m!) n social choice rules. But how many of these rules are strategy-proof? Classified by the size of their range, there are three kinds of strategy-proof rules. (1) Rules with |Range(f )| = 1. There arem such constant rules. (2) Rules with |Range(f )| > 2. Such rules are necessarily dictatorial by the Gibbard–Satterthwaite theorem. So for each such range there are n rules, one for each possible dictator. Of the 2m − 1 non-empty subsets of X , i.e., possible ranges,m are singletons and m 2  = m(m−1) 2 are pairs, so there are 2 m − m − m(m−1) 2 − 1 possible ranges of three or more alternatives. Altogether then, there are n[2m − m − m(m−1) 2 − 1] strategy-proof rules in this category. ∗ Corresponding author. E-mail address: [email protected] (J.S. Kelly). 0165-4896/$ – see front matter© 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.mathsocsci.2012.05.001 (3) Rules with |Range(f )| = 2. There are, as noted earlier, m(m−1) 2 possible ranges of two alternatives. Suppose that the range of rule f is {x, y}. If f is strategy-proof, then the value of f at profile u is entirely determined byhow individuals order just x and y at u. If any other part of an individual’s ordering affects the social choice, that individual could manipulate the social choice rule. Accordingly, f is completely characterized by the collection C of winning coalitions for x against y, those coalitions at which f (u) = x when x is preferred to y by exactly the members of that coalition i.e., f (u) = x if and only if {i/x≻i y} ∈ C. The best known example, for an odd number of individuals, is simple majority voting between a pair of alternatives. In this strategy-proof case, C is the collection of all coalitions of cardinality greater than n/2. It is important to observe that this collection satisfies the superset property: J ∈ C and J ⊆ J ⊆ N implies J ∈ C. This generalizes: A rulewith range {x, y} is strategy-proof if and only if the collectionC of coalitionswinning for x against y satisfies the above superset property. Now suppose that we are given a collection of coalitions satisfying the superset property, and define rule f by the condition f (u) = x if and only if {i/x≻i y} ∈ C. Then f is certainly strategy-proof, but does not necessarily have range {x, y}. For example, if C = 2N , then the f determined by C would be a constant rule always selecting x. At the other extreme, if C is empty, so there is no coalition winning for x, the f determined by C would be a constant rule always selecting y. By a chain we mean a collection of coalitions, C, that satisfies the superset property. For all chains except the empty collection or C = 2N , the 264 D.E. Campbell et al. / Mathematical Social Sciences 64 (2012) 263–265 rule defined by f (u) = x if and only if {i/x≻i y} ∈ C, is a strategyproof rule with a range of two. Counting the strategy-proof rules with range of two is equivalent to counting chains. Because a chain satisfies the superset property, it is entirely characterized by its minimal coalitions, where C is a minimal coalition of C if (1) C is an element of C but (2) no proper subset of C is an element of C. The collection M , of minimal coalitions for a strategy-proof rule satisfies the property: If A, B ∈ Mand A ≠ B, then neither A ⊆ B nor B ⊆ A. (∗) (In addition to serving as a determinant of strategy-proof rules, minimal coalitions, according to Riker’s size principle (Riker, 1962), are the winning coalitions we expect to actually observe as they evolve from political competition.) Any collection M of subsets of N satisfying condition (∗) is called an antichain of N . And any collection C of subsets of N satisfying the superset property is called a chain of N . But just as there are chains of N that do not determine a strategy-proof rule of range two, namely the empty collection and 2N so there are two antichains that do not determine strategy-proof rules of range two, namely the empty collection,which yields the constant rule always selecting y, and {∅}, which yields the rule always selecting x. Summarizing, then, the number of strategy-proof rules on two alternatives with a range of both alternatives is M(n) − 2 where M(n) is the number of antichains on N . To illustrate, if n = 3, the list of 20 antichains of {1, 2, 3} is