Acyclic sets of linear orders

Abstract.

 A set of linear orders on {1,2, ℕ, n} is acyclic if no three of its orders have an embedded permutation 3-cycle {abc, cab, bca}. Let f (n) be the maximum cardinality of an acyclic set of linear orders on {1,2, ℕ, n}. The problem of determining f (n) has interested social choice theorists for many years because it is the greatest number of linear orders on a set of n alternatives that guarantees transitivity of majority preferences when every voter in an arbitrary finite set has any one of those orders as his or her preference order. This paper gives improved lower and upper bounds for f (n). We note that f (5)=20 and that all maximum acyclic sets at n=4, 5 are generated by an “alternating scheme.” This procedure becomes suboptimal at least by n=16, where a “replacement scheme” overtakes it. The presently-best large-n lower bound is approximately f (n)≥(2.1708)ⁿ.