Smallest Tournaments Not Realizable by 23-Majority Voting

Define the predictability number α(G) of a tournament T to be the largest supermajority threshhold 1 2 < α ≤ 1 for which T could represent the pairwise voting outcomes from some population of voter preference orders. We establish that the predictability number always exists and is rational. Only acyclic tournaments have predictability 1; the Condorcet voting paradox tournament has predictability 2 3 ; Gilboa (4) found a tournament on 54 alternatives (i.e. vertices) that has predictability less than 23 , raising the question of whether a smaller such tournament exists. We exhibit an 8-vertex tournament that has predictability 13 20 , and prove that it is the smallest tournament with predictability < 2 3 . Our methodology is to formulate the problem as a finite set of 2-person 0-sum games, employ the minimax duality and linear programming basic solution theorems, and solve using rational arithmetic.