Institute for Mathematical Behavioral Sciences

Gilboa in 1989 found the first explicit example of a set of pairwise contest results on more than 50 alternatives that cannot represent the outcomes of 2/3-majority voting. The existence of such examples has been proved probabilistically. Gilboa asked whether there existed a smaller example. In joint work with Dylan Shepardson, we find a smallest possible example, which requires only 8 alternatives. The example was found and validated via a 2-person zero-sum game formulation of the realization problem. The associated linear program gives some clues as to why it may be difficult to find elegant necessary and sufficient conditions for realization.