On the emptiness of the stability set of order d

We know from Li's theorem (1993) that the stability set of order d may be empty for some preference profiles. However, one may wonder whether such situations are just rare oddities or not. In this paper, we partially answer this question by considering the restrictive case where the number of alternatives is the smallest compatible with an empty stability set. More precisely, we provide an upper bound on the probability for having an empty stability set of order d for the majority game under the Impartial Weak Ordering Culture assumption. This upper bound is already extremely low for small population and tends to zero as the number of individuals goes to infinity.