On the equilibrium of voting games with abstention and several levels of approval

We study the core of “(j, k) simple games”, where voters choose one level of approval from among j possible levels, partitioning the society into j coalitions, and each possible partition facing k levels of approval in the output (Freixas and Zwicker in Soc Choice Welf 21:399–431, 2003). We consider the case of (j, 2) simple games, including voting games in which each voter may cast a “yes” or “no” vote, or abstain (j = 3). A necessary and sufficient condition for the non-emptiness of the core of such games is provided, with an important application to weighted symmetric (j, 2) simple games. These results generalize the literature, and provide a characterization of constitutions under which a society would allow a given number of candidates to compete for leadership without running the risk of political instability. We apply these results to well-known voting systems and social choice institutions including the relative majority rule, the two-thirds relative majority rule, the United States Senate, and the United Nations Security Council.