A core of voting games with improved foresight

This paper considers voting situations in which the vote takes place iteratively. If a coalition replaces the status quo aa with a contestant bb, then bb becomes the new status quo, and the vote goes on until a candidate is reached that no winning coalition is willing to replace. It is well known that the core, that is, the set of undominated alternatives, may be empty. To alleviate this problem, Rubinstein [Rubinstein, A., 1980. Stability of decision systems under majority rule. Journal of Economic Theory 23, 150–159] assumes that voters look forward one vote before deciding to replace an alternative by a new one. They will not do so if the new status quo is going to be replaced by a third that is less interesting than the first. The stability set, that is, the set of undominated alternatives under this behavior, is always non-empty when preferences are strict. However, this is not necessarily the case when voters’ indifference is allowed. Le Breton and Salles [Le Breton, M., Salles, M., 1990. The stability set of voting games: Classification and generecity results. International Journal of Game Theory 19, 111–127], Li [Li, S., 1993. Stability of voting games. Social Choice and Welfare 10, 51–56] and Martin [Martin, M., 1998. Quota games and stability set of order dd. Economic Letters 59, 145–151] extend the sophistication of the voters by having them look dd votes forward along the iterative process. For dd sufficiently large, the resulting set of undominated alternatives is always non-empty even if indifference is allowed. We show that it may be unduly large. Next, by assuming that other voters along a chain of votes are also rational, that is, they also look forward to make sure that the votes taking place later on will not lead to a worst issue for them, we are able to reduce the size of this set while insuring its non-emptiness. Finally, we show that a vote with sufficient foresight satisfies a no-regret property, contrarily to the classical core and the stability set.