Finessing a point: augmenting the core (Review)

The competition between two election candidates is modelled in a policy space, where the voters are represented by their ideal points and the candidates by their position on the policy. A voter will support the candidate whose programme is closer to his ideal point, using Euclidean distance. The candidate supported by at least q voters, where q is the quota, is elected. For simple majority q is simply the smallest integer that is at least half of the number of voters.


It is implicitly assumed that the candidates’ objective is to get elected and are willing to chose a policy that ensures that the other candidate does not win. It is natural to look at this problem as a coalitional game and so we say that the core is empty in case no majority coalition would benefit from another candidate, that is, if no alternative policy can successfully challenge the incumbent.

Unfortunately the core is often empty and then the chaos theorem [McKelvey, 1979] predicts that such an alternative proposal can lead to a campaign (positive or negative) that can lead to a Pareto inferior proposal being accepted. One of the reasons why we do not see this in election campaigns is the cost of changing one’s policy by a positive or the opponent’s by a negative campaign. This cost is proportional to the policy change measured by Euclidean distance, so the losing candidate will look for the smallest policy change that turns a sufficient number of voters away from the winning candidate. The focus is therefore on minimal winning coalitions: subsets of voters whose support is sufficient to elect a candidate, but not any more as soon as a single voter leaves the coalition.

How to choose a policy wisely? A candidate will choose the finesse point, where the cost of turning its winning position into losing is the highest. When campaigns face budget constraints, this is the policy that is most likely to be left unchallenged. The finesse point is well defined, although not necessarily unique. A geometric approach is provided including a number of properties. The finesse point is compared to related concepts, such as the yolk, the uncovered set and the Finagle point.

On the other hand the problem has parallels in coalitional games. Approximate cores [Shubik, Wooders, 1983], such as the weak and strong ε cores accommodate precisely the same idea: coalitions must bear a cost ε in order to block. The least core [Maschler, Peleg, Shapley, 1979] is the ε-core with the lowest value of . The finesse point minimises ε over all possible TU games that can be generated from the given ideal points of the voters and an arbitrary candidate from the policy space.

References
Maschler, M.; Peleg, B.; Shapley, L. S. Geometric properties of the kernel, nucleolus, and related solution concepts. Math. Oper. Res. 4 (1979), no. 4, 303–338.
McKelvey, Richard D. General conditions for global intransitivities in formal voting models. Econometrica 47 (1979), no. 5, 1085–1112.
Saari, Donald G.; Asay, Garrett R. Finessing a point: augmenting the core. Soc Choice Welf  34 (2010), 121-143.
Shubik, Martin; Wooders, Myrna Holtz. Approximate cores of replica games and economies. I. Replica games, externalities, and approximate cores. Math. Social Sci. 6 (1983), no. 1, 27–48.