The Shapley–Owen Value and the Strength of Small Winsets: Predicting Central Tendencies and Degree of Dispersion in the Outcomes of Majority Rule Decision-Making

In a “king of the hill” agenda, there is a prevailing alternative and in pairwise fashion, some new alternative (proposal) is matched against the present “king of the hill.” If the new alternative fails to receive a majority against the present king of the hill, then the process continues with a second, third, etc. alternative being proposed. If the new alternative defeats the present king of the hill, then it becomes the new king of the hill, and the process continues Either the agenda for this process is finite, e.g., a given status quo which enters the last vote, or there is some procedure for invoking cloture, so that voters can stop the process once they find an “acceptable” king of the hill.

The work of Schofield we have previously cited uses a solution concept called the heart, which seems very appropriate for weighted voting coalition games with a limited number of players, such as multiparty cabinet formation games, where ideal points are to a large extent a matter of common knowledge. Schofield (1999) shows that the uncovered set is a subset of the heart. In this essay we focus on committee voting games rather than coalition games, and we will draw our comparisons to the uncovered set rather than the heart.

Note that points outside of the Pareto Set are unlikely to be outcomes of these types of processes even when they have relatively small winsets, and the strong point can sometimes be near the boundary of the Pareto Set. Consequently, we expect that situations where the strong point is close to the boundary of the Pareto Set will be exceptions to our general expectation that the strong point will be central among the outcomes.

Consider, for example, the strong point (shown in black) in Fig. 10. It is very close to the mean location of the outcomes (shown in pink) when we think of closeness relative to the spread of the voter ideal points. The Pareto set in these situations is the convex figure that is enclosed by all the lines between the voter ideal points. For any point outside the Pareto set, the voters always unanimously prefer some other point inside the Pareto set. Consequently, voters generally have little reason to ever propose alternatives outside of the Pareto set, and they rarely do so. Consequently, the effectiveness of prediction should be considered with respect to proposals in the Pareto set (shaded yellow in Fig. 10).

It is important to recognize that, unless there is a core to the voting game, it need not be true that points that beat other points have smaller winsets than the points they beat. At the start of the process, when the points are relatively far out from the strong point, there is a tendency for the process to move inward. However, once the status quo is further in toward the strong point, there is no necessary expectation that further points will have smaller winsets. In fact, nearly all the points in the winsets of points close to the strong point have winsets larger than the strong point itself—consequently, if the process does not stop at the strong point, it necessarily moves to points with a larger winset than the strong point itself. Nonetheless, if the outcome is a point near the strong point this will be a point with a relatively small winset.

On the other hand, there are models of spatially embedded coalition formation games and of party competition games that do generate empirically testable models that garnered considerable empirical support. Trying to reconcile the theoretical and empirical findings on committee voting, coalition formation, and party competition, however, takes us into issues well beyond the scope of this paper.

To the extent that voters can develop a sense of the preferences of other players, points perceived as “more fair” may be more likely to be proposed and accepted as the final outcome, or perhaps, points that are perceived to be likely to defeat other alternatives, e.g., points on the boundary of a minimum winning coalition, may be more likely to be proposed (cf. the notion of the competitive solution in Mckelvey et al. 1978).

The uncovered set consists of points with small winsets because points with large winsets are likely to be covered by some other point with a smaller winset (see Miller 2007).

Bianco et al. (2004, 2006, 2008) note that when the uncovered set is large, the uncovered set can include most of the points in the Pareto Set, so predictions based on the uncovered set are not that specific, though they predict far better than chance. They are equally well aware that, in the unusual situations where the uncovered set is small, e.g. when there is a core, then some observed outcomes in experimental voting will not lie exactly at the core and thus will fall outside the uncovered set.

Bounds on the uncovered set are often stated in terms of the center of the yolk (in two dimensions, the yolk is the smallest circle touching all median lines), e.g., in classic work, Mckelvey (1986) shows that the uncovered set must lie with 4 yolk radii of the center of the yolk, and this bound has been tightened by others (Feld et al. 1987; Miller 2007). Thus, it is natural to ask about the relationship between the center of the yolk and winset size, on the one hand, and the relationship between the center of the yolk and the strong point, on the other. Craig Tovey (personal communication, 2009), investigating a conjecture by Scott Feld, has recently proved a result closely related to the Shapley and Owen result bounding the size of winsets by the size of and distance to the strong point, namely that bounds on the size of the winset of any point can be stated in terms of the size of the winset of the strong point and the distance of the point to the center of the yolk. As a corollary, he also shows that the strong point can be no more than 2.16 yolk radii from the center of the yolk. Although that is the tightest bound known, we have not found any situations where the strong point is more than one yolk radius from the center of the yolk, and we believe it must be within the yolk in the three voter case. In the games reported on here, the yolk is considerably closer to the strong point than 2.16 yolk radii.

Similarly, if voters focus their attention on one dimension at a time, resulting in an outcome at the generalized median, i.e., at the location of the respective median voter along each of two dimensions (Shepsle and Weingast 1981; cf. Feld and Grofman 1988), the generalized median cannot be very far away from the strong point.