For claims problems, compromising between the proportional and constrained equal awards rules

Abstract

For the problem of adjudicating conflicting claims, we define a family of two-claimant rules that offer a compromise between the proportional and constrained equal awards rules. We identify the members of the family that satisfy particular properties. We generalize the rules to general populations by requiring “consistency”: The recommendation made for each problem should be “in agreement” with the recommendation made for each reduced problem that results when some claimants receive their awards and leave. We identify which members of the two-claimant family have consistent extensions, and we characterize these extensions. Here too, we identify which extensions satisfy particular properties. Finally, we propose and study a “dual” family.

Appendix

As mentioned in the body of the paper, the family of rules in ${\mathcal {G}}^{N}$ that satisfy composition down but not claims monotonicity is complex. We only give examples to illustrate the various ways in which the family of rules satisfying these axioms (Theorem 1) is enlarged by dropping this second property. Each of the examples is defined by delimiting subregions in awards space and specifying what to do in these regions. The complexity of the family comes from the choices we have in specifying the boundary between these regions and, for points that belong to the boundaries of two adjacent regions, in deciding whether they should be thought of as belonging to one region or as belonging to the other. These choices cannot be made independently however.

Let $L^1$ be the region of awards space above the 45$^\circ$ line. Figure 9a shows a rule for which $L^1$ is partitioned into two cones, $R^1$ (shaded) and $R^2$. The boundary ray shared by these cones is denoted r. A typical branch of the tree in $R^1$ consists of a segment in $\Lambda$ emanating from the origin followed by a vertical segment to r, excluding the point at which r is reached. A typical branch of the tree in $R^2$ consists of a segment in r followed by an unbounded vertical segment.

Fig. 9

Composition down. Illustrating different possibilities if claims continuity is not imposed. a A simple partitioning of $L^1$ into two cones. All branches of the tree consist of two segments. The loci of kinks are rays. b Here, we have added a cone in which branches are rays. c Here, the loci of kinks are not rays, but visible curves from the origin, $D^1$ and $D^2$. bf d Here, some of the branches are bounded segments, ${\mathrm{seg}}[(0, 0), b[$, for example

Figure 9b shows a rule for which $L^1$ is partitioned into three cones, $R^1$, $R^2$ (shaded) and $R^3$. The boundary ray shared by $R^1$ and $R^2$ is denoted $r^1$. The boundary ray shared by $R^2$ and $R^3$ is denoted $r^2$. A typical branch of the tree in $R^1$ consists of a segment in $\Lambda$ emanating from the origin followed by a vertical segment to r, excluding the point at which $r^1$ is reached. A typical branch of the tree in $R^2$ consists of a ray from the origin. A typical branch of the tree in $R^3$ consists of a segment in $r^2$ followed by an unbounded vertical segment.

Figure 9c shows a rule for which $L^1$ is partitioned into two cones, $R^1$ (shaded) and $R^2$. The boundary ray shared by these cones is denoted r. In $R^1$, there is a downward-sloping continuous curve $D^1$ (the segment $\mathrm{seg}[b,a]$) that is visible from below from the origin. A typical branch of the tree in $R^1$ consists of a segment to $D^1$ emanating from the origin followed by a vertical segment to r, excluding the point at which r is reached. In $R^2$, there is a downward-sloping continuous curve $D^2$ (the segment $\mathrm{seg}[d,b]$) that is visible from below from the origin. A typical branch of the tree in $C^2$ consists of a segment to $D^2$ emanating from the origin followed by an unbounded vertical segment.

Figure 9d shows a rule for which $L^1$ is partitioned into two regions, $R^1$ (shaded) and $R^2$, defined as follows. There is a point $a \in L^1$ such that $R^1$ consists of the union of the cone with boundary rays $\Lambda$ and the ray through a as well as all points in $L^1$ whose abscissa is at least $a_1$. There is a downward slope curve D from a to $\Lambda$. Region $R^2$ is the complement. The unbounded vertical half line V with lowest point is a and is part of the boundary of $R^2$. A typical branch of the tree in $R^1$ consists of a segment to $D^1$ followed by an unbounded vertical segment. A typical branch of the tree in $R^2$ consists of a segment to V.

There is also a point b on the vertical segment V such that all branches in $R^2$ whose limit point is in ${\mathrm{seg}}[a, b]$ do not contain their upper limit point, and all branches in $R^2$ whose limit point is in the vertical half line with lowest end point is b do contain their upper limit point. Then, ${\mathrm{seg}}[0, a]\cup {\mathrm{seg}}[a, b]$ is a branch of the tree.

Figure 10 shows a rule that exhibits all of these features described in Figs. 9a–d. It illustrates the general definition.

Fig. 10

Composition down. Illustrating another possibility