Abstract
We consider the adjudication of conflicting claims over a resource. By mapping such a problem into a bargaining problem à la Nash, we avail ourselves of the solution concepts developed in bargaining theory. We focus on the solution to two-player bargaining problems known as the “equal area solution.” We study the properties of the induced rule to solve claims problems. We identify difficulties in extending it from two claimants to more than two claimants, and propose a resolution.
In homage to Leo Hurwicz, who laid the foundations of the theory of economic design. I thank Patrick Harless for his extensive comments.
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- 1.
- 2.
A family of rules are introduced by Young (1987) under the name of “equal sacrifice” rules.” Our solution is not a member of this family.
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An application of the idea to classical fair allocation problems is proposed and studied by Velez and Thomson (2012).
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Incidentally, this property is necessary and sufficient condition for a rule to be obtainable as the composition of two mappings: one is O’Neill’s mapping from claims problems to transferable utility coalitional games; the other is a solution for this class of games (Curiel et al., 1987).
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We denote by \(\mathbb {R}^N_+\) the cartesian product of |N| copies of \(\mathbb {R}_+\) indexed by the members of N. The superscript N may also indicate some object pertaining to the set N. Which interpretation is the right one should be clear from the context. We allow the equality ∑i ∈ N c i = E for convenience.
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A subset S of \(\mathbb {R}^N_+\) is comprehensive if for each x ∈ S and each 0 ≤ y ≤ x, y ∈ S.
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The usual specification of a bargaining game includes a disagreement point, and our formulation amounts to assuming that it is the origin. This assumption is justified if the theory is required to be independent of the choice of origin for the utility functions that are used to represent the opportunities available to the agents.
- 8.
The bound is introduced by Moreno-Ternero and Villar (2004) under the name of “securement.” Order preservation is introduced by Aumann and Maschler (1985), and order preservation under endowment variations by Dagan et al. (1997) under the name of “supermodularity.” Linked claims-endowment monotonicity appears in connection with a discussion of the duality operator in Thomson and Yeh (2008), and bounded gain under claim increase is introduced by Kasajima and Thomson (2012) together with a variety of other monotonicity properties. Claims truncation invariance is introduced by Curiel et al. (1987) and minimal rights first by the same authors under the name of the “minimal rights property.” Composition down is introduced by Moulin (1987), composition up by Young (1988), and duality notions, including self-duality, by Aumann and Maschler (1985).
- 9.
The typical path of awards of this rule for a claims vector (c 1, c 2) > 0 contains the same initial segment as the equal area rule, a segment that is symmetric with respect to the half claims vector, and a vertical or horizontal segment connecting these two objects, depending upon whether c 1 ≤ c 2 or c 2 ≤ c 1.
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Thomson, W. (2019). Equal Area Rule to Adjudicate Conflicting Claims. In: Trockel, W. (eds) Social Design. Studies in Economic Design. Springer, Cham. https://doi.org/10.1007/978-3-319-93809-7_14
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