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Endowment additivity and the weighted proportional rules for adjudicating conflicting claims

Abstract

We propose and study a new axiom, restricted endowment additivity, for the problem of adjudicating conflicting claims. This axiom requires that awards be additively decomposable with respect to the endowment whenever no agent’s claim is filled. For two-claimant problems, restricted endowment additivity essentially characterizes weighted extensions of the proportional rule. With additional agents, however, the axiom is satisfied by a great variety of rules. Further imposing versions of continuity and consistency, we characterize a new family of rules which generalize the proportional rule. Defined by a priority relation and a weighting function, each rule aims, as nearly as possible, to assign awards within each priority class in proportion to these weights. We also identify important subfamilies and obtain new characterizations of the constrained equal awards and proportional rules based on restricted endowment additivity.

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Notes

  1. 1.

    See Thomson (2003, 2014, 2015a) for surveys.

  2. 2.

    See Moreno-Ternero and Roemer (2006), for example.

  3. 3.

    See, for example, Moulin (1987) and Ju et al. (2007).

  4. 4.

    Shapley (1953) provides the classic axiomatization invoking additivity and Moulin (2013) discusses recent applications.

  5. 5.

    See Bergantiños and Méndez-Naya (2001) and the discussion in Thomson (2014).

  6. 6.

    As described in Remark 4 of Moulin (1987), this characterization is a corollary of his Theorem 5 which applies this property to surplus sharing problems.

  7. 7.

    Restricted versions of joint additivity in claims and the endowment lead to additional rules (Bergantiños and Vidal-Puga 2004, 2006). With other axioms, notions along these lines lead to the “minimal overlap” rule (Alcalde et al. 2008; Marchant 2008).

  8. 8.

    As another example, some cultural norms call for the highest status or wealthiest person to bear the cost of a communal meal or celebration. Although paying for a large banquet may reverse the ex-post wealth ranking, responsibility for future expenses depends on the original ranking and the same individual will pay, provided her wealth is not entirely exhausted.

  9. 9.

    From a claims perspective, these might be two funding agencies from which researchers have requested funds or departments with overlapping responsibilities for reimbursing costs included by individuals.

  10. 10.

    A class of “baseline extension operators” generalize the logic of composition and define new rules from existing rules by first making a baseline award and then applying the original rule to an updated auxiliary problem (Hougaard et al. 2012, 2013). Similar observations apply to these rules as well.

  11. 11.

    Continuity is typically required jointly for the endowment and claims, something we do not impose. This accords our interpretation of claims as intrinsic features not assumed to be decomposable.

  12. 12.

    As with weights, it is important that the priorities be over agent-claim pairs. This allows, for example, the same agent to be in different tax brackets when her income changes. Similarly, two creditors with the same numerical claims, perhaps representing an individual and a corporation, may be in different priority classes.

  13. 13.

    For another case in which endowment monotonicity is implied by the combination of consistency and other axioms, none of which directly imply the property, see Young (1987b).

  14. 14.

    Aumann and Maschler (1985) introduce this property and Young (1987b) also uses it to characterize the proportional rule. Via duality, our results identify and characterize the “dual” families of our rules (Remark 4).

  15. 15.

    Let \(\varvec{\mathbb {N}}\) denote the natural numbers and let \({\mathbb {R}}\), \({\mathbb {R}}_{+}\), and \({\mathbb {R}}_{++}\) denote respectively the real, nonnegative real, and positive real numbers.

  16. 16.

    For comparison and future reference, we state (unrestricted) endowment additivity and the composition axioms together in parallel fashion: For each \(N\subseteq \mathcal {N}\), each \((c,E)\in \mathcal {C}^N\), and each pair \(E^1,E^2\in \mathbb {R}_+\) with \(E=E^1+E^2\),

    $$\begin{aligned} \mathbf Endowment additivity: \quad&\varphi (c,E^1+E^2) = \varphi (c,E^1) + \varphi (c,E^2). \\ \mathbf Composition up: \quad&\varphi (c,E^1+E^2) = \varphi (c,E^1) + \varphi \big (c-\varphi (c,E^1),E^2\big )\\&\quad = \varphi \big (c-\varphi (c,E^2),E^1\big ) + \varphi (c,E^2). \\ \mathbf Composition down: \quad&\varphi (c,E^1) = \varphi \big (\varphi (c,E^1+E^2),E^1\big ) \text { and } \varphi (c,E^2) = \varphi \big (\varphi (c,E^1+E^2),E^2\big ). \end{aligned}$$

    The axioms are similar in that they each decompose a given problem into subproblems, but differ in how they treat claims. In contrast with restricted endowment additivity, these axioms apply to all claims problems.

  17. 17.

    See Thomson (2012) for a thorough normative analysis of the consistency principle and Thomson (2015b) for a survey on its applications.

  18. 18.

    Thomson (2014) mentions this property in passing, and Ju and Moreno-Ternero (2014) introduce an adapted version for an augmented model with exchange.

  19. 19.

    The formal requirement is: For each \(N\subseteq \mathcal {N}\), each \((c,E)\in \mathcal {C}^N\), and each \(i\in N\), if \(c_i=0\), then \(\varphi _{-i}(c,E) = \varphi (c_{-i},E)\).

  20. 20.

    A previous version of this paper considered a weaker requirement, parallel to full compensation consistency, which avoids this overlap. Details are available from the author.

  21. 21.

    See, for example, Young (1987b).

  22. 22.

    A similar, slightly stronger condition requires that in each problem with a positive endowment, those individuals with positive claims receive positive awards. This axiom is studied as “positive share” by Moulin (2002) and “positive awards” by Flores-Szwagrzak (2015).

  23. 23.

    Curiel et al. (1987) introduce this property to distinguish rules which correspond to cooperative game solutions and Dagan (1996) studies the axiom under the name “independence of irrelevant claims.” It is implied by “contraction independence” studied by Kıbrıs (2012, 2013), and Stovall (2014b).

  24. 24.

    For each \(c\in \mathbb {R}_+^N\), the path of awards of \(\varvec{\varphi }\) for \(\varvec{c}\) consists of the locus of awards vectors as the endowment varies: \(\{\varphi (c,E): 0\le E\le \sum _N c_i\}\)

  25. 25.

    Another leading rule, the constrained equal losses (CEL) rule, is not a W-proportional rule. It is the “dual” of CEA and is defined for each \(N\subseteq \mathcal {N}\) and each \((c,E)\in \mathcal {C}^N\) by \({\textit{CEL}}(c,E)=c-{\textit{CEA}}(c,\sum _N c_i-E)\).

  26. 26.

    A complete proof appears in the working paper version of this paper available from the author.

  27. 27.

    A rule \(\varphi \) is: claims monotonic if an increase in an agent’s claim never leads to a decrease in the agent’s award; claims continuous if it is continuous in the vector of claims; and homogeneous if after scaling the claims and endowment by the same constant, the awards vector is scaled by the same constant.

  28. 28.

    We depart from the standard notation for the dual of a rule, \(\varphi ^d\), to avoid confusion that may arise with either \(P^{ud}\) or \((P^u)^d\).

  29. 29.

    Aumann and Maschler (1985) introduced this property for claims problems.

  30. 30.

    A similar result characterizes the proportional rule by self-duality together with either composition axiom (Young 1987a). The proportional rule is also singled out by the full strength of endowment additivity (Moulin 1987; Chun 1988), by applying order preservation to groups (Chambers and Thomson 2002), as well as by strategic considerations (Ju et al. 2007).

  31. 31.

    Also required is that for each \(c_0\in \mathbb {R}_+\), \(\lim _{\lambda \rightarrow -\infty } f(\cdot ,\lambda )=0\) and \(\lim _{\lambda \rightarrow \infty } f(c_0,\lambda )=c_0\).

  32. 32.

    The difference between the families is whether homogeneity is also imposed. Since a weak form of homogeneity follows from restricted endowment additivity, the difference is insubstantial for comparison with our rules.

  33. 33.

    The exception is the proportional rule itself.

  34. 34.

    This axiom is similar in spirit to and implies minimal sharing. It requires that in each problem with a positive endowment, all agents with positive claims receive positive awards. The main theorem in Flores-Szwagrzak (2015) characterizes “priority-augmented” extensions of WCEA rules by the first three axioms.

  35. 35.

    A related approach leads to an “egalitarian rule,” which is also not a PW-proportional rule (Giménez-Gómez and Peris 2014).

  36. 36.

    This rule is introduced by Aumann and Maschler (1985) and generalized variously by Hokari and Thomson (2003), Moreno-Ternero and Villar (2006), and Thomson (2008).

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Correspondence to Patrick Harless.

Additional information

I thank Karol Flores-Szwagrzak and Juan Moreno-Ternero, and three anonymous referees for comments. I am grateful to William Thomson for comments, encouragement, and guidance.

Appendix: Omitted proofs

Appendix: Omitted proofs

Proof of Lemma 1

Proof

For reference in the proof, we call the condition described in the lemma the “scaling property.” First suppose that \(\varphi \) satisfies the scaling property. Let \(N\subseteq \mathcal {N}\) and \((c,E)\in \mathcal {C}^N\) be such that \(\varphi (c,E)\ll c\) and let \(E^1,E^2\in \mathbb {R}_+\) be such that \(E^1+E^2=E\). Let \(\alpha \equiv \frac{E^1}{E}\) so \(1-\alpha =\frac{E^2}{E}\). Then \(\alpha \in [0,1]\), so by the scaling property, \(\varphi (c,E^1) = \varphi (c,\alpha E) = \alpha \varphi (c,E)\) and \(\varphi (c,E^2) = \varphi (c,(1-\alpha ) E) = (1-\alpha )\varphi (c,E)\). Therefore,

$$\begin{aligned} \varphi (c,E^1) + \varphi (c,E^2) = \alpha \varphi (c,E) + (1-\alpha )\varphi (c,E) = \varphi (c,E). \end{aligned}$$

Therefore, \(\varphi \) satisfies restricted endowment additivity.

Conversely, suppose that \(\varphi \) satisfies restricted endowment additivity. Let \(N\subseteq \mathcal {N}\) and \((c,E)\in \mathcal {C}^N\) be such that \(\varphi (c,E)\ll c\). First we show that \(\varphi (c,\cdot )\) is continuous on \(\{E':\varphi (c,E')\ll c\}\). Let \(\varepsilon \in \mathbb {R}_{++}\) and let \(E^1,E^2\in \{E':\varphi (c,E')\ll c\}\) be such that \(E^1\le E^2\) and \(|E^1-E^2|<\frac{\varepsilon }{|N|}\). By restricted endowment additivity, \(\varphi (c,E^2) = \varphi (c,E^1) + \varphi (c,E^2-E^1)\). Also, for each \(i\in N\), \(0\le \varphi _i(c,E^2-E^1)\le E^2-E^1\), so

$$\begin{aligned} \big | \varphi _i(c,E^2) - \varphi _i(c,E^1) \big | = \varphi _i(c,E^2-E^1) \le E^2-E^1 < \frac{\varepsilon }{|N|}. \end{aligned}$$

Then \(\Vert \varphi (c,E^2) - \varphi (c,E^1) \Vert < \sum _N \frac{\varepsilon }{|N|}= \varepsilon \). Therefore, \(\varphi (c,\cdot )\) is continuous on \(\{E':\varphi (c,E')\ll c\}\).

Now we verify the scaling property. By restricted endowment additivity, \(\varphi (c,E) = \varphi (c,\frac{E}{2})+\varphi (c,\frac{E}{2}) = 2\varphi (c,\frac{E}{2})\). By repeated application of restricted endowment additivity, for each \(k\in \mathbb {N}\), \(\varphi (c,E)=k\varphi (c,\frac{E}{k})\). Similarly, \(\varphi (c,\frac{2E}{k})=\varphi (c,\frac{E}{k})+\varphi (c,\frac{E}{k})=2\varphi (c,\frac{E}{k})\). Again by repeated application of restricted endowment additivity, for each \(l\in \mathbb {N}\) with \(l\le k\), \(\varphi (c,E) = k\varphi (c,\frac{E}{k}) = \frac{k}{l}\varphi (c,\frac{l E}{k})\). That is, for each \(q\in \mathbb {Q}_+\cap [0,1]\), \(q\varphi (c,E) =\varphi (c,q E)\). Then by continuity on \(\{E':\varphi (c,E')\ll c\}\), for each \(\alpha \in [0,1]\), \(\alpha \varphi (c,E) =\varphi (c,\alpha E)\). \(\square \)

Proof of Theorem 2

Proof

Each PW-proportional rule satisfies the axioms of the proposition, so we prove the converse. Let \(\varphi \) be a rule satisfying the axioms of the proposition. Let \(N\subseteq \mathcal {N}\) with \(|N|=2\) and \(c\in \mathbb {R}_+^N\). If either agent’s claim is zero, then the awards are completely specified by feasibility and all rules coincide, so suppose \(0\ll c\). To calibrate the candidate PW-proportional rule, let \(c_0\equiv \frac{1}{2} \min _N c_i>0\) and \(w(N,c)\equiv \frac{\varphi (c,c_0)}{c_0}\). That is, w(Nc) represents the fractions of the endowment awarded to each agent at \((c,c_0)\). By construction, \(\sum _N w_i(N,c)=1\) and \(\varphi (c,c_0)\ll c\). Labeling the agents i and j, we choose \(P^{\prec ,u}\) so that: (i) if \(0\ll w(N,c)\), then \((i,c_i)\sim (j,c_j)\), \(u(i,c_i)=w_i(N,c)\), and \(u(i,c_i)=w_i(N,c)\); and (ii) if \(w_i(N,c)=0\) so \(w_j(N,c)=1\), then \((i,c_i)\prec (j,c_j)\). By construction, \(\varphi (c,c_0)=P^{\prec ,u}(c,c_0)\).

We show that \(\varphi \) coincides with \(P^{\prec ,u}\). Let \(\bar{E}\equiv \sup \{E'\in \mathbb {R}_+: \varphi (c,E')\ll c\}\). By feasibility and definition of \(c_0\), \(0<c_0<2c_0\le \bar{E}\). Let \(E\in \mathbb {R}_+\) with \(E\le \sum _N c_i\) so \((c,E)\in \mathcal {C}^N\). If \(E\le c_0\), then by Lemma 1,

$$\begin{aligned} \varphi (c,E) = \varphi (c,\tfrac{E}{c_0} c_0) = \tfrac{E}{c_0}\varphi (c,c_0) = w(N,c)E = \tfrac{E}{c_0}\, P^{\prec ,u}(c,c_0) = P^{\prec ,u}(c,E). \end{aligned}$$

If \(c_0<E<\bar{E}\), then \(\varphi (c,E)\ll c\) and \(P^{\prec ,u}(c,E)\ll c\) so again by Lemma 1,

$$\begin{aligned} \tfrac{c_0}{E}\, \varphi (c,E)= & {} \varphi (c,\tfrac{c_0}{E} E) = \varphi (c,c_0) = w(N,c)c_0 = P^{\prec ,u}(c,c_0) = P^{\prec ,u}\left( c,\tfrac{c_0}{E} E\right) \\= & {} \tfrac{c_0}{E}\, P^{\prec ,u}(c,E) \end{aligned}$$

and \(\varphi (c,E)=P^{\prec ,u}(c,E)\). By endowment continuity, \(\varphi (c,\cdot )\) is continuous at \(\bar{E}\), so \(\varphi (c,\bar{E}) = w(N,c)\bar{E} = P^{\prec ,u}(c,\bar{E})\) as well. Therefore, \(\varphi (c,\cdot )\) coincides with \(P^{\prec ,u}(c,\cdot )\) on \([0,\bar{E}]\).

Finally, suppose \(\bar{E}<E\). Then \(\bar{E}<\sum _N c_i\). By endowment continuity, there is \(i\in N\) such that \(\varphi _i(c,\bar{E})=c_i=P^{\prec ,u}(c,\bar{E})\). Let \(j\in N\backslash \{i\}\). Then \(P^{\prec ,u}(c,E)=(c_i,E-c_i)\). Suppose by way of contradiction that \(\varphi (c,\bar{E})\ne (c_i,E-c_i)\). By definition of \(\bar{E}\), \(\varphi (c,E)\not \ll c\), so \(\varphi (c,E)=(E-c_j,c_j)\). Let \(\hat{E}\equiv \inf \{E'\in [\bar{E},E]: \varphi (c,E')=(E'-c_j,c_j)\}\). Now \(\bar{E}<\hat{E}\), so \(\varphi (c,\hat{E})\in \{(c_i,\hat{E}-c_i),(\hat{E}-c_j,c_j)\}\). But \((c_i,\hat{E}-c_i)\ne (\hat{E}-c_j,c_j)\), so this violates endowment continuity. Instead, \(\varphi (c,E)=(c_i,E-c_i)=P^{\prec ,u}(c,E)\). \(\square \)

Proof of Theorem 1

Proof

Each collection of fixed-population PW-proportional rules satisfies restricted endowment additivity and endowment continuity. If the weights are path-consistent, then the weights are proportional in all pairs of problems meeting the hypothesis of full compensation consistency, so each path-consistent collection of PW-proportional rules satisfies full compensation consistency as well.

For the converse, let \(\varphi \) be a rule satisfying the axioms of the theorem. By Proposition 2, \(\varphi \) coincides with a fixed-population PW-proportional rule for each two-claimant problem. First we show that once an agent’s claim is filled, it remains filled as the endowment increases. With this fact in hand, we show that the rule coincides with a PW-proportional rule for each fixed population and verify path consistency.

Step 1: For each \(\varvec{N\subseteq \mathcal {N}}\), each \(\varvec{(c,E)\in \mathcal {C}^N}\), and each \(\varvec{E'\in \mathbb {R}_+}\) with \(\varvec{E'\le E}\), \(\varvec{\{i\in N:\varphi (c,E')=c_i\}\subseteq \{i\in N:\varphi (c,E)=c_i\}}\). We proceed by induction on the number of agents. By Proposition 2, the assertion is true for all populations of size at most two. Let \(N\subseteq \mathcal {N}\) and suppose that the assertion is true for all populations smaller than N: for each \(\hat{N}\subseteq \mathcal {N}\) with \(|\hat{N}|<|N|\), each \((\hat{c},\hat{E})\in \mathcal {C}^{\hat{N}}\), and each \(\hat{E}'\in \mathbb {R}_+\) with \(\hat{E}'\le \hat{E}\), \(\{i\in N:\varphi (\hat{c},\hat{E}')\}\subseteq \{i\in N:\varphi (\hat{c},\hat{E})\}\).

Let \((c,E)\in \mathcal {C}^N\) and \(E'\in \mathbb {R}_+\) with \(E'<E\). Let \(N_0\equiv \{i\in N:\varphi (c,E)=c_i\}\) and \(N_0'\equiv \{i\in N:\varphi (c,E')=c_i\}\) and suppose by way of contradiction that \(N_0'\not \subseteq N_0\). Then \(|N|>2\), \(N_0'\ne \emptyset \), and since \(E'<E\le \sum _N c_i\), \(N_0'\ne N\). Let \(\bar{E}\equiv \inf \big \{\hat{E}\in [E',E]: \{i\in N:\varphi _i(c,\hat{E})=c_i\} \ne N_0' \big \}\) and let \(\bar{N}_0\equiv \{i\in N:\varphi (c,\bar{E})=c_i\}\). By endowment continuity, \(\varphi (c,\cdot )\) is continuous at \(\bar{E}\), so \(E'\le \bar{E}<E\) and \(\bar{N}_0=N_0'\). Also, by endowment continuity and the definition of \(\bar{E}\), there is \(E''\in [\bar{E},E]\) such that \(\{i\in N: \varphi (c,E'')=c_i\}\ne \bar{N}_0\) and for each \(j\in N\backslash \bar{N}_0\), \(\varphi _j(c,\bar{E})-\varphi _j(c,E'') < c_j - \varphi _j(c,\bar{E})\). Let \(N_0''\equiv \{i\in N: \varphi (c,E'')=c_i\}\), \(N''\equiv N\backslash N_0''\), and \(E_0''\equiv \sum _{N_0''} c_j\). Then \(N_0'' \subseteq \bar{N}_0\). By full compensation consistency,

$$\begin{aligned} \varphi _{N''}(c,\bar{E})&= \varphi \left( c_{N''},\bar{E} - \sum _{N\backslash N''} \varphi _j(c,\bar{E}) \right) = \varphi \left( c_{N''},\bar{E} - \sum _{N_0''} c_j \right) \\&=\varphi (c_{N''},\bar{E}-E_0'') \text { and } \\ \varphi _{N''}(c,E'')&= \varphi \left( c_{N''},E'' - \sum _{N\backslash N''} \varphi _j(c,E'') \right) = \varphi \left( c_{N''},E'' - \sum _{N_0''} c_j \right) \\&= \varphi (c_{N''},E''-E_0''). \end{aligned}$$

Now \(\bar{E}<E''\), so by hypothesis,

$$\begin{aligned} \{j\in N'': \varphi _j(c_{N''},\bar{E}-E_0'')=c_j\}\subseteq \{j\in N'': \varphi _j(c_{N''},E''-E_0'')=c_j\}. \end{aligned}$$

But \(N_0''\subseteq \bar{N}_0\) and \(N_0''\ne \bar{N}_0\), so there is \(i\in \bar{N}_0\backslash N_0''\). Then \(i\in N''\) and

$$\begin{aligned} \varphi _i(c_{N''},E''-E_0'') = \varphi _i(c,E'') < c_i = \varphi _i(c,\bar{E}) = \varphi _i(c_{N''},\bar{E} - E_0''). \end{aligned}$$

That is, \(i\in \{j\in N'': \varphi _j(c_{N''},\bar{E}-E_0'')=c_j\}\) and \(i\not \in \{j\in N'': \varphi _j(c_{N''},E''-E_0'')=c_j\}\), which contradicts \(\{j\in N'': \varphi _j(c_{N''},\bar{E}-E_0'')=c_j\}\subseteq \{j\in N'': \varphi _j(c_{N''},E''-E_0'')=c_j\}\). Instead, \(N_0'\subseteq N_0\).

Step 2: For each \(\varvec{N\subseteq \mathcal {N}}\), \(\varvec{\varphi }\) is a fixed-population PW-proportional rule. First, agents with zero claims are fully compensated in each problem. By full compensation consistency, the awards of agents with positive claims are unchanged when these agents are excluded, so it suffices to consider problems in which all claims are positive. To calibrate the candidate PW-proportional rule, for each \(N\subseteq \mathcal {N}\) and each \(c\in \mathbb {R}_{++}^N\), let \(c_0\equiv \frac{1}{2} \min _N c_i\) and \(w(N,c)\equiv \frac{\varphi (c,c_0)}{c_0}\). Then for each \(i\in N\), \(w_i(N,c)\) is the fraction of the endowment that \(\varphi \) awards to agent i at \((c,c_0)\). By construction, \(\sum _N w_i(N,c)=1\) and \(\varphi (c,c_0)\ll c\).

We argue by induction on the number of agents. By Proposition 2, the assertion is true for populations of size at most two. Now let \(N\subseteq \mathcal {N}\) with \(|N|>2\) and suppose that \(\varphi \) coincides with a fixed-population PW-proportional rule for each population smaller than N: For each \(\hat{N}\subseteq \mathcal {N}\) with \(|\hat{N}|<|N|\), there are \(\prec \in \Pi \) and \(u\in \mathcal {U}\) such that for each \((\hat{c},\hat{E})\in \mathcal {C}^{\hat{N}}\), \(\varphi (\hat{c},\hat{E}) = P^{\prec ,u}(\hat{c},\hat{E})\).

Let \(c\in \mathbb {R}_{++}^N\). Define \(\bar{E}\equiv \inf \{E\in \mathbb {R}_+:\exists i\in N,\, \varphi _i(c,E)=c_i\}\) and \(\bar{N}\equiv \{i\in N: \varphi _i(c,\bar{E})=c_i\}\). If \(\bar{N}=N\), then \(\bar{E}=\sum _N c_i\) and for each \(i\in N\), \(w_i(N,c)=\frac{c_i}{\sum _N c_i}\) and \(\varphi \) coincides with the proportional rule: For each \(E\in \mathbb {R}_+\) with \(E\le \sum _N c_i\), \(\varphi (c,E)=P(c,E)\). Suppose instead that \(\bar{N}\ne N\) and let \(E_0\equiv \sum _{\bar{N}} c_i\).

By Lemma 1, for each \(E\in \mathbb {R}_+\) with \(E<c_0\), \(\varphi (c,E) = \varphi (c,\frac{E}{c_0} c_0) = \frac{E}{c_0}\varphi (c,c_0) = w(N,c)E\). Similarly, for each \(E\in \mathbb {R}_+\) with \(c_0<E<\bar{E}\), \(\varphi (c,E)\ll c\), so again by Lemma 1, \(\frac{c_0}{E}\varphi (c,E)=\varphi (c,\frac{c_0}{E} E) = \varphi (c,c_0) = w(N,c)c_0\). Also, by endowment continuity, \(\varphi (c,\bar{E})=w(N,c)\bar{E}\). Altogether, for each \(E\in \mathbb {R}_+\) with \(E\le \bar{E}\), \(\varphi (c,E) = w(N,c)E\).

By hypothesis, there are \(\prec \in \Pi \) and \(u\in \mathcal {U}\) such that \(\varphi \) coincides with \(P^{\prec ,u}\) for each \((\hat{c},\hat{E})\in \mathcal {C}^{N\backslash \bar{N}}\). Rescaling if necessary, we may suppose that \(\sum _{N\backslash \bar{N}} u(i,c_i) = \sum _{N\backslash \bar{N}} w_i(N,c)\). We now extend \(P^{\prec ,u}|_{N\backslash \bar{N}}\) from \({N\backslash \bar{N}}\) to N. For each \(i\in \bar{N}\), let \(u(i,c_i)\equiv w_i(N,c)\) and let \(\prec \) be such that for each \(j\in N\), if \(w_j(N,c)>0\), \((i,c_i)\sim (j,c_j)\) and if \(w_j(N,c)=0\), then \((i,c_i)\prec (j,c_j)\). Now let \(E\in \mathbb {R}_+\) with \(E\le \sum _N c_i\). By construction, if \(E\le \bar{E}\), then \(\varphi (c,E)=P^{\prec ,u}(c,E)\). Suppose instead that \(\bar{E}<E\). By Step 1, \(\varphi _{\bar{N}}(c,E)=c_{\bar{N}}=P_{\bar{N}}(c,E)\). It remains to consider \(N\backslash \bar{N}\).

By full compensation consistency, \(\varphi _{N\backslash \bar{N}}(c,E) = \varphi (c_{N\backslash \bar{N}},E-E_0)=P^{\prec ,u}(c_{N\backslash \bar{N}},E-E_0)\). Now \(\varphi (c_{N\backslash \bar{N}},\bar{E}-E_0\big ) \ll c_{N\backslash \bar{N}}\) and \(\varphi (c_{N\backslash \bar{N}},\bar{E}-E_0) = \varphi _{N\backslash \bar{N}}(c,\bar{E})\), so

$$\begin{aligned} w(N\backslash \bar{N},c_{N\backslash \bar{N}}) (\bar{E}-E_0) = \varphi (c_{N\backslash \bar{N}},\bar{E}-E_0) = \varphi _{N\backslash \bar{N}}(c,\bar{E}) = w_{N\backslash \bar{N}}(N,c) \bar{E}. \end{aligned}$$

Since \(E_0=\sum _{\bar{N}} \varphi _i(c,\bar{E}) = \sum _{\bar{N}} w_i(N,c) \bar{E}\) and \(\sum _N w_i(N,c) = \sum _{N\backslash \{\bar{N}\}} w_i(N\backslash \bar{N},c_{N\backslash \bar{N}})\), this implies that for each pair \(i,j\in N\backslash \bar{N}\),

$$\begin{aligned} \frac{w_i(N\backslash \bar{N},c_{N\backslash \bar{N}})}{w_i(N,c)} = \frac{\bar{E}-E_0}{\bar{E}} = \frac{w_j(N\backslash \bar{N},c_{N\backslash \bar{N}})}{w_j(N,c)}. \end{aligned}$$

Furthermore, by the definition of \(P^{\prec ,u}\),

$$\begin{aligned} \frac{u(i,c_i)}{u(j,c_j)} = \frac{\varphi _i(c_{N\backslash \bar{N}}, \bar{E}-E_0)}{\varphi _j(c_{N\backslash \bar{N}}, \bar{E}-E_0)} = \frac{w_i(N\backslash \bar{N}, c_{N\backslash \bar{N}})}{w_j(N\backslash \bar{N}, c_{N\backslash \bar{N}})}. \end{aligned}$$

By our normalization, \(\sum _{N\backslash \bar{N}} u(i,c_i) = \sum _{N\backslash \bar{N}} w_i(N,c)\), so in fact for each \(i\in N\backslash \bar{N}\), \(u(i,c_i)=w_i(N,c)\). Therefore, for each \(E\in \mathbb {R}_+\) with \(E\le \sum _N c_i\), \(\varphi (c,E)=P^{\prec ,u}(c,E)\).

Step 3: \(\varvec{\varphi }\) is a path-consistent collection of PW-proportional rules. By Step 2, \(\varphi \) is a collection of fixed-population PW-proportional rules. To see that the collection is path-consistent, let \(N\subseteq \mathcal {N}\), \((c,E)\in \mathcal {C}^N\), and \(i\in N\). Suppose that \(\varphi _i(c,E)=c_i\) and let \(P^{\prec ,u}\) and \(P^{\hat{\prec },\hat{u}}\) be the components of \(\varphi \) associated with N and \(N\backslash \{i\}\), respectively. Then by full compensation consistency,

$$\begin{aligned} P^{\prec ,u}_{-i}(c,E) = \varphi _{-i}(c,E) = \varphi \big (c_{-i},E-\varphi _i(c,E)\big ) = P^{\hat{\prec },\hat{u}}\big (c_{-i},E-\varphi _i(c,E)\big ). \end{aligned}$$

Let \(j,k\in N\backslash \{i\}\). Then \(P^{\prec ,u}_j(c,E)=0\) if and only if \(P^{\hat{\prec },\hat{u}}_j\big (c_{-i},E-\varphi _i(c,E)\big )=0\) and \(P^{\prec ,u}_k(c,E)=0\) if and only if \(P^{\hat{\prec },\hat{u}}_k\big (c_{-i},E-\varphi _i(c,E)\big )=0\), so \((j,c_j)\prec (k,c_k)\) if and only if \((j,c_j)\hat{\prec } (k,c_k)\). Also, if \(P^{\prec ,u}_j(c,E)>0\), then

$$\begin{aligned} \frac{u(k,c_k)}{u(j,c_j)} = \frac{P^{\prec ,u}_k(c,E)}{P^{\prec ,u}_j(c,E)} = \frac{P^{\hat{\prec },\hat{u}}_k\big (c_{-i}, E-\varphi _i(c,E)\big )}{P^{\hat{\prec },\hat{u}}_j\big (c_{-i}, E-\varphi _i(c,E)\big )} = \frac{\hat{u}(k,c_k)}{\hat{u}(j,c_j)}. \end{aligned}$$

Therefore, u and \(\hat{u}\) are related by a rescaling on \((N\backslash \{i\},c_{-i})\). Since this is true for each population and each problem, the collection of fixed-population PW-proportional rules is path-consistent. \(\square \)

Proof of Theorem 2

Proof

We have seen that each PW-proportional rule satisfies the axioms, so let \(\varphi \) be a rule satisfying the axioms of the theorem. By Theorem 1, \(\varphi \) is a path-consistent collection of fixed-population PW-proportional rules. Let \(N\subseteq \mathcal {N}\), \(N'\subseteq N\). Let \(P^{\prec ,u}\) and \(P^{\prec ',u'}\) be the fixed-population PW-proportional rules which coincide with \(\varphi \) on N and \(N'\) respectively.

Step 1: \(\varvec{\prec |_{N'}=\prec '|_{N'}}\). Let \(c\in \mathbb {R}_+^N\) and \(j,k\in N'\). First suppose that \((j,c_j)\prec ^N (k,c_k)\). Then there is \(E\in \mathbb {R}_+\) with \(E\le \sum _N c_i\) such that \(\varphi _j(c,E)>0=\varphi _k(c,E)\). By repeated application of consistency, \(\varphi _j\big (c_{N'},E-\sum _{N\backslash N'} c_i\big )=\varphi _j(c,E)>0\) and \(\varphi _k\big (c_{N'},E-\sum _{N\backslash N'} c_i\big )=\varphi _k(c,E) = 0\). Therefore, \((j,c_j)\prec ^{N'} (k,c_k)\). If instead \((j,c_j)\sim ^N (k,c_k)\). Then there is \(E\in \mathbb {R}_+\) with \(E\le \sum _N c_i\) such that \(0<\varphi _j(c,E)<c_j\) and \(0<\varphi _k(c,E)<c_k\). By repeated application of consistency, \(\varphi _j\big (c_{N'},E-\sum _{N\backslash N'} c_i\big )=\varphi _j(c,E)\) and \(\varphi _k\big (c_{N'},E-\sum _{N\backslash N'} c_i\big )=\varphi _k(c,E)\), so \(0<\varphi _j\big (c_{N'},E-\sum _{N\backslash N'} c_i\big )<c_j\) and \(0<\varphi _k\big (c_{N'},E-\sum _{N\backslash N'} c_i\big )<c_k\). Therefore, \((j,c_j)\sim ^{N'} (k,c_k)\). Altogether, \(\prec '|_{N'}=\prec |_{N'}\).

Step 2: \(\varvec{u|_{N'}}\) and \(\varvec{u'|_{N'}}\) are proportional. Let \(c\in \mathbb {R}_+^N\) and \(j,k\in N'\) with \((j,c_j)\sim ^N (k,c_k)\). By Step 1, \((j,c_j)\sim ^{N'} (k,c_k)\) as well. There is \(E\in \mathbb {R}_+\) with \(E\le \sum _N c_i\) such that \(0<\varphi _j(c,E)<c_j\) and \(0<\varphi _k(c,E)<c_k\). Then \(\frac{\varphi _j(c,E)}{\varphi _k(c,E)}=\frac{u(j,c_j)}{u(k,c_k)}\). By repeated application of consistency, \(\varphi _j\big (c_{N'},E-\sum _{N\backslash N'} c_i\big )=\varphi _j(c,E)\) and \(\varphi _k\big (c_{N'},E-\sum _{N\backslash N'} c_i\big )=\varphi _k(c,E)\), so \(0<\varphi _j\big (c_{N'},E-\sum _{N\backslash N'} c_i\big )<c_j\) and \(0<\varphi _k\big (c_{N'},E-\sum _{N\backslash N'} c_i\big )<c_k\). Then \(\frac{\varphi _j\big (c_{N'}, E-\sum _{N\backslash N'} c_i\big )}{\varphi _j\big (c_{N'},E-\sum _{N\backslash N'} c_i\big )} = \frac{u'(j,c_j)}{u'(k,c_k)}\). Combining results, \(\frac{u(j,c_j)}{u(k,c_k)} = \frac{u'(j,c_j)}{u'(k,c_k)}\) and so \(u|_{N'}\) and \(u'|_{N'}\) are proportional.

By Steps 1 and 2, \(\prec ^{N'}\) and \(u'\) may be replaced by \(\prec ^N\) and u without changing any awards. Moreover, this is true for each pair \(N,N'\subseteq \mathcal {N}\) with \(N'\subseteq N\). In general, let \(N,N'\subseteq \mathcal {N}\) and consider \(N''\equiv N\cup N'\). Then by Steps 1 and 2, replacing \(\prec ^{N'}\) and \(\prec ^{N''}\) with \(\prec ^N\) and replacing \(u'\) and \(u''\) with u does not change any awards. Continuing in this fashion, we conclude that there are \(\prec \in \Pi \) and \(u\in \mathcal {U}\) such that for each \(N\subseteq \mathcal {N}\), \(\prec ^N|_N=\prec |_N\) and \(u^N|_N\) is proportional to \(u|_N\). Thus, \(\varphi =P^{\prec ,u}\). \(\square \)

Proof of Theorem 3

Proof

We have seen that each W-proportional rule satisfies the axioms of the theorem, so let \(\varphi \) be a rule satisfying these axioms. By Theorem 2, \(\varphi \) is a PW-proportional rule: There are \(\prec \in \Pi \) and \(u\in \mathcal {U}\) such that \(\varphi =P^{\prec ,u}\). To show that \(\varphi \) is a W-proportional rule, let \(N\subseteq \mathcal {N}\) and \((c,E)\in \mathcal {C}^N\). Let \(i,j\in N\) and suppose that \(c_i>0\), \(c_j>0\), and \(E>0\). Since \(E>0\), there is \(k\in N\) such that \(\varphi _k(c,E)>0\). Then by minimal sharing, \(\varphi _i(c,E)>0\) and \(\varphi _j(c,E)>0\). Therefore, \((i,c_i)\sim (j,c_j)\). Since this is true for each pair of agents with positive claims and in each problem, \(\prec \) is the complete indifference relation and \(\varphi =P^{\prec ,u}=P^u\). \(\square \)

Proof of Proposition 3

Proof

The constrained equal awards rule satisfies the axioms, so let \(\varphi \) be a rule satisfying the axioms of the proposition.

Step 1: Coincidence for small endowments. Let \(N\subseteq \mathcal {N}\) and \(c\in \mathbb {R}_+^N\). By consistency, the presence of agents with zero claims has no bearing on the awards of the remaining agents, so we may suppose \(c\in \mathbb {R}_{++}^N\). Let \(c_0\equiv \frac{1}{2}\min _N c_i\). By claims truncation invariance and equal treatment of equals,

$$\begin{aligned} \varphi (c,c_0) = \varphi \big ( (c_0,\cdots ,c_0),c_0\big ) = \big ( \tfrac{c_0}{|N|},\cdots ,\tfrac{c_0}{|N|} \big ) = {\textit{CEA}}(c,c_0). \end{aligned}$$

Let \(E\in \mathbb {R}_+\) with \(E\le \sum _N c_i\). First suppose that \(\varphi (c,E)\ll c\). If \(E\le c_0\), then by Lemma 1, \(\frac{E}{c_0}\,\varphi (c,c_0) = \varphi \big (c,\frac{E}{c_0}\,c_0\big ) = \varphi (c,E)\). If \(c_0<E\), then by Lemma 1, \(\frac{c_0}{E}\,\varphi (c,E) = \varphi \big (c,\frac{c_0}{E}\,E\big ) = \varphi (c,c_0)\). Therefore, for each \(E\in \mathbb {R}_+\) such that \(\varphi (c,E)\ll c\), \(\varphi (c,E) = \frac{E}{c_0}\,\varphi (c,c_0) = \frac{E}{c_0}\,{\textit{CEA}}(c,c_0) = {\textit{CEA}}(c,E)\). In particular, this is true for each \(E\in \mathbb {R}_+\) such that \(E<|N|\min _N c_i\).

Step 2: Coincidence for two-claimant problems. Let \(N\subseteq \mathcal {N}\) with \(|N|=2\) and \(c\in \mathbb {R}_+^N\) and label the agents so that \(c_1\le c_2\). By Step 1, for each \(E\in \mathbb {R}_+\) such that \(\varphi (c,E)\ll c\), \(\varphi (c,E)={\textit{CEA}}(c,E)\). Suppose by way of contradiction that there is \(E\in \mathbb {R}_+\) with \(E\le c_1+c_2\) such that \(\varphi (c,E)\ne {\textit{CEA}}(c,E)\). Then \(\varphi (c,E)\not \ll c\) and \(E\ge 2c_1\). If \(c_1=c_2\), then \(E=2c_1\) and by feasibility, \(\varphi (c,E)=(c_1,c_1)={\textit{CEA}}(c,E)\). Instead, \(c_1<c_2\). Then \(\varphi (c,E)\ne {\textit{CEA}}(c,E)=(c_1,E-c_1)\), so \(\varphi (c,E)=(E-c_2,c_2)\). Since \(E-c_2<c_1\), there is \(k\in \mathbb {N}\) such that \(\frac{E}{k}<c_1+c_2-E\). Let \(N'\subseteq \mathcal {N}\) with \(|N'|=k\) and for each \(i\in N'\), let \(c_i'=c_1\). By consistency and equal treatment of equals,

$$\begin{aligned} \varphi \big ((c_1,c_2,c_{-12}'),c_2+(k+1)(E-c_2)\big ) = (E-c_2,c_2,E-c_2,\cdots ,E-c_2). \end{aligned}$$

By choice of k, \(E < k(c_1+c_2-E)\), so \(c_2+(k+1)(E-c_2)<k c_1\). But then by Step 1,

$$\begin{aligned} \varphi \big ((c_1,c_2,c_{-12}'),c_2+(k+1)(E-c_2)\big )= & {} {\textit{CEA}}\big ((c_1,c_2,c_{-12}'),c_2+(k+1)(E-c_2)\big )\\= & {} \big (\tfrac{E}{k+2},\cdots ,\tfrac{E}{k+2}\big ), \end{aligned}$$

a contradiction. Instead, \(\varphi (c,E)=(c_1,E-c_1)={\textit{CEA}}(c,E)\). Therefore, \(\varphi \) coincides with CEA on the domain of two-claimant problems. Since CEA is the unique consistent rule with this property, \(\varphi ={\textit{CEA}}\). \(\square \)

Proof of Theorem 4

Proof

We prove statement (i); since the properties in the statements are dual, (ii) follows immediately by duality. The proportional rule satisfies the properties of statement (i), so let \(\varphi \) be a rule satisfying these properties. By self-duality, \(\dot{\varphi }\) satisfies restricted endowment additivity as well. Let \(N\subseteq \mathcal {N}\) and \(c\in \mathbb {R}_+^N\). By null claims consistency, the presence of agents with zero claims has no bearing on the awards of the remaining agents, so we may suppose \(c\in \mathbb {R}_{++}^N\). By self-duality, \(\varphi \big (c,\frac{1}{2}\,\sum _N c_i\big ) = \frac{c}{2}\). Then by Lemma 1, for each \(\alpha \in [0,1]\),

$$\begin{aligned} \varphi \left( c,\frac{\alpha }{2}\, \sum _N c_i\right)&= \alpha \varphi \left( c,\frac{1}{2}\,\sum _N c_i\right) = \frac{\alpha c}{2} \text { and } \\ \dot{\varphi }\left( c,\frac{\alpha }{2}\, \sum _N c_i\right)&= \alpha \dot{\varphi }\left( c,\frac{1}{2}\,\sum _N c_i\right) = \frac{\alpha c}{2}. \end{aligned}$$

Then furthermore, for each \(\alpha \in [0,\frac{1}{2}]\),

$$\begin{aligned} \varphi \left( c,\alpha \sum _N c_i \right)&= \alpha c, \\ \dot{\varphi }\left( c,\alpha \sum _N c_i \right)&= \alpha c, \text { and } \\ \varphi \left( c,(1-\alpha )\sum _N c_i \right)&= c - \dot{\varphi }\left( c,\alpha \sum _N c_i\right) = c - \alpha c = (1-\alpha ) c. \end{aligned}$$

Combining results, for each \(\alpha \in [0,1]\), \(\varphi \big (c,\alpha \sum _N c_i\big ) = \alpha \varphi \big (c,\sum _N c_i\big )=\alpha c\). That is, \(\varphi \) is the proportional rule. \(\square \)

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Harless, P. Endowment additivity and the weighted proportional rules for adjudicating conflicting claims. Econ Theory 63, 755–781 (2017). https://doi.org/10.1007/s00199-016-0960-9

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Keywords

  • Claims problem
  • restricted endowment additivity
  • Weighted proportional rule
  • Priority-augmented weighted proportional rule

JEL Classification

  • D63
  • D70
  • D71