Abstract
Applying the solutions defined in the axiomatic bargaining theory to actual bargaining problems is a challenge when the problem is not described by its Utility Possibility Set (UPS) but as a social choice environment specifying the set of alternatives and utility profile underlying the UPS. It requires computing the UPS, which is an operational challenge, and then identifying at least one alternative that actually achieves the bargained solution’s outcome. We introduce the axioms of Independence of NonStronglyEfficient Alternatives (resp. Weakly) and Independence of Redundant Alternatives. A solution satisfying these axioms can be applied to a simplified problem based on any reduced set of alternatives generating the strong (resp. weak) Pareto frontier of the initial problem, without changing the outcome, making the application of bargaining solutions to actual problems easier. We compare our axioms to usual independence axioms, and discuss their consistency with usual bargaining solutions. Then, we introduce monotonicity conditions corresponding to the existence of an interest group, i.e., agents ranking the alternatives in the same order. For such monotonic social choice environments, we provide a parameterized family of alternatives that generates the Pareto frontier of the bargaining problem, in line with our previous results. Our analysis illustrates that an axiomatic approach can be useful to foster the application of bargaining solutions, in complement to usual computational methods.
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Notes
An exception is De Clippel (2015) who recovers Nash’s solution using axioms expressed in an economic environment with lotteries, and discusses his results with respect to the two streams of the literature.
Characterizing a particular solution in the economic environment framework is more demanding than on the unrestricted domain of outcomes (Roemer, 1988). Roemer (1990) shows that, when expressed properly over economic environments, Nash’s axioms “hardly restrict the behavior of the bargaining solution at all” (p. 296). Nieto (1992) characterizes a resource egalitarian solution corresponding to the lexicographic extension of the maximin criterion defined on economic environments. Chen and Maskin (1999) enrich the economic environment by considering the possibility of production, focusing on the egalitarian solution. Ginés and Marhuenda (2000) consider the provision of public goods. Yoshihara (2003) considers the fairness of different resource allocation mechanisms when agents have unequal skills. In the alternativespreferences framework, Rubinstein et al. (1992) characterize Nash’s bargaining solution and its extensions to nonexpected utility preferences. Valenciano and Zarzuelo (1994) show that dropping the symmetry axiom in this framework does not characterize the weighted Nash solution as in the classical bargaining framework, but that the solution is obtained assuming the strong property of Independence to Isomorphic Transformations, which requires the bargaining solution to overlook the physical nature of the alternatives and focus exclusively on their positions in terms of the preferences of agents. Grant and Kajii (1995) study other nonexpected utility preferences. Hanany (2007) examines under what conditions on agents’ preferences it is possible to axiomatize group preferences (i.e., define a Social Welfare Function), linking the bargaining theory literature with the issues of social choice. Sprumont (2013) studies preference aggregation rules too, to characterize Relative egalitarianism.
See, e.g., Nicolò and Perea (2005), who study the role of monotonicity axioms in the classical and alternativespreferences frameworks, and characterize a solution based on a gradual expansion of the set of alternatives.
A preliminary, shorter version of the analysis in this paper was included in the working paper Martinet et al. (2019), entitled “Bargaining with intertemporal maximin payoffs” (CESifo 74712019), which has been split in two distinct, extended papers for clarity.
A classical bargaining problem \(({\mathcal {U}},u^d)\), where \({\mathcal {U}}\subset {{\mathbb {R}}}^H\) is the UPS and \(u^d\) the disagreement outcome, could be described in this framework by defining alternatives as the possible utility vectors in \({\mathcal {U}}\), and preferences as the identity function \(U(a)=a\).
For example, \({\mathbb {A}}\) can be a compact subset of \({{\mathbb {R}}}^n\) when \(n \in {{\mathbb {N}}^*}\), or a compact subset of \({{\mathbb {R}}}^{{\mathbb {N}}}\) (the set of all sequences in \({{\mathbb {R}}}\)).
For all \(h=1,\ldots ,H\), the function \(U_h: {\mathbb {A}}\rightarrow {{\mathbb {R}}}\) is upper semicontinuous if, for all \(a\in {\mathbb {A}}\) and for all \(a_k \rightarrow a\), one has \(\limsup _{k \rightarrow \infty } ~U_h(a_k) \le U_h(a)\).
Formally, in the classical framework, a bargaining solution is a correspondence \(\mu\) that maps a problem described by a pair \(({\mathcal {U}},u^d)\) to a subset \(\mu ({\mathcal {U}},u^d)\subset {\mathcal {U}}\). A solution can reduce to a mapping defining a single utility vector \(\mu ({\mathcal {U}},u^d)=u^*\in {\mathcal {U}}\).
Solutions in the classical bargaining theory are usually defined on a very large domain corresponding to the unrestricted class of pairs \(({\mathcal {U}},u^d)\) where \({\mathcal {U}}\) is some set in \({{\mathbb {R}}}^n\) and \(u^d\) is a point in this set. The choice of such a generic domain makes the characterization of solutions less demanding (see the discussion in Roemer, 1988). Rubinstein et al. (1992, p. 1173) emphasize that “specifying the domain of a solution is a delicate issue” when considering bargaining problems over alternatives, as a given UPS can be generated by many set of alternatives. Several authors impose strict restrictions on the domain in which they characterize solutions. Roemer (1990) examines problems of commodity sharing in which “the utility functions all measure the same kind of utility or outcome, and (...) the social alternatives all involve the distribution of the same kind of goods” (p. 290). Valenciano and Zarzuelo (1994) restrict their analysis to domains in which the set of alternatives and the disagreement alternative are fixed, and the set of alternatives is a compact topological space. Nicolò and Perea (2005) consider a domain in which alternatives are subsets of \({{\mathbb {R}}}^n\). De Clippel (2015) emphasizes the importance of the domain when considering axioms that relate the solution of different problems, such as monotonicity or independence axioms.
Note that, in the general analysis of this paper, the disagreement alternative plays no explicit role. It could, however, play a role in practical applications (e.g., in the application to dynamic bargaining problems with intertemporal maximin payoffs discussed in the conclusion), and it is thus of interest to include it in our definitions.
Independence axioms specify that a solution does not change when the UPS—or the set of alternatives—changes, whereas monotonicity axioms specify how the solution evolves under such changes (Thomson & Myerson, 1980) Monotonicity axioms are often imposed or strengthened when independence axioms are relaxed (see, e.g., (Kalai & Smorodinsky, 1975; Imai, 1983; De Clippel, 2015).
We here consider the following definition of IIA. A solution \(\mu\) is IIA if, whenever \({\mathbb {A}}'\subset {\mathbb {A}}\) and \(\mu (\xi )\subset {\mathbb {A}}'\), it follows that \(\mu (\xi ')=\mu (\xi )\).
The ideal point (sometimes called utopia) is the virtual vector of payoffs in which the utility of each agent corresponds to the maximal utility he/she could get from the bargaining.
The proofs are similar to the previous ones, changing \({\mathbb {A}}^s\) by \({\mathbb {A}}^w\), and INSEA by INWEA.
See also the Exhaustivity axiom in De Clippel (2015).
This cost mainly involves an invariance to isomorphic transformations (ISO).
Note that monotonicity here refers to the characteristics of the social choice environment in terms of the relationship that links alternatives to outcomes, i.e., to the type of bargaining problem, and not to the characteristics of the solution, i.e., to a monotonicity axiom as in Roemer (1988) or Yoshihara (2003).
The case in which these agents have higher utility for “higher” alternatives is symmetric. We shall mention how the results change in this case in footnotes.
\(({{{\mathcal {H}}}}^i,{{{\mathcal {H}}}}^o)\) is a partition of \({{{\mathcal {H}}}}\) if \({{{\mathcal {H}}}}^i\) and \({{{\mathcal {H}}}}^o\) are subsets of \({{{\mathcal {H}}}}\), \({{{\mathcal {H}}}}^i\ne \emptyset \ne {{{\mathcal {H}}}}^o\), \({{{\mathcal {H}}}}^i\cup {{{\mathcal {H}}}}^o= {{{\mathcal {H}}}}\), and \({{{\mathcal {H}}}}^i\cap {{{\mathcal {H}}}}^o= \emptyset\).
Respectively, “nondecreasing” if the utility profile of the interest group is nondecreasing in the alternatives; mutatis mutandis in the mathematical conditions.
As we only require \({\mathbb {A}}\) to be equipped with a preorder \(\preceq\), there might be several distinct minima. If \(\preceq\) is an order, the minimum is unique.
Respectively, “a maximum” if the utility profile of the interest group is nondecreasing in the alternatives; mutatis mutandis in the mathematical conditions.
Respectively, “higher” if the utility profile of the interest group is nondecreasing in the alternatives.
Respectively, “highest” if the utility profile of the interest group is nondecreasing in the alternatives.
Replace the \(\min\) by a \(\max\) in Eq. (3) if the utility profile of the interest group is nondecreasing in the alternatives.
Respectively, “highest” if the utility profile of the interest group is nondecreasing in the alternatives.
Respectively, “nondecreasing” if the utility profile of the interest group is nondecreasing in the alternatives.
See the second part (from section 4 of Martinet et al. 2019). This working paper gathered a previous, shorter version of the present analysis along the analysis of dynamic problems. It has been split in two papers for clarity.
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Acknowledgements
We are thankful to Geir Asheim, Marc Fleurbaey, and participants of numerous seminars and conferences where we have presented previous drafts of the paper. We also thank anonymous referees for comments which helped improving the paper.
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This research was supported by international and national grants, through the STICMATH AmSud cooperation programs MIFIMA and 18MATH05, ECOSCONICYT C07E03, and FONDECYT N 1200355.
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Appendix A: Proofs of propositions
Appendix A: Proofs of propositions
1.1 A.1 Proof of Proposition 1
Before proving the proposition, we provide technical results on the characterization of stronglyefficient alternatives.
Lemma 1
Given a social choice environment \(\xi =\langle {\mathbb {A}} \rangle\), the set \({\mathbb {A}}^s\) of stronglyefficient alternatives, as in Definition 6, is characterized as follows
Furthermore, one has that
Finally, the following equalities hold true
Proof of Lemma 1
Observe that if \(A \subset {\mathbb {A}}\) is such that \(U(A) \subset {\mathcal {E}}^s\big (U({\mathbb {A}})\big )\), then \(A \subset {\mathbb {A}}^s\) from Definition 6 of \({\mathbb {A}}^s\). On the other hand, if \(A \subset {\mathbb {A}}\) is such that \(U({\mathbb {A}}\backslash A) \cap {\mathcal {E}}^s\big (U({\mathbb {A}})\big ) = \emptyset\), one can directly check that \({\mathbb {A}}^s \subset A\). Thus, implication \(\Rightarrow\) in (5) is proven. The other implication \(\Leftarrow\) is straightforward.
For proving (6), let \(A \subset {\mathbb {A}}\) be such that \({\mathcal {E}}^s\big (U({\mathbb {A}})\big ) \subset U(A)\). Therefore, any element of \({\mathcal {E}}^s\big (U({\mathbb {A}})\big )\) can be written as \(U(a)\), where \(a\in A\). Now, by the definition of \({\mathbb {A}}^s\) (Definition 6), we deduce that \(a\in {\mathbb {A}}^s\). We conclude that \(a\in A \cap {\mathbb {A}}^s\) and that (6) holds true.
Equation (7) is made of two equalities. First, we prove that \({\mathcal {E}}^s\big (U({\mathbb {A}})\big ) = U({\mathbb {A}}^s)\) by two inclusions. On the one hand, by the definition of \({\mathbb {A}}^s\) (Definition 6), we have that \(U({\mathbb {A}}^s) \subset {\mathcal {E}}^s\big (U({\mathbb {A}})\big )\). On the other hand, the reverse inclusion \({\mathcal {E}}^s\big (U({\mathbb {A}})\big ) \subset U({\mathbb {A}}^s)\) is a consequence of (6) with \(A= {\mathbb {A}}\). Now we prove that \(U( {\mathbb {A}}\backslash {\mathbb {A}}^s ) = U( {\mathbb {A}}) \backslash {\mathcal {E}}^s\big (U({\mathbb {A}})\big )\) by two inclusions. On the one hand, any element in \(U( {\mathbb {A}}\backslash {\mathbb {A}}^s )\) of course belongs to \(U( {\mathbb {A}})\). If this element is also in \({\mathcal {E}}^s\big (U({\mathbb {A}})\big )\) we obtain a contradiction with (5), because in this case \(U( {\mathbb {A}}\backslash {\mathbb {A}}^s ) \cap {\mathcal {E}}^s\big (U({\mathbb {A}})\big ) \ne \emptyset\). Therefore, we obtain \(U( {\mathbb {A}}\backslash {\mathbb {A}}^s ) \subset U( {\mathbb {A}}) \backslash {\mathcal {E}}^s\big (U({\mathbb {A}})\big )\). On the other hand, consider \(a\in {\mathbb {A}}\) such that \(U(a) \not \in {\mathcal {E}}^s\big (U({\mathbb {A}})\big )\). By the definition of \({\mathbb {A}}^s\) (Definition 6), we deduce that \(a\not \in {\mathbb {A}}^s\). Therefore, \(a\in {\mathbb {A}}\backslash {\mathbb {A}}^s\) and \(U(a) \in U( {\mathbb {A}}\backslash {\mathbb {A}}^s )\). We conclude that \(U( {\mathbb {A}}) \backslash {\mathcal {E}}^s\big (U({\mathbb {A}})\big ) \subset U( {\mathbb {A}}\backslash {\mathbb {A}}^s )\).
Hence, we have obtained (7).
We now provide the proof of Proposition 1.
In the following, for two vectors \(u= (u_1,\ldots ,u_H)\) and \(u' = (u_1',\ldots ,u_H')\) in \({{\mathbb {R}}}^H\), we will use the distance \(\delta (u,u')= \sum _{h=1}^Hu_h u_h'\).
The proof is in two steps. In the first part, we prove that, for every \(u\in U({\mathbb {A}})\), there exists \(a^* \in {\mathbb {A}}\) such that \(u^*=U(a^*) \in U({\mathbb {A}}) \cap (u+ {{\mathbb {R}}}^H_+)\) and
For this purpose, we consider a sequence \((u^k)_{k \in {{\mathbb {N}}^*}}\) in \(U({\mathbb {A}}) \cap (u+ {{\mathbb {R}}}^H_+)\) such that
Since \(u^k \in U({\mathbb {A}}) \cap (u+ {{\mathbb {R}}}^H_+)\), then, for each \(k \in {{\mathbb {N}}^*}\), there exist \(a^k \in {\mathbb {A}}\) and \(v^k = (v^k_1, \ldots ,v^k_H) \in {{\mathbb {R}}}^H_+\) such that
As the set \({\mathbb {A}}\) is metric compact, there exist \(a^* \in {\mathbb {A}}\) and a subsequence \((a^{k_j})_{j \in {{\mathbb {N}}^*}}\) such that
We put
and we now prove that \(u^* \in U({\mathbb {A}}) \cap (u+ {{\mathbb {R}}}^H_+)\) and solves (8). Observe that, by (12), we have that \(u^* \in U({\mathbb {A}})\). We also have that \(u^* \in (u+ {{\mathbb {R}}}^H_+)\). Indeed, from (11) and the upper semicontinuity of the functions \(U_h: {\mathbb {A}}\rightarrow {{\mathbb {R}}}\), for \(h=1,\ldots ,H\), we obtain that
Therefore, by (10) where \(v^k = (v^k_1, \ldots ,v^k_H) \in {{\mathbb {R}}}^H_+\), we deduce that
for all \(h= 1, \ldots , H\), that is, \(u^* \in (u+ {{\mathbb {R}}}^H_+)\).
Thus, we obtain that \(\delta (u,u^*) \ge \alpha _u= \sup _{u' \in U({\mathbb {A}}) \cap (u+ {{\mathbb {R}}}^H_+)} \delta (u,u')\). Now, as \(u^* = U(a^*) \in U({\mathbb {A}}) \cap (u+ {{\mathbb {R}}}^H_+)\) by (13), we conclude that (8) holds true.
In the second part, we prove that, for any \(u\in U({\mathbb {A}})\) and \(u^* \in U({\mathbb {A}}) \cap (u+ {{\mathbb {R}}}^H_+)\) such that \(\alpha _u= \delta (u,u^*)\) (\(\alpha _u\) defined in (8)), then \(u^* \in {\mathcal {E}}^s\big (U({\mathbb {A}})\big )\).
The proof is obtained by contradiction. Suppose \(u^* \notin {\mathcal {E}}^s\big (U({\mathbb {A}})\big )\). Then, there exists \(\tilde{u}\in U({\mathbb {A}}) \cap (u^* + {{\mathbb {R}}}^H_+{\setminus } \{0\})\). Expressing that \(u^* \in (u+ {{\mathbb {R}}}^H_+)\) and that \({{\tilde{u}}} \in U({\mathbb {A}}) \cap (u^* + {{\mathbb {R}}}^H_+{\setminus } \{0\})\), we get that \(u^* = u+ v\) and \(\tilde{u}= u^* + w\) for some \(v=(v_1,\ldots ,v_H) \in {{\mathbb {R}}}^H_+\) and \(w = (w_1,\ldots ,w_H) \in {{\mathbb {R}}}^H_+{\setminus } \{0\}\) (i.e., \(w_h> 0\) for some \(h\in \{1,\ldots ,H\}\)). We easily deduce that \({{\tilde{u}}} \in U({\mathbb {A}}) \cap (u+ {{\mathbb {R}}}^H_+)\). Now, we have that \(\alpha _u= \delta (u,u^*) = \sum _{h=1}^Hv_h < \sum _{h=1}^H(v_h + w_h) = \delta (u,\tilde{u})\), which contradicts the definition of \(\alpha _u\) (Eq. 8).
We have shown that, for every \(u\in U({\mathbb {A}})\), there exists \(a^* \in {\mathbb {A}}\) such that \(U(a^*) = u^* \in {\mathcal {E}}^s\big (U({\mathbb {A}})\big ) \cap (u+ {{\mathbb {R}}}^H_+)\). As a consequence, the set \({\mathbb {A}}^s=\left\{ a\in {\mathbb {A}}\mid U(a)\in {\mathcal {E}}^s\big (U({\mathbb {A}})\big ) \right\}\) is not empty.
1.2 A.2 Proof of Proposition 2
The proof of Proposition 2 will be based on the following results that characterize the efficient alternatives of reduced social choice environments.
Lemma 2
Let \((\xi ', \xi )\) be a couple of social choice environments such that
Suppose that the set \({\mathbb {A}}\) of alternatives is a compact metric space and the utility profile \(U: {\mathbb {A}}\rightarrow {{\mathbb {R}}}^H\) is upper semicontinuous. Then, we have that \({\mathbb {A}}'^s \subset {\mathbb {A}}^s\) and \(U({\mathbb {A}}'^s)=U({\mathbb {A}}^s)\).
Proof of Lemma 2
Consider a couple \((\xi ', \xi )\) of social choice environments such that \({\mathbb {A}}' \subset {\mathbb {A}}\) and \({\mathcal {E}}^s\big (U({\mathbb {A}})\big ) \subset U({\mathbb {A}}')\). First, we prove that
From (7), we have that \({\mathcal {E}}^s(U({\mathbb {A}}))=U({\mathbb {A}}^s)\). As, by assumption (16), \({\mathcal {E}}^s(U({\mathbb {A}}))\subset U({\mathbb {A}}')\), we deduce from (6) that \({\mathcal {E}}^s(U({\mathbb {A}}))\subset U({\mathbb {A}}'\cap {\mathbb {A}}^s)\). Linking these two results, we obtain \(U({\mathbb {A}}^s) = {\mathcal {E}}^s(U({\mathbb {A}})) \subset U({\mathbb {A}}'\cap {\mathbb {A}}^s) \subset U({\mathbb {A}}^s)\), and we conclude that all terms are equal, so that \(U({\mathbb {A}}'\cap {\mathbb {A}}^s)= U({\mathbb {A}}^s)\).
Second, we prove that
For this purpose, we will use the following property, which follows from the definition of Pareto sets:
We have
since \({\mathbb {A}}' \subset {\mathbb {A}}\), by assumption (15), hence \(U({\mathbb {A}}') \subset U({\mathbb {A}})\). Therefore, we conclude that \(U({\mathbb {A}}^s) \subset {\mathcal {E}}^s\big (U({\mathbb {A}}')\big ) =U({\mathbb {A}}'^s)\) by (7).
Third, for concluding the proof, we need to show the inclusions \({\mathbb {A}}'^s \subset {\mathbb {A}}^s\) and \(U({\mathbb {A}}'^s)\subset U({\mathbb {A}}^s)\). For this purpose, it is sufficient to prove that \({\mathbb {A}}'^s={\mathbb {A}}'\cap {\mathbb {A}}^s\), an equality that we will deduce using the equivalence (5), that requires three conditions. We establish that these three conditions hold true.

1.
We establish that \({\mathbb {A}}'\cap {\mathbb {A}}^s \subset {\mathbb {A}}'\). This is obvious.

2.
We establish that \(U({\mathbb {A}}'\cap {\mathbb {A}}^s) \subset {\mathcal {E}}^s\big (U({\mathbb {A}}')\big )\). Indeed,
$$\begin{aligned} U({\mathbb {A}}'\cap {\mathbb {A}}^s)&= U({\mathbb {A}}^s) \,\hbox { by (18)}\, \\&\subset U({\mathbb {A}}'^s) \,\hbox { by (19)}\, \\&= {\mathcal {E}}^s\big (U({\mathbb {A}}')\big ) \,\hbox { by (7).}\, \end{aligned}$$ 
3.
We establish that \(U({\mathbb {A}}'\backslash ({\mathbb {A}}'\cap {\mathbb {A}}^s)) \cap {\mathcal {E}}^s\big (U({\mathbb {A}}')\big ) = \emptyset\). This is where the assumptions that the set \({\mathbb {A}}\) of alternatives is compact and that the utility profile \(U: {\mathbb {A}}\rightarrow {{\mathbb {R}}}^H\) is upper semicontinuous play a role, as they make it possible to use Proposition 1. Consider an alternative \(a\in {{\mathbb {A}}'\backslash ({\mathbb {A}}'\cap {\mathbb {A}}^s)}\). By Proposition 1, there exists an alternative \({{\tilde{a}}} \in {\mathbb {A}}\) such that \(U({{\tilde{a}}})\gneq U(a)\), with \(U({{\tilde{a}}}) \in {\mathcal {E}}^s(U({\mathbb {A}}))\). As \({\mathcal {E}}^s(U({\mathbb {A}}))\subset U({\mathbb {A}}')\), we deduce that there exists an alternative \({{\bar{a}}} \in {\mathbb {A}}'\) such that \(U({{\bar{a}}})=U({{\tilde{a}}}) \gneq U(a)\). As a consequence, \(a\not \in {\mathcal {E}}^s\big (U({\mathbb {A}}')\big )\).
We have thus proven that \(U({\mathbb {A}}'\backslash ({\mathbb {A}}'\cap {\mathbb {A}}^s)) \cap {\mathcal {E}}^s\big (U({\mathbb {A}}')\big ) = \emptyset\).
We now provide the proof of Proposition 2.
Consider a couple \((\xi , \xi ')\) of social choice environments such that \({\mathbb {A}}' \subset {\mathbb {A}}\) and \({\mathcal {E}}^s\big (U({\mathbb {A}})\big ) \subset U({\mathbb {A}}')\). We use the corresponding stronglyefficient reduced environments \(\xi ^s=\langle {\mathbb {A}}^s \rangle\) and \({\xi '}^s=\langle {\mathbb {A}}'^s \rangle\) (Definition 6). The assumptions of Lemma 2 are satisfied, as \({\mathbb {A}}' \subset {\mathbb {A}}\) and \({\mathcal {E}}^s\big (U({\mathbb {A}})\big ) \subset U({\mathbb {A}}')\) correspond to equations (15) and (16). From Lemma 2, we obtain that \({\mathbb {A}}'^s\subset {\mathbb {A}}^s\) and \(U({\mathbb {A}}'^{s})=U({\mathbb {A}}^s)\). As the solution \(\mu\) is IRA, we deduce from Axiom 2.3 that
Since the solution \(\mu\) is INSEA and \({\mathbb {A}}^s \subset {\mathbb {A}}\), we deduce from Axiom 2.3 that
Repeating the argument with the environment \({\xi '}^s\), we obtain that
Therefore, from (20), (21), and (22), we conclude that \(U\big (\mu (\xi )\big ) = U\big (\mu (\xi ^s)\big ) =U\big (\mu (\xi '^s)\big ) = U\big (\mu (\xi ')\big )\).
1.3 A.3 Proof of the weak version of Proposition 2
Consider a couple \((\xi , \xi ')\) of social choice environments such that \({\mathbb {A}}' \subset {\mathbb {A}}\) and \({\mathcal {E}}^w\big (U({\mathbb {A}})\big ) \subset U({\mathbb {A}}')\). We aim at proving that, if a bargaining solution \(\mu\) satisfies the INWEA and IRA axioms, then \(U\big (\mu (\xi )\big ) = U\big (\mu (\xi ')\big )\).
For this purpose, we consider the corresponding weaklyefficient reduced social choice environments, \(\xi ^w=\langle {\mathbb {A}}^w \rangle\) and \({\xi '}^w=\langle {\mathbb {A}}'^w \rangle\) (Definition 5), and shall establish the following equalities

\(\mu (\xi ^w)=\mu (\xi )\) (from the INWEA axiom), and thus \(U(\mu (\xi ^w))=U(\mu (\xi ))\).

\(\mu (\xi '^w)=\mu (\xi ')\) (from the INWEA axiom), and thus \(U(\mu (\xi '^w))=U(\mu (\xi '))\).

\(U({\mathbb {A}}^w)=U({\mathbb {A}}'^w)\) which implies, from the IRA axiom, that \(U(\mu (\xi ^w))=U(\mu (\xi '^w))\).
The two first equalities are direct consequences of the definition of the INWEA axiom. To prove the last equality, we need to show that the conditions of the IRA axiom are satisfied.
We first prove that \({\mathbb {A}}'^w\subset {\mathbb {A}}^w\). Consider a weaklyefficient alternative of \(\xi '\), i.e., \(a\in {\mathbb {A}}'^w \subset {\mathbb {A}}' \subset {\mathbb {A}}\). Assume that \(a\notin {\mathbb {A}}^w\). Then, by the definition of the set of weaklyefficient alternatives \({\mathbb {A}}^w\), there would be an alternative \(b\in {\mathbb {A}}\) such that \(U(b) > U(a)\). Given the existence of stronglyefficient alternatives in \({\mathbb {A}}\) (Proposition 1), there would then exist an alternative \(b^s\in {\mathbb {A}}^s \subset {\mathbb {A}}^w\) such that \(U(b^s)\ge U(b) > U(a)\). As \({\mathcal {E}}^w\big (U({\mathbb {A}})\big ) \subset U({\mathbb {A}}')\), one would have \(U(b^s)\in U({\mathbb {A}}')\), meaning that \(a\) would be strongly dominated in \({\mathbb {A}}'\), and thus not part of \({\mathbb {A}}'^w\), a contradiction. We thus conclude that any alternative in \({\mathbb {A}}'^w\) is also in \({\mathbb {A}}^w\), meaning that \({\mathbb {A}}'^w\subset {\mathbb {A}}^w\).
We now prove that \(U({\mathbb {A}}'^w)=U({\mathbb {A}}^w)\), by a double inclusion. From the previous step, we deduce \(U({\mathbb {A}}'^w)\subset U({\mathbb {A}}^w)\). To prove the opposite inclusion, consider an outcome \(u\in U({\mathbb {A}}^w)\), i.e., \(u\in {\mathcal {E}}^w\big (U({\mathbb {A}})\big )\). From the assumption \({\mathcal {E}}^w\big (U({\mathbb {A}})\big ) \subset U({\mathbb {A}}')\), one has \(u\in U({\mathbb {A}}')\). There is thus an alternative \(a\in {\mathbb {A}}'\) such that \(U(a)=u\). Assume that this alternative is not weaklyParetoefficient in \({\mathbb {A}}'\), i.e., \(a\notin {\mathbb {A}}'^w\). Then, there would be an alternative \(b\in {\mathbb {A}}'\) such that \(U(b)>U(a)\). As \({\mathbb {A}}'\subset {\mathbb {A}}\), the alternative b would be in \({\mathbb {A}}\), meaning that \(u< U(b)\), a contradiction with the assumption that \(u\) is weaklyefficient in \({\mathbb {A}}\). We thus have \(u\in U({\mathbb {A}}^w) \Rightarrow u\in U({\mathbb {A}}'^w)\), i.e., \(U({\mathbb {A}}^w)\subset U({\mathbb {A}}'^w)\). We conclude that \(U({\mathbb {A}}^w)=U({\mathbb {A}}'^w)\). The two conditions of the IRA axiom are thus satisfied, proving the third equality.
Combining the three established equalities, we obtain \(U(\mu (\xi ))=U(\mu (\xi ^w))=U(\mu (\xi '^w))=U(\mu (\xi '))\), proving the Proposition.
1.4 A.4 Proof of Proposition 5
We need a preliminary Lemma.
Lemma 3
Consider a given \((i,o)\)monotonic social choice environment \(\xi =\langle {\mathbb {A}} \rangle\), as in Definition 7. Then, for all \(( u^{o}, {\bar{u}}^{o} ) \in {\mathcal {U}}^{o} \times {\mathcal {U}}^{o}\) and all \({{\bar{a}}} \in {\mathbb {A}}\), we have
Proof of Lemma 3
Let \(( u^{o}, {\bar{u}}^{o} )\in {\mathcal {U}}^{o} \times {\mathcal {U}}^{o}\) and \({{\bar{a}}} \in {\mathbb {A}}\).

1.
Suppose that \(u^{o} \le U^{o}({{\bar{a}}})\). By Eq. (3) in Definition 8, defining the set \(a^{o}(u^{o})\) composed by minima of the set \(\{ a\in {\mathbb {A}}\mid U^{o}(a) \ge u^{o} \}\) (see (1) in Definition 7), we deduce that \(a^* \preceq {{{\bar{a}}}}\) for all \(a^* \in a^{o}(u^{o})\). Now, the mapping \(U^{i}: {\mathbb {A}}\rightarrow {{{\mathbb {R}}}^{i}}\) is nonincreasing by item I in the Definition 7 of MonE. We deduce that \(U^{i}({{\bar{a}}}) \le U^{i}\big (a^*\big )\), for all \(a^* \in a^{o}(u^{o})\). Hence, (23a) is proven.

2.
Use (23a) with \({{\bar{a}}} = {{\bar{a}}}^* \in a^{o}({{\bar{u}}}^{o})\) to derive (23b).

3.
Let \(u^{o} \le {\bar{u}}^{o}\). As \({{\bar{u}}}^{o} \in {\mathcal {U}}^{o}\), by definition of the set \({\mathcal {U}}^{o}\) of feasible payoffs for outsiders (Eq. 2), and by Eq. (3), for \({{\bar{a}}}^* \in a^{o}({{\bar{u}}}^{o})\) we have that \({{\bar{u}}}^{o} \le U^{o}\big ( {{\bar{a}}}^* \big )\). Therefore, we get that \(u^{o} \le U^{o}\big ({{\bar{a}}}^* \big )\) and we use (23b) to obtain (23c).
Proof of Proposition 5
First, we prove that
By definition of \(U({\mathbb {A}})\), there exists an alternative \({{\bar{a}}} \in {\mathbb {A}}\), such that \(\big ( u^{i}, u^{o} \big ) = u= U({{\bar{a}}}) = \big ( U^{i}({{\bar{a}}}), U^{o}({{\bar{a}}}) \big )\). On the one hand, as \(u^{o} = U^{o}({{\bar{a}}})\), we deduce from item II in the Definition 7 of MonE (Eq. 2) that \(u^{o} \in {\mathcal {U}}^{o}\), and that \(u^{i} = U^{i}( {{\bar{a}}} ) \le U^{i}\big (a^*\big )\) for all \(a^* \in a^{o}(u^{o})\) by (23a). On the other hand, by definition of the set \({\mathcal {U}}^{o}\) of feasible payoffs for outsiders (Eq. 2), and by Eq. (3), we obtain that \(u^{o} \in {\mathcal {U}}^{o} \Rightarrow u^{o} \le U^{o}\big (a^*\big )\) for all \(a^* \in a^{o}(u^{o})\).
Second, we prove that \({\mathcal {E}}^s\big (U({\mathbb {A}})\big ) \subset U({\mathbb {A}}^{o})\). Let \(u\in {\mathcal {E}}^s\big (U({\mathbb {A}})\big )\). By the definition of a Pareto set, we have that \(u\in {\mathcal {E}}^s\big (U({\mathbb {A}})\big ) \subset U({\mathbb {A}})\). By (24), we obtain that \(u^{i} \le U^{i}\big (a^*\big )\) and that \(u^{o} \le U^{o}\big ( a^*\big )\) for all \(a^* \in a^{o}(u^{o})\). Putting \({\bar{u}} = \big ( U^{i}\big (a^*\big ), U^{o}\big ( a^* \big ) \big )\) for some (fixed) \(a^* \in a^{o}(u^{o})\), we deduce that \(u\le {\bar{u}}\). As \({\bar{u}} \in U({\mathbb {A}})\) and \(u\in {\mathcal {E}}^s\big (U({\mathbb {A}})\big )\), we obtain that \({\bar{u}} = u\), by definition of a (strong) Pareto set. By definition of \({\mathbb {A}}^{o}\) (Eq. 4), we conclude that \(u= {\bar{u}}= \big ( U^{i}\big (a^*\big ), U^{o}\big ( a^* \big ) \big ) \in U({\mathbb {A}}^{o})\).
Third, we prove that \(U({\mathbb {A}}^{o}) \subset {\mathcal {E}}^w\big (U({\mathbb {A}})\big )\). The proof is obtained by contradiction. Let \(u\in U({\mathbb {A}}^{o})\) and suppose that there exists \({\bar{u}} \in U({\mathbb {A}})\) such that \(u< {\bar{u}}\) (i.e, a strict inequality component by component). On the one hand, by definition of \({\mathbb {A}}^{o}\) (Eq. 4), there exists \({\hat{u}}^{o} \in {\mathcal {U}}^{o}\) such that \(u= U\big ({{\hat{a}}} \big )\) for some \(\hat{a}\in a^{o}({\hat{u}}^{o})\). On the other hand, we have that \({{\bar{u}}} \le U\big (\hat{a}\big )\), by (24). We deduce that
As already seen, we have that \({\hat{u}}^{o} \le U^{o}\big ({{\hat{a}}} \big )\). Together with inequality (25), this yields \({\hat{u}}^{o} < U^{o}\big ( \bar{a}^*\big )\) for all \({{\bar{a}}}^* \in a^{o}({{\bar{u}}}^{o})\). Now fix \({{\bar{a}}}^* \in a^{o}({{\bar{u}}}^{o})\). By (23b), we deduce that \(U^{i}\big ( {{\bar{a}}}^*\big ) \le U^{i}\big ({{\hat{a}}}\big )\). Now, using inequality (25), we get that \({{\bar{u}}}^{i} \le U^{i}\big (\bar{a}^*\big ) \le U^{i}\big ({{\hat{a}}}\big ) = u^{i}\). However, this contradicts \(u< {\bar{u}}\). Therefore, no such \({\bar{u}}\) exists and we conclude that \(u\in {\mathcal {E}}^w\big (U({\mathbb {A}})\big )\).
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Martinet, V., Gajardo, P. & De Lara, M. Bargaining on monotonic social choice environments. Theory Decis 96, 209–238 (2024). https://doi.org/10.1007/s11238023099459
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DOI: https://doi.org/10.1007/s11238023099459