Characterization of the lexicographic egalitarian solution in the two-person bargaining problem

Economics Letters 159 (2017) 7–9

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Economics Letters journal homepage: www.elsevier.com/locate/ecolet

Characterization of the lexicographic egalitarian solution in the two-person bargaining problem Osamu Mori Jobu University, 634-1 Toyazuka-cho, Isesaki, Gunma 372-8588, Japan

highlights • We consider the two-person bargaining problem. • We introduce two new axioms, Strict Suppes-Sen Proofness and Restricted Equity. • We provide a new characterization of the lexicographic egalitarian solution.

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Article history: Received 24 April 2017 Received in revised form 18 June 2017 Accepted 30 June 2017 Available online 13 July 2017 JEL classification: C78 Keywords: Bargaining theory Axiomatic characterization The lexicographic egalitarian bargaining solution Independence of common monotone transformations Impartiality Equity

a b s t r a c t In this study, we provide a new characterization of the lexicographic egalitarian solution in the twoperson bargaining problem using the Independence of Common Monotone Transformations axiom introduced by Nielsen (1983). We introduce two new axioms, Strict Suppes-Sen Proofness and Restricted Equity. Strict Suppes-Sen Proofness, which is analogous to Mariotti’s (1999) Suppes-Sen Proofness, represents impartiality in the use of the strong Pareto optimality. Restricted Equity represents the ethical notion that the more equitable distribution of utility gains relative to the disagreement point should be preferred if the total gain is fixed. Then, we show that the lexicographic egalitarian solution is characterized by Strict Suppes-Sen Proofness, Restricted Equity, and Independence of Common Monotone Transformations. © 2017 Elsevier B.V. All rights reserved.

1. Introduction In the two-person bargaining problem, Roth (1979) characterized the lexicographic egalitarian bargaining solution by Strong Individual Rationality, Strong Pareto Optimality, Independence of Irrelevant Alternatives, and Independence of Ordinal Transformations Preserving Interpersonal Comparisons (Theorem 13). Nielsen (1983) replaced the last axiom with a simpler one, Independence of Common Monotone Transformations (ICMT). This axiom implies that the bargaining outcome is invariant under any monotone increasing transformation of utility gain, which is common for all persons. We provide a characterization of the lexicographic egalitarian solution in the two-person bargaining problem by replacing the axioms (with the exception of ICMT) with two new axioms, Strict Suppes-Sen Proofness (SSSP) and Restricted Equity (RE). In contrast E-mail address: [email protected]. http://dx.doi.org/10.1016/j.econlet.2017.06.044 0165-1765/© 2017 Elsevier B.V. All rights reserved.

to Roth (1979) and Nielsen (1983), we do not assume that the bargaining solution should be single-valued nor that the set of Pareto optimal points is connected. Mariotti (1999) introduced an axiom based on the Suppes-Sen dominance, Suppes-Sen Proofness, to characterize the Nash (1950) bargaining solution. The Suppes-Sen Proofness axiom is regarded as representing the property of impartiality in the use of the weak Pareto optimality. In contrast, we use Strict Suppes-Sen Proofness (SSSP), which combines impartiality with the strong Pareto optimality, to characterize the lexicographic egalitarian solution. We also introduce another new axiom, which we call Restricted Equity (RE). The RE axiom requires that given a solution utility vector such that the utility gains are not equal between two persons, there be no feasible utility vector that has the same total utility gain as, and is more equitable than, the solution utility vector. We show that the lexicographic egalitarian solution in the twoperson bargaining problem is characterized by SSSP, RE, and ICMT.

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O. Mori / Economics Letters 159 (2017) 7–9

2. Preliminaries and axioms The following vector notations are used throughout. For s, r ∈ R2 , s ≥ r if si ≥ ri for each i, s > r if si ≥ ri for all i and s ̸ = r, and s ≫ r if si > ri for each i. A two-person bargaining problem is a pair (S , d) where S ⊂ R2 is a set of feasible utilities, and d ∈ S is a disagreement point, which is the utility vector that results when no agreement is reached. It is assumed that S is closed, bounded above, and contains a point s such that s ≫ d. Let B be the class of twoperson bargaining problems satisfying this condition. A bargaining solution is a nonempty correspondence F : B → R2 such that for every (S , d) ∈ B, F (S , d) ⊂ S. We call u ∈ F (S , d) a solution utility vector or a solution vector. A solution vector is a utility vector that is assigned to the outcome of the bargaining. The lexicographic egalitarian solution is the correspondence E : B → R2 that is defined as follows: E(S , d) = {e ∈ S | s ̸ ∈ S if min{e1 − d1 , e2 − d2 } < min{s1 − d1 , s2 − d2 }, or min{e1 − d1 , e2 − d2 } = min{s1 − d1 , s2 − d2 } and max{e1 − d1 , e2 − d2 } < max{s1 − d1 , s2 − d2 }}. Note that E(S , d) ̸ = ∅ since S is closed and bounded above. The properties that can be imposed on a solution F are as follows: Strict Suppes-Sen Proofness (SSSP). For any u ∈ F (S , d), if (s1 − d1 , s2 − d2 ) > (u1 − d1 , u2 − d2 ) or (s2 − d2 , s1 − d1 ) > (u1 − d1 , u2 − d2 ) , then s ̸ ∈ S. SSSP requires that there be no utility vector s in the feasible set such that s − d strictly Suppes-Sen dominates u − d. Restricted Equity (RE). For u ∈ F (S , d) such that u1 − d1 ̸ = u2 − d2 , if s1 − d1 + s2 − d2 = u1 − d1 + u2 − d2 and min{u1 − d1 , u2 − d2 } < si − di < max{u1 − d1 , u2 − d2 }, i = 1, 2, then s ̸ ∈ S. When both s1 −d1 and s2 −d2 are between u1 −d1 and u2 −d2 , we can say that s − d is more equitable than u − d. Therefore, RD states that given a solution utility vector u such that u1 − d1 ̸ = u2 − d2 , there should be no feasible utility vector s such that u and s have the same total utility gains and s is more equitable than u. The RE axiom can be regarded as representing the ethical notion that the more equitable distribution of utility gains relative to the disagreement point should be preferred if the total gain is fixed. Independence of Common Monotone Transformations (ICMT) (Nielsen, 1983). Let τ : R → R be a strictly increasing continuous function with τ (0) = 0, and let S ′ = {(d′1 + τ (s1 − d1 ), d′2 + τ (s2 − d2 )) | s ∈ S }. For (S , d), (S ′ , d′ ) ∈ B, if u ∈ F (S , d), then u′ = (d′1 + τ (u1 − d1 ), d′2 + τ (u2 − d2 )) ∈ F (S ′ , d′ ). ICMT implies that the outcome of the bargaining is invariant under any change of utilities, which preserves the order of utility gains regarding interpersonal and intrapersonal comparisons of gains.1 3. Result We show that E(S , d) is characterized by SSSP, RE, and ICMT. Theorem. A bargaining solution F (S , d) satisfies SSSP, RE, and ICMT if and only if F (S , d) = E(S , d). Proof. (If) It is certain that E(S , d) satisfies SSSP, RE, and ICMT. (Only if) Let F (S , d) satisfy all axioms in the theorem. By ICMT, we can put d = 0 = (0, 0) for simplicity. We suppose by contradiction that F (S , 0) ̸ ⊂ E(S , 0). Consider u and e such that u ∈ F (S , 0), u ̸ ∈ E(S , 0), and e ∈ E(S , 0). Then, there are four cases for u and e : u1 = min{u1 , u2 } and e1 = min{e1 , e2 }, u2 = min{u1 , u2 } and e1 = min{e1 , e2 }, u1 = min{u1 , u2 } and e2 = min{e1 , e2 }, and u2 = min{u1 , u2 } and e2 = min{e1 , e2 }. It suffices to consider just one case since the same method can be applied to the remaining 1 See Nielsen (1983, p.220.).

three cases also. Let u1 = min{u1 , u2 } and e1 = min{e1 , e2 }. Then, since u ̸ = e, by SSSP and the definition of E, it must be true that u1 < e1 ≤ e2 < u2 . Note that 0 < e1 since, by assumption, there exists s such that s ≫ d = 0 and min{s1 , s2 } ≤ e1 . Furthermore, this case can be subdivided into four cases. Let α = e1 − u1 > 0, β = u2 − e2 > 0, and γ = e2 − e1 ≥ 0. Case 1: α > β, γ ≥ 0. Note that e1 + e2 > u1 + u2 . We define τ : R → R as follows:

τ (x) =

⎧ x ⎪ ⎪ ⎨α

if x < e2 ,

α x+ 1− ⎪ β β ⎪ ⎩ x+α−β

(

) e2

if e2 ≤ x < u2 , if u2 ≤ x.

Let S = {(τ (s1 ), τ (s2 )) | s ∈ S }. By construction, it follows that τ (e1 ) + τ (e2 ) = τ (u1 ) + τ (u2 ) and τ (u1 ) < τ (e1 ) ≤ τ (e2 ) < τ (u2 ). However, by ICMT, we have (τ (u1 ), τ (u2 )) ∈ F (S ′ , 0) and (τ (e1 ), τ (e2 )) ∈ S ′ , which contradicts RE. Case 2: α < β, γ > 0. Note that e1 + e2 < u1 + u2 . We define τ : R → R as follows: ⎧ if x < e1 , ⎪x( ⎪ ) ⎪ ⎪ ⎪ β −α β −α ⎪ ⎪ +1 x− e1 if e1 ≤ x < e2 , ⎨ γ γ ) ( τ (x) = ⎪ α α ⎪ ⎪ u2 if e2 ≤ x < u2 , x+ 1− ⎪ ⎪ β β ⎪ ⎪ ⎩ x if u2 ≤ x. ′

Similar to Case 1, letting S ′ = {(τ (s1 ), τ (s2 )) | s ∈ S }, by ICMT, we have (τ (u1 ), τ (u2 )) ∈ F (S ′ , 0) and (τ (e1 ), τ (e2 )) ∈ S ′ , which contradicts RE. Case 3: α < β, γ = 0. We define τ : R → R as follows:

⎧ x ⎪ ⎪ ( ⎪ ⎪ ⎪ β ⎪ ⎪ ⎨αx + 1 − ( τ (x) = ⎪ α ⎪ ⎪ x + 1− ⎪ ⎪ β ⎪ ⎪ ⎩

if x < u1 ,

) β u1 α ) α u2 β

x

if u1 ≤ x < e1 = e2 , if e2 ≤ x < u2 , if u2 ≤ x.

Similar to the cases above, letting S ′ = {(τ (s1 ), τ (s2 )) | s ∈ S }, by ICMT, we have (τ (u1 ), τ (u2 )) ∈ F (S ′ , 0) and (τ (e1 ), τ (e2 )) ∈ S ′ , which contradicts RE. Case 4: α = β, γ ≥ 0. In this case, since e1 + e2 = u1 + u2 , it is certain that the assumption contradicts RE. □ 4. Independence of axioms We show the independence of three axioms by pointing out that each of the following solutions violates one axiom while satisfying the remaining two. The maximin solution, Mmin (S , d) = argmaxs∈S min{s1 − d1 , s2 − d2 }, violates SSSP. The leximax solution, Lmax (S , d) = {z ∈ S | s ̸ ∈ S if max{z1 − d1 , z2 − d2 } < max{s1 − d1 , s2 − d2 }, or max{z1 − d1 , z2 − d2 } = max{s1 − d1 , s2 − d2 } and min{z1 − d1 , z2 − d2 } < min{s1 − d1 , s2 − d2 }}, violates RE. Finally, the Nash solution, N(S , d) = argmaxs∈S , s≥d (s1 − d1 ) (s2 − d2 ), violates ICMT. It is easy to demonstrate that N(S , d) satisfies RE. Let w = v − d for v ∈ N(S , d) such that v1 −d1 ̸ = v2 −d2 . Suppose λw+(1−λ)w ˜ ∈ S for some λ such that 0 < λ < 1 where w ˜ = (w2 , w1 ). Note that λw + (1 − λ)w ˜ has the same total utility gain as w and is more equitable than w . Comparing w with λw + (1 − λ)w ˜ , we obtain (λw1 + (1 − λ)w2 )(λw2 + (1 − λ)w1 )

= w1 w2 + λ(1 − λ)(w1 − w2 )2 > w1 w2 .

O. Mori / Economics Letters 159 (2017) 7–9

This result contradicts v ∈ N(S , d). Hence, RE is a common property of N(S , d) and E(S , d). Acknowledgments I am deeply grateful to an anonymous referee for helpful comments. Also, I would like to thank Enago for the English language review. I acknowledge support from Jobu University.

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References Mariotti, M., 1999. Fair bargains: distributive justice and Nash bargaining theory. Rev. Econom. Stud. 66, 733–741. Nash, J.F., 1950. The bargaining problem. Econometrica 18, 155–162. Nielsen, L.T., 1983. Ordinal interpersonal comparisons in bargaining. Econometrica 51, 219–221. Roth, A.E., 1979. Axiomatic Models of Bargaining, No. 170. Springer-Verlag, Berlin, New York.