Robustness of intermediate agreements for the discrete Raiffa solution

First via a counter example it is shown that the Proposition 3 of Anbarci & Sun (2013) is false. Then a gap and a mistake in their proof are identified. Finally, a modified version of their Proposition 3 is stated and proved. ∗Counterexample and correction for Proposition 3 in N. Anbarci and C.-j. Sun (2013): Robustness of Intermediate Agreements and Bargaining Solutions, Games and Economic Behavior, 77, 367-376 ∗∗ I thank Ching-Jen Sun for a helpful communication. Financial support from the DFG under grant TR 120/15-1 is gratefully acknowledged. 1 Basic Definitions and Axioms This section is mainly an extract of relevant parts of the respective section in Anbarci and Sun (2013) supplemented by some remarks and an axiom from Salonen (1988). For simplicity I consider only the case n = 2. That suffices for the counter example. The extension of my Proposition to general n ∈ N is straight forward. 1.1 Basic Definitions An 2-person (bargaining) problem is a pair (S, d), where S ⊂ R is the set of utility possibilities that the players can achieve through cooperation and d ∈ S is the disagreement point, which is the utility allocation that results if no agreement is reached. For all S, let IR(S, d) := {x ∈ S|x ≥ d} be the set of individually rational utility allocations. Let ∑ be the class of all 2-person problems satisfying the following: (1) The set S is compact, convex and comprehensive. (2) x > d for some x ∈ S Denote the ideal point of (S, d) ∈ ∑ as b(S, d) = (bi(S, d))i=1,...,n where bi(S, d) := max{xi ∈ R|x ∈ IR(S, d)}; the midpoint of (S, d) ∈ ∑ is m(S, d) := 1/2 (b(S, d) + d). A solution on ∑ is a function f : ∑ −→ R such that for all (S, d) ∈ ∑ we have f(S, d) ∈ S. Consider any bargaining problem (S, d) ∈ ∑ . The game (H, d) ∈ ∑ defined by H := co {d, b1(S, d) e1, b2(S, d) e2} with ei, i = 1, 2, the canonical unit vectors of R, is the “largest individually rational hyperplane game contained” in (S, d). Given any bargaining problem (S, d) ∈ ∑ and a solution f : ∑ −→ R the disagreement point set D(S, d, f) := {d′ ∈ IR(S, d) | f(S, d′) = f(S, d)} collects all possible disagreement points d′ that leave the solution f(S, d) unaffected when (S, d) is replaced by (S, d′) ∈ ∑ .