Cooperative bargaining: independence and monotonicity imply disagreement Shiran Rachmilevitch September 23, 2012 Abstract A unique bargaining solution satisfies restricted monotonicity, independence of irrelevant alternatives, independence of equivalent utility representations, and symmetry the disagreement solution. It is also the unique bargaining solution that satisfies independence of irrelevant alternatives, continuity, and strong disagreement monotonicity. Keywords: Bargaining; Disagreement solution. JEL Classification: C78; D74. 1 Introduction A bargaining problem is a pair (S, d), where S R n is the feasible set, representing all possible (v-n.m) utility agreements among the n bargainers (henceforth players), and d S, the disagreement point, is a point that specifies their utilities in case they do not reach a unanimous agreement on some point of S. The following assumptions are made on (S, d): Department of Economics, University of Haifa, Mount Carmel, Haifa, 31905, Israel. shiranrach@econ.haifa.ac.il Web: http://econ.haifa.ac.il/ shiranrach/ Email: 1
S is compact and convex; d < x for some x S; 1 For all x S and y R n : d y x y S. Denote by B the collection of all such pairs (S, d). A solution is any function µ: B R n that satisfies µ(s, d) S for all (S, d) B. Given a feasible set S, the weak Pareto frontier of S is W P (S) {x S : y > x y / S} and the strict Pareto frontier of S is P (S) {x S : y x y / S}. The best that player i can hope for in the problem (S, d), given that each j i obtains at least d j utility units, is a i (S, d) max{x i : x S d }, where S d {x S : x d}. The point a(s, d) = (a 1 (S, d),, a n (S, d)) is the ideal point of the problem (S, d). The Kalai-Smorodinsky solution, KS, due to Kalai and Smorodinsky (1975), is defined by KS(S, d) W P (S) [d; a(s, d)]. 2 The Nash solution, N, due to Nash (1950), is defined to be the unique maximizer of Π i (x i d i ) over S d. Consider the following axioms, in the statements of which (S, d) is an arbitrary problem. Weak Pareto Optimality (WPO): µ(s, d) W P (S). Individual Rationality (IR): µ(s, d) S d. Let F A denote the set of positive affine transformations from R to itself. 3 1 Vector inequalities: xry if and only if x i Ry i for all i, R {>, }; x y if and only if x y & x y. 2 Given two vectors x and y, the segment connecting them is denoted [x; y]. 3 i.e., the set of functions f of the form f(x) = αx + β, where α > 0. 2
Independence of Equivalent Utility Representations (IEUR): f = (f 1,, f n ) (F A ) n f µ(s, d) = µ(f S, f d). 4 Let Π denote the set of permutations on {1,, n}. Symmetry (SY): i, j {1,, n}: [(π S = S)&(π d = d) π Π] µ i (S, d) = µ j (S, d). Say that a solution is admissible if it satisfies WPO, IR, IEUR, and SY. The Nash solution is the only admissible solution that satisfies the following axiom, in the statement of which (S, d) and (T, e) are arbitrary problems. Independence of Irrelevant Alternatives (IIA): [S T ]&[d = e]&[µ(t, e) S] µ(s, d) = µ(t, e). In words, IIA says that deleting unchosen options should not change the bargaining s outcome. Consider the 2-person case, namely n = 2. Given x S d, let gi S (x j ) be the maximal possible payoff for i in S given that j s payoffs is x j. Consider the following axiom, in the statement of which (S, d) and (T, d) are arbitrary 2-person problems with a common disagreement point. Individual Monotonicity (IM): 4 If f i : R R for each i, x R n, and A R n, then: (f 1,, f n ) x (f 1 (x 1 ),, f n (x n )) and (f 1,, f n ) A {(f 1,, f n ) a : a A}. 3
[a j (S, d) = a j (T, d)]&[g S i (x j ) g T i (x j ) x S d T d ] µ i (S, d) µ i (T, d). In words, IM says that if the feasible set stretches in the i-th direction, then player i should not get hurt. Kalai and Smorodinsky (1975) proved that for n = 2, KS is the unique admissible solution that satisfies IM. 5 In the n 3 case, the idea of individual monotonicity can be formulated in several ways; the following in the statement of which (S, d) and (T, d) are arbitrary n-person problems with a common disagreement point is a common and transparent one (see Thomson (1994)). Restricted Monotonicity (RM): If a(s, d) = a(t, d) and S T, then µ(s, d) µ(t, d). The Kalai-Smorodinsky solution is the only admissible solution that satisfies RM. Since N KS, it follows that under admissibility, IIA and RM or, to use slightly more general terms, independence and monotonicity are not compatible. Can we make them compatible by compromising on efficiency? The following result shows that this compromise does not take us very far. Theorem 1. There exists a unique bargaining solution that satisfies independence of equivalent utility representations, independence of irrelevant alternatives, symmetry, and restricted monotonicity. This is the disagreement solution: µ(s, d) d. Proof. Obviously the disagreement solution satisfies the axioms. Conversely, let µ be an arbitrary solution that satisfies them. For each x (0, 1], let Q x = conv hull{0, e i, x 1 : i = 1,, n}. 6 First, I argue that if µ(q x, 0) = 0 for some x (0, 1], then µ(s, d) d. To see this, let (S, d) be an arbitrary problem and suppose that µ(q x, 0) = 0 for some x (0, 1]. 5 IR is redundant: the other axioms suffice for the characterization. 6 0 (0,, 0), 1 (1,, 1), and e i is R n s i-th unit vector. 4
By IEUR we may assume d = 0. By IEUR, µ(m Q x, 0) = M µ(q x, 0) = M 0 = 0 for every M > 0. Fix a sufficiently large M such that there exists a symmetric feasible set T such that T R n + = M Q x and T S. By SY, µ(t, 0) = z 1 for some z. By RM and the fact that T M Q x, z 0. Then, it must be that z = 0 otherwise, the combination of µ(m Q x, 0) = 0 with the fact z 1 M Q x would contradict IIA. Therefore z = 0. By IIA, µ(s, 0) = 0. I argue that if µ is different than the disagreement solution then µ(q x, 0) = x 1 for all x (0, 1]. To see this, note that, by SY, µ(q x, 0) = y 1, for some y x. If y < x, then there is a λ (0, 1) such that y 1 λ Q x, and therefore, by IIA, µ(λ Q x, 0) = y 1. On the other hand, by IEUR, µ(λ Q x, 0) = λy 1, hence y = 0, in contradiction to the fact that µ is not the disagreement solution. Let B {(S, d) B : P (S) = W P (S)}. I will prove that if µ is different than the disagreement solution then it coincides with the Kalai-Smorodinsky solution on B. This would be a contradiction, because KS violates IIA on B. Suppose then that µ is not the disagreement solution and let (S, d) B. By IEUR, we can assume d = 0 and a(s, d) = 1. Let T = conv hull{0, e i, KS(S, d) : i = 1,, n}. Since T = Q x for x = KS 1 (S, d) = = KS n (S, d), 7 and since µ is not the disagreement solution, it follows from the previous paragraph that µ(t, 0) = KS(S, d). By RM, µ(s, d) µ(t, 0) = KS(S, d). Since W P (S) = P (S) and KS(S, d) W P (S), µ(s, d) = KS(S, d). Remark 1 : Given a nonzero vector p, the corresponding proportional solution (due to Kalai (1977)) is defined by µ p (S, d) d + a p, where a > 0 is the maximum such that the aforementioned expression is in S. Corresponding to p = 1 is the egalitarian solution and corresponding to p = e i is the i-th dictatorial solution. The former satisfies all the axioms from Theorem 1 besides IEUR, and the latter all of them but SY. 7 Since both the disagreement and ideal points have equal coordinates, all the players enjoy the same payoffs under KS. 5
Remark 2 : Note that IR is not invoked in Theorem 1. Remark 3 : If IR is added to the axioms listed in Theorem 1, then a much shorter proof obtains. Roth (1980) showed that when one deletes WPO from the list of Nash s axioms, the only solutions that satisfy the remaining axioms are the Nash solution and the disagreement solution. Then, with IR added to the axioms list, it follows that only the disagreement solution satisfies the axioms, because the Nash solution violates RM. IR is important for Roth s theorem; here is an example of a solution that satisfies all of Nash s axioms besides WPO and IR, and is different from the disagreement solution and the Nash solution the anti Nash solution, that picks for every (S, d) the maximizer of the Nash product over {x S : x d}. RM imposes a monotonicity condition on the feasible sets. The following axiom expresses a similar requirement from disagreement-point-changes. In its statement, (S, d) and (T, e) are arbitrary problems and i is an arbitrary player. Strong Disagreement Point Monotonicity (S.DIM): If T = S, a(s, d) / S, e i > d i, and e i = d i, then µ i (T, e) > µ i (S, d). 8 The Nash solution satisfies IIA but not S.DIM, and in the 2-person case the Kalai- Smorodinsky solution satisfies S.DIM but not IIA. This is another example of the informal intuition that the Nash solution is independent, the Kalai-Smorodinsky solution is monotonic, and these two principles are not compatible. I end this paper with a result that formalizes the (essential) incompatibility of IIA and S.DIM. This result is stated for the subclass of B that satisfy the following: for every feasible set S, 8 The first disagreement monotonicity axiom to be introduced to the literature, which is due to Thomson (1987), is stated in terms of a weak inequality. Livne (1989) was the first to consider strict-inequality disagreement monotonicity. 6
if {x 1,, x K } S, then y S, where y i = min{x 1 i,, x K i }. Call this domain B. Additionally, I will make use of the following axiom, in the statement of which (S, d) is an arbitrary problem and {(S n, e)} is an arbitrary sequence of problems sharing the same disagreement point. Continuity (CONT): If e = d and {S n } converges to S in the Hausdorff topology, then µ(s n, e) µ(s, d). Theorem 2. There exists a unique bargaining solution on B that satisfies independence of irrelevant alternatives, strong disagreement monotonicity, and continuity. This is the disagreement solution: µ(s, d) d. Proof. Obviously the disagreement solution satisfies the axioms. Conversely, let µ be an arbitrary solution that satisfies them. Let (S, d). By CONT, we may assume that a(s, d) / S. Assume by contradiction that x d, where x µ(s, d). Wlog, suppose that x 1 d 1. Case 1: x 1 > d 1. Let ɛ x 1 d 1. Recall that d S, there is some s S with s > d, and S is convex and comprehensive; therefore, there is a sufficiently small δ (0, ɛ ) such that s i d + δ e i S for all i = 1,, n. Let T {y S : y x} and T conv hull(t {d, s i : i = 1,, n}). By IIA, µ(t, d) = x. By S.DIM, µ 1 (T, s 1 ) > x 1, which is impossible. Case 2: x 1 < d 1. Let Q {y S : y 1 x 1 }. By IIA, µ(q, d) = x. Let ɛ (0, d 1 x 1 ) and let e d ɛ e 1. By S.DIM, µ 1 (Q, e) < x 1, which is impossible. 9 Acknowledgments: I am grateful to Ehud Kalai, Al Roth, Joel Sobel, and William Thomson for their helpful comments. 9 The restriction to B guarantees that e Q, therefore (Q, e) is a well-defined problem. 7
References Kalai, E. (1977) Proportional solutions to bargaining situations: interpersonal utility comparisons, Econometrica, 45, 1623-1630. Kalai, E. and Smorodinsky, M. (1975), Other solutions to Nash s bargaining problem, Econometrica, 43, 513-518. Livne, Z.A (1989), Axiomatic characterizations of the Raiffa and the Kalai-Smorodinsky solutions to the bargaining problem, Operations Research, 37,972-980. Nash, J. F. (1950), The bargaining problem, Econometrica, 18, 155-162. Roth, A.E. (1980), The Nash solution as a model of rational bargaining. In: Extremal Methods and Systems Analysis, A.V. Fiacco and K.O. Kortanek, eds., Springer Verlag, 306-311. Thomson, W. (1987), Monotonicity of bargaining solutions with respect to the disagreement point, Journal of Economic Theory, 42,50-58. Thomson, W. (1994), Cooperative models of bargaining. In: Handbook of game theory, vol 2., Aumann R.J., Hart S. eds., North-Holland, Amsterdam 1237-1284. 8