The Properly Efficient Bargaining Solutions

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1 The Properly Efficient Bargaining Solutions By Jingang Zhao * Department of Economics Iowa State University 260 Heady Hall Ames, Iowa jingang@iastate.edu Fax: (515) Tel: (515) Abstract: This note is motivated by the similarity between multiobjective programming (MOP) and bargaining theory. It uses the known MOP concepts to establish two new bargaining results: (1) the efficiency of Kalai- Smorodinsky (KS) and Nash (NA) solutions for large classes of bargaining problems can be refined to the sharper concept of proper efficiency; (2) the independence of irrelevant alternatives (IIA) of NA solution in convex bargaining can be upgraded to strong IIA, which is a form of sensitivity analysis in MOP. These results show that bargaining theory and MOP can gain insights from each other. Keywords: Bargaining, multiobjective programming, proper efficiency Running Head: Properly Efficient Bargaining Solutions MAC Number: 90C29, 90D12. * my own. I would like to thank Rick Percy and David Schmeidler for valuable comments. All errors, of course, are

2 1. Introduction Multiobjective programming (MOP) and bargaining theory have been studied as two separate disciplines 1. In bargaining, a point is chosen from a utility set to increase players' payoffs; in MOP, a point is chosen from the feasible set to maximize a vector of functions. These two problems coincide when the objectives are players payoffs, therefore the two disciplines could gain insights from each other. By applying known concepts from MOP, this note establishes two new properties of bargaining solutions. First, it shows that efficiency in large classes of bargaining problems can be refined to the sharper concept of proper efficiency in MOP (Geoffrion, 1968): Nash and Kalai-Smorodinsky solutions both are properly efficient in large classes of bargaining problems. Second, bargaining axioms like monotonicity and independence of irrelevant alternatives can be understood as sensitivity analysis in MOP (i.e., how a solution changes when the environment changes). It shows that independence of irrelevant alternatives (IIA) of Nash solution (i.e., it remains unchanged when the feasible set shrinks) in convex bargaining can be upgraded to strong IIA, which allows both the feasible set and the disagreement-point to change. In addition, linear MOP leads to linear bargaining, whose solutions are alternative answers to the core-selection problem in coalitional TU (transferable utility) games. Finally, the duality theory in MOP leads to a dual result between the cost-sharing and surplus-sharing problems. It is the author s hope that the known bargaining axioms might be applied to MOP in the near future. The rest of the paper is organized as follows: Section 2 describes the problem. 1 For surveys, see Zeleny (1984) on MOP, Zhao (1991) and Oliveira and Ferreira (2000) on multiobjective games, and Thomson (1994) on bargaining. MOP is also called MOMP (multiobjective mathematical programming), MCDM (multiple criterion decision making), or Vector Max/Min. 2

3 Section 3 studies bargaining efficiency, Section 4 studies the strong IIA of Nash solution, and Section 5 studies linear bargaining and the duality between cost- and surplus-sharing problems. Section 6 concludes, and the appendix provides all proofs. 2. Description of the problem Let N = {1, 2,..., n} be a finite set of players. A utility bargaining problem is a pair (d, X) of a d-point and a feasible set satisfying: (i) d R n +, X Rn, X, and X is compact; (ii) d + x for all x X; and (iii) there is some x X such that x >>d. This 2 requires neither d X nor comprehensiveness (see Figures 1-2). (Figures 1 and 2 about here) Let U be the set of the above bargaining problems. A bargaining solution is a function g: U R n + such that g(d, X) X. Let e = (1,..., 1) be a vector of one's in Rn, Arg-Max {h(x) + x X} denote the set of optimal solutions for Max {h(x) x X}, and X * e = {x X {z X z>x}= } denote the efficient frontier of X. Definition 1: Given (d, X). (i) Its egalitarian solution is ES = d+λ E e, where λ E = Max {λ d+λe X}; (ii) its Kalai-Smorodinsky solution is KS = d+λ KS t KS, where λ KS = Max {λ d+λt KS X}, and t KS is its optimal direction; (iii) its Nash solution is NA = Arg-Max {Π(x j d j ) x X}; and (iv) its proportional solution is given by P = (1+λ P )d, where λ P = Max {λ (1+λ)d X}. The "optimal direction" t KS is defined by 2 The inequalities are defined as: d x d i x i, all i; d < x d x, d x; d << x d i < x i, all i. (d, X) is d-comprehensive if for any x X, and x y d, one has y X, see Thomson (1994). 3

4 (1) t i KS = t i KS (d, X) = Max {(x i-di) x X}, i = 1,, n. In other words, the i th component of t KS is the difference between i's maximum feasible payoff and his d-value di. As illustrated in Figure 1-a and Figure 2, the above solutions have a geometric interpretation (except the NA solution): one starts at the d-point, walks along an optimal direction (such as t KS ) towards the efficient frontier X * e, and stops at a solution on the frontier. Such a geometric intuition is also shared by algorithms like the gradient method in MOP and mathematical programming, which is examined in Section 4. A bargaining problem (d, X) is called a convex bargaining problem if X is a convex set. Axiomatic properties of the above solutions have been well established in convex bargaining: The NA solution is the only solution satisfying efficiency, symmetry, scale invariance, and IIA (Nash, 1950); the KS solution is the only solution satisfying efficiency, symmetry, scale invariance, and monotonicity (Kalai and Smorodinsky,1975) The proper efficiency of bargaining solutions A multiobjective programming problem is defined as: (2) MOP: Max F(x) = {f 1 (x);... ; f k (x)}; k > 1 x X ; where X is a non-empty feasible set in R n, and fi: X R, i =1,..., k, are the objective functions to 3 A solution z(d, X) satisfies: (1) Εfficiency if z X * E ; (2) Symmetry if z i(d, X) = z j (d, X) for i j when (d, X) is invariant under all permutations of N; (3) Scale invariance if z(t(d), T(X)) = T(z(d, X)) for any scale transformation T defined by T(x) = {a 1 x 1,, a n x n } and T(X) ={T(x) x X} for some a >>0; (4) IIA if z(d, X ) = z(d, X) for any X X and z(d, X) X ; (5) Monotonicity if z i (d, X) z i (d, X ) for any enlargement X of X in i s favor, where X ( X) is an enlargement of X in i s favor if tks i (d, X ) = Max{(x i-d i ) x X } > tks i j (d, X), and tks (d, X ) = 4

5 be maximized. A linear MOP problem has the following form: Max F(x) = Cx (3) LMOP: x X = {y R n + Ay b}; where C = C k n is a k n matrix. When C = I n n is the identity matrix, (3) is equivalent to a class of linear bargaining problems defined below: Definition 2: A linear bargaining problem is defined by (d; A, b) R n+mn+m, which is a utility bargaining problem (d, X) whose feasible set is given by (4) X = {x R n Ax b}, + where d is the earlier d-point, A = Α m n is an m n matrix, and b = b m 1 R m. A linear bargaining problem is clearly a convex bargaining problem. A bargaining problem (d, X) can now be treated as a MOP problem {F(x) = x x X} with an opportunity cost d: one picks a point x X trying to maximize each of the n objectives fi(x) = xi, and the objectives will take the low values fi(d) = di, all i, if no agreement is reached. Such a connection suggests that one discipline could shed light in better understanding the other. This section examines bargaining efficiency using the following efficiency results in MOP: Definition 3: (i) The weakly efficient, and efficient solutions for the MOP problem defined in (2) are: (5) X * we = {x X {y X F(y) >> F(x) } = }; (6) X * e = {x X {y X F(y) > F(x)} = }. (ii) (Geoffrion, 1968) An efficient solution x 0 is properly efficient if there is M>0 such that, for any i and x X with f i(x) > f i (x 0 ), there is a j i satisfying f j (x) < f j (x 0 ) and f i (x)-f i (x 0 ) f j (x 0 )-f j (x) M. tks j (d, X) for all j i. See Nash (1950), Kalai and Smorodinsky (1975). 5

6 Let the set of all properly efficient solutions be denoted as (7) X pe * = {x X x is a properly efficient solution of (2)}. (Figure 3 about here) It is straightforward to see X pe * X* e X we * (see Figure 3). If x 0 X is not weakly efficient, then there is x X satisfying F(x) >> F(x 0 ). If x 0 is not efficient, then there is x X satisfying F(x) > F(x 0 ). Finally, if x 0 is not properly efficient (like B and C in Figure 3), then for any M > 0, there is an i and x X with f i (x) > f i (x 0 ) such that [f i (x)-f i (x 0 )] > [f j (x 0 )-f j (x)]m for all j with f j (x 0 ) > f j (x). By letting M + and by the fact that there are only a finite number of objectives, one sees that a non-properly efficient solution implies the existence of some objective i * whose marginal gain can be made arbitrarily large relative to each of the marginal losses in all other objectives. By the above observation, the main weakness of a non-properly efficient solution is its extreme unfairness: The increase in one player s payoff can be made arbitrarily large relative to a small loss in another player's payoff. In finite allocation problems, the main difference between the two concepts is that an efficient solution allows one to get all and the remaining players all get zero, this unfair situation is ruled out by a properly efficient solution. An example of such nonproper efficiency is the two endpoints of the efficient frontier in a two-person surplus-sharing problem with risk-averse preference as shown in Figure 3. Similarly, a non-properly efficient bargaining outcome implies the existence of some player i* whose payoff can be increased by an arbitrarily large amount relative to small losses by other players. One would expect a good 6

7 bargaining solution to rule out non-properly efficient outcomes. This is indeed the case: the NA and KS solutions are properly efficient in large classes of bargaining problems defined below. Definition 4: (i) Given (d, X) and S N, let x S = {x i i S}. A feasible point x 0 X is called an S-efficient solution if it is an efficient solution for Max {F S (x) = x S x X}. (ii) A bargaining problem (d, X) is non-degenerate if for any S N and any S-efficient solution x, (x-d) µt KS for all µ (- <µ< ), where t KS is the optimal direction (1). (iii) (d, X) is a ball-shaped bargaining problem if (a) X is convex; (b) For any S N and any S-efficient solution x, one has x i = d i for all i S. The S-efficiency coincide with efficiency if S = N (see Dubey, 1986; Zhao, 1991). A ball-shaped bargaining problem is clearly non-degenerate (see Figure 4-a for a degenerate problem). Note that ball-shaped bargaining problem is a very large set of bargaining problems, it includes all problems satisfying comprehensiveness, it also includes all problems given by X = {x R n + x d, f(x) f(x 0 )} and x 0 >>d, where f is a convex and increasing function (see Claim 2 in the appendix for a proof). The efficient frontier in two player ball-shaped bargaining (i.e., n = 2) is a non-increasing and concave curve linking the lines x 2 = d 2 and x 1 = d 1. Therefore, surplus sharing problem defined by (10) in Section 5 is a ball-shaped problem (see Figure 2). (Figure 4 about here) Proposition 1: (i) the KS solution in non-degenerate and convex bargaining is the only solution satisfying proper efficiency, symmetry, scale invariance, and monotonicity; (ii) the NA solution is properly efficient in linear bargaining and in ball-shaped bargaining. Since proper efficiency is a property of MOP in which a convex feasible set is a basic assumption, it will be extremely difficult (likely impossible) to establish a properly efficient bargaining solution for non-convex problems. The difference between efficiency and proper 7

8 efficiency will become more significant if one pushes bargaining theory from the current riskneutral assumption to risk-averse (-seeking) assumptions (i.e., a player s payoff x i be replaced by a nonlinear concave (convex) utility function f i (x i )). 4. The strong IIA of Nash solution This section is based on the similarity between monotonicity in bargaining and sensitivity analysis in MOP: they both study how a solution will change in response to changes in primitive parameters. It shows that the NA solution in convex bargaining remains as a constant either the feasible set shrinks without losing the NA solution or the d-point moves towards the efficient frontier on the line connecting the original d-point and the NA solution. Given (d, X), t R n +, t 2 = Σ (t i) 2 =1, let U(t; d, X) be the set of bargaining problems (d', X') satisfying: (i) d' = d + µ t for some µ 0; and (ii) X' = {x X d' x }. Definition 5: Given a solution g = g(d, X) for (d, X), let t = (g-d)/ g-d. (i) g satisfies homogeneity if it satisfies strong individual rationality (SIR) and g(d', X') g(d, X) for all (d', X') U(t; d, X); (ii) g satisfies SIIA (strong independence of irrelevant alternatives) if it satisfies homogeneity and IIA. SIR means x>>d. It has been shown that SIR and IIA together imply efficiency (See Roth, 1977; Zhou, 1997). In ball-shaped bargaining problems, one can show that SIR plus IIA (or homogeneity plus weak efficiency or SIR plus weak efficiency) imply proper efficiency (see Claim 1 in the appendix). Proposition 2: In a convex bargaining problem (d, X), its NA solution is the only solution that satisfies scale invariance, SIIA, and symmetry. Note that homogeneity changes the d-point, IIA changes the feasible set, and SIIA 8

9 changes both the d-point and the feasible set. Therefore, g(d, X) satisfies SIIA if g(d', X') = g(d, X) for any (d', X') such that g(d, X) X' X, d' = d + µt >>d (0 µ µ 0 = g-d ), and x d' for all x X'. It remains to be seen if SIIA imply symmetry (or scale invariance). The appendix provides a non-trivial proof for the homogeneity of a NA solution in general convex bargaining problems. One surprising consequence of homogeneity arises from the KS solution. Since KS solution has the geometric intuition that it is obtained by walking along the optimal direction from the d-point towards the efficient frontier, one would expect it to satisfy homogeneity. However, as shown below, the KS solution does not satisfy homogeneity in a general convex problem (see Figure 4-b). Let d = (2, 1), X = {2 x 1 6, 1 x 2 4, x 1 +4x 2 18}, one gets t KS = (4, 3), KS = (5, 3.25). Note that the unit direction is t = (4/5, 3/5). For d = d + (5/4)t = (3, 1.75), X' = {x X d' x} = {3 x 1 6, 1.75 x , x 1 +4x 2 18}, KS = KS(d', X') = (57/11, 141/44) (5.2, 3.2) KS = (5, 3.25). Note that the KS solution satisfies a weak homogeneity defined below. Assume d X and let t and U(t; d, X) be as described in Definition 5. Then, the KS solution is a constant for all (d', X) such that d' = d + µ t (µ 0) and d' X. 5. Linear bargaining and the duality between cost-and surplus-sharing problems This section first discuss two remarks based on the similarity between linear MOP and linear bargaining. It then provides a preliminary result on the duality between cost-sharing and surplus-sharing problems, which suggests future research in the spirit of the duality theory in MOP. Consider the core selection problem for a coalitional TU game Γ = {N, v(. )}, where for each S N, v(s) is the payoff of S. A payoff vector is any x R n + giving x i ( 0) to player i. 9

10 An allocation of v(n) is any x R n + such that Σx i = v(n). Definition 6: The core of a game Γ = {N, v(. )} is given by (8) C(Γ) = {x R n + Σ i S x i v(s) for all S N, and Σx i = v(n)}. In other words, the core 4 is the set of all allocations of the grand coalition s payoff v(n) satisfying the coalitional rationality for all coalitions. Its selection problem is resolved by bargaining solutions given below: Remark 1: Given Γ= {N, v(. )}, assume C(Γ). Define a utility bargaining problem (d, X) by d i = v(i) for all i, and X = C(Γ). Then, any non-empty solution g(d, X) is a solution to the core selection problem. Similarly, core selection in coalitional NTU (non-transferable utility) games can alternatively be solved by solutions in non-linear bargaining. Two other classes of linear bargaining are of special interest: the rectangular bargaining problem and the surplus sharing problem. The feasible set of a rectangular bargaining problem is (9) X = {x R n + d x u}, which can be given as a vector (d, u) R 2n + (i) of the following Remark 2. such that d<<u, and its solutions are provided in part A surplus sharing (SS) problem is a vector (r; d) R n+1 + with r >Σ d i or (d, X) with (10) X = {x R n + x d, Σ x i r} = {x R n + Ax b}, where r is the joint return, d is the d-point, and the matrices are: 4 The core is dual to the base polyhedron of a set function. See Fujishige (1991) for references. 10

11 1 1 K 1 r d e 1 A = Α (n+1) n = K r, = b = b( n + 1) 1 = =. I K K K K d K 0 0 K 1 dn An SS problem represents situations like networks, where returns exhibit scale economies. A network provider i alone generates a revenue of d i, while full cooperation generates a joint return r >Σd i. Solutions of an SS problem are given in part (ii) of Remark 2 below. Remark 2: (I) The solutions of a rectangular problem (d, u) are: (i) KS = NA = u; (ii) the ES solution is efficient u-d = λe for some λ>0 ES = KS = NA = u; (iii) the P solution is efficient u = λd for some λ > 0 P = KS = NA = u. (II) The solutions of an SS problem (r; d) are: (11) ES = KS = NA = d + λ E e, λ E = λ KS = λ N = (r Σ d j) / n; (12) P = (1 + λ P ) d, λ P = (r / Σ d j) 1. The ES and P solutions might not exist (see Figure 1-b). In an SS problem, the uniform solution (or U solution with U i = U j ) might not exist, and the ES, KS and NA solutions always coincide (see Figure 2). Remark 3: Bargaining theory has grown without considering computation. It is nontrivial and useful in future works to design practical methods in finding the KS and other solutions (except NA) 5 in linear (non-linear) bargaining. In an SS problem (r; d), the unit optimal directions and travel distances for ES, KS, NA and P solutions are (see Figure 2): (13) ES= KS= NA= d+µ ES t ES, t ES = t KS = t NA = e/ n, µ ES = µ KS = µ NA = (r Σ d j)/ n; (14) P = d + µ p t p, t P = d / d, µ P = [(r/σd j ) 1] d. Finally, homogeneity and duality theory in MOP lead to a preliminary dual property of the 5 The NA solution can be solved by the known optimization methods. 11

12 SS problems 6. By homogeneity, a solution remains unchanged when the d-point moves along its optimal direction. What happens if the d-point of an SS problem moves beyond the efficient line along an optimal direction? One obtains a cost sharing (CS) problem, defined by an (n+1) vector (L; D) R n+1 + such that Σ D i > L, or a linear dual bargaining problem (D; Y) whose feasible set is (15) Y = {y R n + y D and Σ y i L}. Such CS problems arise from situations like carpooling or municipal cost-sharing (Young, 1985), where costs exhibit scale or scope economies: It costs a commuter D j to drive alone, while it costs only L < Σ D j if all commuters carpool in a single vehicle. Note the economic duality between the SS and CS problems: no player in an SS problem will accept a return less than his d-point, which is below the efficient line; while no player in a CS problem will pay a cost greater than his d-point, which is above the efficient line. Such relation is similar to the duality between Max- and Min-MOP s, which suggests the need to seek the general duality between CS and SS problems. The next proposition is a preliminary step forward along this line of future research. (Figure 5 about here) 6 This is related to the d-monotonicity (Thomson, 1987). Let e(i) be the direction of the i th axis (i.e., e j (i) = 1 if j = i, = 0 if j i). Given (d, X), let d' = d+λe(i) for some small λ > 0, then x(d, X) satisfies d-monotonicity if x i (d', X) x i (d, X), and strong d-monotonicity if x j (d', X) x j (d, X) for all j i. See Thomson (1987) for more discussions. A dual bargaining problem (D; Y) satisfies (see Zhao, 1999): (i) D R n +, Y Rn, Y, and Y is + compact; (ii) y D for all y Y; (iii) there is y Y such that y << D. Its KS and NA solutions are: y NA = Arg- Max {Π(D i -y i ) y Y}; y KS = D+λ KS t KS Y, where t KS is given by t i KS = Min {(y i-d i ) y Y} for all i, and λ KS = Max {λ D+λt KS Y}. 12

13 Proposition 3: Consider an SS problem (r; d) with Σ d i < r: (i) there is a unique CS problem (L'; D') such that y P (L'; D') = P(r; d) and µ ' P = µ P, where L '= r, and D'= (-1+2r/Σ d j)d; (ii) There exists a unique CS problem (L*; D*) such that y KS (L*; D*) = y NA (L*; D*) = y ES (L*; D*) = KS(r; d), µ * KS = µ * NA = µ * ES = µ KS, where (16) L*= r, and D*= d+(2(r Σ d j)/n)e. In other words, the P and ES solutions in a SS problem each defines a unique CS problem whose solution (and its optimal direction and travel distance) coincide with those in the original SS problem (see Figure 5). However, the reverse of Proposition 3 is not true, because the set of CS problems is larger than that of SS problems. 6. Concluding remarks This note has established, by applying concepts in MOP, two new properties of bargaining solutions: (i) the KS and NA solutions are properly efficient in large classes of convex bargaining problems; (ii) the NA solution satisfies strong IIA, which can be used to replace its efficiency and IIA, while the KS solution doesn t satisfy homogeneity. These results illustrate that new insights on bargaining can be obtained from MOP. From this perspective, this note provides more questions for future research than the above two results: it is non-trivial to find a practical algorithm for computing the KS and other solutions (except the NA solution), and it is challenging to establish general duality between cost- and surplus-sharing problems in the spirit of the duality theory in MOP. On the other hand, new insights on MOP might also be obtained from bargaining theory. For example, it is useful to check if the dual Vector-Min problem (i.e., dual to the Vector-Max problem for utility bargaining) can be interpreted as a bargaining problem, and it is useful to apply 13

14 the bargaining axioms (such as scale invariance or monotonicity) to the stability and the sensitivity analysis of MOP solutions. Appendix Before we prove Propositions 1 and 2, we first provide the following six claims. Recall that efficiency and scale invariance are summarized in Footnote 3. Claim 1 (Geoffrion, 1968): Consider the MOP (2), let X * e, X * pe, X * we be given in Definition 3, k + ={λ Rk + Σ λ j = 1} and k ++ ={λ Rk ++ Σλ j =1} denote the k-1 simplex and its relative interior. For each λ k +, let f λ (x) = Σ {λ jf j (x)}, and X * (λ) = Arg-Max {f λ (x) x X}= {x 0 X f λ (x 0 ) f λ (x), x X}. Suppose all f i (x) are continuous and concave, X is a convex and compact set, then U (A1) X we * = X * ( λ ), and λ k + U (A2) X pe * = X * ( λ ). λ k ++ Claim 2: If there is a convex and increasing function f(x) and x 0 >>d such that X = {x R n + x d, f(x) f(x 0 )}, then for any S N and S-efficient solution x 0, xi 0 = d i for all i S. Proof of Claim 2: By the convexity of f, X = {x R n + x d, f(x) f(x 0 )} is a convex set. Therefore, the MOP problem Max { F T (x) = x T x X} is a convex problem for any coalition T. Let x 0 = x(s) be an S-efficient solution for some S N. By Claim 1, it must be a solution of Max {Σ j S (λ j x j ) x X} for some λ S = {λ j j S}such that λ j 0 and Σ j S λ j = 1. Assume by way of contradiction that x i 0 > d i for some i S. Since f is increasing in all arguments, the value of Σ j S {λ j x0 j } can be increased by reducing xi 0 to d i and increasing any x j with λ j > 0, which 14

15 contradicts the optimality of the above maximization problem. Hence, x i 0 = d i for all i S. Q.E.D Claim 3: Homogeneity implies weak efficiency. Proof of Claim 3: Let (d, X) be a bargaining problem, and z = z(d, X) = d + µ 0 t be a solution satisfying homogeneity, where t = (z-d)/ z-d, µ 0 = z-d. Homogeneity leads to (A3) µ 0 = Max {µ d + µ t X}. Otherwise, if there is some µ > µ 0 such that d + µ t X, one sees that z(d', X') d + µ 0 t for d' = d + [µ 0 +(µ-µ 0 )/2] t, and X' = {x X x d' }. By (A3), z(d, X) X * we. Q.E.D Claim 4: (i) In ball-shaped bargaining, homogeneity implies proper efficiency. (ii) In convex bargaining, homogeneity and IIA imply efficiency. Proof of Claim 4: Part (i). Let (d, X) be a ball-shaped bargaining problem. By Claim 3, z(d, X) X we *, where z(d, X) is the same as in Claim 3. The MOP problem Max {F(x) = x x X} is clearly a convex problem. If z(d, X) X pe *, Claim 1 leads to (A4) z(d, X) X * (λ) = Arg-Max {f λ (x) x X} for some λ k +, with at least one i such that λ i = 0. Therefore, z(d, X) is an S-efficient solution for some S N and i S. By Definition 5, one has z i (d, X) = d i, which contradicts SIR or z i (d, X) > d i. Hence, z(d, X) X pe *. Part (ii) follows from Roth (1997) because homogeneity requires SIR. Q.E.D Claim 5: In a linear bargaining problem, the Nash solution satisfies homogeneity. Proof of Claim 5: Let X = {x R n + Ax b}, NA = Arg-Max{f(x,d)= Π(x i-d i ) x X} be the Nash solution. First observe that NA >> d (If NA i = d i for some i, then NA would not be 15

16 a solution of Max {Π(x i -d i ) x X}. Since f(x,d) is strictly quasi-concave and X is a linear convex set, the Nash solution is uniquely determined by the Kuhn-Tucker conditions: (A5.1) f - λ A 0, x ( f - λ A) = 0; Ax-b 0, λ (Ax-b) = 0; λ R m +, x Rn + ; where f(x,d)= { f/ x 1,, f/ x n }. Let J {1,, m} be the set of binding constraints, and A j be the j-th row of A. Then, for j J, λ j > 0 and A j x-b j = 0; and for j J, λ j = 0 and A j x- b j < 0. Since NA satisfies SIR and IIA, NA is the unique feasible x satisfying (A5.2) f - λ A = 0, λ j > 0 and A j x-b j = 0 for all j J. Now, let t = (NA-d)/ NA-d denote the characteristic direction of Nash solution, µ 0 = NA-d, d' = d + µt (0 µ µ 0 ), and X' = {x X d' x}, we need to show NA = NA(d, X) is identical to the new solution (A6) NA(d', X') = Arg-Max{f(x, d') x X'}. It follows from (A5) and f(na, d') = {Π i 1 (NA i - d' i ),, Π i n (NA i- d' i )} = {Π i 1 (NA i - d i ),, Π i n (NA i- d i )}[(µ 0 -µ)/ µ 0 ] (n-1) = [(µ 0 -µ)/ µ 0 ] (n-1) f(na, d) that NA = NA(d, X) satisfies the Kuhn-Tucker conditions (i.e., (A5.2)) for the new Max problem (A6). Since the new NA solution is uniquely determined by its Kuhn-Tucker conditions, one must have NA(d, X) = NA(d', X'). Q.E.D (Figure 6 about here) Claim 6: The Nash solution in a convex bargaining problem satisfies homogeneity. Proof of Claim 6: It follows from Berge s maximum principal and the uniqueness of the Nash solution that NA(d, X) is continuous in (d, X). Since the compact and convex feasible set X is the limit of a sequence of linear feasible sets (see Figure 6), NA(d, X) is the limit of a sequence of Nash solutions for linear bargaining. Therefore, NA(d, X) must satisfy homogeneity. Q.E.D 16

17 Proof of Proposition 1: Part (i). Suppose a solution g(d, X) satisfies proper efficiency, symmetry, scale invariance, and monotonicity. By X pe * X* e, one has g = KS (see Kalai and Smorodinsky, 1975). The symmetry, scale invariance and monotonicity of KS solution are shown in Kalai and Smorodinsky (1975), we therefore only need to show the proper efficiency. Suppose by way of contradiction that KS X pe *. By Claim 1 and the efficiency of KS solution, one has Therefore, one must have U KS X * e X we * = X * ( λ ), and KS X pe * = X * ( λ ). λ k + KS X * (λ) = Arg-Max {f λ (x) x X} U λ k ++ for some λ k +, with at least one i such that λ i = 0. By Claim 1, KS is S-efficient for some S N and i S, which contradicts the non-degeneracy. Hence, KS X * pe. Part (ii). In linear MOP, one has X pe * = X* e. The efficiency of NA solution therefore leads to its proper efficiency in linear bargaining. Now consider a ball-shaped bargaining (d, X). Similar to the proof of part (i), one can show that NA solution must be an S-efficient for some S N and i S. By Definition 4, NA i = d i, which contradicts the optimality of Max {Π(x i -d i ) x X}> 0. Hence, NA X * pe. Q.E.D Proof of Proposition 2: It follows form Claim 6 and Nash (1950) that Nash solution satisfies scale invariance, SIIA and symmetry. Since SIIA implies efficiency and IIA, any solution satisfying scale invariance, SIIA and symmetry must coincide with the Nash solution. Q.E.D Proof of Remark 2: Part (I) Given X = {x R n + d x u}, one can check that the optimal direction of KS solution is given by t KS = (u-d), which leads to KS = u. The rest of part (I) can be similarly proved by using the features of rectangular bargaining. Part (II) Given X = {x R n + x d, Σ x i r}, the optimal direction of KS solution is t KS = (r Σ d j)e, so KS = d + [(r Σ d j)/n] e. The rest of the proof is straightforward. Q.E.D 17

18 Proof of Proposition 3: We only prove part (ii). By D*= d+(2(r Σ d j)/n)e and L* = r, one has (Σ D * j L*) = (r Σ d j). Therefore, y ES (L*; D*) = D* ((Σ D * j L*)/n)e = d+((r Σ d j)/n)e = ES(r; d). The proof for y ES (L*; D*) = y KS (L*; D*) = y NA (L*; D*) is straightforward (see Zhao, 1999). The uniqueness follows from the requirement µ KS * = µ NA * = µ ES * = µ KS. The proof for part (i) is similar. Q.E.D REFERENCES C. Berge (1963), Topological Spaces. New York: Macmillan. P. Dubey (1986), Inefficiency of Nash Equilibria, Mathematics of Operations Research, Vol. 11, 1-8. S. Fujishige (1991), Submodular functions and optimization theory, Annals of Discrete Mathematics, No. 47. Amsterdam: North Holland. A. Geoffrion (1968), Proper efficiency and the theory of vector maximization. Journal of Math. Anal. and Appl., Vol. 22(3). E. Kalai and M. Smorodinsky (1975), Other solutions to Nash's bargaining problem, Econometrica, Vol. 43, J. Nash (1950), The bargaining problem, Econometrica, Vol. 18, S. Oliveira and P. Ferreira (2000), Bi-objective optimisation with multiple decisionmakers: a convex approach to attain majority solutions, Journal of the Operational Research Society, Vol. 51, A. Roth (1977), Individual rationality and Nash's solution to the bargaining problem, Mathematics of Operations Research, Vol. 2, W. Thomson (1987), Monotonicity of bargaining solutions with respect to disagreement point, Journal of Economic Theory, Vol. 42, W. Thomson (1994), Cooperative Models of Bargaining, Chapter 35 in Handbook of Game Theory, Edited by R. Aumann and S. Hart. Amsterdam: Elsevier Science 18

19 Publishers. M. Zeleny (1984), MCDM: past decade and future trends, Greenwich, Conn: JAI Press. H. P. Young (1985), Cost allocation, in Fair Allocation, H.P. Young, Ed., AMS Short Course Lecture Notes, Vol. 33. Providence: American Mathematical Society. J. Zhao (1991), The equilibria of multiple objective games, The International Journal of Game Theory, Vol. 20, J. Zhao (1999), Dual bargaining and the Talmud bankruptcy problem, Working Paper #99-05, Economics Department, Ohio State University. L. Zhou (1997), Nash bargaining theory and the revealed preference theory for non-convex problems, Econometrica, Vol. 65,

20 x 2 x 2 P u = KS = NA KS= NA ES X d= (d 1, d 2 ) d= (d 1, d 2 ) O Fig 1-a x 1 O Fig 1-b x 1 Figure 1. (a) Solutions in a rectangular bargaining problem, whose ES and P solutions are not efficient; (b) A linear bargaining problem whose ES, P and U solutions do not exist. x 2 P ES =KS= NA d = (d 1, d 2 ) x 1 +x 2 =r x 1 O Figure 2. Solutions in an SS problem (d; r), whose U solution does not exist, and whose ES, KS and NA solutions coincide. 20

21 x 2 Properly efficient set = Curve BC without the points B and C; Efficient set = Curve BC; Weakly efficient set = Curve AD. A B C D x 1 Figure 3. The difference between efficient and properly efficient solutions. x 2 x 2 KS= NA X 4 (6, 4) KS = (5, 3.25) (6, 3.75) KS ' = (5.2, 3.2) O d'= (3, 1.75) d= (d 1, d 2 ) 1 d = (2, 1 ) Fig 4-a x 1 2 Fig 4-b 4 6 x 1 O Figure 4. Counter examples. (a) An example of degenerate bargaining, where for S = (1) or (2), an S-efficient solution x coincides with the maximum vector; (b) A bargaining problem in which homogneity fails to hold in the KS solution. 21

22 x 2 x 2 (D', Y') D' D* (D*, Y*) P P (d, X ) d = (d 1, d 2 ) ES =KS = NA (d, X ) d = (d 1, d 2 ) x 1 +x 2 =r x 1 +x 2 =r O Fig 5-a x 1 O Fig 5-b x 1 Figure 5. (a) The P solution in an SS problem uniquely defines a CS problem with an identical P solution; (b) The ES (= NA = KS ) solution in an SS problem uniquely defines a CS problem with an identical ES (= NA = KS) solution. x 2 O d = (d 1, d 2 ) x 1 Figure 6. A feasible set in convex bargaining is a compact and convex set in R n, which can be approximated by a sequence of linear compact and convex sets. 22