ECONOMICS SERIES SWP 2012/7. Robustness of Intermediate Agreements and Bargaining Solutions. Nejat Anbarci and Ching-jen Sun

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1 Faculty of Business and Law School of Accounting, Economics and Finance ECONOMICS SERIES SWP 2012/7 Robustness of Intermediate Agreements and Bargaining Solutions Nejat Anbarci and Ching-jen Sun The working papers are a series of manuscripts in their draft form. Please do not quote without obtaining the author s consent as these works are in their draft form. The views expressed in this paper are those of the author and not necessarily endorsed by the School or IBISWorld Pty Ltd.

2 Robustness of Intermediate Agreements and Bargaining Solutions Nejat Anbarci and Ching-jen Sun October 2012 Abstract Most real-life bargaining is resolved gradually. During this process parties reach intermediate agreements. These intermediate agreements serve as disagreement points in subsequent rounds. We identify robustness criteria which are satisfied by three prominent bargaining solutions, the Nash, Proportional (and as a special case to the Egalitarian solution) and Discrete Raiffa solutions. We show that the robustness of intermediate agreements plus additional well-known and plausible axioms, provide novel axiomatizations of the above-mentioned solutions. Hence, we provide a unified framework for comparing these solutions bargaining theories. JEL classification: C78; D74 Keywords: Nash s bargaining problem, robustness, intermediate agreements, the Discrete Raiffa solution, the Nash solution, Proportional solutions. 1 Introduction Nash s bargaining problem is a pair (S, d), where S R n is a convex and compact utility possibility set and d is the disagreement point, the utility allocation that results if no agreement is reached by the parties. A bargaining solution f associates each problem (S, d) with a unique point in S. Since Nash s (1950) seminal solution and axioms, various other solutions and axioms have been proposed. School of Accounting, Economics and Finance, Deakin University, 70 Elgar Road, Burwood, VIC 3125, Australia. nejat.anbarci@deakin.edu.au and cjsun@deakin.edu.au. 1

3 One prominent axiom is the Step-by-Step Negotiation (SSN) axiom (Kalai, 1977). SSN requires that the bargaining outcome be invariant under decomposition of the bargaining process into stages: if parties know that they will face two nested sets in sequence, first a set S and then a superset T, then the solution outcome of S can function as an intermediate agreement for T. Kalai (1977) emphasized the advantage of SSN as follows: This principle is observed in actual negotiations (e.g., Kissinger s step-by-step), and it is attractive since it makes the implementation of a solution easier. It is also attractive because we can view every bargaining situation that we encounter in life as a first step in a sequence of predictable or unpredictable bargaining situations that may still arise. Thus, the outcome of the current bargaining situation will be the threat point for the future ones. Indeed, most real-life bargaining is resolved gradually. During this process parties reach intermediate agreements, and these intermediate agreements serve as new disagreement points and pave the way for subsequent negotiations. Cooperative bargaining solutions ignore these dynamics and can therefore yield accurate predictions only if they are robust to the specifications of these dynamics. Given two bargaining problems (S, d) and (T, e) with d = e, an intermediate agreement, d, is considered robust if moving the disagreement point from d to d has no effect to the bargaining outcome in either (S, d) or (T, e). As such, SSN does provide a substantial and specific robustness test. When parties face first (S, d) and then (T, e) with d = e and S T, SSN then suggests that they can reach an intermediate agreement at f(s, d), as moving the disagreement point from d to f(s, d) has no effect to the bargaining outcome in (T, e). In this paper we consider alternatives to SSN s robustness test by taking into account additional facts about the relationship between S and T. 1. S is included in T and f(s, d) is not required to be an intermediate agreement. 2. Requirement 1 and S and T have the same ideal payoffs i.e., each party s highest payoff subject to the other party receiving her disagreement payoff is the same in both S and T. 3. Requirement 1 and the parties expect to receive the same relative gains in S and T. 2

4 Thus, in our robustness tests, certain points will serve as intermediate agreements under different circumstances: In Test 1 above, whenever the sets S and T are nested, but any such intermediate agreement is not necessarily the solution outcome of S; in Test 2, whenever S and T are nested and share the same ideal point; in Test 3, whenever S and T are nested and the parties expect to have the same relative gains. The above robustness tests will be termed Robustness of Intermediate Agreements (RIA) axioms. Each of these RIA axioms when combined with some other well-known and plausible axioms will lead to the n-person axiomatizations of the Nash, Proportional and Discrete Raiffa solutions. In that sense we provide a unified framework to obtain general (n-person) characterizations of these solutions using different types of sets of intermediate agreements and thereby compare these solutions bargaining theories in that regard. Our results can be briefly summarized as follows: (1) The Nash solution is characterized by Midpoint Domination (MD), an RIA axiom, and Continuity (CONT). (2) The Proportional solutions (and as a special case to the Egalitarian solution) are characterized by an RIA axiom, Translation Invariance (TI), CONT, Weak Pareto Optimality (WPO) and Strong Individual Rationality (SIR). (3) The Discrete Raiffa solution is characterized by MD, an RIA axiom, and Independence of Non-Midpoint Dominating Alternatives (INMD). 2 Relevant Literature The significance and fundamental role of bargaining were recognized as early as 1881 by Edgeworth, but for a very long time it was deemed to lack a clear solution. Nash (1950) provided the first axiomatic derivation of a bargaining solution, characterized by four axioms namely WPO, Symmetry (SYM), Scale Invariance (SI), and Independence of Irrelevant Alternatives (IIA). Raiffa (1953) soon proposed another solution which essentially described a discrete bargaining process, but did not provide an axiomatic characterization of his solution. Kalai and Smorodinsky (1975) characterized a new solution which, like the Discrete Raiffa solution, emphasized the parties ideal payoffs. Initially, all characterizations employed an independence or monotonicity axiom pertaining to changes in the feasible set (pioneered by Nash, 1950, and Kalai and Smorodinsky, 1975, respectively). The second generation characterizations then shifted the focus to changes in 3

5 the disagreement payoffs, as well as to considerations of uncertain disagreement points - the latter have been first considered by Chun and Thomson (1990). In the literature of arbitration games, properties - such as convexity - of the disagreement point sets played an explicit role; e.g., Tijs and Jansen (1982). In cooperative bargaining, changes in disagreement payoffs have been first considered by Peters (1986) and Thomson (1987). The axioms used in these two generations, however, typically did not refer to any bargaining process. Later, MD (Sobel, 1981) and Step-by-Step Negotiation (SSN) (Kalai, 1977) utilized a bargaining process by reaching intermediate agreements that help eliminate the most lop-sided and/or ineffi cient portions of a utility possibility set S, which at least one of the parties would find undesirable. Moulin (1983) used MD to characterize the Nash solution, and Kalai (1977) used SSN to characterize the Proportional solutions. Rachmilevitch (2012) recently used an axiom, Interim Improvement, which refers to a bargaining process, to characterize the Proportional solutions. There has been a recent revival of interest in Raiffa s work. Livne (1989), Peters (1986), and Peters and van Damme (1991) provided characterizations of a continuous version of the Raiffa solution; based on disagreement point axioms, Peters (1986) also provided a unified framework characterizing different solutions. More recently, Trockel (2009) provided an axiomatic characterization of the Discrete Raiffa solution, in which he uses SYM, WPO, SI as well as an axiom that repeatedly uses the other three axioms to improve the disagreement point iteratively. Dishkin, Koppel and Samet (2011) - using the concept of interim agreements - came up with an axiomatic characterization of a family of stepwise bargaining solutions that connects the Discrete Raiffa solution and the Continuous Raiffa solution. 3 Basic Definitions, Axioms and Solutions An n-person (bargaining) problem is a pair (S, d), where S R n 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. Given x, y R n, x > y if x i > y i for each i, and x y if x i y i for each i. For all S, let IR(S, d) {x S x d} be the set of individually rational utility allocations. Let W P O(S) {x S x R n, x > x x / S} be the set of weakly Pareto optimal points, and P O(S) {x S x R n, x x and 4

6 x x x / S} be the set of strongly Pareto optimal points. Let Σ be the class of all n-person problems satisfying the following: (1) The set S is compact, convex and comprehensive; S is comprehensive if x, z S implies that y S for all x y z. (2) x > d for some x S. Denote the ideal point of (S, d) as b(s, d) (b 1 (S, d),..., b n (S, d)), where b i (S, d) = max{x i x IR(S, d)}; the midpoint of (S, d) is m(s, d) 1 n b(s, d) + (1 1 n )d. A solution is a function f : Σ R n such that for all (S, d) Σ, f(s, d) S. The next concept will be crucial for our axioms. The inverse set of f with respect to (S, d), D(S, d, f) {d IR(S, d) f(s, d )= f(s, d)}, is the set of all points d in S (weakly) dominating d, such that if we replace d with d as an interim agreement and keep the utility feasibility set S unchanged, we still reach the same solution outcome. D(S, d, f) will be a key element in our analysis. Next, we list some basic axioms in the cooperative bargaining literature. The first three axioms are Nash s original axioms and are well-known in the literature. Weak Pareto Optimality (WPO): For all (S, d) Σ, f(s, d) W P O(S). Symmetry (SYM): For all (S, d) Σ and for all permutations π, if d = π(d) and S = π(s), then f 1 (S, d) =... = f n (S, d). Scale Invariance (SI): For all (S, d) Σ, G : R n R n is a positive affi ne transformation if G(x 1,..., x n ) = (a 1 x 1 + c 1, a 2 x 2 + c 2,..., a n x n + c n ) for some a R n ++ and c R n. We require that for such a transformation G, f(g(s), G(d)) = G(f(S, d)). Translation Invariance (TI): For all (S, d) Σ, f(s + t, d + t) = f(s, d) + t for all t R n. The following axiom requires that the solution outcome is not outside of the individually rational set of (S, d). Individual Rationality (IR): For all (S, d) Σ, f(s, d) d. The next axiom is stronger than IR; it requires that both parties should obtain a payoff higher than their disagreement payoff. Strong Individual Rationality (SIR): For all (S, d) Σ, f(s, d) > d. The following axiom states that if two problems have the same individually rational set, then they should have the same solution outcome. Independence of Non-individually Rational Alternatives (INIR): For all (S, d) Σ, f(s, d) = f(ir(s, d), d). The next two axioms are innocuous technical conditions that almost every known solution concept satisfies. Disagreement Point Continuity (DCONT): For all (S, d) Σ and every 5

7 sequence d 1, d 2,... in S, if lim k d k = d S, then lim k f(s, d k ) = f(s, d). Continuity (CONT): For all (S, d) Σ and every sequence {(S k, d k )}, if (S k, d k ) converges to (S, d) in the Hausdorff topology, then lim k f(s k, d k ) = f(s, d). The following axiom requires that the agreement Pareto dominates the outcome of the random dictatorship. Midpoint Domination (MD): For any (S, d) Σ, f(s, d) m(s, d). Note that the relationship between MD and the next axiom is like the relationship between the IR and INIR axioms. IR and INIR are based on d, while MD and INMD are based on m. Independence of Non-Midpoint-Dominating Alternatives (INMD): For all (S, d), (T, d) Σ, if IR(S, m(s, d)) = IR(T, m(t, d)), then f(s, d) = f(t, d). If the hypothesis IR(S, m(s, d)) = IR(T, m(t, d)) holds, then m(s, d) = m(t, d) and b(s, d) = b(t, d). INMD states that parties should focus on the alternatives dominating the midpoint. Here we will consider three prominent solution concepts: The Nash solution N: For each (S, d) Σ, N(S, d) = arg max{π n i=1(x i d i ) x IR(S, d)}. The Discrete Raiffa solution DR: For each (S, d) Σ, consider a nondecreasing sequence {m t } S with m 0 = m(s, d) and m t = m(s, m t 1 ); then DR(S, d) = lim t m t. The Proportional solutions P q : For each (S, d) Σ, there is a positive vector q such that f(s, d) = d + λ(s, d)q, where q = (q 1, q 2,..., q n ) and λ(s, d) = max{t tq S d}. 4 The Robustness of Intermediate Agreements (RIA) Axioms Consider the following axiom of Kalai (1977): Step-by-step Negotiations (SSN): For all (S, d), (T, d) Σ, such that S T and (T f(s, d), 0) Σ, f(t, d) = f(s, d) + f(t f(s, d), 0). SSN is a strong requirement and Kalai (1977) demonstrated that, when combined with WPO and SIR, it is suffi cient to uniquely characterize the Proportional solutions. 6

8 To link our axioms to SSN conceptually, we restate SSN as follows: Given two problems (S, d) and (T, d) Σ, whenever S T, f(s, d) d and (T, f(s, d)) Σ, then D(T, d, f) {f(s, d)}. 1 For a given problem (T, d), SSN requires that D(T, d, f) is not only non-empty, but also contains f(s, d) for all problems (S, d) Σ, with S T and (T, f(s, d)) Σ. In other words, f(s, d) can serve as an intermediate agreement in reaching f(t, d). Next, we propose three fairly intuitive axioms that are closely related to SSN. As SSN, all axioms pertain to the situation where parties first face (S, d) and then (T, d) such that S T. They are in the same spirit as SSN: step-by-step improvements - in Kalai (1977) s words - are "attractive because we can view every bargaining situation that we encounter in life as a first step in a sequence of predictable or unpredictable bargaining situations that may still arise." 1. In our first axiom, as in SSN, the bargaining sets S and T are nested. Robustness of Intermediate Agreements with Inclusion (RIA-Inclusion): For all (S, d), (T, d) Σ such that S T, D(S, d, f) D(T, d, f). The main difference between SSN and the RIA-Inclusion is that RIA-Inclusion does not require the solution outcome, f(s, d), of the smaller set S to be an intermediate agreement; f(s, d) may, however, be one of the possible intermediate agreements. RIA-Inclusion only states that in this case the parties will be willing to reach any point d D(S, d, f) as an intermediate agreement instead of sticking to the status quo d. RIA-Inclusion is weaker than SSN. To see that pick any d D(S, d, f) (if it exists). SSN implies that f(t, d) = f(t, f(s, d)) = f(t, f(s, d)) = f(t, d). 2. Now, consider parties facing the bargaining problems (S, d) and (T, d) with S T such that (S, d) and (T, d) have the same ideal point. Robustness of Intermediate Agreements in the (d, b)-box (RIA-Box): For all (S, d), (T, d) Σ such that S T and b(s, d) = b(t, d), (D(S, d, f) {f(s, d)}) (D(T, d, f) {f(t, d)})\{d}. RIA-Box requires that each pair of nested problems with the same ideal point have a non-empty intersection of their sets of disagreement points union the final 1 Kalai (1977) defined SSN and characterized the Proportional solutions in the class of 0- normalized problems. SSN and our restatement of it are equivalent in the more general domain if Translation Invariance (TI) is also satisfied, since SSN and TI imply that f(t, d) = f(s, d)+ f(t f(s, d), 0) = f(t, f(s, d)). Hence D(T, d, f) {f(s, d)}. 7

9 outcomes. Here too, our axiom does not require the solution outcome, f(s, d), of the smaller set S to be an intermediate agreement; f(s, d) may be one of the intermediate agreements. Note that RIA-Box is satisfied by the Proportional solutions (including both the Egalitarian and Dictatorial solutions), as well as by the Discrete Raiffa solution. 3. Our third robustness criterion concerns the concept of bargaining power. If both parties have the same relative bargaining power in both (S, d) and (T, d) with S T, then we should expect them to reach an intermediate agreement easily. It may not be clear ex-ante what kinds of economic and non-economic factors will determine a party s relative bargaining. Nevertheless, it should be clear ex-post that one party s gain in a negotiation relative to the other s must increase monotonically with respect to that party s relative bargaining power. Let (S, d) Σ and suppose that the solution outcome is such that f(s, d) > d. Then f(s,d) d measures the normalized relative gains in bargaining. f(s,d) d Robustness of Intermediate Agreements with Identical Relative Gains (RIA-Gains): For all (S, d), (T, d) Σ such that S T, if (i) f(s, d) IR(S, d)\{d} f(s,d) d and f(t, d) IR(T, d)\{d} and (ii) = f(t,d) d, then D(S, d, f) f(s,d) d f(t,d) d D(T, d, f)\{d}. Beginning with the same disagreement point d, if parties perceive (correctly) that they will receive the same relative gains in two problems (S, d) and (T, d) with S T, then there exists at least one allocation d that is agreeable to both parties as an intermediate agreement. Any such d therefore needs to be a common intermediate agreement in D(S, d, f) D(T, d, f). In RIA-Gains too, the axiom does not require the solution outcome, f(s, d), of the smaller set S to necessarily be an intermediate agreement; f(s, d) may be one of the intermediate agreements. It is easy to see that RIA-Gains is satisfied by the Proportional solutions, as well as the Nash solution. 5 Characterizations of the Nash, Proportional and Discrete Raiffa Solutions 5.1 The Nash Solution Recall that the RIA-Gains axiom is satisfied by all Proportional solutions and by the Nash solution. It is also known that MD is satisfied by the Nash, Kalai/Smorodinsky, 8

10 Discrete Raiffa, Equal Area, and Average Payoff solutions, while CONT, which is even more innocuous, is satisfied by almost all known solutions. RIA-Gains is closely related to the axiom of Disagreement Point Convexity introduced by Peters and Van Damme (1991): Disagreement Point Convexity (d-conv): f(s, αd + (1 α)f(s, d)) = f(s, d) for all α (0, 1). Let l[x, y] be the line segment connecting x and y. d-conv requires that D(S, d, f) l[d, f(s, d)). If the premises of RIA-Gains hold, then d-conv implies that D(S, d, f) D(T, d, f) l[d, min{f(s, d), f(t, d)}). d-conv therefore implies RIA-Gains. However, the following example shows that RIA-Gains does not imply d-conv. Consider the two-person ɛ-egalitarian solution, E ɛ, defined as follows: (1) if E 1 (S, d) d 1 = E 2 (S, d) d 2 ɛ, then E ɛ (S, d) = E(S, d), where ɛ > 0 and E(S, d) stands for the Egalitarian solution; (2) if E 1 (S, d) d 1 = E 2 (S, d) d 2 < ɛ, then E ɛ (S, d) = d. E ɛ satisfies RIA-Gains but violates d-conv. Nevertheless, RIA-Gains is simply a technical but not an intuitive improvement over d-conv, as d-conv and RIA-Gains take different intuitive perspectives. RIA-Gains is also closely related to the axiom of Weak Decomposability introduced by Chun (1988): Weak Decomposability (W.DEC): For all (S, d), (T, e) Σ such that S T, if (i) e = f(s, d) and (ii) f(s,d) d = f(t,e) e, then f(t, d) = f(t, e). f(s,d) d f(t,e) e W.DEC, while intuitively appealing, is not suitable to generate a family of axioms to characterize other solutions. The reason is that, as SSN does, W.DEC requires the solution outcome of the smaller set S, f(s, d), to be an intermediate agreement for the larger set T. Our RIA axioms are designed to require the possible existence of an intermediate agreement without specifying it. 2 The proofs of our results and independence of axioms in our characterizations are provided in the Appendix. Proposition 1 N is the unique solution satisfying CONT, MD and RIA-Gains. Thus, given CONT and MD, if parties, whenever they face first (S, d) and then (T, d) with S T, are willing to reach intermediate agreements provided that they receive the same relative utility gains over these two possible problems, then the 2 Moreover, we can easily weaken RIA-Gains by requiring a non-empty intersection of disagreement point sets union the final outcomes ((D(S, d, f) {f(s, d)}) (D(T, d, f) {f(t, d)})\{d} ). W.DEC is stronger than this revised version of RIA-Gains. We can characterize the Nash solution by replacing RIA-Gains with this revised version in Proposition 1. 9

11 solution must be N. (Note that CONT can be replaced by DCONT for two-person bargaining problems.) 5.2 Proportional Solutions TI, SIR and WPO are satisfied by all known solution concepts. As was mentioned earlier, RIA-Inclusion is weaker than SSN. By imposing CONT, TI, SIR, WPO and RIA-Inclusion together, we obtain the following result: Proposition 2 P q is the only class of solutions satisfying CONT, TI, SIR, WPO and RIA-Inclusion. Given CONT, TI, SIR and WPO, if parties, whenever they face first (S, d) and then (T, d) such that S T, are willing to reach intermediate agreements, then the solution must be P q. While it seems unessential at first sight, the distinction between RIA-Inclusion and SSN is quite subtle. SSN explicitly requires the bargaining outcomes of smaller sets to be the intermediate agreements of a larger set. RIA-Inclusion alone, on the other hand, does not guarantee the existence of intermediate agreements (D(S, d, f)\{d} could be empty). Indeed, a great portion of the proof of Proposition 2 is devoted to establish the existence of intermediate agreements using an advanced technique in Homotopy Theory. Having said that, Proposition 2 is not a major improvement over Kalai s (1977) characterization; however, that proposition is useful for us in developing comparative characterizations of the Nash, Proportional and Discrete Raiffa solutions in terms of RIA axioms. 5.3 The Discrete Raiffa Solution As was mentioned earlier, RIA-Box is satisfied by all Proportional solutions, as well as by the Discrete Raiffa solution. MD is satisfied by a significant number of solutions, as was mentioned above. INMD is satisfied by the Nash, Kalai/Smorodinsky, Discrete Raiffa and Dictatorial solutions. Proposition 3 DR is the unique solution satisfying INMD, MD and RIA-Box. Thus, given INMD and MD, if parties, whenever they face first (S, d) and then (T, d) such that S T with the same ideal point, are willing to reach intermediate agreements, then the bargaining outcome must be DR. 10

12 6 Conclusion Our RIA axioms are closely related to the SSN axiom of Kalai (1977). In the robustness tests that we consider as alternatives to SSN, we consider additional facts about the relationship between S and T. As in SSN, these tests pertain to a situation where parties face first (S, d) and then (T, e) with S T and d = e. Each test requires the parties to be able to reach an intermediate agreement when S and T conform to a specific relationship. Below is the summary of our results: N RIA-Gains MD CONT P q RIA-Inclusion CONT TI+WPO+SIR DR RIA-Box MD INMD That is, in addition to MD and RIA-Gains, the Nash solution s characterization likewise involves only one more axiom, CONT. Beside RIA-Inclusion, the Proportional solutions characterizations use WPO, CONT, TI and SIR. Beside MD and RIA-Box, the Discrete Raiffa solution s axiomatic characterization uses only one more axiom, INMD. Although there have been earlier non-unified attempts to bring the bargaining process into Nash s bargaining problem (via the SSN axiom of Kalai, 1977, and the MD axiom of Sobel, 1981), previous characterizations of solutions have typically relied on crucial axioms entailing changes in the utility possibility set and in the disagreement point, and have not described any bargaining process. In this paper, we highlight, in a unified way, the crucial role played by the robustness of the intermediate agreements (and the resulting bargaining process) which we propose here. By describing plausible circumstances under which such robust intermediate agreements can be obtained, our Robustness of Intermediate Agreements (RIA) axioms portray a bargaining process for each bargaining solution and thereby help compare these solutions bargaining theories. 7 Appendix 7.1 Proofs Proof of Proposition 1. It is obvious that N satisfies these three axioms. We 11

13 will show that if f satisfies these three axioms, then f = N. 3 First we give some definitions. The equation of the hyperplane H(a, x 0 ) passing through the point x 0 with normal vector a > 0 is given by a (x x 0 ) = 0. Given y with a (y x 0 ) < 0, define H(a, x 0, y) = {x R n x y, a (x x 0 ) 0}. Given a = (a 1,..., a n ) > 0, define 1 = ( 1 a a 1, 1 a 2,..., 1 a n ). We state a simple lemma without proof: Lemma 1 Given a > 0 and c > 0, a c min n i a i c i iff c = r 1 a positive number r. for some strictly (i) First we show that there exists a strictly increasing sequence {a m } l(d, f(s, d)) such that a m D(S, d, f) for each m. Let (S, d) Σ. By MD, f(s, d) > 1 d. Consider first the problem (T, d), where T = H(, (1 1 )d + 1 f(s, d), d). f(s,d) d n n T S. By MD, (i) f(t, d) = (1 1 )d + 1 f(s, d), which in turn implies that n n f(t,d) d = f(s,d) d, and (ii) D(T, d, f) = l[d, (1 1 )d + 1 f(s, d)). By RIA- f(t,d) d f(s,d) d n n Gains, at least one point a 1 l(d, (1 1 )d + 1 f(s, d)) is in the inverse set of f n n with respect to (S, d). Starting with a 1 as a new disagreement point and repeating the same argument gives us a strictly increasing sequence {a m } l(d, (1 1 )d + 1 f(s, d)) such that n n am D(S, d, f) for each m. By RIA-Gains and CONT, lim m a m = (1 1 )d + 1 f(s, d). By CONT, (1 1 )d + 1 f(s, d) D(S, d, f). n n n n Taking (1 1 )d + 1 f(s, d) as a new disagreement point and iteratively invoking n n the same argument as above completes the proof. (ii) By (i) and MD, it can be readily seen that f(s, d) P O(S) for each (S, d) Σ. (iii) Every (S, d) Σ can be approximated by a sequence of smooth, strictly comprehensive problems. Then, by CONT, it is suffi cient to show f = N for every smooth, strictly comprehensive problem. Let (S, d) be such a problem. By (ii), f(s, d) P O(S). Denote by c > 0 the normal vector of the supporting hyperplane of S at f(s, d). By (i), there exists a strictly increasing sequence {a k } l(d, f(s, d)) such that a k D(S, d, f) for each k. Each a k can be uniquely represented by α k d + (1 α k )f(s, d) for some α k (0, 1) with lim k α k = 0. By MD, we have 3 To highlight the role of RIA-Gains serving as an elaboration of a bargaining process, following a referee s suggestion, we give a direct proof here. Alternatively, one may give an indirect proof by replacing step (iii) with de Clippel (2007) s result which invokes Harsanyi s characterization of the Nash solution. 12

14 for each i = 1,..., n and each k, Accordingly But 1 b i (S, a k ) a k i lim k n f i (S, d) a k i 1 n = lim αk 0 = 1 n j i f i (S, d) 1 n b i(s, a k ) + (1 1 n )ak i 1 1 b i (S, a k ) a k i. n f i (S, d) a k i 1 b i (S, a k ) a k i lim k n f i (S, d) a k i 1. b i (S, α k d + (1 α k )f(s, d)) [α k d i + (1 α k )f i (S, d)] f i (S, d) [α k d i + (1 α k )f i (S, d)] b i (S,f(S,d)) x j (f j (S, d) d j ) + f i (S, d) d i. f i (S, d) d i Note that when α k 0, b i (S, a k ) b i (S, f(s, d)) and satisfies the equation j i c j(x j f j (S, d)) + c i (b i (S, f(s, d)) f i (S, d)) = 0. Hence b i(s,f(s,d)) x j = c j c i. Then = 1 n lim k 1 b i (S, a k ) a k i n f i (S, d) a k i c j c i (f j (S, d) d j ) + f i (S, d) d i. f i (S, d) d i j i 1 b Therefore lim i (S,a k ) a k i k n f i 1 requires that (S,d) a k i 1 c j (f j (S, d) d j ) c i (f i (S, d) d i ) i. n j r By Lemma 1, for the inequality to hold, we have c = for some r > 0. Let f(s,d) d r T = H(c, f(s, d), d) = H(, f(s, d), d). By MD, f(t, d) = f(s, d) = N(T, d). f(s,d) d As T IR(S, d) and N(T, d) = f(s, d) S, N(S, d) = N(T, d) = f(s, d). Proof of Proposition 2. It can easily be seen that these five axioms are satisfied by P q. To show that P q is the unique class of solutions satisfying TI, CONT, SIR, WPO and RIA-Inclusion, we first show that CONT, SIR, WPO and RIA- Inclusion imply SSN, and the proof is complete by invoking Kalai (1977) s classical 13

15 characterization. 4 Since every (S, d) Σ can be approximated by a sequence of strictly comprehensive problems, by CONT it is suffi cient to restrict ourselves to the class of strictly comprehensive problems. We first make the following claim: Claim 1 Let f satisfy CONT, SIR, and WPO. For every strictly comprehensive problem (S, d), there exists a sequence {a m } with a m D(S, d, f) such that lim m a m = f(s, d). According to Claim 1, we can find a sequence {a k } with a k D(S, d, f) such that lim k a k = f(s, d). Then, by RIA-Inclusion, a k D(T, d, f) for all k; CONT completes the proof. Proof of Claim 1. Let f be a solution that satisfies CONT, SIR, and WPO. Assume that for every strictly comprehensive problem (S, d) there is an a > d with a D(S, d, f). It is suffi cient to prove that for each ɛ > 0 there is an a > d such that a D(S, d, f) and a f(s, d) < ɛ. Suppose this were not true, then e := inf{ɛ > 0 there is an a > d with a D(S, d, f) and a f(s, d) < ɛ} > 0. Take a decreasing sequence {ɛ k } converging to e and for each k an a k > d such that a k D(S, d, f) and a k f(s, d) < ɛ k. By compactness we may assume that a k converges to some ã. By CONT, f(s, ã) = f(s, d). Now take a > ã with f(s, a ) = f(s, ã) = f(s, d). Since ã f(s, d) = e we have a f(s, d) < e (by SIR) but this contradicts the definition of e. Hence, it is suffi cient to establish the existence of the first disagreement point a > d with a D(S, d, f). Depending on the number of players, we divide the proof into three subcases: n = 2, n = 3, and n 4. For the subcase n = 2, we use a simple topological property, connectedness, to show that there exists some a > d with a D(S, d, f). Connectedness is not suffi cient to prove the existence of such an a when n 3. We need to invoke another topological property, the fundamental group, to establish the existence of such an a when n = 3. The fundamental group approach, however, is not suffi cient to show the existence of such an a when n 4. We appeal to another topological property, homotopy groups, to complete the proof when n 4. n = 2. Consider f(s, ) on the restricted domain SIR(S, d) {d}\w P O(S). By CONT, SIR and WPO, f is a continuous mapping from SIR(S, d) {d}\w P O(S) 4 Kalai (1977) showed that P q is the unique solution that satisfies WPO, SIR, and SSN for a class of 0-normalized bargaining problems. When extended to a more general domain as we do here, TI is required in the characterization. Also, Roth (1979) showed that WPO is redundant in Kalai s characterization. We do still need WPO in our characterization, as without WPO RIA-Inclusion does not imply SSN. 14

16 to SIR(S, d) W P O(S). Suppose to the contrary that there exists no a > d such that f(s, a) = f(s, d). Then removing {d} from the domain and {f(s, d)} from the range, f is still a continuous mapping from SIR(S, d)\w P O(S) to SIR(S, d) W P O(S)\{f(S, d)}, a disconnected one-dimensional manifold. However, SIR(S, d)\w P O(S) is connected and f is continuous implies that SIR(S, d) W P O(S)\{f(S, d)} is connected, a contradiction. Therefore, there exists a > d such that f(s, a) = f(s, d). n = 3. Consider f(s, ) on the restricted domain SIR(S, d) {d}\w P O(S). Define f(s, ) to be an extension of f(s, ) from SIR(S, d) {d}\w P O(S) to SIR(S, d) {d} with f(s, x) = f(s, x) for each x SIR(S, d) {d}\w P O(S), and f(s, x) = x for each x SIR(S, d) W P O(S). By CONT, SIR and WPO, f is a retraction of SIR(S, d) {d} onto SIR(S, d) W P O(S). Suppose to the contrary that there exists no a > d such that f(s, a) = f(s, d). Then, for the extended mapping f, the preimage of {f(s, d)} is {f(s, d), d}. By removing {f(s, d), d} from the domain, f is still a retraction of SIR(S, d)\{f(s, d)} onto SIR(S, d) W P O(S)\{f(S, d)} (and SIR(S, d) W P O(S)\{f(S, d)} is a retract of SIR(S, d)\{f(s, d)}). Recall the following standard definitions in Topology: A loop based at x 0 in a space X is a continuous map g : I X such that g(0) = g(1) = x 0, where I = [0, 1]. Two loops g and h based at x 0 are path homotopic if there is a continuous map (path homotopy) F : I 2 X such that F (s, 0) = g(s), F (s, 1) = h(s) and F (0, t) = F (1, t) = x 0. Given two loops g and h based at x 0, the path product g h : I X of g and h is a loop based at x 0 such that g h(s) = g(2s) for s [0, 1/2], and g h(s) = h(2s 1) for s [1/2, 1]. Let X be a space and x 0 a point of X. The set of path homotopy classes of loops based at x 0, with the operation, is the fundamental group of X relative to the based point x 0. It is denoted by π 1 (X, x 0 ), and we use the notation [g] to denote the path-homotopy equivalence class of g in π 1 (X, x 0 ). Let κ : (X, x 0 ) (Y, y 0 ) be a continuous map with κ(x 0 ) = y 0. Define κ : π 1 (X, x 0 ) π 1 (Y, y 0 ) by the equation κ ([g]) = [κ g]. The map κ is called the homomorphism induced by κ. We have the following lemma (p. 348, Munkres, 2000): Lemma 2 If A is a retract of X, then the homomorphism of fundamental groups induced by inclusion j : A X is injective. 15

17 By Lemma 2, the homomorphism of fundamental groups induced by inclusion j : SIR(S, d) W P O(S)\{f(S, d)} SIR(S, d)\{f(s, d)} is injective. Note that SIR(S, d)\{f(s, d)} is star convex (but not necessary convex), 5 and hence is simply connected, i.e., it is path-connected and π 1 (SIR(S, d)\{f(s, d)}, x 0 ) is the trivial (one-element) group for some x 0 SIR(S, d)\{f(s, d)}. On the other hand, SIR(S, d) W P O(S)\{f(S, d)} is homeomorphic to the punctured open unit disc, which has a non-trivial fundamental group (π 1 (SIR(S, d) W P O(S)\{f(S, d)}, x 0 ) = Z, i.e., π 1 (SIR(S, d) W P O(S)\{f(S, d)}, x 0 ) is isomorphic to the additive group of integers). Therefore there exists no injective homomorphism from π 1 (SIR(S, d) W P O(S)\{f(S, d)}, x 0 ) to π 1 (SIR(S, d)\{f(s, d)}, x 0 ). This contradicts Lemma 2; then the preimage of {f(s, d)} under f cannot be just {f(s, d), d}. Hence there exists a > d such that f(s, a) = f(s, d). n 4. Again, following the construction in n = 3 case, let f be a retraction of SIR(S, d) {d} onto SIR(S, d) W P O(S). Suppose to the contrary that there exists no a > d such that f(s, a) = f(s, d). Then, by removing {f(s, d), d} from the domain, f is still a retraction of SIR(S, d)\{f(s, d)} onto SIR(S, d) W P O(S)\{f(S, d)}. Next we will define homotopy groups. But before doing so, we need to denote the k-dimensional unit cube by I k. The boundary of I k, I k, is the subspace consisting of points with at least one coordinate equal to 0 or 1. For a space X with basepoint x 0 X, define the k-th homotopy group π k (X, x 0 ) to be the set of homotopy classes of maps g : (I k, I k ) (X, x 0 ), where the homotopy F : I k+1 X is required to satisfy F (t, I k ) = x 0 for all t. The induced homomorphism for homotopy groups can be defined analogously. The following straightforward extension of Lemma 2 is observed: Lemma 3 If A is a retract of X, then the homomorphism of homotopy groups induced by inclusion j : A X is injective. We also need the following theorem to complete the proof (p. 366, Hatcher, 2001): Lemma 4 (Hurewicz theorem) For a simply-connected space X, the first nonzero homotopy group π k (X, x 0 ), with k > 0, is isomorphic to the first nonzero homology group H k (X, x 0 ). 5 X R n is star convex if there exists x X such that the line segment from x to any point in X is contained in X. 16

18 By Lemma 3, the homomorphism of homotopy groups induced by inclusion j : SIR(S, d) W P O(S)\{f(S, d)} SIR(S, d)\{f(s, d)} is injective. Let n 4 be the dimension of the bargaining problems. Note that SIR(S, d)\{f(s, d)} is path-connected and π k (SIR(S, d)\{f(s, d)}, x 0 ) is trivial for each k. On the other hand, SIR(S, d) W P O(S)\{f(S, d)} is homeomorphic to the punctured (n 1)-dimensional open unit ball, B n 1 {0}, where B n 1 = {x R n 1, x < 1}. As B n 1 {0} deformation-retracts to the (n 2)-dimensional sphere S n 2, π n 2 (SIR(S, d) W P O(S)\{f(S, d)}, x 0 ) = π n 2 (S n 2 ) = Z. The last equality comes from Lemma 4 and the fact that the first non-zero homology group of S n 2 is H n 2 (S n 2, x 0 ) = Z. Therefore there exists no injective homomorphism from π n 2 (SIR(S, d) W P O(S)\{f(S, d)}, x 0 ) to π n 2 (SIR(S, d)\{f(s, d)}, x 0 ). This contradicts Lemma 3; then the preimage of {f(s, d)} under f cannot be just {f(s, d), d}. Hence there exists a > d such that f(s, a) = f(s, d). Proof of Proposition 3. It is obvious that DR satisfies these three axioms. Suppose that f satisfies INMD, MD and RIA-Box. We show that f = DR. Let (S, d) Σ. By the convexity of S, m(s, d) S. If m(s, d) P O(S), then by MD f(s, d) = m(s, d) = DR(S, d). Now suppose that m(s, d) / P O(S). By the convexity of S again, m(s, d) / W P O(S), and hence, (S, m(s, d)) Σ. To show that f(s, d) = DR(S, d) in this case, it is suffi cient to show that f(s, d) = f(s, m(s, d)). Consider the problem (T, d), where T is the convex hull of {d, (b 1 (S, d), d 2,..., d n ), (d 1, b 2 (S, d), d 3,..., d n ),..., (d 1,..., d n 1, b n (S, d))} S. By MD, (i) f(t, d) = m(s, d), and (ii) D(T, d, f) = l[d, m(s, d)). By RIA-Box, there exists a common intermediate agreement a l[d, m(s, d)) {m(s, d)} such that f(s, d) = f(s, a). INMD excludes all points below m(s, d) from being a common intermediate agreement. Hence, a = m(s, d). 7.2 Independence of Axioms In the following, we provide the independence of axioms for each solution: The Nash bargaining solution N RIA-Gains The Discrete Raiffa solution DR MD The Proportional solutions P q CONT Consider the following (slightly revised) example by de Clippel (2007). Denote a bargaining set T = conv{(0, 0, 0), (15, 0, 0), (0, 15, 0), (10, 10, 0), (0, 0, 10)}. 17

19 Let f(s, d) = (5, 5, 5) if (S, d) = (T, λ(0, 0, 0) + (1 λ)(5, 5, 5)) for some λ [0, 1] and f(s, d) = N(S, d) otherwise. The Proportional solutions P q RIA-Inclusion The Nash bargaining solution N CONT While it can be readily seen from our characterization (proof) of the Proportional solutions that CONT is indispensable, the interplay between TI and RIA-Inclusion makes it diffi cult to construct an example that satisfies the other four axioms but violates CONT. We leave constructing such an example an open question. TI Consider two-person problems. Define h(x) = x+sin x/2 for every x R. h(x) is strictly increasing in x. The family of functions {h(x) + k k R} covers R 2. For any (x 1, x 2 ) R 2, there exists a unique k(x 1, x 2 ) such that x 2 = h(x 1 ) + k(x 1, x 2 ). For each two-person problem (S, d), let f(s, d) = max{(x 1, x 2 ) S x 2 = h(x 1 )+k(d 1, d 2 )}, the maximum element on the curve h(x)+k passing through d in set S. D(S, d, f) is the points on h(x 1 )+k(d 1, d 2 ) with d 1 x 1 < f 1 (S, d). Therefore D(S, d, f) D(T, d, f) whenever S T, and RIA-Inclusion is satisfied. It can be easily seen that CONT, WPO and SIR are satisfied as well. f(s, d) violates TI, as f(s + t, d + t) f(s, d) + t for horizontal shifts t. WPO Let f(s, d) = 1E(S, d) + 1 d, where E(S, d) stands for the Egalitarian 2 2 solution. Note that D(S, d, f)\{d} is always empty and RIA-Inclusion is trivially satisfied. SIR The Dictatorial solutions The Discrete Raiffa solution DR RIA-Box The Nash bargaining solution N MD The Dictatorial solutions INMD For each (S, d) Σ, consider a non-decreasing sequence {m t } S with m 0 = 1 2 m(s, d) and m t = m(s, m t 1 ). Let f(s, d) = lim t m t. Acknowledgements We would like to thank Mei-Fang Chiang, Hulya Eraslan, Ehud Kalai, Cem Karayalcin, Ozgur Kibris, Ehud Lehrer, Devashish Mitra, Herve Moulin, Shiran 18

20 Rachmilevitch, Al Roth, Randy Silvers and Peter Thompson, as well as conference and seminar participants at the 2009 International Conference on Game Theory (Stonybrook), the 2009 Australasian Economic Theory Workshop (Auckland), and University of Adelaide, University of Melbourne, Northwestern University and Ohio State University for their valuable comments and suggestions. In particular, we are grateful to an anonymous Associate Editor and two anonymous referees, as well as Faruk Gul and William Thomson for giving us many invaluable detailed comments and suggestions; in addition, William Thomson gave us the initial impetus for starting this project. References [1] Chun, Y., Nash Solution and Timing of Bargaining. Econ. Lett. 28, [2] Chun, Y., Minimal Cooperation in Bargaining. Econ Lett. 34, [3] Chun, Y., Thomson, W., Bargaining with Uncertain Disagreement Points. Econometrica 58, [4] de Clippel, G., An Axiomatization of the Nash Bargaining Solution. Soc. Choice Welf. 29, [5] Dishkin, A., Koppel, M., Samet, D., Generalized Raiffa Solutions. Games Econ. Behav. 73, [6] Edgeworth, F. Y., Mathematical Psychics. Kegan Paul Publishers, London. Reprinted in 2003, P. Newman (ed.) F. Y. Edgeworth s Mathematical Psychics and Further Papers on Political Economy. Oxford University Press. [7] Hatcher, A., Algebraic Topology, Cambridge University Press, New York, NY. [8] Kalai, E., Proportional Solutions to Bargaining Situations: Interpersonal Utility Comparisons. Econometrica 45, [9] Kalai, E., and Smorodinsky, M., Other Solutions to Nash s Bargaining Problem. Econometrica, 43,

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