Nash Bargaining for Log-Convex Problems

Transcription

1 Nash Bargaining for Log-Convex Problems Cheng-Zhong Qin Shuzhong Shi Guofu Tan February 6, 202 Abstract We introduce log-convexity for bargaining problems. With the requirement of some basic regularity conditions, log-convexity is shown to be necessary and sufficient for Nash s axioms to determine a unique single-valued bargaining solution up to choices of bargaining powers. Specifically, we show that the single-valued asymmetric Nash solution is the unique solution under Nash s axioms on the class of all regular and log-convex bargaining problems, but this is not true on any class of bargaining problems containing the class of all regular and log-convex bargaining problems as a proper subclass. We apply our results to a class of duopoly bargaining problems which are long-convex but not convex. We compare the Nash bargaining solution for log-convex bargaining problems with some of its extensions in the literature. KEYWORDS: Bargaining problem, non-convexity, log-convexity, Nash bargaining solution, Nash product. Corresponding author, Department of Economics, University of California, Santa Barbara, CA qin@econ.ucsb.edu. Tel: ; fax: Guanghua School of Management, Beijing University. Department of Economics, University of Southern California, Los Angeles, CA guofutan@usc.edu.

2 Introduction The bargaining theory introduced in the seminal papers of Nash (950, 953) postulates that a group of players chooses a payoff allocation from a set of feasible payoff allocations. The implementation of a payoff allocation requires unanimous agreement among the players. In case of disagreement, the players end up getting some prespecified payoff allocation known as the status quo or the threat point. A bargaining problem (henceforth, a problem in short) in the sense of Nash is thus represented by a pair consisting of a choice set in the payoff space and a threat point. A solution on a class of problems is a rule assigning a feasible payoff allocation to each problem in the class. Nash adopted an axiomatic approach to resolve bargaining: One states as axioms several properties that it would seem natural for the solution to have and then one discovers that the axioms actually determine the solution uniquely (Nash 953, p. 29). He postulated a set of axioms that are deemed to be natural for a solution to satisfy. These axioms are Pareto Optimality (PO), Symmetry (SYM), Invariance to Equivalent Utility Representations (INV), and Independence of Irrelevant Alternatives (IIA). It is remarkable that these axioms determine a unique rule, to be called the Nash rule, on the class of all compact and convex problems. Furthermore, the Nash rule assigns to each problem the unique maximizer of the (symmetric) Nash product over the choice set. Nash bargaining model has become one of the most fruitful paradigms in game theory. When SYM is removed, Kalai (977) showed that an asymmetric Nash rule unique up to (henceforth unique) specifications of payoff allocations for a normalized problem is characterized by the remaining axioms on the class of compact convex problems. A payoff allocation for the normalized problem has been customarily interpreted as representing players bargaining powers. 2 Furthermore, given players bargaining powers, the asymmetric Nash rule assigns to each problem the unique maximizer of the Nash product weighted by the bargaining powers. The Nash product weighted by players Thomson (2009) provides an excellent survey of the literature on the axiomatic approach to bargaining as pioneered by Nash. An alternative approach is to model bargaining explicitly as a noncooperative game. Nash (953, p. 29) argued that the two approaches are complementary; each helps to justify and clarify the other. See Binmore, Rubinstein and Wolinsky (986) for a formal establishment of the relationship between the Nash axiomatic and the sequential non-cooperative approaches to bargaining. 2 This interpretation can be traced to Shubik (959, p. 50). 2

3 bargaining powers is also known as the generalized Nash product. 3 These properties consist of the standard unique axiomatic characterization of the asymmetric Nash rule on the class of compact convex problems. For applications, the uniqueness of the solution up to specifications of players bargaining powers is useful, since there is no a priori reason why players bargaining powers are not case-specific. In addition, the choice set is not directly given, but often derived from more primitive information. As a result, the choice set needs not to be convex. Indeed, the non-convexity of the choice set of feasible profit shares arising from duopolies with asymmetric constant marginal costs and concave profit functions has been well recognized in the industrial organization literature. 4 The non-convexity is usually removed by allowing for randomized payoff allocations and by assuming that players are expected utility maximizers. Randomized allocations can cause conflicts of interest at the post-realization stage and may not be realistic in some applications. It is therefore desirable to analyze the extent to which the Nash bargaining theory can be extended to allow for non-convexity. Our purposes in this paper are threefold. First, we characterize the complete class of regular problems allowing for non-convexity, on which the axioms of IIA, INV, and SIR uniquely determine the asymmetric Nash rule as in the case with convex and compact problems. Second, we apply our results to analyze non-convex problems arising from duopoly models with asymmetric constant marginal costs. approach with two extensions of Nash bargaining theory in the literature. Third, we compare our A notion of log-convexity plays a key role in characterizing problems in our class. With arbitrary bargaining powers, our first main result shows that the maximizer of the generalized Nash product over the choice set of a regular problem is unique if and only if the problem is log-convex (Theorem ). Regularity of a problem in this paper means that the threat point is strictly dominated and the choice set is closed, comprehensive relative to the threat point, and bounded above. Log-convexity, on the other hand, requires that the strict Pareto frontier of the set of the natural logtransformed allocations of positive payoff gains be strictly concave. The regularity requirement is mild while the log-convexity of a problem is strictly weaker than the convexity of the choice set. Indeed, we show that duopoly problems with asymmetric 3 By the Lemma in Roth (977, p. 65), a bargaining solution is Pareto optimal whenever the solution satisfies IIA, INV, and strict individual rationality (SIR). Thus, IIA, INV, and PO can be replaced by IIA, INV, and SIR. Since bargaining problems in our class satisfy all the assumptions in Roth (977) except for the convexity, we will consider IIA, INV, and SIR as the Nash s axioms for the asymmetric Nash bargaining solution. 4 See, for example, Bishop (960), Schmalensee (987), and Tirole (988, p. 242, 27). 3

4 constant marginal costs and concave monopoly profit function for the efficient firm are log-convex but not convex. With convexity, the establishment of the axiomatic characterization of the asymmetric Nash rule relies on the following separation property: the unique maximizer of the generalized Nash product over a convex choice set can be separated from any other feasible point by the tangent line to the indifference curve of the generalized Nash product at the maximizer. With log-convexity, however, this separation is not always valid. Nevertheless, our second main result shows that each non-maximizer is separable from the unique maximizer by the tangent line to the indifference curve of the generalized Nash product at some feasible point, which may depend on but differ from the non-maximizer (Lemma ). As shown by our third main result (Theorem 2), this separation, interesting in its own right, turns out to be strong enough for IIA, INV, and SIR to uniquely determine the asymmetric Nash rule on our class of all regular and log-convex bargaining problems. Moreover, our class is complete in the sense that the three axioms do not uniquely determine single-valued bargaining solutions on any class containing ours as a proper subclass (Theorem 3). We establish these results for 2-person problems. However, it follows from the bilateral stability of the Nash rule that parallel results hold for the n-person case. We provide a brief discussion along this line at the end of Section 3.2. Zhou (997) showed that on the class of all regular problems any single-valued rule satisfying IIA, INV, and SIR must be a selection of the maximizers of the generalized Nash product. However, such single-valued rule on this broach class of problems is not unique. Denicolo and Mariotti (2000) associated a social welfare ordering with each single-valued rule on a class of problems, which includes those with the comprehensive hulls of two-point sets as choice sets. Their work introduces techniques from social choice theory useful for extending Nash bargaining theory. Peters and Vermeulen (2007) established new techniques to characterize all multi-valued bargaining solutions satisfying Nash s three axioms on a class of problems, including problems with choice sets consisting of finitely many payoff allocations, and they showed that a multi-valued rule satisfying Nash s axioms can be found by an iterated maximization of generalized Nash products. 5 Results in the preceding papers are based on large classes of problems that are not 5 They replaced PO with weak Pareto optimality (WPO). See Peters and Vermeulen (2007, p ) for discussions about their refinements of multi-valued extensions in Kaneko (980), Mariotti (998a), and Xu and Yoshihara (2006). For further references and discussions on non-convex bargaining problems, see the recent survey by Thomson (2009). 4

5 all log-convex. For example, the comprehensive hull of two payoff allocations is not log-convex. Intuitively, solutions are subject to stronger requirements on a larger class of problems than on a smaller one. Thus, although a rule on a larger class always induces a rule on a smaller class satisfying the same axioms, such induced solutions do not necessarily exhaust all possible solutions on the smaller class. Consequently, the existing results do not imply the unique axiomatic characterization of the asymmetric Nash rule by IIA, INV, and SIR on our class of all regular and log-convex problems, nor do they imply that our class is the complete one on which the three axioms characterize a unique single-valued solution. Conley and Wilkie (996) established a single-valued extension of the Nash rule by extending Nash s axioms. Their approach involves a two-step procedure for finding the solution. First, convexify the choice set via randomization and consider the Nash rule for the convexified problem. Second, use the intersection point of the original Pareto frontier with the segment between the threat point and the Nash solution for the convexified problem as the solution. 6 The uniqueness and single-valuedness of their extension enables an interesting comparison with the asymmetric Nash rule for logconvex problems. Herrero (989) introduced an equal distance property to characterize her multi-valued extension with equal bargaining powers. When adjusted by players bargaining powers, this property turns out to be also closely related to the Nash product maximization for log-convex problems. The rest of the paper is organized as follows. The next section briefly reviews Nash axioms and Nash rule with convexity. Section 3 introduces log-convexity and presents our main results. Section 4 applies our results to log-convex duopoly problems. Section 5 compares our approach with some other extensions. Section 6 concludes. 2 Nash Bargaining with Convexity A 2-person problem is composed of a choice set S R 2 of feasible payoff allocations the players can jointly achieve with agreement, and a threat point d S the players end up getting in case of disagreement. A solution on a class B of problems is a rule f assigning a feasible allocation f(s, d) = (f (S, d), f 2 (S, d)) S to each problem (S, d) B. A positive affine transformation for player i s payoff is a mapping τ i : R R such that for some two real numbers a i > 0 and b i, τ i (u i ) = a i u i + b i for all u i R. Given τ and τ 2, τ(u) = (τ (u ), τ 2 (u 2 )) for all u R 2. 6 See Mariotti (998b) for an alternative characterization of Conley and Wilkie s (996) extension. 5

6 Nash s axioms can now be specified as follows (see footnote 3 for the replacement of PO with SIR). SIR: For any (S, d) B, f i (S, d) > d i, i =, 2. SYM : For any (S, d) B with d = d 2 and (u 2, u ) S whenever (u, u 2 ) S, f (S, d) = f 2 (S, d). INV : For any (S, d) B and for any positive affine transformation τ : R 2 R 2, f(τ(s), τ(d)) = τ(f(s, d)). IIA: For any (S, d), (S, d) B with S S, f(s, d) S implies f(s, d) = f(s, d). When B is composed of compact convex problems with strictly Pareto dominated threat points, there is a unique rule satisfying the above four axioms such that the solution for all (S, d) B is represented by See Nash (953) and Roth (977) for details. f(s, d) = arg max u S,u d (u d )(u 2 d 2 ). () When SYM is removed, Kalai (977) showed that for the same class of compact convex bargaining problems, there is also a unique rule satisfying IIA, INV, SIR such that the solution for all (S, d) B is represented by f(s, d) = arg max u S,u d (u d ) α (u 2 d 2 ) α, (2) with (α, α) as the solution for the normalized bargaining problem: S = { u R 2 u + u 2 } and d = (0, 0). (3) For later references in the paper, we call a rule f on B with (S, d ) B an asymmetric Nash rule with bargaining powers α (0, ) for player and ( α) for player 2, if f satisfies IIA, INV, SIR, and f(s, d ) = (α, α); we say that the asymmetric Nash rule on B is unique if it is unique up to choices of the players bargaining powers. 6

7 3 Nash Bargaining with Log-Convexity The convexity of problems plays a crucial role in the establishment of the uniqueness and representation of the rule under Nash axioms. When the convexity is dropped, the rules under the axioms of IIA, INV, and SIR can still be represented by the Nash product maximization, but may fail to be unique. In this section, we characterize the complete class of bargaining problems allowing for non-convexity, on which the uniqueness as well as the representation of the asymmetric rule under the three Nash axioms holds. 3. Log-Convexity We first introduce log-convexity of a bargaining problem. Fix a problem (S, d) and take the natural logarithmic transformation of the positive payoff gains to get V (S, d) = v R2 u S : Then, the maximization (2) is equivalent to u > d, u 2 > d 2, v ln(u d ), v 2 ln(u 2 d 2 ).. max αv + ( α)v 2. (4) v V (S,d) Since ln(u i d i ) is increasing and concave over u i (d i, ) for i =, 2, the convexity of S implies the convexity of V (S, d). For the rest of the paper we consider (S, d) that satisfies the following regularity condition: Regularity: S is closed, bounded above, d comprehensive (i.e. u S whenever d u u for some u S), and d is strictly Pareto dominated in S. The following log-convexity of (S, d) is critical for the uniqueness of the asymmetric Nash rule: Log-Convexity: For any two different points v and v on the strict Pareto frontier of V (S, d), λv + ( λ)v is in the interior of V (S, d) for all 0 < λ <. Before presenting our main results, we make the following observations about logconvexity that will be useful later. First, since V (S, d) is comprehensive, log-convexity 7

8 implies that V (S, d) is convex. Second, log-convexity of V (S, d) is equivalent to the conditions that V (S, d) is convex and it does not contain segments with normal vectors (θ, θ) for all θ (0, ). Third, let (S, d) and (S, d) be two regular problems with the same threat point d. Then V (S S, d) = V (S, d) V (S, d). This implies that (S S, d) is regular and log-convex whenever both (S, d) and (S, d) are. 7 Theorem Let (S, d) be a regular problem. Then, (4) has a unique solution for all α (0, ) if and only if (S, d) is log-convex. For α (0, ), the log-convexity of (S, d) implies that solution for (4), if exists, must be unique. Next, the existence of a solution for (4) is guaranteed by the regularity condition of (S, d). This establishes the sufficiency. We prove the necessity in the Appendix. The essential part of the proof is the convexity of V (S, d). Our idea is to show that the closure of the convex hull cov (S, d) of V (S, d) equals V (S, d). When V (S, d) = C R 2 + for some compact set C R 2 (i.e., when V (S, d) is compactly generated), we can apply Caratheodory theorem to first show that the closure of cov (S, d) = coc R 2 +. Next, by applying separating hyperplane theorem to the boundary points of the closure of coc R 2 +, the condition that problem (4) has a unique solution implies that they must be contained in C. 8 Thus, cov (S, d) = coc R 2 + = C R 2 + = V (S, d), implying that V (S, d) is convex. However, V (S, d) is not necessarily compactly generated, although V (S, d) is closed, comprehensive, and bounded from above. For example, V = {v R 2 v < 0, v 2 /v } is closed, comprehensive, and bounded above, but V C R 2 + for any compact set C R 2. Our proof proceeds as follows. First, we apply the separating hyperplane theorem to each boundary point of the convex hull of V (S, d). Next, we approximate the boundary point by a sequence of points in the convex hull and apply Caratheodory theorem to represent these points as convex combinations of points in V (S, d). Finally, by considering the limits of the representing points, the uniqueness of the solution for 7 Indeed, it is straightforward to verify that (S S, d) is regular and V (S S, d) V (S, d) V (S, d). On the other hand, take any v V (S, d) V (S, d), u S, and u S such that u i > d i, u i > d i, v i ln(u i d i ), and v i ln(u i d i) for i =, 2. Set u i = min{u i, u i }. Then, v i ln(u i d i ) and d << u u, u. By the regularity, u S S. This shows v V (S S, d). Hence, V (S S, d) V (S, d) V (S, d). 8 Since V (S, d) is comprehensive, it suffices to consider boundary points that are Pareto optimal. 8

9 (4) enables us to show that the boundary point is either on the boundary of V (S, d) or is Pareto dominated in V (S, d). 3.2 Log-Convexity and Uniqueness We now show that IIA, INV, and SIR uniquely determine the asymmetric Nash rule on the class of all regular and log-convex bargaining problems. Theorem 2 Under the axioms of IIA, INV, and SIR, there exists a single-valued rule unique up to choices of bargaining powers on the class of all regular and log-convex problems. Furthermore, the rule is represented by the asymmetric Nash product maximization. To prove this theorem, we need the following separation result. Lemma Suppose (T, d ) is regular and log-convex. Let x = arg max u T :u d u α u α 2 with α (0, ). Then, for any y T with y d and y x, there exists z T such that z y and αz 2 x + ( α)z x 2 z z 2 (5) and αz 2 y + ( α)z y 2 < z z 2. (6) Proof. Since (T, d ) satisfies regularity and log-convexity, it follows from Theorem that x = arg max u T :u d u α u α 2 is uniquely determined. Let y T be such that y d and y x. Then, from the uniqueness of the maximizer, x α x α 2 > y α y α 2, or equivalently, α(ln y ln x ) < ( α)(ln x 2 ln y 2 ). Assume without loss of generality x < y (the other case can be analogously proved). The locations of x and y are illustrated in Figure (a). Notice with x < y, the previous inequality implies α α < ln x 2 ln y 2 ln y ln x. (7) Since x, y T and d = (0, 0), it follows from the log-convexity t(ln x, ln x 2 ) + ( t)(ln y, ln y 2 ) = (ln x t y t, ln x t 2y t 2 ) V (T, d ) 9

10 for t [0, ]. This is the segment between (ln x, ln x 2 ) and (ln y, ln y 2 ) in Figure (b). Thus, by the definition of V (T, d ), there exists u(t) T such that (ln x t y t, ln x t 2y2 t ) (ln u (t), ln u 2 (t)), or equivalently, (x t y t, x t 2y2 t ) (u (t), u 2 (t)). Since x, y d implies (x t y t, x t 2y2 t ) d, it follows from the regularity that z(t) (x t y t, x t 2y t 2 ) T for all t [0, ] as illustrated in Figure (c). is For t (0, ), the absolute value of the slope of the tangent line of u α u α 2 at z(t) αx t 2y t 2 ( α)x t y t and that of the segment between z(t) and y is x t 2y2 t y 2 y x t y t. Thus, the tangent line of u α u α 2 at z(t) is flatter than the segment between z(t) and y if and only if 9 αx t 2y t 2 ( α)x t y t < xt 2y t 2 y 2 y x t y t αxt 2 ( α)x t < xt 2 y2 t. y t x t By L Hôpital s rule and (7), the above inequalities hold as t 0. t (0, ) such that the tangent line of u α u α 2 at point z(t), Now choose αz 2 (t)u + ( α)z (t)u 2 = z (t)z 2 (t), (8) is flatter than the segment between z(t) and y. Since x is Pareto optimal, x < y necessarily implies x 2 > y 2. Thus, z (t) < y and z 2 (t) > y 2. Consequently, y is below the tangent line in (8). On the other hand, by definition of x, x cannot lie below the indifference curve of u α u α 2 at z(t) (see Figure (c) for an illustration). Thus, (5) and (6) are established by setting z = z(t). Lemma extends the separation between the maximizer of the generalized Nash product over a convex choice set and the rest of the choice set. Specifically, with convexity the maximizer is separable from any other point of the choice set by the 9 Since x < y, the condition is equivalent to y lying below the tangent line. 0

11 tangent line of the indifference curve of the generalized Nash product through the maximizer. In the case with a regular and log-convex choice set, the separation in Lemma is weaker in the sense that it may be point-dependent (i.e., z may change as point y changes in the choice set), but it is nonetheless strong enough for IIA, INV, and SIR together to determine a unique rule. Proof of Theorem 2: Let B be the class of all regular and log-convex problems. By Theorem, given players bargaining powers, (2) has a unique solution f(s, d) for each (S, d) B. As noted before, f(s, d) satisfies IIA, INV, and SIR. This establishes the existence. We prove the uniqueness by contradiction. Let g be a rule on B satisfying IIA, INV, SIR, and g(s, d ) = (α, α). Suppose g f. Then, there exists (T, d ) B such that g(t, d ) arg max u T :u d uα u α 2. (9) Set y = g(t, d ). By SIR, y >> d. By Lemma and (9), there exists z T such that z y and y is below the tangent line of u α u α 2 at z: αz 2 u + ( α)z u 2 = z z 2, u R 2. (0) Let S = {u R 2 + αz 2 u + ( α)z u 2 z z 2 }. Clearly, (S, d ) is regular and log-convex and y S (see Figure (c)). By INV and g(s, d ) = (α, α), we have 0 g(s, d ) = z. () Since (S T, d ) is regular and log-convex and since z S T, it follows from IIA and () that g(s T, d ) = z. On the other hand, since y S T and y = g(t, d ), it follows from IIA that g(s T, d ) = y. This shows y = z, which is a contradiction. 0 Consider affine transformation τ : R 2 R 2 : τ(u) = ( α z u, α z 2 u 2 ), u R 2. Then, by (0), τ(s) = S. Since g(s, d ) = (α, α), g(τ(s), τ(d )) = (α, α). By INV, τ(g(s, d )) = g(τ(s), τ(d )). Hence, g(s, d ) = τ (α, α) = z.

12 Our next result shows that the class of all regular and log-convex problems is the complete class for IIA, INV, and SIR to uniquely determine the asymmetric Nash rule. Theorem 3 Under the axioms of IIA, INV, and SIR, single-valued rules on any class of regular problems containing the class of all regular and log-convex problems as a proper subclass are not unique up to choices of bargaining powers. Proof. Let B be the class of all regular and log-convex probelms and B be a class of regular problems such that B B. Then, there exist (S, d ) B which is not log-convex. By Theorem, (S, d ) has multiple generalized Nash product maximizers for some bargaining power α (0, ) for player. Now, consider f and f 2 on B with player s bargaining power equal to α, where for (S, d) B, f (S, d) = arg max u, u N (S,d) f 2 (S, d) = arg max u 2, u N (S,d) and N (S, d) = arg max u S:u d u α u α 2. By the multiplicity of N (S, d ), f (S, d ) f 2 (S, d ). Moreover, by construction, the usual proof implies that f and f 2 are singlevalued rules satisfying IIA, INV, SIR, and f (S, d ) = f 2 (S, d ) = (α, α ). Since the class of regular and log-convex problems is a proper subset of the regular problems, Theorem 3 implies that solutions under IIA, INV, and SIR cannot be unique on the broad class of bargaining problems in Zhou (997). The notions of regularity and log-convexity and Theorem can all be naturally generalized to the n-person case. The asymmetric Nash rule for the n-person case satisfies bilateral stability (consistency), which states that the solution for a n-person problem agrees with the solution for each 2-person subproblem obtained by taking the other players utility levels in the original solution as given. The 2-person subproblems of a n-person regular and log-convex problem are again regular and log-convex. Thus, by applying Lemma to 2-person subproblems, it can be shown that Theorem 2 can be generalized to n-person regular and log-convex problems. Finally, the method of proof of Theorem 3 is directly applicable to n-person regular and log-convex problems. It follows that Theorem 3 can also be generalized to the n-person case. Bilateral stability of the n-person symmetric Nash rule was introduced in Harsanyi (959). Bilateral stability also holds for the asymmetric Nash rule. See Lensberg (988) for general results on multilateral as well as bilateral stabilities of the symmetric Nash rule. 2

13 3.3 A Subclass of Log-Convex Problems For Nash product maximization with generic bargaining powers to have a unique solution, log-convexity quantifies how non-convex the individually rational (IR) portion of a bargaining problem can be allowed. When the frontier is smooth, log-convexity is implied by a simple condition as shown in the following proposition. Proposition Let (S, d ) be a regular problem such that the IR frontier of S is given by a C 2 function u j = ψ(u i ), u i [0, ū i ] for some number ū i < with ψ(ū i ) = 0 and for i j. If (R) ψ (u i ) < 0 and e(u i ) u iψ (u i ) ψ(u i ) is strictly decreasing over (0, ū i ), then (S, d ) is log-convex. Proof. Notice first for any v = (v i, v j ) in the frontier of V (S, d ) there exists u i (0, ū i ) such that v i = ln(u i ) and v j = ln[ψ(u i )]. That is, the frontier of V (S, d ) is parameterized by u i (0, ū i ). Thus, the slope of the frontier at v is given by e(u i ) = u iψ (u i ) ψ(u i ). Notice ψ (u i ) < 0 implies that the frontier of V (S, d ) is downward sloping. Hence, it is strictly concave if e(u i ) is strictly decreasing. Note that e(u i ) in Proposition is the elasticity of the IR frontier, u j = ψ(u i ), of a bargaining problem with respect to u i. Thus, in applications it is relatively easier to check (R) than the log-convexity. A stronger sufficient condition is either ψ is log-concave, or u i ψ (u i ) is decreasing, in addition to ψ being decreasing. With a generic threat point d, (R) takes the following form, ψ (u i ) < 0 and e(u i ) (u i d i )ψ (u i ) ψ(u i ) d j is strictly decreasing over (d i, ū i ). Given ψ < 0 over (d i, ū i ), (S, d) satisfying the above condition implies that (S, d ) also satisfies it for all d S with d d such that d j < ψ(ū i ). Moreover, as d varies in S, the log-convexity of (S, d) puts further restrictions on S. A natural question is whether S is necessarily convex when (S, d) is log-convex for all strictly Pareto dominated threat points d S. Proposition 2 in the next section 3

14 shows a class of problems that is not convex but log-convex for all threat points that are strictly Pareto dominated. 4 An Application: Bargaining Problems in Duopoly In this section we apply our results to a class of duopoly bargaining (or collusion) problems. 2 Suppose there are two firms in an industry supplying a homogeneous product with a market demand function D(p). constant marginal cost of production given by c i, i =, 2. Each firm has zero fixed costs and Let p = inf{p : D(p) = 0}. We make the following assumptions: (i) 0 c < c 2 < p ; (ii) D(p) is twice continuously differentiable and D (p) < 0 on (0, p ); and (iii) firm s monopoly profit function π (p) (p c )D(p) is strictly concave on (0, p ) and lim p p π (p) = 0. Assumption (iii) implies that there exists a unique monopoly price p that maximizes π (p) and that c < p < p. To make bargaining non-trivial, we also assume (iv) c 2 < p. Suppose the two firms bargain over their production outputs (or market shares) without side-payments. 3 We first describe the feasible set of profit allocations for the firms, which is denoted as Π. Assuming d-comprehensiveness of Π, we only need to characterize the Pareto frontier of Π, which we denote by PF. Let profit allocation π = (π, π 2 ) PF be resulted from an output agreement (q, q 2 ). Since π i = q i (D (q + q 2 ) c i ) for i =, 2, π PF satisfies π p c + π 2 p c = D(p) (2) for some price p 0 such that D(p) = q + q 2. That is, feasible profit allocations on PF can be parameterized by a range of prices instead of output allocations. We shall narrow down the price range that completely characterizes PF. Notice that profit allocations associated with prices p c 2 or p p are Pareto dominated by those with prices in (c 2, p ). Given p (c 2, p ) and π 2, it follows from (2) that h(p, π 2 ) π (p) π 2(p c ) p c 2 (3) 2 Binmore (2007, p ) illustrated the use of Nash bargaining solution as a tool for studying collusive behavior in a Cournot duopoly with asymmetric constant marginal costs. However, the profit choice set without side-payments is drawn as if it is convex (Figure 6.3(a)), which is not with respect to the assumed demand and cost functions. 3 Explicit transfers or briberies between firms may be too risky due to antitrust scrutiny. 4

15 is the profit for firm should they agree to let firm 2 receive profit π 2. Thus, π PF if and only if π = max h(p, π 2). c 2 <p<p The supporting price, p(π 2 ), is determined by the first-order condition π (p) + (c 2 c )π 2 (p c 2 ) 2 = 0 and hence, the Pareto frontier is represented by π = h(p(π 2 ), π 2 ) for π 2 [0, π 2 ], where π 2 is the maximum monopoly profit for firm 2. Note that the second-order condition is satisfied due to Assumption (iii). To illustrate non-convexity of Π, consider a simple example in which D(p) = 20 p, c = 0, and c 2 = 0. In this case, the Pareto frontier can be easily computed and is given by π = 00 π 2 3(5π 2 ) 2 3 for 0 π Simple calculation shows that the frontier is a strictly convex curve. Consequently, the feasible choice set of the resulting bargaining problem is not convex. Intuitively, without side-payments, production arrangements are necessarily inefficient in order for firm 2, the inefficient one, to produce positive quantities and receive positive profits, so that one unit of profit gain by firm 2 results in more than one unit of profit loss for firm. We next show that under our assumptions the Pareto frontier curve π = h(p(π 2 ), π 2 ) is strictly decreasing and convex in π 2 [0, π 2 ]. 4 of profit allocations satisfies log-convexity. Nevertheless, the feasible choice set Proposition 2 Suppose (i)-(iv) hold. Then, Π is non-convex, but (Π, d) is regular and log-convex for all d 0 that are strictly Pareto dominated in Π. Proposition 2 and Theorem 2 together imply that the duopoly problem (Π, d) has a unique Nash bargaining solution under the assumed conditions. The uniqueness of the solution makes it easier for one to perform comparative static analysis of the Nash solution with respect to bargaining powers, marginal costs, threat points, and demand parameters. 4 The non-convexity of Π was shown in Tirole (988, p. 242, 27) under one extra assumption that the monopoly profit function for firm 2 is concave. We will show the non-convexity without such an assumption. 5

16 5 Comparison We now compare our extension of Nash bargaining with the two other extensions by Conley and Wilkie (996) and Herrero (989), respectively. To simplify our comparison, we focus on log-convex problems (S, d ) satisfying condition (R) in Section 3.3. Under Conley and Wilkie s (996) extension, the solution is determined by the intersection of the Pareto frontier of the choice set with the segment connecting the threat point and the Nash solution for the convexified problem. Although they only consider extensions of the symmetric Nash solution, their method can be also applied to extend the asymmetric Nash solution. Given (S, d) and α (0, ), we denote by f N (S, d α) the asymmetric Nash solution with bargaining powers α for player and α for player 2 and by f CW (S, d α) the Conley-Wilkie s extension. Proposition 3 Let (S, d ) be a regular problem such that the IR frontier of S is given by a C 2 function u 2 = ψ(u ), u [0, ū ] for some number ū < with ψ(ū ) = 0. Suppose ψ satisfies (R) for i = and ψ > 0. 5 Then, there exists ˆα (0, ) such that (a) f N (S, d ˆα) = f CW (S, d ˆα), (b) f N (S, d α) < f CW (S, d α) and f2 N (S, d α) > f2 CW (S, d α) for α < ˆα, and (c) f N (S, d α) > f CW (S, d α) and f2 N (S, d α) < f2 CW (S, d α) for α > ˆα. Proposition 3 implies that the Nash solution without convexification is more responsive to the relative bargaining powers than the Conley-Wilkie solution. The intuition appears to be that the Nash solution without convexification utilizes more local curvature of the Pareto frontier of the choice set than the Conley-Wilkie solution does. For the class of duopoly problems with c < c 2 discussed in Section 4, we can further show that ˆα > /2, implying that in the presence of asymmetric marginal costs and equal bargaining powers, the Nash solution favors the less efficient firm as compared to the Conley-Wilkie solution. Conditions (R) together with the regularity condition also guarantees a geometric property of the asymmetric Nash solution given in Proposition 4 below. Given α (0, ), let I (α) and I 2 (α) denote the horizontal and vertical intercepts of the tangent line to the Pareto frontier of S at f N (S, d α), respectively. We use to denote the Euclidean norm. 5 If ψ is concave, then S is convex in which case both extended Nash solutions reduce to the standard Nash solution. 6

17 Proposition 4 Let (S, d ) be a regular problem such that the IR frontier of S is given by a C function u 2 = ψ(u ), u [0, ū ] for some number ū < with ψ(ū ) = 0. Suppose ψ satisfies (R) for i =. Then, for any α (0, ), α f N (S, d α) I 2 (α) = ( α) f N (S, d α) I (α). Proposition 4 states that the distances between the asymmetric Nash solution and the two intercepts are proportional to the associated bargaining powers. In the case of equal bargaining powers, this geometric (equal distance) property is the main feature in Herrero s (989) characterization of the multi-valued solution under a different set of axioms and has also been noted in Mas-Colell, Whinston and Green (995, Example 22.E.3, p. 842) for compact convex problems. The proportional distance property of the asymmetric Nash solution in Proposition 4 represents an extension of the equal distance property to asymmetric cases and to a large class of regular problems satisfying (R). 6 Conclusion It is well-known that on the class of compact convex bargaining problems, Nash axioms without that of symmetry uniquely determine the asymmetric Nash rule, which is parameterized by players bargaining powers. The parametrization by players bargaining powers is useful in applications, for it offers flexibility in fitting data by choosing suitable bargaining powers. In this paper, we have extended the standard axiomatic characterization of the unique asymmetric Nash rule to the class of all regular and logconvex problems. We have shown that this is the complete class for such an extension to hold. Bargaining between duopolists with asymmetric constant marginal costs and concave monopoly profit function for the efficient firm results in non-convex problems, which are regular and log-convex. The uniqueness and single-valuedness of the asymmetric Nash rule make it possible to conduct comparative static analysis with respect to bargaining powers, threat points, and determinants of the choice sets such as the marginal costs for duopoly problems. They also enable interesting comparisons with other single-valued rules appeared in the literature. 7

18 Before proceeding with the proof of Theorem, we make the following observations. Appendix: Proofs First, the logarithmic transformation (v, v 2 ) = (ln(u d ), ln(u 2 d 2 )) is a homeomorphism from S {(u, u 2 ) u > d, u 2 > d 2 } onto V (S, d). It follows that the boundary V (S, d) of V (S, d) must be the homeomorphic image of that of S {(u, u 2 ) u > d, u 2 > d 2 }; that is, V (S, d) = { ( ln(u d ), ln(u 2 d 2 ) ) u S : u > d, u 2 > d 2. Thus, V (S, d) is closed. Second, because S is bounded above, V (S, d) is bounded and V (S, d). Third, for any (u, u 2 ) S {(u, u 2 ) u > d, u 2 > d 2 }, the intervals ((d, u 2 ), (u, u 2 )] and ((u, d 2 ), (u, u 2 )] are transformed into ((, ln(u 2 d 2 )), (ln(u d ), ln(u 2 d 2 ))] and ((ln(u d ), ), (ln(u d ), ln(u 2 d 2 ))] in V (S, d), respectively. As a result, V (S, d) = V (S, d) R 2 +. Thus, by moving the origin properly, we can assume, without loss of generality, that for some constant h > 0, }. V (S, d) (h, h) R 2 +, (A) so that the closed cone generated by V (S, d) and any line passing through the origin with a normal vector in R 2 + have no point in common other than the origin. Lemma 2 Let U R 2 be closed, convex, and comprehensive (i.e. U = U R 2 +). If the boundary U of U is non-empty, then U = U R 2 +. Proof. Notice U R 2 + U R 2 + = U is automatic. Conversely, let u be any interior point of U. Then, there exists a number ɛ > 0 such that (u i +δ, u j ) U for i j and for all δ [0, ɛ). Set δ i = sup{δ (u i + δ, u j ) U}. Since U is convex and comprehensive, δ i < for at least one i, for otherwise U = R 2 which contradicts U. Assume without loss of generality δ <. Then, u = (u + δ, u 2 ) U because U is closed. It follows that u = u (δ, 0) U R 2 +. This shows U U R 2 +. Proof of Theorem : As observed earlier, log-convexity is equivalent to conditions that V (S, d) is convex and it does not contain segments with normal vectors (α, α) for all α (0, ). To prove the necessity, we only need to show that V = V (S, d) is convex because the necessity of the rest is obvious. 8

19 Let V denote the closure of the convex hull of V. Notice V also satisfies V = V R 2 +; hence, from Lemma 2, V = V R 2 +. For any boundary point v V, it follows from the separation theorem and the comprehensiveness of V that there exists a R 2 + such that a v = max v V a v = max a v. (A2) v V Without loss of generality, we may take a = (θ, θ) for some θ [0, ]. There exist sequences {v k (n)} n in V for k =, 2, 3, {λ(n)} n in R 3 +, and {ɛ(n)} n in R such that 6 v = λ (n)v (n) + λ 2 (n)v 2 (n) + λ 3 (n)v 3 (n) + ε(n), λ (n) + λ 2 (n) + λ 3 (n) =, ε(n) 0. (A3) (A4) (A5) By the non-negativity of λ(n) and (A4), we may assume λ(n) λ R 3 + with λ + λ 2 + λ 3 =. (A6) By (A), the sequences {λ k (n)v k (n)} n, k =, 2, 3, are bounded above and by (A3), (A5), and (A6), they are also bounded below. Hence, we may assume λ k (n)v k (n) w k R 2, k =, 2, 3 v = w + w 2 + w 3. (A7) Suppose first λ k > 0 for k =, 2, 3. Then, since V is closed, (A3) and (A5) (A7) imply v k (n) v k V, k =, 2, 3, v = λ v + λ 2 v 2 + λ 3 v 3. (A8) Suppose now λ k = 0 for some k but λ k > 0 for k k. Without loss of generality, assume λ = 0, λ 2 > 0, and λ 3 > 0. In this case, v = w + λ 2 v 2 + λ 3 v 3, λ 2, λ 3 > 0, λ 2 + λ 3 =. (A9) 6 The reason is as follows. Since v is in V, there exists a sequence { v(n)} n in the convex hull of V such that v(n) v. By Carathéodory Theorem, for each n, there exist v (n), v 2 (n), v 3 (n) in V and λ(n) R 3 + satisfying (A4) such that v(n) = λ (n)v (n) + λ 2 (n)v 2 (n) + λ 3 (n)v 3 (n). Finally, (A3) and (A5) are established by setting ɛ(n) = v v(n) for all n. 9

20 By (A2) and (A7), a w = lim n a λ (n)v (n) = lim n λ (n)a v (n) lim n λ a v = 0. On the other hand, by (A2) and (A9), a w = a v [ λ 2 (a v 2 ) + λ 3 (a v 3 )] 0. It follows that a w = 0. This shows that w lies on the line having normal vector a = (θ, θ) R 2 + and passing though the origin. Furthermore, as the limit of {λ (n)v (n)}, w is in the closed cone generated by V. Thus, by (A), we must have w = 0. Suppose finally λ k = for some k. Assume without loss of generality λ 3 0. In this case, a similar proof as before shows w = w 2 = 0. In summary, by letting v k be arbitrary element in V when λ k = 0, the preceding analysis establishes v = λ v + λ 2 v 2 + λ 3 v 3. (A0) When λ k > 0, (A2) and (A0) together imply a v k = max v V a v, which implies v k V. If a R 2 ++, then the above equality and the assumption that (4) has a unique solution imply v k = v whenever λ k > 0. Hence, v V V. If a = (, 0), then v = a v = a v k = v k. Assume without loss of generality λ > 0 and v 2 = max{ v k 2 λ k > 0}. Then, by (A0), v 2 = λ v 2 + λ 2 v λ 3 v 3 2 v 2. Thus, v v. This shows v V. If a = (0, ), then a similar proof establishes v V. We have shown that v V for any v V. Thus, by Lemma 2, V V = V R 2 + V R 2 + = V. This concludes V = V which implies that V is convex. Proof of Proposition 2: To show the non-convexity of Π, notice that the PF of Π 20

21 is given by the function π = ψ(π 2 ) = h(p(π 2 ), π 2 )), π 2 [0, π 2 ]. By (3), and h 2 (p, π 2 ) = p c p c 2, h 22 (p, π 2 ) = 0, h 2 (p, π 2 ) = c 2 c (p c 2 ) 2, (A) h (p, π 2 ) = π (p) 2(c 2 c )π 2 (p c 2 ) 3, p (π 2 ) = h 2(p(π 2 ), π 2 ) h (p(π 2 ), π 2 ). It follows from the envelope theorem, (A), and (A2) that (A2) ψ (π 2 ) = h 2 (p(π 2 ), π 2 ), (A3) and ψ (π 2 ) = h 2 (p(π 2 ), π 2 )p (π 2 ) = [h 2(p(π 2 ), π 2 )] 2 > 0, (A4) h (p(π 2 ), π 2 ) where the strict inequality holds since h (p, π 2 ) < 0 due to the strict concavity of π (p). Next, we show that (Π, d) is log-convex for all d 0 that are strictly Pareto dominated in Π. Without loss of generality, assume d = d. By Proposition, it suffices to show that ψ (π 2 ) < 0 and π 2 ψ (π 2 ) is decreasing in π 2. By (A) and (A3), ψ (π 2 ) < 0 if p(π 2 ) > c 2. Suppose p(π 2 ) < p. The strict concavity of π (p) implies that π (p) > 0 for p [p(π 2 ), p ], which in turn implies h (p, π 2 ) > 0 for p [p(π 2 ), p ]. Consequently, p(π 2 ) cannot be Pareto optimal. Thus, p(π 2 ) p > c 2. By the strict concavity of π (p) and (A2), h (p, π 2 ) > 2(c 2 c )π 2 (p c 2 ) 3 > 0 (A5) 2

22 for any π 2 (0, π 2 ). It then follows from (A)-(A5) that (π 2 ψ (π 2 )) = ψ (π 2 ) + π 2 ψ (π 2 ) [h 2 (p(π 2 ), π 2 )] 2 = h 2 (p(π 2 ), π 2 ) + π 2 h (p(π 2 ), π 2 ) < p(π 2) c c 2 c + p(π 2 ) c 2 2[p(π 2 ) c 2 ] = 2p(π 2) c c 2 2[p(π 2 ) c 2 )] < 0 for any π 2 (0, π 2 ), where the last inequality holds since p(π 2 ) p > c 2 > c. Proof of Proposition 3: (S, d ) is regular and by Proposition, it is also log-convex. Thus, by Theorem 2, there is a unique Nash solution given by (2), where the first-order condition is given by u ψ (u ) ψ(u ) = α α. (A6) The Nash solution (f N (S, d α), f N 2 (S, d α)) is determined by (A6) and u 2 = ψ(u ). It follows that the ratio of the two payoff gains is R N (α) f 2 N (S, d α) f N (S, d α) = α α ψ (f N (S, d α)). Moreover, the left-hand side of (A6) is strictly increasing in u by (R) and the righthand side is strictly increasing in α. It follows that f N (S, d α) is strictly increasing in α for any interior solution. The frontier of the convex hull of S is a straight line given by u 2 = ū2 ū (ū u ). Applying Nash solution on the convex hull yields a solution u = (αū, ( α)ū 2 ). This solution and the Conley-Wilkie s extended Nash solution have the same ratio of the payoff gains for the two players, which is given by R CW (α) f 2 CW (S, d α) f CW (S, d α) = α ū 2. α ū 22

23 Comparing the two solutions yields R N (α) R CW (α) = ψ (f N (S, d α)). ū 2 /ū The strict convexity of ψ and the monotonicity of f N (S, d α) at the interior solution imply that ψ (f N (S, d α)) is strictly decreasing in α. Thus, there exists ˆα (0, ) such that (a) R N (ˆα) = R CW (ˆα), (b) R N (α) > R CW (α) if α < ˆα, and (c) R N (α) < R CW (α) if α > ˆα. The claims then follow since both solutions are on the frontier. Proof of Proposition 4: Notice first that (S, d ) is both regular and log-convex. Set z = f N (S, d α). Then, by Theorem, z is the only tangent point between the indifference curve of Nash product u α u α 2 and the Pareto frontier of S. The corresponding tangent line is represented by u 2 = ψ(z ) + ψ (z )(u z ). It intersects the horizontal axis at I = (z + ψ(z )/ψ (z ), 0) and it intersects the vertical axis at I 2 = (0, ψ(z ) z ψ (z )). The distances z I from z to I and z I 2 from z and I 2 are given by and z I 2 = (ψ(z )/ψ (z )) 2 + (ψ(z )) 2 = (ψ(z )) 2 [ + (ψ (z )) 2 ] (ψ (z )) 2 z I 2 2 = (z ) 2 + (z ψ (z )) 2 = (z ) 2 [ + (ψ (z )) 2 ]. It follows that z I 2 2 z I 2 = ( ) z ψ 2 (z ). (A7) ψ(z ) Recall that the proportional distance property states that z I 2 z I = α α. By (A7), the preceding equality is equivalent to ( α α z ) ( ψ (z ) α ψ(z ) α + z ) ψ (z ) = 0. ψ(z ) 23

24 Since ψ (z ) < 0, the above equality in turn is equivalent to (A6), which is automatic because it is the first-order condition for Nash bargaining solution z = f N (S, d α). 24

25 Figure : Separation between Maximizer x of Nash Product u 2 u over Choice Set T and Point y on the Frontier of T by the Tangent Line of Nash Product at Point T t z z ) ( with y t z ) (. x y z T u 2 2 x x u u x 2 u 2 2 z z u u (c) (a) (b) 2 2 x x u u 2 u v 2 v u T y ) ln, (ln 2 y y ) ln, (ln 2 x x ), ( d S V 25

26 Acknowledgements: We gratefully acknowledge helpful comments from Kim Border, Juan Carrillo, Bo Chen, Matthew Jackson, Joel Sobel, Walter Trockel, Simon Wilkie, Adam Wong, Lin Zhou, and seminar and conference participants at Beijing University, Ohio State University, Southwest Economic Theory Conference, the Third Congress of the Game Theory Society, Shanghai University of Finance and Economics, University of Arizona, University of Bielefeld, University of California at Irvine, Riverside, and San Diego, University of Southern California, and Zhejiang University. References [] Binmore, K. (2007), Playing for Real, A Text on Game Theory, Oxford University Press. [2] Binmore, K., A. Rubinstein, and A. Wolinsky (986), The Nash Bargaining Solution in Economic Modelling, RAND Journal of Economics 7: [3] Bishop, R. (960), Duopoly: Collusion or Warfare? American Economic Review 50: [4] Conley, J. and S. Wilkie (996), An Extension of the Nash Bargaining Solution to Nonconvex Problems, Games and Economic Behavior 3: [5] Denicolò, V. and M. Mariotti (2000), Nash Bargaining Theory, Nonconvex Problems and Social Welfare Orderings, Theory and Decision 48: [6] Harsanyi, J. (959), A Bargaining Model for the Cooperative n-person Game, in Contributions to the Theory of Games IV, Annals of Mathematical Studies, No 40 (A. W. Tucker and R. D. Luce, Eds.), Princeton University Press, Princeton, NJ, pp [7] Herrero, M. (989), The Nash Program: Non-convex Bargaining Problems, Journal of Economic Theory 49: [8] Kalai, E. (977), Nonsymmetric Nash Solutions and Replications of Two-Person Bargaining, International Journal of Game Theory 6:

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