Parametrized bargaining solutions
Transkript
Parametrized bargaining solutions
Parametrized bargaining solutions Shiran Rachmilevitch∗ January 11, 2015 Abstract I introduce a new bargaining solution, the endogenous dictatorship solution, and a parametrized family of solutions the endpoints of which are the egalitarian and the endogenous dictatorship solutions. The Kalai-Smorodinsky solution is the “midpoint” of this family. This is analogous to the fact that the Nash solution is the “family midpoint” of the constant elasticity bargaining solutions. I also derive results about a parametrized family of solutions the endpoints of which are the egalitarian and equal-loss solutions. It is shown that the KalaiSmorodinsky solution is more oriented towards egalitarianism than towards the equal-loss principle. Keywords: Constant elasticity solutions; Endogenous dictatorship; Kalai-Smorodinsky solution; Parametrization. JEL Codes: C71; C78; D61; D63. ∗ Department of Economics, University of Haifa, Mount Carmel, Haifa, 31905, Israel. Email: shiranrach@econ.haifa.ac.il Web: http://econ.haifa.ac.il/∼shiranrach/ 1 1 Introduction A bargaining problem is a compact, convex, and comprehensive set S ⊂ R2+ that contains the origin as well as some x such that x > 0 ≡ (0, 0).1 For convenience, I will restrict my attention to problems S such that (x1 + x2 ) is maximized at a unique point x ∈ S. Let B denote the collection of these problems. A bargaining solution is a selection—a function that assigns a unique point of S for every S ∈ B. The interpretation of this model is this: two players need to choose a point of S; if they agree on x then the bargaining situation is resolved and each player i receives the utility payoff xi , while failure to reach agreement leads to the implementation of the status quo payoffs, 0. The assumption S ∩ R2++ 6= ∅ guarantees that there are strict incentives to avoid disagreement. A solution is a concise description of the outcome that would result from the players’ behavior, in every possible problem; alternatively, one can think of a solution as representing a third party—an impartial arbitrator—who provides recommendations regarding what is the right agreement in every conceivable problem. The family of constant elasticity solutions, or CES (Bertsimas et al. (2012), Haake and Qin (2013)), is parametrized by a single number, ρ ∈ (−∞, 1]: the solution corresponding to ρ, σ ρ , is defined by: 1 σ ρ (S) ≡ argmaxx∈S [xρ1 + xρ2 ] ρ . The “endpoints” of this family, the ones corresponding to ρ = 1 and to the limit ρ → −∞, are the utilitarian and egalitarian solutions, respectively. The egalitarian solution, due to Kalai (1977), assigns to each S the intersection point of its north-east boundary with the 45◦ -line, hereafter denoted E(S) = (e(S), e(S)). The utilitarian solution, U , assigns to each S the maximizer of the utility-sum over S. As the parameter ρ increases, the corresponding solution, informally speaking, assigns 1 Vector inequalities are as follows: xRy iff xi Ryi for all i, for both R ∈ {≥, >}. Comprehensive- ness of S means that if x ∈ S then y ∈ S, for every y that satisfies 0 ≤ y ≤ x. 2 less importance to fairness—in the sense of egalitarianism—and more importance to efficiency—as expressed by utilitarianism. The limit ρ → 0 is a special case: it corresponds to the Nash solution (Nash, 1950). This solution, denoted hereafter by N , assigns to each S the maximizer of x1 · x2 over x ∈ S. It is (i) the unique CES solution that satisfies scale invariance, and (ii) the unique CES solution that satisfies midpoint domination. The former property means that for every S and every pair of positive linear transformation l = (l1 , l2 ), the solution, call it σ, satisfies σ(l ◦ S) = l ◦ σ(S); the latter property, due to Sobel (1981), means that σ(S) ≥ 21 a(S) for every S, where ai (S) ≡ max{si : s ∈ S}.2 These are well-known properties (or axioms), which, for the sake of brevity, I will not discuss (the interested reader is referred to Thomson (1994) for a discussion). Traditionally, utilitarianism has been viewed as the central ethical criterion to compete with (or be contrasted with) egalitarianism.3 The CES family parametrizes an entire spectrum between these opposing criteria. There are, however, other principles that are “opposite” to egalitarianism, not only utilitarianism. In particular: 1. Dictatorship. It is opposite to egalitarianism in the sense that it maximizes inequality, rather then minimizing it. Formally, the dictatorial bargaining solutions are D1 (S) ≡ (a1 (S), 0) and D2 (S) ≡ (0, a2 (S)). 2. The equal-loss principle. It is opposite to egalitarianism in the sense that the welfare-measure that it equates between the players is losses, not gains. Formally, the equal-loss solution (Chun, 1988), EL, selects for each S the point of its frontier, x, such that a1 (S) − x1 = a2 (S) − x2 .4 I seek to find parametrized families of solutions—one for the case of dictatorship, 2 The point a(S) is called the ideal point of S. See, e.g., Fleurbaey et al. (2008). 4 I use the term “egalitarianism” to denote equality of gains, as is customary in the bargaining 3 literature. The equal-loss principle is, of course, also egalitarian, in the sense that it implements equality (of losses). 3 one for the case of equal-losses—that will be analogous to the CES family. Namely, two families, each of which parametrizes a spectrum between egalitarianism and one of its alternative opposing principles. The merits of such a parametrization can be understood in two contexts: when both endpoints of the parameter space describe appealing principles, and when one of these endpoints is the focus of attention. Examples for the former context are provided by the CES family and by the “equal-gains-to-equal-losses” spectrum. In each of these cases the parametrization formalizes a tradeoff between desirable but mutually inconsistent objectives: fairness versus efficiency in the former case, and a tradeoff between two alternative notions of fairness in the latter case. The “egalitarianism-to-dictatorship” spectrum exemplifies the second context. Here, the framework lends itself, for example, to the analysis of questions such as how far can we move away from dictatorship, and towards egalitarianism, without compromising on scale invariance? Ideally, it would be nice to obtain parameterizations that parallel CES: ones where the parameter space has an “anchor” that corresponds to an appealing bargaining solution, which, moreover, can be singled out from the parametrized family with the help of a meaningful axiom. I describe such a parametrization for the case of dictatorship. Since there are two possible dictators, a preliminary issue that needs to be addressed is the determination of the dictator. I determine it endogenously, by declaring player i to be the dictator in the problem S if ai (S) > aj (S) (when the ideal payoffs are the same there is no endogenous dictator). The endogenous dictatorship solution coincides with Di when i is the endogenous dictator, and coincides with E when there is no endogenous dictator. The proposed parametrization is such that the parameter space is Θ = [0, 1) with θ = 0 corresponding to the egalitarian solution and the limit θ → 1 corresponding to the endogenous dictatorship solution. The parameter space’s midpoint, θ = 12 , corresponds to the Kalai-Smorodinsky solution (Kalai and Smorodinsky, 4 1975), which assigns to each S the point λa(S), where λ is the maximum possible. The Kalai-Smorodinsky solution is the only member of this parametrized family that satisfies scale invariance, and the only member of this family that satisfies midpoint domination. This is analogous to the fact that N is the unique CES solution that satisfies any of these axioms. Deriving a parametrization for equal-losses turns out to be more challenging. I consider two alternatives that are, informally speaking, natural, and show that both result in an impossibility. Specifically, in both cases the parameter space is Θ = [0, 1], with θ = 0 corresponding to the egalitarian solution and θ = 1 corresponding to the equal-loss solution. However, for any 0 < θ < 1 the following holds: (i) under the first parametrization the corresponding solution does not make its selection from the part of S’s frontier that is between E(S) and EL(S),5 and (ii) under the second parametrization the solution coincides with E if θ < 12 , it coincides with EL if θ > 12 , and is undetermined for θ = 12 . I therefore settle for describing a midpoint between E and EL, not an entire spectrum. I do so by building on the CES structure. Specifically, I apply the CES operator to the functions of the individual utilities that EL and E seek to maximize. Namely, 1 I consider the maximization of [(min{a1 (S) − x1 , a2 (S) − x2 })ρ + (min{x1 , x2 })ρ ] ρ . Interestingly, it turns out that the maximizer is independent of ρ. Moreover, this maximizer turns out to be related, in a non-trivial way, to endogenous dictatorship. The rest of the paper is organized as follows. Section 2 concerns the parametrized family that stretches from egalitarianism to (endogenous) dictatorship; Section 3 proposes a characterization of the endogenous dictatorship solution; Sections 4 and 5 concern equal losses. 5 The requirement that a “midpoint between the solutions ψ and φ” makes a selection from the part of S’s boundary that is between ψ(S) and φ(S) is a central geometrical feature of CES; namely, σ ρ (S) is between E(S) and U (S) for every S ∈ B and ρ ∈ (−∞, 1]. For more on such betweenness properties in bargaining, see Cao (1982), Marco et al. (1995), Naeve-Steinweg (2004), and Rachmilevitch (2014a,b,c). 5 2 From egalitarianism to dictatorships: positive results Let the endogenous dictatorship solution, ED, be defined as follows: Di (S) if a (S) > a (S) i j ED(S) ≡ E(S) if a (S) = a (S). 1 2 Given θ ∈ [0, 1), let µθ be the solution that assigns to each S the point of its frontier, x ∈ W P (S),6 that satisfies: θ x2 a2 (S) 1−θ =[ ] . x1 a1 (S) Note that µ0 = E, that the limit θ → 1 corresponds to ED, and that the solution moves continuously from the former to the latter as θ increases from zero to one. 1 The “midpoint” of this family, the solution µ 2 , is the Kalai-Smorodinsky solution, hereafter denoted KS. This is analogous to the fact that the CES family is “centered” around the Nash solution. Proposition 1 below is analogous to the fact that N is the unique CES solution that satisfies scale invariance. Proposition 1. A solution in {µθ : θ ∈ [0, 1)} is scale invariant if and only if it is the Kalai-Smorodinsky solution. Proof. It is well known that KS is scale invariant. Conversely, consider µθ for an arbitrary θ ∈ [0, 1). Let S be a triangle with a(S) = (1, b) for some b > 0, and consider the linear transformations l = (l1 , l2 ), where l1 is the identity and l2 (a) = λa, for some λ > 1. Suppose that µθ (S) = (x, y), which means that y x θ = b 1−θ . Therefore, if this solution satisfies scale invariance, the application of l to S would imply 6 λy x θ θ = (λ) 1−θ b 1−θ , which implies θ = 12 . W P (S) ≡ {x ∈ S : y > x ⇒ y ∈ / S} is the weak Pareto frontier of S; P (S) ≡ {x ∈ S : (y ≥ x)&(y ∈ S) ⇒ y = x} is its strict Pareto frontier. 6 Next, Proposition 2 parallels the fact that N is the unique CES solution that satisfies midpoint domination. Proposition 2. A solution in {µθ : θ ∈ [0, 1)} satisfies midpoint domination if and only if it is the Kalai-Smorodinsky solution. Proof. It is well known that KS satisfies midpoint domination. Conversely, consider µθ for an arbitrary θ ∈ [0, 1). Let S be a triangle with a(S) = (1, b) for some b > 0. It is easy to see that µθ (S) = 12 a(S) iff θ = 12 . 3 A characterization of endogenous dictatorship Nash (1950) characterized N on the basis of scale invariance (SINV) and the following three axioms; in their statements, S and T are arbitrary elements of B: • Weak Pareto optimality: σ(S) ∈ W P (S). • Symmetry (SY): If S is such that [(a, b) ∈ S ⇔ (b, a) ∈ S], then σ1 (S) = σ2 (S). • Independence of irrelevant alternatives (IIA): σ(T ) ∈ S ⊂ T implies σ(S) = σ(T ). Below I characterize ED by four axioms. For simplicity, I derive the axiomatization on the domain of strictly comprehensive problems: those S ∈ B such that P (S) = W P (S); call this domain C.7 Relatively to Nash’s theorem, I (i) leave WPO intact, (ii) strengthen SY, (iii) weaken IIA, and (iv) replace SINV. The strengthening of SY is this: ideal point order (IPO) requires σi (S) ≥ σj (S) whenever ai (S) ≥ aj (S). It is easy to see that IPO implies SY, and, additionally, that IPO captures much of ED’s essence. Let B i ≡ {S ∈ B : ai (S) > aj (S)}. The strengthening of IIA is this: constrained 7 The characterization extends to B with an appropriate continuity axiom. 7 IIA imposes the requirement of IIA only on pairs of problems (S, T ) for which there is an i such that S, T ∈ B i . This weakening of IIA is essentially a stronger version of homogeneous ideal IIA (Dubra (2001), Rachmilevitch (2014d)), which imposes the requirement of IIA only on pairs of problem (S, T ) such that a(S) = ra(T ) for some r ≤ 1.8 The qualification “essentially” in the preceding sentence concerns continuity; specifically, on the domain of continuous solutions, constrained IIA implies homogeneous ideal IIA.9 Finally, I utilize strong ideal point monotonicity (S.IPM), which requires that if S ⊂ T , aj (S) = aj (T ), ai (S) < ai (T ) and σi (S) > 0, then σi (S) < σi (T ). Proposition 3. There is a unique solution on C that satisfies weak Pareto optimality, constrained independence of irrelevant alternatives, strong ideal point monotonicity, and the ideal point order property: it is the endogenous dictatorship solution. Proof. It is easy to check that ED satisfies the axioms. Conversely, let σ be an arbitrary solution that satisfies them. Let S ∈ C. If a1 (S) = a2 (S) then by WPO and IPO σ(S) = E(S) = ED(S). Suppose, then, wlog, that a1 (S) > a2 (S). Assume by contradiction that x ≡ σ(S) 6= ED(S). Given r > 0, let Qr ≡ {s ∈ S : s1 ≤ r}. Obviously we can find α and β such that x1 < α < β < a1 (S) and such that Qα , Qβ ∈ B 1 . By constrained IIA, σ(Qα ) = σ(Qβ ). This contradicts S.IPM. The axioms in Proposition 3 are independent. The solution λED for some λ ∈ (0, 1) satisfies all of them but WPO; the solutions KS and EL satisfy all of them but constrained IIA; Di satisfies all of them but IPO; the solution that assigns to each S ∈ B i the maximal point of the form λ[2ei + ej ] and coincides with E on B \ (B 1 ∪B 2 ) 8 Homogeneous ideal IIA is a strengthening of Roth’s (1977) restricted IIA, which imposes, in addition to the requirement of homogeneous ideal IIA, r = 1. Dubra (2001) used the term “restricted IIA” to what is called above “homogeneous ideal IIA.” I introduced the latter term in Rachmilevitch (2014d) in order to distinguish it from Roth’s axiom. 9 A solution σ is continuous if whenever {Sn } converges to the problem S in the Hausdorff topology, it is true that σ(Sn ) → σ(S). 8 satisfies all the axioms but S.IPM.10 The obvious drawback of ED is—pardon the triviality—its dictatorial character. One way to fix this unappealing feature, but keep in place the favorable treatment of the endogenous dictator, is expressed by the following solution, the midpoint robust endogenous dictatorship solution, hereafter denoted as mED. For S ∈ B i , this solution assigns to the endogenous dictator (player i) his maximum possible payoff, subject to the constraint that player j receives 12 aj (S); on B \ (B 1 ∪ B 2 ), the solution coincides with E. Interestingly, a version of mED has been studied in a non-cooperative setting by Sertel (1992). Sertel’s (1992) model is as follows: two players face a triangular bargaining problem, in which player 1’s ideal payoff is 1 and player 2’s ideal payoff is α > 1. Before the bargaining stage, each player can commit to pre-donate a fraction of his would-be payoff to the other player. After pre-donations are made, the Nash solution is applied to the resulting problem. Sertel showed that the equilibrium of this two-stage game is such that player 1—the one who is not the endogenous dictator (in the original problem)—receives half of his ideal payoff, and player 2 receives his maximal payoff subject to the constraint that player 1 receives half of his ideal payoff. Finally, it is worth noting that there are other ways to express the idea underlying ED. For example, in a richer model where a solution is defined to be an assignment of a lottery over agreements for each problem, the following variant of ED presents itself: a (stochastic) solution that coincides with Di on B i and selects each element of {D1 (S), D2 (S)} with equal probability for S ∈ B \ (B 1 ∪ B 2 ). Exploring such alternative notions of endogenous dictatorship is beyond the scope of the present paper. 10 1 e = (1, 0), e2 = (0, 1). 9 4 From egalitarianism to equal-losses: difficulties Consider the following parametrization: given θ ∈ [0, 1] let x ∈ W P (S) be such that: θ[a1 (S) − x1 ] + (1 − θ)x1 = θ[a2 (S) − x2 ] + (1 − θ)x2 . (1) Clearly, θ = 0 corresponds to E and θ = 1 corresponds to EL. Unfortunately, however, for 0 < θ < 1 and S such that a1 (S) 6= a2 (S), there does not exist an x ∈ W P (S) that satisfies (1) and lies “between” E(S) and EL(S). Proposition 4. Let S be a problem such that a1 (S) 6= a2 (S) and let θ ∈ (0, 1). Then there does not exist a point x ∈ {s ∈ W P (S) : si ≥ min{e(S), ELi (S)}∀i} that satisfies (1). Proof. Let S and θ be as above. Wlog, suppose that a1 (S) > a2 (S) (so EL(S) is to the right of E(L)). Case 1: θ ≤ 21 . For x = E(S), the LHS of (1) exceeds the RHS. The derivative of the LHS wrt x1 and of the RHS wrt x2 is (1 − 2θ) ≥ 0, so when we move from E(S) to EL(S) the LHS weakly increases and the RHS weakly decreases. Therefore, there is no x ∈ W P (S) between E(S) and EL(S) that satisfies (1). Case 2: θ > 12 . For x = EL(S), the LHS of (1) exceeds the RHS. The derivative of the LHS wrt x1 and of the RHS wrt x2 is (1 − 2θ) < 0, so when we move from EL(S) to E(S) the LHS increases and the RHS decreases. Therefore, there is no x ∈ W P (S) between E(S) and EL(S) that satisfies (1). All the solutions mentioned in this paper are related to maximization problems. The obvious case in point is the utilitarian solution, but the same is true for the egalitarian, dictatorial, and equal-loss solutions: given a problem S, the point E(S) is a maximizer of min{x1 , x2 } over x ∈ S, Di (S) is a maximizer of xi over x ∈ S, and EL(S) maximizes min{a1 (S) − x1 , a2 (S) − x2 } over x ∈ S. 10 The following parametrization therefore presents itself as a sensible candidate for describing a spectrum between E and EL: given θ ∈ [0, 1], consider the maximizer of W (S|θ) ≡ θmin{a1 (S) − x1 , a2 (S) − x2 } + (1 − θ)min{x1 , x2 }. Unfortunately, this parametrization has very limited implications on the solution. Proposition 5. Let S be a problem such that a1 (S) 6= a2 (S). Then E(S) is the unique maximizer of W (S|θ) for θ < 21 , and EL(S) is the unique maximizer of W (S|θ) for θ > 21 . Proof. Fix such θ and S. Wlog, suppose that a1 (S) > a2 (S). On the part of W P (S) between E(S) and EL(S), the objective W (S|θ) assumes the form θ(a2 (S)−x2 )+(1− θ)x2 , hence its derivative wrt x2 is x2 (1 − 2θ) on the aforementioned domain. If θ < the unique maximizer is E(S) while if θ > 1 2 1 2 the unique maximizer is EL(S).11 Propositions 4 and 5 imply that an “equal-gains-to-equal-losses spectrum” does not lend itself to an obvious parametrization. This is unfortunate, because such parameterizations pop out naturally in discussions about distributive justice.12 5 From egalitarianism to equal-losses: a midpoint approach If obtaining a parametrized family that “ranges from E to EL” turns out to be nonobvious, one way to go is to embark on a more modest task, and look for a “midpoint” between E and EL, not an entire spectrum. Since E maximizes min{x1 , x2 } and EL maximizes min{a1 (S) − x1 , a2 (S) − x2 }, a maximizer of an appropriately selected mixture of these objectives can be viewed as a midpoint between their respective maximizers. The objective W from above fails to be an appropriate such mixture, since its linearity implies corner solutions. The following smoothening of W presents 11 12 Any point between E(S) and EL(S) is a maximizer of W (S| 12 ). See, e.g., Moreno-Ternero and Villar (2006). 11 itself as a candidate for fixing this problem: given a parameter ρ ∈ (−∞, 0) ∪ (0, 1], consider the following: 1 ν ρ (S) ≡ argmaxx∈S [(min{a1 (S) − x1 , a2 (S) − x2 })ρ + (min{x1 , x2 })ρ ] ρ . Note that ν 1 = W (.| 21 ). It is easy to check that ν ρ (S) is between E(S) and EL(S) for every ρ, and therefore, in particular, for S such that a1 (S) = a2 (S) we have ν ρ (S) = E(S) = EL(S) = KS(S).13 Interestingly, the solution point is independent of ρ also for problems with non-identical ideal payoffs—it is the point which is selected by the midpoint robust endogenous dictatorship solution, mED. Proposition 6. ν ρ = mED for every ρ ∈ (−∞, 0) ∪ (0, 1). Proof. Clearly ν ρ (S) = mED(S) for S such that a1 (S) = a2 (S). Consider then an S such that, wlog, a1 (S) > a2 (S). Consider first ρ > 0. Here, the solution makes its selection from between E(S) and EL(S) in order to maximizes (a2 (S)−x2 )ρ +xρ2 . The derivative of this expression wrt x2 is −ρ(a2 (S)−x2 )ρ−1 +ρxρ−1 2 , which is equal to zero at x2 = a2 (S) . 2 It is easy to check that the second derivative at this point is negative. For ρ < 0, the analogous arguments apply (the solution minimizes (a2 (S) − x2 )ρ + xρ2 and the second derivative is positive at at optimum). As for ρ ∈ {−∞, 0}, it is easy to see that limρ→−∞ ν ρ (S) = mED(S) for any S, and that the product function (min{a1 (S) − x1 , a2 (S) − x2 }) · (min{x1 , x2 }) (which is analogous to ρ = 0 in the case of CES) is maximized at mED(S). Finally, it is easy to see that for every S such that E(S) 6= EL(S) the solution mED splits the parts of W P (S) between E(S) and EL(S) into two regions, and that KS(S) is in the region that is closer to E(S). 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