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:
[email protected] 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). Hence, since mED can be viewed as a
“midpoint between equal-gains and equal-losses,” one is led to the view that KS is
more oriented towards egalitarianism than towards the equal-loss principle.
13
This is analogous to µθ (S) = E(S) = EL(S) = KS(S) for all θ ∈ [0, 1), when S is as above.
12
Acknowledgments: I wish to thank Emin Karagözoglu and Yasemin Dede for stimulating conversations and comments that contributed to the research reported in this
paper.
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