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Bargaining under Uncertainty and the Monotone Path Solutions
GAMES AND ECONOMIC BEHAVIOR ARTICLE NO.
14, 173–189 (1996)
0047
Bargaining under Uncertainty and the Monotone Path Solutions∗ Walter Bossert† and Ed Nosal Department of Economics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
and Venkatraman Sadanand Department of Economics, University of Guelph, Guelph, Ontario N1G 2W1, Canada Received January 26, 1993
Uncertainty with respect to the feasible set of utility vectors is introduced in an axiomatic bargaining model. Given a criterion for nonprobabilistic decision-making under uncertainty, a natural efficiency requirement can be imposed on a bargaining solution. Using the maximin ordering, the strictly monotone path solutions (generalizations of the egalitarian solution) to the bargaining problem are characterized as the only continuous solutions that satisfy this efficiency axiom. If the maximin criterion is replaced by the maximax ranking or a strict convex combination of the maximin and the maximax criterion, imposing our efficiency axiom and continuity leads to the dictatorial solutions. Journal of Economic Literature Classification Number: C78. © 1996 Academic Press, Inc.
1. INTRODUCTION In this paper, a generalization of Nash’s (1950) axiomatic model of bargaining is considered by incorporating uncertainty with respect to the feasible set of utility vectors (see Chun and Thomson, 1990a,b,c, for discussions of uncertainty ∗ An earlier version of the paper was presented at the 1992 Buffalo–Cornell–Rochester Economic Theory conference in Buffalo. We thank Nejat Anbarci, Patrick Legros, William Thomson, and a referee for their comments. Financial support through grants from the Social Sciences and Humanities Research Council of Canada, the University of Guelph, and the University of Waterloo is gratefully acknowledged. †
FAX: (519) 725-0530. E-mail: [email protected]. 173 0899-8256/96 $18.00 Copyright © 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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with respect to the disagreement point). Assuming the disagreement point to be fixed, a bargaining problem with uncertainty can be described by a vector of possible feasible sets that may occur. There is a straightforward way of extending a bargaining solution for problems without uncertainty to solve problems under uncertainty. For each of the possible feasible sets, if the state of the world represented by this feasible set occurs, the corresponding outcome according to the solution without uncertainty is defined as the contingent outcome. For some solutions, however, this outcome might not be very satisfying. For example, there may exist another vector of contingent outcomes that can make all agents better off compared to the one prescribed by the bargaining solution. Thus, a desirable property for a bargaining solution is to be efficient under uncertainty in the sense that the resulting vector of contingent outcomes is not Pareto dominated by another vector of outcomes that are feasible for the respective states of the world. Of course, in order to find solutions with such efficiency properties, we have to specify how agents evaluate vectors of contingent outcomes. If probabilities can be assigned to different feasible sets that can occur, a reasonable condition to impose on a bargaining solution is to require that all agents benefit from reaching an early agreement on the “expected” bargaining problem. This property of bargaining solutions is desirable, because waiting until the uncertainty is resolved can lead to a Pareto-dominated situation. In Myerson (1981), a concavity condition with respect to the feasible set is used to deal with this problem. Related conditions include Perles and Maschler’s (1981) superadditivity and the weak linearity axiom used in Peters (1986). Riddell (1981) considers a situation of bargaining under uncertainty where agents can form probability assessments. When agents bargain after the uncertainty is resolved, they apply Nash’s (1950) axioms to determine the solution for the state-contingent bargaining problem. Agents can assess the value of this problem by simply calculating the expected utility of the state-contingent bargaining solutions. If, however, agents bargain over allocations before the uncertainty is resolved, their expected utilities can serve as the disagreement point and a new feasible set can be constructed. The solution to this bargaining problem is obtained by applying the axioms to the new feasible set. Thus, if agents can anticipate the bargaining solution in each state, then the solution to the bargaining problem under uncertainty is obtained by “applying the Nash axioms twice”: first to each state-contingent feasible set and then to the set that results when bargaining occurs before the state of nature is revealed. If, on the other hand, the agents cannot assign (objective or subjective) probabilities to the occurrence of different states of the world, standard expected utility arguments cannot be applied. Consequently, in these situations (and these are the situations we want to analyze in this paper), nonprobabilistic models of choice under uncertainty have to be used. In particular, we will investigate
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the consequences of applying the maximin criterion, the maximax criterion, and combinations thereof. For the maximin ranking, imposing efficiency under uncertainty and continuity characterizes the class of strictly monotone path solutions (see Thomson and Myerson, 1980; Thomson, 1994), a generalization of Kalai’s (1977) proportional (or weighted egalitarian) solutions. Adding symmetry to the above axioms leads, of course, to the egalitarian solution. If the maximax decision rule or a strict convex combination of maximin and maximax (see Arrow and Hurwicz, 1972) is used, the implications of imposing efficiency under uncertainty are not very satisfying: only the dictatorial solutions are efficient and continuous in these cases. To prove our results, it will be sufficient to consider cases where there are two possible states of the world (represented by two possible feasible sets). Given continuity (which we will assume in all our results), our efficiency axiom with two possible states of the world is equivalent to an analogous formulation involving any number of possible states. Because the two-state version is easier to relate to other axioms commonly used in bargaining theory, and because it simplifies the exposition of our results, we have chosen to use it. However, given the above-mentioned equivalence, this choice does not involve any loss of generality.
2. BARGAINING PROBLEMS AND SOLUTIONS The set of positive integers is denoted by N. Let R (R+ , R++ ) denote the set of (nonnegative, positive) real numbers, and let Rn (Rn+ , Rn++ ) be the (nonnegative, positive) Euclidean n-orthant, n ∈ N. The origin of Rn is denoted by 0. We use ei ∈ Rn+ to denote the ith unit vector. Our notation for vector inequalities is ≥, >, À. For a function ψ with range Rn+ , we let ψi = ei · ψ, i = 1, . . . , n. An n-person bargaining problem is described by a set S ⊆ Rn+ , which is interpreted as the set of feasible utility vectors of the agents. The set of agents is N = {1, . . . , n}. Throughout the paper, we assume that n ≥ 2 is fixed. S ⊆ Rn+ is comprehensive (strictly comprehensive) if and only if, for all x ∈ S, for all y ∈ Rn+ , x > y ⇒ y ∈ S (and ∃z ∈ S : z À y). The following class of bargaining problems is considered in this paper. DEFINITION 1. 6 is the set of bargaining problems S ⊆ Rn+ such that S is convex, compact, comprehensive, and there exists x ∈ S such that x À 0.
˜ We denote the subclass of 6 of strictly comprehensive problems by 6. Dropping convexity in the above definition would not change our characterization results—their proofs do not make use of this assumption. We include convexity in the definition of 6, since this is the domain usually considered in
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axiomatic models of bargaining. Note that the agents’ utilities are not interpreted as von Neumann–Morgenstern utilities (we consider nonprobabilistic decision rules under uncertainty), and therefore, if convexity is required, it has to be motivated in a different way. One possibility is to assume that the agents have concave utility functions defined on a set of physical outcomes, which implies that the feasible set of utility vectors is convex without invoking arguments that involve randomization over outcomes. In the above definition, we implicitly assume that utilities are translation invariant, which allows us to choose, without loss of generality, the disagreement point (the utility vector that results if the agents fail to reach an agreement) to be the origin—see, for instance, Chun and Peters (1991), Blackorby et al. (1994), Thomson (1996) for details. In other words, we could interpret S as the set of feasible vectors of gains of the agents over the disagreement outcome. By restricting attention to subsets of Rn+ as feasible sets, we exclude the possibility that nonindividually rational points are taken into account. For one of our proofs, it will be convenient to consider the subclass of 6 of budget problems, defined by DEFINITION 2. A bargaining problem S ∈ 6 is a budget problem if and only if there exist p ∈ Rn++ and m ∈ R++ such that S = {x ∈ Rn+ | p · x ≤ m}. The budget problem defined by ( p, m) ∈ Rn+1 ++ is denoted by T ( p, m).
For S ∈ 6, the ideal utility of S for agent i is given by ai (S) = max{xi | x ∈ S} for all i ∈ N . The convex and comprehensive hull of A ⊆ Rn+ is denoted by cch(A). A bargaining solution (on 6) is a mapping F: 6 7→ Rn+ such that F(S) ∈ S for all S ∈ 6. The following axioms are used in this paper. All of them are well-established in the literature and do not require any explanation. Weak Pareto optimality (WPO). For all S ∈ 6, for all y ∈ Rn+ , y À F(S) ⇒ y ∈/ S. The set of weakly Pareto optimal points in S ∈ 6 is denoted by WP(S), that is, WP(S) = {x ∈ S |6 ∃y ∈ S such that y À x}
∀S ∈ 6.
Continuity (CONT). For all sequences {S ν } in 6, for all S ∈ 6, lim S ν = S (in the Hausdorff topology) ⇒ lim F(S ν ) = F(S).
ν→∞
ν→∞
Let 5 be the set of all permutations π of N . For π ∈ 5 and S ⊆ Rn , let π(S) = {y ∈ Rn | ∃x ∈ S such that y = π(x)}.
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Symmetry (SY). For all S ∈ 6, π(S) = S
∀π ∈ 5 ⇒ Fi (S) = Fj (S)
∀i, j ∈ N .
Strong monotonicity (SM). For all S, S 0 ∈ 6, S 0 ⊆ S ⇒ F(S 0 ) ≤ F(S). The solutions that are of particular interest in this paper are the strictly monotone path solutions (see Thomson and Myerson, 1980; Thomson, 1996). To define this class of solutions, we first introduce the notion of a strictly monotone path (see Thomson, 1996, Chapter 2). n DEFINITION 3. A strictly Pn monotone path (in R+ ) is the image G of a function n ψ: R+ 7→ R+ such that i=1 ψi (t) = t for all t ∈ R+ , ψ(t) ≥ ψ(s) whenever t ≥ s, and, for all i ∈ N , ψi (t) = 0 whenever t 6= s and ψi (t) = ψi (s).
The strictly monotone path solution relative to the strictly monotone path G, E G , is defined by letting, for each S ∈ 6, E G (S) be the maximal point in S along G. Choosing ψi (t) = t/n for all i ∈ N , t ∈ R+ in Definition 3 leads to the egalitarian solution E—see Kalai (1977). The ith dictatorial solution D i , i ∈ N , is defined by D i (S) = ai (S) · ei for all S ∈ 6. Clearly, dictatorial solutions are special cases of strictly monotone path solutions. Other well-known bargaining solutions (which are not discussed in this paper) include, among others, the Nash (1950) solution and the Kalai–Smorodinsky (1975) solution. A result which will turn out to be very useful for our analysis is the following characterization of the strictly monotone path solutions—see Thomson (1996, Chapter 2). For completeness, we sketch a proof of this result. THEOREM 1 (Thomson, 1996). A bargaining solution F satisfies (WPO), (CONT), and (SM) if and only if F is a strictly monotone path solution.
Proof. That the strictly monotone path solutions satisfy (WPO), (CONT), and (SM) is easy to verify. suppose F satisfies these axioms. For PConversely, n xi ≤ t}, and define ψ(0) = 0 and ψ(t) = F(S t ) t > 0, let S t = {x ∈ Rn+ | i=1 for t > 0. Then the image of ψ is G = {F(S t ) | t > 0} ∪ {0}. By (WPO), P n i=1 ψi (t) = t for all t > 0, and by (SM), ψ(t) ≥ ψ(s) whenever t ≥ s. ˜ Let Next, we show that P F(S) = E G (S) for all S ∈ 6. Suppose first S ∈ 6. n xi . First, consider the case x À 0. Choose ε > 0 x = E G (S) and t = i=1
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such that the set Sε = cch({x, (x1 + ε) · e1 , . . . , (xn + ε) · en }) satisfies Sε ⊆ S ∩ S t . Clearly, such an ε exists, and Sε ∈ 6. By definition, F(S t ) = x. Since Sε ⊆ S t , (SM) implies F(Sε ) ≤ x, and by (WPO), F(Sε ) = x. Again using (SM), Sε ⊆ S implies F(S) ≥ F(Sε ) = x. From (WPO) and the strict comprehensiveness of S, it follows F(S) = x = E G (S). If xi = 0 for some i ∈ N , the definition of Sε can be modified to Sε = cch({x} ∪ {ε · ei | i ∈ N such that xi = 0}), and the same argument as above can be applied. If S is not strictly comprehensive, we can appeal to (CONT). To complete the proof that G is a strictly monotone path, suppose, by way of contradiction, thereP exist i ∈ N and t, s ∈ R+ such that t 6= s and ψi (t) = n ψi (u) = u for all u ∈ R+ and t 6= s, it follows ψi (s) > 0. Because i=1 ψ(t) 6= ψ(s). Let S ∈ 6 be such that {ψ(t), ψ(s)} ⊆ WP(S). Furthermore, let {T ν }, {S ν } be two sequences of strictly comprehensive elements of 6 such that ψ(t) ∈ WP(T ν ), ψ(s) ∈ WP(S ν ) for all ν ∈ N, and limν→∞ T ν = limν→∞ S ν = S. It follows F(T ν ) = ψ(t) and F(S ν ) = ψ(s) for all ν ∈ N, and by (CONT), F(S) = ψ(t) and F(S) = ψ(s), contradicting ψ(t) 6= ψ(s). In the presence of (WPO) and (CONT), (SM) could be replaced by other conditions such as domination—see Thomson and Myerson (1980) for details.
3. EFFICIENT BARGAINING UNDER UNCERTAINTY Suppose there is some uncertainty concerning the feasible set of a bargaining problem. We assume that there are two possible states of the world, represented by two possible feasible sets. Hence, a bargaining problem under uncertainty can be defined as a pair (S, S 0 ) ∈ 6×6. A natural efficiency requirement for bargaining solutions in the presence of uncertainty is that, for any bargaining problem under uncertainty (S, S 0 ), the pair of contingent outcomes (F(S), F(S 0 )) cannot be Pareto improved by some other pair of feasible contingent outcomes. Clearly, to formalize such an efficiency notion, we have to specify how agents rank different pairs of contingent outcomes. From the viewpoint of agent i ∈ N , bargaining problems under uncertainty can be evaluated in terms of the utility he or she enjoys in each possible state according to F, that is, individual i ∈ N evaluates pairs of contingent outcomes (F(S), F(S 0 )) on the basis of the pair (Fi (S), Fi (S 0 )). We assume that the agents’ evaluations of contingent outcomes are based on the minimal and the maximal utility they can achieve (examples for such
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nonprobabilistic decision rules are convex combinations of the minimal and maximal utilities—see the definition of the Arrow–Hurwicz criterion below). Hence, we restrict attention to rankings of pairs of contingent utilities that can be represented by a function of the minimal and maximal component of a pair of utilities. Formally, we consider functions g: R2+ 7→ R such that g is nondecreasing in both arguments and increasing in at least one argument, and define an associated ranking ºg of pairs of contingent utilities in the following way. DEFINITION 4. Let g: R2+ 7→ R be nondecreasing in both arguments and increasing in at least one argument. Then the ordering ºg of contingent utility pairs is defined by
∀z, z 0 ∈ R2+ ,
z ºg z 0 ⇔ g(min{z 1 , z 2 }, max{z 1 , z 2 }) ≥ g(min{z 10 , z 20 }, max{z 10 , z 20 }).
Hence, the first (second) argument of g is the minimum (maximum) of a pair of contingent utilities, and g specifies how the minimum and the maximum are combined according to the criterion defined by this function. The strict preference relation corresponding to ºg is denoted by Âg , and ∼g is the corresponding indifference relation. Special cases of ºg that are of particular interest in this paper are the maximin criterion, the maximax criterion, and the Arrow–Hurwicz α-criterion (which is a convex combination of maximin and maximax). The maximin criterion is obtained by using the function gmin , defined by gmin (u) = u 1
∀u ∈ R2+ ;
the maximax criterion corresponds to gmax , which is defined by gmax (u) = u 2
∀u ∈ R2+ ;
and the α-criterion for 0 < α < 1 (see Arrow and Hurwicz, 1972) is obtained by choosing gα , where gα (u) = α · u 1 + (1 − α) · u 2
∀u ∈ R2+ .
Given a function g with the above monotonicity properties (and therefore, given a decision criterion ºg as defined in Definition 4), weak efficiency under uncertainty with respect to g requires that F selects contingent outcomes that cannot be Pareto improved according to ºg . Formally, this axiom is defined as Weak efficiency under uncertainty with respect to g (WEg ). ∀S, S 0 ∈ 6 with S 6= S 0 , 6 ∃(x, x 0 ) ∈ S × S 0
such that (xi , xi0 ) Âg (Fi (S), Fi (S 0 )) ∀i ∈ N .
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A stronger version of this efficiency condition could be obtained by replacing the part “Âg . . . ∀i ∈ N ” by “ºg . . . ∀i ∈ N with at least one strict preference for some agent in N ” in the above definition. This requirement is equivalent to ˜ If problems that are comprehensive but not (WEg ) for the restriction of F to 6. necessarily strictly comprehensive are included in the domain of F (as is the case in this paper), this strengthening of (WEg ) would turn our characterizations into impossibility results. For any choice of g satisfying the above monotonicity requirements, (WEg ) implies (WPO). This is stated in LEMMA 1. Let g: R2+ 7→ R be nondecreasing in both arguments and increasing in at least one argument. If a bargaining solution F satisfies (WEg ), then F satisfies (WPO).
Proof. Suppose F satisfies (WEg ), but violates (WPO). Then there exist S ∈ 6 and x ∈ S such that x À F(S). Let x 0 ∈ S be such that x 0 À F(S) and x À x 0 (clearly, such a point x 0 exists by the definition of 6). Let S 0 = cch({x 0 }). By definition of x 0 , S 0 ∈ 6. Now it follows min{Fi (S), Fi (S 0 )} < xi0 < xi
∀i ∈ N
max{Fi (S), Fi (S 0 )} ≤ xi0 < xi
∀i ∈ N
and
(see Fig. 1 for an illustration in the case n = 2). By the monotonicity properties of g, it follows (xi , xi0 ) Âg (Fi (S), Fi (S 0 ))
∀i ∈ N ,
contradicting (WEg ).
4. THE MAXIMIN CRITERION In this section, we use the maximin rule to provide a characterization of the strictly monotone path solutions. Before stating and proving this result, we present a lemma that will be used in the proof. LEMMA 2. If a bargaining solution F satisfies (CONT) and (WEgmin ), then F satisfies (SM).
Proof. Suppose F satisfies (CONT) and (WEgmin ). By way of contradiction, suppose (SM) is violated. Then there exist S, S 0 ∈ 6, i ∈ N such that S 0 ⊆ S and Fi (S 0 ) > Fi (S).
(1)
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FIGURE 1
˜ From (WPO)—see Lemma 1—and the strict compreFirst, suppose S, S 0 ∈ 6. hensiveness of S and S 0 , it follows that there exists j ∈ N such that Fj (S) > Fj (S 0 ).
(2)
Define N s , N 0 ⊆ N by k ∈ N s ⇔ Fk (S) > Fk (S 0 ), k ∈ N 0 ⇔ Fk (S 0 ) > Fk (S) for all k ∈ N . From (1) and (2), it follows that N s and N 0 are nonempty. Furthermore, Fk (S) > 0 for all k ∈ N s , and Fk (S 0 ) > 0 for all k ∈ N 0 . By the strict comprehensiveness of S and S 0 , we can find x ∈ S, x 0 ∈ S 0 such that Fk (S) > xk > Fk (S 0 ) Fk (S) < xk Fk (S 0 ) > xk0 > Fk (S) Fk (S 0 ) < xk0
∀k ∀k ∀k ∀k
∈ ∈ ∈ ∈
Ns, N \N s , N 0, N \N 0 .
By construction, min{xk , xk0 } > min{Fk (S), Fk (S 0 )} for all k ∈ N , and hence, (WEgmin ) is violated.
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FIGURE 2
If S or S 0 is not strictly comprehensive, we can use (CONT) to obtain the desired conclusion. See Fig. 2 for an illustration of Lemma 2 in the two-person case. Clearly, if we restrict attention to strictly comprehensive problems, (CONT) is not needed in the above result. Similar results concerning the equivalence of (WEgmin ) and other independence or monotonicity conditions (given (WPO) and (CONT)) could be formulated—see Thomson and Myerson (1980). The main result of this section is THEOREM 2. A bargaining solution F satisfies (CONT) and (WEgmin ) if and only if F is a strictly monotone path solution.
Proof. That the strictly monotone path solutions satisfy the required axioms can be verified easily. To prove the converse, recall that, by Lemma 1, (WEgmin ) implies (WPO), and combine Theorem 1 and Lemma 2. If symmetry is added to the axioms in Theorem 2, the egalitarian solution is characterized. This follows from the observation that E is the only symmetric monotone path solution. Hence, we obtain
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183
COROLLARY 1. A bargaining solution F satisfies (CONT), (SY), and (WEgmin ) if and only if F = E.
5. THE MAXIMAX CRITERION If we impose weak efficiency under uncertainty given the maximax criterion, only a rather unattractive subclass of the strictly monotone path solutions satisfy this requirement and continuity, namely the dictatorial solutions. THEOREM 3. A bargaining solution F satisfies (CONT) and (WEgmax ) if and only if F is a dictatorial solution.
Proof. That the dictatorial solutions satisfy (CONT) and (WEgmax ) can be verified by the reader. Suppose now F satisfies these two axioms. We first show ˜ there exists i ∈ N such that F(S) = ai (S) · ei . Suppose this that, for any S ∈ 6, ˜ i, j ∈ N such that is not the case. By (WPO)—see Lemma 1—there exist S ∈ 6, Fi (S) > 0 and Fj (S) > 0. By the strict comprehensiveness of S, Fk (S) < ak (S) for all k ∈ N . ˜ be such that cch(F(S)) ⊆ S 0 ⊆ S and Let S 0 ∈ 6 W P(cch(F(S))) ∩ W P(S 0 ) = W P(S 0 ) ∩ W P(S) = {F(S)}. If F(S 0 ) = F(S), let x ∈ S, x 0 ∈ S 0 be such that x1 > F1 (S) and xk0 > Fk (S 0 ) for all k ∈ N \{1}. If F(S 0 ) 6= F(S), define N 0 ⊆ N by k ∈ N 0 ⇔ Fk (S 0 ) > Fk (S) for all k ∈ N . Choose x ∈ S, x 0 ∈ S 0 such that xk > Fk (S 0 ) ∀k ∈ N 0
and
xk0 > Fk (S) ∀k ∈ N \N 0 .
In either case, it follows max{xk , xk0 } > max{Fk (S), Fk (S 0 )} for all k ∈ N , ˜ there exists i ∈ N contradicting (WEgmax ). This proves that, for each S ∈ 6, such that F(S) = ai (S) · ei . See Fig. 3 for an illustration of the above argument for n = 2. ˜ and i, j ∈ N such that i 6= j, Fi (S) = ai (S) · ei , If there exist S, S 0 ∈ 6 and Fj (S 0 ) = a j (S 0 ) · e j , we obtain a contradiction to (CONT). Hence, F is ˜ and by (CONT), on 6. dictatorial on 6, 6. THE ARROW–HURWICZ CRITERION The characterization of the dictatorial solutions presented in the previous section can also be obtained for criteria other than maximax. If instead of minimax
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FIGURE 3
we use a strict convex combination of maximin and maximax (see Arrow and Hurwicz, 1972), we still obtain the dictatorial solutions as the only continuous solutions satisfying the corresponding efficiency axiom. This is shown in THEOREM 4. Let α ∈ (0, 1). A bargaining solution F satisfies (CONT) and (WEgα ) if and only if F is a dictatorial solution.
Proof. Clearly, the dictatorial solutions satisfy (WEgα ). Suppose F satisfies (CONT) and (WEgα ). First, we show S ∈ 6 is a budget problem ⇒ ∃i ∈ N such that F(S) = ai (S) · ei .
(3)
Suppose (3) is false. By (WPO), there exist ( p, m) ˜ ∈ Rn+1 ++ and i, j ∈ N such ˜ > 0 and Fj ( S) ˜ > 0, where S˜ = T ( p, m). that Fi ( S) ˜ By (CONT), there exists ˜ Fj (S) < Fj ( S), ˜ m ∈ R++ such that, with S = T ( p, m), we have Fi (S) < Fi ( S), and (i) Fi (S) > 0
or
(ii) Fj (S) > 0.
If (i) is true, let p 0 ∈ Rn++ be such that pi0 / p 0j < pi / p j and pk0 = pk for all k ∈ N \{i, j}. By (CONT), p 0 can be chosen such that Fi (S 0 ) > Fi (S) and
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Fj (S 0 ) > Fj (S), where S 0 = T ( p 0 , m). ˜ Now define xi = Fi (S) − ε, x j = Fj (S) +
x˜i = Fi (S 0 )+
pi · ε, xk = Fk (S) ∀k ∈ N \{i, j}, pj
α p0 α ·ε, x˜ j = Fj (S 0 )− i0 · ·ε, x˜k = Fk (S 0 ) ∀k ∈ N \{i, j}, 1−α pj 1 − α
where ε ∈ R++ is chosen sufficiently small so that xi > 0 and x˜ j > x j . Clearly, x ∈ S and x˜ ∈ S 0 . We obtain α · xi + (1 − α) · x˜i = α · Fi (S) + (1 − α) · Fi (S 0 ) and
" 0
α · x j + (1 − α) · x˜ j = α · Fj (S) + (1 − α) · Fj (S ) + α · ε ·
pi p0 − i0 pj pj
#
> α · Fj (S) + (1 − α) · Fj (S 0 ). Therefore, (xi , x˜i ) ∼gα (Fi (S), Fj (S 0 )) and (x j , x˜ j ) Âgα (Fj (S), Fj (S 0 )). Furthermore, since xk = Fk (S) and x˜k = Fk (S 0 ) for all k ∈ N \{i, j}, (xk , x˜k ) ∼gα (Fk (S), Fk (S 0 ))
∀k ∈ N \{i, j}.
Since S 0 is a budget problem, S 0 is strictly comprehensive. Let δ, δ˜ ∈ R++ , and define x 0 ∈ Rn by x j0 = x˜ j − δ, xk0 = x˜k + δ˜
∀k ∈ N \{ j}.
By the strict comprehensiveness of S 0 , δ and δ˜ can be chosen such that x j0 > Fj (S 0 ) and x 0 ∈ S 0 . Combined with the above observations, it follows (xk , xk0 ) Âgα (Fk (S), Fk (S 0 ))
∀k ∈ N ,
contradicting (WEgα ). Case (ii) is analogous. Therefore, (3) is true.
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FIGURE 4
Analogous to the argument in the proof of Theorem 3, (CONT) implies that the restriction of F to the class of budget problems must be dictatorial. ˆ ⊆ 6 be the set of all bargaining problems S with the property that there Let 6 exists a budget problem S 0 such that S ⊆ S0
and
W P(S) ∩ W P(S 0 ) = {F(S 0 )}.
(4)
ˆ and let S 0 be a budget problem such that (4) is satisfied. Suppose Let S ∈ 6, F(S) 6= F(S 0 ). Then there exists x 0 ∈ S 0 such that x 0 À F(S). Furthermore, let x = F(S 0 ) ∈ S. See Fig. 4 for an illustration in the two-person case. Since α ∈ (0, 1), it follows (xi , xi0 ) Âgα (Fi (S), Fi (S 0 ))
∀i ∈ N ,
ˆ must be dictatorial. Now contradicting (WEgα ). Hence, the restriction of F to 6 (CONT) can be applied again to conclude that F must be dictatorial on 6, which completes the proof.
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7. NONLINEAR CRITERIA An obvious question that arises after the above characterizations of dictatorial solutions is whether nonlinear combinations of the maximin and the maximax criterion lead to dictatorships as well. In general, this is not the case. Consider, for example, the function g∗ defined by √ √ g∗ (u) = u 1 + u 2 ∀u ∈ R2+ . Clearly, the dictatorial solutions satisfy (WEg∗ ) (the dictatorial solutions obviously satisfy (WEg ) for any choice of g satisfying the usual monotonicity requirements). However, there also exist nondictatorial continuous solutions that satisfy (W E g∗ ). Define the solution F ∗ by ( ) n X √ ∗ xi | x ∈ S ∀S ∈ 6. F (S) = argmax i=1 ∗
Clearly, F is a well-defined bargaining solution (note that the above maximizer exists and is unique for all S ∈ 6). F ∗ is nondictatorial and continuous, and furthermore satisfies (WEg∗ ), as shown in THEOREM 5.
F ∗ satisfies (WEg∗ ).
Proof. Suppose F ∗ violates (WEg∗ ). Then there exist S, S 0 ∈ 6, x ∈ S, x ∈ S 0 such that q p p √ xi + xi0 > Fi∗ (S) + Fi∗ (S 0 ) ∀i ∈ N . 0
Adding these inequalities for all i ∈ N , we obtain q ¶ X n µ n ³p ´ X p √ xi + xi0 > Fi∗ (S) + Fi∗ (S 0 ) . i=1
(5)
i=1
Since x ∈ S and x 0 ∈ S 0 , we obtain, by definition of F ∗ , n p X
Fi∗ (S) ≥
i=1
n X √ i=1
xi
and
n p X i=1
Fi∗ (S 0 ) ≥
n q X
xi0 .
i=1
Adding these inequalities, it follows q ¶ n ³p n µ ´ X X p √ Fi∗ (S) + Fi∗ (S 0 ) ≥ xi + xi0 , i=1
i=1
contradicting (5). Therefore, the characterizations of the dictatorial solutions in general do not remain true if we consider criteria that are not linear combinations of the maximin and the maximax criterion.
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8. CONCLUDING REMARKS Our characterization results based on efficiency under uncertainty are, of course, sensitive with respect to the assumptions we make concerning how agents rank vectors of contingent outcomes. Intuitively, the crucial feature of Theorem 2 is that, if F is a monotone path solution, there is a “common-worst” state of the world for any pair of contingent outcomes (S, S 0 ). By a common-worst state of the world, we mean that, for a given (S, S 0 ), we must have either min{Fi (S), Fi (S 0 )} = Fi (S) for all i ∈ N or min{Fi (S), Fi (S 0 )} = Fi (S 0 ) for all i ∈ N . This ensures that no Pareto improvement is possible. If the maximax criterion is used, it is not too surprising that only dictatorial solutions are efficient. One agent has to receive her or his maximal possible (ideal) utility, and continuity implies that this must be the same agent for all problems in 6. It is, however, quite remarkable that this dictatorship result is robust in the sense that any strict convex combination of maximin and maximax leads to the dictatorial solutions as well. As soon as outcomes other than the worst possible outcome are given any positive weight in evaluating contingent utility vectors, it is possible that mutually beneficial “trades” can occur, unless the solution is dictatorial. Another question that is related to the problem analyzed here is the following. Suppose a given bargaining solution which is not a monotone path solution is used for bargaining problems without uncertainty, given the maximin criterion. If one still wants to achieve outcomes that are efficient under uncertainty, a vector of contingent outcomes cannot be given by the respective outcomes selected by this solution in this case. Hence, a more complex notion of a solution to bargaining problems under uncertainty has to be employed. See Nosal and Sadanand (1991) for an example of how to construct efficient outcomes based on the Nash solution in such a model. In this paper, we have analyzed bargaining problems under uncertainty, given specific nonprobabilistic decision rules. There are several related issues of interest which could be addressed in future work. For example, one could investigate the question of efficiency under uncertainty if the agents use different decision rules. Furthermore, other nonprobabilistic decision rules such as minimax regret could be analyzed. REFERENCES Arrow, K. J., and Hurwicz, L. (1972). “An Optimality Criterion for Decision-Making under Ignorance,” in Uncertainty and Expectations in Economics: Essays in Honour of G.L.S. Shackle, (C. F. Carter and J. C. Ford, Eds.), pp. 1–11. Oxford: Blackwell. Blackorby, C., Bossert, W., and Donaldson, D. (1994). “Generalized Ginis and Cooperative Bargaining Solutions,” Econometrica 62, 1161–1178.
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Chun, Y., and Peters, H. (1991). “The Lexicographic Equal-Loss Solution,” Mathematical Social Sciences 22, 151–161. Chun, Y., and Thomson, W. (1990a). “Bargaining with Uncertain Disagreement Points,” Econometrica 58, 951–959. Chun, Y., and Thomson, W. (1990b). “Nash Solution and Uncertain Disagreement Points,” Games Econ. Behav. 2, 213–223. Chun, Y., and Thomson, W. (1990c). “Egalitarian Solutions and Uncertain Disagreement Points,” Econ. Lett. 33, 29–33. Kalai, E. (1977). “Proportional Solutions to Bargaining Situations: Interpersonal Utility Comparisons,” Econometrica 45, 1623–1630. Kalai, E., and Smorodinsky, M. (1975). “Other Solutions to Nash’s Bargaining Problem,” Econometrica 43, 513–518. Myerson, R. B. (1981). “Utilitarianism, Egalitarianism and the Timing Effect in Social Choice Problems,” Econometrica 49, 883–897. Nash, J. F. (1950). “The Bargaining Problem,” Econometrica 18, 155–162. Nosal, E., and Sadanand, V. (1991). “Bargaining under Pure Uncertainty,” working paper 91-05, University of Waterloo, Department of Economics. Perles, M. A., and Maschler, M. (1981). “The Super-additive Solution for the Nash Bargaining Game,” Int. J. Game Theory 10, 163–193. Peters, H. (1986). “Simultaneity of Issues and Additivity in Bargaining,” Econometrica 54, 153–169. Riddell, W. C. (1981). “Bargaining under Uncertainty,” Amer. Econ. Rev. 71, 579–590. Thomson, W. (1996). Bargaining Theory: The Axiomatic Approach, New York: Academic Press, forthcoming. Thomson, W., and Myerson, R. B. (1980). “Monotonicity and Independence Axioms,” Int. J. Game Theory 9, 37–49.
14, 173–189 (1996)
0047
Bargaining under Uncertainty and the Monotone Path Solutions∗ Walter Bossert† and Ed Nosal Department of Economics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
and Venkatraman Sadanand Department of Economics, University of Guelph, Guelph, Ontario N1G 2W1, Canada Received January 26, 1993
Uncertainty with respect to the feasible set of utility vectors is introduced in an axiomatic bargaining model. Given a criterion for nonprobabilistic decision-making under uncertainty, a natural efficiency requirement can be imposed on a bargaining solution. Using the maximin ordering, the strictly monotone path solutions (generalizations of the egalitarian solution) to the bargaining problem are characterized as the only continuous solutions that satisfy this efficiency axiom. If the maximin criterion is replaced by the maximax ranking or a strict convex combination of the maximin and the maximax criterion, imposing our efficiency axiom and continuity leads to the dictatorial solutions. Journal of Economic Literature Classification Number: C78. © 1996 Academic Press, Inc.
1. INTRODUCTION In this paper, a generalization of Nash’s (1950) axiomatic model of bargaining is considered by incorporating uncertainty with respect to the feasible set of utility vectors (see Chun and Thomson, 1990a,b,c, for discussions of uncertainty ∗ An earlier version of the paper was presented at the 1992 Buffalo–Cornell–Rochester Economic Theory conference in Buffalo. We thank Nejat Anbarci, Patrick Legros, William Thomson, and a referee for their comments. Financial support through grants from the Social Sciences and Humanities Research Council of Canada, the University of Guelph, and the University of Waterloo is gratefully acknowledged. †
FAX: (519) 725-0530. E-mail: [email protected]. 173 0899-8256/96 $18.00 Copyright © 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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with respect to the disagreement point). Assuming the disagreement point to be fixed, a bargaining problem with uncertainty can be described by a vector of possible feasible sets that may occur. There is a straightforward way of extending a bargaining solution for problems without uncertainty to solve problems under uncertainty. For each of the possible feasible sets, if the state of the world represented by this feasible set occurs, the corresponding outcome according to the solution without uncertainty is defined as the contingent outcome. For some solutions, however, this outcome might not be very satisfying. For example, there may exist another vector of contingent outcomes that can make all agents better off compared to the one prescribed by the bargaining solution. Thus, a desirable property for a bargaining solution is to be efficient under uncertainty in the sense that the resulting vector of contingent outcomes is not Pareto dominated by another vector of outcomes that are feasible for the respective states of the world. Of course, in order to find solutions with such efficiency properties, we have to specify how agents evaluate vectors of contingent outcomes. If probabilities can be assigned to different feasible sets that can occur, a reasonable condition to impose on a bargaining solution is to require that all agents benefit from reaching an early agreement on the “expected” bargaining problem. This property of bargaining solutions is desirable, because waiting until the uncertainty is resolved can lead to a Pareto-dominated situation. In Myerson (1981), a concavity condition with respect to the feasible set is used to deal with this problem. Related conditions include Perles and Maschler’s (1981) superadditivity and the weak linearity axiom used in Peters (1986). Riddell (1981) considers a situation of bargaining under uncertainty where agents can form probability assessments. When agents bargain after the uncertainty is resolved, they apply Nash’s (1950) axioms to determine the solution for the state-contingent bargaining problem. Agents can assess the value of this problem by simply calculating the expected utility of the state-contingent bargaining solutions. If, however, agents bargain over allocations before the uncertainty is resolved, their expected utilities can serve as the disagreement point and a new feasible set can be constructed. The solution to this bargaining problem is obtained by applying the axioms to the new feasible set. Thus, if agents can anticipate the bargaining solution in each state, then the solution to the bargaining problem under uncertainty is obtained by “applying the Nash axioms twice”: first to each state-contingent feasible set and then to the set that results when bargaining occurs before the state of nature is revealed. If, on the other hand, the agents cannot assign (objective or subjective) probabilities to the occurrence of different states of the world, standard expected utility arguments cannot be applied. Consequently, in these situations (and these are the situations we want to analyze in this paper), nonprobabilistic models of choice under uncertainty have to be used. In particular, we will investigate
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the consequences of applying the maximin criterion, the maximax criterion, and combinations thereof. For the maximin ranking, imposing efficiency under uncertainty and continuity characterizes the class of strictly monotone path solutions (see Thomson and Myerson, 1980; Thomson, 1994), a generalization of Kalai’s (1977) proportional (or weighted egalitarian) solutions. Adding symmetry to the above axioms leads, of course, to the egalitarian solution. If the maximax decision rule or a strict convex combination of maximin and maximax (see Arrow and Hurwicz, 1972) is used, the implications of imposing efficiency under uncertainty are not very satisfying: only the dictatorial solutions are efficient and continuous in these cases. To prove our results, it will be sufficient to consider cases where there are two possible states of the world (represented by two possible feasible sets). Given continuity (which we will assume in all our results), our efficiency axiom with two possible states of the world is equivalent to an analogous formulation involving any number of possible states. Because the two-state version is easier to relate to other axioms commonly used in bargaining theory, and because it simplifies the exposition of our results, we have chosen to use it. However, given the above-mentioned equivalence, this choice does not involve any loss of generality.
2. BARGAINING PROBLEMS AND SOLUTIONS The set of positive integers is denoted by N. Let R (R+ , R++ ) denote the set of (nonnegative, positive) real numbers, and let Rn (Rn+ , Rn++ ) be the (nonnegative, positive) Euclidean n-orthant, n ∈ N. The origin of Rn is denoted by 0. We use ei ∈ Rn+ to denote the ith unit vector. Our notation for vector inequalities is ≥, >, À. For a function ψ with range Rn+ , we let ψi = ei · ψ, i = 1, . . . , n. An n-person bargaining problem is described by a set S ⊆ Rn+ , which is interpreted as the set of feasible utility vectors of the agents. The set of agents is N = {1, . . . , n}. Throughout the paper, we assume that n ≥ 2 is fixed. S ⊆ Rn+ is comprehensive (strictly comprehensive) if and only if, for all x ∈ S, for all y ∈ Rn+ , x > y ⇒ y ∈ S (and ∃z ∈ S : z À y). The following class of bargaining problems is considered in this paper. DEFINITION 1. 6 is the set of bargaining problems S ⊆ Rn+ such that S is convex, compact, comprehensive, and there exists x ∈ S such that x À 0.
˜ We denote the subclass of 6 of strictly comprehensive problems by 6. Dropping convexity in the above definition would not change our characterization results—their proofs do not make use of this assumption. We include convexity in the definition of 6, since this is the domain usually considered in
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axiomatic models of bargaining. Note that the agents’ utilities are not interpreted as von Neumann–Morgenstern utilities (we consider nonprobabilistic decision rules under uncertainty), and therefore, if convexity is required, it has to be motivated in a different way. One possibility is to assume that the agents have concave utility functions defined on a set of physical outcomes, which implies that the feasible set of utility vectors is convex without invoking arguments that involve randomization over outcomes. In the above definition, we implicitly assume that utilities are translation invariant, which allows us to choose, without loss of generality, the disagreement point (the utility vector that results if the agents fail to reach an agreement) to be the origin—see, for instance, Chun and Peters (1991), Blackorby et al. (1994), Thomson (1996) for details. In other words, we could interpret S as the set of feasible vectors of gains of the agents over the disagreement outcome. By restricting attention to subsets of Rn+ as feasible sets, we exclude the possibility that nonindividually rational points are taken into account. For one of our proofs, it will be convenient to consider the subclass of 6 of budget problems, defined by DEFINITION 2. A bargaining problem S ∈ 6 is a budget problem if and only if there exist p ∈ Rn++ and m ∈ R++ such that S = {x ∈ Rn+ | p · x ≤ m}. The budget problem defined by ( p, m) ∈ Rn+1 ++ is denoted by T ( p, m).
For S ∈ 6, the ideal utility of S for agent i is given by ai (S) = max{xi | x ∈ S} for all i ∈ N . The convex and comprehensive hull of A ⊆ Rn+ is denoted by cch(A). A bargaining solution (on 6) is a mapping F: 6 7→ Rn+ such that F(S) ∈ S for all S ∈ 6. The following axioms are used in this paper. All of them are well-established in the literature and do not require any explanation. Weak Pareto optimality (WPO). For all S ∈ 6, for all y ∈ Rn+ , y À F(S) ⇒ y ∈/ S. The set of weakly Pareto optimal points in S ∈ 6 is denoted by WP(S), that is, WP(S) = {x ∈ S |6 ∃y ∈ S such that y À x}
∀S ∈ 6.
Continuity (CONT). For all sequences {S ν } in 6, for all S ∈ 6, lim S ν = S (in the Hausdorff topology) ⇒ lim F(S ν ) = F(S).
ν→∞
ν→∞
Let 5 be the set of all permutations π of N . For π ∈ 5 and S ⊆ Rn , let π(S) = {y ∈ Rn | ∃x ∈ S such that y = π(x)}.
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Symmetry (SY). For all S ∈ 6, π(S) = S
∀π ∈ 5 ⇒ Fi (S) = Fj (S)
∀i, j ∈ N .
Strong monotonicity (SM). For all S, S 0 ∈ 6, S 0 ⊆ S ⇒ F(S 0 ) ≤ F(S). The solutions that are of particular interest in this paper are the strictly monotone path solutions (see Thomson and Myerson, 1980; Thomson, 1996). To define this class of solutions, we first introduce the notion of a strictly monotone path (see Thomson, 1996, Chapter 2). n DEFINITION 3. A strictly Pn monotone path (in R+ ) is the image G of a function n ψ: R+ 7→ R+ such that i=1 ψi (t) = t for all t ∈ R+ , ψ(t) ≥ ψ(s) whenever t ≥ s, and, for all i ∈ N , ψi (t) = 0 whenever t 6= s and ψi (t) = ψi (s).
The strictly monotone path solution relative to the strictly monotone path G, E G , is defined by letting, for each S ∈ 6, E G (S) be the maximal point in S along G. Choosing ψi (t) = t/n for all i ∈ N , t ∈ R+ in Definition 3 leads to the egalitarian solution E—see Kalai (1977). The ith dictatorial solution D i , i ∈ N , is defined by D i (S) = ai (S) · ei for all S ∈ 6. Clearly, dictatorial solutions are special cases of strictly monotone path solutions. Other well-known bargaining solutions (which are not discussed in this paper) include, among others, the Nash (1950) solution and the Kalai–Smorodinsky (1975) solution. A result which will turn out to be very useful for our analysis is the following characterization of the strictly monotone path solutions—see Thomson (1996, Chapter 2). For completeness, we sketch a proof of this result. THEOREM 1 (Thomson, 1996). A bargaining solution F satisfies (WPO), (CONT), and (SM) if and only if F is a strictly monotone path solution.
Proof. That the strictly monotone path solutions satisfy (WPO), (CONT), and (SM) is easy to verify. suppose F satisfies these axioms. For PConversely, n xi ≤ t}, and define ψ(0) = 0 and ψ(t) = F(S t ) t > 0, let S t = {x ∈ Rn+ | i=1 for t > 0. Then the image of ψ is G = {F(S t ) | t > 0} ∪ {0}. By (WPO), P n i=1 ψi (t) = t for all t > 0, and by (SM), ψ(t) ≥ ψ(s) whenever t ≥ s. ˜ Let Next, we show that P F(S) = E G (S) for all S ∈ 6. Suppose first S ∈ 6. n xi . First, consider the case x À 0. Choose ε > 0 x = E G (S) and t = i=1
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such that the set Sε = cch({x, (x1 + ε) · e1 , . . . , (xn + ε) · en }) satisfies Sε ⊆ S ∩ S t . Clearly, such an ε exists, and Sε ∈ 6. By definition, F(S t ) = x. Since Sε ⊆ S t , (SM) implies F(Sε ) ≤ x, and by (WPO), F(Sε ) = x. Again using (SM), Sε ⊆ S implies F(S) ≥ F(Sε ) = x. From (WPO) and the strict comprehensiveness of S, it follows F(S) = x = E G (S). If xi = 0 for some i ∈ N , the definition of Sε can be modified to Sε = cch({x} ∪ {ε · ei | i ∈ N such that xi = 0}), and the same argument as above can be applied. If S is not strictly comprehensive, we can appeal to (CONT). To complete the proof that G is a strictly monotone path, suppose, by way of contradiction, thereP exist i ∈ N and t, s ∈ R+ such that t 6= s and ψi (t) = n ψi (u) = u for all u ∈ R+ and t 6= s, it follows ψi (s) > 0. Because i=1 ψ(t) 6= ψ(s). Let S ∈ 6 be such that {ψ(t), ψ(s)} ⊆ WP(S). Furthermore, let {T ν }, {S ν } be two sequences of strictly comprehensive elements of 6 such that ψ(t) ∈ WP(T ν ), ψ(s) ∈ WP(S ν ) for all ν ∈ N, and limν→∞ T ν = limν→∞ S ν = S. It follows F(T ν ) = ψ(t) and F(S ν ) = ψ(s) for all ν ∈ N, and by (CONT), F(S) = ψ(t) and F(S) = ψ(s), contradicting ψ(t) 6= ψ(s). In the presence of (WPO) and (CONT), (SM) could be replaced by other conditions such as domination—see Thomson and Myerson (1980) for details.
3. EFFICIENT BARGAINING UNDER UNCERTAINTY Suppose there is some uncertainty concerning the feasible set of a bargaining problem. We assume that there are two possible states of the world, represented by two possible feasible sets. Hence, a bargaining problem under uncertainty can be defined as a pair (S, S 0 ) ∈ 6×6. A natural efficiency requirement for bargaining solutions in the presence of uncertainty is that, for any bargaining problem under uncertainty (S, S 0 ), the pair of contingent outcomes (F(S), F(S 0 )) cannot be Pareto improved by some other pair of feasible contingent outcomes. Clearly, to formalize such an efficiency notion, we have to specify how agents rank different pairs of contingent outcomes. From the viewpoint of agent i ∈ N , bargaining problems under uncertainty can be evaluated in terms of the utility he or she enjoys in each possible state according to F, that is, individual i ∈ N evaluates pairs of contingent outcomes (F(S), F(S 0 )) on the basis of the pair (Fi (S), Fi (S 0 )). We assume that the agents’ evaluations of contingent outcomes are based on the minimal and the maximal utility they can achieve (examples for such
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nonprobabilistic decision rules are convex combinations of the minimal and maximal utilities—see the definition of the Arrow–Hurwicz criterion below). Hence, we restrict attention to rankings of pairs of contingent utilities that can be represented by a function of the minimal and maximal component of a pair of utilities. Formally, we consider functions g: R2+ 7→ R such that g is nondecreasing in both arguments and increasing in at least one argument, and define an associated ranking ºg of pairs of contingent utilities in the following way. DEFINITION 4. Let g: R2+ 7→ R be nondecreasing in both arguments and increasing in at least one argument. Then the ordering ºg of contingent utility pairs is defined by
∀z, z 0 ∈ R2+ ,
z ºg z 0 ⇔ g(min{z 1 , z 2 }, max{z 1 , z 2 }) ≥ g(min{z 10 , z 20 }, max{z 10 , z 20 }).
Hence, the first (second) argument of g is the minimum (maximum) of a pair of contingent utilities, and g specifies how the minimum and the maximum are combined according to the criterion defined by this function. The strict preference relation corresponding to ºg is denoted by Âg , and ∼g is the corresponding indifference relation. Special cases of ºg that are of particular interest in this paper are the maximin criterion, the maximax criterion, and the Arrow–Hurwicz α-criterion (which is a convex combination of maximin and maximax). The maximin criterion is obtained by using the function gmin , defined by gmin (u) = u 1
∀u ∈ R2+ ;
the maximax criterion corresponds to gmax , which is defined by gmax (u) = u 2
∀u ∈ R2+ ;
and the α-criterion for 0 < α < 1 (see Arrow and Hurwicz, 1972) is obtained by choosing gα , where gα (u) = α · u 1 + (1 − α) · u 2
∀u ∈ R2+ .
Given a function g with the above monotonicity properties (and therefore, given a decision criterion ºg as defined in Definition 4), weak efficiency under uncertainty with respect to g requires that F selects contingent outcomes that cannot be Pareto improved according to ºg . Formally, this axiom is defined as Weak efficiency under uncertainty with respect to g (WEg ). ∀S, S 0 ∈ 6 with S 6= S 0 , 6 ∃(x, x 0 ) ∈ S × S 0
such that (xi , xi0 ) Âg (Fi (S), Fi (S 0 )) ∀i ∈ N .
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A stronger version of this efficiency condition could be obtained by replacing the part “Âg . . . ∀i ∈ N ” by “ºg . . . ∀i ∈ N with at least one strict preference for some agent in N ” in the above definition. This requirement is equivalent to ˜ If problems that are comprehensive but not (WEg ) for the restriction of F to 6. necessarily strictly comprehensive are included in the domain of F (as is the case in this paper), this strengthening of (WEg ) would turn our characterizations into impossibility results. For any choice of g satisfying the above monotonicity requirements, (WEg ) implies (WPO). This is stated in LEMMA 1. Let g: R2+ 7→ R be nondecreasing in both arguments and increasing in at least one argument. If a bargaining solution F satisfies (WEg ), then F satisfies (WPO).
Proof. Suppose F satisfies (WEg ), but violates (WPO). Then there exist S ∈ 6 and x ∈ S such that x À F(S). Let x 0 ∈ S be such that x 0 À F(S) and x À x 0 (clearly, such a point x 0 exists by the definition of 6). Let S 0 = cch({x 0 }). By definition of x 0 , S 0 ∈ 6. Now it follows min{Fi (S), Fi (S 0 )} < xi0 < xi
∀i ∈ N
max{Fi (S), Fi (S 0 )} ≤ xi0 < xi
∀i ∈ N
and
(see Fig. 1 for an illustration in the case n = 2). By the monotonicity properties of g, it follows (xi , xi0 ) Âg (Fi (S), Fi (S 0 ))
∀i ∈ N ,
contradicting (WEg ).
4. THE MAXIMIN CRITERION In this section, we use the maximin rule to provide a characterization of the strictly monotone path solutions. Before stating and proving this result, we present a lemma that will be used in the proof. LEMMA 2. If a bargaining solution F satisfies (CONT) and (WEgmin ), then F satisfies (SM).
Proof. Suppose F satisfies (CONT) and (WEgmin ). By way of contradiction, suppose (SM) is violated. Then there exist S, S 0 ∈ 6, i ∈ N such that S 0 ⊆ S and Fi (S 0 ) > Fi (S).
(1)
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MONOTONE PATH SOLUTIONS
FIGURE 1
˜ From (WPO)—see Lemma 1—and the strict compreFirst, suppose S, S 0 ∈ 6. hensiveness of S and S 0 , it follows that there exists j ∈ N such that Fj (S) > Fj (S 0 ).
(2)
Define N s , N 0 ⊆ N by k ∈ N s ⇔ Fk (S) > Fk (S 0 ), k ∈ N 0 ⇔ Fk (S 0 ) > Fk (S) for all k ∈ N . From (1) and (2), it follows that N s and N 0 are nonempty. Furthermore, Fk (S) > 0 for all k ∈ N s , and Fk (S 0 ) > 0 for all k ∈ N 0 . By the strict comprehensiveness of S and S 0 , we can find x ∈ S, x 0 ∈ S 0 such that Fk (S) > xk > Fk (S 0 ) Fk (S) < xk Fk (S 0 ) > xk0 > Fk (S) Fk (S 0 ) < xk0
∀k ∀k ∀k ∀k
∈ ∈ ∈ ∈
Ns, N \N s , N 0, N \N 0 .
By construction, min{xk , xk0 } > min{Fk (S), Fk (S 0 )} for all k ∈ N , and hence, (WEgmin ) is violated.
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FIGURE 2
If S or S 0 is not strictly comprehensive, we can use (CONT) to obtain the desired conclusion. See Fig. 2 for an illustration of Lemma 2 in the two-person case. Clearly, if we restrict attention to strictly comprehensive problems, (CONT) is not needed in the above result. Similar results concerning the equivalence of (WEgmin ) and other independence or monotonicity conditions (given (WPO) and (CONT)) could be formulated—see Thomson and Myerson (1980). The main result of this section is THEOREM 2. A bargaining solution F satisfies (CONT) and (WEgmin ) if and only if F is a strictly monotone path solution.
Proof. That the strictly monotone path solutions satisfy the required axioms can be verified easily. To prove the converse, recall that, by Lemma 1, (WEgmin ) implies (WPO), and combine Theorem 1 and Lemma 2. If symmetry is added to the axioms in Theorem 2, the egalitarian solution is characterized. This follows from the observation that E is the only symmetric monotone path solution. Hence, we obtain
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COROLLARY 1. A bargaining solution F satisfies (CONT), (SY), and (WEgmin ) if and only if F = E.
5. THE MAXIMAX CRITERION If we impose weak efficiency under uncertainty given the maximax criterion, only a rather unattractive subclass of the strictly monotone path solutions satisfy this requirement and continuity, namely the dictatorial solutions. THEOREM 3. A bargaining solution F satisfies (CONT) and (WEgmax ) if and only if F is a dictatorial solution.
Proof. That the dictatorial solutions satisfy (CONT) and (WEgmax ) can be verified by the reader. Suppose now F satisfies these two axioms. We first show ˜ there exists i ∈ N such that F(S) = ai (S) · ei . Suppose this that, for any S ∈ 6, ˜ i, j ∈ N such that is not the case. By (WPO)—see Lemma 1—there exist S ∈ 6, Fi (S) > 0 and Fj (S) > 0. By the strict comprehensiveness of S, Fk (S) < ak (S) for all k ∈ N . ˜ be such that cch(F(S)) ⊆ S 0 ⊆ S and Let S 0 ∈ 6 W P(cch(F(S))) ∩ W P(S 0 ) = W P(S 0 ) ∩ W P(S) = {F(S)}. If F(S 0 ) = F(S), let x ∈ S, x 0 ∈ S 0 be such that x1 > F1 (S) and xk0 > Fk (S 0 ) for all k ∈ N \{1}. If F(S 0 ) 6= F(S), define N 0 ⊆ N by k ∈ N 0 ⇔ Fk (S 0 ) > Fk (S) for all k ∈ N . Choose x ∈ S, x 0 ∈ S 0 such that xk > Fk (S 0 ) ∀k ∈ N 0
and
xk0 > Fk (S) ∀k ∈ N \N 0 .
In either case, it follows max{xk , xk0 } > max{Fk (S), Fk (S 0 )} for all k ∈ N , ˜ there exists i ∈ N contradicting (WEgmax ). This proves that, for each S ∈ 6, such that F(S) = ai (S) · ei . See Fig. 3 for an illustration of the above argument for n = 2. ˜ and i, j ∈ N such that i 6= j, Fi (S) = ai (S) · ei , If there exist S, S 0 ∈ 6 and Fj (S 0 ) = a j (S 0 ) · e j , we obtain a contradiction to (CONT). Hence, F is ˜ and by (CONT), on 6. dictatorial on 6, 6. THE ARROW–HURWICZ CRITERION The characterization of the dictatorial solutions presented in the previous section can also be obtained for criteria other than maximax. If instead of minimax
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FIGURE 3
we use a strict convex combination of maximin and maximax (see Arrow and Hurwicz, 1972), we still obtain the dictatorial solutions as the only continuous solutions satisfying the corresponding efficiency axiom. This is shown in THEOREM 4. Let α ∈ (0, 1). A bargaining solution F satisfies (CONT) and (WEgα ) if and only if F is a dictatorial solution.
Proof. Clearly, the dictatorial solutions satisfy (WEgα ). Suppose F satisfies (CONT) and (WEgα ). First, we show S ∈ 6 is a budget problem ⇒ ∃i ∈ N such that F(S) = ai (S) · ei .
(3)
Suppose (3) is false. By (WPO), there exist ( p, m) ˜ ∈ Rn+1 ++ and i, j ∈ N such ˜ > 0 and Fj ( S) ˜ > 0, where S˜ = T ( p, m). that Fi ( S) ˜ By (CONT), there exists ˜ Fj (S) < Fj ( S), ˜ m ∈ R++ such that, with S = T ( p, m), we have Fi (S) < Fi ( S), and (i) Fi (S) > 0
or
(ii) Fj (S) > 0.
If (i) is true, let p 0 ∈ Rn++ be such that pi0 / p 0j < pi / p j and pk0 = pk for all k ∈ N \{i, j}. By (CONT), p 0 can be chosen such that Fi (S 0 ) > Fi (S) and
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Fj (S 0 ) > Fj (S), where S 0 = T ( p 0 , m). ˜ Now define xi = Fi (S) − ε, x j = Fj (S) +
x˜i = Fi (S 0 )+
pi · ε, xk = Fk (S) ∀k ∈ N \{i, j}, pj
α p0 α ·ε, x˜ j = Fj (S 0 )− i0 · ·ε, x˜k = Fk (S 0 ) ∀k ∈ N \{i, j}, 1−α pj 1 − α
where ε ∈ R++ is chosen sufficiently small so that xi > 0 and x˜ j > x j . Clearly, x ∈ S and x˜ ∈ S 0 . We obtain α · xi + (1 − α) · x˜i = α · Fi (S) + (1 − α) · Fi (S 0 ) and
" 0
α · x j + (1 − α) · x˜ j = α · Fj (S) + (1 − α) · Fj (S ) + α · ε ·
pi p0 − i0 pj pj
#
> α · Fj (S) + (1 − α) · Fj (S 0 ). Therefore, (xi , x˜i ) ∼gα (Fi (S), Fj (S 0 )) and (x j , x˜ j ) Âgα (Fj (S), Fj (S 0 )). Furthermore, since xk = Fk (S) and x˜k = Fk (S 0 ) for all k ∈ N \{i, j}, (xk , x˜k ) ∼gα (Fk (S), Fk (S 0 ))
∀k ∈ N \{i, j}.
Since S 0 is a budget problem, S 0 is strictly comprehensive. Let δ, δ˜ ∈ R++ , and define x 0 ∈ Rn by x j0 = x˜ j − δ, xk0 = x˜k + δ˜
∀k ∈ N \{ j}.
By the strict comprehensiveness of S 0 , δ and δ˜ can be chosen such that x j0 > Fj (S 0 ) and x 0 ∈ S 0 . Combined with the above observations, it follows (xk , xk0 ) Âgα (Fk (S), Fk (S 0 ))
∀k ∈ N ,
contradicting (WEgα ). Case (ii) is analogous. Therefore, (3) is true.
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FIGURE 4
Analogous to the argument in the proof of Theorem 3, (CONT) implies that the restriction of F to the class of budget problems must be dictatorial. ˆ ⊆ 6 be the set of all bargaining problems S with the property that there Let 6 exists a budget problem S 0 such that S ⊆ S0
and
W P(S) ∩ W P(S 0 ) = {F(S 0 )}.
(4)
ˆ and let S 0 be a budget problem such that (4) is satisfied. Suppose Let S ∈ 6, F(S) 6= F(S 0 ). Then there exists x 0 ∈ S 0 such that x 0 À F(S). Furthermore, let x = F(S 0 ) ∈ S. See Fig. 4 for an illustration in the two-person case. Since α ∈ (0, 1), it follows (xi , xi0 ) Âgα (Fi (S), Fi (S 0 ))
∀i ∈ N ,
ˆ must be dictatorial. Now contradicting (WEgα ). Hence, the restriction of F to 6 (CONT) can be applied again to conclude that F must be dictatorial on 6, which completes the proof.
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7. NONLINEAR CRITERIA An obvious question that arises after the above characterizations of dictatorial solutions is whether nonlinear combinations of the maximin and the maximax criterion lead to dictatorships as well. In general, this is not the case. Consider, for example, the function g∗ defined by √ √ g∗ (u) = u 1 + u 2 ∀u ∈ R2+ . Clearly, the dictatorial solutions satisfy (WEg∗ ) (the dictatorial solutions obviously satisfy (WEg ) for any choice of g satisfying the usual monotonicity requirements). However, there also exist nondictatorial continuous solutions that satisfy (W E g∗ ). Define the solution F ∗ by ( ) n X √ ∗ xi | x ∈ S ∀S ∈ 6. F (S) = argmax i=1 ∗
Clearly, F is a well-defined bargaining solution (note that the above maximizer exists and is unique for all S ∈ 6). F ∗ is nondictatorial and continuous, and furthermore satisfies (WEg∗ ), as shown in THEOREM 5.
F ∗ satisfies (WEg∗ ).
Proof. Suppose F ∗ violates (WEg∗ ). Then there exist S, S 0 ∈ 6, x ∈ S, x ∈ S 0 such that q p p √ xi + xi0 > Fi∗ (S) + Fi∗ (S 0 ) ∀i ∈ N . 0
Adding these inequalities for all i ∈ N , we obtain q ¶ X n µ n ³p ´ X p √ xi + xi0 > Fi∗ (S) + Fi∗ (S 0 ) . i=1
(5)
i=1
Since x ∈ S and x 0 ∈ S 0 , we obtain, by definition of F ∗ , n p X
Fi∗ (S) ≥
i=1
n X √ i=1
xi
and
n p X i=1
Fi∗ (S 0 ) ≥
n q X
xi0 .
i=1
Adding these inequalities, it follows q ¶ n ³p n µ ´ X X p √ Fi∗ (S) + Fi∗ (S 0 ) ≥ xi + xi0 , i=1
i=1
contradicting (5). Therefore, the characterizations of the dictatorial solutions in general do not remain true if we consider criteria that are not linear combinations of the maximin and the maximax criterion.
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8. CONCLUDING REMARKS Our characterization results based on efficiency under uncertainty are, of course, sensitive with respect to the assumptions we make concerning how agents rank vectors of contingent outcomes. Intuitively, the crucial feature of Theorem 2 is that, if F is a monotone path solution, there is a “common-worst” state of the world for any pair of contingent outcomes (S, S 0 ). By a common-worst state of the world, we mean that, for a given (S, S 0 ), we must have either min{Fi (S), Fi (S 0 )} = Fi (S) for all i ∈ N or min{Fi (S), Fi (S 0 )} = Fi (S 0 ) for all i ∈ N . This ensures that no Pareto improvement is possible. If the maximax criterion is used, it is not too surprising that only dictatorial solutions are efficient. One agent has to receive her or his maximal possible (ideal) utility, and continuity implies that this must be the same agent for all problems in 6. It is, however, quite remarkable that this dictatorship result is robust in the sense that any strict convex combination of maximin and maximax leads to the dictatorial solutions as well. As soon as outcomes other than the worst possible outcome are given any positive weight in evaluating contingent utility vectors, it is possible that mutually beneficial “trades” can occur, unless the solution is dictatorial. Another question that is related to the problem analyzed here is the following. Suppose a given bargaining solution which is not a monotone path solution is used for bargaining problems without uncertainty, given the maximin criterion. If one still wants to achieve outcomes that are efficient under uncertainty, a vector of contingent outcomes cannot be given by the respective outcomes selected by this solution in this case. Hence, a more complex notion of a solution to bargaining problems under uncertainty has to be employed. See Nosal and Sadanand (1991) for an example of how to construct efficient outcomes based on the Nash solution in such a model. In this paper, we have analyzed bargaining problems under uncertainty, given specific nonprobabilistic decision rules. There are several related issues of interest which could be addressed in future work. For example, one could investigate the question of efficiency under uncertainty if the agents use different decision rules. Furthermore, other nonprobabilistic decision rules such as minimax regret could be analyzed. REFERENCES Arrow, K. J., and Hurwicz, L. (1972). “An Optimality Criterion for Decision-Making under Ignorance,” in Uncertainty and Expectations in Economics: Essays in Honour of G.L.S. Shackle, (C. F. Carter and J. C. Ford, Eds.), pp. 1–11. Oxford: Blackwell. Blackorby, C., Bossert, W., and Donaldson, D. (1994). “Generalized Ginis and Cooperative Bargaining Solutions,” Econometrica 62, 1161–1178.
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Chun, Y., and Peters, H. (1991). “The Lexicographic Equal-Loss Solution,” Mathematical Social Sciences 22, 151–161. Chun, Y., and Thomson, W. (1990a). “Bargaining with Uncertain Disagreement Points,” Econometrica 58, 951–959. Chun, Y., and Thomson, W. (1990b). “Nash Solution and Uncertain Disagreement Points,” Games Econ. Behav. 2, 213–223. Chun, Y., and Thomson, W. (1990c). “Egalitarian Solutions and Uncertain Disagreement Points,” Econ. Lett. 33, 29–33. Kalai, E. (1977). “Proportional Solutions to Bargaining Situations: Interpersonal Utility Comparisons,” Econometrica 45, 1623–1630. Kalai, E., and Smorodinsky, M. (1975). “Other Solutions to Nash’s Bargaining Problem,” Econometrica 43, 513–518. Myerson, R. B. (1981). “Utilitarianism, Egalitarianism and the Timing Effect in Social Choice Problems,” Econometrica 49, 883–897. Nash, J. F. (1950). “The Bargaining Problem,” Econometrica 18, 155–162. Nosal, E., and Sadanand, V. (1991). “Bargaining under Pure Uncertainty,” working paper 91-05, University of Waterloo, Department of Economics. Perles, M. A., and Maschler, M. (1981). “The Super-additive Solution for the Nash Bargaining Game,” Int. J. Game Theory 10, 163–193. Peters, H. (1986). “Simultaneity of Issues and Additivity in Bargaining,” Econometrica 54, 153–169. Riddell, W. C. (1981). “Bargaining under Uncertainty,” Amer. Econ. Rev. 71, 579–590. Thomson, W. (1996). Bargaining Theory: The Axiomatic Approach, New York: Academic Press, forthcoming. Thomson, W., and Myerson, R. B. (1980). “Monotonicity and Independence Axioms,” Int. J. Game Theory 9, 37–49.