BARGAINING UNDER UNCERTAINTY: A NEW APPROACH BASED ON LOSSES

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1 BARGAINING UNDER UNCERTAINTY: A NEW APPROACH BASED ON LOSSES JAUME GARCÍA-SEGARRA, MIGUEL GINÉS-VILAR Abstract.. In the era of Articial intelligence the need for normative approaches to problem solving regains its relevance. Several real world problems dealt with by the Articial intelligence literature require inputs from bargaining problems that treat results as losses from the utopia point. A particular application to astrodynamics is proposed as a potential eld for future research. When the agents are bargaining under uncertainty, strong Pareto optimality and restricted monotonicity provide a unique family of solutions based on sharing losses. We name this family the weighted losses solutions. Adding scale invariance and restricting the uncertainty, we obtain and characterize a new family of solutions, the weighted proportional losses solutions. This family is the normalized version of the weighted losses solutions. The weighted proportional losses solution satises individual rationality even for three or more agents. We also formalize the relationship between concavity and super-additivity in ecient solutions. Bargaining solutions, Axiomatization, Uncertainty. 1. Introduction Traditionally, bargaining theory has been mainly applied to the social sciences, and more concretely, to economics. However, the eld of applications of the axiomatic bargaining theory is much broader. In fact, some mathematical procedures already used in engineering and computer science 1 belong to the concepts used by the axiomatic bargaining theory. A better understanding of the underline mechanisms could yield tools with a higher degree of applicability 2. For example, based on our weighted proportional losses solutions, Juan S. Senent and J. Garcia-Segarra[18] The authors wish to thank William Thomson, Herve Moulin, Carmen Herrero, M. Carmen Marco, Josep Peris and Nikos Georgantzis for their useful comments. Financial support by the Ministery of Science and Innovation (projects ECO /ECON and ECO ), UJI-Fundación Bancaja (project P1-1B ), and the Junta de Andalucía ( project P07-SEJ-03155) is gratefully acknowledged. PREDOC/2007/28 Fundación Bancaja, E Pla de Promoció de la Investigació de la UJI. 1 Especially, after the rise of articial intelligence. 2 One criticism of the axiomatic bargaining theory is that some times the solution concepts do not t with the applied models. This is, among others, why we provide solutions without the assumption of symmetry between agents. 1

2 2 JAUME GARCÍA-SEGARRA, MIGUEL GINÉS-VILAR address a well-known problem in the eld of astrodynamics 3. NASA engineers calculating two-impulse transfers 4 among orbits, usually choose the minimization of the sum of speed increments that engines should undergo in order for their orbit to change. For the environment of astrodynamics, this solution is equivalent to the utilitarian solution of the canonical bargaining theory. NASA engineers understand the solution's outcome but not its axiomatic implications. Another interesting issue is the computability of the solutions. In outer space, the computation power required to converge to the solution matters 5. The minimization of a sum needs huge computing resources, especially for problems of n-dimensions. In addition, only local optima can be identied through optimization. Thus, the starting point of the process matters. The solution concepts that we show in this paper can be easily computed since the solutions consist of convex combinations between two points on the R n space. This way, in few interpolations it is possible to nd the intersection between a line and a convex closed set. Regarding the social sciences, the well-known Nash bargaining solution (Nash 1950 [11]) has been mechanically applied to solve most problems that involve bargaining among economic agents. Several authors, like Gerber and Upmann (2006)[4], Pita and Torregrosa (2003)[15], among others, point out that the Nash bargaining solution is used almost exclusively in economic applications. They propose that the routine application of Nash solution should be revised. Especially, when the choice of dierent solution concepts implies opposing comparative static eects. For these cases, the use of a misspecied model may lead to seriously awed policy implications. Some authors use the Nash solution without further contemplation. They argue that it is easier to calculate in comparison to other solutions, like for example, the Kalai-Smorodinsky solution. These authors also point out that the Nash bargaining solution can be implemented from a non cooperative bargaining game. The rst statement assumes that the utility function is not very complicated. We can refute 3 Astrodynamics is the term used to describe the application of Newtonian mechanics to man-made objects in space, such as rockets and spacecraft. It is a subeld of celestial mechanics. 4 Two-impulse transfers imply 2-agent problems, this paper provides the extension for n-agents problems that can be applied to n-impulse transfers. 5 The spacecrafts have radioactivation proof hardware with relatively low computing power. If a problem with the communications arises, it is very interesting to have an algorithm that converges to the solution in few interpolations.

3 BARGAINING UNDER PROBABILISTIC UNCERTAINTY 3 this assumption. For example, imagine that the Nash solution is applied to a complicated engineering problem, where the utility function is a n-degree polynomial with roots. In this case, as it happens with our astrodynamics example, it is easier to compute the Kalai-Smorodinsky solution or other dened by the intersection of the convex combination among two points with the bargaining set. The second statement relies on the fact that the Nash solution coincides with the subgame-perfect Nash equilibrium outcome of Rubinstein's(1982)[17] game. This is the reason why it can be derived from a strategic analysis in the sense of non-cooperative game theory. We point out that other solutions can also be applied from a non-cooperative point of view. That is, when selsh and rational agents are trying to maximize their own utility. Moulin (1984)[9], shows that under some conditions, the Kalai-Smorodinsky solution could be implemented from a non-cooperative point of view. In our opinion, all these arguments supporting the use of the Nash solution without other considerations are insucient. Sometimes, when economists choose a bargaining solution to explain their models, critical details of the models are omitted. In addition, Kalai and Smorodinsky (1975)[7] point out that the Nash solution does not satisfy a very natural axiom in terms of fairness, individual monotonicity. Individual monotonicity says that if the opportunities of one agent improve, this agent should not be harmed. There are several wage bargaining models that deal with uncertainty over the future states of nature. However, in some cases the concavity axiom is not satised by the solution. Concavity says that if bargaining is taking place now, but the feasible set will be known only later, it would be preferable for the agents to reach a compromise now instead of waiting for the resulting set. For this reason, concavity should be satised when agents are bargaining under probabilistic uncertainty. Myerson (1981) [10], Perles and Mashler (1981)[12], and Thomson (1994)[20] point out that neither the Nash solution nor the Kalai-Smorodinsky solution satisfy concavity. For this reason, the Nash solution should not be used when agents are bargaining under probabilistic uncertainty. Moreover, Peters (1986)[13] points out that, there is no solution for 2-agents problems satisfying strong Pareto optimality, scale invariance, Pareto continuity, super-additivity, and symmetry. If in addition we think that any kind of monotonicity

4 4 JAUME GARCÍA-SEGARRA, MIGUEL GINÉS-VILAR should be satised, then, by Kalai (1977)[6] and Myerson (1981)[10] we can state that there is no solution for 2-agent problems satisfying strong Pareto optimality, scale invariance, monotonicity, and concavity. In this paper we propose and characterize two families of solutions, the weighted losses and the weighted proportional losses solutions, that should be used in bargaining problems under uncertainty. In section II, we introduce denitions and notation. In section III, we analyze the logical implications of concavity and super-additivity. In section IV, we introduce the weighted losses solutions and its characterization. In section V, we require scale invariance. Thus, we propose and characterize an alternative family of solutions, the weighted proportional losses solutions. In section VI, we provide also a characterization of the weighted proportional losses solutions for n-agent problems. In section VII, we present the concluding remarks. 2. Definitions and Notations Let x, y R 2. The following notation is used: y x if y 1 x 1 and y 2 x 2 ; y x if y 1 > x 1 and y 2 > x 2 or y 1 = x 1 and y 2 = x 2 ; y > x if y x and y x; y x if y 1 > x 1 and y 2 > x 2. A set is comprehensive when for each x S if y x, then y S. A subset S R 2 is called a bargaining domain if it is convex, compact and comprehensive. The convex comprehensive hull of a set S R 2, cch(s), is the smallest convex and comprehensive set containing S. Let cch{m(s)} be convex comprehensive hull of the point m(s). The comprehensive hull of a point x R 2, comp(x) is the smallest comprehensive set containing x. The disagreement point, d, is the outcome that the agents get if they do not reach an agreement, furthermore d S. A pair (S, d) is called a bargaining problem if S is a bargaining domain and d is the disagreement point. Denote the set of regular bargaining pairs by Σ. A solution is any function f : Σ R 2 satisfying f(s, d) S for every (S, d) Σ. The point f(s, d) is the solution point of the bargaining situation represented by the pair (S, d). Without loss of generality, we work with the class of problems whose disagreement point is zero (Σ 0 ). Translation invariance says that f(s + t, d + t) = f(s, d) + t for all t R n. For problems Σ d we can make the following transformation by means of

5 BARGAINING UNDER PROBABILISTIC UNCERTAINTY 5 translation invariance, if S Σ d, then S = S {d} = {x R n x = s d; s S} Σ 0. For simplicity, we write (S) every pair (S, d) Σ 0. By individual rationality, f(comp(d)) = d. Furthermore, by translation invariance, f(comp(0)) = 0. A solution satises individual rationality if f(s, d) d. Given S and d = 0, we restrict our attention to sets contained in R 2 +. For simplicity, we omit the subscript +. For a given S, the utopia point, m(s), is a point dened by the maximum outcome for each player i on the set S. The strong and weak Pareto frontiers of a bargaining domain S are the sets SPO(S) = {x S if y > x then y / S} and WPO(S) = {x S if y x then y / S}, respectively. To measure asymmetries we dene α n 1 = {β R n + : n i=1 β i = 1}, that reect relative agent i's dierences from the rest. 3. Concavity and super-additivity We focus on the canonical framework of bargaining. This is, on convex, closed and comprehensive sets. As it happens in Nash (1950)[11], Kalai-Smorodinsky (1975)[7], Kalai (1977b)[6], we restrict our attention to the alternatives contained in R n +. Then, our solutions only consider individually rational alternatives 6. For our characterizations, we use the following axioms: Strong Pareto optimality: For every S Σ 0, then f(s) SP O(S). Strong Pareto optimality implies eciency. It says that the solution will be in the frontier of the set where none of the agents can improve her utility without a loss of utility for the other agent. The scale invariance axiom deals with the possibility that the bargaining set could be normalized by means of a positive ane transformation; formally: Scale invariance: For each S Σ 0 and p R n +, pf(s) = f(ps). Scale invariance was introduced by Nash (1950)[11] and is one of the four axioms that characterize the Nash solution. When one solution satises scale invariance, the bargaining set S can be normalized in another with disagreement point d = (0, 0,..., 0) and utopia point m(s) = (1, 1,..., 1). Then, the solution on this set is in relative terms equivalent to the solution outcome in the original set. In other words, 6 Regarding this point, our work is dierent from Peters (1986)[13]. In Peters (1986), the alternatives that are not included in R n + are taken into account to calculate the utopia point, in contrast, we consider that the utopia point should be given by the set of all individually rational alternatives. This point explains all contradictions that arise among his results and ours.

6 6 JAUME GARCÍA-SEGARRA, MIGUEL GINÉS-VILAR it does not matter if agents change the scale of units in which they are bargaining. Hence, scale invariance implies interpersonal comparison of utility, which can usually be observed when agents try to reach compromises. The concept of monotonicity is very clear in bargaining situations. It is very convincing to argue that, if the feasible set increases, the payo of no agent should decrease. In this type of axioms we have: Restricted monotonicity: For each pair S, T Σ 0, if S T and m(s) = m(t ); then f(s) f(t ). Restricted monotonicity is a weaker axiom than individual monotonicity 7 and it was introduced by Roth (1979)[16]. This axiom says that if there is an expansion of the feasible set, leaving unaected the utopia point, no agent might lose out. We are also interested in contexts with uncertainty. There are several kinds of uncertainty in the literature 8. We focus in probabilistic uncertainty, that is, when we can relate each event with a probability that this event happens. Myerson(1981)[10] introduced concavity, whose meaning is closely related to bargaining under probabilistic uncertainty. When concavity is satised, the agents are willing to reach an agreement before the uncertainty is resolved because all of them benet from this early agreement (see Thomson's survey[20]). Concavity: For each λ [0, 1] and for each pair S, T Σ 0, then f(λs + (1 λ)t ) λf(s) + (1 λ)f(t ). Concavity says that if bargaining takes place now but the feasible set will be known only later, it would be preferable for the agents to reach a compromise now instead of waiting for the uncertainty to be resolved. A necessary condition for this is that both agents benet from an early agreement. Bargaining on several problems simultaneously is dened as a multi-issue problem. In this context, Perles and Maschler (1981)[12] introduce the super-additivity axiom. When a solution satises super-additivity, the agents prefer to consolidate the problems into a single problem and solving it, rather than solving the problems separately. 7 Individual monotonicity: For each pair S, T Σ 0, and for each j i, if S T and m j (S) = m j (T ); then f i (S) f i (T ). Individual monotonicity was used by Kalai and Smorodinsky (1975). This axiom says that if there is an expansion of the possibilities of one agent, this agent might not lose out. 8 Probabilistic and non probabilistic uncertainty.

7 BARGAINING UNDER PROBABILISTIC UNCERTAINTY 7 Super-additivity: For each S, T Σ 0, f(s + T ) f(s) + f(t ). Homogeneity says that the solution remains invariant with respect to choices of units in which the agents are bargaining. Homogeneity: For each λ R and λ > 0, f(λs) = λf(s). Lemma 1. For n-agent problems, if a solution satises weak Pareto optimality and super-additivity, then it also satises homogeneity. Proof. We prove homogeneity of S, rst for natural numbers; then for rational numbers, nally for real numbers. Let λ N. It is easy to prove that if S is a convex set, ns n i=1 S. By super-additivity, f(ns) = f(s + S S) f(s) + f((n 1)S) nf(s). Since 1 f(ns) S and f(s) S, by weak Pareto optimality, it is not possible that n 1 f(ns) f(s). Thus, 1 f(ns) = f(s). n n Let λ Q and n, p, q N be, such that λ = p q. Now, f(s) = f(n 1 n S) = nf( 1 n S). Thus f( 1 n S) = 1 n f(s). Hence, f(p q S) = pf(s q ) = p q f(s). Let λ R\Q. Let p q and {p, q} R. By super additivity and individual rationality, f(ps) = f((p q)s + qs) f((p q)s) + f(qs) f(qs). Let ( pn q n ) n 1 be a decreasing sequence of rational numbers whose limit is λ. Let {p n, q n } Q. Since pn q n λ, then f( pn q n S) f(λs). By homogeneity of f with rational numbers, p n q n f(s) f(λs). Taking limits, lim n p n Since f(λs) W P O(λS), then f(λs) = λf(s). q n f(s) f(λs). Thus λf(s) f(λs). Lemma 2. For n-agents problems, if a solution satises weak Pareto optimality and concavity, then it also satises homogeneity. Proof. Let f be a solution that satises weak Pareto optimality and concavity. Let S Σ 0, that is, the disagreement of S is equal to 0. Thus, λs +(1 λ)comp(d) λs and f(comp(d)) = 0. Since f satises concavity, f(λs + (1 λ)comp(d)) λf(s) + (1 λ)f(comp(d)) = λf(s). Then, for each λ ]0, 1[, f(λs) λf(s). Suppose 1 f(λs) f(s). Since f(s) W P O(S), this a contradiction. Then, for λ each λ ]0, 1[, f(λs) = λf(s). Let r R + and r > 1, f(s) = f( 1rS) 1 f(rs). Suppose rf(s) f(rs). Since r r f(rs) W P O(rS), this is a contradiction. Then, for each r R +, rf(s) = f(rs).

8 8 JAUME GARCÍA-SEGARRA, MIGUEL GINÉS-VILAR Theorem 1. For n-agent problems. A weakly Pareto optimal solution satises super-additivity if and only if it satises concavity. Proof. (a) First, we prove that if f satises weak Pareto optimality and superadditivity, then it also satises concavity. Let S, T Σ 0 and λ [0, 1]. Suppose f(λs + (1 λ)t ) f(λs) + f((1 λ)t ). By Lemma 1, f(λs) + f((1 λ)t ) = λf(s) + (1 λ)f(t ). (b) Next, we prove that if f satises weak Pareto optimality and concavity, then it also satises super-additivity. Suppose f(s + T ) = f(λ S T + (1 λ) ) λ 1 λ λf(s T ) + (1 λ)f( ). By Lemma 2, λ 1 λ λf( S T ) + (1 λ)f( ) = λ f(s) + (1 λ) f(t ) = f(s) + f(t ). λ 1 λ λ 1 λ The only formal result that relates concavity and super-additivity is provided by Peters (1985)[14], who states that super-additivity and homogeneity implies concavity. Our result provides a formal equivalence for ecient solutions. 4. Weighted Losses Chun (1988)[1] proposes a solution that equalizes among the agents the losses from the ideal point. This solution was named the equal losses solution. The equal losses solution is dened as the intersection between the line with a x slope reecting the relative bargaining power among the agents, and the frontier of the set. For n 3 agents, this line might not intersect the set. Thus, equal losses solution does not satisfy individual rationality for n-agents problems. We adapt the equal losses solution proposed by Chun (1988)[1] to a weighted version, the weighted losses solution. Figure 1 shows the weighted losses solution for 2-agent problems. Formally: Denition 1. Given an α, the weighted losses solution for all S Σ 0 with weights α is dened by: (4.1) EL α (S) = max{x S α 1 (m 1 (S) x 1 ) = (1 α 1 )(m 2 (S) x 2 )} as: Let k R +, and α 1 0 or 1. The weighted losses solution 9 can also be dened 9 Note that the weighted losses solution is in fact a family of solutions. There is a solution for each α. For simplicity we use the singular voice to avoid misunderstandings with other solutions that do not belong to this family.

9 BARGAINING UNDER PROBABILISTIC UNCERTAINTY 9 Figure 1. The weighted losses solution for 2 agents' bargaining problem. (4.2) EL α (S) = max{x S x 1 = m 1 (S) k α 1, x 2 = m 2 (S) k (1 α 1 ) } This weighted losses solution does not satisfy individual rationality when n 3 agents 10. However, for 2-person bargaining problems it does. Herrero and Marco (1993)[5] characterize the rational equal-losses that ensures individual rationality. But, all these types of lexicographical solutions cannot deal with any type of probabilistic uncertainty. Theorem 2. For n = 2. A solution satises strong Pareto optimality, restricted monotonicity, and concavity, if and only if it is a weighted losses solution. Proof. Existence Let α. It is clear that, EL α satises strong Pareto optimality and restricted monotonicity. We show that EL α also satises concavity. 10 We restrict our attention to the individually rational solutions. This is the reason why we do not provide a characterization for n-agents problems. Note that an extension of the weighted losses solution for n-agent problems is trivial following the same steps as thus used to characterize the weighted proportional losses solution for n-agent problems.

10 10 JAUME GARCÍA-SEGARRA, MIGUEL GINÉS-VILAR Let C λs + (1 λ)t. The EL α solution satises concavity if and only if EL α i (C) λel α i (S) + (1 λ)el α i (T ). Clearly, for each i {1, 2}, m i (C) = λm i (S) + (1 λ)m i (T ). Let k R. We prove that λel α i (S)+(1 λ)el α i (T ) {x R 2 x i = m i (C) k α i }. By denition of EL α, for each i {1, 2}, EL α i (S) = m i (S) k α i. Then, for each i {1, 2}, λ(m i (S) k α i ) + (1 λ)(m i (T ) k α i ) = m i (C) λk+(1 λ)k. Therefore, λel α i (S) + (1 λ)el α i (T ) belongs to the same line of EL α i (C). α i Moreover, by denition of C, λel α i (S) + (1 λ)el α i (T ) C. Since, EL α i (C) C and it is the maximum point in C on this line (EL α (C)). Then EL α i (C) λel α i (S) + (1 λ)el α i (T ). For the case when α 1 = 0 or 1 is trivial to prove existence. Uniqueness. First we dene a sub-domain Ω 0 of Σ 0 and prove uniqueness on Ω 0. Then, we extend the proof to the entire domain. Step 1. Let B Σ 0. Let cch(b) {x R 2 + x 1 m 1 + x 2 m 2 1} B. Let cch{m(b)} {x R 2 + x m(b)} B be the convex comprehensive hull of B containing m(b). Let Ω 0 Σ 0 be dened as follow: Ω 0 (B) {K R 2 λ [0, 1) with K = λcch{m(b)} + (1 λ)cch(b)}. Let K λcch{m(b)} + (1 λ)cch(b). Now we prove that K = {x R 2 + x < m(b) and x 1 m 1 + x 2 m 2 < 1 + λ}. If x cch{m(b)}, then x 1 m 1 + x 2 m 2 < m 1 m 1 + m 2 m 2 < 2. Let t cch(b) and s cch{m(b)}. Let x K. Note that, x = λs + (1 λ)t. Dividing each x i by m i, we have x 1 m 1 = λ s 1 m 1 + (1 λ) t 1 m 1 and x 2 m 2 = λ s 2 m 2 + (1 λ) t 2 m 2. Then x 1 m 1 + x 2 m 2 < λ2 + (1 λ) < 1 + λ. Step 2. Uniqueness on Ω 0. Let f be a solution satisfying strong Pareto optimality, restricted monotonicity, and concavity. Let P Ω 0. Let P λp + (1 λ)cch(b). By strong Pareto optimality, f(p ) P and f(p ) SP O(P ). Let β be such that f(cch(b)) = (βm 1, (1 β)m 2 ). We know that λf(p ) + (1 λ)f(cch(b)) P. By the denition of Ω 0, λf(p ) + (1 λ)f(cch(b)) SP O(P ). By concavity, and since f(p ) SP O(P ) and λf(p ) + (1 λ)f(cch(b)) SP O(P ), then f(p ) λf(p ) + (1 λ)f(cch(b)). Moreover, f(p ) and λf(p ) + (1 λ)f(cch(b)) are lined up with m(p ). By the denition of Ω 0, f(p ) = λf(p ) + (1 λ)f(cch(b)). Thus, f(p ) λf(p )+(1 λ)f(cch(b)). In Figure 2 we show an illustration of Theorem 2. To conclude the proof of uniqueness on Ω 0, we prove that λf(p )+(1 λ)f(cch(b)) L(f(cch(B)), m(b)). Let T Ω 0. Let t SP O(T ) be such that t < m(b). Then, t = (t 1, m 2 (B)) or (m 1 (B), t 2 ). Suppose by contradiction, that λf(p ) + (1

11 BARGAINING UNDER PROBABILISTIC UNCERTAINTY 11 Agent 2 m(t) T EL m( S + (1- ) T = m(s) + (1- )m(t) f f(s) + (1- )f(t) f S + (1- )T m(s) S + (1- )T f d = (0,0) S Agent 1 Figure 2. Illustration of the Theorem 2. λ)f(cch(b)) L(f(cch(B)), t). By concavity, f(t ) = t. Thus, there is G Ω 0 such that G T, g G, and g > t. By restricted monotonicity, f(g) f(t ) = t. Then, either (m 1 (B), f 2 (G)) (m 1 (B), t 2 ) or (f 1 (G), m 2 (B)) (t 1, m 2 (B)). Finally, L(f(cch(B)), t) L(f(cch(B)), f(g)). Step 3. Extending the proof to the entire domain. Let B Σ 0. Let T Ω 0 be such that EL α (B) T. We have B T B and B T T. By restricted monotonicity, f(b T ) f(b) and f(b T ) f(t ) = EL α (T ) = EL α (B). Since f(b T ) SP O(B T ) and EL α (B) B T, then f(b T ) = EL α (T ) = EL α (B). By restricted monotonicity, EL α (B) = f(b T ) f(b). Since EL α (B) SP O(B), then f(b) = EL α (B). All axioms of Theorem 2 are tight. (1) Dropping strong Pareto optimality. The proportional solutions (E α ), introduced by Kalai (1977b)[6], become admissible. (2) Dropping concavity. The Kalai-Smorodinsky solution (KS), introduced by Kalai and Smorodinsky (1975) [7], becomes admissible.

12 12 JAUME GARCÍA-SEGARRA, MIGUEL GINÉS-VILAR (3) Dropping restricted monotonicity. The utilitarian solution, becomes admissible. Theorem 3. For n = 2. A solution satises strong Pareto optimality, restricted monotonicity, and super-additivity if and only if it is a weighted losses solution. Proof. Existence. Let α. Clearly, EL α satises strong Pareto optimality and restricted monotonicity. We show that EL α also satises super-additivity. It is clear that for each i {1, 2}, m i (S + T ) = m i (S) + m i (T ). Thus, there is k R such that EL α i (S + T ) = m i (S + T ) k α i = m i (S) + m i (T ) k α i and EL α i (S + T ) S + T. Thus, there are k, k R such that EL α i (S) + EL α i (T ) = m i (S) k α i + m i (T ) k α i = m i (S) + m i (T ) k+k α i and EL α i (S) + EL α i (T ) S + T. Since k k + k, then EL α i (S + T ) EL α i (S) + EL α i (T ). For the case when α 1 = 0 or 1 is trivial to prove existence. Uniqueness. Let α. Let f be a solution satisfying strong Pareto optimality, restricted monotonicity, and super-additivity. By Theorem 1, f satises concavity. By Theorem 2, the weighted losses is the only solution that satises strong Pareto optimality, restricted monotonicity and super-additivity. The axioms of Theorem 3 are tight. All solutions used in the independence of the axioms of Theorem 2 applies to Theorem 3. A special case occurs when α 1 = (1 α 1 ) = 1. Then, the weighted losses solution 2 yields a particular outcome. Whenever the agents have equal bargaining power, the solution becomes symmetric. Then, the weighted losses solution becomes equal losses solution. Therefore, the equal losses solution also satises concavity and super-additivity. We provide a generalization of the equal losses solution and two characterizations. One on the basis of concavity, due to which the weighted losses solution should be applied when two agents are bargaining under probabilistic uncertainty. For example, in the standard bilateral monopoly model 11. Thus, if the agents are bargaining under probabilistic uncertainty, we should use one solution that satises concavity. In addition, if we also want the solution to satisfy strong Pareto optimality and any type of monotonicity 12, the unique solution that we can apply is the weighted losses solution. If we have a multi-issue problem, the weighted losses solution also satises 11 Typically applied to the context of Labor Market. 12 In this case, restricted or individual monotonicity.

13 BARGAINING UNDER PROBABILISTIC UNCERTAINTY 13 super-additivity. We also point out that the equal losses solution satises concavity and super-additivity, yielding the unique symmetric solution that is strong Pareto optimal for two agents, restricted monotone and either concave or super-additive. Unfortunately, the weighted losses solution does not satisfy scale invariance except for the cases where one agent has all the bargaining power. That is, when α i = 1. Corollary 1. The only two solutions satisfying strong Pareto optimality, restricted monotonicity, scale invariance, and concavity or super additivity, are the weighted losses solutions EL (1,0) and EL (0,1). To give all bargaining power to one agent implies a type of dictatorial solutions. These dictatorial solutions are strong Pareto optimal. In this sense, EL (1,0) and EL (0,1) are less "tyrannical" that the proportional solution 13 when all bargaining power is given to one agent. The reason is because the dictator grants utility to the other agent always that the other agent can improve her pay-o without cost for the dictator 14. In the next section, we introduce a straightforward transformation of concavity, that it is a weaker condition than the usual version. Thus, we relax concavity and add scale invariance. Kalai-Smorodinsky solution. We propose a new solution that is a generalization of the In the line of mutual concessions philosophy, we propose a new solution for asymmetric bargaining problems that we name weighted proportional losses solution. This weighted proportional losses solution is graphically represented in Figure Weighted Proportional Losses Let N = {1, 2, 3,..., n} be the set of all agents. Let dist be the distance in the Hausdor metric topology. Let f : Σ R n. Given an α n 1 = {γ R n + : n i=1 γ i = 1}. Let m(s) be the utopia point and edr(s) the endogenous reference point for each i N given by edr i (S) = α i m i (S). Denition 2. The weighted proportional losses solution with weights α n 1 is dened by setting, for each S Σ 0 : (5.1) P L α (S) = max{x S x = (1 λ)edr(s) + λm(s)}. 13 Proposed by Kalai 1977[6]. 14 For n agents, the solution could not intersect the set of utility, then the solution does not yield a lexicographic version of the dictatorial solution.

14 14 JAUME GARCÍA-SEGARRA, MIGUEL GINÉS-VILAR This point results from the distribution of bargaining power on the smallest convex and comprehensive set given an utopia point. There are many solutions based on reference points, for example: Salonen (1985)[?] and (1987)[?], Gupta and Livne (1988)[?], and Anbarci (1995)[?]. Our weighted proportional losses is dened as a convex combination between an endogenous reference point and the utopia point. In terms of losses, we can dene the weighted proportional losses solution 15 for 2-agent problems as follows: (5.2) P L α (S) = max{x S α 1(m 1 (S) x 1 ) m 1 (S) = (1 α 1)(m 2 (S) x 2 ) } m 2 (S) It is a normalized version of weighted losses solution, in the sense that the solution outcome is proportional to the utopia point of the bargaining set and for this reason the solution also satises scale invariance. We provide a characterization for n-agent problems that does not involve this axiom. However, note that the property is still satised by P L α. This happens because for 2-agent problems, the generalization of the equal losses solution 16 and also the symmetric case 17 are individually rational. Therefore, we need to include scale invariance in the characterization to achieve uniqueness in the proof of Theorem 1. In contrast, with n 3, both the equal losses and the weighted losses solutions do not satisfy individual rationality. Then, we do not need to add scale invariance in the characterization to get uniqueness. We will see later that the P L α solution satises strong Pareto optimality for 2- agent problems and not for n-agent problems as is the case for the Kalai-Smorodinsky solution. In this context, as well as in the KS solution, we are interested in properties that remain unaected for a given utopia point. As in the case of monotonicity, we introduce a straightforward transformation of concavity that is a weaker condition than the usual version. This transformation is related to the concept of probabilistic uncertainty for sets with the same utopia point. Therefore, concavity is desirable when the agents are bargaining under restricted probabilistic uncertainty. 15 Note that the weighted proportional losses solution is also a family of solutions. There is a solution for each α. For simplicity we use the singular voice to avoid misunderstandings with other solutions that do not belong to this family. 16 The weighted losses solution. 17 The equal losses solution.

15 BARGAINING UNDER PROBABILISTIC UNCERTAINTY 15 Figure 3. The weighted proportional losses solution for 2 agents bargaining problem. The P L α solution also satises proportional concavity. Thus, if there are changes in all utopia points but they are in the same proportion, the P L α solution maintains the concavity property. Restricted concavity: For each λ [0, 1] and for each pair S, T Σ 0, such that m(s) = m(t ), f(λs + (1 λ)t ) λf(s) + (1 λ)f(t ). The Restricted concavity axiom says the following: Suppose that bargaining takes place now but the feasible set will be known only later. In addition, we know that the utopia point of all possible sets are the same. Then, it would be preferable for the agents to reach a compromise now instead of waiting for the uncertainty to be resolved. A necessary condition for this is that both agents benet from an early agreement. Theorem 4. For n = 2. A solution satises strong Pareto optimality, scale invariance, restricted monotonicity, and restricted concavity, if and only if it is a weighted proportional losses solution.

16 16 JAUME GARCÍA-SEGARRA, MIGUEL GINÉS-VILAR Proof. Existence. Let α n. We show that P L α satises the axioms of the theorem. First, we show that P L α is an individually monotonic solution as dened by Peters and Tijs (1985)[14]. Let = conv{(1, 0), (0, 1), (1, 1)}. Let (C) be the following property: For all s, t [1, 2] with s t, γ(s) γ(t), and γ 1 (s) + γ 2 (s) = s. Let γ : [1, 2] and τ [1, 2]. Thus, γ(τ) = (2 τ)(α, 1 α) + (τ 1)(1, 1) is a map satisfying the (C) property. Let Λ be the family of maps satisfying (C). Then γ P L α Λ. By Peters and Tijs (1985)[14], the proportional losses solution is an individually monotonic solution. Then, P L α satises strong Pareto optimality, scale invariance and restricted monotonicity. Next, we know that P L α also satises restricted concavity. Let S, T Σ 0 and C(S, T, λ) (λs +(1 λ)t ). We know that λp L α (S)+(1 λ)p L α (T ) C(S, T, λ) and P L α (λs+(1 λ)t ) C(S, T, λ). By restricted concavity, P L α (λs+(1 λ)t ) λp L α (S) + (1 λ)p L α (T ). Suppose by contradiction that P L α (λs + (1 λ)t ) < λp L α (S)+(1 λ)p L α (T ). By denition of P L α, if x S, there is no y S such that y > x. Thus, if λp L α (S)+(1 λ)p L α (T ) C(S, T, λ), then P L α (λs+(1 λ)t, α) λp L α (S) + (1 λ)p L α (T ). Uniqueness. Let f be a solution satisfying the axioms of Theorem 4. By Peters and Tijs (1985)[14], f is individually monotonic. Then, there is a monotonic curve γ : [1, 2] such that γ Λ. We prove that γ is a line. Let S Σ 0 be such that m(s) = (1, 1), f(s) is the intersection of the strong Pareto boundary of S and γ. Then, there is t [1, 2] such that f(s) = γ(t). Let V t conv{(0, 0), (1, 0), (1, t 1), (t 1, 1), (0, 1)}. Thus, f(v t ) = f(s) = γ(t). Let λ [0, 1]. Since f satises restricted concavity, f(λcch(v t ) + (1 λ)v t ) λf(cch(v t )) + (1 λ)f(v t ). By the denition of V t, f(λcch(v t ) + (1 λ)v t ) = λf(cch(v t )) + (1 λ)f(v t ). This implies that γ = L(f(cch(V t )), f(v t )). Let s, t R be such that 2 s t 1. Let V s conv{(0, 0), (1, 0), (1, s 1), (s 1, 1), (0, 1)}. Analogously to what we showed for V t, it is clear that, for each s < 1, the curve that connects f(v s ) and f(cch(v s )) is a line. The axioms of Theorem 4 are tight. (1) Dropping strong Pareto optimality. The weighted Kalai-Smorodinsky solutions (KS α ), introduced by Thomson (1994) [20], become admissible.

17 BARGAINING UNDER PROBABILISTIC UNCERTAINTY 17 (2) Dropping scale invariance. The equal losses solution, characterized by Chun (1988) [1], becomes admissible. (3) Dropping restricted concavity. The asymmetric Kalai-Smorodinsky solutions (lks α ), introduced by Dubra (2001) [2], become admissible. (4) Dropping restricted monotonicity. An utilitarian solution appears dened as follows: Let S Σ 0, UT (S) = argmax{ x 1 m 1 + x 2 m 2 x S}. This version of the utilitarian solution satises scale invariance. The reason is because the outcome is the tangent point with a straight line parallel to the weak Pareto frontier of (cchs). Thus, it also satises restricted concavity. In some cases, the utilitarian solution yields a set. In these cases, we dene a tie break rule. Thus, lut (S) = argmin{dist(x, P L α (S)) x UT (S)}, if UT (S) is a single point, then lut (S) = UT (S). Clearly, this solution becomes admissible. A special case occurs when α i = 1. If the agents have equal bargaining power, n the solution becomes symmetric and then the weighted proportional losses solution becomes the Kalai-Smoradinsky solution. There is a weaker axiom than symmetry to reect this fact, midpoint domination. (Moulin (1983) [8]; also see Thomson (1994)[20], Sect 4.1). Midpoint domination says that, for each problem, each agent's payo should be at least as large as the average of its dictatorial outcomes. This average can be interpreted as an equalprobability lottery over these outcomes. Midpoint domination is formally dened in the R 2 domain as follows. Midpoint domination: For each S Σ 0, f(s) [ 2 i=1 f D i (S)]/2 where f D i (S) is the dictatorial non lexicographic solution for player i. The next corollary shows an alternative characterization of the Kalai-Smorodinsky solution in terms of restricted probabilistic uncertainty. Thus, the KS solution satises restricted concavity (however, it does not satisfy concavity. Thomson (1994) [20]). Corollary 2. The Kalai-Smorodinsky solution is the only solution that satises strong Pareto optimality, midpoint domination, restricted monotonicity, and restricted concavity. There are two cases where the weighted losses solution satises concavity, P L (1,0) and P L (0,1).

18 18 JAUME GARCÍA-SEGARRA, MIGUEL GINÉS-VILAR Corollary 3. For 2 agents, P L (1,0) EL (1,0) and P L (0,1) EL (0,1). This phenomenon only happens for two agents. The reason is because the comprehensive hull of the utopia point 18 intersects the axes x, y in each vertex of the individually rational simplex 19. In contrast to EL α, when one agent has all the bargaining power 20, P L (α) denes a type of the dictatorial solutions even for n 3 agents. The weighted proportional losses solution belongs to a family of solutions that are all lines on the feasible set starting on the utopia point and nishing on the cch boundary of the set. The solution is the intersection of these lines with the strong Pareto frontier. We named this family dual reference functions solutions. Conceptually, our dual reference function solutions are related with Thomson's[19] reference function solutions, that are all lines on the feasible set starting on the disagreement point. The Kalai-Smorodinky solution is itself a dual function. Thus, it could be considered as proportional gains from the disagreement point, or as proportional losses from utopia point. The weighted Kalai-Smorodinky solution (KS α ), introduced by Thomson (1994) [20], is viewed in the sense of proportional gains from the disagreement point. The KS α solution satises restricted concavity. However, it does not satisfy strong Pareto optimality. The asymmetric Kalai-Smorodinsky solution (lks α ), introduced by Dubra (2001) [2], satises strong Pareto optimality. However, it does not satisfy restricted concavity. In contrast, the weighted proportional losses solution focuses on the sense of proportional concessions from the utopia point. Both the Kalai-Smorodinsky solution (KS) and the weighted proportional losses solution (P L α ) satises strong Pareto optimality 21 and restricted monotonicity. For these reasons, the P L α is a non symmetric Kalai-Smorodinsky version that ts accurately in terms of eciency and restricted probabilistic uncertainty. In contrast to the EL α solution, the P L α solution satises individual rationality even for n 3 agents. In the next section, we extend the model for n-agent problems. We also provide a characterization of the weighted proportional losses solution for n-agent problems. We replace strong Pareto optimality by weak Pareto optimality. We also drop scale invariance. Finally, we add utopia continuity. Utopia continuity 18 EL (1,0) and EL (0,1) are the rays that denes the comprehensive hull of the utopia point. 19 P L (1,0) and P L (0,1) are rays that connect the utopia point with each vertex of the individually rational simplex. 20 Her α i = For 2-agent problems.

19 BARGAINING UNDER PROBABILISTIC UNCERTAINTY 19 was introduced by Garcia-Segarra and Gines-Vilar [3]. Utopia continuity says that if the Hausdor metric distance among a sequence of sets and the utopia point tend to zero, then the Hausdor metric distance among the solutions of each set and the utopia point should also tend to zero. Let{S k } k 1 be a sequence of sets picking elements of Σ 0. We formally dene: Utopia continuity: Given {S k } k 1, such that dist(s k, m(s)) 0 in the Hausdor topology, then dist(f(s k ), m(s)) 0. Theorem 5. For n agents. A solution satises weak Pareto optimality, restricted monotonicity, utopia continuity, and restricted concavity axioms if and only if it is a weighted proportional losses solution. Proof. Existence. Let α n 1. Clearly, P L α satises Weak Pareto optimality, restricted monotonicity, and utopia continuity. We prove that it also satises Restricted concavity. Suppose by contradiction that λp L α (S) + (1 λ)p L α (T ) (λs +(1 λ)t ) and P L α (λs +(1 λ)t ) (λs +(1 λ)t, ). By restricted concavity, P L α (λs + (1 λ)t ) λp L α (S) + (1 λ)p L α (T ). Suppose P L α (λs + (1 λ)t ) < λp L α (S) + (1 λ)p L α (T ). By denition of P L α, there is no y S such that y > x. Then, if λp L α (S) + (1 λ)p L α (T ) (λs + (1 λ)t ), P L α (λs + (1 λ)t ) λp L α (S) + (1 λ)p L α (T ). Uniqueness. Let S Σ 0. Let V t = {x R n + n i=1 x i m i t}. Let s, t [1, n] and cch(v t ) = cch(v s ). By restricted concavity, f(λcch(v t ) + (1 λ)v t ) λf(cch(v t ) + (1 λ)f(v t ). By denition of (λcch(v t ) + (1 λ)v t ), f(λ) = λf. Suppose this line is not aligned with the utopia point of S. Then, there is a point x m(s) on this line such that for some j N, x j = m j. Moreover, there is k such that f(v k ) = x. Let p > k. By restricted monotonicity, f(v p ) f(v k ). If f(v p ) = f(v k ), this is a violation of utopia continuity. If there is an i j such that f i (V p ) > f i (V k ). This is a violation of restricted concavity. The axioms of Theorem 5 are tight. (1) Dropping weak Pareto optimality. Let S Σ 0. If S = cch(s), then, f(s) = d 22 ; otherwise, f(s) KS(S). Clearly, KS(S) satises all axioms of Theorem 5. However, if S = cch(s), the solution yields the disagreement point that is an interior point of W P O(S). 22 Note that if S Σ 0, then, d = 0.

20 20 JAUME GARCÍA-SEGARRA, MIGUEL GINÉS-VILAR KS PL KS EL Figure 4. Solutions for 3 agents bargaining problem. (2) Dropping restricted concavity. The asymmetric Kalai-Smorodinsky solutions (lks α ), introduced by Dubra(2001)[2], satisfy the remaining axioms of Theorem 5. (3) Dropping restricted monotonicity. The same utilitarian solution that we use in the independence of axioms for 2-agent problems in Theorem 4, satisfy the remaining axioms of Theorem 5. (4) Dropping utopia continuity. The weighted Kalai-Smorodinsky solutions (KS α ), introduced by Thomson (1994) [20], satisfy the remaining axioms of Theorem 5. Figure 4 shows several solutions for 3-agent problems. We can see several interesting things. The rst one is that P L α is a solution based on losses that satises individual rationality for n-agent problems. The cch(s) for n = 3 agents, is the triangular surface. Then, the P L α solution, is the intersection of the W P O(S) and the convex combination between an endogenous reference point (interior point of S) and m(s). This is the reason why P L α will always intersect the bargaining set for

21 BARGAINING UNDER PROBABILISTIC UNCERTAINTY 21 n-agent problems in an individually rational point. We provide a characterization that does not include scale invariance. However, the property is still satised for n 3 agents. For some bargaining solutions, the following phenomenon could happen. Let S Σ 0. Suppose, that we expand the whole frontier of S maintaining the same utopia point, m(s). Then, the frontier is closer to m(s) in the Hausdor metric distance, but the solution outcome might not change. Therefore, these solutions might not capture any advantage to be closer of the same utopia point. This phenomenon is called the stagnation eect (Garcia-Segarra and Gines-Vilar (2011)[3]). The weighted Kalai-Smorodinsky solutions, the egalitarian solution, the proportional solution among others, could suer from this stagnation eect. Figure 4 also shows that the weighted Kalai-Smorodinsky solution suers from the stagnation eect. An interesting feature of these bargaining solutions is that they have a desirable property in terms of fairness. Note that if we have a distribution of the bargaining power, given an utopia point, if the feasible set is expanded inside the comprehensive hull of this utopia point, then, the "tension" among the agents decreases. This happens because the relative dierences among the agents are reduced when the utility set has been expanded maintaining the same utopia point. 6. Concluding remarks We show a deep analysis around the concavity and super-additivity combined with eciency and monotonicity. For example: in the canonical bargaining domain for 2- agent problems, there is no solution satisfying concavity (or super-additivity), strong Pareto optimality and monotonicity. The Perles-Maschler solution fails to satisfy any type of monotonicity; the egalitarian solution fails to satisfy strong Pareto optimality. But the weighted losses solution satises concavity, strong Pareto optimality, and individual monotonicity 23. We recommend this solution when facing bilateral bargaining situations with uncertainty between the potential sets of utility The weighted losses solution fails to satisfy monotonicity but it satises a mild type of monotonicity that reects partially the philosophy of the original axiom. 24 As it happens in several Labor market models.

22 22 JAUME GARCÍA-SEGARRA, MIGUEL GINÉS-VILAR If we want a scale invariant solution for all possible distributions of bargaining power 25, then, we must ask for restricted uncertainty 26. This way, we can face bargaining situations even for n-agent problems. Concavity and super additivity. To reect the idea of uncertainty, we study the concavity axiom combined with monotonicity properties. We also propose a straightforward transformation of concavity, restricted concavity. This axiom is a weaker version of concavity. In order to face bargaining over multi-issue problems, we study the super-additivity axiom and the interaction that this axiom has with concavity. We prove the equivalence between supper-additivity and concavity for weak Pareto optimal solutions. Weighted Losses. We provide a generalization of equal losses solution and two characterizations. One, on the basis of concavity, which is why the weighted losses solution should be applied where two agents are bargaining under probabilistic uncertainty. For example, in a standard bilateral monopoly that usually is used in labor market models, if the agents are bargaining under probabilistic uncertainty, we should use one solution that satises concavity. In addition, if we also want the solution to satisfy strong Pareto optimality and some type of monotonicity (in this case, restricted or individual monotonicity), the unique solution that we can apply is the weighted losses solution. If we have a multi-issue problem, the weighted losses solution also satises super-additivity. We also point out that the equal losses solution satises concavity and super-additivity, yielding the unique symmetric solution that is strong Pareto optimal for two agents, restricted monotone and either concave or super-additive. As literature points out, there are no solutions for 2-person problems, satisfying strong Pareto optimality, monotonicity and concavity. But the weighted losses solution is the only solution that satises strong Pareto optimality, restricted monotonicity and concavity. Unfortunately, this solution does not satisfy individual rationality for 3 or more agents. Weighted Proportional Losses. We propose and characterize the weighted proportional losses solution on the basis of restricted concavity. A consequence of 25 For all possible distributions of α i. 26 This way, we can extend the weighted losses solutions for n-agent problems. Note that, as it happens with the Kalai-Smorodinsky solution, this family of solutions does not satisfy strong Pareto optimality for n agent problems. However, the weighted proportional solutions satisfy utopia continuity that implies a mild kind of eciency.

23 BARGAINING UNDER PROBABILISTIC UNCERTAINTY 23 this result is an alternative characterization of the Kalai-Smorodinsky solution which includes contexts with some kind of uncertainty. We show that the weighted proportional losses solution satises desirable properties as are strong Pareto optimality for 2 agents and restricted concavity also fullled by the Kalai-Smorodinsky solution, that are not satised either by weighted[20] or asymmetric[2] Kalai-Smorodinsky solutions. This way, the weighted proportional solution reects a very interesting non-symmetric version of the Kalai-Smorodinsky solution. As literature points out, there is no solution that can satisfy strong Pareto optimality, monotonicity, concavity, and scale invariance for 2-agent problems. But the weighted proportional losses solution is the only one that satises strong Pareto optimality, restricted monotonicity, restricted concavity and scale invariance for 2-agent problems. This solution satises individual rationality even for n-agent problems and we also provide a characterization for this domain. References [1] Y. Chun. The equal-loss principle for bargaining problems. Economics Letters, 26:103106, [2] J. Dubra. An asymmetric Kalai-Smorodinsky solution. Economics Letters, 73:131136, [3] J. Garcia-Segarra and M. Gines-Vilar. The stagnation eect and the individually strict monotonic path solutions. Working paper series UJI MIMEO, [4] A. Gerber and T. Upmann. Bargaining solutions at work: Qualitative dierences in policy implications. Mathematical Social Sciences, 52:162175, [5] C. Herrero and M.C. Marco. Rational equal-loss solution for bargaining problems. Mathematical Social Science, 26:273286, [6] E. Kalai. Proportional solutions to bargaining situations: interpersonal utility comparison. Econometrica, 45: , [7] E. Kalai and M. Smorodinsky. Other solutions to Nash's bargaining problem. Econometrica, 43:513518, [8] H. Moulin. Le choix utilitariste. Ecole Polytechnique, [9] H. Moulin. Implementing the Kalai-Smorodinsky bargaining solution. Journal of Economic Theory, 33:3245, [10] R.B. Myerson. Utilitarianism, egalitarianism, and the timing eect in social choice problems. Econometrica, 49:883897, [11] J. F. Nash. The bairgainig problem. Econometrica, 28:155162, [12] M.A. Perles and M. Maschler. The super-additive solution for the Nash bargaining game. International Journal of Game Theory, 10:163193, [13] H Peters. Simultaneity of issues and additivity in bargaining. Econometrica, 54:153169, 1986.