Risk aversion for losses and the Nash bargaining solution

We call a decision maker risk averse for losses if that decision maker is risk averse with respect to lotteries having alternatives below a given reference alternative in their support. A two-person bargaining solution is called invariant under risk aversion for losses if the assigned outcome does not change after correcting for risk aversion for losses with this outcome as pair of reference levels, provided that the disagreement point only changes proportionally. We present an axiomatic characterization of the Nash bargaining solution based on this condition, and we also provide a decision-theoretic characterization of the concept of risk aversion for losses.


Introduction
In this paper, we propose a variation on the concepts of loss aversion and of risk aversion, called risk aversion for losses. A decision maker is risk averse for losses if this decision maker downgrades payoffs below a given reference level by a nondecreasing concave transformation.
We first formulate and apply this concept in the context of the two-person Nash bargaining model (Nash, 1950), as follows. Given a bargaining solution, assume that the bargainers regard the assigned payoffs as their reference levels. If they are risk averse for losses, then the problem (feasible set) is corrected by applying the associated concave transformations. We call a bargaining solution invariant under risk aversion for losses if after this correction the assigned outcome does not Thanks are due to Horst Zank and to a reviewer for their useful comments.
& Hans Peters h.peters@maastrichtuniversity.nl change, provided that the disagreement point only changes proportionally, i.e., the new, corrected, disagreement point is still on the straight line through the original disagreement point and the bargaining solution outcome. This last restriction is reasonable, and without it the axiom would be overly demanding. Invariance under risk aversion for losses is satisfied by many well-known bargaining solutions, including the Nash bargaining solution and the Kalai-Smorodinsky solution (Kalai & Smorodinsky, 1975). We next concentrate on the Nash bargaining solution and provide an axiomatic characterization with invariance under risk aversion for losses as one of the axioms. The other axioms are Pareto optimality, symmetry, covariance, and expansion independence. This last condition requires the solution not to change if we add only payoff pairs exceeding the utopia payoff (i.e., maximally possible payoff) of one of the bargainers. We also show that the axioms are logically independent.
We finally present a decision-theoretic characterization of the new concept of risk aversion for losses, closely related to Yaari's (1969) characterization of comparative risk aversion.
Our results are closely related to many other results in the literature, both on bargaining and on risk and loss aversion, but we postpone discussion of this literature until the relevant parts of the paper.
Section 2 introduces the Nash bargaining model, and Sect. 3 the concept of risk aversion for losses within this model. Sect. 4 presents the characterization of the Nash bargaining solution based on this concept. In Sect. 5, we provide the decisiontheoretic characterization of risk aversion for losses.

Bargaining
A (two-person bargaining) problem is a convex and compact set S R 2 such that for the disagreement point dðSÞ ¼ ðminfx 1 j x 2 Sg; minfx 2 j x 2 SgÞ, we have (i) x [ dðSÞ for some x 2 S, and (ii) y 2 S whenever y 2 R 2 and dðSÞ y x for some x 2 S. 12 The set of all problems is denoted by B.
The point hðSÞ ¼ ðmaxfx 1 2 R j x 2 Sg; maxfx 2 2 R j x 2 SgÞ is the utopia point of S 2 B. The set WðSÞ ¼ fx 2 S j y 6 [ x for all y 2 Sg is the weakly Pareto optimal subset of S and Note that this implies that d 1 ðSÞ ¼ d 2 ðSÞ and h 1 ðSÞ ¼ h 2 ðSÞ.
A (bargaining) solution is a map u : B ! R 2 such that uðSÞ 2 S for all S 2 B. The following possible properties of u are standard.
The Nash solution N assigns to each problem S the point where the product ðx 1 À d 1 ðSÞÞðx 2 À d 2 ðSÞÞ is maximized over S. Nash (1950) showed that N is the unique solution satisfying, besides Weak Pareto Optimality, Symmetry, and Covariance, the following property. 3 Independence of irrelevant alternatives uðTÞ ¼ uðSÞ for all S; T 2 B such that dðSÞ ¼ dðTÞ, S T, and uðTÞ 2 S.
Independence of irrelevant alternatives can be interpreted as follows. The solution uðTÞ of the bargaining problem T can be seen as the best compromise available in T: in this respect, it beats all other alternatives in T. But if this is the case, then it certainly beats all alternatives in a smaller set S, given that the disagreement point is still the same. 4 On a similar ground, however, the condition can also be criticized: uðTÞ may be a good compromise in T but that does not necessarily imply that it is a good compromise in S, for instance, because the utopia point h(T), the pair of highest available payoffs in T, may be different from h(S), the utopia point of S. This led Kalai & Smorodinsky (1975) to propose the following alternative solution. The Kalai-Smorodinsky solution K assigns to each problem S the unique point of P(S) on the line segment connecting d(S) and h(S). This solution had been studied earlier in Raiffa (1953). Kalai & Smorodinsky (1975) showed that K is the unique solution satisfying, besides Weak Pareto Optimality, Symmetry, and Covariance, the following property. 5 Individual monotonicity u i ðTÞ ! u i ðSÞ for all i 2 f1; 2g and S; T 2 B such that dðSÞ ¼ dðTÞ, S T, and h j ðSÞ ¼ h j ðTÞ for j 6 ¼ i.

Loss aversion in bargaining
Loss aversion (Kahneman & Tversky, 1979) is the often observed phenomenon (e.g., Kahneman et al., 1990;Tversky & Kahneman, 1992) that people tend to downgrade utilities or payoffs that are below some reference level. If one expects to receive one Euro, then 99 cents is perceived as worse than the same 99 cents if one expects to receive less than 99 cents. This 'expectation' is usually called the 'reference level'. In the context of game theory, including bargaining, loss aversion was first introduced by Shalev (2000Shalev ( , 2002, who assumed 'linear loss aversion': see below for a detailed discussion. As usual, a decision maker with utility function v (say, on R) is called more risk averse than a decision maker with utility function u if there is a nondecreasing concave function k such that v ¼ k u (Arrow, 1971;Pratt, 1964;Yaari, 1969). The effect of increased risk aversion in bargaining has been studied extensively: the literature includes Kannai (1977), Kihlstrom et al. (1981), Peters & Tijs (1981), Roth & Rothblum (1982), de Koster et al. (1983, van Damme (1986), Safra et al. (1990), and more.
Here, we consider a property which is closely related to the concept of increased risk aversion, adapted to the bargaining context. Formally, let S be a bargaining problem. Bargainer i is risk averse for losses if for each r i 2 ½d i ðSÞ; h i ðSÞ there is a nondecreasing concave function k i ½r i : ½d i ðSÞ; h i ðSÞ ! R such that k i ½r i ðx i Þ ¼ x i for all x i 2 ½r i ; h i ðSÞ. Here, r i is the reference level. Note that, if r i \h i ðSÞ, this definition implies that k i ½r i ðx i Þ x i for all x i 2 ½d i ðSÞ; r i . Thus, if bargainer i is risk averse for losses, then i's payoffs below a reference level r i are downgraded in a way consistent with increased risk aversion. We call k i ½r i bargainer i's loss function at r i .
The concept of linear loss aversion (Shalev, 2000(Shalev, , 2002Köszegi & Rabin, 2006, 2007see Peters (2012), for a preference foundation) is a special case of risk aversion for losses, obtained by taking Thus, linear loss aversion is stronger than risk aversion for losses. In turn, risk aversion for losses is a special case of (thus, stronger than) some of the loss aversion formulations in the literature: see Sect. 5, where we provide a preference foundation of risk aversion for losses in the spirit of Yaari (1969). Shalev (2002) considered the following property based on the concept of linear loss aversion in the Nash bargaining model. Suppose a bargaining solution u is used, and consider a problem S. Suppose that the players are linearly loss averse with pair of loss aversion coefficients k ¼ ðk 1 ; k 2 Þ. If they regard the payoffs r i ¼ u i ðSÞ, i ¼ 1; 2, as their reference levels, then after correction for linear loss aversion the problem becomes Sðk; rÞ ¼ fðx 1 ðk 1 ; r 1 Þ; x 2 ðk 2 ; r 2 ÞÞ j ðx 1 ; x 2 Þ 2 Sg, with Shalev (2002) showed that the Nash bargaining solution is invariant under such a linear loss aversion correction, i.e., we have NðSðk; NðSÞÞ ¼ NðSÞ.
In the present paper, we consider risk aversion for losses instead of linear loss aversion. For a bargaining problem S, a pair of reference levels r ¼ ðr 1 ; r 2 Þ 2 ½d 1 ðSÞ; h 1 ðSÞ Â ½d 2 ðSÞ; h 2 ðSÞ, and a pair of loss functions k½r ¼ ðk 1 ½r 1 ; k 2 ½r 2 Þ, we denote by Sðk½rÞ ¼ fðk 1 ½r 1 ðx 1 Þ; k 2 ½r 2 ðx 2 ÞÞ j ðx 1 ; x 2 Þ 2 Sg the problem corrected for risk aversion for losses. Now the condition uðSðk½uðSÞÞÞ ¼ uðSÞ for a bargaining solution u is much more demanding than for linear loss aversion, and indeed the Nash bargaining solution does not satisfy it, as the following example shows.
Þ for for all 0 x 1 \ 1 2 , hence bargainer 1 is not linearly loss averse. / We will weaken the condition that the solution u does not change after correcting for risk aversion for losses, by imposing it only when the disagreement point changes proportionally, as follows. For a; b 2 R 2 , let [a, b] denote the convex hull of a and b, i.e., the line segment with endpoints a and b.
Invariance under risk aversion for losses uðSðk½uðSÞÞÞ ¼ uðSÞ for all S 2 B such that uðSÞ\hðSÞ, and loss function pairs k½uðSÞ such that dðSÞ 2 ½dðSðk½uðSÞÞÞ; uðSÞ.
It is not difficult to see that the Nash bargaining solution is invariant under risk aversion for losses. Geometrically, the Nash bargaining solution picks the point on the Pareto boundary of a bargaining problem S at which there is a supporting line with slope equal to minus the slope of the line through this point and the disagreement point d(S) (e.g., Lemma 2.2 in Peters, 1992). Clearly, if we correct S for risk aversion for losses, then the original supporting line of S at N(S) is still a supporting line at N(S) for the corrected problem, and if the disagreement point of the corrected problem is on the straight line through d(S) and N(S), then N(S) is still the Nash bargaining solution outcome of the corrected problem. For ease of reference, we formulate this observation as a lemma.
Lemma 3.2 The Nash bargaining solution is invariant under risk aversion for losses.
The above geometrical argument might suggest that invariance under risk aversion for losses is tailor-made for the Nash bargaining solution, but the following lemma shows that this is not quite true.  S is the triangle with vertices (0, 0), (1, 0), and (0, 1). The problem corrected for risk aversion for losses, T, is the shaded area losses. In the next section, we will add another property to single out the Nash bargaining solution.

A characterization of the Nash bargaining solution
We consider the following possible property of a solution u.
Expansion independence uðTÞ ¼ uðSÞ for all S; T 2 B such that dðSÞ ¼ dðTÞ, S T, and there is i 2 f1; 2g for which u i ðSÞ\h i ðSÞ\x i for all x 2 T n S.
This property says the following (cf. Fig. 2). Suppose the solution of a problem S is a 'true compromise' on the part of bargainer i in the sense that i receives less than the maximally achievable payoff, i.e., the utopia payoff h i ðSÞ. If we now extend S to a problem T by adding only points that are better for bargainer i than this utopia payoff, then the solution should not change. Another way of stating this is that, if we cut off, from a problem T, payoffs exceeding a certain level for bargainer i, and the solution of the resulting problem S is below this level for bargainer i, then the solution of the original problem T should be equal to the solution of S. This property is a kind of dual of independence of irrelevant alternatives. It is a very weak version of the condition of 'independence of irrelevant expansions' used by Thomson (1981) in a characterization of the Nash bargaining solution. It also resembles the condition of 'independence of irrelevant claims' in Albizuri et al. (2020). Of course, it can be criticized on similar grounds as the independence of irrelevant alternatives condition.
Observe that, for the situation in the definition of expansion independence, any supporting line of S at uðSÞ is still a supporting line of T at uðSÞ. Therefore, by the same geometric characterization of the Nash bargaining solution as used to prove Lemma 3.2, we obtain: The announced characterization of the Nash bargaining solution is as follows.
Theorem 4.2 The Nash bargaining solution is the unique solution satisfying Pareto optimality, symmetry, covariance, invariance under risk aversion for losses, and expansion independence.
Proof The results of Nash (1950) and Lemmas 3.2 and 4.1 imply that N has the five properties stated in the theorem. 6 Now, let u be a solution with these properties, and let S 2 B. We prove that uðSÞ ¼ NðSÞ.
By covariance, we may assume without loss of generality that dðSÞ ¼ ð0; 0Þ and NðSÞ ¼ ð1; 1Þ. This also implies that the straight line through the points (2, 0) and (0, 2) is a supporting line of S at (1, 1).
problem T 2 B 0 , T 6 ¼ S, as in the definition of expansion independence, does not exist. This leads us to define the bargaining solution u : B 0 ! R 2 by uðSÞ ¼ KðSÞ for all S 2 B 0 0 and uðSÞ ¼ NðSÞ for all S 2 B 0 n B 0 0 . Then, u satisfies all five conditions in Theorem 4.2 on B 0 , but it is not the Nash bargaining solution. The characterization in the theorem can be restored to hold for B 0 if the expansion independence condition is strengthened by replacing 'for all x 2 T n S' by 'for all x 2 PðTÞ n S' in its formulation. We leave verification of these claims to the reader.

A preference foundation for risk aversion for losses
In the approach of Nash (1950), a bargaining problem arises as the set of expected utility payoff pairs from an underlying set of alternatives and associated lotteries. In this context, it is indeed meaningful to study the impact of risk aversion and loss aversion, as we did in the preceding sections. In this section, we provide a characterization of our concept of risk aversion for losses of a single decision maker. More precisely, we define the concept of risk aversion for losses in terms of preferences, and then show that this is equivalent to the same concept in terms of the specific payoff transformation as introduced earlier in the bargaining context.
We assume that any preference # under consideration in this section can be represented by a von Neumann-Morgenstern 8 utility function u, i.e., ' # ' 0 , Euð'Þ ! Euð' 0 Þ for all '; ' 0 2 L, where EuðÁÞ denotes expected utility: if lottery ' results with probability p j in alternative a j for j ¼ 1; . . .; k for some k 2 N, then Euð'Þ ¼ P k j¼1 uða j Þ. Under this assumption, it is in particular sufficient to know the representing function u on the set of (riskless) alternatives A, i.e., to know u : A ! R. See Herstein and Milnor (1953) for an axiomatic foundation of this assumption. Such a representation is unique up to positive linear transformations, i.e., v also represents # if and only if there are a; b 2 R with a [ 0 such that vðaÞ ¼ auðaÞ þ b for all a 2 A.
For a preference # and an alternative a 2 A, we denote by L 1 ðaÞ ¼ f' 2 L j '1ag the strict preference set of # with respect to a. 9 For r 2 A, define the set L #;r by L #;r ¼ f' 2 L j a # r for all a 2 A in the support of '}. We call a (decision maker with) pair of preferences ð#; # r Þ risk averse for losses at r if L 1 r ðaÞ L 1 ðaÞ and L 1 r ðaÞ \ L #;r ¼ L 1 ðaÞ \ L #;r for all a 2 A. In words: first, if such a decision maker strictly prefers a lottery ' over an alternative a at # r , then the decision maker 8 Von Neumann and Morgenstern (1947). 9 If we replace 1 by # in this definition, then we obtain Yaari's (1969) 'acceptance set'. also strictly prefers ' over a at #; and, second, for any lottery ' of which the decision maker prefers every alternative in its support over r at # r , for any alternative a the decision maker strictly prefers ' over a at # r if and only if the decision maker strictly prefers ' over a at #. The first part of this definition is (a slight variation on) the usual definition of (a decision maker with preference) # r being more risk averse than (a decision maker with preference) #. The second part adds to this that lotteries involving no losses with respect to the reference alternative r are treated equally under # r and #.
Theorem 5.1 Let r 2 A, let ð#; # r Þ be a pair of preferences, and let u : A ! R represent #. Assume that there exist a; a 2 A such that uðAÞ ¼ ½uðaÞ; uð aÞ. Then, the following two assertions are equivalent: (a) ð#; # r Þ is risk averse for losses at r. (b) There is a nondecreasing concave function k : uðAÞ ! R with kðuðaÞÞ ¼ uðaÞ for all a 2 A with a # r, such that v : A ! R, defined by vðaÞ ¼ kðuðaÞÞ for all a 2 A, represents # r .
If uðrÞ ¼ uð aÞ then the proof of (b) is complete. Now assume uðrÞ\uð aÞ, and let a 2 A such that uðrÞ\uðaÞ\uð aÞ, hence uðaÞ ¼ kuðrÞ þ ð1 À kÞuð aÞ for some 0\k\1. Let ' 2 L #;r be the lottery resulting in r with probability k and in a with probability 1 À k. Then, uðaÞ ¼ Euð'Þ, and vðaÞ ¼ kðuðaÞÞ ¼ kðkuðrÞ þ ð1 À kÞuð aÞÞ ! kkðuðrÞÞ þ ð1 À kÞkðuð aÞÞ ¼ kvðrÞ þ ð1 À kÞvð aÞ, where the inequality follows from concavity of k. Suppose that this inequality is strict. Then let k 0 \k such that still vðaÞ ! k 0 vðrÞ þ ð1 À k 0 Þvð aÞ, and let ' 0 denote the lottery resulting in r with probability k 0 and in a with probability 1 À k 0 . Then uðaÞ\k 0 uðrÞ þ ð1 À k 0 Þuð aÞ ¼ Euð' 0 Þ. Hence, ' 0 2 L 1 ðaÞ, ' 0 6 2 L 1 r ðaÞ, and ' 0 2 L #;r , in contradiction with L 1 r ðaÞ \ L #;r ¼ L 1 ðaÞ \ L #;r . Therefore, k is linear on ½uðrÞ; uð aÞ, and in particular kðuðaÞÞ ¼ uðaÞ for all a 2 A with a # r. This completes the proof of the implication [(a) ) (b)]. h A few further remarks on this result are in order. First, the theorem can be extended to cases where u(A) is a general subset of the real numbers, that is, not necessarily bounded, closed, or convex, by using, in particular, a result of Peters and Wakker (1987). For the purpose of the present paper, however, we do not need this. For a bargaining game S, we implicitly assume that the payoff pairs in S arise as expected utilities of lotteries on an underlying set of alternatives A, such that for each weakly Pareto optimal point of S there is an alternative in A resulting in that point. Note that, in Theorem 5.1, if uðrÞ\uð aÞ, then the function k satisfies, additionally, that kðuðaÞÞ uðaÞ whenever uðaÞ uðrÞ-this follows since k is identity on ½uðrÞ; uð aÞ, and k is concave. In our bargaining application, the condition uðrÞ\uð aÞ is satisfied since in the definition of invariance under risk aversion for losses, it is required that the reference point is below the utopia point.
Also, similarly as in Peters (2012), Theorem 5.1 may be extended to characterize objects of the form ð#; # r Þ r2A , in particular, to establish existence of a function k[r] for every r 2 A, possibly with relations between these functions depending on the imposed axioms.
As mentioned earlier, linear loss aversion is a special case of our risk aversion for losses concept. On the other hand, the latter is a special case of some of the loss aversion concepts proposed in the literature. Bowman et al. (1999) and Blavatskyy (2011) propose quite general definitions of loss aversion, of which ours is a special case. Also, Köszegi & Rabin (2007) propose a rather general definition but focus on linear loss aversion. For still other formulations of loss aversion, see Neilson (2002), Köbberling & Wakker (2005), and Schmidt and Zank (2005).
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creativecommons.org/licenses/by/4.0/.