Individual disagreement point concavity and the bargaining problem

In this study, we provide a characterization of the class of proportional bargaining solutions introduced by Kalai. Our result differs from earlier axiomatizations in that we use a property that we label individual disagreement point concavity. This property is a weakening of disagreement point concavity used by Chun and Thomson. An application illustrates the potential usefulness of our new property in a strategic setting.

past, these contributions have been complemented by an increasing number of approaches that explicitly address the behavior of bargaining solutions with respect to the disagreement point. There are numerous studies of monotonicity properties with respect to the disagreement outcome such as those of Thomson (1987), Wakker (1987), Livne (1989), and Bossert (1994), but for our purposes, models that allow for uncertainty in the exact location of the disagreement point are of more relevance. Properties that pertain to changes of disagreement points and their consequences are examined by Livne (1988), Chun (1989), Chun and Thomson (1990a,b,c), Peters and van Damme (1991), Bossert and Peters (2002), Peters (2010), and, among these, the present paper is most closely related to Chun and Thomson (1990a). Chun and Thomson (1990a) characterize the class of proportional solutions introduced by Kalai (1977). In addition to some well-established conditions, they employ a concavity property with respect to the disagreement point. Suppose that the exact location of the disagreement point is unknown and the agents involved in a bargaining situation have two possible procedures of dealing with the resulting choice problem under uncertainty. The first of these possibilities is to agree on the expected bargaining outcome. This option is not very attractive because usually these expected payoffs are Pareto dominated. A second way of approaching the problem is to calculate the expected value of the disagreement point and solve the problem that results. This ensures that the solution outcome is undominated but it may be the case that the solution thus obtained is worse for some agents than the outcome generated by the first procedure. Disagreement point concavity ensures that this latter shortcoming is excludedonly outcomes that are at least as good as the expected bargaining outcome are permitted to be selected by the solution. The present paper provides an alternative characterization of the same class by weakening disagreement point concavity to an agent-by-agent variant and using a different system of additional axioms.
The following section introduces our basic definitions, along with a discussion of the axioms that are relevant for this paper. Section 3 is devoted to the statement and proof of our main result. We conclude the paper with Section 4 where we discuss a possible application of our new property. In particular, we illustrate that individual disagreement point concavity may be used to guarantee the existence (and, in some cases, the efficiency) of Nash equilibria in a setting where disagreement positions can be chosen strategically.

| PRELIMINARIES
There is a fixed finite set of agents N n = {1, …, } with n 2 ≥ . The sum of a subset S n  ⊆ and a point x n  ∈ is defined as S x s x s S + = { + } | ∈ . For any two vectors x and y in n  , we use the A bargaining problem is a pair S d ( , ), where S is the feasible set and d S ∈ is the disagreement pointthe utility distribution that results if no agreement is reached. The set of all bargaining problems is denoted by n  . This domain of bargaining problems is the same as that used in Chun and Thomson (1990a), except that we allow for the disagreement point to be on the boundary of S. For a feasible set S, let denote the weakly Pareto optimal subset of S. Observe that W S ( ) is equal to the boundary of S. A bargaining solution is a mapping F: n Bargaining solutions that are of particular interest in this paper are the proportional solutions, introduced by Kalai (1977) An important special case is the egalitarian solution, which is obtained for Examples of proportional solutions that are at the opposite extreme from the viewpoint of distributional justice are the dictatorial solutions: if p is the k th unit vector, the entire potential surplus from the negotiations is allocated to the dictator, agent k, and all other agents receive their disagreement utility in all possible bargaining problems S d ( , ). We denote the proportional solution that corresponds to p by E p , and E is the egalitarian solution.
The following four properties of a bargaining solution are well known from the existing literature and do not require much discussion.
Weak Pareto optimality ensures that no strictly dominated outcomes are selected by a solution.
Translation invariance is an information invariance condition. As a consequence of this property, what matters for determining a bargaining outcome are the individual gains from the disagreement outcome.
Translation invariance. For all S d ( , ) Individual rationality ensures that everyone's utility at a solution is not below the disagreement level.
. ≥ Independence of irrelevant alternatives is a collective rationality property: if the set of feasible options shrinks while the chosen outcome remains feasible (with the disagreement point unchanged), the options that are removed are treated as irrelevantthe previously selected utility distribution continues to be the choice recommended by the solution. Independence An additional property with an intuitive interpretation is the axiom of disagreement point sensitivity. It rules out situations in which the solution is not responsive to changes in the disagreement point that are advantageous to at least one agent and disadvantageous to at least one other agent.

Disagreement point sensitivity. For all
. ≠ Chun and Thomson (1990a) provide a characterization of the proportional solutions that employs, in addition to some well-established properties, the following axiom of disagreement point concavity.
This property has an intuitive interpretation in the context of bargaining under uncertainty. Suppose that there are two possible disagreement points, one of which materializes once the uncertainty is resolved. If the agents agree on waiting until the uncertainty is resolved, then the expected outcome may be (and often is) Pareto dominated. As an alternative, the agents could proceed by replacing the pair of possible disagreement points with its expected value and agree on an outcome for the resulting problem instead. In the latter case, assuming that the bargaining solution is at least weakly Pareto optimal, the solution outcome is not Pareto dominated (at least not strictly), but it may make some of the agents worse off as compared to the expected (ex ante) solution. The axiom of disagreement point concavity ensures that the only outcomes that can be selected weakly Pareto dominate the expected solution method. See Chun and Thomson (1990a) for a detailed discussion. Chun and Thomson's (1990a) main result shows that the conjunction of disagreement point concavity, weak Pareto optimality, independence of non-individually rational alternatives, and feasible set continuity characterizes the class of proportional solutions. Independence of nonindividually rational alternatives requires that the chosen outcome does not depend on points that do not weakly dominate the disagreement point, and feasible set continuity demands that if a sequence of feasible sets converges to a set S (in the Hausdorff metric), then the corresponding sequence of solution outcomes (with a given disagreement point d) converges to the solution outcome F S d ( , ), provided that all of the problems in the sequence as well as S d ( , ) are in n  .
In this paper, we weaken disagreement point concavity to a property that we call individual disagreement point concavity. The milder axiom is an agent-by-agent variant of the original, defined as follows.
Individual disagreement point concavity.
Along with the first five properties introduced earlier, we use this axiom to provide an alternative characterization of the proportional solutions.

| A CHARACTERIZATION
In this section, we state and prove our main result, which is a characterization of the proportional solutions, introduced by Kalai (1977), and relies on the new property of individual disagreement point concavity. To prove this result, we first show that the restriction of F to the problem of which the weakly Pareto optimal set is the hyperplane of points with sum of coordinates equal to one is a proportional solution. For this, we (only) use that F satisfies weak Pareto optimality, translation invariance, individual rationality, and individual disagreement point concavity.
Lemma 1. Let a bargaining solution F: n n   → satisfy weak Pareto optimality, translation invariance, individual rationality, and individual disagreement point Proof. In this proof, with a slight abuse of notation, we write Our first step is to prove the following claim.
Proof of Claim. For i j = the claim follows from individual disagreement point concavity. For i j ≠ , we obtain Here, the first and next-to-last equalities follow from translation invariance; observe that the applied translations have sums of coordinates equal to zero and therefore leave the feasible set S intact. The inequality follows from individual disagreement point concavity. This completes the proof of the claim.
With d η , , and λ as in the claim, we have for all i j N , where the equalities follow from weak Pareto optimality, and the inequality follows from the claim (with ℓ instead of i); therefore, Thus, for all d η , , and λ as in the claim, and all i j N , ∈ , we obtain by the claim and by (1) that Finally, define p F = (0). By weak Pareto optimality and individual rationality, (as earlier, the translation is over a vector with coordinate sum equal to 0, so that the feasible set remains S). This completes the proof of the lemma. □ For a feasible set S, a point x S ∈ , and a bargaining solution F , we define the disagreement point set (of x under F in S) as The following lemma is a consequence of Lemma 1.
Proof. Let p be as in Lemma 1. By independence of irrelevant alternatives, The proof is now completed by invoking translation invariance. □ The above auxiliary results do not make use of disagreement point sensitivity. We now add this axiom to obtain the final lemma to be used in the proof of the theorem.
Proof. This proof follows immediately from Lemma 2 and disagreement point sensitivity. □ We are now ready to prove our characterization result.
Proof of Theorem 1. It is easy to see that every proportional solution satisfies the first five axioms in the statement of the theorem. Chun and Thomson (1990a) prove that these solutions satisfy disagreement point concavity which, in turn, implies individual disagreement point concavity.
Conversely, assume that F satisfies the six axioms. Let p be as in Lemma 3. We prove

| AN APPLICATION
Disagreement point concavity can be interpreted as requiring that in the case where the disagreement point is uncertain, it is in the interest of all players to reach an agreement on the basis of the expected disagreement point, rather than wait for the uncertainty to resolve; see Chun and Thomson (1990a) for a detailed discussion. Individual disagreement point concavity has a similar interpretation, but now for the case where each player faces uncertainty with respect to the own disagreement payoff.
We now describe another situation in which individual disagreement point concavity is a convenient property. Consider a bargaining problem between a labor union and an employer on a contract that specifies the wages and the employment rate, in the spirit of McDonald and Solow (1981), and suppose this problem is repeated (occurs twice). If the union insists on a combination of high wages and a high employment rate this year, then this may require a longterm strike and thus a large budget to maintain such a strike so that, as a consequence, there may only be a small strike budget available for next year. Similarly, if the employer insists on a combination of low wages and a low employment rate this year, which may require to endure a long strike and thus a large profit loss, then it may have a much weaker position next year.
In a simplified version, this situation can be represented by two feasible sets S a and S b such that where C is a compact and convex subset of + We also fix the "disagreement budget" for each player to be δ δ , (0,1] 1 2 ∈ . Let F be a weakly Pareto optimal and individually rational two-player bargaining solution. We define a two-player non-cooperative game as follows. Each player i {1, 2} ∈ chooses a pair of personal disagreement levels d d ( , ) clearly constitutes a Nash equilibrium but, unless C is a triangle, this equilibrium is inefficientthat is, there are feasible total payoffs that are higher for both players than the equilibrium payoffs. We now focus on interior Nash equilibria, that is, strategy pairs such that d d δ 0 < , < If, in addition, we narrow down our focus on the egalitarian solution E, then it is again not hard to see that at an interior Nash equilibrium 2 . This implies, in particular, that the equilibrium is efficient. Thus, individual disagreement point concavity makes it easy to characterize interior Nash equilibria in this game. Moreover, applying the egalitarian solution guarantees that these equilibria are efficient.