Stagnation proofness and individually monotonic bargaining solutions. Jaume García-Segarra Miguel Ginés-Vilar 2013 / 04

Stagnation proofness and individually monotonic bargaining solutions Jaume García-Segarra Miguel Ginés-Vilar 2013 / 04

Stagnation proofness and individually monotonic bargaining solutions Jaume García-Segarra LEE-Universitat Jaume I Department of Economics jagarcia@eco.uji.es Miguel Ginés-Vilar LEE-Universitat Jaume I Department of Economics mgines@eco.uji.es 2013 / 04 Abstract The aim of this paper is to generalize the results of Peter and Tijs [10] and [9]. We propose a characterization that holds both for n-agent problems and for a larger class of problems. The axioms used in our characterization are weaker because they are implied by the characterization in the aforementioned references. We analyze a phenomenon known as stagnation effect, which takes place when a bargaining solution remains unchanged facing all possible expansions of the bargaining set not affecting the utopia point. Our main result relies on a very weak axiom, stagnation proofness. Whenever a bargaining solution satisfies this axiom, such solution does not suffer from the stagnation effect. Keywords: Bargaining, axiomatization, n-agent problem, monotone paths JEL classification: C78; D74

Stagnation proofness and individually monotonic bargaining solutions Jaume García-Segarra Miguel Ginés-Vilar Abstract The aim of this paper is to generalize the results of Peter and Tijs [10] and [9]. We propose a characterization that holds both for n-agent problems and for a larger class of problems. The axioms used in our characterization are weaker because they are implied by the characterization in the aforementioned references. We analyze a phenomenon known as stagnation eect, which takes place when a bargaining solution remains unchanged facing all possible expansions of the bargaining set not aecting the utopia point. Our main result relies on a very weak axiom, stagnation proofness. Whenever a bargaining solution satises this axiom, such solution does not suer from the stagnation eect. Keywords: Bargaining, Axiomatization, n-agent problem, Monotone paths. JEL classication: C78; D74. Introduction Many bargaining solutions are dened based upon monotone paths. Some examples are the egalitarian solution or the Kalai-Smorodinski solution. There are two interesting generalizations in the literature of solutions generated by monotone paths. The rst one is a characterization for n-agent problems provided by Thomson and Myerson [15]. It focusses on the environment of solutions where there is interpersonal comparability of utility. The egalitarian solution, for example, belongs to this environment. The second one includes two characterizations provided by Peters and Tijs in [10] and [9] respectively. One of them holds for 2-agent problems [10], while the other one holds for n-agent problems [9]. Both characterizations focus on the environment of solutions in which there is no interpersonal comparability of utility. The Kalai-Smorodinsky solution belongs to this environment. The behavior of the solutions characterized in [10] and [9] is more appealing than the behavior of those characterized in [15], when thinking of the way the benets from cooperation are distributed among the agents. This is so because the results introduced in [10] and [9] rely on the axiom of strong Pareto optimality, according to which all gains from cooperation should be exhausted. However, in order to provide a characterization for n-agent problems in [9], a restriction on the bargaining domain is required. For this reason, it seems interesting to look for alternative properties that may allow us to deal with the whole class of bargaining problems. Considering all potential expansions of the possibilities of a bargaining problem not involving a change in the utopia point, we found that, in some cases, there are bargaining solutions providing a better payo for at least one agent, while there are other bargaining solutions that remain unchanged. This is what we call the stagnation eect. Based on this concept, we introduce a new very weak axiom that we call stagnation proofness. Relying on the stagnation proofness axiom, we provide a characterization that holds for n-agents and for the whole class of problems. We remark that our characterization is based on weaker axioms than those introduced in [10] and [9] 1. The article is organized in ve sections. Section 1 contains denitions and notation. In Section 2, we introduce the stagnation eect and the stagnation proofness axiom. In Section 3, we state our main result. In Section 4, we prove that the axioms shown in the main result are implied by the axioms introduced by [10] and [9]. In Section 5, with some remarks, we close the paper. Jaume García-Segarra Miguel Ginés-Vilar Department of Economics, Universitat Jaume I de Castelló, E-mail: jagarcia@eco.uji.es and mgines@eco.uji.es 1 In the sense that they are implied by the axioms introduced in [10] and [9].

2 Jaume García-Segarra, Miguel Ginés-Vilar 1 Denitions and Notation Let N = {1,..., n} be the set of all agents indexed by i. A set S R n is comprehensive when for each x S if y 2 x, then y S. A set S is called a bargaining set if it is convex, closed and comprehensive. The disagreement point, d, is the outcome that the agents get if they do not reach an agreement. A pair (S, d) is called a bargaining problem if S is a bargaining set and d S is the disagreement point. Let Σ be the space of all bargaining problems. A solution is a function f : Σ R n 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). For a given S, the utopia point m(s) is a point dened by the maximum outcome reachable by each player i on the individually rational part of the set S. Formally: m(s, d) max x S x d x 1,..., max x S x d Given a point d R n and a set S R n, S is d comprenhensive if d x y and y S implies x S. The comprehensive hull of a set is formally dened as follows: x n. comp(s, d) {x R n d x s, s S}. The convex hull of a set is dened as follows: { } m m con(s) x R n x = λ k s k where λ k = 1, λ k 0, s k S, k {1,..., m}, m N. k=1 k=1 Let cch{z 1,..., z m } be the convex and d-comprehensive hull of {z 1,..., z m }, where {z 1,..., z m } R n and z k d for each k {1,..., m}. In other words, cch{z 1,..., z m } is the smallest convex and d-comprehensive subset of R n containing z 1,..., z m. Let n 1 (S, d) be the (n 1)-dimensional simplex 3 of S. Formally: n 1 (S, d) { s S, s d n i=1 s i m i (S, d) = 1 Without loss of generality, we work with the class of problems whose disagreement point is zero (Σ 0 ) 4. For simplicity, we will denote 5 by S every pair (S, d) Σ 0. The strong and weak Pareto frontiers of a bargaining set S are respectively the sets SPO(S) {x S if y > x then y / S} and WPO(S) {x S if y x then y / S}. Let int(s) be the set of interior points of S, that is, the points of S that do not belong to the strong or the weak Pareto frontier. Let H and H + be closed half spaces below and above the hyperplane H. }. 2 The stagnation eect Some bargaining solutions may remain unchanged when facing all possible expansions of the bargaining set that do not aect the utopia point. We name this phenomenon the stagnation eect. Bargaining solutions such as the weighted Kalai-Smorodinsky solutions [14], the egalitarian solution, the family of solutions generated by strictly monotone paths [15], and the proportional solutions [6] suer from this stagnation eect. In Figure 1, we represent two paths that generate weak Pareto optimal solutions. However, their behaviour is dierent when the bargaining set expands without aecting the utopia point. The path connecting the points AD 6 provides the solution at point C, and an expansion in the bargaining set without aecting the utopia point D 2 Let x, y R n. The following notation is used: y x if y 1 x 1,..., y n x n; y > x if y x and y x; y x if y 1 > x 1,..., y n > x n. Reversals are dened analogously. 3 Hereinafter just simplex. 4 The axiom of translation invariance says that f(s + t, d + t) = f(s, d) + t for all t R n. For problems Σ d, we can use the following transformation by using translation invariance: If S Σ d, then, S = S {d} = {x R n x = s d; s S} Σ 0. Our results do not depend on translation invariance. 5 Analogously, we will denote by m(s), comp(s), and n 1 (S) the concepts dened above. 6 This is the Kalai-Smorodinsky solution.

Stagnation proofness and individually monotonic bargaining solutions 3 B D C A Fig. 1: The stagnation eect. improves the outcome of some agents. In contrast, the solution generated by the path connecting AB remains stagnant at point B, irrespective of any expansion of the bargaining set 7. Below we provide a formal denition of the stagnation eect, Denition 1 A solution suers from the stagnation eect if and only if S Σ 0, with m(s) / S, such that for each S S with m(s ) = m(s), f(s ) = f(s). Alternatively, we introduce an axiom such that prevents a solution from suering the stagnation eect. This is what we call stagnation proofness. Stagnation proofness: For each S Σ 0, with m(s) / S, there is at least one S S with m(s ) = m(s) and one agent j N, such that f j (S ) > f j (S). Stagnation proofness says that for each bargaining set S, where the utopia point m(s) is not feasible, there is at least one superset S with the same utopia point m(s ) = m(s), such that, for at least one agent j, the solution in the superset S is greater than the solution in the bargaining set S. In other words, this axiom says that the solution has the capability to improve upon the payo of at least one agent when the bargaining set is expanded, and the utopia point 8 remains unchanged. We think that the fullment or not of the stagnation eect may reect substantial dierences in the behavior of the bargaining solutions. We would like to draw the reader's attention to analyze the dierences between weak Pareto optimal solutions that satisfy stagnation proofness and weak Pareto optimal solutions that do not. The example below illustrates it. Consider the following bargaining sets S 1 = cch{(2, 1, 0), (1000, 0, 0), (1000, 0, 1), (2, 1, 1)}, S 2 = cch{(500, 1, 0), (1000, 0, 0), (1000, 0, 1), (500, 1, 1)} and, S 3 = comp{(1000, 1, 1)}. Note that m(s 1 ) = m(s 2 ) = m(s 3 ). As we can see, the possibilities of agent 1 have considerably increased when moving from S 1 to S 2 and S 3. The egalitarian solution 9 is a weak Pareto optimal solution not satisfying stagnation proofness. This solution recommends the same payos E(S 1 ) = E(S 2 ) = E(S 3 ) = (1, 1, 1) for S 1, S 2 and, S 3. In this way, despite the improved possibilities of agent 1, the outcome remains stagnant at the point (1, 1, 1). In contrast, the Kalai-Smorodinsky solution 10 recommends the following payos: KS(S 1 ) = (500.5, 0.5005, 0.5005), KS(S 2 ) = ( 2000 3, 2 3, 2 3 ), and 7 We illustrate the stagnation eect by using a 3-agent problem because we think that it is more intuitive than a 2-agent problem. For two agents this phenomenon could be easily confused with weak Pareto optimality. 8 The solution improves the payo of at least one agent for at least one of the possible expansions of the bargaining set maintaining the same utopia point. 9 The egalitarian solution of a bargaining set E(S) is the maximal point of S of equal coordinates. 10 The Kalai-Smorodinsky solution of a bargaining set KS(S) is the maximal point of S that belongs to the segment connecting the disagreement point with the utopia point.

4 Jaume García-Segarra, Miguel Ginés-Vilar KS(S 3 ) = (1000, 1, 1). Although the Kalai-Smorodinsky solution is also a weak Pareto optimal solution, it satises stagnation proofness. We appreciate in the example how this solution can improve upon the payos of agent 1 for certain expansions of her possibilities. Stagnation proofness is a very weak axiom 11, and there are many solutions that satisfy it. This is the case of the individually monotonic solutions[10] and [9], the Nash solution[8], the utilitarian solution, the Kalai-Smorodinsky solution[7], the asymmetric KS solution (Dubra[4]), the leximin solution (Imai[5]), the lexicographic monotone path solutions (Chun and Peters[2]), the partially monotonic solutions (Salonen[13]), and the asymmetric leximin solution (Driesen[3]). In Section 3, we introduce a generalization of Peters and Tijs[10] and [9]. Based on the stagnation proofness axiom, we provide a characterization for n-agents and for a larger class of problems than those there considered. 3 A general characterization of the individually monotonic bargaining solutions. In addition to stagnation proofness, we will consider the following well-known axioms: Strong Pareto optimality: For every S Σ 0, f(s) SP O(S). Strong Pareto optimality says that all gains from cooperation should be exhausted. Weak Pareto optimality: For every S Σ 0, f(s) W P O(S). Weak Pareto optimality says that no opportunity that benets all agents is ignored. Another well-known axiom deals with the possibility of normalizing the bargaining set by means of a positive transformation. Let p R n ++, ps {p 1 s 1,..., p n s n ) s S}, and pf(s) {p 1 f 1 (S),..., p n f n (S)}. Formally, this axiom states that: Scale invariance: For each S Σ 0, pf(s) = f(ps). When a solution satises scale invariance, the bargaining set S can be normalized in another set with utopia point m(s) = (1,..., 1). Therefore, the scale of units does not matter. 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 version of the traditional monotonicity axiom, which says that, if the feasible set is expanded, nobody should lose utility. However, there is a trade-o between monotonicity and strong Pareto optimality. Even for 2-agent problems, there is no solution satisfying both axioms. Rosenthal[11] introduced the restricted monotonicity axiom, which asks for no losing utility whenever the expansion in the bargaining set does not alter the utopia point 12. This axiom was used by Peters and Tijs[10] and [9] to characterize the family of individually monotonic solutions for two and for n-agent 13 problems respectively. Restricted contraction independence: For each pair S, T Σ 0, with S T and m(s) = m(t ), if f(t ) S, then f(s) = f(t ). The restricted contraction independence 14 axiom was introduced by Roth[12]. This axiom is a weaker and straightforward version of the well-known contraction independence 15 introduced by Nash[8]. The restricted version of contraction independence is compatible with restricted monotonicity. In [10] and [9], Peters and Tijs characterize the individually monotonic bargaining solutions 16 for two and for n-agent problems respectively. In [10], the main result states that a solution satis- es strong Pareto optimality, scale invariance, and restricted monotonicity if and only if it is an individually monotonic solution. This characterization holds for the whole class of convex, closed and d-comprehensive 2-agent bargaining problems. Unfortunately, this characterization does not hold for the whole class of n-agent problems. In [9], a characterization for n-agent problems based 11 We can dene stagnation proofness even in a weaker way: For each S Σ 0, with m(s) / S, there is at least one S S with m(s ) = m(s) such that f(s ) f(s). However, this version of the axiom loses economic motivation, since it also holds when the possibilities of at least one agent increases and at least one agent loses utility while the other remains unchanged. Thus, we cannot dene this version of the axiom as a desirable property. Nevertheless, it can be seen that replacing stagnation proofness by this alternative version of that axiom, all results of this paper still holds. We thank Hans Peters for suggesting us this alternative and weaker version of stagnation proofness. 12 For 2-agent problems this axiom is compatible with strong Pareto optimality. 13 With a restriction on the bargaining domain for the case of n-agent problems. 14 This axiom is commonly called restricted independence of irrelevant alternatives. 15 This axiom is commonly called independence of irrelevant alternatives. 16 That is, the bargaining solutions generated by a monotone path connecting the simplex of a bargaining set and its utopia point.

Stagnation proofness and individually monotonic bargaining solutions 5 ψ 1 ψ 2 ψ 5 ψ 3 ψ 6 ψ 4 ψ 7 Fig. 2: The solutions generated by monotone paths for n-agent problems. on the same axioms is provided. But such characterization requires a restrictive condition 17 on the class of problems. Figure 1 shows a bargaining set that fails to satisfy condition C2. Thus, the set in Figure 1 does not belong to the domain dened in [9]. The aim of this paper is to provide a generalization of [10] and [9] for n-agents problems and for the whole class of bargaining sets. Based on the original denition by Peters and Tijs [9], we introduce a small change in the denition of the solution. We dene a monotonic curve as follows: Let p, q R with {p, q} [1, n]. Given any ψ : [1, n] R n + such that ψ 1 (p) +... + ψ n (p) = p, we say that ψ is a monotonic curve if p q implies ψ(p) ψ(q). Let us denote the family of monotonic curves by Ψ. Denition 2 Given a path ψ, the family of the individually monotonic bargaining solutions for n-agent problems is dened as: h ψ (S) = max{ψ(p) S p [1, n]}. (1) By denition, S is a bounded from above set and ψ is a continuous monotonic curve. Therefore, h ψ (S) is well dened. We characterize the family of the individually monotonic bargaining solutions for n-agent problems and for the whole bargaining domain 18. The departure point is the characterization introduced in [10] and [9]. Next, the strong Pareto optimality axiom is weakened to weak Pareto optimality, and, nally, we add the restricted contraction independence and the stagnation proofness axioms, respectively. Theorem 1 For n-agent problems. A solution f satises weak Pareto optimality, scale invariance, restricted monotonicity, restricted contraction independence, and stagnation proofness if and only if there is a ψ Ψ such that f = h ψ. Proof. Existence. The solution is dened as the maximum point on an individually monotonic path. Thus, it clearly satises all axioms of Theorem 1. Uniqueness. Let S Σ 0 and t [1, n]. By scale invariance, m(s) = 1. Let V t = {x comp(1,..., 1) n i=1 x i t}. Let f be a solution satisfying the axioms. Now, we dene the following path: ψ : [1, n] R n +, ψ(t) = f(v t ). By stagnation proofness and restricted contraction 17 They called this condition C2. Let e i be the vector in R n with i th coordinate equal to 1 and the other coordinates equal to 0. Condition C2 states that for each x S and each i N, if x / SP O(S) and x i < m i (S), then there is an ɛ > 0 such that x + ɛe i S. 18 Which is a larger class of problems than those considered in [9].

6 Jaume García-Segarra, Miguel Ginés-Vilar lex(s) m lex(t) = K(T) K(S) S T d Fig. 3: The leximin solution does not satisfy restricted monotonicity. independence, ψ Ψ. Let h ψ be the solution associated with the path ψ. Suppose f(s) h ψ (S). Take a p [1, n] such that f(s) SP O(V p ). Since f(s) S and f(s) V p, f(s) S V p. By restricted contraction independence, f(s V p ) = f(s). By restricted monotonicity, f(s) = f(s V p ) f(v p ). However, f(s) SP O(V p ), therefore, f(s) = f(v p ). Let k be a real number such that f(v k ) = max{ψ(t) S t [1, n]}. Suppose, f(s) = f(v p ) f(v k ). By restricted monotonicity, f(s) = f(v p ) f(v k ). By denition, f(v k ) S V k. By restricted contraction independence, f(s V k ) = f(v k ). By restricted monotonicity, f(v k ) = f(s V k ) f(s), this implies that f(s) = f(v k ) = h ψ (S). The axioms of Theorem 1 are tight: 1. Dropping weak Pareto optimality. Let S Σ 0. If S = n 1 (S), then f(s) = d 19 = 0, otherwise, f(s) KS(S). Clearly, KS(S) satises all axioms of Theorem 1. However, if S = n 1 (S), then f(s) = d int W P O(S). In words, if the bargaining set S is the simplex of S, this solution recommends the disagreement point, which is an interior point of the weak Pareto optimal frontier of S. Thus, the bargaining solution dened above fails to satisfy weak Pareto optimality. 2. Dropping scale invariance. For 2-agent problems, the equal losses solution, introduced by Chun [1], becomes admissible. When there are three or more agents, the equal losses solution does not satisfy individual rationality 20. We propose an alternative solution that satises all remaining axioms. Let S Σ 0. We dene a solution in two steps. When the utopia point of S is m(s) = (1,..., 1), then, f(s) KS(S) 21, otherwise, f(s) h ψ (S). 3. Dropping restricted monotonicity. The Nash solution [8], becomes admissible. 4. Dropping restricted contraction independence. We can dene a solution that satises the remaining axioms. Fix a point A in a normalized problem 22 (since the solution satises scale invariance) such that A = (x 1..., 1,..., x n ). Thus, A is a point of S where the agent i attains her maximal payo. Let S Σ 0 and V t {x comp(1,..., 1) n i=1 x i t}. By scale invariance, m(s) = (1,..., 1). Let t 1 be a real number such that A SP O(V t 1). Thus, t 1 1, we dene the curve ψ(t) as a composition of two segments, one connecting the disagreement point d with the point A, and the other connecting the point A with the utopia point m(s) = (1,..., 1): 19 Note that if S Σ 0, then, d = 0. 20 Individual rationality is a desirable property. This axiom implies that no agents lose utility by reaching an agreement. 21 The Kalai-Smorodinsky solution over the set S. 22 A problem S with m(s) = (1,..., 1).

Stagnation proofness and individually monotonic bargaining solutions 7 If 0 t t 1, then ψ(t) = t t A + (1 t 1 t )d. If t 1 t n, then ψ(t) = (1 t t1 1 n t )A + t t1 1 n t m(s). 1 We dene the next auxiliary function: ˆf(S) = max{ψ(t) S 1 t n}. Given S, let t 2 be the maximum t such that ψ(t) S, thus, ˆf(S) = ψ(t 2 ). We dene a solution that satises the remaining axioms of Theorem 1. If A S, then, f(s) = ψ(t 1 + t2 t 1 n t (t 2 t 1 )), otherwise 23, f(s) = ˆf(S). Clearly, f satises 1 weak Pareto optimality, scale invariance, restricted monotonicity, and stagnation proofness but it does not satisfy restricted contraction independence. 5. Dropping stagnation proofness. The weighted Kalai-Smorodinsky solutions (KS α ) 24, introduced by Thomson [14], satisfy the remaining axioms of Theorem 1. It can be seen that solutions such as the asymmetric Kalai-Smorodinsky solution (Dubra [4]) are particular cases of our result, since these solutions can be described as solutions generated by a monotone path. Other solutions such as the scale invariance leximin solution (Imai [5]) fail to satisfy restricted monotonicity. A counterexample in Figure 3 illustrates this fact 25. 4 Comparison of the axioms In this section we analyze the relationship between some combinations of axioms in order to see how weak is our characterization in comparison with other results in the literature. Lemma 1 For n-agents. If a solution satises strong Pareto optimality, then it also satises weak Pareto optimality and stagnation proofness. Proof. It is obvious that strong Pareto optimality qualies as weak Pareto optimality 26. Next step is to prove that strong Pareto optimality it also implies stagnation proofness. Let S Σ 0 with m(s) / S. By the strong separating hyperplane theorem there is an hyperplane H such that S int(h ) and m(s) int(h + ). Let S comp(m(s)) H. Since f(s ) SP O(S ), there is a j N such that f j (S ) > f j (S). Suppose by contradiction that for each i N, f i (S ) f i (S), thus, by comprehensiveness of S, f(s ) S int(h ). This is a contradiction with the fact that f(s ) SP O(S ). In Lemma 1 above, we prove that the strong Pareto optimality axiom is stronger than the combination of weak Pareto optimality and stagnation proofness axioms. As we mentioned, there are bargaining solutions such as the Kalai-Smorodinsky solution, that satisfy weak Pareto optimality and stagnation proofness and that fail to satisfy strong Pareto optimality. This feature holds even if we only focus on 2-agent problems, an example is the bargaining solution shown in Section 3 when we drop restricted contraction invariance. Lemma 2 For 2-agents. If a solution satises strong Pareto optimality and restricted monotonicity, then it also satises restricted contraction independence. Proof. Let S, T Σ 0 with m(s) = m(t ); and S T such that f(t ) S. By restricted monotonicity, f(s) f(t ). Since f(s) SP O(S) and f(t ) S, then, f(s) = f(t ). Proposition 1 For 2-agents. If a solution satises strong Pareto optimality and restricted monotonicity, then it also satises weak Pareto optimality, restricted contraction independence, and stagnation proofness. Proof. This proof is straightforward from Lemmas 1 and 2. In terms of how weak each result is, we should distinguish between the 2-agent and the n-agent case. For 2-agent problems, we can compare Theorem 1 with the characterization provided in [10], since both are stated for the whole class of bargaining problems. Relying on Proposition 1, it can 23 If A / S. 24 Note that KS α is a family of solutions because there is a solution for each α. 25 The leximin solution characterized by Imai [5] satises strong Pareto optimality and symmetry for n-agent problems. Then, by the impossibility result of Roth [12], the leximin solution does not satisfy restricted monotonicity. 26 But not all weak Pareto optimal allocations are necessarily strong Pareto optimal.

8 Jaume García-Segarra, Miguel Ginés-Vilar be seen that Theorem 1 is implied by [10]. Nevertheless, the main result in [10] is not implied by Theorem 1. Thus, when we focus on 2-agent and the whole class of problems, Theorem 1 is based on weaker axioms than those provided in [10]. Regarding the n-agent case, we can not compare Theorem 1 with the result introduced in [9], since both results are stated for dierent class of problems. However, if we focus only on the class of problems demanding the C2 condition 27, Lemma 2 and Proposition 1 can be extended for n-agent problems. It is is easy to see that, focusing on such class of problems, Lemma 1 and the extensions for n-agent problems of Lemma 2 and Proposition 1 still hold. That is why a version of Theorem 1, on the class of problems demanding C2, is also implied by the characterization provided in Peters and Tijs [9]. 5 Concluding remarks The stagnation eect prevents the solutions suering from this phenomenon to (at least) partially compensate agents whose bargaining possibilities increase. Hence, the fact that a bargaining solution remains stagnant has important consequences on the stability of the solution and its capabilities of being the result of binding agreements among the agents. In order to avoid the stagnation eect, we introduced the stagnation proofness axiom, an extremely weak axiom fullled by a large amount of bargaining solutions. We also claried the relationship between stagnation proofness and dierent versions of Pareto optimality. The main result of this article is stated in Theorem 1. This result provides new axiomatic underpinnings to explain how the individually monotonic bargaining solutions behave. These ndings contribute to a better understanding of all bargaining solutions generated by monotone paths. References 1. Chun Y (1988) The equal-loss principle for bargaining problems. Economics Letters 26:103106 2. Chun Y, Peters H (1989) Lexicographic monotone path solutions. Operation Research Spektrum 11:4347 3. Driesen B (2012) The asymmetric leximin solution. Working Papers, University of Heidelberg 0523 4. Dubra J (2001) An asymmetric kalai-smorodinsky solution. Economics Letters 73:131136 5. Imai H (1983) Individual monotonicity and lexicographic maxmin solution. Econometrica 51:513518 6. Kalai E (1977b) Proportional solutions to bargaining situations: interpersonal utility comparison. Econometrica 45:16231630 7. Kalai E, Smorodinsky M (1975) Other solutions to nash's bargaining problem. Econometrica 43:513518 8. Nash JF (1950) The bairgaining problem. Econometrica 28:155162 9. Peters H, Tijs S (1984) Individually monotonic bargaining solutions for n-person bargaining games. Methods of Operations Research 51:377384 10. Peters H, Tijs S (1985) Characterization of all individually monotonic bargaining solutions. International Journal of Game Theory 14:219228 11. Rosenthal RW (1976) An arbitration model for normal-form games. Mathematics of Operations Research 1:8288 12. Roth AE (1979) An impossibility result concerning n-person bargaining games. International Journal of Game Theory 8:129132 13. Salonen H (1987) Partially monotonic bargaining solutions. Social Choice and Welfare 4:18 14. Thomson W (1994) Cooperative models of bargaining. Handbook of the Game Theory (R Aumann and S Hart) Chapter 35:12371287 15. Thomson W, Myerson R (1980) Monotonicity and independence axioms. International Journal of Game Theory 9:3749 27 That is, the same class of problems considered in [9].