A Concessions-Based Procedure for Meta-Bargaining Problems

Abstract

In 1950, Nash’s seminal paper introduced the axiomatic approach to the analysis of bargaining situations. Since then, many bargaining solutions have been proposed and axiomatically analyzed. The fact that agents, when face with a bargaining problem, can come up with different solution concepts (that is, different notions of fairness and equity) was first introduced by van Damme (J Econ Theory 38:78–100, 1986) with the meta-bargaining model. In this paper we present and axiomatically analyze a procedure for solving meta-bargaining problems, which we call Unanimous–Concession. As an example, we show that the Nash solution is the result of the meta-bargaining process we define, when agents have dual egalitarian criteria. Finally, we compare, from an axiomatic point of view, our proposal with other meta-bargaining procedures.

Introduction

The multiplicity of reasonable criteria to determine a solution in a bargaining problem leads to a dilemma. If the agents agree on one of them, this should be applied. But what happens if the agents’ criteria are different? The following situation, analyzed in experiments by Roth and Murnighan (1982), shows how, in real bargaining, the agents support different criteria.

“Two subjects are given one hundred lottery tickets to divide between them. Each subject’s chance of winning a prize is proportional to the number of lottery tickets he receives in the bargaining, but the money values of prizes are different for the two players: $\$20$ for the first one and $\$5$ for the second.”

The experimental results show that there are two focal proposals: either to split the expected prize equally [(20 and 80 tickets respectively, which is the egalitarian solution (Kalai 1977)], or to split the lottery tickets equally [50 and 50 tickets, namely the Nash solution (Nash 1950)]. The final outcome lies within the range between these two options. In Roth and Murnighan’s words:

“The observed results suggest that theories which depend only on the feasible set and on the status quo are insufficiently powerful to capture the complexity of this kind of bargaining.”

Although the traditional bargaining problem has been enriched in different ways (see, for instance, Chun and Thomson 1992), van Damme (1986) was the first author who considered the fact that the agents could support different concepts of fairness and equity by means of a meta-bargaining model. In order to solve these situations, he proposed a mechanism consisting of eliminating from the feasible set those alternatives exceeding agents’ previous demands. By using this procedure, van Damme provided strategic foundations supporting the Nash bargaining solution.

From van Damme’s work, different mechanisms have been defined: Chun (1985), Anbarci and Yi (1992), Marco et al. (1995), Naeve-Steinweg (1999), Naeve-Steinweg (2002), Naeve-Steinweg (2004) and Trockel (2002). These papers show that the formulation of the procedure is indeed relevant, since, from an strategic point of view, modifications of the procedure considered yield different conclusions. Our work is focused on the axiomatic point of view.

Our proposal follows a negotiation process where agents proceed stepwise through the bargaining set. This idea, formalized in the step-by-step negotiation axiom (Kalai 1977), already appears in the discrete Raiffa (1953), which was extended in Diskin et al. (2011). John and Raith (1999) develop a strategic bargaining game which incorporates multistage negotiations. As mentioned, we are interested in the axiomatic approach. In this context, in a recent paper, Trockel (2015) axiomatically analyzes the Raiffa solution.

The paper is organized as follows. In Sect. 2 we introduce the notation and preliminary notions. We present our procedure, which we call Unanimous–Concession, in Sects. 3 and 4 characterizes it axiomatically. The last two sections are devoted to apply our procedure to a particular family of bargaining situations and compare it with different mechanisms in the literature.

Preliminaries

A two-person bargaining problem is defined by a pair (S, d), where $S \subseteq {\mathbb {R}}^{2}$ and $d\in S$. The points in S represent the feasible utility levels that the individuals can reach if they agree. When this is not the case, they end up at the disagreement point d. We will denote by $\Sigma ^{2}$ the class of two-person bargaining problems (S, d) such that:¹

• 1. S is convex, closed and bounded from above.

• 2. S satisfies strict comprehensiveness: if $x\in$ S and $x \ge y,$ then $y\in$ int(S).

• 3. There exists $x\in S$ such that $x > d$.

A bargaining solution is a single-valued function $f: \Sigma ^{2}\rightarrow {\mathbb {R}}^{2}$ such that for all (S, d) $\in \Sigma ^{2},$ $f(S,d)\in S.$

Since Nash’s solution many other bargaining solutions have been introduced in the literature, and most of them have been axiomatically characterized in terms of a set of properties that they satisfy. Some of these properties are shared for most of the solutions, while others are specific for each one of these solutions. Let F be an arbitrary set of bargaining solutions and call its elements F-admissible solutions.

Definition 1

A two-person meta-bargaining problem on F is given by a triple $\left[ (S,d);f,g\right]$ where $(S,d) \in \Sigma ^{2}$ and $f,g \in F.$

The interpretation is that $f, \, g$ are two F-admissible bargaining solutions supported by agents one and two, respectively.

Definition 2

For any non-empty set F of bargaining solutions on $\Sigma ^{2}$ a meta-bargaining procedure on F is a (single valued) mapping

$$\begin{aligned} \varPhi :F\times F\rightarrow F,\text { such that, for all } f,g\in F, \, \varPhi (f,g)=\varPhi _{fg}\in F. \end{aligned}$$

The Unanimous–Concession Procedure

Definition 3

The Unanimous–Concession procedure ${\mathbf {U}}$ assigns to each pair of bargaining solutions $f,g \in F$ a new solution ${\mathbf {U}}_{fg}$ defined in the following way:

Let (S, d) $\in \Sigma ^{2}$ and let $\left\{ d^{k}\right\}$ be the sequence,

      $d^{1}=d,$

and for all $k\in {\mathbb {N}},k>1,$

      $d^{k}=\left\{ \begin{array}{lll} \inf \left\{ f\left( S,d^{k-1}\right) ,g\left( S,d^{k-1}\right) \right\} &{} \text { if } &{} d^{k-1}\in int\left( S\right) , \\ \\ d^{k-1} &{} \text { if } &{} d^{k-1}\in \partial \left( S\right) . \end{array} \right.$

Then,

$$\begin{aligned} {\mathbf {U}}_{fg}\left( S,d\right) =\lim \limits _{k\rightarrow \infty }d^{k} = d^*. \end{aligned}$$

Figure 1 shows how the Unanimous–Concession procedure is defined. The interpretation of this procedure is as follows: each agent has his own proposal, given by the solutions f and g; thus, for a problem $\left( S,d\right) \in \Sigma ^{2}$, the solution outcome is the result of a step-by-step meta-bargaining process, beginning at the disagreement point. In the first step, each agent concedes a certain amount to the other, according to a unanimous criterion: the maximum amount on which both agents agree. So these concessions are a natural expression of the idea that what is not actually in dispute should be conceded to the opponent. “What is not actually in dispute” is determined by the solutions proposed by the individuals. This allows the definition of a new disagreement point, and then a new bargaining problem, in which the same idea can be applied. The point of convergence of this iterative process determines the agents’ agreement for the meta-bargaining problem.

Fig. 1

The Unanimous–Concession procedure

Remark 1

In order to define a procedure for meta-bargaining problems, a crucial point is to choose the appropriate domain of criteria to be supported by the agents; that is, the family F of admissible solutions. This is important for two reasons: on the one hand, it should contain the solutions which, in general, can be considered fair for bargaining problems; and, on the other hand, this domain must contain only those solutions for which the existence of the procedure can be guaranteed. For instance, the domain of solutions in which van Damme’s mechanism is well defined consists of the family of solutions that satisfy Pareto optimality, scale invariance, symmetry and risk sensitivity. Moreover, he assumes that each agent’s proposal gives more utility to himself than his opponent’s does. However, for the existence of our procedure, we just need to restrict the class of admissible solutions supported by the agents to those individually rational (IR); that is, solutions which give to any agent at least the disagreement utility point. This condition, under strict comprehensiveness² of S, ensures the existence and uniqueness of the point $d^{*}$ in Definition 3 since, if the solutions satisfy (IR), the sequence $\{d^{k}\}$ is increasing, by construction, and bounded, because the set S is bounded from above, so it converges to some point in S.

Apart from the above mentioned rationality condition (IR), to ensure efficiency of the meta-bargaining solution outcome, we consider the following conditions:

(SIR):

Strong individual rationality:

$$\begin{aligned} \text {For all } (S,d)\in \Sigma ^{2}, \quad f(S,d)>d. \end{aligned}$$

(Cont):

Continuity:³

If a sequence $\left\{ (S^{n},d^{n})\right\} \subseteq \Sigma ^{2}$ converges to $(S,d)\in \Sigma ^{2}$ in the Haussdorf topology, the sequence $\left\{ f(S^{n},d^{n})\right\}$ converges to f(S, d).

From here on, we will consider the following family of admissible solutions

$$\begin{aligned} F^{*}=\{\text{bargaining solutions satisfying } \textbf{(SIR)} \text{ and } \textbf{(Cont)} \} \end{aligned}$$

Next result shows that the Unanimous–Concession procedure is well defined in the family $F^{*}.$

Proposition 1

Whenever $f,g\in F^{*},$ then ${\mathbf {U}}_{fg}\in F^{*}$.

Proof

By observing that ${\mathbf {U}}_{fg}\left( S,d\right) \ge \inf \{f(S,d),g(S,d)\}$ and being both $f(S,d)>d$ and $g(S,d)>d,$ (SIR) is satisfied, and only the continuity axiom remains to be proven. However, this property is a direct consequence of the way in which the solution has been defined, since, when $f,g\in F^{*},$ for all (S, d),  ${\mathbf {U}} _{fg}(S,d)=\lim \nolimits _{k\rightarrow \infty }d^{k},$ and the infimum function is continuous. $\square$

The following result proves that the Unanimous–Concession procedure, when defined in the class $F^{*},$ proposes a Pareto optimal point for any bargaining situation.

Proposition 2

For any pair of solutions $f,g\in F^{*}$ and any bargaining problem $(S,d)\in \Sigma ^{2},$ the Unanimous–Concession procedure yields a Pareto optimal point.

Proof

Let us show that the point $d^{*}$, introduced in Definition 3, belongs to PO(S). If we suppose that $d^{*}=\left( d_{1}^{*},d_{2}^{*}\right) \in$ int$\left( S\right)$, since

$$\begin{aligned} d_{1}^{*}= & {} \lim \limits _{k\rightarrow \infty }\min \left( f_{1}\left( S,d^{k}\right) ,g_{1}\left( S,d^{k}\right) \right) \\ d_{2}^{*}= & {} \lim \limits _{k\rightarrow \infty }\min \left( f_{2}\left( S,d^{k}\right) ,g_{2}\left( S,d^{k}\right) \right) , \end{aligned}$$

there is a subsequence such that, without loss of generality,

$$\begin{aligned} d_{1}^{*}=\lim \limits _{k\rightarrow \infty }f_{1}\left( S,d^{k}\right) . \end{aligned}$$

If we consider the sequence $\left\{ (S,d^{k})\right\}$, which converges to $(S,d^{*})$, by (Cont) we obtain $d_{1}^{*}=f_{1}\left( S,d^{*}\right) ,$ contradicting that f satisfies (SIR). $\square$

The following property provides a complete description of the Unanimous–Concession procedure.

Definition 4

A meta-bargaining procedure, $\varPhi :F\times F\rightarrow F,$ satisfies Independence of Conceded Alternatives (ICA) if for each $f,g\in F$ and each $(S,d)\in \Sigma ^{2},$ if $d^{\prime }=\inf \{f(S,d),g(S,d)\},$

• 1. $d^{\prime }\notin PO(S)\Rightarrow \varPhi _{fg}(S,d)=\varPhi _{fg}(S,d^{\prime }).$

• 2. $d^{\prime }\in PO(S)\Rightarrow \varPhi _{fg}(S,d)=d^{\prime }.$

Remark 2

Part (1) of (ICA) can be seen as a modification of a property used in van Damme (1986), in which it is required that a meta-bargaining procedure be independent of those alternatives that are not demanded by the agents. Here we require the independence of those alternatives that are simultaneously dominated by the proposals of the agents, so that the problem may be solved in two ways: (i) directly, by applying the procedure to the initial problem; or (ii) by giving each agent the maximum amount on which both players agree, $\inf \{f(S,d),g(S,d)\},$ which can be viewed as “natural concessions”, and applying the procedure to the problem where the new disagreement point is such a unanimous agreement. The fact that both methods should give the same result is related to the conditions analyzed in the classical bargaining theory (Kalai 1977; Myerson 1977), called step by step negotiation and composition. Part (2) says that whenever both agents agree on a Pareto optimal point, such a point should be the solution of the problem.

The following result shows that on $F^{*}$, the Unanimous–Concession procedure is completely described by (ICA).

Proposition 3

The Unanimous–Concession procedure ${\mathbf {U}}$ is the only procedure on $F^{*}$ satisfying $\mathbf {(ICA).}$

Proof

It is quite simple to prove that ${\mathbf {U}}$ satisfies (ICA). Let $\varPhi$ be a meta-bargaining procedure, $\varPhi :F^{*}\times F^{*}\rightarrow F^{*}$ satisfying (ICA), and let $(S,d)\in \Sigma ^{2},$ $f,g\in F^{*}.$ Then, if $f(S,d)=g(S,d)\in PO(S),$

$$\begin{aligned} \varPhi _{fg}(S,d)={\mathbf {U}}_{fg}(S,d)=\inf \{f(S,d),g(S,d)\}=d^{\prime }. \end{aligned}$$

In other case,

$$\begin{aligned} \varPhi _{fg}(S,d)=\varPhi _{fg}(S,d^{\prime }) \text { and } {\mathbf {U}}_{fg}(S,d)= {\mathbf {U}}_{fg}(S,d^{\prime }). \end{aligned}$$

We can now apply the above argument to the problem $(S,d^{\prime })$ and, if $f(S,d^{\prime })=g(S,d^{\prime })\in PO(S)$

$$\begin{aligned} \varPhi _{fg}(S,d^{\prime })={\mathbf {U}}_{fg}(S,d^{\prime }). \end{aligned}$$

In other case, by denoting $\inf \{f(S,d^{\prime }),g(S,d^{\prime })\}=d^{\prime \prime },$

$$\begin{aligned} \varPhi _{fg}(S,d^{\prime })=\varPhi _{fg}(S,d^{\prime \prime }) \text { and } {\mathbf {U}} _{fg}(S,d^{\prime })={\mathbf {U}}_{fg}(S,d^{\prime \prime }). \end{aligned}$$

By repeating this process, we obtain the required result. $\square$

Axiomatic Characterization

In order to characterize the Unanimous–Concession procedure, we introduce new properties that analyze the behavior of a meta-bargaining procedure with regard to the solution concepts used by the agents.

Our first property says that all opinions should be taken into account equally, regardless of whose the opinion is. This idea is easily written as $\varPhi _{fg}=\varPhi _{gf}$ whenever the solutions supported by the agents, f,  g, satisfy Pareto optimality. If the solutions may provide non Pareto optimal outcomes, then the idea is that only the values proposed to any agent are relevant, regardless of who is proposing each value; that is, the final agreement only depends on the values

$$\begin{aligned} \left\{ f_1(S,d),g_1(S,d)\right\} \, \text { and } \, \left\{ f_2(S,d),g_2(S,d)\right\} . \end{aligned}$$

Formally, this condition is established in the following way.

Definition 5

A meta-bargaining procedure $\varPhi :F\times F\rightarrow F$ satisfies Impartiality (Im) if for each $f,g, f', g' \in F,$ such that for all $(S,d)\in \Sigma ^{2}$

$$\begin{aligned} \left\{ f_1(S,d),g_1(S,d),f_2(S,d),g_2(S,d)\right\} =\left\{ f_1^{\prime }(S,d),g_1^{\prime }(S,d),f_2^{\prime }(S,d),g_2^{\prime }(S,d)\right\} , \end{aligned}$$

then $\varPhi _{fg}=\varPhi _{f^{\prime }g^{\prime }}.$

It is clear that the Unanimous–Concession procedure satisfies this property, since the sequence $d^k$ obtained with the bargaining solutions f, g coincides with the one obtained with the bargaining solutions $f'$ and $g'$.

Our second requirement is related to the usual condition of unanimity: whenever the agents agree on a proposed outcome, this will be the solution.

Definition 6

A meta-bargaining procedure $\varPhi :F\times F\rightarrow F$ satisfies Unanimity (Un) if $f(S,d)=g(S,d)$ for some $(S,d)\in \Sigma ^{2}$ implies

$$\begin{aligned} \varPhi _{fg}\left( S,d\right) =f(S,d)=g(S,d). \end{aligned}$$

Note that, in general, this axiom is not compatible with Pareto optimality of the procedure: if we allow the agents to propose non Pareto optimal solutions, then $f(S,d)=g(S,d) = d' \in \text {int}(S)$ yields, by unanimity, $\varPhi _{fg}\left( S,d\right) = d' \notin PO(S).$ The following property is a suitable modification of unanimity that allows us to make it compatible with efficiency.

Definition 7

A meta-bargaining procedure $\varPhi :F\times F\rightarrow F$ satisfies Improvement of dominated solutions (IDS) if for each $(S,d)\in \Sigma ^{2},$

• 1. $f(S,d)=g(S,d)\in PO(S)$ implies $\varPhi _{fg}\left( S,d\right) =f(S,d) = g(S,d)$;

• 2. $f(S,d)\leqq g(S,d)$ implies $\varPhi _{fg}\left( S,d\right) =\varPhi _{fg}(S,f\left( S,d\right) )$;

• 3. $g(S,d)\leqq f(S,d)$ implies $\varPhi _{fg}\left( S,d\right) =\varPhi _{fg}(S,g\left( S,d\right) )$.

This condition means that, when one of the proposals is dominated by the other, the result of the initial problem should be the same if we take the dominated proposal as an initial agreement and then distribute the remainder. When the agents propose Pareto optimal points, this condition coincides with unanimity. By the way in which the Unanimous–Concession procedure is defined, it is immediate that ${\mathbf {U}}$ satisfies (IDS).

The following property establishes that neither agent can lose by increasing the outcome he demands for himself (and not hurting the outcome he proposes for the other agent). Note that if the proposals of agents are Pareto optimal, then this property has no requirement, so it is necessarily satisfied.

Definition 8

A meta-bargaining procedure $\varPhi :F\times F\rightarrow F$ satisfies Monotonicity (Mon) if

• 1. For all f, h bargaining solutions such that $f_{1}\ge h_{1}$, and $f_{2} = h_{2}$ then

  $$\begin{aligned} (\varPhi _{fg})_{1}\ge (\varPhi _{hg})_{1}, \text { for all } g \text { such that } g_{1}\le h_{1}; \, \text { and symmetrically,} \end{aligned}$$

• 2. For all g, h bargaining solutions such that $g_{2}\ge h_{2}$, and $g_{1} = h_{1}$ then

  $$\begin{aligned} (\varPhi _{fg})_{2}\ge (\varPhi _{fh})_{2}, \text { for all } f \text { such that } f_{2}\le h_{2}. \end{aligned}$$

It is clear that the Unanimous–Concession procedure satisfies this property, since the sequence $d^k$ obtained in both cases coincides. The following theorem provides a characterization of our procedure in the class $F^{*}$ of bargaining solutions.

Theorem 1

The Unanimous–Concession procedure ${\mathbf {U}}$ is the only procedure in the class $F^{*}$ satisfying $\mathbf {(Im)}$, $\mathbf {(Mon)}$ and $\mathbf {(IDS)}$.

Proof

As previously mentioned, ${\mathbf {U}}$ satisfies the required axioms. In order to prove its uniqueness, let $\varPhi$ be any meta-bargaining procedure satisfying these properties. First note that for each bargaining problem (S, d) the proposals made by the bargaining solutions f, g are the same as the proposals made by the bargaining solutions $f'$ and $g'$ defined by

$$\begin{aligned} f^{\prime }(S,d)= & {} \left( \max \left\{ f_1(S,d), g_1(S,d)\right\} , \min \left\{ f_2(S,d), g_2(S,d)\right\} \right) \\ g^{\prime }(S,d)= & {} \left( \min \left\{ f_1(S,d), g_1(S,d)\right\} , \max \left\{ f_2(S,d), g_2(S,d)\right\} \right) \end{aligned}$$

that is,

$$\begin{aligned} \{f_1(S,d),g_1(S,d),f_2(S,d),g_2(S,d)\}=\{f_1^{\prime }(S,d),g_1^{\prime }(S,d),f_2^{\prime }(S,d),g_2^{\prime }(S,d)\}, \end{aligned}$$

so, by (Im), $\varPhi _{fg}=\varPhi _{f^{\prime }g^{\prime }}.$

Let us consider now the auxiliary bargaining solution $h\in F^{*}$ defined for all $(S,d)\in \Sigma ^{2}$ as:

$$\begin{aligned} h(S,d)= & {} \inf \{f'(S,d),g'(S,d)\} \\= & {} \left( \min \left\{ f_1(S,d), g_1(S,d)\right\} , \min \left\{ f_2(S,d), g_2(S,d)\right\} \right) . \end{aligned}$$

Then, we can apply (Mon) to the bargaining solutions $f',h$ and $g'$ so the following inequality must be fulfilled:

$$\begin{aligned} (\varPhi _{f'g'})_{1}\ge (\varPhi _{hg'})_{1}. \end{aligned}$$

By construction, $h(S,d)\leqq g'(S,d),$ so (IDS) implies

$$\begin{aligned} \varPhi _{hg'}\left( S,d\right) =\varPhi _{hg'}(S,h\left( S,d\right) ). \end{aligned}$$

Now, it is possible to apply again the (IDS) property to the new problem $(S,d^{2})\in \Sigma ^{2},$ where $d^{2}=h(S,d)=\inf \{f'(S,d),g'(S,d)\},$ since, by construction, $h(S,d^{2})\leqq g'(S,d^{2}),$ and we then obtain

$$\begin{aligned} \varPhi _{hg'}\left( S,d\right) =\varPhi _{hg'}\left( S,d^{2}\right) =\varPhi _{hg'}\left( S,h\left( S,d^{2}\right) \right). \end{aligned}$$

By naming $d^{3}=h(S,d^{2})=\inf \{f'(S,d^{2}),g'(S,d^{2})\}$ and successively applying this property, we obtain a sequence of problems (S, d),  $(S,d^{2}),$ $(S,d^{3}),...$ that converges to a point in the Pareto boundary of S, which coincides, when $f,g\in F^{*}$, with ${\mathbf {U}}_{f'g'}\left( S,d\right) =\lim \limits _{k\rightarrow \infty }d^{k};$ so we obtain:

$$\begin{aligned} (\varPhi _{fg})_{1}\left( S,d\right) = (\varPhi _{f'g'})_{1}\left( S,d\right) \ge (\varPhi _{hg'})_{1}\left( S,d\right) =({\mathbf {U}}_{fg})_{1}\left( S,d\right) . \end{aligned}$$

With an analogous argument, we obtain that

$$\begin{aligned} (\varPhi _{fg})_{2}\left( S,d\right) = (\varPhi _{f'g'})_{2}\left( S,d\right) \ge (\varPhi _{f'h})_{2}\left( S,d\right) =({\mathbf {U}}_{fg}) _{2}\left( S,d\right) , \end{aligned}$$

and as ${\mathbf {U}}_{fg}\left( S,d\right)$ is a Pareto optimal point,

$\varPhi _{fg}(S,d)={\mathbf {U}}_{fg}(S,d).$□

The following examples establish the independence of the properties that characterize the Unanimous–Concession procedure. First, given a bargaining problem (S, d) the ideal point a(S, d) is defined by:

$$\begin{aligned} a_{1}(S,d)=\max \{x : \, (x,d_{2})\in S\}, \quad a_{2}(S,d)=\max \{x : \, (d_{1},x)\in S\}. \end{aligned}$$

Example 1

The procedure which provides, for any $f,g\in F^{*}$, the bargaining solution, $\varPhi _{fg}\left( S,d\right) =\frac{3}{4}d+\frac{1}{4}a(S,d)$, for all (S, d) does not satisfy (IDS), but does satisfy (Im) and (Mon).

Example 2

Next procedure satisfies all the properties except (Mon),

$$\begin{aligned} \varPhi _{fg}\left( S,d\right) =\left\{ \begin{array}{lll} {\mathbf {U}}_{fg}\left( S,d\right) &{} &{} \text {if one proposal is dominated by the other} \\ \\ \frac{3}{4}d+\frac{1}{4}a(S,d) &{} &{} \text {otherwise} \end{array} \right. \end{aligned}$$

Example 3

The following procedure does not satisfy (Im), but does satisfy (IDS) and (Mon),

$$\begin{aligned} \varPhi _{fg}\left( S,d\right) =\left\{ \begin{array}{lll} {\mathbf {U}}_{fg}\left( S,d\right) &{} &{} \text {if one proposal is dominated by the other} \\ \\ {\mathbf {U}}_{ff}\left( S,d\right) &{} &{} \text {otherwise} \end{array} \right. \end{aligned}$$

A Particular Case: Linear Bargaining Problems

If we consider the subclass of problems $\Sigma _{L}^{2}$ in which the feasible sets S are linear half-spaces of ${\mathbb {R}}^{2},$

$$\begin{aligned} S=\{(x,y)\in {\mathbb {R}}^{2} : \, \alpha x+ \beta y \le m \}, \quad \alpha , \beta , m \in {\mathbb {R}}_{++} \end{aligned}$$

then the Nash solution N(S, d) (Nash 1950) is obtained when we apply the Unanimous–Concession procedure for agents supporting dual egalitarian solution concepts: one supports the egalitarian solution E(S, d) (Kalai 1977) and the other proposes the equal-loss solution EL(S, d) (Chun 1988). These solutions are formally defined in the following way:

$$\begin{aligned} \begin{array}{lll} N(S,d) = x^* \in S &{} : &{} x^* = \arg \max \left\{ (x_1-d_1)( x_2 -d_2); \, x \in S, x \ge d \right\} \\ &{} \\ E(S,d) = y \in \partial (S) &{} : &{} y_1 - d_1 = y_2 - d_2 \\ &{} \\ EL(S,d) = z \in \partial (S) &{} : &{} a_1(S,d) - z_1 = a_2(S,d) - z_2\\ \end{array} \end{aligned}$$

In this context we obtain the following result.

Proposition 4

For any $(S,d)\in \Sigma _{L}^{2},$ if f, g are the egalitarian and the equal-loss solutions, then

$$\begin{aligned} {\mathbf {U}}_{fg}(S,d)=N(S,d). \end{aligned}$$

Proof

It is clear that, in the class $\Sigma _{L}^{2},$ the Nash solution is given by

$$\begin{aligned} N(S,d)=d+\frac{1}{2}(a-d), \end{aligned}$$

where $a=a(S,d)$ denotes the ideal point of the problem (S, d). In order to prove that the Unanimous–Concession procedure gives the same proposal, we observe (see Fig. 2) that the sequence $\{d^{k}\}$ that defines U$_{fg}$ satisfies:

$$\begin{aligned} d^{2}= d+v^{1}\Rightarrow a^{2}=a(S,d^{2})=a-v^{1}, \hspace{4cm} \\ d^{3}= d^{2}+v^{2}=d+v^{1}+v^{2}\Rightarrow a^{3}=a(S,d^{3})=a-v^{1}-v^{2}, \end{aligned}$$

and so on. By construction, both sequences $\{d^{k}\}$ and $\{a^{k}\}$ converge to the same point $d^{*}={\mathbf {U}}_{fg}(S,d)$, so that

$$\begin{aligned} d^{*}=d+\sum \limits _{i}v^{i}=a-\sum \limits _{i}v^{i} \end{aligned}$$

and then, $\sum \nolimits _{i}v^{i}=\frac{1}{2}(a-d)$, so ${\mathbf {U}} _{fg}(S,d)=d+\frac{1}{2}(a-d).$ $\square$

Fig. 2

Unanimous–Concession procedure in linear problems

Remark 3

Note that, in the class of linear bargaining problems, $\Sigma _{L}^{2},$ the Nash solution always coincides with the Kalai-Smorodinsky solution (Kalai and Smorodinsky 1975) defined as

$$\begin{aligned} \begin{array}{lll} KS(S,d) = x \in \partial (S)&:&x_2 - d_2 = \dfrac{a_2(S,d) - d_2}{a_1(S,d) - d_1} \left( x_1 - d_1\right) \end{array} \end{aligned}$$

Possible Extensions

The result in Proposition 4 cannot be extended to more general two-person bargaining problems (S, d) (even for piece-linear sets), or to other solutions f, g, different from the dual egalitarian ones, as shown in the following examples.

Example 4

Let us consider the (piece-linear) bargaining problem $d = (0,0)$

$$\begin{aligned} S = \left\{ \left( x_1,x_2\right) \in {\mathbb {R}}^2 : x_1 + 3x_2 \le 18, \, x_1 + x_2 \le 8 \right\} . \end{aligned}$$

Computing the egalitarian solutions for (S, d) we obtain $E(S,d) = (4,4)$ and $EL(S,d) = (5,3)$, so $d^2 = (4,3)$. We now compute the solutions in the new problem $(S,d^2)$ obtaining $E(S,d^2) = (4.5,3.5)$ and $EL(S,d^2) = (4.5,3.5)$, so ${\mathbf {U}} _{fg}(S,d)= (4.5,3.5) \ne (4,4) = N(S,d).$

Example 5

Let us consider the (linear) bargaining problem $d = (0,0)$

$$\begin{aligned} S = \left\{ \left( x_1,x_2\right) \in {\mathbb {R}}^2 : x_1 + 2x_2 \le 8 \right\} . \end{aligned}$$

Let f the egalitarian solution, $f(S,d) = E(S,d)$ and $g(S,d) = KS(S,d)$, the Kalai-Smorodinsky solution. Then, $f(S,d) = \left( 8/3,8/3\right)$ and $g(S,d) = (4,2)$, and $d^2 = \left( 8/3,2\right)$. Being f and g SIR bargaining solutions, the construction of the Unanimous–Concession procedure implies $\left( {\mathbf {U}} _{fg}(S,d)\right) _2 > (d^2)_2 = 2$. Therefore, ${\mathbf {U}} _{fg}(S,d) \ne (4,2) = N(S,d).$

An Illustration

In order to illustrate the result in Proposition 4 and the behavior of the Unanimous–Concession procedure, we consider the following problem concerning the optimal provision and cost-sharing of a public good.

Two agents, indexed by $i=1,2,$ consume one public good x, and share its cost. The technology for producing the public good is owned by both agents and uses as input the private good y. Agent i’s consumption set is $\mathbb {\ R}_{+}\times Y_{i}$ , $Y_{i}\subset {\mathbb {R}}_{+}$, with elements $(x,y_{i}).$ Here x is a level of public good and $y_{i}\in Y_{i}$ is agent i’s net loss of his private good, i.e. $y_{i}\ge 0$ means that agent i contributes $y_{i}$ units of private good to the production of the public good, so that no private transfers are allowed.

The production function is $x=f(y),$ where $y=y_{1}+y_{2}$, and the cost function is $y=c(x)=f^{-1}(x)$. The agents’ utility functions are assumed to take the form $u_{i}(x,y_{i})=h_{i}(x)-a_{i}y_{i},$ $a_i > 0,$ $i=1,2.$ Individual decisions for the provision of the public good, $z_{i}^{*},$ are defined by

$$\begin{aligned} z_{i}^{*} = \underset{x\ge 0}{\arg \max }\{u_{i}(x,c(x))\}=\underset{x\ge 0}{\arg \max }\{h_{i}(x)-a_{i}c(x)\}, \quad i=1,2. \end{aligned}$$

Under standard assumptions, once the optimal level of public good $x^{*}$ is determined, the cooperative game associated with this situation is the bargaining problem (S, d), where

$$\begin{aligned} S= \,& {} \left\{ (u_{1},u_{2}) : \, \, \, \frac{a_{2}}{a_{1}}u_{1}+u_{2}\le h_{2}(x^{*})+\frac{ a_{2}}{a_{1}}h_{1}(x^{*})-a_{2}c(x^{*}) \right\} \, \text { and}\\ d_{i}= &\, {} u_{i}(z_{i}^{*},c(z_{i}^{*})), \quad i=1,2. \end{aligned}$$

Therefore, agents face a linear bargaining problem. In particular, if we fix

$$\begin{aligned} f(y)=\frac{1}{2}y \qquad u_{1}(x,y_{1})=x-\frac{2}{3}y_{1}\qquad u_{2}(x,y_{2})=2\sqrt{x} -y_{2} \end{aligned}$$

the bargaining problem is given by

$$\begin{aligned} S=\left\{ (u_{1},u_{2}): \, \, \, \frac{3}{2}u_{1}+u_{2}\le \dfrac{7}{8} \right\} \qquad d=\left( 0,\frac{1}{2}\right) . \end{aligned}$$

Consider now that two agents, both supporting egalitarian criteria, do not agree on what is relevant: either gains from the disagreement point, or losses from the ideal point (that is, one supports the egalitarian solution and the other proposes the equal-loss solution. In such a case, what we have is in fact a meta-bargaining problem. Let us see how the Unanimous–Concession procedure solves this problem.

Agent one proposes the utility distribution $\left( \frac{3}{20},\frac{13}{20}\right) ,$ which implies gains of $\frac{3}{20}$ from the disagreement point, $(0,\frac{1 }{2}),$ for both agents. The agents’ contributions are are, respectivelt, $\frac{3}{20}$ and $\frac{7}{20}$ units of the private good.

Agent two proposes the utility distribution $\left( \frac{ 1}{10},\frac{29}{40}\right) ,$ which implies equal losses of $\frac{3}{20}$ from the ideal point $\left( \frac{ 1}{4},\frac{7}{8}\right)$ and contributions of $\frac{9}{40}$ and $\frac{11}{40}$ units of the private good.

Therefore, the maximum amount of utility distribution on which both agents agree is

$$\begin{aligned} \inf \left\{ \left( \frac{3}{20},\frac{13}{20} \right) ,\left( \frac{1}{10},\frac{29}{10}\right) \right\} =\left( \frac{1}{10},\frac{13}{20}\right) . \end{aligned}$$

This point has been denoted by $d^{2}$ in the sequence $\{d^{k}\}$ used to define the Unanimous–Concession procedure. The next term of this sequence is obtained by setting the infimum of the solution outcomes of the bargaining problem $(S,d_{2}),$ and so on. Then, the utility distribution provided by the Unanimous–Concession procedure is $(\frac{1}{8},\frac{11}{16})$, which coincides with the Nash bargaining solution to the problem (S, d). The contributions of the agents are $\frac{1}{16}$ and $\frac{5}{16}$ units of the private good for agent one and agent two respectively.

A Comparison with Other Meta-Bargaining Mechanisms

From the meta-bargaining mechanisms introduced by van Damme (1986) (modified in Naeve-Steinweg (1999)), and Chun (1985) (modified in Naeve-Steinweg (2002)), other procedures have been discussed in the literature: the Minimal Agreement procedure⁴ (Anbarci and Yi 1992), the Meta-Bargaining Game (Trockel 2002), and the Averaging mechanism (Naeve-Steinweg 2004).

The mechanisms of van Damme and Chun were introduced in order to support, from an strategic point of view Nash and Kalai-Smorodinsky solutions, respectively. The Meta-bargaining Game also supports Nash solution. The first work that considers an axiomatic approach to meta-bargaining problems is Marco et al. (1995). Since then, this methodology has been applied to the analysis of other procedures (see Naeve-Steinweg 1999, 2002, 2004). As Naeve-Steinweg (2002) points out, from a non-cooperative approach, “we can interpret meta-bargaining theory as a means to give a justification for a certain bargaining solution. [...] Based on van Damme’s paper it seems that the meta-bargaining approach singles out the Nash solution. However, it has to be stressed that the optimality strongly depends on the mechanism we use. In particular, we have seen that also the Kalai and Smorodinsky (1975) solution is optimal for the appropriate mechanism.”⁵ That is why we are interested in the meta-bargaining process from a normative point of view, in which the agents have different equity criteria (represented by the bargaining solutions they support) and, as mentioned in Naeve-Steinweg (2004), “this leads to the interpretation of a mechanism as an arbitration scheme”.

In order to compare the Unanimous Concession procedure with the other above mentioned mechanisms, we will show which properties they satisfy. Additionally to the ones introduced in Sects. 3 and 4, we use the following properties.⁶

Definition 9

A meta-bargaining procedure $\varPhi :F\times F\rightarrow F$ satisfies Generalized Midpoint Domination (GMD) if for each $f,g\in F,$ and all $(S,d)\in \Sigma ^{2},$ $\varPhi _{fg}(S,d)\ge \frac{1}{2}f(S,d)+\frac{1}{2}g(S,d).$

Definition 10

A meta-bargaining procedure $\varPhi :F\times F\rightarrow F$ satisfies Step-by-step Bargaining (STEP) if for each $f,g\in F,$ and all $(S,d)\in \Sigma ^{2},$ such that there is $T\subseteq S$ satisfying $(T,d)\in \Sigma ^{2},$ $f(S,d)=f(T,d),$ $g(S,d)=g(T,d)$ and the segment joining f(T, d) and g(T, d),  belongs to PO(T),  then

$$\begin{aligned} \varPhi _{fg}(S,d)=\varPhi _{fg}(S,\varPhi _{fg}(T,d)). \end{aligned}$$

Next table summarizes the properties that are fulfilled by each meta-bargaining mechanism, where⁷

UC: Unanimous–Concession procedure (Marco et al. 1995).

Min Agr: Minimal Agreement procedure (Anbarci and Yi 1992).

v-Damme: van Damme’s mechanism (modified in Naeve-Steinweg (1999)).

Chun: Chun’s mechanism (modified in Naeve-Steinweg (2002)).

M-Game: Meta-Bargaining Game (Trockel 2002).

Av: Averaging mechanism (Naeve-Steinweg 2004).

	${UC }$	${Min Agr }$	${v-Damme }$	${Chun }$	M-Game	${Av }$
(PO)	$\checkmark$	$\times$ $(^{*},^{**})$	$\times$ $(^{*})$	$\times$ $(^{*})$	$\times$	$\checkmark$
(Im)	$\checkmark$	$\times$	$\checkmark$	$\checkmark$	$\times$	$\checkmark$
(Un)	$\times$ $(^{*})$	$\checkmark$	$\checkmark$	$\checkmark$	$\checkmark$	$\times$ $(^{*})$
(IDS)	$\checkmark$	$\times$	$\times$	$\times$	$\times$	$\times$
(Mon)	$\checkmark$	$\checkmark$ $(^{**})$	$\times$	$\times$	$\times$	$\checkmark$
(GMD)	$\times$	$\times$	$\times$	$\times$	$\times$	$\checkmark$
(STEP)	$\times$	$\times$	$\times$	$\times$	$(^{***})$	$\checkmark$
(ICA)	$\checkmark$	$\times$	$\times$	$\times$	$\times$	$\times$

1. $(^{*})$ If the bargaining solutions f,  g are required to be PO, this property is fulfilled by the mechanism
2. $(^{**})$ Whenever this mechanism is well defined
3. $(^{***})$ Not applicable