On restricted bargaining sets

Abstract

In this paper we analyze the behavior of bargaining sets in continuum economies when there are restrictions on the formation of coalitions. We provide several characterizations of Vind’s (J Math Econ 21:89–97, 1992) bargaining set in terms of its restricted versions, where not all the coalitions are formed. Moreover, we show that these equivalences do not hold for Mas-Colell’s (J Math Econ 18:129–139, 1989) bargaining set. These findings highlight the different nature of both notions of bargaining sets. Finally, we illustrate the impossibility of extrapolating our results to a more general setting.

Appendix

Proof of Lemma 3.1

Since $(T,h)$ is a counter-objection to $(S,g)$ we have that $h$ is a feasible allocation, it is also attainable for $T$ and every $t \in T$ prefers $h(t)$ rather than $g(t).$ By Lyapunov’s convexity theorem, for every $\varepsilon >0,$ there exists a coalition $H_{\varepsilon } \subset T,$ such that $h$ is attainable for $H_{\varepsilon }$ and then $(H_{\varepsilon }, h)$ is also a counter-objection to $(S,g)$ (see the proof in Schmeidler (1972) for further details).

As in Grodal (1972) let $(t_{n}, n \in \mathbb {N})$ be a dense subset in $H_{\varepsilon }.$ Define $H^{1}_{\varepsilon } = H_{\varepsilon } \cap B(t_{1}, \varepsilon /2)$ and $H^{n}_{\varepsilon } = H_{\varepsilon } \cap B(t_{n}, \varepsilon /2) \setminus \cup _{k=1}^{n-1} H_{\varepsilon }^{k},$ for $n >1.$ Let $h_{n}= \int _{H^{n}_{\varepsilon }} \left( h(t) - \omega (t)\right) d \! \mu (t).$ Define $\mathcal{C}$ as the convex hull of the set $\Gamma =\{ h_{n} | n \in N\},$ where $N= \{n \in \mathbb {N}| \mu ( H^{n}_{\varepsilon }) >0 \}.$ Now let $\chi$ be the smallest affine subspace containing $\mathcal{C}.$ Note that $\sum _{n \in N} h_{n} =0$ and 0 belongs to the interior of $\mathcal{C}$ relative to $\chi \subset \mathbb {R}^{\ell }.$ By Caratheodory’s theorem, there is $A \subset N$ with at most $\ell +1$ elements such that we can write $0= \sum _{j \in A} \alpha _{j} h_{j},$ with $\alpha _{j} \in [0,1]$ and $\sum _{j \in A} \alpha _{j}=1.$ Lyapunov’s theorem allows us to guarantee that for each $j \in A$ there exists $H_{j} \subset H^{j}_{\varepsilon }$ such that $\mu (H_{j}) = \alpha _{j} \mu (H^{j}_{\varepsilon })$ and $\int _{H_{j}} \left( h(t) - \omega (t)\right) d \! \mu (t) = \alpha _{j} h_{j}.$ Therefore, by construction, the coalition $H=\cup _{j \in A} H_{j}$ belongs to $\mathcal{S}_{\varepsilon }$ and counter-objects $(S,g)$ via $h$. $\square$

Proof of Theorem 3.1

Let us first prove that $B^{*}(\mathcal{E}) \subset B^{*}_{\varepsilon }(\mathcal{E})$. Indeed, assume that $f$ is a feasible allocation such that $f \in B^{*}(\mathcal{E}) \setminus B^{*}_{\varepsilon }(\mathcal{E})$. Since $f \notin B^{*}_{\varepsilon }(\mathcal{E})$, there exists a global objection $(S,g)$ to $f$, with $S \in \mathcal{S}_{\varepsilon }$, for which there is no counter-objection $(C,y)$ such that $C \in \mathcal{S}_{\varepsilon },$ and by Lemma 3.1, this is the same as saying that it does not exist any counter-objection to $(S,g)$. This means that $f$ is global objected but not counter-objected, so $f \notin B^{*}(\mathcal{E})$.

We will now prove that $B^{*}_{\varepsilon }(\mathcal{E}) \subset B^{*}(\mathcal{E})$. Let $f \in B^{*}_{\varepsilon }(\mathcal{E})$, and let $(S,g)$ be a global objection to $f$. By Lyapunov’s and Caratheodory’s theorems (see the proof of the Lemma 3.1) we can guarantee the existence of $K\subset S$ such that $K \in \mathcal{S}_{\varepsilon }$ and $\int _{K} g(t) d \!\mu (t) \le \int _{K} \omega (t) d \!\mu (t)$, meaning that $(K,g)$ is also a global objection to $f$. Since $f \in B^{*}_{\varepsilon }(\mathcal{E})$, $(K,g)$ necessarily has a counter-objection $(C,y)$, with $C \in \mathcal{S}_{\varepsilon },$ which is straightforward also a counter-objection to $(S,g)$, concluding that $f \in B^{*}(\mathcal{E})$. $\square$

Proof of Lemma 3.2

Since $(T,h)$ is a counter-objection to $(S,g)$, the following holds:

• (i) $\int _T h(t) d\! \mu (t) \le \int _T \omega (t) d\! \mu (t)$ and

• (ii) $h(t) \succ _{t} g(t)$ for almost all $t \in T.$

If $\alpha < \mu (T),$ consider the measure $\nu (A)= \left( \mu (A), \int _{A} (h(t) -\omega (t)) d\! \mu (t) \right)$ restricted to measurable subsets of the coalition $T.$ By Lyapunov’s convexity theorem we obtain that there exists $T_{\alpha } \subset T,$ with $\mu (T_{\alpha })= \alpha ,$ that blocks the allocation $g$ via the same $h$.

Consider $\alpha > \mu (T).$ Let the measure $\eta (A)= \left( \mu (A), \int _{A} (h(t) -g(t)) d\! \mu (t) \right)$ restricted to subsets of $T.$ Applying Lyapunov’s convexity theorem we obtain that for any $\beta \in (0,1)$ there exits $A \subset T$ such that $\mu (A)= \beta \mu (T)$ and $\int _{A} (h(t) -g(t)) d\! \mu (t) = \beta \int _{T} (h(t) -g(t)) d\! \mu (t).$ By continuity and measurability, there exist $\tilde{h}$ and $\delta > 0$ such that $\int _{A} \tilde{h} (t) d\! \mu (t) = \int _{A} h(t) d\! \mu (t) - \delta$ and $\tilde{h}(t) \succ g(t)$ for every $t \in A.$

Let the allocation $z$ defined as follows:

$$\begin{aligned} z(t)= \left\{ \begin{array}{ll} \tilde{h}(t) &{} \hbox {if }\quad t \in A, \\ g(t) + \frac{\delta }{\mu (T\setminus A)} &{} \hbox {if }\quad t \in T\setminus A. \end{array} \right. \end{aligned}$$

Note that $\int _{T} z(t) d\!\mu (t) = \int _{T} \left( \beta h(t) + (1 - \beta ) g(t) \right) d \!\mu (t)$ and $z(t) \succ g(t)$ for every $t \in T.$ As before, there exists $\gamma >0$ and $\tilde{z}$ such that $\int _{T} \tilde{z}(t) d\!\mu (t) = \int _{T} z(t) d\!\mu (t) - \gamma$ and $\tilde{z}(t) \succ g(t)$ for every $t \in T.$ Applying Lyapunov’s theorem again to the above measure $\nu$ restricted to $I \setminus T,$ we have that there exists $B \subset I \setminus T$ such that $\mu (B)= (1 - \beta ) \mu (I\setminus T)$ and $\int _{B} \left( g(t) - \omega (t) \right) d\! \mu (t) = (1- \beta ) \int _{ I \setminus T} \left( g(t) - \omega (t) \right) d\! \mu (t)$.

The coalition $C= T \cup B$ blocks $g$ via de allocation $y$ given by

$$\begin{aligned} y(t)= \left\{ \begin{array}{ll} \tilde{z}(t) &{} \hbox {if}\;\quad t \in T, \\ g(t) + \frac{\gamma }{\mu (B)} &{} \hbox {if} \;\quad t \in B \; \mathrm{and }\\ \omega (t) &{} \hbox { otherwise.} \; \;\end{array} \right. \end{aligned}$$

By construction $y$ is a feasible allocation and $\int _{C} y(t) d \! \mu (t) \le \int _{C} \omega (t) d \! \mu (t).$ Taking $\beta = ( 1 - \alpha ) / (1 - \mu (T)),$ we conclude that $\mu (C)= \alpha$. $\square$

Proof of Theorem 3.2

We follow the same proof as in Theorem 3.1. To prove that $B^{*}(\mathcal{E}) \subset \alpha -B^{*}_{\varepsilon }(\mathcal{E}),$ we apply Lemma 3.2 instead of Lemma 3.1, and to show that $\alpha -B^{*}_{\varepsilon }(\mathcal{E}) \subset B^{*}(\mathcal{E})$ it is important to remark that if $(S,g)$ is a global objection to $f,$ then there exists $K \in \mathcal{S}_{\varepsilon }$ such that $(K, g)$ is also a global objection to $f.$ That is, both coalitions $S$ and $K$ use the same feasible allocation $g$ which is crucial in the proof. $\square$

Proof of Lemma 3.3

Let $(S,g)$ be a full objection to $f.$ Since $g$ has no counter-objection, $g$ is in the core. Let $p$ a price system such that $(p,g)$ is a competitive equilibrium for the economy $\mathcal{E}.$ Suppose that there exist a coalition $T$ and a number $\alpha \in (0,1]$ such that $g$ is dominated in the economy $\mathcal{E}(T, g, \alpha ),$ that is, there exists an allocation $h$ which is feasible in the perturbed economy $\mathcal{E}(T, g, \alpha )$ and $h(t) \succ _{t} g(t)$ for almost all $t \in I.$ Then, we have that $p \cdot h(t) > p \cdot \omega (t) \ge p \cdot g(t),$ for almost all agent $t \in I.$

Multiplying the above inequalities by $(1-\alpha )$ and by $\alpha ,$ respectively, we obtain $p \cdot h(t) > p \cdot ((1- \alpha ) \ \omega (t) + \alpha \ g(t))$ for almost all agent $t \in T.$ Therefore, we have $\int _{I} p \cdot h(t) \ d\!\mu (t) > \int _{I\setminus T} p \cdot \omega (t) d\!\mu (t) \ + \ \int _{T} p \cdot \left( (1-\alpha ) \omega (t) + \alpha g(t) \right) \ d\!\mu (t) = \int _{I} p \cdot \omega (T,g,\alpha )(t) \ d\!\mu (t),$ which is a contradiction with the feasibility of $h$ in the economy $\mathcal{E}(T,g, \alpha ).$

To show the converse, let $g$ a non-dominated allocation for every economy $\mathcal{E}(A,g,\alpha ).$ Assume that $(S,g)$ has a counter-objection, namely $(T,z).$ Arguing as in the proof of Lemma 3.2, we can take $z$ such that $\int _{T} z(t) d\! \mu (t) \le \int _{T} \omega (t) d\! \mu (t) - \delta ,$ with $\delta > 0$ and moreover, given any $\alpha \in (0,1),$ there exists an allocation $y: T \rightarrow \mathbb {R}^{\ell }_{+}$ such that $\int _{T} y(t) d\! \mu (t) = \int _{T} (\alpha z(t) + (1-\alpha ) g(t)) d\! \mu (t)$ and $y(t) \succ _{t} f(t)$ for every $t \in T.$ Let us consider the allocation $h: I \rightarrow \mathbb {R}^{\ell }_{+}$ given by

$$\begin{aligned} h(t)= \left\{ \begin{array}{ll} y(t) &{} \hbox {if }\quad t \in T \; \mathrm{and }\\ g(t) + \frac{\alpha }{\mu (I\setminus T)} \ \delta &{} \hbox {if }\quad t \in I \setminus T. \end{array} \right. \end{aligned}$$

By construction, we can deduce $\int _{I} (h(t) - \omega (I \setminus T,g, \alpha )(t)) d\! \mu (t) = (1 -\alpha ) \int _{I} ( g(t) - \omega (t)) d\! \mu (t) \le 0.$ Therefore, the grand coalition blocks $g$ via $h$ in the economy $\mathcal{E}(I\setminus T, g, \alpha ),$ which is a contradiction with the fact that $g$ is a non dominated allocation for every economy $\mathcal{E}(A, g, \alpha )$. $\square$

Proof of Theorem 3.3

It is enough to apply Lemma 3.3 and follow the same proof as in Theorem 3.1 and 3.2. $\square$