Value allocation under ambiguity

Abstract

We consider a pure exchange economy with asymmetric information where individual behavior exhibits ambiguity aversion along the line of maximin expected utility decision making. For such economies, we introduce different notions of maximin value allocations. We also introduce a strong notion of incentive compatibility. We prove the existence and incentive compatibility of the maximin value allocation. We conclude that unlike the Bayesian value allocation approach in Krasa Yannelis (Econometrica 62(4):881–900, 1994), incentive compatibility is related to efficiency rather than to direct exchange of information.

Appendix

Proof of Theorem 1

The proof follows the one of Krasa and Yannelis (1996). For the sake of completeness we repeat all the steps.

Step I

By hypothesis $(A_1)$ of the theorem, $L_{X_{i}}$ is convex, norm closed and bounded from below and furthermore $e_{i}\in L_{X_{i}}$, so that $L_{X_{i}}\ne \emptyset$ for every $i\in I$. Hence, $L_X\ne \emptyset$ is nonempty as well. By hypothesis $(A_{2})$ of the theorem, the concavity of $u_{i}$ implies the concavity of $\upsilon _{i}$ for each $i\in I$. Also, by Theorem 2.8 of Balder and Yannelis (1993), since $u_{i}$ is continuous, concave and bounded, it follows that $\upsilon _{i}$ is weakly upper semi continuous on $L_{X_{i}}$ for each $i\in I$. Let $\mathcal {A}$ be the (same for all the agents) set of all the finite dimensional subspaces of their $L^1(\Omega ,\mathcal {F}, \mu _i^{\star };B)$, each one of these subspaces containing all the agents’ projections of initial endowments $e_{i}$. Then, for each $\alpha \in \mathcal {A}$ we define the $L_{X_{i}}^{\alpha }=L_{X_{i}}\cap \alpha$ be the restriction of the consumption set, $\upsilon _{i}^{\alpha }=\upsilon _{i}\cap \alpha$ is the restriction of the utility function and $e_{i}^{\alpha }=e_{i}\cap \alpha$ the restriction of the endowment of the $ith$ agent on the finite dimensional subspace $\alpha \in \mathcal {A}$. We can construct therefore the $\alpha$-finite dimensional commodity space economy

$$\begin{aligned} \mathcal {E}^{\alpha }=\{(L^{\alpha }_{X_{i}}, \upsilon _{i}^{\alpha }, e_{i}^{\alpha }): i=1,2,\ldots ,n\}. \end{aligned}$$

(16)

Step II

Certainly, the standard results of existence of a cardinal value allocation apply to $\mathcal {E}^{\alpha }$, as long as $\mathcal {E}^{\alpha }$ satisfies all the assumptions of Emmons and Scafuri (1985). Continuity of $\upsilon ^{\alpha }_{i}$ is the only one which is not immediate. However, it follows by the hypothesis $(A_2)$ of the theorem via the Dominated Convergence Theorem, and because $L^{\alpha }_{X_{i}}$ is a finite dimensional space, that $\upsilon ^{\alpha }_{i}: L^{\alpha }_{X_{i}}\times 2^{I}\rightarrow \mathfrak {R}_+$ is continuous on $L^{\alpha }_{X_{i}}$ for each $S\in 2^{I}$. Hence, there exists a cardinal (in our case, ex ante maximin) value allocation for $\mathcal {E}^{\alpha }$, i.e., there exists $x^{\alpha }\in L^{\alpha }_X=\prod \nolimits _{i=1}^n L^{\alpha }_{X_{i}}$ such that:

• (i) $\sum _{i\in I}x^{\alpha }_{i}=\sum _{i\in I}e_{i}^{\alpha }$,

• (ii) there exists a $\lambda ^{\alpha }_{i}\ge 0$ for every agent $i$, with $\sum _{i\in I}\lambda ^{\alpha }_{i}=1$, such that $\lambda ^{\alpha }_{i}$ $\upsilon ^{\alpha }_{i}(x^{\alpha }_{i},I)=Sh_{i}(V_{\lambda ^{\alpha },\upsilon ^{\alpha }})$, for all $i\in I$, where $Sh_{i}(V_{\lambda ^{\alpha },\upsilon ^{\alpha }})$ is the Shapley value of the $ith$ agent, derived from the TU game $(I,V_{\lambda ^{\alpha },\upsilon ^{\alpha }})$, whose characteristic function $V_{\lambda ^{\alpha },\upsilon ^{\alpha }}$ is defined for every coalition $S\subseteq I$ as follows:

  $$\begin{aligned} V_{\lambda ^{\alpha },\upsilon ^{\alpha }}(S)=\left\{ \max \limits _{x^{\alpha }\in \prod _{i\in S}L^{\alpha }_{X_{i}}}\sum \limits _{i\in S}\lambda ^{\alpha }_{i} \upsilon _{i}^{\alpha }(x_{i}^{\alpha },S): \sum \limits _{i\in S}x_{i}^{\alpha }=\sum \limits _{i\in S}e_{i}^{\alpha }\right\} . \end{aligned}$$

  (17)

By denoting $\sum _{i\in I}e_{i}=e\ge 0$, we have that $0\le \sum _{i\in I}x^{\alpha }_{i}=\sum _{i\in I}e_{i}^{\alpha }$, which implies that for each $\mathcal {E}^{\alpha }$ economy and for each cardinal value allocation $x^{\alpha }\in L^{\alpha }_X$ of this economy, every $x^{\alpha }_{i}$ lies in the order interval $[0,e]$ of $\sum _{i\in I}L_{X_{i}}\subset L^1(\Omega ,\mathcal {F}, \mu _i^{\star }$; $B), i\in I$. We direct the set $\mathcal {A}$ by inclusion so that $\{(x^{\alpha }_1,\ldots ,x^{\alpha }_n,\lambda ^{\alpha }_1,\ldots ,\lambda ^{\alpha }_n) : \alpha \in \mathcal {A}\}$ forms a net in $C=\prod \nolimits _{i=1}^n[0,e] \times \Delta$, where $\Delta$ is the $(n-1)$-dimensional simplex. Since $B$ is an ordered Banach space so is $L^1(\Omega ,\mathcal {F}, \mu _i^{\star }$; $B), i\in I$, which implies (see, for example, in Aliprantis and Burkinshaw 1985) that this space has weakly compact order intervals. Hence, the order interval $[0,e]$ is weakly compact, and therefore, $C$ is compact. It follows that there is a subnet (still indexed by $\mathcal {A}$ for notational simplicity) $\{x^{\alpha },\lambda ^{\alpha }\}_{\alpha \in \mathcal {A}}$, which converges to a point $(\bar{x},\bar{\lambda })=(\bar{x}_1,\ldots ,\bar{x}_n,\bar{\lambda }_1,\ldots ,\bar{\lambda }_n)$ in $C$. We shall show that this limit constitutes an ex ante maximin value allocation for the infinite dimensional commodity space economy $\mathcal {E}^{ea}=\{(L_{X_{i}}, \upsilon _{i}, e_{i}): i=1,2,\ldots ,n\}$.

Step III

First notice that since $L_{X_{i}}$ is convex and norm closed, by Mazur’s Theorem, it is weakly closed as well; hence, the weak limit $\bar{x}_{i}$ lies actually in $L_{X_{i}}$, for each $i\in I$, so we conclude that $\bar{x}\in L_X$. Moreover, since for each $\alpha \in \mathcal {A}$ we have $\sum _{i\in I} x^{\alpha }_{i}=\sum _{i\in I} e^{\alpha }_{i}$, by taking weak limits we conclude that

$$\begin{aligned} \sum _{i\in I} \bar{x}_{i}=\sum _{i\in I} e_{i}. \end{aligned}$$

(18)

Step IV

In order to define the ex ante maximin value allocation of the original economy, we verify first that the TU game $(I,V_{\bar{\lambda },\upsilon })$, whose characteristic function $V_{\bar{\lambda },\upsilon }$ is given for every coalition $S\subseteq I$ as in (10), is well defined. Indeed, for every $S\subseteq I$, since $[0,\sum _{i\in S}e_{i}]$ is weakly compact and $\upsilon _{i}$ is weakly upper semi continuous on this set it follows that $V_{\bar{\lambda },\upsilon }(S)$ is well defined and so is the Shapley value $Sh_{i}(V_{\bar{\lambda },\upsilon })$ of each individual $i\in I$. Next, let $x^{\star }\in \prod _{i\in S}L_{X_{i}}$ be such that $\sum _{i\in S}x^{\star }_{i}=\sum _{i\in S}e_{i}$ and $V_{\bar{\lambda },\upsilon }(S)=\sum _{i\in S}\bar{\lambda }_{i}\upsilon _{i}(x^{\star }_{i},S)$. We have that the sequence $\{V_{\lambda ^{\alpha },\upsilon ^{\alpha }}(S)\}_{\alpha \in \mathcal {A}}$ is bounded, so we may assume, by passing to a subnet if necessary, that it converges. Since $\sum _{i\in S}x^{\star }_{i}=\sum _{i\in S}e_{i}$, by denoting $x^{\star \alpha }_{i}=x^{\star }_{i}\cap \alpha \in \prod _{i\in S}L^{\alpha }_{X_{i}}$, we have that: $x^{\star \alpha }_{i}\rightarrow x^{\star }$ weakly and $\sum _{i\in S}x^{\star \alpha }_{i}=\sum _{i\in S}e^{\alpha }_{i}$. Consequently, for each $\alpha \in \mathcal {A}$, $V_{\lambda ^{\alpha },\upsilon ^{\alpha }}(S)\ge \sum _{i\in S}\lambda ^{\alpha }_{i}\upsilon ^{\alpha }_{i}(x^{\star \alpha }_{i},S)$. Further, by the definition of the weak upper semicontinuity of $\upsilon _i$ at $x^{\star }_i$, for every $\epsilon >0$, there exists an $\alpha _{\epsilon }$, such that for every $\alpha >\alpha _{\epsilon }$ we have

$$\begin{aligned} \sum _{i\in S}\lambda _i\upsilon _i(x_i^{\star },S)\le \sum \limits _{i\in S}\lambda ^{\alpha }_i\upsilon _i^{\alpha }(x^{\star \alpha }_{i},S) + \epsilon . \end{aligned}$$

By considering a net $\epsilon \rightarrow 0$, we can extract a subnet (still indexed by $\alpha$) such that $\liminf _{\alpha }\sum _{i\in S}\lambda ^{\alpha }_i\upsilon _i^{\alpha }(x^{\star \alpha }_{i},S)\ge \sum _{i\in S}\lambda _i\upsilon _i(x_i^{\star },S)$. Hence, we conclude that

$$\begin{aligned} \lim _{\alpha }V_{\lambda ^{\alpha },\upsilon ^{\alpha }}(S)&= \liminf _{\alpha }V_{\lambda ^{\alpha },\upsilon ^{\alpha }}(S) \nonumber \\&\ge \liminf _{\alpha }\sum _{i\in S}\lambda ^{\alpha }_{i}\upsilon ^{\alpha }_{i}(x^{\star \alpha }_{i},S) \nonumber \\&\ge \sum _{i\in S}\bar{\lambda }_i\upsilon _i(x_i^{\star },S) \nonumber \\&= V_{\bar{\lambda },\upsilon }(S). \end{aligned}$$

(19)

On the other hand, let $y^{\alpha }\in \prod _{i\in S}L^{\alpha }_{X_{i}}$, for each $\alpha \in \mathcal {A}$, be such that it satisfies $\sum _{i\in S}y^{\alpha }_{i}=\sum _{i\in S}e^{\alpha }_{i}$ and $\sum _{i\in S}\lambda ^{\alpha }_{i}\upsilon ^{\alpha }_{i}(y^{\alpha }_{i},S)= V_{\lambda ^{\alpha },\upsilon ^{\alpha }}(S)$. Certainly $y^{\alpha }$ converges weakly to $y\in \prod _{i\in S} L_{X_{i}}$ and $\sum _{i\in S}y_{i}=\sum _{i\in S}e_{i}$. It follows that

$$\begin{aligned} \lim _{\alpha }V_{\lambda ^{\alpha },\upsilon ^{\alpha }}(S)&= \limsup _{\alpha }V_{\lambda ^{\alpha },\upsilon ^{\alpha }}(S) \nonumber \\&= \limsup _{\alpha }\sum _{i\in S}\lambda ^{\alpha }_{i}\upsilon ^{\alpha }_{i}(y^{\alpha }_{i},S) \nonumber \\&= \sum _{i\in S}\bar{\lambda }_{i}\limsup _{\alpha }\upsilon ^{\alpha }_{i}(y^{\alpha }_{i},S) \nonumber \\&\le \sum _{i\in S}\bar{\lambda }_{i}\upsilon _{i}(y_{i},S)\nonumber \\&\le V_{\bar{\lambda },\upsilon }(S), \end{aligned}$$

(20)

where the fourth line follows from the weak upper semi continuity of $\upsilon _{i}$ and the last from the fact that $y\in L_{X_{i}}$ is feasible and the definition of $V_{\bar{\lambda },\upsilon }(S)$. Therefore, from (19) and (20), we conclude that $\lim _{\alpha }V_{\lambda ^{\alpha },\upsilon ^{\alpha }}(S)= V_{\bar{\lambda },\upsilon }(S)$, and therefore,

$$\begin{aligned} \lim _{\alpha }Sh_{i}(V_{\lambda ^{\alpha },\upsilon ^{\alpha }})= Sh_{i}(V_{\bar{\lambda },\upsilon }), \forall i\in I. \end{aligned}$$

Step V

We finally prove that $\bar{\lambda }_{i}\upsilon _{i}(\bar{x}_{i},I)= Sh_{i}(V_{\bar{\lambda },\upsilon })$, for all $i\in I$.

By definition of the (ex ante maximin) value allocation for each $\alpha \in \mathcal {A}$, we have:

$$\begin{aligned} \lambda ^{\alpha }_{i}\upsilon ^{\alpha }_{i}(x^{\alpha }_{i},I)&= Sh_{i}(V_{\lambda ^{\alpha },\upsilon ^{\alpha }}), \forall i\in I, \\ \sum _{i\in S}x^{\alpha }_{i}&= \sum _{i\in S}e^{\alpha }_{i}. \end{aligned}$$

Therefore, $\lim _{\alpha }\sup \lambda ^{\alpha }_{i}\upsilon ^{\alpha }_{i}(x^{\alpha }_{i},I)=\lim _{\alpha }\sup Sh_{i}(V_{\lambda ^{\alpha },\upsilon ^{\alpha }})$.

It follows that $\bar{\lambda }_{i}\lim _{\alpha }\sup \upsilon ^{\alpha }_{i}(x^{\alpha }_{i},I)=\lim _{\alpha }\sup Sh_{i}(V_{\lambda ^{\alpha },\upsilon ^{\alpha }})$. By the weak upper semi continuity of $\upsilon _{i}$, since $x^{\alpha }_{i}\rightarrow \bar{x}_{i}$ weakly, it follows that $\bar{\lambda }_{i}\upsilon _{i}(\bar{x}_{i},I)\ge \bar{\lambda }_{i}\lim _{\alpha }\sup \upsilon ^{\alpha }_{i}(x^{\alpha }_{i},I)$, $\forall i\in I$. Hence, we conclude that $\bar{\lambda }_{i}\upsilon _{i}(\bar{x}_{i},I)\ge \lim _{\alpha } \sup Sh_{i}(V_{\lambda ^{\alpha },\upsilon ^{\alpha }})$ and further, by the previous step, we infer that $\bar{\lambda }_{i}\upsilon _{i}(\bar{x}_{i},I)\ge Sh_{i}(V_{\bar{\lambda },\upsilon }), \forall i\in I$. Suppose that the above inequality is strict for some $i\in I$. Summing up over $i\in I$, we have $\sum _{i\in I}\bar{\lambda }_{i}\upsilon _{i}(\bar{x}_{i},I)>\sum _{i\in I}Sh_{i}(V_{\bar{\lambda },\upsilon })$ and further, since $\sum _{i\in I}Sh_{i}(V_{\bar{\lambda },\upsilon })=V_{\bar{\lambda },\upsilon }(I)$, it follows that

$$\begin{aligned} \sum _{i\in I}\bar{\lambda }_{i}\upsilon _{i}(\bar{x}_{i},I)>V_{\bar{\lambda },\upsilon }(I). \end{aligned}$$

(21)

However, by (18) and the definition of $V_{\bar{\lambda },\upsilon }(I)$, it must be $\sum _{i\in I}\bar{\lambda }_{i}\upsilon _{i}(\bar{x}_{i},I)\le V_{\bar{\lambda },\upsilon }(I)$, which together with (21) imply $V_{\bar{\lambda },\upsilon }(I)\ge \sum _{i\in I}\bar{\lambda }_{i}\upsilon _{i}(\bar{x}_{i},I)>V_{\bar{\lambda },\upsilon }(I)$, which is a clear contradiction. Therefore, it must be

$$\begin{aligned} \bar{\lambda }_{i}\upsilon _{i}(\bar{x}_{i},I)=Sh_{i}(V_{\bar{\lambda },\upsilon }), \forall i\in I, \end{aligned}$$

(22)

as desired¹⁰. Now, (18) along with (22) prove the claim of the theorem. $\square$

Proof of Theorem 2

Let $x\in L_{X}$ be an interim maximin value allocation for $\mathcal {E}^{i}$. According to remark 6, $x$ is a Pareto optimal allocation. Suppose that $x$ is not transfer (interim) maximin coalitional incentive compatible. Then, there exist a coalition $S\subset I$, two different states $a$ and $b$ of $\Omega$ and $z\in \mathfrak {R}^{mS}_{+}$, where $\sum _{i\in S}z_{i}=\sum _{i\in S}[e_{i}(a)+x_{i}(b)-e_{i}(b)]$, such that

• (i) ${\mathcal K}_{i}^{I}(a)={\mathcal K}_{i}^{I}(b)$ if and only if $i\in I{\setminus }S$,

• (ii) $u_{i}(x_{i}(a))=u_{i}(x_{i}(b))$, for all $i\in I{\setminus }S$,

• (iii) ${\underline{u}}_{i}(a,y_{i},I)>{\underline{u}}_{i}(a,x_{i},I)$ for all $i\in S$, where for all $i\in S$

  $$\begin{aligned} y_{i}(\omega )={\left\{ \begin{array}{ll}z_{i}, &{}\quad \text{ if }~ \omega =a\\ x_{i}(\omega ), &{}\quad \text{ otherwise }.\end{array}\right. } \end{aligned}$$

  (23)

We will construct a feasible allocation $w\in L_{X}$ for $\mathcal {E}^{i}$ that Pareto improves upon $x$ at the state $a$, which is a contradiction. To this end, for each $i\in I{\setminus }S$ define the assignment $w_i$, such that

$$\begin{aligned} w_{i}(\omega )= {\left\{ \begin{array}{ll}e_{i}(a)+x_{i}(b)-e_{i}(b), &{}\quad \text{ if }~ \omega =a\\ x_{i}(\omega ), &{}\quad \text{ otherwise },\end{array}\right. } \end{aligned}$$

(24)

and for each $i\in S$ define the assignment $w_i$, such that

$$\begin{aligned} w_{i}(\omega )= {\left\{ \begin{array}{ll}z_{i}, &{}\quad \text{ if }~ \omega =a\\ x_{i}(\omega ), &{}\quad \text{ otherwise }.\end{array}\right. } \end{aligned}$$

(25)

By construction, we have that $w_{i}\in L_{X_{i}}$ for all $i\in S$. Moreover, by the measurability of the endowments and (i) above, we have for all $i\in I{\setminus }S$ that $e_{i}(a)=e_{i}(b)$, so

$$\begin{aligned} w_{i}(a)=e_{i}(a)+x_{i}(b)-e_{i}(b)=x_{i}(b). \end{aligned}$$

(26)

Therefore, $w_{i}\in L_{X_{i}}$ for all $i\in I{\setminus }S$ as well, so that finally $w\in L_{X}$. For this allocation we have at the state $a$: $w_{i}(a)=e_{i}(a)+x_{i}(b)-e_{i}(b)$, for all $i \in I{\setminus }S$ and $w_{i}(a)=z_i$, for all $i\in S$. Therefore,

$$\begin{aligned} \sum _{i\in I}w_{i}(a)&= \sum _{i\in I{\setminus }S}[e_{i}(a)+x_{i}(b)-e_{i}(b)]+\sum _{i\in S}z_{i} \\&= \sum _{i\in I\setminus S}[e_{i}(a)+x_{i}(b)-e_{i}(b)]+\sum _{i\in S}e_{i}(a)+\sum _{i\in S}x_{i}(b)-\sum _{i\in S}e_{i}(b) \\&= \sum _{i\in I}e_{i}(a)+\sum _{i\in I}x_{i}(b)-\sum _{i\in I}e_{i}(b) \\&= \sum _{i\in I}e_{i}(a) \end{aligned}$$

i.e., $w$ is feasible in the state $a\in \Omega$, which is an arbitrary state, so $w$ is feasible for all other states as well. We conclude therefore that $w$ is a feasible allocation for $\mathcal {E}^i$.

Next, by (26) and condition (ii) above, it follows that $u_{i}(w_{i}(a))=u_{i}(x_{i}(b))=u_{i}(x_{i}(a))$, for all $i\in I{\setminus }S$. This in turn implies the following for all $i\in I{\setminus }S$:

$$\begin{aligned} \underline{u}_{i}(a, w_{i},I)&= \min \limits _{\omega \in \mathcal {K}_{i}^{I}(a)}u_{i}(w_{i}(\omega )) \\&= \min \limits _{\omega \in \mathcal {K}_{i}^{I}(a){\setminus }\{a\}}u_{i}(x_{i}(\omega )) \\&\ge \min \limits _{\omega \in \mathcal {K}_{i}^{I}(a)}u_{i}(x_{i}(\omega )) \\&= \underline{u}_{i}(a, x_{i},I). \end{aligned}$$

Therefore, $\underline{u}_{i}(a, w_{i},I) \ge \underline{u}_{i}(a, x_{i},I)$, for all $i\in I{\setminus }S$, while by (iii) above $\underline{u}_{i}(a, w_{i}, I) > \underline{u}_{i}(a, x_{i}, I)$, for all $i\in S$, which (according to Definition (3)) contradicts the Pareto optimality of the interim maximin value allocation $x$.$\square$