Separate aggregation of beliefs and values under ambiguity

Abstract

Maximin expected utility model for individual decision making under ambiguity prescribes that the individual posits independently a utility function and a set of probability distributions over events to represent the values and belief, respectively. It assumes that individual evaluates each act on the basis of its minimum expected utility over this class of distributions. In this paper, we attempt to generalize the model to social decision making. It is assumed that the society’s belief is formed through a linear aggregation of individual beliefs and society’s values through a linear aggregation of individual values. We propose principles which characterize such separate aggregation procedures. We also generalize Choquet expected utility model, which posits a nonadditive measure over events and a utility function to represent belief and values, respectively. We prove that the only aggregation procedures that respect our principles are the separate linear aggregations of beliefs and values.

Appendices

Appendix 1: A general affine aggregation theorem

In this section, we prove a simple and general theorem on linear aggregation, which provides the basic tools to prove our theorems. This theorem can be regarded as a generalization of De Meyer and Mongin (1995).

Consider a non-empty convex set $\mathscr {F}$. Let $V=(V_0,V_1,\ldots ,V_I):\mathscr {F}\rightarrow \mathrm {R}^{I+1}$ be a vector-valued function. Notice that the range of V may not be convex. The problem we explore is to study the following property.

Condition P: for each pair of finite sequences $\{f_m\}_{m=1}^M$ and $\{g_l\}_{l=1}^L$ in $\mathscr {F}$, and every numbers $\alpha _m\ge 0$ and $\beta _l\ge 0$ such that $\sum _{m=1}^M\alpha _m=1$ and $\sum _{l=1}^L\beta _l=1$,

$$\begin{aligned} \sum _{m=1}^M \alpha _m V_i(f_m)\ge \sum _{l=1}^L \beta _l V_i(g_l)\quad \forall 1\le i\le I\Longrightarrow \sum _{m=1}^M \alpha _m V_0(f_m)\ge \sum _{l=1}^L \beta _l V_0(g_l). \end{aligned}$$

If each $V_i$ is a representation function, then Condition P is equivalent to CEPC. We say a vector-valued function V is regular if (i) each $V_i$ is homogenous of degree one, (ii) normalized; i.e., there is a $f^*\in \mathscr {F}$ such that $V(f^*)=0$ and satisfies shift; i.e., $V(f+f^*)=V(f)$.

Theorem 5

Suppose V is regular. Then Condition P holds if and only if there are nonnegative numbers $\lambda _1,\ldots ,\lambda _I$ such that for all $f\in \mathscr {F}$

$$\begin{aligned} V_0(f)=\sum ^n_{i=1}\lambda _iV_i(f) \end{aligned}$$

(7)

The sufficiency part of the theorem is straightforward. We only prove the necessity. To this end, we first define the (finite) addition set of vector-valued functions by

$$\begin{aligned} W=\Big \{\sum ^M_{m=1}V(f_m): f_m\in \mathscr {F}\text { and } M\in \mathbb {N}\Big \}\subset \mathbb {R}^{I+1}. \end{aligned}$$

The proof consists of several lemmas and is organized as follows: We first prove that W is actually a convex set. We then define a set $W^-=\{v=w-w': w,w'\in W\}$ and consider the subspace spanned by $W^-$, to be denoted by ${{\mathrm{span}}}{W^-}$. Second, we consider a set R (defined later), which is implied by the Condition P. We show that R is a convex set and disjoint to ${{\mathrm{span}}}(W^-)$. Finally, applying the Separating Theorem, expression as in (7) is obtained.

Lemma 1

W is convex.

Proof

Let $w\in W$ and $\alpha \in [0,1]$. We first show that $\alpha w\in W$. By definition of W, there exists a sequence of acts $(f_m)_{m=1}^M$ in $\mathscr {F}$ such that

$$\begin{aligned} w=\sum ^M_{m=1}V(f_m). \end{aligned}$$

By regularity, for $0\le i\le I$,

$$\begin{aligned} V_i(\alpha f_m+(1-\alpha )f^*)=\alpha V_i(f_m). \end{aligned}$$

So, $\alpha w=\sum ^M_{m=1}V(\alpha f_m+(1-\alpha )f^*)$. By convexity of $\mathscr {F}$, $\alpha f_m+(1-\alpha )f^*\in \mathscr {F}$. Hence, we have $\alpha w\in W$. Now let $w,v\in W$. To complete the proof, it suffices to show that $w+v\in W$. However, it follows directly from the definition of W. Hence, W is a convex set. $\square$

Let $W^-$ denote the set $\{w-w': w,w'\in W\}$. We define

$$\begin{aligned} R=\{r\in \mathbb {R}^{I+1}: r_0<0\text { and }r_i\ge 0\text { for every }i\in \mathcal {I}\}, \end{aligned}$$

and let $\bar{R}$ be the closure of the set R. We show next that the sets R and $W^-$ are disjoint.

Lemma 2

$R\cap W^-=\emptyset$.

Proof

Suppose not. There exist two sequences of acts $(f_m)^M_{m=1}$ and $(g_l)_{l=1}^L$ in $\mathscr {F}$ such that

$$\begin{aligned} \sum ^M_{m=1}V_i(f_m)-\sum ^L_{l=1}V_i(g_l)\ge & {} 0\qquad \text { for } i\in \mathcal {I}, \text {and}\\ \sum ^M_{m=1}V_0(f_m)-\sum ^L_{l=1}V_0(g_l)< & {} 0. \end{aligned}$$

These imply that

$$\begin{aligned}&\frac{1}{M}\sum ^M_{m=1}V_i\left( \frac{M}{M+L}f_m+\frac{L}{M+L}f^*\right) -\frac{1}{L}\sum ^L_{l=1}V_i\left( \frac{L}{M+L}g_l+\frac{M}{M+L}f^*\right) \nonumber \\&\quad \ge 0\quad \text {for } i\in \mathcal {I}, \text {and}\\&\frac{1}{M}\sum ^M_{m=1}V_0\left( \frac{M}{M+L}f_m+\frac{L}{M+L}f^*\right) -\frac{1}{L}\sum ^L_{l=1}V_0\left( \frac{L}{M+L}g_l+\frac{M}{M+L}f^*\right) <0 \end{aligned}$$

Obviously, these contradict the Condition P. Hence, sets R and $W^-$ are disjoint. $\square$

Let ${{\mathrm{span}}}{W^-}$ denote the vector space spanned by $W^-$.

Lemma 3

$R\cap {{\mathrm{span}}}{W^-}=\emptyset$.

Proof

Now suppose there is a vector $w\in R \cap {{\mathrm{span}}}{W^-}$. Clearly w is a nonzero vector. Then there must exist a sequence of elements $w_1,w_2,\ldots ,w_M$ in $W^-$ and a sequence of numbers $\alpha _1,\ldots ,\alpha _M$ such that $w=\sum ^M_{m=1}\alpha _m w_m$. Since $W^-$ is convex and symmetric with respect to vector zero, each $\alpha _i$ can be taken nonnegative. Therefore, $(\sum ^M_{m=1}\alpha _m)^{-1}w\in R\cap W^-$, which contradicts the fact that sets R and $W^-$ are disjoint.

$\square$

For $0\le i\le I$, let $e_i$ denote the ith vector of the canonical basis. We denote by $\bar{R}-e_0$ the set $\{r-e_0: r\in \bar{R}\}$.

Lemma 4

There exist nonnegative numbers $\lambda _1,\ldots ,\lambda _I$ such that for every $f\in \mathscr {F}$

$$\begin{aligned} V_0(f)=\sum _{i=1}^I \lambda _i V_i(f). \end{aligned}$$

Proof

Notice that both sets, $\bar{R}-e_0$ and ${{\mathrm{span}}}(W^-)$, are polyhedral, non-empty and mutually disjoint. Applying the separation theorem, there exists a nonzero vector $v=(v_0,v_1,\ldots ,v_I)\in \mathbb {R}^{I+1}$ such that

$$\begin{aligned} \left<v,r-e_0\right>>\left<v,w\right>\text { for all }r\in \bar{R} \hbox { and } w\in {{\mathrm{span}}}{W^-}. \end{aligned}$$

The vector space property of ${{\mathrm{span}}}{W^-}$ implies that

$$\begin{aligned} \left<v,w\right>=0\text { for all }w\in {{\mathrm{span}}}{W^-}. \end{aligned}$$

(8)

Applying the above inequality to $r=0$, we conclude that $v_0<0$. Thus, Eq (8) as restricted to the set $\{V(f)-V(g): f,g\in \mathscr {F}\}\subset W^-$ can be written as: For every $f,g\in \mathscr {F}$,

$$\begin{aligned} V_0(f)-V_0(g)=\sum ^I_{i=1}\lambda _i[V_i(f)-V_i(g)], \end{aligned}$$

where $\lambda _i=-\frac{v_i}{v_0}$ for $i\in \mathcal {I}$.

To see $\lambda _i$ is nonnegative, first note that for every $i\in \mathcal {I}$ and $\alpha >0$, we have $\alpha e_i\in \bar{R}$, whence

$$\begin{aligned} \left<v,\alpha e_i\right>>\left<v,e_0\right>. \end{aligned}$$

Thus, $v_i\ge 0$ follows by dividing by $\alpha$ and letting $\alpha$ goes to infinity. Hence, for every $f\in \mathscr {F}$, $V_0(f)=\sum ^I_{i=1}\lambda _i V_i(f)+\mu$ for some real number $\mu$. Since each $V_i$ is homogeneous of degree one, we have $\mu =0$. $\square$

This result can be further implied to many ambiguity models, which are developed based on Anscombe-Aumann framework. For instance, if the representation functions of both individuals and society are regular, then CEPC principle implies that society’s representation function is a weighted sum of individuals’ functions.

Appendix 2: Proofs

In this section, we apply the result of Theorem 5 to prove the theorems in Sects. 3 and 4.

Proof of Theorem 1

We first show that conditions (i) and (ii) are equivalent. Note that CEPC is equivalent to Condition P if both individuals and society satisfy the conditions of the MEU model. Since individuals have identical values, we can renormalize each $V_i$ such that for some $x\in X$, $V_i(x)=0$ for each $i\in \mathcal {I}$. Furthermore, it is well known that each $V_i$ is homogeneous of degree one and satisfies certainty independence. Therefore, $V=(V_0,V_1,\ldots ,V_I)$ is regular. Hence, along with the assumption $u_i=u_j$ for all $0\le i,j\le I$, conditions (i) and (ii) are equivalent following directly from the results of Theorem 5.

We prove now that conditions (ii) and (iii) are equivalent.

Assume that (ii) holds and prove (iii). Let $\hat{K}=\sum ^I_{i=1}\lambda _i K_i$ and let $\hat{V}(f)=\min _{p\in \hat{K}}\int u(f)\mathrm {d}p$ for all $f\in \mathscr {F}$. It is easy to see that $\hat{K}$ is a closed and convex set of priors. Therefore, $\hat{V}$ is a well-defined MEU function on $\mathscr {F}$. If $f\in \mathscr {F}$ and $i\in \mathcal {I}$, let $p_i^f\in K_i$ be such that $\int u(f)\mathrm {d}p_i^f=V_i(f)$ and let $\hat{p}_i^f\in K_i$ be such that $\int u(f)\mathrm {d}(\sum ^I_{i=1}\lambda _i\hat{p}_i^f)=\hat{V}(f)$, and define $\hat{p}=\sum ^I_{i=1}\lambda _i\hat{p}_i^f$. Therefore,

$$\begin{aligned} \hat{V}(f)\le \int u(f)\mathrm {d}\left( \sum \lambda _ip_i^f\right) =\sum \lambda _iV_i(f)=V_0(f). \end{aligned}$$

Conversely,

$$\begin{aligned} V_0(f)=\sum \lambda _iV_i(f) \le \sum \lambda _i\int u(f)\mathrm {d}\hat{p}^f_i =\hat{V}(f) \end{aligned}$$

Hence, $V_0(f)=\hat{V}(f)$. Uniqueness of MEU representation implies $K_0=\hat{K}$.

Next, assume that (iii) holds and prove (ii). Suppose that $V_0(f)\ne \sum \lambda _iV_i(f)$ for some $f\in \mathscr {F}$. Then there must exist $p_i$ and $\hat{p}_i$ such that $\int u(f)\mathrm {d}p_i\ne \int u(f)\mathrm {d}\hat{p}_i$. Therefore, either $V_i(f)>\int u(f)\mathrm {d}\hat{p}_i$ or $V_0(f)>\int u(f)\mathrm {d}(\sum _{j\ne i}\lambda _j\hat{p}_j+\lambda _i p_i)$. Hence, $V_0(f)=\sum \lambda _iV_i(f)$. $\square$

Proof of Theorem 2

The necessity part is clear. To see the sufficiency part, notice that RPC and CPC conditions imply that society’s preference agree with individual preferences on $\mathscr {G}$. Again uniqueness of MEU representation implies that society’s belief is also K. $\square$

Proof of Theorem 3

It is straightforward that condition (ii) implies (i). We only prove that condition (i) implies (ii).

Assume that RCEPC and CPC hold. We prove first that the conditions imply that $K_0$ is an linear combination of $\{K_i\}_{i=1}^I$, with coefficients summing to one. Next, we show that these imply also that $u_0$ is an linear combination of $\{u_i\}_{i=1}^I$, with coefficients summing to one.

By assumption, $u_i(\underline{x})=0$ and $u_i(\overline{x})=1$ for $0\le i\le I$. Therefore, $\{u_i\}_{i=0}^I$ agree on $[\underline{x},\overline{x}]$. It follows directly from Theorem 1 that $V_0$ restricted to the set $\mathscr {G}$ is an linear combination of $\{V_i\}_{i=1}^I$. Equivalently, there is $\lambda \in \Delta (\mathcal {I})$ such that $K_0=\sum ^I_{i=1}\lambda _i K_i$.

Now, we show that $u_0$ is an linear combination of $\{u_i\}_{i=1}^I$, with coefficient summing to one. Since CPC holds, according to Harsanyi’ Theorem, we conclude that there is $\gamma \in \Delta (\mathcal {I})$ such that $u_0=\sum _{i=1}^I\gamma _i u_i$. $\square$

Proof of Theorem 4

It is straightforward that (ii) implies (i). We show that (i) implies (ii). We assume that RCEPC and CPC hold. According to Harsanyi’s Theorem, CPC implies that society’s utility function is a weighted sum of individual utility functions. Hence, the rule as in (6) holds.

We prove that rule (5) holds. First notice that $V=(V_0,V_1,\ldots ,V_I)$ where each $V_i$ is CEU function is regular restricted on set $\mathscr {G}$. According to the properties of CEU model, each $V_i$ is homogeneous of degree one and satisfies certainty independence. Since $V_i(\underline{x})=0$ for every i, vector function V is regular. Condition RCEPC is equivalent to Condition P restricted on set $\mathscr {G}$. By Theorem 5, there exist $\alpha \in \Delta (\mathcal {I})$ such that for every $f\in \mathscr {G}$,

$$\begin{aligned} V_0(f)=\sum ^I_{i=1}\alpha _i\ V_i(f). \end{aligned}$$

If $E\in \mathscr {A}$, $\overline{x}E\underline{x}$ denote a binary act which yields $\overline{x}$ if E occurs and $\underline{x}$ otherwise. Therefore, for every $E\in \mathscr {A}$,

$$\begin{aligned} \mu _0(E)= & {} V_0(\overline{x}E\underline{x})\\= & {} \sum ^I_{i=1}\alpha _i V_i(\overline{x}E\underline{x})\\= & {} \sum ^I_{i=1}\alpha _i\mu _i(E) \end{aligned}$$

$\square$