CAHIERS ECO & MATHS

94.27

ASYMMETRIC INFORMATION, NON-ADDITIVE
EXPECTED UTILITY AND THE INFORMATION REVEALED
BY PRICES : A SIMPLE EXAMPLE

Jean-Marc TALLON

MAD
# Asymmetric information, non-additive expected utility and the information revealed by prices: a simple example

Jean-Marc Tallon*

November 1994

## Abstract

In this note we develop a simple example of a model in which agents have asymmetric information, and preferences that are represented by a non-additive expected utility function. Even in the finite case, that is when the equilibrium price function is fully revealing, it can be worth for an *a priori* uninformed agent to buy some private information, if he is more confident in that information than in the one revealed by the price system.

## Résumé

Il est proposé un exemple simple d'un modèle dans lequel les agents ont une information asymétrique et dont les préférences sont représentées par une fonction d'utilité espérée par rapport à une mesure de probabilité non-additive. Même lorsque le système de prix est parfaitement révélateur, les agents ont une incitation à acquérir de l'information privée s'ils jugent celle-ci plus fiable que celle révélée par les prix d'équilibre.

## 1 Introduction

General equilibrium models of asymmetric information have the “bad property” of yielding, in the case of a finite state space, equilibrium at which all the private information is revealed by the price system<fnref n="1" />. This gives rise to well-known paradoxes, such as the Grossman-Stiglitz paradox<fnref n="2" /> that, since all the relevant information is revealed at equilibrium, no agent will have a private incentive to acquire it. A way of solving this type of paradox is to go to state space of larger dimensionality<fnref n="3" />

<footnote>* M.A.D.-C.N.R.S. Address : M.A.D., Université Paris I, 90 rue de Tolbiac, 75634 Paris Cedex 13, France. e-mail: jmtallon@univ-paris1.fr</footnote>

<footnote n="1">¹See for example, Allen [1981], Grossman [1981], Allen [1984].</footnote>

<footnote n="2">²Grossman and Stiglitz [1976].</footnote>

<footnote n="3">³Another way would be to question the relevance of the rational expectations hypothesis in this type of models, see Dubey, Geanakoplos and Shubik [1987].</footnote>
(Allen [1986] for a survey). This is essentially the approach followed in finance, through the gimmick of noise traders (see Shleifer and Summers [1990]). Still another way, exposed in this paper, is to abandon the expected utility framework and to go to the more general setup in which agents' preferences can be represented by a non-additive expected utility.

Indeed, the argument at the heart of this full revelation result is that agents know the equilibrium price function, and therefore if it is one to one, know the state that occured. However, what is meant by the knowledge of the state is not all that clear. As Morris [1992] argues, knowledge could (should) be defined as a property of agent's preferences under uncertainty. All the literature on rational expectations uses some kind of standard expected utility representation of preferences. Then, knowledge of a state means that it is assigned probability one, and its complement is assigned probability zero.

On the other hand, if agent's preferences cannot be represented by a standard expected utility, but rather by a non-additive expected utility, then different notions of knowledge can be contemplated (again, see Morris [1992] for details). In particular, it is not the same to put zero probability on an event and to say that its complement has probability one.

In this paper, a simple example shows that if agents' preferences are of the non-additive type, then an agent can assign probability zero to a particular state when he observes the (equilibrium) price while he does not put probability one on its complement. Thus, if there are only two states, $\alpha$ and $\beta$, he can put probability zero on $\beta$ when he observes $p(\alpha)$, without necessarily putting probability one on $\alpha$. This behavior reflects a lack of confidence in the price system as a good aggregator of private information. Consequently, even though the price function is one to one, the agent would be willing (under some conditions) to pay something to acquire some private information, which has the same “objective” content as the price system, but that the agent trusts more. Thus, an agent could at the same time believe that financial markets are efficient, and read audit of firms before buying their shares.

Observe that the price system keeps its informative role, without excluding that private information be worth acquiring. To which degree agents “believe” in the price system as a good information aggregator depends on their preferences.

The setup of the model is exposed in section 2 while the equilibrium is computed in section 3. Section 4 deals with the issue of whether private information can be valuable to an *a priori* uninformed agent when the equilibrium price function is one
to one (i.e. when the equilibrium is fully revealing).

## 2 The model

There are two goods in each state, $x$ and $y$, whose price are denoted respectively by $p$ and $q$. Prices are normalized state by state so that $p + q = 1$. Each of the two agents ($h = 1, 2$), has constant-across-states endowments of one unit of each good, so that his wealth in each state is one.

Here, we include the (normalized) price in the definition of a state. Thus, the state space is defined to be: $S = \{\{\alpha, p(\alpha)\}, \{\alpha, p(\beta)\}, \{\beta, p(\alpha)\}, \{\beta, p(\beta)\}\}$. $\alpha$ and $\beta$ are descriptions of the “fundamentals” (i.e. the coefficients in the utility function), while $p(\alpha)$ and $p(\beta)$ are the associated (equilibrium) prices.

Implicit in the definition of this state space is the fact that agents are certain that the price will be either $p(\alpha)$ or $p(\beta)$. We assume that these two prices are different so that the equilibrium is of the “revealing” type.

We now define the informational structure of this economy, in relation to agents' preferences. It is assumed that agent 1 is fully informed of the state that occured. Suppose $\alpha$ occured, then his preferences are represented by the following utility function: $u_1^\alpha(x_1, y_1) = a_1(\alpha) \log x_1 + b_1(\alpha) \log y_1$. Similarly, if $\beta$ occurs, his preferences are represented by: $u_1^\beta(x_1, y_1) = a_1(\beta) \log x_1 + b_1(\beta) \log y_1$

Agent 2's preferences can be represented by a non-additive probability measure, together with state dependent utility functions. More precisely, define $\pi$ in the following way<fnref n="4" />:

(i) $\pi(\{s, p(s)\}) < 1$ for $s = \alpha, \beta$

(ii) $\pi(\{s, p(s')\}) = 0$ for $s \neq s'$

(iii) $\pi(\{s, p(s)\} \cup \{s, p(s')\} \cup \{s', p(s')\}) < 1$ for $s \neq s'$

(iv) $\pi(\{s, p(s)\} \cup \{s', p(s)\} \cup \{s, p(s')\}) = \pi(\{s, p(s)\} \cup \{s', p(s)\})$ for $s \neq s'$

together with the usual requirements that $\pi(\emptyset) = 0$, $\pi(S) = 1$ and $A \subset B \Rightarrow \pi(A) \le \pi(B)$. Notice that this probability measure is not additive, hence reflecting “uncertainty aversion” or lack of confidence in one’s probabilistic assessments (see Schmeidler [1989] and Dow and Werlang [1992]b). Indeed, either $\pi(\{\alpha, p(\alpha)\}) + \pi(\{\beta, p(\beta)\})$ is strictly less than one and then $\pi(S) > \pi(\{\alpha, p(\alpha)\}) + \pi(\{\alpha, p(\beta)\}) + \pi(\{\beta, p(\alpha)\}) + \pi(\{\beta, p(\beta)\})$, or it is equal to one, but then $\pi(\{\alpha, p(\alpha)\} \cup \{\alpha, p(\beta)\} \cup \{\beta, p(\beta)\}) < \pi(\{\beta, p(\beta)\})$

<footnote n="4">⁴To keep notation to a minimum we'll omit the subscript corresponding to agent 2 and write his probability measure $\pi$ instead of $\pi_2$.</footnote>
$\pi(\{\alpha, p(\alpha)\}) + \pi(\{\alpha, p(\beta)\}) + \pi(\{\beta, p(\beta)\})$. In both cases, the probability of the union of several disjoint events is less than the sum of their probabilities.

Assume that 2's ("von Neumann - Morgenstern") utility function depends<fnref n="5" /> on the realisation of $s = \alpha, \beta$, and not (directly) on prices. Let $u_2^\alpha(x_2, y_2) = a_2(\alpha) \log x_2 + b_2(\alpha) \log y_2$ and $u_2^\beta(x_2, y_2) = a_2(\beta) \log x_2 + b_2(\beta) \log y_2$. Assume without loss of generality that $a_h(s) + b_h(s) = 1$ for $h = 1, 2$ and $s = 1, 2$.

We need now to define what agent 2's decision problem is upon observing, say, $p(\alpha)$. A natural extension of Bayes' law for non-additive probability measures is the Dempster and Shafer rule which specifies that:

$$ \pi(A/B) = \frac{\pi(A \cup B^c) - \pi(B^c)}{1 - \pi(B^c)} $$

where $B^c$ is the complement of $B$. This rule reduces to Bayes' law if the measure is additive, and can be derived axiomatically as a "pessimistic" updating rule<fnref n="6" />.

Applying that rule to the measure defined above, one gets a non-additive measure on $S' = \{\{\alpha, p(\alpha)\}, \{\beta, p(\alpha)\}\}$. Agent 2, when he observes $p(\alpha)$, places the following probability on the occurrence of $\alpha$:

$$ \pi(\{\alpha, p(\alpha)\} | \{\alpha, p(\alpha)\} \cup \{\beta, p(\alpha)\}) = \frac{\pi(\{\alpha, p(\alpha)\} \cup \{\alpha, p(\beta)\} \cup \{\beta, p(\beta)\}) - \pi(\{\alpha, p(\beta)\} \cup \{\beta, p(\beta)\})}{1 - \pi(\{\alpha, p(\beta)\} \cup \{\beta, p(\beta)\})} $$

To keep notation simple, we'll write this conditional probability $\pi(\alpha/p(\alpha))$.

Observe that this probability is less than 1 given assumption (iii) on the measure $\pi$. This means that when he sees $p(\alpha)$, the second agent is not "sure" that $\alpha$ occured, in the sense that he does not place probability one on the state $\{\alpha, p(\alpha)\}$. However, given assumption (iv) on the measure $\pi$, agent 2 places a zero probability on the occurrence of state $\{\beta, p(\alpha)\}$ given that he observes $p(\alpha)$<fnref n="7" />.

Following Morris [1992] terminology, the "knowledge" 2 has of $\alpha$ could be called negative knowledge: an event is negatively known if its complement is assigned probability zero, whereas it is positively known if it is assigned probability one. In the case of preferences represented through non-additive probability measures, these

<footnote n="5">⁵The axiomatization of non-additive expected utility by, e.g., Schmeidler [1989] yields a von Neumann-Morgenstern utility function that is independent of the state. Morris [1992] however, uses state-dependent preferences in this framework. We'll adopt this approach here.</footnote>

<footnote n="6">⁶See Gilboa and Schmeidler [1993] and Gilboa and Schmeidler [1992] for an axiomatization. For an application of this type of updating, see Dow and Werlang [1992]a. This rule however, is not the only plausible one, as discussed in the two papers by Gilboa and Schmeidler cited above.</footnote>

<footnote n="7">⁷As Dow and Werlang [1991] explain, "if a set has probability zero, that does not mean its complement is of probability one. But if the complement of a zero-probability set has positive probability, it has relatively infinitely more chance of happening than the original set."</footnote>
two notions need not be the same. Hence, agent 2 knows $\alpha$ when he sees $p(\alpha)$ in the sense that he assigns zero probability to its complement, $\beta$. He is therefore never proven wrong (he never wrongly associates the occurence of $p(\alpha)$ with $\beta$), in the spirit of the rational expectations hypothesis.

Be as it may, the second agent's preferences are represented by the (non-additive) expected utility conditional on the price he observes. The expected value with respect to a non-additive measure $v$ of a function $f$ defined over a discrete space $\Omega = \{1, ..., \Omega\}$ such that $f(1) \ge f(2) \ge ... \ge f(\Omega)$ is equal to:

$$E_v f = \sum_{\omega=1}^{\Omega} (f(\omega) - f(\omega + 1)) v(\{1, ..., \omega\}) + f(\Omega)$$

It is thus necessary to order the outcomes to compute this expected value.

The expected utility of agent 2 if he observes $p(\alpha)$ is consequently equal to:

$$u^{\beta}(x, y) + \pi(\alpha/p(\alpha)) \cdot [u^{\alpha}(x, y) - u^{\beta}(x, y)]$$

if $u^{\alpha}(x, y) > u^{\beta}(x, y)$.

### 3 Computation of the equilibrium

A (revealing) equilibrium in this framework is defined in the same way as usual. It is a price vector $(p(\alpha), p(\beta))$ such that agents maximize their (non-additive) expected utility conditionally on their private information as well as the information revealed by the price system, and markets clear.

Agent 1 being fully informed, solves a different problem, depending on whether $\alpha$ or $\beta$ occured. This leads to the demand functions:

$$x_1(s, p) = \frac{a_1(s)}{p} \quad \text{and} \quad y_1(s, p) = \frac{b_1(s)}{1-p} \quad s = \alpha, \beta$$

Assuming for the moment that $p(\alpha) \ne p(\beta)$ (this condition will be shown to hold at equilibrium), agent 2 solves the following program, if he observes say, $p(\alpha)$ and if his utility in state $\beta$ is less than in state $\alpha$ (this condition will also be shown to hold at equilibrium):

$$\begin{array}{ll} \max_{x,y} & u_2^{\beta}(x,y) + \pi(\alpha/p(\alpha)) (u_2^{\alpha}(x,y) - u_2^{\beta}(x,y)) \\ \text{s.t.} & px + qy = 1 \end{array}$$

Demand functions are then given by:

$$x_2(p(\alpha)) = \frac{(1 - \pi(\alpha/p(\alpha))) a_2(\beta) + \pi(\alpha/p(\alpha)) a_2(\alpha)}{p(\alpha)}$$

$$y_2(p(\alpha)) = \frac{(1 - \pi(\alpha/p(\alpha))) b_2(\beta) + \pi(\alpha/p(\alpha)) b_2(\alpha)}{1 - p(\alpha)}$$
An equilibrium is then a price vector $(p(\alpha), p(\beta))$, $p(\alpha) \neq p(\beta)$ such that:

$$ 
\begin{aligned}
\frac{a_1(\alpha)}{p(\alpha)} + \frac{(1 - \pi(\alpha/p(\alpha))) a_2(\beta) + \pi(\alpha/p(\alpha)) a_2(\alpha)}{p(\alpha)} &= 2 \\
\frac{a_1(\beta)}{p(\beta)} + \frac{(1 - \pi(\beta/p(\beta))) a_2(\alpha) + \pi(\beta/p(\beta)) a_2(\beta)}{p(\beta)} &= 2
\end{aligned}
$$

which gives:

$$ 
\begin{aligned}
p^*(\alpha) &= \frac{a_1(\alpha) + (1 - \pi(\alpha/p^*(\alpha))) a_2(\beta) + \pi(\alpha/p^*(\alpha)) a_2(\alpha)}{2} \\
p^*(\beta) &= \frac{a_1(\beta) + (1 - \pi(\beta/p^*(\beta))) a_2(\alpha) + \pi(\beta/p^*(\beta)) a_2(\beta)}{2}
\end{aligned}
$$

Obviously, these prices are different for almost all choice of the parameters.

What remains to be checked is that the utilities of agent 2 are indeed ranked as we assumed, i.e. $u_2^\alpha[x(\alpha, p^*(\alpha)), y(\alpha, p^*(\alpha))] > u_2^\beta[x(\alpha, p^*(\alpha)), y(\alpha, p^*(\alpha))]$. This condition is equivalent to:

$$ (a_2(\alpha) - a_2(\beta)) [b_1(\alpha) [(1 - \pi(\alpha/p^*(\alpha)))a_2(\beta) + \pi(\alpha/p^*(\alpha))a_2(\alpha)] - a_1(\alpha)] > 0 $$

Similarly, the condition if $\beta$ occured is:

$$ (a_2(\beta) - a_2(\alpha)) [b_1(\beta) [(1 - \pi(\beta/p^*(\beta)))a_2(\alpha) + \pi(\beta/p^*(\beta))a_2(\beta)] - a_1(\beta)] > 0 $$

It is easy to check that both these conditions can hold simultaneously. Suppose for example that $a_1(\alpha) = b_1(\beta) = 0$ and that $b_1(\alpha) = a_1(\beta) = 1$. Then these two conditions hold for any choice of $a_2(\alpha) > a_2(\beta)$. Therefore, by continuity, these conditions hold for an open set of parameters around these values.

# 4 Is it worth acquiring private information?

In the traditional framework – i.e. with preferences that can be represented by an additive probability measure and state-dependent utility functions – the rational expectations equilibrium gives rise to the Grossman-Stiglitz paradox that, at a fully revealing equilibrium, nobody has an incentive to acquire private information.

Having computed the equilibrium prices, and observed that they are revealing in the sense that prices are different for different realisations of $s = \alpha, \beta$, one can now ask whether agent 2 would have, in the present framework, an incentive to acquire some private information concerning which state occured.

Suppose that agent 2 has access to another source of information, modeled by a signal $\sigma^\alpha, \sigma^\beta$, and that this signal is not “ambiguous” in the sense that the agent's
preferences can be represented by an additive probability measure $\tilde{\pi}$ and the same state dependent utility function as above. In particular, one has:

$$ \tilde{\pi}(\{\alpha, \sigma^{\alpha}\}) + \tilde{\pi}(\{\beta, \sigma^{\beta}\}) = 1 $$

This yields a conditional probability (computed according to Bayes' rule) having the following properties:

$$ \tilde{\pi}(s/\sigma^s) = \frac{\tilde{\pi}(\{s, \sigma^s\})}{\tilde{\pi}(\{s, \sigma^s\} \cup \{s', \sigma^s\})} = 1 \quad \text{for } s \neq s' $$

and

$$ \tilde{\pi}(s'/\sigma^s) = 0 \quad \text{for } s \neq s' $$

The signal $\sigma$ is consequently more “informative” than the price system, not that it truly reveals more information, but rather because the agent is more confident in his private information than in the information revealed by the price system<fnref n="8" />.

It is easy to compute the equilibrium in the economy where both agents are fully informed of the state (in the sense that they both place probability one on the state that occured and zero on the other one). It is given by the price system:

$$ \begin{aligned} \bar{p}(\alpha) &= \frac{a_1(\alpha) + a_2(\alpha)}{2} \\ \bar{p}(\beta) &= \frac{a_1(\beta) + a_2(\beta)}{2} \end{aligned} $$

What remains to be done is to see whether agent 2 has an incentive to use (buy) this private information. He will do so if the ex-post utility he gets at equilibrium, in each state, is larger when he has access to a private source of information than when he learns which state occured through the price system, that is if:

$$ u_2^s[x_2(\bar{p}(s)), y_2(\bar{p}(s))] > u_2^s[x_2(p^*(s)), y_2(p^*(s))] \quad \text{for } s = \alpha, \beta $$

These conditions have no simple expression but it can be checked that they hold, e.g. if $a_1(\alpha) = b_1(\beta) = 0$ and $b_1(\alpha) = a_1(\beta) = 1$, a condition on the parameters we already met. By continuity, these conditions are also true for values close to the one above. Therefore, there exists an open set of parameters for which the uninformed agent has an incentive to acquire some private information despite the fact that the equilibrium price system is fully revealing.

<footnote n="8">⁸See, in a different set-up Caballe and Sakovics [1994], who have a similar assumption, stating that traders are over-confident in their private information.</footnote>
Thus, the second agent is ready to pay a strictly positive amount to acquire this private information, even though the price system “objectively” contains all the information he needs. In other words, were we to give him the additional information in the form of the signal $\sigma^s$, he would not tend to disregard it, as in the additive case, where private information is redundant with respect to the one revealed by the price system. Observe that the agent is willing to buy the information no matter what it is. In both states he is better off (ex post) having bought the information.

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