# Attitude towards risk, prospect variability, and the value of imperfect information.

I. Introduction

Choice in decision making under uncertainty often includes the option of choosing to utilize an information system, a set of potential messages that may improve current decisions and resulting outcomes. In decision theory the demand price for an information system is defined as the maximum non-stochastic cost, payable from initial wealth prior to the receipt of any message, that makes the decision maker (DM) indifferent between purchasing the system or not. The determinants of the value of information are important to both the buyers and the sellers in the marketplace for information products, and economists study the subject with both theoretical and practical motivations.

Theoretical analyses seek to discover the characteristics of the economic and/or statistical environment that allow for definitive qualitative statements about the value of information. The most famous result, Blackwell's Theorem |1; 2~, states the necessary and sufficient conditions under which any potential user values one information system more than another. Unfortunately, most results are of the negative variety: there is no general monotonic relationship between the information value and the degree of aversion towards risk |5~, the amount of statistical information transmitted |8~, the level of initial wealth |6~, or the Rothschild-Stiglitz variability of the prior distribution on the state |4~.

In practical situations, the buyer or the producer of an information system may desire to utilize quantitative estimates of the value of information. Complete quantitative analysis requires knowledge of the statistical characteristics of the information, identification of the payoff function for the decision problem at hand, and assessment of the information user's utility function for wealth. This last factor, the utility function, is specific to each individual DM and is unlikely to be known by the producer of the information. Unless it can be assumed that all potential buyers are neutral towards risk and hence have an easily assessed linear utility function, the seller of the information product (e.g., a forecasting service with many clients) faces the problem of estimating demand without knowledge of the demanders' utility functions. Policy makers, seeking to assess the value of publicly collected and disseminated information, face a similar problem |12~.

Blair and Romano |3~, henceforth B-R, recently have introduced a new problem-specific approach to information valuation that has both theoretical and practical importance. The B-R idea is to use the Rothschild-Stiglitz variability of the DM's alternative prospects (i.e., probability distributions on terminal wealth) as a criterion for qualitative results concerning the role of the DM's attitude towards risk in determining the demand price for information. Specifically, B-R present a criterion sufficient to ensure that no risk averter values information by more than the benchmark risk neutral DM. The B-R analysis is most useful for the comparative valuation of perfect information (the case of unequivocal identification of the state), since the optimal informed prospects are then identical for all DMs regardless of attitude towards risk. In this case there is no need to assume any specifics about the utility function of the risk averter.

This paper investigates the value of imperfect information as it depends upon the DM's attitude towards risk. When the information system is not perfect, testing for the satisfaction of the B-R criterion requires knowledge of each potential user's specific non-linear utility function and the corresponding optimal decision rules. The primary result of this paper is an extension of the B-R approach that provides definitive conditions, requiring only risk neutral assessments and computations, under which no risk averse DM values imperfect information more than the risk neutral counterpart. On the theoretical level, the analysis intends to contribute to the understanding of attitude towards risk as a determinant of the value of imperfect information. In practice, satisfaction of the condition may be useful to the producer of the information, as it sets an upper bound on the demand price of any risk averse potential user facing the given decision problem.

Section II reviews the theory of the value of information, comparing the prospects chosen by risk neutral and risk averse DMs. When valuing information, the risk neutral DM cares only about the impact on expected wealth that results from the use of the information. The risk averter cares also about changes in prospect variability, that is, changes in riskiness in the sense of Rothschild and Stiglitz |11~. Accordingly, the comparison of the value of imperfect information between risk averse and risk neutral DMs depends critically on the comparative impacts the use of the information has both on expected wealth and on prospect variability. Section III presents three theorems that study the relative value of information in a systematic way. Theorem 3 presents definitive results by comparing the Rothschild-Stiglitz variability of two relatively easy to assess prospects: the risk neutral prior optimal prospect and the risk neutral prior distribution of the posterior mean wealth, which is here a random variable since it is conditioned on the realization of a random information signal. Theorem 3 contains a sufficiency condition, verifiable using a standard test for second order stochastic dominance between the two risk neutral prospects, under which no risk averter values information more than the benchmark risk neutral DM. Section IV presents a simple parameterized model of imperfect information that indicates the role statistical informativeness plays in achieving the sufficiency condition of Theorem 3. Section V contains concluding remarks.

II. Prospects and the Value of Information

This paper concerns the demand price for imperfect information in decision problems under uncertainty with the following characteristics, definitions, assumptions, and notation:

1) A set a of available actions a; the DM chooses a |element of~ a.

2) A set X of mutually exclusive state descriptions x, one x |element of~ X ultimately obtains according to a known unconditional (prior) probability measure on X, denoted p (x).

3) The actions and states may be vectors, and the monetary payoff of the endeavor when state x obtains and action a is chosen is quantified in a real-valued payoff function |omega~(x, a) defined on X x a.

4) A set W of potential terminal wealths that may result from the decision problem; the outcome of the decision problem is that fixed and known initial wealth ||omega~.sub.0~ changes to terminal wealth W according to

W = ||omega~.sub.0~ + |omega~(x, a). (1)

It is assumed that all actions in a remain feasible regardless of the level of wealth.

5) A Von Neumann-Morgenstern utility function on W such that u (W) is continuous, strictly increasing, and weakly concave in W.

6) The option of utilizing an information system I comprised of:

a) a set Y of potential signals or messages that can emanate from the system; one y |element of~ Y is received by the DM prior to the choice of action, and

b) a joint probability measure p(x,y) on X x Y, with the unconditional measures p(x) and p(y) being proper and strictly greater than zero.

7) The DM uses as the basis for choice the maximization of expected utility, with all required expectations finite and optimal actions unique.

The methodology of this paper focuses on the evaluation and comparison of prospects--probability distributions over terminal wealth determined by the choice of action. For each a |element of~ a, the payoff function (1) defines a function from X to W that serves as the basis for a change of variables. When the DM chooses an action a |element of~ a (or "accepts the lottery" a), this induces ownership of a prospect denoted |Mathematical Expression Omitted~. The advantage of working with prospects is being able to deal directly with univariate probability measures over cash amounts.

In the uninformed situation, without the utilization of an information system, the DM chooses the action that is optimal under the prior measure p(x). The DM solves the problem

|Mathematical Expression Omitted~

where the subscript on the expectations operator indicates the random variable with respect to which the expectation is taken. In general, this choice of optimal prior action depends upon the DM's specific utility function. Denote the prior optimal action given any concave utility function u as ||a.sup.u~.sub.0~, defined by

|Mathematical Expression Omitted~

In the uninformed decision, the choice of action ||a.sup.u~.sub.0~ gives the optimizing DM the prospect |Mathematical Expression Omitted~, where it is implicit in the notation that this prospect is induced both by the choice of optimal action ||a.sup.u~.sub.0~ and the original probability measure p(x). The expected utility from owning this prospect is written

|Mathematical Expression Omitted~

and is equal to the amount stated in (3).

Especially important are two summarizing cash amounts associated with a prospect: the reservation price and the mean wealth. The reservation price, or certainty equivalent, of the prior decision is the cash equivalent terminal wealth that the DM would accept as a sale price to walk away from all risk inherent in the optimal solution to the decision problem. The prior reservation price ||R.sup.u~.sub.0~ is defined implicitly as the solution to

|Mathematical Expression Omitted~

The mean of the optimal prior prospect

|Mathematical Expression Omitted~

is the expected terminal wealth from the prior decision. As is well known (and can be shown by applying Jensen's inequality to (5)), the concavity of the utility function ensures that

|Mathematical Expression Omitted~

The results of this paper derive from comparing the prospects chosen by any risk averse DM with those chosen by or associated with the risk neutral benchmark DM having a utility function linear in W. When it is important to distinguish, the notation replaces the superscript u with A to indicate any risk averter, and N to identify the risk neutral counterpart. For the risk neutral DM, (7) holds with equality and the prior reservation price is the mean of the optimal prior prospect:

|Mathematical Expression Omitted~

Useful inequalities for proving the subsequent theorems arise by comparing optimal prospects with specific suboptimal ones. Denoting the prior optimal actions under risk aversion and risk neutrality by ||a.sup.A~.sub.0~ and ||a.sup.N~.sub.0~ respectively, the optimality of ||a.sup.A~.sub.0~ guarantees that

|Mathematical Expression Omitted~

Similarly, with ||a.sup.N~.sub.0~ being optimal under risk neutrality,

|Mathematical Expression Omitted~

or, using (6),

|Mathematical Expression Omitted~

Combining (11) with (8) and (7) obtains the summarizing relationship between the uninformed reservation prices and means:

|Mathematical Expression Omitted~

Assume now that the DM opts to utilize the information system I, paying $|psi~

|is greater than or equal to~ 0 from initial wealth, receiving message y |element of~ Y, and viewing the relevant probability measure on X as the conditional (posterior) distribution |Mathematical Expression Omitted~. If the DM chooses action based upon initial wealth being reduced by $|psi~, then the optimal action given y is denoted

|Mathematical Expression Omitted~

as defined by

|Mathematical Expression Omitted~

As a matter of notation, if the information system were free the subscript |psi~ in statements like (13) is omitted. Except in special cases,

|Mathematical Expression Omitted~

The primary exception is when the DM's preferences exhibit constant absolute risk aversion (i.e., when the DM has either the risk neutral linear or the concave-exponential utility function). Otherwise, as the information bill |psi~ differs, so does ||omega~.sub.0~ - |psi~, which in turn changes the DM's absolute risk aversion and thereby the relative desirability of the alternative actions available for choice. Thus, the payment of $|psi~ has both an indirect and a direct impact upon the DM's well-being: indirectly by affecting the optimal action chosen, and directly by reducing all potential terminal wealths by the amount of the bill.

The optimal posterior prospect is denoted |Mathematical Expression Omitted~. Here the subscript y does double duty, indicating that the prospect is induced both by the choice of action |Mathematical Expression Omitted~ and by the conditional measure |Mathematical Expression Omitted~ on the state space. The conditional posterior reservation price is the solution for |Mathematical Expression Omitted~ in the equation

|Mathematical Expression Omitted~

and the mean of the optimal conditional prospect is

|Mathematical Expression Omitted~

For each y |element of~ Y and any |psi~ |is greater than or equal to~ 0 the DM determines the optimal response |Mathematical Expression Omitted~. However, the DM in deciding whether to purchase and utilize the information system cannot know beforehand the actual y; the received message is itself a random variable, distributed according to p(y). The preposterior expected utility from the best use of the information system is the expectation of (14) with respect to p(y):

|Mathematical Expression Omitted~

The collection of optimal decision rules writes succinctly as

|Mathematical Expression Omitted~

Each posterior prospect |Mathematical Expression Omitted~ has probability p(y) of applying; the preposterior optimal informed prospect from the system I is a mixture distribution

|Mathematical Expression Omitted~

For example, when Y is finite and there are n distributions |Mathematical Expression Omitted~ and another distribution p(|y.sub.j~), then |Mathematical Expression Omitted~ is a distribution over W that assigns probability

|Mathematical Expression Omitted~

to terminal wealth W |element of~ W.

The expected utility of the optimal informed prospect derives by taking the expectation of the right hand side of (16) with respect to p(y) and using (20):

|Mathematical Expression Omitted~

The reservation price of the informed decision with information cost |psi~, |Mathematical Expression Omitted~, is defined from (22) as

|Mathematical Expression Omitted~

Using the results of (22) and then the definition (17), the overall preposterior mean

|Mathematical Expression Omitted~

is simply the mean of the conditional means. Jensen's inequality ensures

|Mathematical Expression Omitted~

with equality holding under risk neutrality.

In addition, under risk neutrality any reduction in initial wealth does not change the DM's choices, so the optimal decision rule in the risk neutral version of (19) is the same as if |psi~ = 0: |Mathematical Expression Omitted~. Hence,

|Mathematical Expression Omitted~

in other words, the preposterior expected wealth shifts down $|psi~ from what it would have been if the information system were free. Finally, analogous to (10), the suboptimality of |Mathematical Expression Omitted~ for any risk averter ensures

|Mathematical Expression Omitted~

Combining (25), (26), and (27) gives the useful summarizing result

|Mathematical Expression Omitted~

The demand price for information is defined as the maximum non-stochastic cost that makes the DM indifferent between owning the system or not. Hence, the value of information is the solution for |psi~ = V that equates the optimal expected utilities in (18) and (3):

|Mathematical Expression Omitted~

Equivalently in terms of prospects,

|Mathematical Expression Omitted~

or, using (23) and (5),

|Mathematical Expression Omitted~

which implies

|Mathematical Expression Omitted~

The existence, uniqueness, and non-negativity of V follows from the assumptions that u is continuous and strictly increasing; chapter 2 of Marschak and Radner |9~ provides a formal proof.

The situation is simplified under risk neutrality because again, for every y, |Mathematical Expression Omitted~. In the risk neutral version of (29b), the solution for |V.sup.N~ in the equation

|Mathematical Expression Omitted~

is, by rearranging and using the definitions of expected wealth,

|Mathematical Expression Omitted~

Equation (32) states that the value of information to a risk neutral DM is simply the before cost difference in expected wealth with and without the information. This assessment requires only one calculation of the optimal decision rule, directly from the payoff function itself.

The assessment of |V.sup.A~, the value of information to a risk averter, requires the specification of the DM's utility function. The purpose of the next section is to investigate relationships between |V.sup.A~ and |V.sup.N~ that utilize less than complete knowledge of the risk averter's specific preferences.

III. Prospect Variability and Comparative Information Value

When comparing |V.sup.A~ to |V.sup.N~, intuition might lead us to hypothesize that risk averters would seek information as a means of risk reduction, and hence tend to value it more than their risk neutral counterparts. This is not necessarily so, as Theorem 1 and the subsequent discussion indicate. LaValle |7, 396~ presents the following upper bound on the value of information to a risk averter:

THEOREM 1. |V.sup.A~ |is less than or equal to~ |V.sup.N~ + ||R.sup.N~.sub.0~ - ||R.sup.A~.sub.0~.

Proof. Use the summarizing equation (28), evaluated at |psi~ = |V.sup.A~ so that (30) also holds:

|Mathematical Expression Omitted~

Rearranging, using the definition of |V.sup.N~ given by (32), and then using the equality of ||R.sup.N~.sub.0~ and |Mathematical Expression Omitted~ shows

|Mathematical Expression Omitted~

From (12), |R.sup.N~.sub.0~ - |R.sup.A~.sub.0~ |is greater than or equal to~ 0, so there is no general sign relationship between |V.sup.A~ and |V.sup.N~ without further conditions on the problem. This bound is nevertheless convenient because the only input needed from the risk averse DM is the prior reservation price; no conditional decision rules need be determined. Ziemba and Butterworth |13~ also present some related bounds useful when the payoff function is concave.

The reason there is no general relationship is that although the use of the information system increases the expected wealth and expected utility for all decision makers, it does not necessarily leave unchanged the riskiness of the preposterior optimal informed prospect compared to the optimal prior prospect. To the risk neutral DM any change in riskiness is inconsequential, but risk averters view altered riskiness as a significant factor that affects their overall valuation of the information.

Example 1. One situation in which Theorem 1 guarantees that |V.sup.A~ |is less than or equal to~ |V.sup.N~ is when there is a risk free action in a that is the optimal prior choice for the risk neutral DM. Consider the decision to enter a new market, with demand in that market the unknown state variable. If the market is entered, terminal wealth depends upon realized demand. If the market is not entered, terminal wealth is fixed at the current level. Suppose the prior probability measure on demand is such that it is optimal for the risk neutral DM not to enter the market. In this case all risk averters would also choose the risk free action, the reservation prices ||R.sup.N~.sub.0~ and ||R.sup.A~.sub.0~ would be the same, and from Theorem 1 no risk averter would value any information system by more than the risk neutral DM.

Economically, even though the prior prospects |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ are identical with |Mathematical Expression Omitted~ and zero riskiness, if the information system is not useless the prospects |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ do have some variability. Comparing these two preposterior prospects, the inequality (27) in the cost-free case ensures |Mathematical Expression Omitted~. (Recall that when |psi~ = 0, the notation drops the |psi~). Regardless of how the risk neutral DM values the information, the risk averter values it by less for two reasons: 1) since |Mathematical Expression Omitted~, the increase in the risk averter's expected wealth |Mathematical Expression Omitted~ cannot be greater than |Mathematical Expression Omitted~, and 2) the risk averter's preposterior prospect is more risky than the degenerate prior, and this is detrimental to overall valuation. In this example the bound |V.sup.A~ |is less than or equal to~ |V.sup.n~ is true not only in the special case when |Mathematical Expression Omitted~, as occurs under perfect information, but a fortiori, also when |Mathematical Expression Omitted~, as may occur under imperfect information.

This discussion suggests the possibility of signing the difference |V.sup.A~ - |V.sup.N~ for a more general case in which the prior prospects are different and nondegenerate. Blair and Romano |3~ show that direct investigation of the change in prospect variability as the DM moves from the uninformed to the informed situations is indeed a fruitful approach.

The formal criterion for comparing the variability of prospects is in Rothschild and Stiglitz |11~. Given two prospects with the same mean, Prospect 1 is Rothschild-Stiglitz more variable (or more risky) than Prospect 2 if and only if the expected utility from Prospect 2 is at least as great as from Prospect 1 for any strictly concave utility function defined on W. Rothschild and Stiglitz show that this partial ordering is equivalent to forming Prospect 1 via a mean preserving spread of Prospect 2.

As an illustration of the Rothschild-Stiglitz approach, consider a variability comparison of the prior decisions. Since |Mathematical Expression Omitted~, yet the DM prefers |Mathematical Expression Omitted~, adjusting the two prospects in (9) to have the same mean shows that

|Mathematical Expression Omitted~

Equation (35) states that every risk averse DM prefers |Mathematical Expression Omitted~ to |Mathematical Expression Omitted~ when the mean of the former prospect is adjusted to |Mathematical Expression Omitted~. The conclusion is that |Mathematical Expression Omitted~ cannot be more Rothschild-Stiglitz variable than |Mathematical Expression Omitted~. The optimality of ||a.sup.A~.sub.0~ reveals that the risk averse DM prefers to accept a reduction in expected wealth by |Mathematical Expression Omitted~ in exchange for less variability.

Blair and Romano compare the variability of the cost-free preposterior prospect of the risk averter with the prior prospect chosen under risk neutrality. In the current notation, their result is:

THEOREM 2. If |Mathematical Expression Omitted~ is Rothschild-Stiglitz more variable than |Mathematical Expression Omitted~, then |Mathematical Expression Omitted~.

The proof of this theorem need not be repeated, but satisfaction of the sufficiency condition ensures |V.sup.A~ |is less than or equal to~ |V.sup.N~ because |Mathematical Expression Omitted~.

Theorem 2 is most easily applied to the perfect information system, denoted |I.sup.*~. Under perfect information the state is precisely identified; there is no risk posterior to the receipt of any message. The prospect |Mathematical Expression Omitted~ is degenerate with all its mass at the same wealth level for any u, hence |Mathematical Expression Omitted~. The application of (20) shows the two preposterior prospects are identical: |Mathematical Expression Omitted~. Suppose the two DMs have different initial prospects |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ and the identical perfectly informed prospect. Since |Mathematical Expression Omitted~, the risk averter is guaranteed to get at least as great an increase in expected wealth as the risk neutral counterpart:

|Mathematical Expression Omitted~

Under the premise of Theorem 2, the perfectly informed prospect |Mathematical Expression Omitted~ is more variable than |Mathematical Expression Omitted~. Hence by transitivity from the result (35), |Mathematical Expression Omitted~ is more variable than |Mathematical Expression Omitted~. In valuing the information the risk neutral DM does not care about this increased risk, but the risk averter's relative valuation involves comparing the greater increase in the mean with the greater increase in the variability. The content of Theorem 2 is that the relative gain in the mean cannot be enough to compensate for the relative increase in the risk averter's prospect variability.

Example 2. To illustrate, consider a simple problem with two states, X = {|x.sub.1~, |x.sub.2~}, two actions a = {|a.sub.1~, |a.sub.2~}, initial wealth ||omega~.sub.0~ = 2, and the outcome in matrix form as

|Mathematical Expression Omitted~

Suppose p(|x.sub.1~) = p(|x.sub.2~) = .5. Then under risk neutrality the optimal prior decision is to choose |a.sub.2~, the prospect |Mathematical Expression Omitted~ is a 50-50 chance of being worth 6 or 4, and hence |Mathematical Expression Omitted~. Given access to cost-free perfect information, DMs of all risk preferences choose |a.sub.1~ if |x.sub.1~, and |a.sub.2~ if |x.sub.2~. Thus all DMs have a 50-50 chance of being worth 8 or 4, |Mathematical Expression Omitted~, and from (32), the value of perfect information under risk neutrality is |V.sup.N~ = 1. No risk averse DM would value the perfect information more because the conditions of Theorem 2 are clearly met and the increased variability of the preposterior informed prospect is detrimental to the risk averter's valuation.

As pointed out by Blair and Romano |3, 394~, testing for the satisfaction of the sufficiency condition of Theorem 2 in the case of imperfect information requires the assessment of a specific risk averse utility function and the corresponding optimal decision rule. Note that under imperfect information and having assessed that the specific |Mathematical Expression Omitted~ is more variable than |Mathematical Expression Omitted~, Theorem 2 states that the bound on |V.sup.A~ becomes tighter by the amount |Mathematical Expression Omitted~, i.e., the expected wealth the risk averter willingly gives up in the informed situation.

The third theorem concerns the relative values |V.sup.A~ and |V.sup.N~ for decision problems under imperfect information in which both the prior and the preposterior prospects can have different means and variabilities, but where |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ need not be assessed. In this case both |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~, so the increase in the risk averter's expected wealth, |Mathematical Expression Omitted~, may or may not be greater than |V.sup.N~.

Definitive results derive from studying the mean of the risk neutral conditional prospect |Mathematical Expression Omitted~. Defined in the risk neutral version of (17), the quantity |Mathematical Expression Omitted~ is the expected wealth of the risk neutral DM who has paid $|psi~ for an information system and has received and responded optimally to the specific message y. Prior to the receipt of y, this quantity is a random variable, distributed according to the marginal distribution p(y). Since each |Mathematical Expression Omitted~ is expressed in the monetary unit, |Mathematical Expression Omitted~ and there is no conceptual problem in viewing 'this random variable as having a probability distribution on W and hence as a prospect. This prospect, the prior distribution of the posterior mean wealth of an optimizing risk neutral DM, is denoted |Mathematical Expression Omitted~. Keep in mind that since the risk neutral DM exhibits constant absolute risk aversion and simply maximizes expected wealth conditional on y, the magnitude of |psi~ affects only the mean of this prospect, not its variability. The mean is |Mathematical Expression Omitted~ as defined in (24), and, in light of (26),

|Mathematical Expression Omitted~

With the notable exception of perfect information, this prospect is hypothetical in the sense that is not actually available to any DM, regardless of attitude towards risk. It does, however, give the risk neutral DM the same expected wealth (and hence expected utility) as the actual preposterior informed prospect. The advantage of focusing on the prospect |Mathematical Expression Omitted~ is that it requires only risk neutral assessments yet plays the major role relating |V.sup.A~ to |V.sup.N~ in Theorem 3.

The analysis begins by showing that any risk averse DM would prefer, if they both were equally costly, to obtain the hypothetical prospect over the actual optimal prospect chosen. This is formalized in the following lemma:

LEMMA 1. |Mathematical Expression Omitted~.

Proof. Since the risk neutral DM maximizes expected wealth conditional on y, the strict increasingness of the utility function ensures that any risk averse decision maker would prefer to receive the larger quantity |Mathematical Expression Omitted~ over |Mathematical Expression Omitted~:

|Mathematical Expression Omitted~

Using the definition of |Mathematical Expression Omitted~ in (17), the application of Jensen's inequality to the right hand side of (38) yields

|Mathematical Expression Omitted~

Taking the expectation of (39) with respect to p(y) and applying (22) shows definitive preference for the hypothetical prospect:

|Mathematical Expression Omitted~

Finally, since under risk neutrality no decision rule depends upon the monetary payment |psi~,

|Mathematical Expression Omitted~

Consider now a variability comparison of the prior optimal prospect |Mathematical Expression Omitted~ with the hypothetical prospect evaluated at |phi~ = |V.sup.N~. Both prospects naturally have the same mean |Mathematical Expression Omitted~: using (37), if the risk neutral DM pays the maximum amount for the information system, |V.sup.N~ defined in (32), the resulting prospect has the mean |Mathematical Expression Omitted~. The basic result is the following theorem.

THEOREM 3. The condition that |Mathematical Expression Omitted~ is Rothschild-Stiglitz more variable than |Mathematical Expression Omitted~ is sufficient but not necessary for |V.sup.A~ |is less than or equal to~ |V.sup.N~.

Sufficiency. If |Mathematical Expression Omitted~ is more variable than |Mathematical Expression Omitted~, then |V.sup.A~ |is less than or equal to~ |V.sup.N~. The proof begins by evaluating the lemma at |psi~ = |V.sup.A~, invoking the definition of |V.sup.A~ given in (29b), and using the inequality (9):

|Mathematical Expression Omitted~

Adding and subtracting the constant |V.sup.N~ to the left hand side yields

|Mathematical Expression Omitted~

The premise of the theorem, that |Mathematical Expression Omitted~ is more variable than |Mathematical Expression Omitted~, holds if and only if for any continuous, increasing, bounded and strictly concave utility function,

|Mathematical Expression Omitted~

For (44) to hold, it must be true that |V.sup.A~ |is less than or equal to~ |V.sup.N~, otherwise (43) is contradicted.

If the risk averter could actually have the hypothetical prospect, the relative valuation is |V.sup.A~ |is less than or equal to~ |V.sup.N~ for exactly the same reason as in Theorem 2. Since any attainable prospect |Mathematical Expression Omitted~ is "worse" in the risk averter's mind, the actual |V.sup.A~ must be even less. Note finally that under perfect information, the sufficiency part of Theorem 3 leads to the same comparison as in Theorem 2.

Nonnecessity. Necessity would mean that |V.sup.A~ |is less than or equal to~ |V.sup.N~ for all strictly concave utility functions implies that |Mathematical Expression Omitted~ is more variable than |Mathematical Expression Omitted~. Example 3 in Section IV proves nonnecessity, with a case of imperfect information in which |V.sup.A~ |is less than or equal to~ |V.sup.N~ but |Mathematical Expression Omitted~ is a mean preserving spread of |Mathematical Expression Omitted~.

The contrapositive of the sufficiency condition of Theorem 3 is of some economic interest, stating a condition that must hold under an assumption that all risk averters value the information more than the risk neutral benchmark. The contrapositive is that if |V.sup.A~ |is greater than or equal to~ |V.sup.N~ for all strictly concave utility functions, then |Mathematical Expression Omitted~ is Rothschild-Stiglitz more variable than |Mathematical Expression Omitted~.

IV. Information Quality and Information Value

This section introduces a simple systematic model of imperfect information for the purpose of illustrating the relations between the theorems. Specifically, the analysis shows that the statistical quality of the information system is a factor in the achievement of the sufficiency condition in Theorem 3, and illustrates by example that necessity does not hold.

In the simple two state model with X = {|x.sub.1~, |x.sub.2~}, consider a sequence of information systems each with a message space Y = {|y.sub.1~, |y.sub.2~} in which each message is a categorical yet imperfectly accurate statement of the form |y.sub.j~ = "state |x.sub.j~ obtains," j = 1, 2. Given the unconditional prior probabilities p(|x.sub.1~) and p(|x.sub.2~), the information quality, or statistical informativeness, of each system is modeled using a parameter |Mathematical Expression Omitted~, that determines the conditional posterior probabilities as

|Mathematical Expression Omitted~

If |Mathematical Expression Omitted~, then |Mathematical Expression Omitted~, the messages are always correct and the information is perfect. At the other extreme, when |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~. Since neither message has any effect on the probabilities of the states, Marschak |8~ terms this case null information. As |Mathematical Expression Omitted~ rises from 0 to 1, the probabilities on the main diagonal of (45) increase; the forecast is more likely to be correct. A convenient property of this model (easily verified by applying the identity |Mathematical Expression Omitted~ to (45)) is that, for any |Mathematical Expression Omitted~, the unconditional probability the system produces message |y.sub.j~ is equal to the unconditional probability of state |x.sub.j~ obtaining: |Mathematical Expression Omitted~.

Example 2 (continued). Combining the outcome matrix given by (36) with the sequence of information systems modelled by (45) illustrates that even if the sufficiency condition holds under perfect information it does not necessarily hold under imperfect information.

Given now that |Mathematical Expression Omitted~, under risk neutrality the optimal prior decision is to choose |a.sub.2~, the prospect |Mathematical Expression Omitted~ is a 50-50 chance of being worth 6 or 4. As before,

|Mathematical Expression Omitted~

Using the information system model (45), conditional on message |y.sub.1~,

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

so the optimal decision rule |Mathematical Expression Omitted~ depends upon |Mathematical Expression Omitted~. Only when |Mathematical Expression Omitted~ is this message informative enough to induce the DM to change the prior optimal action from |a.sub.2~ to |a.sub.1~. On the other hand, given message |y.sub.2~, the optimal action always remains |a.sub.2~, leading to expected wealth |Mathematical Expression Omitted~. Thus,

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

Taking the expectation with respect to p(y), the overall expected wealth is

|Mathematical Expression Omitted~

Figure 1 graphs the optimal prospects under the cost free use of an information system with |Mathematical Expression Omitted~. The value of information is the difference between (51) and (46):

|Mathematical Expression Omitted~

The value of information is non-linear in |Mathematical Expression Omitted~; the system must achieve a certain amount of statistical informativeness before it can alter actions and hence payoffs. Marschak |8~ uses the term useless information to describe the problem-specific situation in which a system is statistically informative but generates no economic value.

Subtracting |V.sup.N~ in (52) from (49) and (50) identifies the prior distribution of the posterior mean wealth, the prospect |Mathematical Expression Omitted~. Figure 2 graphs this prospect as a function of the parameter |Mathematical Expression Omitted~. As |Mathematical Expressions Omitted~ gets larger the distribution |Mathematical Expression Omitted~ spreads out from being degenerate at null information (with all mass at the prior optimal mean |Mathematical Expression Omitted~) to being most variable at perfect information (the non-hypothetical prospect offering even odds the DM winds up being worth 7 or 3, having paid |V.sup.N~ = 1 for perfect information). It is interesting to compare this spreading with the behavior of the prior distribution of the posterior mean that arises in statistical decision theory |10, 104-109~.

Figure 2 also graphs the optimal prior prospect |Mathematical Expression Omitted~. As the figure shows, the spreading prospect |Mathematical Expression Omitted~ reaches the point of being a mean preserving spread of |Mathematical Expression Omitted~ at |Mathematical Expression Omitted~. Thus Theorem 3 assures that for any information system with informativeness parameter |Mathematical Expression Omitted~, |V.sup.A~ |is less than or equal to~ |V.sup.N~ for every risk averse decision maker. Notice also that under perfect information, the comparison in Theorem 3 is the same as in Theorem 2 (the B-R comparison described in Section III, after adjusting the prospect |Mathematical Expression Omitted~ to the same mean as |Mathematical Expression Omitted~.

Example 3. This example illustrates that the necessity part of Theorem 3 does not hold: the prior prospect |Mathematical Expression Omitted~ can be more variable than |Mathematical Expression Omitted~, yet |V.sup.A~ |is less than or equal to~. If the unconditional probabilities in example 2 change to p(|x.sub.1~) = p(|y.sub.1~) = .75 and p(|x.sub.2~) = p(|y.sub.2~) = .25, then the risk neutral DM's prior choice is the action |a.sub.1~ with its relatively wide ranging potential outcomes. The same analysis as in example 2 shows that the prior prospect is the more variable for every |Mathematical Expression Omitted~, |Mathematical Expression Omitted~, and that

|Mathematical Expression Omitted~

Let us suppose a risk averter has the specific utility function |Mathematical Expression Omitted~. Then for this DM, ||a.sup.A~.sub.0~ = |a.sub.2~ and

|Mathematical Expression Omitted~

The prior reservation price solves |Mathematical Expression Omitted~ and is ||R.sup.A~.sub.0 = 5.4174. Hence for this specific utility function the bound given by Theorem 1,

|Mathematical Expression Omitted~

holds for any information system.

Using the information system model (45), the cost-free informed expected utility is

|Mathematical Expression Omitted~

When |Mathematical Expression Omitted~ the application of the definition (29a) gives the value of information as the solution for |V.sup.A~ in the equation

|Mathematical Expression Omitted~

which solves to

|Mathematical Expression Omitted~

Figure 3 graphs both |V.sup.A~ and |V.sup.N~ as they depend upon |Mathematical Expression Omitted~. The equality of (53) and (58) occurs at |Mathematical Expression Omitted~; when |Mathematical Expression Omitted~, |V.sup.A~ |is less than or equal to~ |V.sup.N~ despite the fact that |Mathematical Expression Omitted~ is more variable than |Mathematical Expression Omitted~.

V. Summarizing Remarks

When comparing the choices made and the prospects accepted by risk averse decision makers with their risk neutral counterparts, the definitive relationships both in the uninformed (equation 12) and the informed (equation 28) situations do not necessarily lead to definitive statements signing the difference |V.sup.A~ - |V.sup.N~. The key is that although the utilization of cost-free, non-useless information increases the expected utility for all DMs, it does not necessarily leave the riskiness of the optimal prospects unchanged; this aspect is relevant only to the risk averter's valuation.

The desire to avoid the need to assess a specific utility function motivates the search for conditions involving the sign of |V.sup.A~ - |V.sup.N~ that use only risk neutral assessments. Section III studies the subject in three steps: 1) the situation with identical risk-free prior prospects and possibly different preposterior informed prospects, 2) the situation with possibly different prior prospects but identical perfectly informed preposterior prospects, and 3) the most general situation in which both the prior and the preposterior prospects may be different. The primary result for the third case is that for |V.sup.A~ |is less than or equal to~ |V.sup.N~, it is sufficient that the prior distribution of the posterior risk neutral mean be Rothschild-Stiglitz more variable than the prior risk neutral optimal prospect (Theorem 3).

The examples in section IV indicate the statistical informativeness of the system may play a role in achieving the conditions of the theorems. Of course, Rothschild-Stiglitz variability is only a partial ordering; the two prospects are not necessarily comparable and when they are not, the relationship between |V.sup.A~ and |V.sup.N~ may depend upon a more complex interaction between the problem structure, the statistical informativeness of the information, and the specific utility function. Further research on this topic could investigate more rigorously the theoretical and practical connections between the statistical informativeness of the information system and the comparative value of information.

References

1. Blackwell, David, "Equivalent Comparisons of Experiments." Annals of Mathematical Statistics, June 1953, 265-73.

2. ----- and M. A. Girshick. Theory of Games and Statistical Decisions. New York: John Wiley, 1954.

3. Blair, Roger D. and Richard E. Romano, "The Influence of Attitudes Toward Risk on the Value of Forecasting." Quarterly Journal of Economics, May 1988, 387-96.

4. Gould, John P., "Risk, Stochastic Preference, and the Value of Information." Journal of Economic Theory, May 1974, 64-84.

5. Hilton, Ronald W., "The Determinants of Information Value: Synthesizing Some General Results." Management Science, January 1981, 57-64.

6. LaValle, Irving H., "On Cash Equivalents and Information Evaluation Under Uncertainty." Journal of the American Statistical Association, March 1968, 252-90.

7. -----. Fundamentals of Decision Analysis. New York: Holt, Rinehart and Winston, 1978.

8. Marschak, Jacob, "Economics of Information Systems." Journal of the American Statistical Association, March 1971, 192-219.

9. Marschak, Jacob and Roy Radner. The Economic Theory of Teams. New Haven: Yale Press, 1972.

10. Raiffa, Howard, and Robert Schlaifer. Applied Statistical Decision Theory. Boston: Division of Research, The Harvard School of Business Administration, 1961.

11. Rothschild, Michael and Joseph E. Stiglitz, "Increasing Risk: I. A Definition." Journal of Economic Theory, September 1970, 225-43.

12. Savage, I. Richard et al. Setting Statistical Priorities. Report of the Panel on Methodology for Statistical Priorities. Washington, D.C.: National Academy of Sciences, 1976.

13. Ziemba, William T. and J. E. Butterworth, "Bounds on the Value of Information in Uncertain Decision Problems." Stochastics, 1975, 361-78.

Choice in decision making under uncertainty often includes the option of choosing to utilize an information system, a set of potential messages that may improve current decisions and resulting outcomes. In decision theory the demand price for an information system is defined as the maximum non-stochastic cost, payable from initial wealth prior to the receipt of any message, that makes the decision maker (DM) indifferent between purchasing the system or not. The determinants of the value of information are important to both the buyers and the sellers in the marketplace for information products, and economists study the subject with both theoretical and practical motivations.

Theoretical analyses seek to discover the characteristics of the economic and/or statistical environment that allow for definitive qualitative statements about the value of information. The most famous result, Blackwell's Theorem |1; 2~, states the necessary and sufficient conditions under which any potential user values one information system more than another. Unfortunately, most results are of the negative variety: there is no general monotonic relationship between the information value and the degree of aversion towards risk |5~, the amount of statistical information transmitted |8~, the level of initial wealth |6~, or the Rothschild-Stiglitz variability of the prior distribution on the state |4~.

In practical situations, the buyer or the producer of an information system may desire to utilize quantitative estimates of the value of information. Complete quantitative analysis requires knowledge of the statistical characteristics of the information, identification of the payoff function for the decision problem at hand, and assessment of the information user's utility function for wealth. This last factor, the utility function, is specific to each individual DM and is unlikely to be known by the producer of the information. Unless it can be assumed that all potential buyers are neutral towards risk and hence have an easily assessed linear utility function, the seller of the information product (e.g., a forecasting service with many clients) faces the problem of estimating demand without knowledge of the demanders' utility functions. Policy makers, seeking to assess the value of publicly collected and disseminated information, face a similar problem |12~.

Blair and Romano |3~, henceforth B-R, recently have introduced a new problem-specific approach to information valuation that has both theoretical and practical importance. The B-R idea is to use the Rothschild-Stiglitz variability of the DM's alternative prospects (i.e., probability distributions on terminal wealth) as a criterion for qualitative results concerning the role of the DM's attitude towards risk in determining the demand price for information. Specifically, B-R present a criterion sufficient to ensure that no risk averter values information by more than the benchmark risk neutral DM. The B-R analysis is most useful for the comparative valuation of perfect information (the case of unequivocal identification of the state), since the optimal informed prospects are then identical for all DMs regardless of attitude towards risk. In this case there is no need to assume any specifics about the utility function of the risk averter.

This paper investigates the value of imperfect information as it depends upon the DM's attitude towards risk. When the information system is not perfect, testing for the satisfaction of the B-R criterion requires knowledge of each potential user's specific non-linear utility function and the corresponding optimal decision rules. The primary result of this paper is an extension of the B-R approach that provides definitive conditions, requiring only risk neutral assessments and computations, under which no risk averse DM values imperfect information more than the risk neutral counterpart. On the theoretical level, the analysis intends to contribute to the understanding of attitude towards risk as a determinant of the value of imperfect information. In practice, satisfaction of the condition may be useful to the producer of the information, as it sets an upper bound on the demand price of any risk averse potential user facing the given decision problem.

Section II reviews the theory of the value of information, comparing the prospects chosen by risk neutral and risk averse DMs. When valuing information, the risk neutral DM cares only about the impact on expected wealth that results from the use of the information. The risk averter cares also about changes in prospect variability, that is, changes in riskiness in the sense of Rothschild and Stiglitz |11~. Accordingly, the comparison of the value of imperfect information between risk averse and risk neutral DMs depends critically on the comparative impacts the use of the information has both on expected wealth and on prospect variability. Section III presents three theorems that study the relative value of information in a systematic way. Theorem 3 presents definitive results by comparing the Rothschild-Stiglitz variability of two relatively easy to assess prospects: the risk neutral prior optimal prospect and the risk neutral prior distribution of the posterior mean wealth, which is here a random variable since it is conditioned on the realization of a random information signal. Theorem 3 contains a sufficiency condition, verifiable using a standard test for second order stochastic dominance between the two risk neutral prospects, under which no risk averter values information more than the benchmark risk neutral DM. Section IV presents a simple parameterized model of imperfect information that indicates the role statistical informativeness plays in achieving the sufficiency condition of Theorem 3. Section V contains concluding remarks.

II. Prospects and the Value of Information

This paper concerns the demand price for imperfect information in decision problems under uncertainty with the following characteristics, definitions, assumptions, and notation:

1) A set a of available actions a; the DM chooses a |element of~ a.

2) A set X of mutually exclusive state descriptions x, one x |element of~ X ultimately obtains according to a known unconditional (prior) probability measure on X, denoted p (x).

3) The actions and states may be vectors, and the monetary payoff of the endeavor when state x obtains and action a is chosen is quantified in a real-valued payoff function |omega~(x, a) defined on X x a.

4) A set W of potential terminal wealths that may result from the decision problem; the outcome of the decision problem is that fixed and known initial wealth ||omega~.sub.0~ changes to terminal wealth W according to

W = ||omega~.sub.0~ + |omega~(x, a). (1)

It is assumed that all actions in a remain feasible regardless of the level of wealth.

5) A Von Neumann-Morgenstern utility function on W such that u (W) is continuous, strictly increasing, and weakly concave in W.

6) The option of utilizing an information system I comprised of:

a) a set Y of potential signals or messages that can emanate from the system; one y |element of~ Y is received by the DM prior to the choice of action, and

b) a joint probability measure p(x,y) on X x Y, with the unconditional measures p(x) and p(y) being proper and strictly greater than zero.

7) The DM uses as the basis for choice the maximization of expected utility, with all required expectations finite and optimal actions unique.

The methodology of this paper focuses on the evaluation and comparison of prospects--probability distributions over terminal wealth determined by the choice of action. For each a |element of~ a, the payoff function (1) defines a function from X to W that serves as the basis for a change of variables. When the DM chooses an action a |element of~ a (or "accepts the lottery" a), this induces ownership of a prospect denoted |Mathematical Expression Omitted~. The advantage of working with prospects is being able to deal directly with univariate probability measures over cash amounts.

In the uninformed situation, without the utilization of an information system, the DM chooses the action that is optimal under the prior measure p(x). The DM solves the problem

|Mathematical Expression Omitted~

where the subscript on the expectations operator indicates the random variable with respect to which the expectation is taken. In general, this choice of optimal prior action depends upon the DM's specific utility function. Denote the prior optimal action given any concave utility function u as ||a.sup.u~.sub.0~, defined by

|Mathematical Expression Omitted~

In the uninformed decision, the choice of action ||a.sup.u~.sub.0~ gives the optimizing DM the prospect |Mathematical Expression Omitted~, where it is implicit in the notation that this prospect is induced both by the choice of optimal action ||a.sup.u~.sub.0~ and the original probability measure p(x). The expected utility from owning this prospect is written

|Mathematical Expression Omitted~

and is equal to the amount stated in (3).

Especially important are two summarizing cash amounts associated with a prospect: the reservation price and the mean wealth. The reservation price, or certainty equivalent, of the prior decision is the cash equivalent terminal wealth that the DM would accept as a sale price to walk away from all risk inherent in the optimal solution to the decision problem. The prior reservation price ||R.sup.u~.sub.0~ is defined implicitly as the solution to

|Mathematical Expression Omitted~

The mean of the optimal prior prospect

|Mathematical Expression Omitted~

is the expected terminal wealth from the prior decision. As is well known (and can be shown by applying Jensen's inequality to (5)), the concavity of the utility function ensures that

|Mathematical Expression Omitted~

The results of this paper derive from comparing the prospects chosen by any risk averse DM with those chosen by or associated with the risk neutral benchmark DM having a utility function linear in W. When it is important to distinguish, the notation replaces the superscript u with A to indicate any risk averter, and N to identify the risk neutral counterpart. For the risk neutral DM, (7) holds with equality and the prior reservation price is the mean of the optimal prior prospect:

|Mathematical Expression Omitted~

Useful inequalities for proving the subsequent theorems arise by comparing optimal prospects with specific suboptimal ones. Denoting the prior optimal actions under risk aversion and risk neutrality by ||a.sup.A~.sub.0~ and ||a.sup.N~.sub.0~ respectively, the optimality of ||a.sup.A~.sub.0~ guarantees that

|Mathematical Expression Omitted~

Similarly, with ||a.sup.N~.sub.0~ being optimal under risk neutrality,

|Mathematical Expression Omitted~

or, using (6),

|Mathematical Expression Omitted~

Combining (11) with (8) and (7) obtains the summarizing relationship between the uninformed reservation prices and means:

|Mathematical Expression Omitted~

Assume now that the DM opts to utilize the information system I, paying $|psi~

|is greater than or equal to~ 0 from initial wealth, receiving message y |element of~ Y, and viewing the relevant probability measure on X as the conditional (posterior) distribution |Mathematical Expression Omitted~. If the DM chooses action based upon initial wealth being reduced by $|psi~, then the optimal action given y is denoted

|Mathematical Expression Omitted~

as defined by

|Mathematical Expression Omitted~

As a matter of notation, if the information system were free the subscript |psi~ in statements like (13) is omitted. Except in special cases,

|Mathematical Expression Omitted~

The primary exception is when the DM's preferences exhibit constant absolute risk aversion (i.e., when the DM has either the risk neutral linear or the concave-exponential utility function). Otherwise, as the information bill |psi~ differs, so does ||omega~.sub.0~ - |psi~, which in turn changes the DM's absolute risk aversion and thereby the relative desirability of the alternative actions available for choice. Thus, the payment of $|psi~ has both an indirect and a direct impact upon the DM's well-being: indirectly by affecting the optimal action chosen, and directly by reducing all potential terminal wealths by the amount of the bill.

The optimal posterior prospect is denoted |Mathematical Expression Omitted~. Here the subscript y does double duty, indicating that the prospect is induced both by the choice of action |Mathematical Expression Omitted~ and by the conditional measure |Mathematical Expression Omitted~ on the state space. The conditional posterior reservation price is the solution for |Mathematical Expression Omitted~ in the equation

|Mathematical Expression Omitted~

and the mean of the optimal conditional prospect is

|Mathematical Expression Omitted~

For each y |element of~ Y and any |psi~ |is greater than or equal to~ 0 the DM determines the optimal response |Mathematical Expression Omitted~. However, the DM in deciding whether to purchase and utilize the information system cannot know beforehand the actual y; the received message is itself a random variable, distributed according to p(y). The preposterior expected utility from the best use of the information system is the expectation of (14) with respect to p(y):

|Mathematical Expression Omitted~

The collection of optimal decision rules writes succinctly as

|Mathematical Expression Omitted~

Each posterior prospect |Mathematical Expression Omitted~ has probability p(y) of applying; the preposterior optimal informed prospect from the system I is a mixture distribution

|Mathematical Expression Omitted~

For example, when Y is finite and there are n distributions |Mathematical Expression Omitted~ and another distribution p(|y.sub.j~), then |Mathematical Expression Omitted~ is a distribution over W that assigns probability

|Mathematical Expression Omitted~

to terminal wealth W |element of~ W.

The expected utility of the optimal informed prospect derives by taking the expectation of the right hand side of (16) with respect to p(y) and using (20):

|Mathematical Expression Omitted~

The reservation price of the informed decision with information cost |psi~, |Mathematical Expression Omitted~, is defined from (22) as

|Mathematical Expression Omitted~

Using the results of (22) and then the definition (17), the overall preposterior mean

|Mathematical Expression Omitted~

is simply the mean of the conditional means. Jensen's inequality ensures

|Mathematical Expression Omitted~

with equality holding under risk neutrality.

In addition, under risk neutrality any reduction in initial wealth does not change the DM's choices, so the optimal decision rule in the risk neutral version of (19) is the same as if |psi~ = 0: |Mathematical Expression Omitted~. Hence,

|Mathematical Expression Omitted~

in other words, the preposterior expected wealth shifts down $|psi~ from what it would have been if the information system were free. Finally, analogous to (10), the suboptimality of |Mathematical Expression Omitted~ for any risk averter ensures

|Mathematical Expression Omitted~

Combining (25), (26), and (27) gives the useful summarizing result

|Mathematical Expression Omitted~

The demand price for information is defined as the maximum non-stochastic cost that makes the DM indifferent between owning the system or not. Hence, the value of information is the solution for |psi~ = V that equates the optimal expected utilities in (18) and (3):

|Mathematical Expression Omitted~

Equivalently in terms of prospects,

|Mathematical Expression Omitted~

or, using (23) and (5),

|Mathematical Expression Omitted~

which implies

|Mathematical Expression Omitted~

The existence, uniqueness, and non-negativity of V follows from the assumptions that u is continuous and strictly increasing; chapter 2 of Marschak and Radner |9~ provides a formal proof.

The situation is simplified under risk neutrality because again, for every y, |Mathematical Expression Omitted~. In the risk neutral version of (29b), the solution for |V.sup.N~ in the equation

|Mathematical Expression Omitted~

is, by rearranging and using the definitions of expected wealth,

|Mathematical Expression Omitted~

Equation (32) states that the value of information to a risk neutral DM is simply the before cost difference in expected wealth with and without the information. This assessment requires only one calculation of the optimal decision rule, directly from the payoff function itself.

The assessment of |V.sup.A~, the value of information to a risk averter, requires the specification of the DM's utility function. The purpose of the next section is to investigate relationships between |V.sup.A~ and |V.sup.N~ that utilize less than complete knowledge of the risk averter's specific preferences.

III. Prospect Variability and Comparative Information Value

When comparing |V.sup.A~ to |V.sup.N~, intuition might lead us to hypothesize that risk averters would seek information as a means of risk reduction, and hence tend to value it more than their risk neutral counterparts. This is not necessarily so, as Theorem 1 and the subsequent discussion indicate. LaValle |7, 396~ presents the following upper bound on the value of information to a risk averter:

THEOREM 1. |V.sup.A~ |is less than or equal to~ |V.sup.N~ + ||R.sup.N~.sub.0~ - ||R.sup.A~.sub.0~.

Proof. Use the summarizing equation (28), evaluated at |psi~ = |V.sup.A~ so that (30) also holds:

|Mathematical Expression Omitted~

Rearranging, using the definition of |V.sup.N~ given by (32), and then using the equality of ||R.sup.N~.sub.0~ and |Mathematical Expression Omitted~ shows

|Mathematical Expression Omitted~

From (12), |R.sup.N~.sub.0~ - |R.sup.A~.sub.0~ |is greater than or equal to~ 0, so there is no general sign relationship between |V.sup.A~ and |V.sup.N~ without further conditions on the problem. This bound is nevertheless convenient because the only input needed from the risk averse DM is the prior reservation price; no conditional decision rules need be determined. Ziemba and Butterworth |13~ also present some related bounds useful when the payoff function is concave.

The reason there is no general relationship is that although the use of the information system increases the expected wealth and expected utility for all decision makers, it does not necessarily leave unchanged the riskiness of the preposterior optimal informed prospect compared to the optimal prior prospect. To the risk neutral DM any change in riskiness is inconsequential, but risk averters view altered riskiness as a significant factor that affects their overall valuation of the information.

Example 1. One situation in which Theorem 1 guarantees that |V.sup.A~ |is less than or equal to~ |V.sup.N~ is when there is a risk free action in a that is the optimal prior choice for the risk neutral DM. Consider the decision to enter a new market, with demand in that market the unknown state variable. If the market is entered, terminal wealth depends upon realized demand. If the market is not entered, terminal wealth is fixed at the current level. Suppose the prior probability measure on demand is such that it is optimal for the risk neutral DM not to enter the market. In this case all risk averters would also choose the risk free action, the reservation prices ||R.sup.N~.sub.0~ and ||R.sup.A~.sub.0~ would be the same, and from Theorem 1 no risk averter would value any information system by more than the risk neutral DM.

Economically, even though the prior prospects |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ are identical with |Mathematical Expression Omitted~ and zero riskiness, if the information system is not useless the prospects |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ do have some variability. Comparing these two preposterior prospects, the inequality (27) in the cost-free case ensures |Mathematical Expression Omitted~. (Recall that when |psi~ = 0, the notation drops the |psi~). Regardless of how the risk neutral DM values the information, the risk averter values it by less for two reasons: 1) since |Mathematical Expression Omitted~, the increase in the risk averter's expected wealth |Mathematical Expression Omitted~ cannot be greater than |Mathematical Expression Omitted~, and 2) the risk averter's preposterior prospect is more risky than the degenerate prior, and this is detrimental to overall valuation. In this example the bound |V.sup.A~ |is less than or equal to~ |V.sup.n~ is true not only in the special case when |Mathematical Expression Omitted~, as occurs under perfect information, but a fortiori, also when |Mathematical Expression Omitted~, as may occur under imperfect information.

This discussion suggests the possibility of signing the difference |V.sup.A~ - |V.sup.N~ for a more general case in which the prior prospects are different and nondegenerate. Blair and Romano |3~ show that direct investigation of the change in prospect variability as the DM moves from the uninformed to the informed situations is indeed a fruitful approach.

The formal criterion for comparing the variability of prospects is in Rothschild and Stiglitz |11~. Given two prospects with the same mean, Prospect 1 is Rothschild-Stiglitz more variable (or more risky) than Prospect 2 if and only if the expected utility from Prospect 2 is at least as great as from Prospect 1 for any strictly concave utility function defined on W. Rothschild and Stiglitz show that this partial ordering is equivalent to forming Prospect 1 via a mean preserving spread of Prospect 2.

As an illustration of the Rothschild-Stiglitz approach, consider a variability comparison of the prior decisions. Since |Mathematical Expression Omitted~, yet the DM prefers |Mathematical Expression Omitted~, adjusting the two prospects in (9) to have the same mean shows that

|Mathematical Expression Omitted~

Equation (35) states that every risk averse DM prefers |Mathematical Expression Omitted~ to |Mathematical Expression Omitted~ when the mean of the former prospect is adjusted to |Mathematical Expression Omitted~. The conclusion is that |Mathematical Expression Omitted~ cannot be more Rothschild-Stiglitz variable than |Mathematical Expression Omitted~. The optimality of ||a.sup.A~.sub.0~ reveals that the risk averse DM prefers to accept a reduction in expected wealth by |Mathematical Expression Omitted~ in exchange for less variability.

Blair and Romano compare the variability of the cost-free preposterior prospect of the risk averter with the prior prospect chosen under risk neutrality. In the current notation, their result is:

THEOREM 2. If |Mathematical Expression Omitted~ is Rothschild-Stiglitz more variable than |Mathematical Expression Omitted~, then |Mathematical Expression Omitted~.

The proof of this theorem need not be repeated, but satisfaction of the sufficiency condition ensures |V.sup.A~ |is less than or equal to~ |V.sup.N~ because |Mathematical Expression Omitted~.

Theorem 2 is most easily applied to the perfect information system, denoted |I.sup.*~. Under perfect information the state is precisely identified; there is no risk posterior to the receipt of any message. The prospect |Mathematical Expression Omitted~ is degenerate with all its mass at the same wealth level for any u, hence |Mathematical Expression Omitted~. The application of (20) shows the two preposterior prospects are identical: |Mathematical Expression Omitted~. Suppose the two DMs have different initial prospects |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ and the identical perfectly informed prospect. Since |Mathematical Expression Omitted~, the risk averter is guaranteed to get at least as great an increase in expected wealth as the risk neutral counterpart:

|Mathematical Expression Omitted~

Under the premise of Theorem 2, the perfectly informed prospect |Mathematical Expression Omitted~ is more variable than |Mathematical Expression Omitted~. Hence by transitivity from the result (35), |Mathematical Expression Omitted~ is more variable than |Mathematical Expression Omitted~. In valuing the information the risk neutral DM does not care about this increased risk, but the risk averter's relative valuation involves comparing the greater increase in the mean with the greater increase in the variability. The content of Theorem 2 is that the relative gain in the mean cannot be enough to compensate for the relative increase in the risk averter's prospect variability.

Example 2. To illustrate, consider a simple problem with two states, X = {|x.sub.1~, |x.sub.2~}, two actions a = {|a.sub.1~, |a.sub.2~}, initial wealth ||omega~.sub.0~ = 2, and the outcome in matrix form as

|Mathematical Expression Omitted~

Suppose p(|x.sub.1~) = p(|x.sub.2~) = .5. Then under risk neutrality the optimal prior decision is to choose |a.sub.2~, the prospect |Mathematical Expression Omitted~ is a 50-50 chance of being worth 6 or 4, and hence |Mathematical Expression Omitted~. Given access to cost-free perfect information, DMs of all risk preferences choose |a.sub.1~ if |x.sub.1~, and |a.sub.2~ if |x.sub.2~. Thus all DMs have a 50-50 chance of being worth 8 or 4, |Mathematical Expression Omitted~, and from (32), the value of perfect information under risk neutrality is |V.sup.N~ = 1. No risk averse DM would value the perfect information more because the conditions of Theorem 2 are clearly met and the increased variability of the preposterior informed prospect is detrimental to the risk averter's valuation.

As pointed out by Blair and Romano |3, 394~, testing for the satisfaction of the sufficiency condition of Theorem 2 in the case of imperfect information requires the assessment of a specific risk averse utility function and the corresponding optimal decision rule. Note that under imperfect information and having assessed that the specific |Mathematical Expression Omitted~ is more variable than |Mathematical Expression Omitted~, Theorem 2 states that the bound on |V.sup.A~ becomes tighter by the amount |Mathematical Expression Omitted~, i.e., the expected wealth the risk averter willingly gives up in the informed situation.

The third theorem concerns the relative values |V.sup.A~ and |V.sup.N~ for decision problems under imperfect information in which both the prior and the preposterior prospects can have different means and variabilities, but where |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~ need not be assessed. In this case both |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~, so the increase in the risk averter's expected wealth, |Mathematical Expression Omitted~, may or may not be greater than |V.sup.N~.

Definitive results derive from studying the mean of the risk neutral conditional prospect |Mathematical Expression Omitted~. Defined in the risk neutral version of (17), the quantity |Mathematical Expression Omitted~ is the expected wealth of the risk neutral DM who has paid $|psi~ for an information system and has received and responded optimally to the specific message y. Prior to the receipt of y, this quantity is a random variable, distributed according to the marginal distribution p(y). Since each |Mathematical Expression Omitted~ is expressed in the monetary unit, |Mathematical Expression Omitted~ and there is no conceptual problem in viewing 'this random variable as having a probability distribution on W and hence as a prospect. This prospect, the prior distribution of the posterior mean wealth of an optimizing risk neutral DM, is denoted |Mathematical Expression Omitted~. Keep in mind that since the risk neutral DM exhibits constant absolute risk aversion and simply maximizes expected wealth conditional on y, the magnitude of |psi~ affects only the mean of this prospect, not its variability. The mean is |Mathematical Expression Omitted~ as defined in (24), and, in light of (26),

|Mathematical Expression Omitted~

With the notable exception of perfect information, this prospect is hypothetical in the sense that is not actually available to any DM, regardless of attitude towards risk. It does, however, give the risk neutral DM the same expected wealth (and hence expected utility) as the actual preposterior informed prospect. The advantage of focusing on the prospect |Mathematical Expression Omitted~ is that it requires only risk neutral assessments yet plays the major role relating |V.sup.A~ to |V.sup.N~ in Theorem 3.

The analysis begins by showing that any risk averse DM would prefer, if they both were equally costly, to obtain the hypothetical prospect over the actual optimal prospect chosen. This is formalized in the following lemma:

LEMMA 1. |Mathematical Expression Omitted~.

Proof. Since the risk neutral DM maximizes expected wealth conditional on y, the strict increasingness of the utility function ensures that any risk averse decision maker would prefer to receive the larger quantity |Mathematical Expression Omitted~ over |Mathematical Expression Omitted~:

|Mathematical Expression Omitted~

Using the definition of |Mathematical Expression Omitted~ in (17), the application of Jensen's inequality to the right hand side of (38) yields

|Mathematical Expression Omitted~

Taking the expectation of (39) with respect to p(y) and applying (22) shows definitive preference for the hypothetical prospect:

|Mathematical Expression Omitted~

Finally, since under risk neutrality no decision rule depends upon the monetary payment |psi~,

|Mathematical Expression Omitted~

Consider now a variability comparison of the prior optimal prospect |Mathematical Expression Omitted~ with the hypothetical prospect evaluated at |phi~ = |V.sup.N~. Both prospects naturally have the same mean |Mathematical Expression Omitted~: using (37), if the risk neutral DM pays the maximum amount for the information system, |V.sup.N~ defined in (32), the resulting prospect has the mean |Mathematical Expression Omitted~. The basic result is the following theorem.

THEOREM 3. The condition that |Mathematical Expression Omitted~ is Rothschild-Stiglitz more variable than |Mathematical Expression Omitted~ is sufficient but not necessary for |V.sup.A~ |is less than or equal to~ |V.sup.N~.

Sufficiency. If |Mathematical Expression Omitted~ is more variable than |Mathematical Expression Omitted~, then |V.sup.A~ |is less than or equal to~ |V.sup.N~. The proof begins by evaluating the lemma at |psi~ = |V.sup.A~, invoking the definition of |V.sup.A~ given in (29b), and using the inequality (9):

|Mathematical Expression Omitted~

Adding and subtracting the constant |V.sup.N~ to the left hand side yields

|Mathematical Expression Omitted~

The premise of the theorem, that |Mathematical Expression Omitted~ is more variable than |Mathematical Expression Omitted~, holds if and only if for any continuous, increasing, bounded and strictly concave utility function,

|Mathematical Expression Omitted~

For (44) to hold, it must be true that |V.sup.A~ |is less than or equal to~ |V.sup.N~, otherwise (43) is contradicted.

If the risk averter could actually have the hypothetical prospect, the relative valuation is |V.sup.A~ |is less than or equal to~ |V.sup.N~ for exactly the same reason as in Theorem 2. Since any attainable prospect |Mathematical Expression Omitted~ is "worse" in the risk averter's mind, the actual |V.sup.A~ must be even less. Note finally that under perfect information, the sufficiency part of Theorem 3 leads to the same comparison as in Theorem 2.

Nonnecessity. Necessity would mean that |V.sup.A~ |is less than or equal to~ |V.sup.N~ for all strictly concave utility functions implies that |Mathematical Expression Omitted~ is more variable than |Mathematical Expression Omitted~. Example 3 in Section IV proves nonnecessity, with a case of imperfect information in which |V.sup.A~ |is less than or equal to~ |V.sup.N~ but |Mathematical Expression Omitted~ is a mean preserving spread of |Mathematical Expression Omitted~.

The contrapositive of the sufficiency condition of Theorem 3 is of some economic interest, stating a condition that must hold under an assumption that all risk averters value the information more than the risk neutral benchmark. The contrapositive is that if |V.sup.A~ |is greater than or equal to~ |V.sup.N~ for all strictly concave utility functions, then |Mathematical Expression Omitted~ is Rothschild-Stiglitz more variable than |Mathematical Expression Omitted~.

IV. Information Quality and Information Value

This section introduces a simple systematic model of imperfect information for the purpose of illustrating the relations between the theorems. Specifically, the analysis shows that the statistical quality of the information system is a factor in the achievement of the sufficiency condition in Theorem 3, and illustrates by example that necessity does not hold.

In the simple two state model with X = {|x.sub.1~, |x.sub.2~}, consider a sequence of information systems each with a message space Y = {|y.sub.1~, |y.sub.2~} in which each message is a categorical yet imperfectly accurate statement of the form |y.sub.j~ = "state |x.sub.j~ obtains," j = 1, 2. Given the unconditional prior probabilities p(|x.sub.1~) and p(|x.sub.2~), the information quality, or statistical informativeness, of each system is modeled using a parameter |Mathematical Expression Omitted~, that determines the conditional posterior probabilities as

|Mathematical Expression Omitted~

If |Mathematical Expression Omitted~, then |Mathematical Expression Omitted~, the messages are always correct and the information is perfect. At the other extreme, when |Mathematical Expression Omitted~ and |Mathematical Expression Omitted~. Since neither message has any effect on the probabilities of the states, Marschak |8~ terms this case null information. As |Mathematical Expression Omitted~ rises from 0 to 1, the probabilities on the main diagonal of (45) increase; the forecast is more likely to be correct. A convenient property of this model (easily verified by applying the identity |Mathematical Expression Omitted~ to (45)) is that, for any |Mathematical Expression Omitted~, the unconditional probability the system produces message |y.sub.j~ is equal to the unconditional probability of state |x.sub.j~ obtaining: |Mathematical Expression Omitted~.

Example 2 (continued). Combining the outcome matrix given by (36) with the sequence of information systems modelled by (45) illustrates that even if the sufficiency condition holds under perfect information it does not necessarily hold under imperfect information.

Given now that |Mathematical Expression Omitted~, under risk neutrality the optimal prior decision is to choose |a.sub.2~, the prospect |Mathematical Expression Omitted~ is a 50-50 chance of being worth 6 or 4. As before,

|Mathematical Expression Omitted~

Using the information system model (45), conditional on message |y.sub.1~,

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

so the optimal decision rule |Mathematical Expression Omitted~ depends upon |Mathematical Expression Omitted~. Only when |Mathematical Expression Omitted~ is this message informative enough to induce the DM to change the prior optimal action from |a.sub.2~ to |a.sub.1~. On the other hand, given message |y.sub.2~, the optimal action always remains |a.sub.2~, leading to expected wealth |Mathematical Expression Omitted~. Thus,

|Mathematical Expression Omitted~

|Mathematical Expression Omitted~

Taking the expectation with respect to p(y), the overall expected wealth is

|Mathematical Expression Omitted~

Figure 1 graphs the optimal prospects under the cost free use of an information system with |Mathematical Expression Omitted~. The value of information is the difference between (51) and (46):

|Mathematical Expression Omitted~

The value of information is non-linear in |Mathematical Expression Omitted~; the system must achieve a certain amount of statistical informativeness before it can alter actions and hence payoffs. Marschak |8~ uses the term useless information to describe the problem-specific situation in which a system is statistically informative but generates no economic value.

Subtracting |V.sup.N~ in (52) from (49) and (50) identifies the prior distribution of the posterior mean wealth, the prospect |Mathematical Expression Omitted~. Figure 2 graphs this prospect as a function of the parameter |Mathematical Expression Omitted~. As |Mathematical Expressions Omitted~ gets larger the distribution |Mathematical Expression Omitted~ spreads out from being degenerate at null information (with all mass at the prior optimal mean |Mathematical Expression Omitted~) to being most variable at perfect information (the non-hypothetical prospect offering even odds the DM winds up being worth 7 or 3, having paid |V.sup.N~ = 1 for perfect information). It is interesting to compare this spreading with the behavior of the prior distribution of the posterior mean that arises in statistical decision theory |10, 104-109~.

Figure 2 also graphs the optimal prior prospect |Mathematical Expression Omitted~. As the figure shows, the spreading prospect |Mathematical Expression Omitted~ reaches the point of being a mean preserving spread of |Mathematical Expression Omitted~ at |Mathematical Expression Omitted~. Thus Theorem 3 assures that for any information system with informativeness parameter |Mathematical Expression Omitted~, |V.sup.A~ |is less than or equal to~ |V.sup.N~ for every risk averse decision maker. Notice also that under perfect information, the comparison in Theorem 3 is the same as in Theorem 2 (the B-R comparison described in Section III, after adjusting the prospect |Mathematical Expression Omitted~ to the same mean as |Mathematical Expression Omitted~.

Example 3. This example illustrates that the necessity part of Theorem 3 does not hold: the prior prospect |Mathematical Expression Omitted~ can be more variable than |Mathematical Expression Omitted~, yet |V.sup.A~ |is less than or equal to~. If the unconditional probabilities in example 2 change to p(|x.sub.1~) = p(|y.sub.1~) = .75 and p(|x.sub.2~) = p(|y.sub.2~) = .25, then the risk neutral DM's prior choice is the action |a.sub.1~ with its relatively wide ranging potential outcomes. The same analysis as in example 2 shows that the prior prospect is the more variable for every |Mathematical Expression Omitted~, |Mathematical Expression Omitted~, and that

|Mathematical Expression Omitted~

Let us suppose a risk averter has the specific utility function |Mathematical Expression Omitted~. Then for this DM, ||a.sup.A~.sub.0~ = |a.sub.2~ and

|Mathematical Expression Omitted~

The prior reservation price solves |Mathematical Expression Omitted~ and is ||R.sup.A~.sub.0 = 5.4174. Hence for this specific utility function the bound given by Theorem 1,

|Mathematical Expression Omitted~

holds for any information system.

Using the information system model (45), the cost-free informed expected utility is

|Mathematical Expression Omitted~

When |Mathematical Expression Omitted~ the application of the definition (29a) gives the value of information as the solution for |V.sup.A~ in the equation

|Mathematical Expression Omitted~

which solves to

|Mathematical Expression Omitted~

Figure 3 graphs both |V.sup.A~ and |V.sup.N~ as they depend upon |Mathematical Expression Omitted~. The equality of (53) and (58) occurs at |Mathematical Expression Omitted~; when |Mathematical Expression Omitted~, |V.sup.A~ |is less than or equal to~ |V.sup.N~ despite the fact that |Mathematical Expression Omitted~ is more variable than |Mathematical Expression Omitted~.

V. Summarizing Remarks

When comparing the choices made and the prospects accepted by risk averse decision makers with their risk neutral counterparts, the definitive relationships both in the uninformed (equation 12) and the informed (equation 28) situations do not necessarily lead to definitive statements signing the difference |V.sup.A~ - |V.sup.N~. The key is that although the utilization of cost-free, non-useless information increases the expected utility for all DMs, it does not necessarily leave the riskiness of the optimal prospects unchanged; this aspect is relevant only to the risk averter's valuation.

The desire to avoid the need to assess a specific utility function motivates the search for conditions involving the sign of |V.sup.A~ - |V.sup.N~ that use only risk neutral assessments. Section III studies the subject in three steps: 1) the situation with identical risk-free prior prospects and possibly different preposterior informed prospects, 2) the situation with possibly different prior prospects but identical perfectly informed preposterior prospects, and 3) the most general situation in which both the prior and the preposterior prospects may be different. The primary result for the third case is that for |V.sup.A~ |is less than or equal to~ |V.sup.N~, it is sufficient that the prior distribution of the posterior risk neutral mean be Rothschild-Stiglitz more variable than the prior risk neutral optimal prospect (Theorem 3).

The examples in section IV indicate the statistical informativeness of the system may play a role in achieving the conditions of the theorems. Of course, Rothschild-Stiglitz variability is only a partial ordering; the two prospects are not necessarily comparable and when they are not, the relationship between |V.sup.A~ and |V.sup.N~ may depend upon a more complex interaction between the problem structure, the statistical informativeness of the information, and the specific utility function. Further research on this topic could investigate more rigorously the theoretical and practical connections between the statistical informativeness of the information system and the comparative value of information.

References

1. Blackwell, David, "Equivalent Comparisons of Experiments." Annals of Mathematical Statistics, June 1953, 265-73.

2. ----- and M. A. Girshick. Theory of Games and Statistical Decisions. New York: John Wiley, 1954.

3. Blair, Roger D. and Richard E. Romano, "The Influence of Attitudes Toward Risk on the Value of Forecasting." Quarterly Journal of Economics, May 1988, 387-96.

4. Gould, John P., "Risk, Stochastic Preference, and the Value of Information." Journal of Economic Theory, May 1974, 64-84.

5. Hilton, Ronald W., "The Determinants of Information Value: Synthesizing Some General Results." Management Science, January 1981, 57-64.

6. LaValle, Irving H., "On Cash Equivalents and Information Evaluation Under Uncertainty." Journal of the American Statistical Association, March 1968, 252-90.

7. -----. Fundamentals of Decision Analysis. New York: Holt, Rinehart and Winston, 1978.

8. Marschak, Jacob, "Economics of Information Systems." Journal of the American Statistical Association, March 1971, 192-219.

9. Marschak, Jacob and Roy Radner. The Economic Theory of Teams. New Haven: Yale Press, 1972.

10. Raiffa, Howard, and Robert Schlaifer. Applied Statistical Decision Theory. Boston: Division of Research, The Harvard School of Business Administration, 1961.

11. Rothschild, Michael and Joseph E. Stiglitz, "Increasing Risk: I. A Definition." Journal of Economic Theory, September 1970, 225-43.

12. Savage, I. Richard et al. Setting Statistical Priorities. Report of the Panel on Methodology for Statistical Priorities. Washington, D.C.: National Academy of Sciences, 1976.

13. Ziemba, William T. and J. E. Butterworth, "Bounds on the Value of Information in Uncertain Decision Problems." Stochastics, 1975, 361-78.

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Author: | Lawrence, David B. |
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Publication: | Southern Economic Journal |

Date: | Oct 1, 1992 |

Words: | 6702 |

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