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

• Author: Lawrence, David B.

1. 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.

2. 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:

   1. 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

   2. 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...