Blackwell Informativeness Criterion

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1 Smeal College of Business Pennsylvania State University Analytical Models: ACCTG 597E Professor Huddart 1. Set-up and Notation Denote the possible states of nature by the set Θ = {θ 1, θ 2,..., θ m } for some m > 1. The actual state of nature is unknown. An experiment is available that will provide some information about the true state of nature. The experiment will result in one observation from the set {z 1, z 2,..., z n } for some n > 1. The probability that observation z i is observed if the true state is θ k is π ik, for i = {1,..., n} and k = {1,..., m}. Let Π denote this n m matrix of probabilities. Suppose a second experiment is also available. Analogous to the first experiment, this experiment will result in one observation from the set {ẑ 1, ẑ 2,..., ẑˆn } for some ˆn > 1. The probability that observation ẑ i is observed if the true state is θ k is ˆπ ik, for i = {1,..., ˆn} and k = {1,..., m}. Let ˆΠ denote this ˆn m matrix of probabilities. 2. The Criterion of Statistical Sufficiency If there exists a matrix, B, of real numbers b ij for i = {1,... n} and j = {1,..., ˆn}, all non-negative and such that: ˆn j=1 b ij = 1 i and ˆπ jk = n π ik b ij, i=1 Prompted by a question from Kimball Chapman. c Steven Huddart,

2 ACCTG 597E then the first experiment is at least as informative as the second experiment. Equivalently, if there exists a matrix B whose rows are probability vectors such that ˆΠ = B Π, (1) then the first experiment is at least as informative as the second experiment. 3. Some Examples Suppose m = 4 so that there are four states. 3.1 Perfect Information The result of this experiment is an element of {z 1, z 2, z 3, z 4 }. The probability that z i is observed given state θ j obtains is the element in the ith row and the jth column of the matrix Π Π = Incomplete Information The result of this experiment is an element of {ẑ 1, ẑ 2, ẑ 3 }. The probability that ẑ i is observed given state θ j obtains is the element in the ith row and the jth column of the matrix ˆΠ. ˆΠ = Page 2

3 ACCTG 597E Note that (1) is satisfied for so Π is at least as informative as ˆΠ B =, Garbled Information The result of this experiment is an element of {z 1, z 2, z 3, z 4}. probability that z i The is observed given state θ j obtains is the element in the ith row and the jth column of the matrix Π. 1 / Π = 1 / 1 2 / / 1 2 / / 2 1 Note that (1) is satisfied for B = transpose(π ). 3.4 Non-comparable Experiments (1) What must be shown to arrive at the conclusion that ˆΠ and Π cannot be ordered in terms of Blackwell s criterion? (2) Is conducting both experiment ˆΠ and Π as informative as conducting experiment Π? (3) Describe an informative experiment that is strictly less informative than ˆΠ. Page 3

4 ACCTG 597E (4) Describe an informative experiment that is strictly less informative than Π. 4. Informativeness Criterion Consider a decision maker whose utility function u(a, θ i ) is defined over A Θ, where A is the set of possible actions and Θ is the set of possible states of nature. An experiment provides the decision maker with an updated set of probabilities over the states of nature. One information system is at least as informative as another if, for all utility functions, the decision maker s expected utility from implementing the optimal action (which may vary with his preferences and with the signal he observes) is at least at least as high with that first information system. 5. Blackwell s Theorem Blackwell s theorem is described precisely and proven in Crémer (1982). A loose statement of it follows. The condition that information system Π is at least as informative as ˆΠ is equivalent to the condition that Π is statistically sufficient for ˆΠ. Page 4

5 ACCTG 597E 6. Application Return to this section after coding up the Scarf s (1994) smokestack/hitech plant problem. In that problem, it is assumed (implicitly) that the decision maker knows the demand for the commodity. (5) In that problem, what are the states? (6) What is the form of the corresponding matrix Π? Suppose instead that the information the decision maker receives takes the following form: He receives a signal z i {z j } 68 j=55 demands of i, i + 1, and i + 2 each have a probability of 1 / 3. indicating that (7) For each signal, what is the expected cost minimizing choice of plants that results in all demand being met? (8) What is the relation of the cost curve for this revised problem to the cost curve for the original problem? E.g., does the revised curve lie below or above the original curve? Everywhere? (9) Can the relationship between the original and revised cost curves be deduced without computing the cost curves? [Hint: Determine the form of the information matrix ˆΠ corresponding to the revised problem, assess whether Π is sufficient for ˆΠ, and apply Blackwell s Theorem.] 7. References Crémer, Jacques, 1982, A Simple Proof of Blackwell s Comparison of Experiments Theorem, Journal of Economic Theory 27, Demski, Joel, 2005, Analytic Modeling in Management Accounting Research, Working paper, University of Florida. Scarf, H. E., 1994, The Allocation of Resources in the Presence of Indivisibilities, Journal of Economic Perspectives 8, Page 5