Notes on Blackwell’s Comparison of Experiments

There is a finite set of states: Ω = {ω1, . . . , ωn} where n ∈ N. An experiment P is an n ×N matrix {Pij}i=1,...,n;j=1,...,N , where N ∈ N, such that Pij ≥ 0 for all i, j and ∑N j=1 Pij = 1 for all i = 1, . . . , n. We shall call such a matrix a “Markov matrix.” The interpretation is that Pij is the probability of observing experiment outcome j in state ωi. The decision maker conducts experiments to solve decision problems in which the loss that the decision maker suffers depends on the true state of the world. The decision maker’s decision problem is given by an action space A which is a subset of Rn. Suppose a ∈ A. Then ai is the loss that the decision maker suffers if she chooses action a and the state turns out to be ωi. If the decision maker first observes the outcome of experiment P and then chooses an action from A, then a decision rule maps outcomes of P into elements of A. We can describe such decision rules using a N×n matrix D where every row of D is an element of A. The interpretation is that the decision maker plans to take the action that is the j-th row of D if she observes outcome j of the experiment. Suppose the experiment is P , and the decision maker chooses decision rule D. Then the diagonal elements of the n × n matrix PD describe the expected loss in each state i of the world. This is called the “risk” in state i. The vector of diagonal elements of PD is b(P,D). We call b(P,D) the “risk vector.” As D varies over all possible decision rules, we obtain the set of all possible risk vectors, denoted by B(P,A). The central definition is: