Fuzzy Decisions Under Risk

Abstract

A decision problem under risk is characterized by three things: (1) the decision maker must choose an action a _i from a finite set of actions A = {a ₁,..., a _m }; (2) a finite set of chance events (also called the states of nature) ε = {e ₁,..., e _n } over which we have no control; and (3) for each a _i and resulting e _j a payoff c(a _i , e _j ) which describes what happens when event e _j occurs after the choice of action a _i . We assume that the payoff c(a _i , e _j ) is measured in dollars and the decision maker wants to maximize his payoff. Sometimes one will use a utility function for the decision maker for the payoff, but we shall not discuss utility theory in this book. What makes this decision making under risk is that we now assume that there is a probability distribution over ε giving the probability of each event e _j . Let the probability of e _j be p _j , 1 ≤ j ≤ n. So the whole decision problem may be described by a m ×n matrix M where; (1) the rows are labeled by the actions a ₁,..., a _m ; (2) the columns are labeled by the events e ₁,..., e _n ; (3) the ij ^th element in M is the payoff c(a _i , e _j ); and (4) the probabilities p _j are placed over the events e _j .