Decision Making under Uncertainty
Caroline Rizza2015, Frontiers research topics
In many real world contexts, a decision maker has make a choice from the set of 'risky' alternatives. As common sense will suggest, an alternative or an act is risky if under it several outcomes are possible -some of the outcomes may be less desirable than the others. For instance, taking an exam is risky act. The possible outcomes can be 'pass' and 'fail'. Choice of a career path is risky in that it can lead to various possible wealth or utility levels over the lifetime. Most of the industrial and commercial projects are also risk. A project may fail totally. Even if it succeeds, the resulting profits can take various possible values. At a basic level, tossing of a coin is a risky act. There are two possible outcomes, i.e., the set of outcomes is {H, T }. Plausibly, risky alternatives are also described as lotteries. The act/experiment of tossing a coin can be written as a lottery, (p H , p T ), where p H is the probability of Head and p T the probability of Tail. For a fair coin the outcomes are equiprobable, i.e., p H = p T = 1 2 . So, tossing of a fair coin can be treated as a lottery denoted by (p H , p T ) = ( 1 2 , 1 2 ). On the other hand, tossing of a biased coin will be a different lottery, say ( 1 3 , 2 3 ). Differently biased coins will generate different lotteries. In general, suppose for a risk alternative the set of possible outcomes is known and finite. Such a risk/alternative can be described as a lottery denoted by a probability tuple/vector whose components are the probabilities. For instance, Let, s denote a state of nature or a possible outcome of a risky alternative. Let Ω denote the set of possible outcomes. Ω is assumed to be non-empty. When there are S possible outcomes, we have Ω = {s 1 , s 2 , ..., s S }. For the experiment involving tossing of a coin Ω = {s 1 , s 2 } = {H, T }. A 'Simple Lottery', L, is a vector (p 1 , ..., p S ), where p s is the probability of the occurrence of outcome s. Moreover, p s ≥ 0 and s (p s ) = 1. For given set of possible outcomes, Ω = {s 1 , s 2 , ..., s S }, let L denote the set of simple lotteries, i.e., L = (p 1 , p 2 , ..., p S )| p i ≥ 0 and For example, for a risky alternative of there are only three possible outcomes, s 1 , s 2 , and s 3 , we have Ω = {s 1 , s 2 , s 3 }. A general lottery L ∈ L is a vector (p 1 , p 2 , p 3 ) such that p i ≥ 0 and p 1 + p 2 + p 3 = 1. Some of the specific simple lotteries are; L 1 = (1, 0, 0), L 2 = (0, 1, 0), L 3 = (0, 0, 1), L 4 = ( 1 2 , 1 2 , 0), L 5 = ( 1 2 , 1 6 , 1 3 ). Note that Ω = {s 1 , s 2 , s 3 }, i.e., a three-component lottery can be represented as a point in equilateral triangle whose * This document is not for circulation. Please bring typos to my attention.
