Advanced Microeconomics/Decision Making Under Uncertainty

Decision Making Under Uncertainty Lotteries A simple lottery is a tuple assigning probabilities to N outcomes such that . A compound lottery assigns probabilities to one or more simple lotteries A reduced lottery may be calculated for any compound lottery, yielding a simple lottery which is outcome equivalent (produces the same probability distribution over outcomes) to the original compound lottery. Consider a compound lottery over the lotteries each of which assigns probabilities to N outcomes. The compound lottery implies a probability distribution over the N outcomes which, for any outcome n, may be calculated as In words, the probability of event n implied by a compound lottery is the probability of event n assigned by each lottery, weighted by the probability of a given lottery being chosen. Example Consider an outcome space . A (fair) six sided dice replicates the simple lottery and a (fair) ten sided dice replicates the simple lottery Now imagine a person randomly draws a dice from an urn known to contain nine six sided dice and one ten sided dice. This draw represents a compound lottery defined over the outcome space. The probability of any outcome and the probability of an outcome . Producing a reduced lottery, Preferences and Uncertain Outcomes Let represent a set of possible outcomes (consumption bundles, monetary payments, et cetera) with a space of compound lotteries .