Decision Making Under Uncertainty
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.
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 .