Decision Making Under Uncertainty[edit]
Lotteries[edit]
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[edit]
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 ![{\displaystyle \in [1,6]={\frac {9}{10}}\cdot {\frac {1}{6}}+{\frac {1}{10}}\cdot {\frac {1}{10}}={\frac {16}{100}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f77c363ea047cef19421ff86f5ad8910888b4c7a)
and the probability of an outcome
.
Producing a reduced lottery, 
Preferences and Uncertain Outcomes[edit]
Let
represent a set of possible outcomes (consumption bundles, monetary payments, et cetera) with a space of compound lotteries
.