# Advanced Microeconomics/Decision Making Under Uncertainty

## Decision Making Under Uncertainty[edit | edit source]

### Lotteries[edit | edit source]

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 | edit source]

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[edit | edit source]

Let represent a set of possible outcomes (consumption bundles, monetary payments, et cetera) with a space of compound lotteries .