Advanced Microeconomics/Decision Making Under Uncertainty

Decision Making Under UncertaintyEdit

LotteriesEdit

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.

ExampleEdit

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 OutcomesEdit

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