# Advanced Microeconomics/Decision Making Under Uncertainty

## Decision Making Under Uncertainty

### Lotteries

A simple lottery is a tuple ${\displaystyle (p_{1},\dots ,p_{N})}$  assigning probabilities to N outcomes such that ${\displaystyle \sum _{n=1}^{N}p_{k}=1}$ .

A compound lottery assigns probabilities ${\displaystyle (\alpha _{1},\dots ,\alpha _{K})}$  to one or more simple lotteries ${\displaystyle L_{1},\dots ,L_{K}}$

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 ${\displaystyle L_{1},\dots ,L_{K}}$  each of which assigns probabilities ${\displaystyle p_{1},\dots ,p_{N}}$  to N outcomes. The compound lottery implies a probability distribution over the N outcomes which, for any outcome n, may be calculated as ${\displaystyle \sum _{k=1}^{K}\alpha _{k}\cdot p_{n}^{k}}$
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 ${\displaystyle \{1,2,3,4,5,6,7,8,9,10\}}$ . A (fair) six sided dice replicates the simple lottery ${\displaystyle \left({\frac {1}{6}},{\frac {1}{6}},{\frac {1}{6}},{\frac {1}{6}},{\frac {1}{6}},{\frac {1}{6}},0,0,0,0\right)}$
and a (fair) ten sided dice replicates the simple lottery ${\displaystyle \left({\frac {1}{10}},{\frac {1}{10}},{\frac {1}{10}},{\frac {1}{10}},{\frac {1}{10}},{\frac {1}{10}},{\frac {1}{10}},{\frac {1}{10}},{\frac {1}{10}},{\frac {1}{10}}\right)}$

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 ${\displaystyle \in [1,6]={\frac {9}{10}}\cdot {\frac {1}{6}}+{\frac {1}{10}}\cdot {\frac {1}{10}}={\frac {16}{100}}}$
and the probability of an outcome ${\displaystyle \in [7,10]={\frac {9}{10}}\cdot 0+{\frac {1}{10}}\cdot {\frac {1}{10}}={\frac {1}{100}}}$ .
Producing a reduced lottery, ${\displaystyle \left({\frac {4}{25}},{\frac {4}{25}},{\frac {4}{25}},{\frac {4}{25}},{\frac {4}{25}},{\frac {4}{25}},{\frac {1}{100}},{\frac {1}{100}},{\frac {1}{100}},{\frac {1}{100}}\right)}$

### Preferences and Uncertain Outcomes

Let ${\displaystyle \mathbf {\mathcal {Z}} }$  represent a set of possible outcomes (consumption bundles, monetary payments, et cetera) with a space of compound lotteries ${\displaystyle \Delta \mathbf {\mathcal {Z}} }$ .