Advanced Microeconomics/Decision Making Under Uncertainty

Decision Making Under Uncertainty

Lotteries

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

A compound lottery assigns probabilities $(\alpha _{1},\dots ,\alpha _{K})$$(\alpha _{1},\dots ,\alpha _{K})$ to one or more simple lotteries $L_{1},\dots ,L_{K}$$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 $L_{1},\dots ,L_{K}$$L_{1},\dots ,L_{K}$ each of which assigns probabilities $p_{1},\dots ,p_{N}$$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 $\sum _{k=1}^{K}\alpha _{k}\cdot p_{n}^{k}$$\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 $\{1,2,3,4,5,6,7,8,9,10\}$$\{1,2,3,4,5,6,7,8,9,10\}$. A (fair) six sided dice replicates the simple lottery $\left({\frac {1}{6}},{\frac {1}{6}},{\frac {1}{6}},{\frac {1}{6}},{\frac {1}{6}},{\frac {1}{6}},0,0,0,0\right)$$\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 $\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)$$\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 $\in [1,6]={\frac {9}{10}}\cdot {\frac {1}{6}}+{\frac {1}{10}}\cdot {\frac {1}{10}}={\frac {16}{100}}$$\in [1,6]={\frac {9}{10}}\cdot {\frac {1}{6}}+{\frac {1}{10}}\cdot {\frac {1}{10}}={\frac {16}{100}}$
and the probability of an outcome $\in [7,10]={\frac {9}{10}}\cdot 0+{\frac {1}{10}}\cdot {\frac {1}{10}}={\frac {1}{100}}$$\in [7,10]={\frac {9}{10}}\cdot 0+{\frac {1}{10}}\cdot {\frac {1}{10}}={\frac {1}{100}}$.
Producing a reduced lottery, $\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)$$\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 $\mathbf {\mathcal {Z}}$$\mathbf {\mathcal {Z}}$ represent a set of possible outcomes (consumption bundles, monetary payments, et cetera) with a space of compound lotteries $\Delta \mathbf {\mathcal {Z}}$$\Delta \mathbf {\mathcal {Z}}$.