Advanced Microeconomics/Decision Making Under Uncertainty

Decision Making Under Uncertainty

Lotteries

A simple lottery is a tuple $(p_1, \dots, p_N)$ assigning probabilities to N outcomes such that $\sum_{n=1}^{N}p_k = 1$.

A compound lottery assigns probabilities $(\alpha_1,\dots,\alpha_K)$ to one or more simple lotteries $L_1, \dots, L_K$

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 $L_1,\dots,L_K$ each of which assigns probabilities $p_1,\dots,p_N$ to N outcomes. The compound lottery implies a probability distribution over the N outcomes which, for any outcome n, may be calculated as $\sum_{k=1}^K\alpha_k\cdot p_n^k$
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 $\{1,2,3,4,5,6,7,8,9,10\}$. A (fair) six sided dice replicates the simple lottery $\left( \frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},\frac{1}{6},0,0,0,0 \right)$
and a (fair) ten sided dice replicates the simple lottery $\left(\frac{1}{10}, \frac{1}{10}, \frac{1}{10}, \frac{1}{10}, \frac{1}{10}, \frac{1}{10}, \frac{1}{10}, \frac{1}{10}, \frac{1}{10}, \frac{1}{10}\right)$

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

Preferences and Uncertain Outcomes

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