This is a tool for modeling decision taking process for Decision Under Risk

Notation:

- $A$ - set of alternatives
- $X = \{x_1, ..., x_n)\}$ - a finite set of consequences
- could be $X \subseteq \mathbb{R}$ - e.g. money, etc

A *simple lottery* $l$ on $X$ is

- a discrete Random Value on $X$
- $l = \{(x_1, p_1), (x_2, p_2), ..., (x_n, p_n) \}$
- $x_i$ is a consequence, $p_i$ is the probability that $x_i$ will happen
- this is a simple model: it depends only on one set of consequences

Visual representation:

But we can also have a lottery over lotteries

A set of lotteries:

- simple lotteries on $X$
- first-order lotteries on simple lotteries
- second-order lotteries on first-order lotteries
- etc

Notation

- Let $L(X)$ denote the set of all lotteries at all finite orders
- $L(X)$ includes all lotteries that correspond to implementation of alternatives from $A$

- $l \in L(X)$ a lottery from $L(X)$ - can be simple or not
- $p_l(x)$ is the probability to face consequence $x$ in lottery $l$

Decision Trees have three kinds of nodes:

- decision nodes
- here the decision maker has to choose which action to implement

- chance nodes
- at a chance node the Nature chooses a branch according to the probability distribution
- this is a lottery of higher order

- terminal nodes
- single lotteries out of $L(X)$

To be able to decide on the decision nodes, a decision maker needs to be able to compare different lotteries

- Expected Values for Lotteries
- Expected Utility Theory
- this way he may rank all possible alternatives (lotteries) of the decision nodes and take the best decision