Decision Tree

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


Lotteries

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


Simple Lotteries

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:

  • simple-lottery.png


Set of Lotteries

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

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 $L(X)$


Comparing Lotteries

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


See Also

Sources

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