4  Decision making under uncertainty

4.1 Set theory and algebras

Consider a set \(S\). A \(\sigma\)-algebra over the set \(S\) is a collection \(\Sigma\) of subsets of \(S\) with the following properties:

  1. \(S \in \Sigma\)
  2. If \(A \in \Sigma\), then \(A^{c} \in \Sigma\).
  3. For any sequence \(\{A\}_{n}\) of elements of \(\Sigma\), \(\cap A_{n} \in \Sigma\)

Examples of why a \(\sigma\)-algebra is a useful set of things to consider.

4.2 Probability measures

A set \(S\) and a \(\sigma\)-algebra \(Sigma\) represent a probability space. A measure is a function \(\mu:\Sigma \rightarrow \mathbb{R}\) with the following properties

  1. \(\mu(\sigma) \in [0,1]\) for all \(\sigma \in \Sigma\)
  2. \(\mu(\cap \sigma_{i}) = \sum_{i}\mu(\sigma_{i})\) if all \(\sigma_{i}\) are disjoint.

4.3 von Neumann-Morgenstern representation

Let \(Z\) be a finite set of prizes. Let \(P\) be the set of all lotteries over \(Z\). One has preference relations over \(P\). Assume that these preference relations satisfy:

  1. \(succsum\) is complete and transitive.
  2. For all \(p,q,r \in P\), and \(\alpha \in (0,1]\), \(p \succsim q\) implies that \(\alpha p + (1-\alpha) r \succsim \alpha q + (1-\alpha) r\). (Independence or substitution)
  3. For all \(p,q,r \in P\), if \(p \succ q \succ r\) then there exist \(a,b \in (0,1)\) such that \(a p + (1-a)r \succ q \succ bp + (1-b)r\) (continuity)
  4. There exists an element \(z^{0}\) such that \(z^{0} \succsim p\) for all \(p \in P\) and another element \(z_{0}\) such that \(p \succsim z_{0}\) for all \(p \in P\).

Theorem 4.1 A binary preference relation \(\succsim\) satisfies the previous three axioms if and only if there exists a function \(u(\cdot)\) such that \(p \succsim q\) if and only if \(\sum p(z)u(z) \ge \sum q(z)u(z)\).

Proof.

Let \(f(p) = a: a\delta_{z^{0}} + (1-a)\delta_{z_{0}} \sim p\). How would you prove that this satisfied the necessary properties?

First we note the following three facts:

  1. \(p \succsim q\) and \(0 \le a < b \le 1\) imply \(bp + (1-b)p \succsim ap + (1-a)q\).
  2. \(p \succsim q \succsim r\) and \(p \succ r\) imply there exists a unique \(a^{*} \in [0,1]\) such that \(q \sim a^{*}p + (1-a^{*})r\)
  3. \(p \sim q\) and \(a \in [0,1]\) imply \(ap + (1-a)r \sim aq + (1-a)r\) for all \(r \in P\).

4.4 Preferences over lotteries

Consider two potential ways to establish preferences over consumption over sets.

Anscombe-Aumann:

An act is a mapping from states of the world to measures over \((S,\Sigma)\). These are called horse-race lotteries.

Savage:

Different.