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 \(\Sigma\), \(\cup_{n} A_{n} \in \Sigma\). (By property 2 and De Morgan’s laws, \(\Sigma\) is then also closed under countable intersections.)

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

4.2 Probability measures

A set \(S\) together with a \(\sigma\)-algebra \(\Sigma\) over \(S\) is called a measurable space. A probability measure on \((S,\Sigma)\) is a function \(\mu:\Sigma \rightarrow \mathbb{R}\) with the following properties

1. \(\mu(\sigma) \ge 0\) for all \(\sigma \in \Sigma\)
2. \(\mu(S) = 1\)
3. \(\mu(\cup_{i} \sigma_{i}) = \sum_{i}\mu(\sigma_{i})\) for any countable collection \(\{\sigma_{i}\}\) of pairwise disjoint elements of \(\Sigma\).

The triple \((S, \Sigma, \mu)\) is called a probability space.

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. \(\succsim\) is complete and transitive.
2. For all \(p,q,r \in P\), and \(\alpha \in (0,1]\), \(p \succsim q\) if and only if \(\alpha p + (1-\alpha) r \succsim \alpha q + (1-\alpha) r\). (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 exist prizes \(z^{0}\) and \(z_{0}\) such that \(\delta_{z^{0}} \succsim p \succsim \delta_{z_{0}}\) for all \(p \in P\), where \(\delta_{z}\) denotes the degenerate lottery that pays \(z\) with probability one.

Note that the independence axiom is stated as a biconditional (this is the standard formulation; see, e.g., Mas-Colell, Whinston, and Green, Definition 6.B.4). The “only if” direction is what allows mixing to preserve strict preference and indifference, which the proof below uses repeatedly.

Theorem 4.1 A binary preference relation \(\succsim\) on \(P\) satisfies the four axioms above if and only if there exists a function \(u: Z \rightarrow \mathbb{R}\) such that \(p \succsim q\) if and only if \(\sum_{z} p(z)u(z) \ge \sum_{z} q(z)u(z)\).

Proof. We first show that the axioms imply the representation. The strategy: calibrate every lottery \(p\) against mixtures of the best and worst lotteries, i.e. find the unique weight \(f(p)\) such that

\[p \sim f(p)\,\delta_{z^{0}} + (1 - f(p))\,\delta_{z_{0}},\]

and then show that \(f\) represents \(\succsim\) and is affine in probabilities, so that \(f(p) = \sum_{z} p(z)u(z)\) with \(u(z) = f(\delta_{z})\).

Preliminaries. Two consequences of independence (axiom 2). First, if \(p \sim q\) then for all \(\alpha \in (0,1]\) and \(r \in P\),

\[\alpha p + (1-\alpha)r \sim \alpha q + (1-\alpha)r,\]

by applying the axiom to \(p \succsim q\) and to \(q \succsim p\). Second, if \(p \succ q\) then

\[\alpha p + (1-\alpha)r \succ \alpha q + (1-\alpha)r.\]

Indeed the “if” direction gives \(\alpha p + (1-\alpha)r \succsim \alpha q + (1-\alpha)r\); and if the two mixtures were indifferent, then \(\alpha q + (1-\alpha)r \succsim \alpha p + (1-\alpha)r\) together with the “only if” direction would give \(q \succsim p\), a contradiction. So mixing preserves \(\succsim\), \(\succ\), and \(\sim\) alike.

We now establish three facts.

Fact 1 (mixture monotonicity). If \(p \succ q\) and \(0 \le a < b \le 1\), then \(bp + (1-b)q \succ ap + (1-a)q\).

Write \(m_{c} = cp + (1-c)q\). Since \(b > 0\), strict independence (mixing \(p \succ q\) with \(r = q\) and weight \(b\)) gives

\[m_{b} = bp + (1-b)q \succ bq + (1-b)q = q.\]

If \(a = 0\) we are done. If \(a > 0\), let \(\lambda = a/b \in (0,1)\) and check the algebra \(m_{a} = \lambda m_{b} + (1-\lambda)q\). Applying strict independence to \(m_{b} \succ q\) with weight \(1 - \lambda\) and third lottery \(m_{b}\),

\[m_{b} = \lambda m_{b} + (1-\lambda)m_{b} \succ \lambda m_{b} + (1-\lambda) q = m_{a}.\]

Fact 2 (unique calibration). If \(p \succsim q \succsim r\) and \(p \succ r\), then there exists a unique \(a^{*} \in [0,1]\) such that \(q \sim a^{*}p + (1-a^{*})r\).

Write \(m_{a} = ap + (1-a)r\). Uniqueness is immediate from Fact 1: since \(p \succ r\), \(a \ne a'\) implies \(m_{a} \not\sim m_{a'}\), so at most one weight can work. For existence, if \(q \sim p\) take \(a^{*} = 1\); if \(q \sim r\) take \(a^{*} = 0\). Otherwise \(p \succ q \succ r\). Let

\[S = \{a \in [0,1] : q \succsim m_{a}\}, \qquad a^{*} = \sup S,\]

which is well defined since \(0 \in S\). By completeness, exactly one of \(m_{a^{*}} \succ q\), \(q \succ m_{a^{*}}\), or \(q \sim m_{a^{*}}\) holds. We rule out the first two.

Suppose \(m_{a^{*}} \succ q\). Note \(a^{*} > 0\) (if \(a^{*} = 0\) then \(m_{a^{*}} = r\), contradicting \(q \succ r\)). By continuity (axiom 3) applied to \(m_{a^{*}} \succ q \succ r\), there exists \(b \in (0,1)\) with

\[b\,m_{a^{*}} + (1-b)r \succ q, \qquad \text{i.e.} \qquad m_{ba^{*}} \succ q,\]

where \(ba^{*} < a^{*}\). But by definition of the supremum there exists \(a \in S\) with \(ba^{*} < a \le a^{*}\), and Fact 1 gives \(m_{a} \succsim m_{ba^{*}} \succ q\), contradicting \(q \succsim m_{a}\).

Suppose instead \(q \succ m_{a^{*}}\). Note \(a^{*} < 1\) (if \(a^{*} = 1\) then \(m_{a^{*}} = p\), contradicting \(p \succ q\)). By continuity applied to \(p \succ q \succ m_{a^{*}}\), there exists \(b \in (0,1)\) with

\[q \succ b\,p + (1-b)m_{a^{*}} = m_{b + (1-b)a^{*}}.\]

But then \(a' = b + (1-b)a^{*} > a^{*}\) belongs to \(S\), contradicting \(a^{*} = \sup S\). Hence \(q \sim m_{a^{*}}\).

Fact 3 (substitution of indifferent lotteries). If \(p \sim q\) and \(a \in [0,1]\), then \(ap + (1-a)r \sim aq + (1-a)r\) for all \(r \in P\). This is the indifference version of independence established in the preliminaries (the case \(a = 0\) is trivial).

The degenerate case. If \(\delta_{z^{0}} \sim \delta_{z_{0}}\), then for every \(p\) we have \(\delta_{z^{0}} \succsim p \succsim \delta_{z_{0}} \sim \delta_{z^{0}}\), so all lotteries are indifferent and any constant function \(u\) delivers the representation. Assume henceforth \(\delta_{z^{0}} \succ \delta_{z_{0}}\).

Construction of \(f\). For any \(p \in P\), axiom 4 gives \(\delta_{z^{0}} \succsim p \succsim \delta_{z_{0}}\) with \(\delta_{z^{0}} \succ \delta_{z_{0}}\), so Fact 2 yields a unique \(f(p) \in [0,1]\) such that

\[p \sim m_{f(p)}, \qquad \text{where } m_{a} = a\,\delta_{z^{0}} + (1-a)\,\delta_{z_{0}}.\]

\(f\) represents \(\succsim\). We claim \(p \succsim q\) if and only if \(f(p) \ge f(q)\). If \(f(p) = f(q)\) then \(p \sim m_{f(p)} = m_{f(q)} \sim q\). If \(f(p) > f(q)\) then Fact 1 (applied to \(\delta_{z^{0}} \succ \delta_{z_{0}}\)) gives \(m_{f(p)} \succ m_{f(q)}\), so

\[p \sim m_{f(p)} \succ m_{f(q)} \sim q.\]

Together these give the “if” direction, and the “only if” direction follows by contraposition: \(f(p) < f(q)\) implies \(q \succ p\).

\(f\) is affine. We claim that for all \(p, q \in P\) and \(\alpha \in [0,1]\),

\[f\left(\alpha p + (1-\alpha)q\right) = \alpha f(p) + (1-\alpha)f(q).\]

Using Fact 3 twice — first substituting \(m_{f(p)}\) for \(p\), then \(m_{f(q)}\) for \(q\) —

\[\alpha p + (1-\alpha)q \;\sim\; \alpha\, m_{f(p)} + (1-\alpha)q \;\sim\; \alpha\, m_{f(p)} + (1-\alpha)m_{f(q)}.\]

The right-hand side is itself a mixture of \(\delta_{z^{0}}\) and \(\delta_{z_{0}}\): collecting probabilities,

\[\alpha\, m_{f(p)} + (1-\alpha)m_{f(q)} = \left[\alpha f(p) + (1-\alpha)f(q)\right]\delta_{z^{0}} + \left[1 - \alpha f(p) - (1-\alpha)f(q)\right]\delta_{z_{0}} = m_{\alpha f(p) + (1-\alpha)f(q)}.\]

So \(\alpha p + (1-\alpha)q \sim m_{\alpha f(p) + (1-\alpha)f(q)}\), and the uniqueness in Fact 2 forces \(f(\alpha p + (1-\alpha)q) = \alpha f(p) + (1-\alpha)f(q)\).

Expected utility form. Define \(u(z) = f(\delta_{z})\) for each \(z \in Z\). Since \(Z\) is finite, any \(p \in P\) can be written \(p = \sum_{z} p(z)\,\delta_{z}\), a finite mixture of degenerate lotteries. Iterating the affinity property (formally, by induction on the number of prizes in the support of \(p\)),

\[f(p) = \sum_{z} p(z) f(\delta_{z}) = \sum_{z} p(z)u(z).\]

Combining with the representation step, \(p \succsim q\) if and only if \(\sum_{z} p(z)u(z) \ge \sum_{z} q(z)u(z)\), which completes the sufficiency direction.

Necessity. Conversely, suppose \(p \succsim q\) if and only if \(U(p) \ge U(q)\), where \(U(p) = \sum_{z} p(z)u(z)\). Completeness and transitivity of \(\succsim\) are inherited from those of \(\ge\) on \(\mathbb{R}\). For independence, note that \(U\) is affine: \(U(\alpha p + (1-\alpha)r) = \alpha U(p) + (1-\alpha)U(r)\). Hence for \(\alpha \in (0,1]\),

\[U(\alpha p + (1-\alpha)r) \ge U(\alpha q + (1-\alpha)r) \iff \alpha U(p) \ge \alpha U(q) \iff U(p) \ge U(q),\]

which is precisely the biconditional in axiom 2. For continuity, if \(p \succ q \succ r\) then \(g(a) = U(ap + (1-a)r) = aU(p) + (1-a)U(r)\) is continuous in \(a\) with \(g(1) = U(p) > U(q) > U(r) = g(0)\), so choosing \(a\) close enough to \(1\) gives \(g(a) > U(q)\) and choosing \(b\) close enough to \(0\) gives \(U(q) > g(b)\). Finally, since \(Z\) is finite, let \(z^{0}\) maximize and \(z_{0}\) minimize \(u\) over \(Z\); then for any \(p\), \(U(\delta_{z_{0}}) = \min_{z} u(z) \le U(p) \le \max_{z} u(z) = U(\delta_{z^{0}})\), delivering axiom 4.

4.4 Subjective uncertainty: preferences over acts

Consider two potential ways to establish preferences when the probabilities themselves are not given.

Anscombe-Aumann:

An act is a mapping from states of the world \(s \in S\) to lotteries over the prize space \(Z\). These are called horse-race lotteries: the state determines which (objective) lottery you face.

Savage:

Different.