Probabilistic decisions: utility and the value of information

From my doctoral course

This note comes from the doctoral course “Multi-Criteria Decision Making”. I rewrote the explanations in my own words and recomputed the numerical examples. Main source of the course: Asgharpour, Multi-Criteria Decision Making (University of Tehran).

Where does this fit?

So far we have talked about several attributes and finite alternatives. Now the reverse: one criterion (profit, say) but the outcome is random. We must choose among several “actions” while not knowing the “state of nature”.

The payoff table

Actions (rows), states of nature (columns), and the payoff of each combination:

	State 1 ($p=0.4$)	State 2 ($p=0.6$)
Action 1	100	−20
Action 2	40	30

Decision criteria

• Without probabilities: maximin, maximax, Hurwicz, minimax regret (Savage), Laplace (equal probabilities). (See Non-compensatory methods.)
• With probabilities: expected monetary value (EMV): $EMV(a)={\sum }_{\theta }p(\theta )u(a,\theta )$. Here $EM{V}_1=28$ and $EM{V}_2=34$; action 2 wins.

Expected value of perfect information (EVPI)

If before deciding we knew the state of nature exactly, in each state we would choose the best action: $E[\max ]=0.4\times 100+0.6\times 30=58$. Its difference from the best EMV:

$EVPI=58-34=24$

That is, no source of information is worth more than 24 units. This is a ceiling; for imperfect information (sampling) the value is lower, and we update probabilities with Bayes’ rule: $P(\theta |e)=\frac{P(e|\theta )P(\theta )}{P(e)}$. (The class example: if we pay to look at a sample item, it is worth it only when the value of the sample’s information exceeds its cost.)

Utility theory

Expected monetary value does not see risk aversion: someone who wins 100 million or zero is not the same as someone who surely gets 50. Von Neumann and Morgenstern (1944) showed that if preferences satisfy a few axioms (completeness, transitivity, continuity, independence), there is a utility function $u$ such that the decision maker in effect maximises expected utility.

Example: a 50-50 gamble between 0 and 100 with $u(x)=\sqrt{x}$: expected utility $0.5\sqrt{100}=5$; the certainty equivalent (the sure amount with the same utility) is $5^2=25$, less than the EMV (50); the person is risk-averse. A convex $u$ means risk-seeking and a linear one risk-neutral. We usually define utility between 0 and 1.

The decision tree

When decisions come in sequence (first sample, then decide), a decision tree lays out the decision (square) and chance (circle) nodes in order and is solved from leaves to root (backward induction).

My note

When I have no data, I make up the “probability” myself, and that always looks suspicious to me. So I compute EVPI first: if the ceiling on the value of information is small, I do not spend time on analysis and just decide; if it is large, then gathering information is a better decision than polishing the model.

Related: Decisions under uncertainty: fuzzy numbers and Introduction: what is multi-attribute decision making?.

probabilistic_decisionsEVPIutility