12.1 Decisions under uncertainty

A decision problem has actions \(a_1, \ldots , a_m\), states of nature \(\theta _1, \ldots , \theta _n\), and a payoff \(P_{ij}\) for action \(a_i\) in state \(\theta _j\). With no probabilities for the states, several criteria are in use, each expressing an attitude to risk:

• maximin (Wald): choose the action whose worst payoff is largest, the pessimist’s rule;
• maximax: choose the action whose best payoff is largest, the optimist’s;
• Hurwicz: maximise \(\alpha \max _jP_{ij} + (1 - \alpha )\min _jP_{ij}\) for a chosen coefficient of optimism \(\alpha \);
• minimax regret (Savage): with the regret \(R_{ij} = \max _kP_{kj} - P_{ij}\), the amount lost by not choosing the best action for state \(j\), minimise the largest regret;
• Laplace: treat the states as equally likely and maximise the average payoff.