Decision Theory — Elements and Decision-Making under Uncertainty

Decision Theory — Elements and Decision-Making under Uncertainty

A decision-making situation in business mathematics has three elements:

• Courses of action (alternatives) — the choices open to the decision-maker, e.g. $A_1,A_2,A_3$ (which product to launch, how much stock to order).
• States of nature — the possible future conditions the decision-maker cannot control, e.g. $S_1,S_2,S_3$ (low, medium, high demand).
• Payoff table — a table of the outcome (profit, cost, etc.) for every combination of action and state of nature.

When the probabilities of the states of nature are not known, the situation is decision-making under uncertainty, and four classical criteria are used to pick an action.

Worked example — payoff table (profit, ₹'000)

1. Maximax criterion (the optimist) — pick the action whose best possible outcome is highest. Row maxima: $A_1=200$, $A_2=300$, $A_3=400$. The highest is 400 → choose $A_3$.

2. Maximin criterion (Wald, the pessimist) — pick the action whose worst possible outcome is least bad. Row minima: $A_1=200$, $A_2=100$, $A_3=50$. The highest is 200 → choose $A_1$.

3. Minimax Regret criterion (Savage) — build a regret (opportunity-loss) table: in each state-of-nature column, regret = (column's best payoff) − (that action's payoff). Column bests are $S_1=200$ (from $A_1$), $S_2=300$ (from $A_2$), $S_3=400$ (from $A_3$):

Pick the action with the smallest maximum regret: $\min(200,100,150)=100$ → choose $A_2$.

4. Laplace (equal-probability) criterion — since nothing is known about likelihood, treat all states as equally likely and pick the action with the highest average payoff:

The highest average is $A_2$ (233.33) → choose $A_2$.

Summary