Decision Criteria

362. Decision Criteria

Five classical criteria for decisions under uncertainty when probabilities are unknown or deeply uncertain (probability estimates aren’t trustworthy). Used when EMV doesn’t apply.

362.1. Setup

Acts $𝑎_{𝑖}$ , states $𝑠_{𝑗}$ , payoff matrix $𝑉_{𝑖 𝑗}$ . No probabilities. How to choose?

Five rules give five different answers, reflecting different attitudes toward uncertainty.

362.2. Maximin (pessimist)

Worst-case thinking. For each act, find the worst possible payoff. Pick the act with the best worst case:

𝑎^{*} = arg max_{𝑖} (min_{𝑗} 𝑉_{𝑖 𝑗})

Very conservative — wars, safety-critical systems, fiduciary obligations.

362.3. Maximax (optimist)

Best-case thinking. For each act, find the best possible payoff. Pick the act with the best best case:

𝑎^{*} = arg max_{𝑖} (max_{𝑗} 𝑉_{𝑖 𝑗})

Aggressive — startup founders, venture capital, lottery players.

362.4. Minimax regret (Savage)

Avoid kicking-yourself thinking. For each (act, state), compute the regret — how much better you’d have done with the best act for that state:

𝑅_{𝑖 𝑗} = max_{𝑘} 𝑉_{𝑘 𝑗} - 𝑉_{𝑖 𝑗}

Then minimize the maximum regret:

𝑎^{*} = arg min_{𝑖} (max_{𝑗} 𝑅_{𝑖 𝑗})

Used when retrospective comparison is the bigger concern than absolute outcome — diplomatic / political decisions.

362.5. Hurwicz criterion (compromise)

Linear combination of optimism and pessimism. Coefficient of optimism $𝛼 \in [0, 1]$ :

𝐻 (𝑎_{𝑖}) = 𝛼 max_{𝑗} 𝑉_{𝑖 𝑗} + (1 - 𝛼) min_{𝑗} 𝑉_{𝑖 𝑗}

𝑎^{*} = arg max_{𝑖} 𝐻 (𝑎_{𝑖})

$𝛼 = 1$ : maximax. $𝛼 = 0$ : maximin. $𝛼 = 0.5$ : balanced.

Calibrate $𝛼$ to the decision-maker’s risk attitude (or context: $𝛼$ high in opportunity-rich environments, low in threat-heavy ones).

362.6. Laplace (insufficient reason)

Pretend each state is equally likely (uniform prior), then apply EMV:

𝐿 (𝑎_{𝑖}) = \frac{1}{𝑛} \sum_{𝑗} 𝑉_{𝑖 𝑗} (simple average over states)

𝑎^{*} = arg max_{𝑖} 𝐿 (𝑎_{𝑖})

Equivalent to Bayesian decision under uniform prior. The first thing to try when you have no information about state probabilities.

362.7. Comparison example

Payoffs:

	$𝑠_{1}$	$𝑠_{2}$	$𝑠_{3}$
$𝑎_{1}$	$50$	$- 10$	$30$
$𝑎_{2}$	$20$	$40$	$10$
$𝑎_{3}$	$0$	$0$	$100$

	Maximin	Maximax	Min Regret	Hurwicz ( $𝛼 = 0.6$ )	Laplace
$𝑎_{1}$	$- 10$	$50$	$0$ (regret table needed)	$0.6 \cdot 50 + 0.4 \cdot (- 10) = 26$	$\frac{1}{3} \cdot (50 - 10 + 30) = 23.3$
$𝑎_{2}$	$10$	$40$		$28$	$23.3$
$𝑎_{3}$	$0$	$100$		$60$	$33.3$

Different criteria, different winners — picks $𝑎_{2}$ (Maximin) vs $𝑎_{3}$ (Maximax / Hurwicz / Laplace).

362.8. Which to use?

No single right answer — depends on:

Stakes (high-stakes one-shot → maximin)
Repeatability (many similar decisions → Laplace/EMV)
Reputation cost (criticism risk → minimax regret)
Risk preference (default to Hurwicz with calibrated $𝛼$ )

362.9. See also

EMV — when probabilities are known
Utility Theory — handles risk aversion under known probabilities
Decision Analysis overview