Maximin criterion (Wald criterion)

Wald's criterion is one of the fundamental methods for decision-making under uncertainty. It is based on the principle of extreme pessimism and aims to select the strategy that ensures the best outcome in the worst-case scenario.

Essence of the Criterion

The criterion assumes that the decision-maker focuses on the most unfavorable outcome for each strategy. For each alternative, its worst possible result is identified. The strategy chosen is the one for which this worst result is the best among all alternatives.

In other words, Wald's criterion is aimed at minimizing maximum losses and provides the greatest protection against adverse conditions.

Applying the Criterion

The application process involves the following steps:

1. For each strategy, the worst possible outcome is determined.
2. The worst outcomes of all strategies are compared.
3. The strategy with the best of the worst outcomes is selected.

This approach is characteristic of extremely cautious individuals who seek to minimize risks regardless of the probability of different outcomes occurring.

Mathematical Formulation

Let the following be given:

• $S=\{s_1,s_2,\dots ,s_m\}$ — the set of available strategies (alternatives) from which the decision-maker (DM) chooses.
• $\mathrm{\Theta }=\{{\theta }_1,{\theta }_2,\dots ,{\theta }_n\}$ — the set of possible states of nature (external conditions, scenarios), which are independent of the DM's choice.
• $u(s_i,{\theta }_j)$ — the payoff (utility) function, representing a numerical evaluation of the outcome when strategy $s_i\in S$ is chosen and state of nature ${\theta }_j\in \mathrm{\Theta }$ occurs. It is often represented as a payoff matrix $A=[a_{ij}]$, where $a_{ij}=u(s_i,{\theta }_j)$.

Wald's criterion (or the maximin criterion) prescribes the following procedure:

1. Finding the minimum payoff for each strategy: For each strategy $s_i$, its worst possible outcome (minimum payoff) across all possible states of nature is determined:

   $u_i^{\min }=\underset{j=1,\dots ,n}{\min }u(s_i,{\theta }_j)=\underset{\theta \in \mathrm{\Theta }}{\min }u(s_i,\theta )$

1. Choosing the strategy with the maximum of the minimum payoffs: The strategy $s_{\text{Wald}}^{*}$ is chosen that provides the highest value among the identified minimum payoffs:

   $s_{\text{Wald}}^{*}=\arg \underset{i=1,\dots ,m}{\max }(u_i^{\min })=\arg \underset{s_i\in S}{\max }(\underset{{\theta }_j\in \mathrm{\Theta }}{\min }u(s_i,{\theta }_j))$

The guaranteed (minimum) payoff level when using Wald's criterion is: $V_{\text{Wald}}=\underset{i=1,\dots ,m}{\max }(u_i^{\min })=\underset{s_i\in S}{\max }(\underset{{\theta }_j\in \mathrm{\Theta }}{\min }u(s_i,{\theta }_j))$

Thus, Wald's criterion implements the maximin principle — maximizing the minimum payoff. It is based on a pessimistic outlook, assuming that whatever decision is made, nature will respond in the least favorable way for the DM.

Note on the Loss Function

If a loss function $L(s_i,{\theta }_j)$, which needs to be minimized, is used instead of a payoff function $u(s_i,{\theta }_j)$, Wald's criterion becomes the minimax loss criterion:

1. For each strategy $s_i$, the maximum loss value is found: $L_i^{\max }=\underset{\theta \in \mathrm{\Theta }}{\max }L(s_i,\theta )$
2. The strategy that minimizes this maximum loss is chosen: $s_{\text{Wald}}^{*}=\arg \underset{s_i\in S}{\min }(\underset{\theta \in \mathrm{\Theta }}{\max }L(s_i,\theta ))$

Advantages and Disadvantages

Advantages:

• Simplicity and clarity of the method.
• Ensures a high degree of safety for decisions under complete uncertainty.

Disadvantages:

• Excessive pessimism: potentially high gains are ignored in favor of minimizing losses.
• May lead to the selection of overly conservative strategies.