--- title: "Minimax regret criterion (Savage criterion)" source: "https://systems-analysis.info/eng/Minimax_regret_criterion_(Savage_criterion)" wiki: "systems-analysis.info/eng" article: "Minimax_regret_criterion_(Savage_criterion)" language: "en" categories: - "Category:Decision theory" - "Category:Decision-making" - "Category:English" - "Category:Science" revision_id: 251 wiki_created_at: 2026-09-06T22:19:32Z wiki_modified_at: 2026-09-06T22:19:32Z downloaded_at: 2026-09-07T22:22:12Z --- # Minimax regret criterion (Savage criterion) **Savage's criterion** (also known as the **minimax regret criterion**) is a method for decision-making under uncertainty. It is applied in situations where the probabilities of different outcomes are unknown, and the goal is to minimize potential losses resulting from making a non-optimal decision. ## General Characteristics Under conditions of uncertainty, the consequences of choosing each strategy are not precisely defined. A number of criteria are used to evaluate possible alternatives, such as the criteria of Wald, Hurwicz, Laplace, and Savage. Savage's criterion is not focused on achieving maximum profit, but on **minimizing the maximum regret** (losses compared to the best possible outcome). **Regret** is a value that reflects the opportunity loss incurred because a non-optimal strategy was chosen for a specific outcome. ## Algorithm for Applying Savage's Criterion - Constructing the payoff matrix: A table is created where rows correspond to possible strategies and columns correspond to possible outcomes (states of nature). The intersection records the expected result for a specific strategy and outcome. - Constructing the regret matrix (risk matrix): For each outcome (column), the maximum payoff value is determined. Then, for each cell, the amount of regret is calculated. - Determining the maximum regret for each strategy: In each row of the regret matrix, the maximum value is selected (the worst-case scenario for that strategy). - Choosing the optimal strategy: The strategy with the minimum maximum regret is chosen. Thus, Savage's criterion implements the principle of minimizing the potential loss from an incorrect decision. ## Mathematical Formulation Let the following be given: - $S = \{ s_{1},s_{2},\ldots,s_{m}\}$ — the set of available strategies (alternatives). - $\Theta = \{\theta_{1},\theta_{2},\ldots,\theta_{n}\}$ — the set of possible states of nature. - $u(s_{i},\theta_{j})$ — the payoff (utility) function for choosing strategy $s_{i}$ when state $\theta_{j}$ occurs. This is often represented by a payoff matrix $A = \lbrack a_{ij}\rbrack$, where $a_{ij} = u(s_{i},\theta_{j})$. Savage's criterion is based on the concept of **regret** or **opportunity loss**. The regret $r(s_{i},\theta_{j})$ for strategy $s_{i}$ under state of nature $\theta_{j}$ is defined as the difference between the maximum possible payoff that could have been obtained for that state of nature $\theta_{j}$ (if the best strategy for that state had been chosen) and the actual payoff from strategy $s_{i}$. The algorithm for applying Savage's criterion: 1. **Calculate the regret (risk) matrix:** **a) Find the maximum payoff for each state of nature (each column of the payoff matrix):** $u_{j}^{\ast} = \max\limits_{k = 1,\ldots,m}u(s_{k},\theta_{j}) = \max\limits_{k = 1,\ldots,m}a_{kj}$ This is the best possible result if state $\theta_{j}$ occurs. **b) Calculate the elements of the regret matrix** $R = \lbrack r_{ij}\rbrack$:\*\* $r_{ij} = r(s_{i},\theta_{j}) = u_{j}^{\ast} - u(s_{i},\theta_{j}) = (\max\limits_{k = 1,\ldots,m}a_{kj}) - a_{ij}$ The element $r_{ij}$ shows how much the payoff from strategy $s_{i}$ is less than the maximum possible payoff under state $\theta_{j}$. All elements $r_{ij} \geq 0$. 1. **Find the maximum regret for each strategy:** For each strategy $s_{i}$ (each row of the regret matrix $R$), its worst possible outcome in terms of regret is determined: $r_{i}^{\max} = \max\limits_{j = 1,\ldots,n}r_{ij} = \max\limits_{j = 1,\ldots,n}\left( (\max\limits_{k = 1,\ldots,m}a_{kj}) - a_{ij} \right)$ 1. **Choose the strategy with the minimum maximum regret (the minimax regret principle):** The strategy $s_{\text{Savage}}^{\ast}$ that minimizes the found maximum regret is chosen: $s_{\text{Savage}}^{\ast} = \arg\min\limits_{i = 1,\ldots,m}(r_{i}^{\max}) = \arg\min\limits_{s_{i} \in S}\left( \max\limits_{\theta_{j} \in \Theta}r(s_{i},\theta_{j}) \right)$ Or, substituting the expression for $r_{ij}$: $s_{\text{Savage}}^{\ast} = \arg\min\limits_{i = 1,\ldots,m}\left( \max\limits_{j = 1,\ldots,n}\left\lbrack (\max\limits_{k = 1,\ldots,m}a_{kj}) - a_{ij} \right\rbrack \right)$ The minimum value of the maximum regret achieved using Savage's criterion is: $V_{\text{Savage}} = \min\limits_{i = 1,\ldots,m}(r_{i}^{\max}) = \min\limits_{i = 1,\ldots,m}\left( \max\limits_{j = 1,\ldots,n}r_{ij} \right)$ Thus, Savage's criterion aims to select the strategy that guarantees the smallest losses relative to the best possible action for each state of nature. Key points in the mathematical formulation: - Definition of regret $r_{ij}$: This is the central concept. It is important to show that it is calculated as the difference between the best outcome in column j and the current outcome a\_{ij}. - Regret matrix $R$: It is explicitly stated how it is constructed. - Finding $r_{i}^{\max}$: The search for the maximum in each row of the regret matrix is shown. - Minimax principle: The choice of strategy is clearly formulated using $\arg\min$ of the $\max$ of regrets. - Notation used: Standard for game theory and decision theory (S, Θ, u, a_ij, r_ij, max, min, arg min). ## Advantages and Disadvantages **Advantages:** - Focuses on minimizing risks. - Particularly effective under conditions of high uncertainty. **Disadvantages:** - Ignores expected profit, focusing only on potential losses. - Can lead to overly conservative decisions. ## Decision Criteria - Hurwicz's Criterion - Laplace's Criterion - Wald's Criterion ## External links - Regret (decision theory) — Wikipedia ## See also - [Bayes minimax criterion](https://systems-analysis.info/eng/Bayes_minimax_criterion "Bayes minimax criterion") - [Bayes criterion](https://systems-analysis.info/eng/Bayes_criterion "Bayes criterion") - [Criterion](https://systems-analysis.info/eng/Criterion "Criterion") - [Hodges-Lehmann criterion](https://systems-analysis.info/eng/Hodges-Lehmann_criterion "Hodges-Lehmann criterion")