---
title: "Hurwicz criterion"
source: "https://systems-analysis.info/eng/Hurwicz_criterion"
wiki: "systems-analysis.info/eng"
article: "Hurwicz_criterion"
language: "en"
categories:
  - "Category:Decision theory"
  - "Category:Decision-making"
  - "Category:English"
  - "Category:Science"
revision_id: 185
wiki_created_at: 2026-09-06T22:18:36Z
wiki_modified_at: 2026-09-06T22:18:36Z
downloaded_at: 2026-09-07T22:21:40Z
---

# Hurwicz criterion

**Hurwicz's Criterion** is one of the methods for decision-making under uncertainty, which offers a balanced approach between extreme optimism and extreme pessimism.

## The Essence of the Criterion

When choosing a strategy in a situation where the outcome of events is unknown, Hurwicz's criterion proposes considering simultaneously:

- the worst possible outcome (pessimistic approach),
- the best possible outcome (optimistic approach).

For this, a special parameter is used—the coefficient of optimism, which takes a value from zero to one. The closer the coefficient is to one, the more attention is given to the best outcomes; the closer it is to zero, the more the worst possible results are taken into account.

### 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. It is often represented by a payoff matrix $A = \lbrack a_{ij}\rbrack$, where $a_{ij} = u(s_{i},\theta_{j})$.

Hurwicz's criterion introduces a **coefficient of optimism** $\alpha$ (alpha), which is chosen by the decision-maker (DM) in the range $0 \leq \alpha \leq 1$. This coefficient reflects the DM's degree of optimism:

- $\alpha = 1$ corresponds to complete optimism (only the best possible outcome is considered).
- $\alpha = 0$ corresponds to complete pessimism (only the worst possible outcome is considered, reducing the criterion to Wald's criterion).
- The value $(1 - \alpha)$ can be interpreted as the **coefficient of pessimism**.

The procedure for applying Hurwicz's criterion is as follows:

1.  **Finding the minimum and maximum payoff for each strategy:** For each strategy $s_{i} \in S$, the following are determined:
    **Worst outcome (minimum payoff):**
    $u_{i}^{\min} = \min\limits_{j = 1,\ldots,n}u(s_{i},\theta_{j}) = \min\limits_{\theta \in \Theta}u(s_{i},\theta)$
    **Best outcome (maximum payoff):**
    $u_{i}^{\max} = \max\limits_{j = 1,\ldots,n}u(s_{i},\theta_{j}) = \max\limits_{\theta \in \Theta}u(s_{i},\theta)$

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1.  **Calculating the Hurwicz value for each strategy:** For each strategy $s_{i}$, a weighted value is calculated that combines the best and worst outcomes, taking into account the coefficient of optimism $\alpha$:
    $H(s_{i},\alpha) = \alpha \cdot u_{i}^{\max} + (1 - \alpha) \cdot u_{i}^{\min}$
    This value represents the expected payoff of strategy $s_{i}$ according to the DM's preferences, as expressed through $\alpha$.

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1.  **Selecting the optimal strategy:** The strategy $s_{\text{Hurwicz}}^{\ast}$ that maximizes the calculated Hurwicz value is chosen:
    $s_{\text{Hurwicz}}^{\ast} = \arg\max\limits_{i = 1,\ldots,m}H(s_{i},\alpha) = \arg\max\limits_{s_{i} \in S}\left( \alpha \cdot u_{i}^{\max} + (1 - \alpha) \cdot u_{i}^{\min} \right)$

The optimal value of the Hurwicz criterion (the guaranteed level given optimism $\alpha$) is: $V_{\text{Hurwicz}} = \max\limits_{i = 1,\ldots,m}H(s_{i},\alpha) = \max\limits_{s_{i} \in S}\left( \alpha \cdot (\max\limits_{\theta \in \Theta}u(s_{i},\theta)) + (1 - \alpha) \cdot (\min\limits_{\theta \in \Theta}u(s_{i},\theta)) \right)$

#### Note on the Loss Function

If a loss function $L(s_{i},\theta_{j})$ is used, which needs to be minimized, then Hurwicz's criterion is applied to minimize the weighted combination of the best (minimum loss) and worst (maximum loss) outcomes:

1.  For each strategy $s_{i}$, the minimum $L_{i}^{\min}$ and maximum $L_{i}^{\max}$ losses are found.
2.  The value is calculated: $H_{L}(s_{i},\alpha) = \alpha \cdot L_{i}^{\min} + (1 - \alpha) \cdot L_{i}^{\max}$
3.  The strategy that minimizes this value is chosen: $s_{\text{Hurwicz}}^{\ast} = \arg\min\limits_{s_{i} \in S}H_{L}(s_{i},\alpha)$

Here, $\alpha$ is still the coefficient of optimism: when $\alpha = 1$, the DM focuses on minimizing the minimum losses (optimistically hoping for the best outcome), and when $\alpha = 0$, on minimizing the maximum losses (pessimistically preparing for the worst).

## Applying the Criterion

The process of applying Hurwicz's criterion includes the following steps:

1.  The minimum and maximum outcomes are determined for each possible strategy.
2.  For each strategy, a final score is calculated, which is a weighted value between its worst and best outcomes, depending on the chosen level of optimism.
3.  The strategy with the highest final score is selected.

Thus, the decision-maker selects a strategy that best accounts for their own attitude toward risk and uncertainty.

## Advantages and Disadvantages

**Advantages:**

- It allows the selection process to be adapted based on the character and preferences of the decision-maker.
- It considers both risk and potential rewards.

**Disadvantages:**

- It requires a subjective choice of the coefficient of optimism, which can affect the objectivity of the decision.

## See also

- [Bayes criterion](https://systems-analysis.info/eng/Bayes_criterion "Bayes criterion")
- [Bayes minimax criterion](https://systems-analysis.info/eng/Bayes_minimax_criterion "Bayes minimax criterion")
- [Criterion](https://systems-analysis.info/eng/Criterion "Criterion")
- [Hodges-Lehmann criterion](https://systems-analysis.info/eng/Hodges-Lehmann_criterion "Hodges-Lehmann criterion")