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:

• $S=\{s_1,s_2,\dots ,s_m\}$ — the set of available strategies (alternatives).
• $\mathrm{\Theta }=\{{\theta }_1,{\theta }_2,\dots ,{\theta }_n\}$ — 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 $A=[a_{ij}]$, 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 $0\le \alpha \le 1$. 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):**

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

   **Best outcome (maximum payoff):**

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

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 $\alpha$.

1. Selecting the optimal strategy: The strategy $s_{\text{Hurwicz}}^{*}$ that maximizes the calculated Hurwicz value is chosen:

   $s_{\text{Hurwicz}}^{*}=\arg \underset{i=1,\dots ,m}{\max }H(s_i,\alpha )=\arg \underset{s_i\in S}{\max }(\alpha \cdot u_i^{\max }+(1-\alpha )\cdot u_i^{\min })$

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

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: $s_{\text{Hurwicz}}^{*}=\arg \underset{s_i\in S}{\min }{H}_L(s_i,\alpha )$

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.