1. **Problem Statement:** We have a payoff matrix for strategies S1, S2, and S3 against states of nature N1, N2, N3, and N4. We need to decide the best strategy using three criteria: Optimistic, Pessimistic, and Minimax Regret. 2. **Matrix:** | Strategy | N1 | N2 | N3 | N4 | |----------|-------|-------|-------|-------| | S1 | 4000 | -100 | 6000 | 18000 | | S2 | 20000 | 5000 | 400 | 0 | | S3 | 20000 | 15000 | -2000 | 1000 | 3. **Optimistic Criterion (Maximax):** - Choose the strategy with the maximum possible payoff. - For each strategy, find the maximum payoff: - $\max(S1) = 18000$ - $\max(S2) = 20000$ - $\max(S3) = 20000$ - The maximum among these is $20000$ for S2 and S3. - So, the decision is either S2 or S3 (both have the same optimistic payoff). 4. **Pessimistic Criterion (Maximin):** - Choose the strategy with the best worst-case payoff. - For each strategy, find the minimum payoff: - $\min(S1) = -100$ - $\min(S2) = 0$ - $\min(S3) = -2000$ - The maximum among these minimums is $0$ for S2. - So, the decision is S2. 5. **Minimax Regret Criterion:** - First, find the best payoff for each state: - $\max(N1) = 20000$ - $\max(N2) = 15000$ - $\max(N3) = 6000$ - $\max(N4) = 18000$ - Calculate regret matrix by subtracting each payoff from the best payoff in that state: - Regret for S1: $[20000-4000=16000, 15000-(-100)=15100, 6000-6000=0, 18000-18000=0]$ - Regret for S2: $[20000-20000=0, 15000-5000=10000, 6000-400=5600, 18000-0=18000]$ - Regret for S3: $[20000-20000=0, 15000-15000=0, 6000-(-2000)=8000, 18000-1000=17000]$ - Find the maximum regret for each strategy: - $\max(S1) = 16100$ - $\max(S2) = 18000$ - $\max(S3) = 17000$ - Choose the strategy with the minimum of these maximum regrets: $16100$ for S1. - So, the decision is S1. **Final decisions:** - Optimistic criterion: S2 or S3 - Pessimistic criterion: S2 - Minimax regret criterion: S1