1. **Problem Statement:** We have three investment alternatives with payoffs under three economic conditions and their probabilities. We want to find the preferred decision using expected value and expected utility approaches. 2. **Expected Value Approach:** The expected value (EV) for each investment is calculated as: $$EV = \sum (\text{payoff} \times \text{probability})$$ Calculate EV for each investment: - Investment A: $$EV_A = 100 \times 0.40 + 25 \times 0.30 + 0 \times 0.30 = 40 + 7.5 + 0 = 47.5$$ - Investment B: $$EV_B = 75 \times 0.40 + 50 \times 0.30 + 25 \times 0.30 = 30 + 15 + 7.5 = 52.5$$ - Investment C: $$EV_C = 50 \times 0.40 + 50 \times 0.30 + 50 \times 0.30 = 20 + 15 + 15 = 50$$ **Preferred decision by expected value:** Investment B with EV = 52.5 3. **Expected Utility Approach:** Given lotteries with payoff $100,000 with probability $p$ and $0$ with probability $(1-p)$, decision makers express indifference probabilities for certain profits. We use these to find utility values. Utility for $100,000 is normalized to 1 and for $0$ is 0. For each decision maker, utility of profit $x$ is equal to the indifference probability $p$: - Decision Maker A utilities: - $75,000: U(75,000) = 0.80$ - $50,000: U(50,000) = 0.60$ - $25,000: U(25,000) = 0.30$ - Decision Maker B utilities: - $75,000: U(75,000) = 0.60$ - $50,000: U(50,000) = 0.30$ - $25,000: U(25,000) = 0.15$ Calculate expected utility (EU) for each investment: For Decision Maker A: $$EU_A = 0.40 \times 1 + 0.30 \times 0.30 + 0.30 \times 0 = 0.40 + 0.09 + 0 = 0.49$$ $$EU_B = 0.40 \times 0.80 + 0.30 \times 0.60 + 0.30 \times 0.30 = 0.32 + 0.18 + 0.09 = 0.59$$ $$EU_C = 0.40 \times 0.60 + 0.30 \times 0.60 + 0.30 \times 0.60 = 0.24 + 0.18 + 0.18 = 0.60$$ For Decision Maker B: $$EU_A = 0.40 \times 1 + 0.30 \times 0.15 + 0.30 \times 0 = 0.40 + 0.045 + 0 = 0.445$$ $$EU_B = 0.40 \times 0.60 + 0.30 \times 0.30 + 0.30 \times 0.15 = 0.24 + 0.09 + 0.045 = 0.375$$ $$EU_C = 0.40 \times 0.30 + 0.30 \times 0.30 + 0.30 \times 0.30 = 0.12 + 0.09 + 0.09 = 0.30$$ **Preferred decisions by expected utility:** - Decision Maker A prefers Investment C (EU=0.60) - Decision Maker B prefers Investment A (EU=0.445) 4. **Explanation for different preferences:** Decision Makers A and B have different risk preferences reflected in their utility functions. Decision Maker A values moderate payoffs more (higher utilities for lower profits), showing risk aversion, while Decision Maker B values high payoffs more selectively, showing different risk tolerance. This leads to different preferred investments under expected utility despite the same payoffs.