Mat540 Homework Week 2 Chapter 121 A
Mat540 Homework Week 2chapter 121 A
A local real estate investor in Orlando is considering three alternative investments; a motel, a restaurant, or a theater. Profits from the motel or restaurant will be affected by gasoline availability and the number of tourists; profits from the theater will be relatively stable under any conditions. The following payoff table shows the profit or loss that could result from each investment:
| Investment | Shortage | Stable | Surplus |
|---|---|---|---|
| Motel | −7,500 | 12,000 | 23,000 |
| Restaurant | 3,000 | 7,000 | 6,500 |
| Theater | 5,000 | 6,000 | 4,000 |
Decision criteria analysis
To determine the best investment, we will evaluate the options under various decision criteria:
- Maximax: Choose the investment with the highest possible payoff in the best-case scenario.
- Maximin: Choose the investment with the best of the worst-case payoffs.
- Minimax regret: Minimize the maximum regret across all states of nature.
- Hurwicz (α = 0.4): Weighted average of the best and worst payoffs with α = 0.4.
- Equal likelihood: Assume all states are equally probable, selecting based on average payoffs.
Decision analysis
Maximax criterion involves identifying the highest payoff within each investment and selecting the maximum among them:
- Motel: 23,000
- Restaurant: 7,000
- Theater: 6,000
Thus, the maximax decision is to invest in the motel, expecting the highest possible profit of $23,000.
Maximin analysis considers the minimum payoff from each investment:
- Motel: -7,500
- Restaurant: 3,000
- Theater: 4,000
The best of these minimums is $4,000 (theater), so the maximin decision recommends investing in the theater.
Minimax regret involves calculating regrets—the difference between the best payoff in each state and the payoff for each investment:
| Investment | Regret (Shortage) | Regret (Stable) | Regret (Surplus) |
|---|---|---|---|
| Motel | 0 (since 23,000 - 23,000) | 0 (12,000 - 12,000) | 0 (23,000 - 23,000) |
| Restaurant | 20,000 (23,000−3,000) | 5,000 (7,000−7,000) | 16,500 (23,000−6,500) |
| Theater | 18,000 (23,000−5,000) | 17,000 (7,000−6,000) | 19,000 (23,000−4,000) |
Largest regrets are for Motel in Shortage (18,000), Restaurant in Surplus (16,500), and Theater in Surplus (19,000). The maximum regrets are 18,000, 17,000, and 19,000 respectively. The minimum of these maximum regrets is 18,000 (for Motel), so the minimax regret decision favors investing in the motel.
The Hurwicz criterion with α = 0.4 weights the best and worst payoffs:
- Motel: 0.4×23,000 + 0.6×−7,500 = 9,200 - 4,500 = 4,700
- Restaurant: 0.4×7,000 + 0.6×3,000 = 2,800 + 1,800 = 4,600
- Theater: 0.4×6,000 + 0.6×4,000 = 2,400 + 2,400 = 4,800
Thus, the Hurwicz criterion favors investing in the theater with a score of $4,800.
Under equal likelihood (assuming each state is equally probable), the expected payoff for each investment is:
- Motel: (−7,500 + 12,000 + 23,000)/3 ≈ 9,833
- Restaurant: (3,000 + 7,000 + 6,500)/3 ≈ 5,500
- Theater: (5,000 + 6,000 + 4,000)/3 ≈ 5,000
Clearly, the motel is the most favorable based on expected value with approximately $9,833.
Conclusion
Different decision criteria lead to varying recommendations: the maximax criterion favors the motel, the maximin and minimax regret point towards the theater or motel respectively, and the equal likelihood approach strongly favors the motel. The choice ultimately depends on the decision-maker’s risk preference. Considering the risk-averse nature, the maximin approach might be more appropriate, favoring the theater, whereas risk-takers may prefer the motel per the maximax rule. The decision analysis demonstrates the importance of explicit criteria and risk considerations in investment decisions.
References
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