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Chapter 9: Problem 1

Given the following payoff matrix, show all work to answer parts (a) and (b). $$ \begin{aligned} &\text { States of Nature }\\\ &\begin{array}{ccccc} \hline & p=0.35 & p=0.3 & p=0.25 & p=0.1 \\ \text { Alternatives } & 1 & 2 & 3 & 4 \\ \hline \text { A } & 1100 & 900 & 400 & 300 \\ \text { B } & 850 & 1500 & 1000 & 500 \\ \text { C } & 700 & 1200 & 500 & 900 \\ \hline \end{array} \end{aligned} $$ a. Which alternative do we choose if our criterion is to maximize the expected value? b. Find the opportunity loss (regret) table and compute the expected opportunity loss (regret) for each alternative. What decision do you make if your criterion is to minimize expected regret?

Short Answer

Expert verified
Choose Alternative B to maximize expected value and minimize expected regret.

Step by step solution

01

Compute Expected Value for Each Alternative

To find the expected value for each alternative, we will use the probabilities \(p\) given and the payoffs to calculate the expected payoff for Alternatives A, B, and C. The formula is:\[E(X) = p_1 \times x_1 + p_2 \times x_2 + p_3 \times x_3 + p_4 \times x_4\]where \( x_i \) are the payoffs and \( p_i \) are the probabilities.1. **Alternative A:**\[E(A) = 0.35 \times 1100 + 0.3 \times 900 + 0.25 \times 400 + 0.1 \times 300\]\[E(A) = 385 + 270 + 100 + 30 = 785\]2. **Alternative B:**\[E(B) = 0.35 \times 850 + 0.3 \times 1500 + 0.25 \times 1000 + 0.1 \times 500\]\[E(B) = 297.5 + 450 + 250 + 50 = 1047.5\]3. **Alternative C:**\[E(C) = 0.35 \times 700 + 0.3 \times 1200 + 0.25 \times 500 + 0.1 \times 900\]\[E(C) = 245 + 360 + 125 + 90 = 820\]
02

Choose Alternative with Maximum Expected Value

Compare the expected values computed in Step 1:- \( E(A) = 785 \)- \( E(B) = 1047.5 \)- \( E(C) = 820 \)The alternative with the maximum expected value is Alternative B with an expected value of 1047.5.
03

Compute Opportunity Loss (Regret) for Each Alternative

To find the opportunity loss, we first determine the best payoff for each state \( (p = 0.35, 0.3, 0.25, 0.1) \) and then calculate the regret for each alternative by subtracting its payoff from this best payoff.For state \( p = 0.35 \): Best payoff is 1100 (A)- Regret(A): \( 1100 - 1100 = 0 \)- Regret(B): \( 1100 - 850 = 250 \)- Regret(C): \( 1100 - 700 = 400 \)For state \( p = 0.3 \): Best payoff is 1500 (B)- Regret(A): \( 1500 - 900 = 600 \)- Regret(B): \( 1500 - 1500 = 0 \)- Regret(C): \( 1500 - 1200 = 300 \)For state \( p = 0.25 \): Best payoff is 1000 (B)- Regret(A): \( 1000 - 400 = 600 \)- Regret(B): \( 1000 - 1000 = 0 \)- Regret(C): \( 1000 - 500 = 500 \)For state \( p = 0.1 \): Best payoff is 900 (C)- Regret(A): \( 900 - 300 = 600 \)- Regret(B): \( 900 - 500 = 400 \)- Regret(C): \( 900 - 900 = 0 \)
04

Compute Expected Opportunity Loss for Each Alternative

Using the regret values from Step 3, compute the expected regret for each alternative.The expected opportunity loss is calculated using the probabilities:1. **Alternative A:**\[E_{regret}(A) = 0.35 \times 0 + 0.3 \times 600 + 0.25 \times 600 + 0.1 \times 600\]\[E_{regret}(A) = 0 + 180 + 150 + 60 = 390\]2. **Alternative B:**\[E_{regret}(B) = 0.35 \times 250 + 0.3 \times 0 + 0.25 \times 0 + 0.1 \times 400\]\[E_{regret}(B) = 87.5 + 0 + 0 + 40 = 127.5\]3. **Alternative C:**\[E_{regret}(C) = 0.35 \times 400 + 0.3 \times 300 + 0.25 \times 500 + 0.1 \times 0\]\[E_{regret}(C) = 140 + 90 + 125 + 0 = 355\]
05

Choose Alternative with Minimum Expected Regret

Compare the expected regrets computed in Step 4:- \( E_{regret}(A) = 390 \)- \( E_{regret}(B) = 127.5 \)- \( E_{regret}(C) = 355 \)The alternative with the minimum expected regret is Alternative B with an expected regret of 127.5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Expected Value
In decision analysis, the expected value is an essential tool for making informed choices. When faced with uncertain outcomes, calculating the expected value helps you determine the most beneficial option on average. This is done by weighing each possible outcome by its probability and the payoff associated with it. For example, consider you have multiple alternatives (like A, B, and C in a payoff matrix) and a set of probable states of nature. Each alternative has a certain payoff for each state.
  • The formula for calculating expected value for an alternative is: \[ E(X) = p_1 \times x_1 + p_2 \times x_2 + p_3 \times x_3 + p_4 \times x_4 \]
  • Here, \( p_i \) represents the probability of each state of nature, and \( x_i \) is the payoff for that state.
  • By multiplying these values and summing them, you get the expected value, which represents the average payoff you can expect.
When deciding which alternative to choose, you generally pick the one with the highest expected value. In the exercise example, Alternative B turned out to be the best choice with a calculated expected value of 1047.5, indicating it has the most favorable average outcome under uncertainty.
Opportunity Loss
Opportunity loss, also known as regret, is a crucial concept when the goal is to minimize potential misses in decision-making. It reflects the cost of not selecting the best possible option for each state of nature.
  • To calculate opportunity loss, you first identify the best possible payoff for each state of nature.
  • Then, for every alternative, subtract its payoff in that state from the best payoff for that state. This shows the potential loss incurred from choosing that alternative.
Once you have quantified the regrets for each alternative across all states, the next step is to calculate the expected opportunity loss. Similar to the expected value, you weight each state's regret by its probability and sum them for an expected loss figure.
  • For instance, Alternative A in the provided exercise had an expected opportunity loss of 390, meaning choosing this option incurs a regret of 390 on average.
  • The decision rule here is to choose the alternative with the smallest expected opportunity loss.
Keeping regret low helps ensure you are not far from the best decision possible as the actual state of nature unfolds. In the exercise, Alternative B, with an expected regret of 127.5, was the optimal choice when aiming to minimize regret.
Payoff Matrix
A payoff matrix is a fundamental tool used in decision analysis to display possible outcomes of different alternatives in a structured manner. It captures the potential payoffs corresponding to each combination of alternative and state of nature.
  • Each row in the matrix represents an alternative decision you can make.
  • Each column addresses a different state of nature or condition the decision faces.
  • Within the cells of the matrix, you find the payoffs, which are the results of coupling a particular alternative with a specific state of nature.
For example, in the provided matrix:
  • Alternative A, when matched with state of nature one (with probability 0.35), has a payoff of 1100.
  • This matrix enables calculation of expected values and opportunity losses, as it offers all necessary input data for such analyses.
  • By examining the payoff matrix, you can visualize the consequences of each choice depending on what actually occurs, aiding in more strategic decision-making.
Using a payoff matrix offers a comprehensive view of the decision problem, supporting sound assessments and more informed decision-making processes.

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Most popular questions from this chapter

An oil company is considering making a bid on a new alternative energy contract to be awarded by the government. The company has decided to bid \(\$ 2.10\) billion. The oil company has a good reputation, and it estimates that it has a \(70 \%\) chance of winning the contract bid. If the oil company wins the contract, the oil company management has decided to pursue one of two options: either design an electric car or develop a new fuel substitute. The development cost of the new design for the electric car is estimated at \(\$ 300\) million. The estimated revenue and probabilities of success associated with developing and marketing the electric car are as follows: $$ \begin{array}{lcc} \hline \text { Event } & \begin{array}{c} \text { Probability of developing } \\ \text { and marketing } \end{array} & \begin{array}{c} \text { Estimated revenue } \\ \text { (in millions) } \end{array} \\ \hline \text { Extremely successful } & 0.7 & \$ 4500 \\ \text { Moderately successful } & 0.2 & \$ 2000 \\ \text { Weakly successful } & 0.1 & \$ 90 \\ \hline \end{array} $$ The fuel substitute development cost is estimated at \(\$ 170\) million. The estimated revenue and probabilities for success are given next: $$ \begin{array}{lcc} \hline \text { Event } & \begin{array}{c} \text { Probability of developing } \\ \text { and marketing } \end{array} & \begin{array}{c} \text { Estimated revenue } \\ \text { (in millions) } \end{array} \\ \hline \text { Extremely successful } & 0.6 & \$ 3000 \\ \text { Moderately successful } & 0.2 & \$ 2000 \\ \text { Weakly successful } & 0.2 & \$ 100 \\ \hline \end{array} $$ Construct a decision tree and determine the oil company's best strategy.

We have engaged in a business venture. Assume the probability of success is \(P(s)=2 / 5 ;\) further assume that if we are successful we make \(\$ 55,000\), and if we are unsuccessful we lose \(\$ 1750\). Find the expected value of the business venture.

Consider the steroid testing in baseball of Problem \(7 .\) Assume new data for the expanded roster has been collected, and our table now is as follows. Build a new decision tree and interpret the results: $$ \begin{array}{lcrr} \hline & \text { Positive results } & \text { Negative results } & \text { Totals } \\ \hline \text { Users } & 18 & 5 & 23 \\ \text { Nonusers } & 49 & 828 & 877 \\ \text { Totals } & 67 & 833 & 900 \\ \hline \end{array} $$

A new energy company, All Green, has developed a new line of energy products. Top management is attempting to decide on both marketing and production strategies. Three strategies are considered and are referred to as \(A\) (aggressive), \(B\) (basic), and \(C\) (cautious). The conditions under which the study will be conducted are \(S\) (strong) and \(W\) (weak) conditions. Management's best estimates for net profits (in millions of dollars) are given in the following table. Build a decision tree to assist the company determine its best strategy. $$ \begin{array}{lcc} \hline \text { Decision } & \text { Strong (with probability } 45 \%) & \text { Weak (with probability 55\%) } \\ \hline A & 30 & -8 \\ B & 20 & 7 \\ C & 5 & 15 \\ \hline \end{array} $$

A big private oil company must decide whether to drill in the Gulf of Mexico. It costs \(\$ 1\) million to drill, and if oil is found its value is estimated at \(\$ 6\) million. At present, the oil company believes that there is a \(45 \%\) chance that oil is present. Before drilling begins, the big private oil company can hire a geologist for \(\$ 100,000\) to obtain samples and test for oil. There is only about a \(60 \%\) chance that the geologist will issue a favorable report. Given that the geologist does issue a favorable report, there is an \(85 \%\) chance that there is oil. Given an unfavorable report, there is a \(22 \%\) chance that there is oil. Determine what the big private oil company should do.
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