# Decision tree analysis by maximizing expected value

The Blackduck Players

The Blackduck Players is a major league baseball expansion team beginning its third year of operation. The Players had losing records in each of its first two years and finished near the bottom of its division. However, the team was young and generally competitive. The team's general manager, Donavan West, and manager, Jackson Kendall, believe that with a few additional good players the Blackduck Players can become a contender for the division title and perhaps even for the pennant. They have prepared several proposals for free-agent acquisitions to present to the team's owner, Kimberly Duck.

In one proposal the team would sign several good available free agents including two pitchers, a good fielding shortstop, and two power-hitting outfielders for $52 million in bonuses and annual salary. The second proposal is less ambitious, costing $20 million to sign a relief pitcher, a solid, good-hitting infielder, and one power-hitting outfielder. The final proposal would be to stand pat with their current team and continue to develop.

General Manager West wants to lay out a possible season scenario for the owner so she can assess the long-run ramifications of each decision strategy. Because the only thing the owner understands is money, Donavan wants this analysis to be quantitative, indicating the money to be made or lost from each strategy. To help develop this analysis, Donavan hired his children, Michael and Brittany, both management science graduates from the University of Pennsylvania.

Michael and Brittany analyzed league data for the previous five seasons for attendance trends, logo sales (i.e., clothing, souvenirs, hats, etc.), player sales and trades, and revenues. In addition, they interviewed several other owners, general managers, and league officials. They also analyzed the free agents that the team was considering signing.

Based on their analysis, Michael and Brittany feel that if the Players do not invest in any free agents, the team will have a 25% chance of contending for the division title and a 75% chance of being out of contention most of the season. If the team is a contender, there is a 0.70 probability that attendance will increase as the season progresses and the team will have high attendance levels (between 1.5 and 2.0 million) with profits of $170 million from ticket sales, concessions, advertising sales, TV and radio sales, and log sales. They estimate a 0.25 probability that the team's attendance will be mediocre (between 1.0 and 1.5 million) with profits of $115 million, and a 0.05 probability that the team will suffer low attendance (less than 1.0 million) with profit of $90 million. If the team is not a contender, Michael and Brittany estimate that there is a 0.05 probability of high attendance with profits of $95 million, a 0.20 probability of medium attendance with profits of $55 million, and a 0.75 probability of low attendance with profits of $30 million.

If the team marginally invests in free agents at a cost of $20 million, there is a 50-50 chance it will be a contender. If it is a contender, then later in the season it can either stand pat with its existing roster or buy or trade for players that could improve the team's chances of winning the division. If the team stands pat, there is a 0.75 probability that attendance will be high and profits will be $195 million. There is a 0.20 probability that attendance will mediocre with profits of $160 million and a 0.05 probability of low attendance and profits of $120 million. Alternatively, if the team decides to buy or trade for players, it will cost $8 million and the probability of high attendance with profits of $200 million will be 0.80. The probability of mediocre attendance with $170 million in profits will be 0.15 and there will be a 0.05 probability of low attendance with profits of $125 million.

If the team is not in contention, then it will either stand pat or sell some of its players, earning approximately $8 million in profit. If the team stands, pat, there is a 0.12 probability of high attendance with profits of $110 million, a 0.28 probability of mediocre attendance with profits of $65 million, and a 0.60 probability of low attendance with profits of $40 million. If the team sells players, the fans will likely lose interest at an even faster rate and the probability of high attendance with profits of $100 million will drop to 0.08, the probability of mediocre attendance with profits of $60 million will be 0.22, and the probability of low attendance with profits of $35 million will be 0.70.

The most ambitious free-agent strategy will increase the team's chances of being a contender to 65%. This strategy will also excite the fans most during the off-season and boost ticket sales and advertising and logo sales early in the year. If the team does contend for the division title, then later in the season it will have to decide whether to invest in more players or not. If the Players stand pat, the probability of high attendance with profits of $210 million will be 0.80, the probability of mediocre attendance with profits of $170 million will be 0.15, and the probability of low attendance with profits of $125 million will be 0.05. If the team buys players at a cost of $10 million, then the probability of having high attendance with profits of $220 million will increase to 0.83, the probability of mediocre attendance with profits of $175 million will be 0.12, and the probability of low attendance with profits of $130 million will be 0.05.

If the team is not in contention, it will either sell some players' contracts later in the season for profits of around $12 million or stand pat. If it stays with its roster, the probability of high attendance with profits of $110 million will be 0.15, the probability of mediocre attendance will be 0.30 with profits of $70 million, and the probability of low attendance with profits of $50 million will be 0.55. If the team sells players late in the season, there will be a 0.10 probability of high attendance with profits of $105 million, a 0.30 probability of mediocre attendance with profits of $65 million, and a 0.60 probability of low attendance with profits of $45 million.

Assist Michael and Brittany in determining the best strategy to follow and its expected value.

A. Prepare your case analysis by performing a decision tree analysis of assisting Michael and Brittany's decision dilemma using expected value, and indicate the appropriate decision with these criteria.

B. Make sure you explain your modeling process.

https://brainmass.com/statistics/central-tendency/24440

#### Solution Summary

In this solution, we perform a decision tree analysis of assisting Michael and Brittany's decision dilemma using expected value, and indicate the appropriate decision with these criteria.

Decision analysis

A financial advisor has recommended two possible mutual funds for investment: Fund A and Fund B. The return that will be achieved by each of these depends on whether the economy is good, fair, or poor. A payoff table has been constructed to illustrate this situation.

(a) Draw a decision tree to represent this situation.

(b) Perform the necessary calculations to determine which of the mutual funds is better. Which one should you choose to maximize the expected value?

(c) Suppose there is a question about the return of Fund A in a good economy. It could be higher or lower than $10,000. What value for this would cause a person to be indifferent between Fund A and Fund B (i.e., the EMVs would be the same)?

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