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Chapter 3: Problem 34

More Retail Discount Wars Your Abercrom B men's fashion outlet has a \(30 \%\) chance of launching an expensive new line of used auto-mechanic dungarees (complete with grease stains) and a \(70 \%\) chance of staying instead with its traditional torn military-style dungarees. Your rival across from you in the mall, Abercrom A, appears to be deciding between a line of torn gym shirts and a more daring line of "empty shirts" (that is, empty shirt boxes). Your corporate spies reveal that there is a \(20 \%\) chance that Abercrom A will opt for the empty shirt option. The following payoff matrix gives the number of customers your outlet can expect to gain from Abercrom A in each situation: Abercrom \(\mathbf{A}\) \(\begin{array}{ll}\text { Torn Shirts } & \text { Empty Shirts }\end{array}\) Mechanics \(\left[\begin{array}{rr}10 & -40 \\ -30 & 50\end{array}\right]\) Abercrom B \(\quad\) Military What is the expected resulting effect on your customer base?

Short Answer

Expert verified
The expected resulting effect on Abercrom B's customer base is a decrease of \(9.8\) customers.

Step by step solution

01

Identifying Probabilities and Payoffs

We are given the following probabilities and payoff matrix: - Probability of Abercrom B choosing Mechanics: \(30\%\) - Probability of Abercrom B choosing Military: \(70\%\) - Probability of Abercrom A choosing Torn Shirts: \(80\%\) (Since the other option is \(20\%\)) - Payoff matrix: \[ \begin{array}{c|c|c} & \text {Torn Shirts } & \text {Empty Shirts} \\ \hline \text {Mechanics} & 10 & -40 \\ \text {Military} & -30 & 50 \end{array} \]
02

Calculating Expected Payoffs for Each Strategy

We can calculate the expected payoffs for Abercrom B for each strategy as follows: 1. Expected payoff if Abercrom B chooses Mechanics: \[E_{\text{Mechanics}} = (10 \times 0.8) + (-40 \times 0.2) = 8 - 8 = 0\] 2. Expected payoff if Abercrom B chooses Military: \[E_{\text{Military}} = (-30 \times 0.8) + (50 \times 0.2) = -24 + 10 = -14\]
03

Calculating the Expected Resulting Effect

To find the expected resulting effect on Abercrom B's customer base, we will apply the probabilities of choosing each strategy: Expected Resulting Effect = (Probability of Mechanics 脳 Expected payoff from Mechanics) + (Probability of Military 脳 Expected payoff from Military) Therefore, Expected Resulting Effect = \((0.3 \times 0) + (0.7 \times (-14)) = -9.8\) The expected resulting effect on Abercrom B's customer base is a decrease of \(9.8\) customers.

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Key Concepts

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

Probability
Understanding the role of probability in game theory is essential for predicting outcomes based on statistical likelihoods. Probabilities, which range from 0 to 1, are indicative of the chance of an event occurring. For instance, if an event has a probability of 0, it means it cannot occur, while a probability of 1 means it is certain.

In the given exercise, we navigate through distinct probabilities when two companies are making strategic decisions. Abercrom B has a 30% chance of choosing Mechanics clothes line and a 70% chance of opting for the Military clothes line. On the other hand, the rival Abercrom A has an 80% chance of choosing Torn Shirts and a 20% chance of going for Empty Shirts. By evaluating these probabilities, we can anticipate potential outcomes and strategize accordingly.

Moreover, probabilities can be combined with the effects of certain actions, which in this scenario are represented as the number of customers gained or lost, to compute an expected value, giving us a clearer picture of likely scenarios.
Expected Value
The expected value is a powerful concept in game theory that represents the average outcome if a certain action is repeated multiple times. In other words, it conveys the typical payoff or result to expect from a risky decision. Computing the expected value involves weighing each possible outcome by its probability and then summing all these values.

In our exercise, Abercrom B calculates the expected value for choosing either the Mechanics or Military line by taking the product of the payoffs for each scenario with their respective probabilities. The final figure represents an average, synthesizing multiple possible outcomes into a single predictive figure. Decisions in business and various other fields can be guided by the expected value as it quantifies the anticipated benefits or losses from certain actions in the context of inherent uncertainties.
Payoff Matrix
Moving on to the payoff matrix, which is a cornerstone of game theory, this tool lets us visualize and analyze the consequences of different decisions made by players in a strategic game. A payoff matrix presents the gains or losses (payoffs) for each combination of strategies executed by the players in a tabular format.

The payoff matrix in our Abercrom B vs. Abercrom A scenario reveals the number of customers Abercrom B expects to gain or lose depending on the clothing lines chosen by both companies. With rows representing Abercrom B鈥檚 strategies and columns for Abercrom A鈥檚 strategies, the matrix effectively summarizes all the potential outcomes. By interpreting a payoff matrix, businesses can assay the potential outcomes of their strategic interactions, thereby informing more calculated, savvy decisions.

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Most popular questions from this chapter

In 1990 the U.S. population, broken down by regions, was \(50.8\) million in the Northeast, \(59.7\) million in the Midwest, \(85.4\) million in the South, and \(52.8\) million in the West. \({ }^{4}\) Between 1990 and 2000, the population in the Northeast grew by \(2.8\) million, the population in the Midwest grew by \(4.7\) million, the population in the South grew by \(14.8\) million, and the population in the West grew by \(10.4\) million. Set up the population figures for 1990 and the growth figures for the decade as row vectors. Assuming that the population will grow by the same numbers from 2000 to 2010 as they did from 1990 to 2000 , show how to use matrix operations to find the population in each region in 2010 .

\mathrm{\\{} M a r k e t i n g ~ Y o u r ~ f a s t - f o o d ~ o u t l e t , ~ B u r g e r ~ Q u e e n , ~ h a s ~ o b - ~ tained a license to open branches in three closely situated South African cities: Brakpan, Nigel, and Springs. Your market surveys show that Brakpan and Nigel each provide a potential market of 2,000 burgers a day, while Springs provides a potential market of 1,000 burgers per day. Your company can finance an outlet in only one of those cities. Your main competitor, Burger Princess, has also obtained licenses for these cities, and is similarly planning to open only one outlet. If you both happen to locate at the same city, you will share the total business from all three cities equally, but if you locate in different cities, you will each get all the business in the cities in which you have located, plus half the business in the third city. The payoff is the number of burgers you will sell per day minus the number of burgers your competitor will sell per day.

If you think of numbers as \(1 \times 1\) matrices, which numbers are invertible \(1 \times 1\) matrices?

What would it mean if the total output figure for a particular sector of an input-output table were equal to the sum of the figures in the row for that sector?

In 1980 the U.S. population, broken down by regions, was \(49.1\) million in the Northeast, \(58.9\) million in the Midwest, \(75.4\) million in the South, and \(43.2\) million in the West. \({ }^{3}\) In 1990 the population was \(50.8\) million in the Northeast, \(59.7\) million in the Midwest, \(85.4\) million in the South, and \(52.8\) million in the West. Set up the population figures for each year as a row vector, and then show how to use matrix operations to find the net increase or decrease of population in each region from 1980 to 1990 .
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