# MATHEMATICS

**Part 1**

The first item you want to highlight is your customer satisfaction record. Over the past 18 months of employment you have worked with 25 clients, and according to the customer satisfaction surveys all clients are asked to take after their contract is completed, 22 of them rated you with the highest level of satisfaction. Use this information to find the best predicted probability of having a new client give the highest level of satisfaction.

However, having a single satisfied customer is not enough. It is very important to G & B Consulting that they maintain high ethical standards and continue to receive the highest rating from the Better Business Bureau consistently. In order to maintain this rating, the company must maintain at least 85% or higher of customers giving you high customer satisfaction ratings. Internal projections predict that G&B Consulting will serve 60 clients over the next year. If you become the project manager and use your prior record as an indicator, find the probability that 85% or more of new clients will give the highest level of customer satisfaction rating. Show all calculations and formulas used. If you use Excel for your calculations include that along with your submission. Use this information to make a convincing argument that you are a good choice for the position.

**Slide 1**

· Calculate the probability of getting a satisfied client based off of your prior work history at G& B Consulting. Show all work.

· Explain how you found your solution.

**Slide 2**

· Using your prior work history, calculate the probability of getting at least 85% of clients giving you high customer satisfaction ratings.

· Explain how you found your solution.

**Slide 3**

· Interpret your results from slide 2 and use them to make an argument on why you would be good for the position.

**Part 2**

The other item you wish to highlight comes from some work done for a local manufacturer. They believed that a competitor was illegally using one of their patents in their own manufacturing process and were considering litigation. This competitor’s product were directly cutting into the manufacturer’s profit, so they wanted to prevent the competitor’s product from being made and recoup those lost profits. Of course, the litigation process is long and costly, and the outcome is not guaranteed and would likely result in a countersuit brought by the competitor. Before continuing with costly legal counsel, the manufacturer hired G&B Consulting to help them determine their best strategy.

This was potentially a very lucrative contract, and a high profile one too since the manufacturer is one of the biggest employers in the area. Therefore, several analysts at G&B Consulting, including you, were asked to tackle the problem independently to help ensure the best possible results. Before compiling a final report to give to the manufacturer, the results were presented to the project lead:

The first solution brought up was given by one of your coworkers. Using the manufacturer’s projections of profits solely, he was able to create the following payoff matrix where each entry is in millions of dollars annually:

**Competitor**

**Manufacturer**

**Sue**

**Don’t Sue**

**Sue**

(5, -5)

(20, -20)

**Don’t Sue**

(-15, 15)

(-10, 10)

This coworker concluded that the manufacturer should always do the opposite of the competitor chooses making this a strictly competitive game.

**Slide 4**

· Using the payoff matrix shown above, determine if the manufacturer has a dominant strategy. Show and explain all steps.

· Using the payoff matrix shown above, determine if the competitor has a dominant strategy. Show and explain all steps.

**Slide 5**

· Find all Nash equilibrium points. Show and explain all steps.

· Identify the optimum strategy of the game.

**Slide 6**

· Do your results match those of your coworkers? Explain why you agree or disagree

**Part 3**

The second solution was brought by a different coworker. She also created a payoff matrix for the scenario but did some independent research to estimate the competitor’s profits in each outcome independent of the manufacturer’s profits and came up with the following payoff matrix, again, the values represent millions of dollars in annual profits.

**Competitor**

**Manufacturer**

**Sue**

**Don’t Sue**

**Sue**

(5, -5)

(20, 10)

**Don’t Sue**

(10, 20)

(15, 15)

This coworker concludes that under these circumstances, the manufacturer should sue 50% of the time and not sue the other 50% of the time, and they should expect their competitor to do the same.

**Slide 7**

· Use the mixed strategy algorithm to find the optimum strategy for the manufacturer. Show all of your steps.

**Slide 8**

· Use the mixed strategy algorithm to find the optimum strategy for the competitor. Show all of your steps.

**Slide 9**

· Do your results match those of your coworker’s? Explain why you agree or disagree.

**Part 4**

You present your own solution that is based on a non-simultaneous game where the manufacturer first has to decide whether or not they wish to pursue litigation. If they do so choose, then their competitor will also have to decide if the wish to file a counter-suit. You utilized the same payoff matrix that the second coworker provided that contains information about both companies’ projected profits

**Slide 10**

· Create the game tree for this scenario, exclude any non-credible threats.

· Explain how you identified the non-credible threat.

**Slide 10**

· Perform the first step of backwards induction. Shown and explain your work.

**Slide 11**

· Perform the second step of backwards induction. Show and explain all your work.

**Slide 12**

· Identify the optimum strategy for the game.