Optimal Strategy With Zero-Sum Game

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Competency

This competency will allow you to demonstrate your ability and skill in analyzing zero-sum games and synthesizing optimal strategies within them.

Instructions

A night club owner has just changed ownership and the new owner has contracted G&B consulting to help decide the direction the club will take. Based on the local music scene, it would make the most sense to book either rock bands or country bands. The new owner is aware that a competing night club is hoping to attract new customers while the new club is getting itself established. Since these are the only two night clubs in town competing to book the same bands, whatever market share is gained by one is lost by the other making this a zero-sum game. The new club owner has hired you to find the optimum strategy for this situation.

What to Submit

To complete this assignment, you must first download the word document.

Your step-by-step breakdown of the problems, including explanations, should be present within the word document provided.

Nash Equilibria Analysis

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Competency

This competency will allow you to demonstrate your ability and skill in applying concepts of Nash equilibrium.

Instructions

A local law firm has hired G&B consulting to help determine a business strategy. There is a competing firm in the area and recently the two firms have begun talks in regards to a merger. The merger would mean there are more shared resources than either had before individually. There would also be less competition for both companies. Both would be able to operate more efficiently, however it would take a strong commitment from both sides for the merger to be successful. The law firm that has hired you wants to know their optimum strategy in this situation.

What to Submit

To complete this assignment, you must first download the word document.

Your step-by-step breakdown of the problems, including explanations, should be present within the word document provided.

PERT

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Binomial Distribution Analysis Binomial Distribution Analysis

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Competency

This competency will allow you to demonstrate your ability and skill in applying counting principles and analyzing binomial distributions.

Instructions

You have just been hired by G&B consulting and given your first assignment. A local independently wealthy citizen who happens to enjoy both gambling and baseball, but not math, has asked you to help him determine the likelihood of some different outcomes for the upcoming season.

What to Submit

To complete this assignment, you must first download the word document.

Your step-by-step breakdown of the problems, including explanations, should be present within the word document provided. If you use Excel for any of your calculations that file must also be included in the drop box.

Project

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All members of the group are to be involved in the computer part as well as in the write-up.You should respond to the questions posed below and write your report in the form of a paper with an introduction and conclusion.  Your report should include at least two pictures.Preliminary step.  Open MATLAB and type pplane8 in the command line.  If this program is on your computer (and in the MATLAB path) then the pplane window will open.  If not,then you need to download the program by visiting http://math.rice.edu/~dfield/ Go  to  this  site  in  a  browser  and  select  ‘pplane8.m’  and  save  the  file  as  pplane8.m  on  your computer.   As  long  as  this  file  is  in  the  MATLAB  path,  you  will  be  able  to  enter  the  word pplane8 in the command line and the program will run.  It produces menu driven windows that are somewhat self-explanatory.When you run pplane8, notice that one of the menus of the setup screen is labeled gallery.You will want to select ’predator prey’ from that menu for the problem below.  Then you enter various parameter values and hit ’proceed’ to go to the next step.  That opens a display screen with its own menus.  Under the heading solutions you will want to try ’plot several solutions’and ’find an equilibrium point’.The problem.  Investigate the non-linear, predator-prey systemx′=0.06x−0.0004yx,y′=−.08y+ 0.0002xy. The system models the populations of rabbitsxand foxesyin a hypothetical nature pre-serve, where timet is measured in months.Begin your report with an introduction of the problem.  In later paragraphs, deal with the following questions and tasks.Step 1.  Produce a picture showing the direction field and phase portrait of the system.  Plota selection of solution curves (trajectories) including the one referred to in Step 2 below.  Also,plot  the  equilibrium  point.   You  will  need  to  experiment  with  the  minimum  and  maximumpopulations in order to obtain a window that shows the equilibrium point and a good view of1

trajectories.  Explain the meanings of the direction arrows, the trajectories, and the equilibriumpoint (in terms of rabbits and foxes).  How can one tell from the picture which population isthe predator and which is the prey?  In other words, how can we tell whether foxes eat rabbitsor rabbits eat foxes?Step 2.  Assume that at time 0, the preserve is stocked with 200 rabbits and 50 foxes.  Fora second picture, obtain a plot of both the rabbit and fox populations versus time (using thegraph* menu of the display screen).  Discuss the behavior of the populations over time.  Yourdiscussion should include estimates of the minimums and maximums of each population andwhen they occur.  What is the period of oscillation of the rabbit and fox populations?  Is itreasonable that such populations would be cyclical?  If, at a low point in the fox population, anillness would wipe out the remaining foxes, what would then happen to the rabbit populationaccording to the model?  Would it still be cyclical?Step 3.  Assume you are managing the preserve discussed in Step 2.  You recognize that therabbit and fox populations are quite unstable, with populations swinging dramatically betweenhigh and low values, so you plan to trap and remove a number of foxes.  Based on the pictures,at what point should you take this step and about how many foxes should you remove in orderto  achieve  the  goal  of  making  the  population  somewhat  more  stable?   Discuss  options  andobserve what happen if the timing or the number is wrong.  At what point on the trajectorywould the removal of foxes have the most destabilizing effect?2

Grading Rubric for ProjectsMathematical Writing assignments involve writing skills as well as mathematical skills, sothe evaluation criteria include both.  The criteria for this assignment are:1.  Comprehensiveness (30%):(a)  all aspects of the assignment are covered;(b)  all questions stated in the assignment are answered;(c)  explanations are thorough;(d)  supporting arguments are included.2.  Mathematical Correctness (30%):(a)  appropriate mathematical and computer methods are used;(b)  mathematical steps are correct;(c)  mathematical logic is clear.3.  Expository Quality (30%):(a)  organization is logical and includes an introduction and conclusion;(b)  transitions from one idea to the next are smooth;(c)  mathematical work and computer output are smoothly integrated into the text;(d)  mathematical statements are presented within complete sentences;(e)  any figures or computer output are described in the exposition;(f)  grammar, spelling and punctuation are correct.4.  Extras (10%):(a)  the overall impact is positive;(b)  creativity and innovation are apparent.

Linear And Nonlinear Programming

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Linear And Nonlinear Programming

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Four problems for linear and nonlinear programming need to be solved

due December 8, New York, 6 p.m

There are four big questions, you have to answer them in English, you have to answer them in detail, either by hand or in electronic format as long as it’s clear enough

Pictures 1 and 2 are the professor’s notes

Math & Stats for Business

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Instructions

1. This assignment is to be done individually . This is not a group assignment.

1. The homework assignment is due in the beginning of the next class. Please bring a hard copy with you to the class.

1. If you have difficulty printing the assignment, please email the assignment to the instructor before the class.

1. Late or redone homework assignments will be accepted only at the final class (up to a total of two).

1. Only explained and detailed answers will be accepted.

1. During one holiday season, the Texas lottery played a game called the Stocking Stuffer. With this game, total instant winnings of $34.8 million were available in 70 million $1 tickets, with ticket pr4izes ranging from $1 to $1,000.

Shown here are the various prizes and the probability of winning each prize. Use these data to compute the expected value of the game, the variance of the game, and the standard deviation of the game. (8 points)

Prize (x) Probability P(x)
$1,000 .00002
100 .00063
20 .00400
10 .00601
4 .02403
2 .08877
1 .10479
0 .77176

2. The following data are the result of a historical study of the number of flaws found in a porcelain cup produced by a manufacturing firm. Use these data and the associated probabilities to compute the expected number of flaws and the standard deviation of flaws. (7 points)

Flaws Probability
0 .461
1 .285
2 .129
3 .087
4 .038

3. Determine the mean, variance, and standard deviation of the following discrete distribution. (5 points)

X P(x)
1 .238
2 .280
3 .177
4 .158
5 .137

Discrete Mathematics

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· The Assessment 1 Context document contains additional information about the concepts and language of sets, functions, sequences, and relations and their application to relational databases.

· Toggle Drawer

Resources

Suggested Resources

The following optional resources are provided to support you in completing the assessment or to provide a helpful context. For additional resources, refer to the Research Resources and Supplemental Resources in the left navigation menu of your courseroom.

Capella Resources

Click the links provided to view the following resources:

· Assessment 1 Context.

SHOW LESS

Capella Multimedia

Click the links provided below to view the following multimedia pieces:

· Functions and Relations | Transcript.

2. This presentation introduces the following topics.

1. Functions.

1. Modulus operator.

1. One to one functions.

1. Onto functions.

1. Composite functions.

1. Sequences.

1. Sigma notation.

1. Relations.

1. Order relations.

Library Resources

The following e-book from the Capella University Library is linked directly in this course:

· Koshy, T. (2004).  Discrete mathematics with applications . Burlington, MA: Elsevier Academic Press.

. Chapters 2, 3, and 7.

Course Library Guide

A Capella University library guide has been created specifically for your use in this course. You are encouraged to refer to the resources in the MAT-FP2051 – Discrete Mathematics Library Guide to help direct your research.

Bookstore Resources

The resource listed below is relevant to the topics and assessments in this course and is not required. Unless noted otherwise, this resource is available for purchase from the Capella University Bookstore. When searching the bookstore, be sure to look for the Course ID with the specific –FP (FlexPath) course designation.

· Johnsonbaugh, R. (2018). Discrete mathematics (8th ed.). New York, NY: Pearson.

. Chapter 3, “Functions, Sequences, and Relations,” is particularly useful for your work in this assessment. In addition to focusing on functions, sequences, and relations, this chapter discusses equivalence relations, matrices of relations, and relational databases.

· Assessment Instructions

Imagine your company asks you to organize a relational database containing employee information. A database has entries represented by rows and fields represented by columns. An entry corresponds to each employee, and each employee has various characteristics depicted by each field. In this database, you will store information for the following six fields for each employee: firstName, lastName, SS#, age, yearsAtCompany, and phoneNumber. Complete the following:

· In general, entries in a relational database can be represented as a set of n-tuples, where n is the number of fields. Provide 3 sample entries from 3 employees in the company. Feel free to devise their personal information in each field. Write out this information as a set of 3 6-tuples.

· Since the database has 6 fields and is represented as a set of 6-tuples, it is considered an n-ary relation where n=6. Let us focus in on two fields of our database: firstName and lastName. Given these two fields, we can define a binary relation, represented by the set of 2-tuples, which includes all entries in our database of these two fields. Note: The first entries in the 2-tuples represent elements from the domain, and the second entries in the 2-tuples represent elements from the range. Assume you have populated this database with all employee information. Do you believe this binary relation would be symmetric, transitive, and/or reflexive? Justify your answers. Feel free to use examples to help justify.

· In general, a function is a specific type of relation such that each element in the domain is related to exactly one element of the range. Given the binary relation composed by the fields firstName and lastName, is this relation a function? Justify your answer.

· In the context of databases, a key is a field in the database with the particular characteristic that each value in this field is related to only 1 entry (row) in the database. It is a unique identifier. Given this database, which field would be the best key? Justify your answer.

· Using 2 of the fields from this database, describe a binary relation such that the relation is a function. Define the domain and the range of this function. Describe this function onto (surjective), one-to-one, (injective), or both (bijective). Justify your answer. Feel free to use an illustration to aid in explanation.