Is this a single-parameter environment? Does the greedy allocation rule maximize social welfare? Prove the claim or construct an explicit counterexample.

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Homework 4

You can earn extra credit by typing up your solutions in LaTeX. I suggest you use Overleaf as a LaTeX editor.

• Exercise adapted from Problem 4.3:
Consider a set M of distinct items. There are n bidders, and each bidder i has a publicly known subset Ti ⊆ M of items that it wants, and a private valuation vi for getting them. If bidder i is awarded a set Si of items at a total price of p, then her utility is vixi − p, where xi is 1 if Si ⊇ Ti and 0 otherwise. Since each item can only be awarded to one bidder, a subset W of bidders can all receive their desired subsets simultaneously if and only if Ti ∩ Tj = ∅ for each distinct i, j ∈ W .

(a) Is this a single-parameter environment? Explain fully.

(b) The allocation rule that maximizes social welfare is well known to be NP hard (as the Knapsack auction was) and so we make a greedy allocation rule. Given a reported truthful bid bi from each player i, here is a greedy allocation rule:
(i) Initialize the set of winners W = ∅, and the set of remaining items X = M.
(ii) Sort and re-index the bidders so that b1 ≥ b2 ≥ · · · ≥ bn.
(iii) For i = 1, 2, 3, . . . , n :

If Ti ⊆ X, then:
– Delete Ti from X. – Add i to W .

(iv) Return W (and give the bidders in W their desired items).
Is this allocation rule monotone (bidder smaller leads to a smaller cost)? If so, find a DSIC auction based on this allocation rule. Otherwise, provide an explicit counterexample.

(c) Does the greedy allocation rule maximize social welfare? Prove the claim or construct an explicit counterexample.
• Exercise 6.4
• Exercise 7.4
• Exercise 9.5

• Exercise 10.5
• Exercise 10.6

Comment on Exercise 9.5 Here is a more clear description of the Random Serial Dictatorship
algorithm:
(i) Each agent submits a ranked list of house preferences.
(ii) The agents are randomly ordered (independent of their ranked list of house preferences).
(iii) The agents are considered in order. When agent i is considered, she receives her top-ranked option that is still availa

Is the correlation you obtained in cell A13 weak, moderate or strong?  Is the correlation you obtained in cell A13 positive or negative?

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Correlation coefficient

  • We would like learn about the correlation coefficient using Excel. Lets look at the following question.
  • To find out how genetically related the cholesterol level is, they measured the cholesterol level of eight mothers and their daughters. They obtained the following data.
  • Mother Her daughter
  • 157                           154
  • 189                           150
  • 201                           184
  • 174                           170
  • 159                           158
  • 213                           192
  • 149                           143
  • 143                           132
  • Evaluate the correlation coefficient and describe the correlation.
  • We will now learn how to use Excel to quickly check our answers.

Open Excel

In cell A1 type Mother

In cell B1 type Her daughter

Below Mother type the data from the exam one by one with pressing Enter in between: 157,189,201,174,159,213,149,143

Below Her Daughter  type the data from the exam one by one with pressing Enter in between: 154,150,184,170,158,192,143,132

In cell A11 type Correlation coefficient

In cell A13 begin typing =Cor for correlation coefficient

After you typed =Cor you will only be offered one choice =CORREL, double click on it.

Then highligh all data under Mother type ,

Then highligh all data under Her daughter

You should obtain the correlation coefficient.  To learn more about correlation coefficient study the textbook.

Exercises

Exercise 1.  Is the correlation you obtained in cell A13 weak, moderate or strong?  State your result in cell A14.

Exercise 2.  Is the correlation you obtained in cell A13 positive or negative?  State your result in cell A15.  Explain what it means in cell A16.

Prove that the greedy algorithm in the proof of Theorem 4.2 always computes an optimal fractional knapsack solution. Prove that the three-step greedy knapsack auction allocation rule in Section 4.2.2 is monotone. Does it remain monotone with the two optimizations discussed in the footnotes?

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Algorithmic Mechanism Design

Exercise 4.1

Consider an arbitrary single-parameter environment, with feasible set X. Prove that the welfare-maximizing allocation rule

x(b) = argmax(x1,…,xn)∈X n∑ i=1 bixi

[Assume that ties are broken in a deterministic and consistent way, such as lexicographically.]

Exercise 4.2

Continuing the previous exercise, restrict now to feasible sets X that contain only 0-1 vectors—that is, each bidder either wins or loses. We can identify each feasible outcome with a “feasible set” of bidders (the winners). Assume that for every bidder i, there is an outcome in which i does not win. Myerson’s payment formula (3.5) dictates that a winning bidder pays her “critical bid”—the infimum
of the bids at which she would continue to win. Prove that, when S∗ is the set of winning bidders under the allocation rule (4.2) and i ∈ S∗, i’s critical bid equals the difference between (1) the maximum social welfare of a feasible set that excludes

i; and (2) the social welfare ∑ j∈S∗\{i} vj of the bidders other than i in the chosen outcome S∗.

[In this sense, each winning bidder pays her “externality”—the welfare loss she imposes on others.]

Exercise 4.3 Continuing the previous exercise, consider a 0-1 single- parameter environment. Suppose you are given a subroutine that, given bids b, computes the outcome of the welfare-maximizing allocation rule (4.2).
(a) Explain how to implement a welfare-maximizing DSIC mechanism by invoking this subroutine n + 1 times, where n is the number of participants.
(b) Conclude that mechanisms that are ideal in the sense of Theorem 2.4 exist for precisely the families of single-parameter environments in which the welfare-maximization problem (given b as input, compute (4.2)) can be solved in polynomial time.

Exercise 4.4

Prove that the greedy algorithm in the proof of Theorem 4.2 always computes an optimal fractional knapsack solution.

Exercise 4.5

Prove that the three-step greedy knapsack auction allocation rule in Section 4.2.2 is monotone. Does it remain monotone with the two optimizations discussed in the footnotes?

Exercise 4.6

Consider a variant of a knapsack auction in which we have two knapsacks, with known capacities W1 and W2. Feasible sets of this single-parameter environment now correspond to subsets
S of bidders that can be partitioned into sets S1 and S2 satisfying∑ i∈Sj wi ≤ Wj for j = 1, 2.
Consider the allocation rule that first uses the single-knapsack greedy allocation rule of Section 4.2.2 to pack the first knapsack, and then uses it again on the remaining bidders to pack the second knapsack. Does this algorithm define a monotone allocation rule? Give either a proof of this fact or an explicit counterexample.

Exercise 4.7 (H)

The revelation principle (Theorem 4.3) states that (direct-revelation) DSIC mechanisms can simulate all other mechanisms with dominant-strategy equilibria. Critique the revelation principle from a practical perspective. Name a specific situation where you might prefer a non-direct-revelation mechanism with a dominant- strategy equilibrium to the corresponding DSIC mechanism, and ex-plain your reasoning.

Problems
Problem 4.1

Consider a variant of a knapsack auction in which both the valuation vi and the size wi of each bidder i are private. A mechanism now receives both bids b and reported sizes a from the bidders. An allocation rule x(b, a) now specifies the amount of capacity allocated to each bidder, as a function of the bids and reported sizes. Feasibility dictates that ∑n i=1 xi(b, a) ≤ W for every b and a, where W is the total capacity of the shared resource. We define the utility of a bidder i as vi − pi(b, a) if she gets her required capacity (i.e., xi(b, a) ≥ wi) and as −pi(b, a) otherwise. This is not a single-parameter environment. Consider the following mechanism. Given bids b and reported sizes a, the mechanism runs the greedy knapsack auction of Sec- tion 4.2.2, taking the reported sizes a at face value, to obtain a subset of winning bidders and prices p. The mechanism concludes by awarding each winning bidder capacity equal to her reported size ai, at a price of pi; losing bidders receive and pay nothing. Is this mechanism DSIC? Prove it or give an explicit counterexample.

Problem 4.2

Section 4.2.2 gives an allocation rule for knapsack auctions that is monotone, guarantees at least 50% of the maximum social welfare, and runs in polynomial time. Can we do better? We first describe a classical fully polynomial-time approximation scheme (FPTAS) for the knapsack problem. The input to the problem is item values v1, . . . , vn, item sizes w1, . . . , wn, and a knapsack capacity W . For a user-supplied parameter ǫ > 0, we consider the following algorithm Aǫ; m is a parameter that will be chosen shortly.
• Round each vi up to the nearest multiple of m, call it v′ i.
• Divide the v′ i’s through by m to obtain integers ̃v1, . . . , ̃vn.
• For item values ̃v1, . . . , ̃vn, compute the optimal solution using a pseudopolynomial-time algorithm.

[You can assume that there exists such an algorithm with run- ning time polynomial in n and maxn
i=1 ̃vi.]

(a) Prove that if we run algorithm Aǫ with the parameter m set to ǫ(maxn i=1 vi)/n, then the running time of the algorithm is polynomial in n and 1 ǫ (independent of the vi’s).

(b) (H) Prove that if we run algorithm Aǫ with the parameter m set to ǫ(maxn i=1 vi)/n, then the algorithm outputs a solution with total value at least 1 − ǫ times the maximum possible.

(c) Prove that if we run algorithm Aǫ with the parameter m set to a fixed constant, independent of the vi’s, then the algorithm yields a monotone allocation rule.

(d) Prove that if we run algorithm Aǫ with the parameter m set as in (a) and (b), then the algorithm need not yield a monotone allocation rule.

(e) (H) Give a DSIC mechanism for knapsack auctions that, for a user-specified parameter ǫ and assuming truthful bids, outputs an outcome with social welfare at least 1−ǫ times the maximum possible, in time polynomial in n and 1

Problem 4.3

Consider a set M of distinct items. There are n bidders, and each bidder i has a publicly known subset Ti ⊆ M of items that it wants, and a private valuation vi for getting them. If bidder i
is awarded a set Si of items at a total price of p, then her utility is vixi − p, where xi is 1 if Si ⊇ Ti and 0 otherwise. This is a single parameter environment. Since each item can only be awarded to one
bidder, a subset W of bidders can all receive their desired subsets simultaneously if and only if if Ti ∩ Tj = ∅ for each distinct i, j ∈ W .

(a) (H) Prove that the problem of computing a welfare-maximizing feasible outcome, given the vi’s and Ti’s as input, is N P-hard.

(b) Here is a greedy algorithm for the social welfare maximization problem, given bids b from the bidders. initialize W = ∅ and X = M sort and re-index the bidders so that b1 ≥ b2 ≥ · · · ≥ bn for i = 1, 2, 3, . . . , n do if Ti ⊆ X then remove Ti from X and add i to W return winning bidders W

54 Algorithmic Mechanism Design Does this algorithm define a monotone allocation rule? Prove it or give an explicit counterexample.

(c) (H) Prove that if all bidders report truthfully and have sets Ti of cardinality at most d, then the outcome of the allocation rule in (b) has social welfare at least 1 d times that of the maximum possible

Compose a well-organized essay discussing the three topics you most enjoyed learning in MTH100.

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Three Topic

ASSIGNMENT

Compose a well-organized essay discussing the three topics you most enjoyed learning in MTH100.

Use the five-paragraph format and make sure the following is contained in your essay:                               • Topic must be explained utilizing words, not math symbols (i.e. use the words greater than, not symbol (>).

• You must explain why you enjoyed the topic.

• Each topic should include an application to real-life.

Devise a sampling strategy that combines both systematic and stratified sampling techniques. Explain this strategy in detail. Devise a sampling strategy that is both random and systematic. Explain your strategy in detail.

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Analysis of Class Database Survey: Types of Data, Sampling and Assessing Bias


1) Consider the survey we have at hand. It was designed to harvest a plethora of measurements to
better understand the metrics of High School Students in Ontario in general.

  • a) Consider the technique I used to sample the members of our class. Again, if the population we
    were intending to study was all Ontario high school students … then what type of sampling
    technique did I use? Justify your Answer. (2)
  • b) Let’s say you now want to use this survey (as is) to sample High School students in Ontario.
    You perform a Multi–stage sample and wind up choosing Loyola as a site for sampling. You contact the school administration and receive the following information to help you determine any and all of the amount of each “type” of student you intend to select:

M:F Ratio = 44:56 Average age: 15.7 yrs
Median Age: 15.5 yrs
# Grade 9’s 271
# Grade 10’s 328 You can estimate any other demographic you
# Grade 11’s 220 like and use this in your sampling technique
# Grade 12’s 138
PIP students 18

Note: You will have access to any and all information and equipment necessary to do this sampling
(provided it can be found at the school). Funding for supplies and a staff of two volunteers will be
provided for you.

  • i. Devise a sampling strategy that combines both systematic and stratified sampling techniques. Explain this strategy in detail. (2)
  • ii. Devise a sampling strategy that is both random and systematic. Explain your strategy in detail (2)
  • iii. A colleague of yours tells you that they would not use any of the above strategies.

Instead they suggest the following:

1. Stratify the population as follows:
Grade Total # of classes Classes Sampled
9 11 5
10 14 5
11 8 5
12 6 5
PIP 2 1

2. Randomly choose the required number of classes by drawing their teacher’s names ‘out of a hat’.

3. Sample the entire class.

4. Have the teachers sort the survey’s alphabetically.

5. Choose every other survey and those remaining surveys become your sample. Explain two problems that exist with this strategy and then describe how you could use their framework to effectively perform a random, stratified, and systematic sample. (4)

2) Before we implement our study we must consider the quality of our survey.

a) We have gathered many different types of data and as such may need to treat each differently.
ON YOUR SURVEY indicate which of the data is Categorical (C), Discrete (D) or Continuous (I).
Place these labels to the left of the question number at the beginning of each statement in
the survey.

b) There is some question as to whether or not the response(s) to question #37 constitutes a discrete or a continuous variable. Choose which one is appropriate and defend your position VIGOROUSLY!!!

c) Many of these questions can solicit incorrectly measurements when the survey is administered
to uninformed students. Outline the flaws in question #1, #4, #34. Without throwing the question out of the survey, repair each question to ensure it generates an effective measurement. (4)

3) There are many sources of error that would act to introduce bias in our information gathering
process. We should look to our survey to ensure we eliminate these ahead of time and ensure we
measure the true patterns that exist in our population.

a) Indicate at least three instances where measurement bias will occur in the gathering of the data
fro this survey. Address leading and loaded questions in the selection of these instances (you
should have one example of each). Explain your reasoning for these choices and either fix them
or suggest how one would go about doing so. (8)

b) Use question 16 – 18 to explain the concept of response bias. (2) THEN, outline two other
questions that would elicit this type of bias and correct for this bias by providing adequate
editing, conditions, and/or controls to ensure the responses to said questions have a decreased
propensity for response bias. (4)

c) Outline strategies that you could reasonably and legally implement to remove the likelihood of
non–response bias (and explain how they would be effective in lessening this problem!). A
minimum of two ideas (with explanations!) is suggested. (4)

Note – You can be creative and you can use any control that an administrator or teacher has at
their disposal. However, you must consider the rights and responsibilities these people have to
students, parents, to the school board, and to their professional/ethical governing institutions!

Identify the variable(s) that you will collect, including how they will be measured. Describe your plan for collecting the necessary data. Describe your plan for analyzing the data you collect.

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Bivariate Quantitative Data

1. Choose your variable(s) and identify a research question (Must be Bivariate Quantitative Data)
2. Devise a plan to collect your data

1. a) Collect and organize data
2. b) Conduct the appropriate analysis
3. c) Write your results in a report

Part 1
a) State the research question you have chosen.
b) Specify the population(s) that will be represented in your study.
c) Identify the variable(s) that you will collect, including how they will be measured.
Part 2
Describe your plan for collecting the necessary data. This should include:
a) A description of where you will obtain your data (website, physical location, etc.)
b) A sampling strategy
c) A copy of your survey if you intend to use one

5. Describe your plan for analyzing the data you collect.

Example of A Typical Project:
Purpose Statement: We want to find out about students’ attitudes about cheating.
Population: College students

Survey
1. Have you ever cheated on a test in college? Yes No
2. Do you think that turning in an assignment or paper that was not your original work is wrong? Yes No
3. What is your GPA?
4. How many hours a week do you spend studying?

For each sub-field, state what you perceive as the one or two most interesting questions or problems in this area. Explain why these sorts of questions interest you.

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Mechanical Engineering

Articulate why the research fields chosen on the previous page are intriguing and exciting to you. For each sub-field, state what you perceive as the one or two most interesting questions or problems in this area. Explain why these sorts of questions interest you. Your responses are shared with mentors.Respond with clarity (0 of 5000 characters)

The first research field I have chosen is Data Science, and the sub-field related to data science is Statistics.

The second research field is Mechanical Engineering, and the sub-field is Robotics. Make some relationship between Robotics and Data science.

Plot the constant value contours for 𝑒(𝑥1, 𝑥2), and explain the magnitude of the error at various locations in the context of the problem.

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Report in R Markdown

Problem 1: Graphing a bivariate function and its second order approximation

Use R to graph a 3–d plot of the function 𝑓(𝑥1, 𝑥2) = cos⁡(𝑥1𝑥2) and its second order Taylor approximation
ℎ(𝑥1, 𝑥2) = 1 − 𝜋2 8 𝑥1

2. Following are the requirements:

i. [1 pt] Write an R function that produces the function 𝑓.

ii. [1 pt] Write an R function that produces the function ℎ.

iii. [3 pts] Use the functions in (i) and (ii) to plot a 3d graph that contains the surfaces for both 𝑓 and ℎ on the same
frame (axes). Color surface of 𝑓 red and surface of ℎ blue. The values of 𝑥1 for your graph should range in the
interval [− 𝜋
4 , ⁡𝜋
4 ] and the values of 𝑥2 must range in the interval [𝜋
4 , ⁡3𝜋
4 ]. You may consider dividing each of the intervals by 30 equal points and evaluate the function on the 30 by 30 grid generated, and then make your plot.

iv. [1 pt] Label the first axis 𝑥1and the second axis 𝑥2. Your graph should have the title “Taylor approximation of
cos⁡(𝑥1𝑥2).” Make sure to show the plot from an angle with a good view of the function.

v. [3 pts] In a separate 3d plot, graph the absolute value error function 𝑒(𝑥1, 𝑥2) = |𝑓(𝑥1, 𝑥2) − ℎ(𝑥1, 𝑥2)|. Use the
same range and grid for 𝑥1 and 𝑥2 as described in (iii).

vi. [1 pt] Label the first axis 𝑥1and the second axis 𝑥2. Title your plot “The error in second order Taylor expansion of
cos⁡(𝑥1𝑥2).”

vii. [3 pts] Plot the constant value contours for 𝑒(𝑥1, 𝑥2), and explain the magnitude of the error at various locations
in the context of the problem.

Problem 2:

Given a 𝑝 × 1vector 𝝁 and a 𝑝 × 𝑝 positive definite matrix Σ, the pdf for a 𝑝–variate normal density at a point
𝒙 = (𝑥1, 𝑥2, … , 𝑥𝑝)𝑇can be written as
𝑓(𝒙) = (2𝜋)−𝑝
2 |Σ|−1/2 exp [− 1
2 (𝒙 − 𝝁)𝑇 Σ−1(𝒙 − 𝝁)] .
Now consider the bivariate normal random variable, where
𝒙 = (𝑥1
𝑥2) ,⁡⁡⁡⁡𝝁 = (𝜇1
𝜇2) ,⁡⁡⁡⁡⁡⁡Σ = (𝜎11 𝜎12
𝜎21 𝜎22).

(a) [3 pts] Write the second order Taylor expansion for 𝑓(𝒙), for the bivariate normal density, around the point
𝒙𝟎 = (𝜇1 /𝜇2).

(b) [3 pts] Graph a 3–d plot of the function 𝑓(𝑥1, 𝑥2) and its second order Taylor expansion for the following parameters
(for each set of parameters, f and its approximation should be on the same frame):
(i)⁡⁡⁡𝝁 = (0
0) ,⁡⁡⁡⁡⁡⁡Σ = ( 1 −0.3
−0.3 1 ) ⁡;⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡(ii)⁡⁡⁡⁡⁡⁡⁡⁡⁡⁡𝝁 = (0
0) ,⁡⁡⁡⁡⁡⁡Σ = ( 1 0.8
0.8 1 )

 

Note that the means and variances for each of the variables in the cases (i) and (ii) are 0 and 1 respectively. This should
guide you an idea for a reasonable range for 𝑥1 and 𝑥2 to consider for your graphs.

 

(c) [3 pts] Graph the constant value contours for 𝑓(𝒙) for the cases (i) and (ii) in part (b). What is the shape of the
constant value contours? What is the center of the constant value contours?

(d) [3 pts] Compute the eigenvalues and eigenvectors for each of the covariance matrices 𝛴 given in part (b).
Superimpose the eigenvectors on each of their corresponding constant value contours that you drew in part (c) and
explain how the eigenvectors and eigenvalues are related to the constant value contours.

Explain why it is important to be able to solve math problems Using different styles? How does this help future teachers become successful in the classroom?

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Solve math problems

In 250 words, Explain why it is important to be able to solve math problems Using different styles? How does this help future teachers become successful in the classroom?

Identify the mathematical concept chosen, and briefly describe its real-life application. Research your topic and include as much information as you reasonably can about your topic. Begin by summarizing what we have learned in this course about the topic, and then explain how it applies to the area that you have chosen to examine.

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Culminating Activity: Characteristics of Functions(Trigonometric Functions)

Each of these topics has a wide variety of real-life applications. For your culminating activity, your task is to choose a topic from this course, and examine one of its applications in the real world. The topic you choose is up to you, and the application you examine is also up to you. Some possible examples are
listed at the bottom of this page. Your completed project should include the following elements:
1) Title Page

2) Clear statement of topic that is being examined
• Identify the mathematical concept chosen, and briefly describe its real-life application.

3) Background Research/Prior Knowledge
• Research your topic and include as much information as you reasonably can about your topic. Begin by summarizing what we have learned in this course about the topic, and then explain how it applies to the area that you have chosen to examine. You will need to cite your sources accurately here (i.e., page #s of textbook), and be sure that you choose reliable sources of information.

4) Sample Problem and Solution
• Create a sample word problem (including an equation) from the area that you have chosen to examine, and include a full, detailed solution of the problem. Your problem and solution should be of a complexity level appropriate to the level of this course and be solved using a method taught in class. Refer to the appropriate Chapter in your textbook or class notes for ideas on application type questions.

5) Conclusion
• Summarize what you have learned about the application you have chosen. Discuss any things which you are still not sure about, or what you would like to learn next about this topic.

6) References
• A complete list of all sources that you have consulted in completing this project. You may use textbooks, websites, and other reliable sources of information. Your list of references should be in APA format.

Examples of Topics:
• Polynomial Functions and Architecture
• Rates of Change and Kinematics
• Trigonometric Functions and Electromagnetic Waves
• Logarithmic Functions and Seismic Activity (i.e., the Richter Scale)
• Exponential Functions and Population Growth
• Any number of others…choose something that is of interest to you!