Game Theory. Due in 2 hours!!!

HW 1: Answer sheet Instructor: Dr. Dai ZUSAI Econ 3519, Temple University

Your name: (Example: Nash, John F.)

Problem 1 Q0 & Q2

Student 2 5 4 3 2 1 0

S tu

d en

t 1

5 4 3 2 1 0

Q1 1) Student 1’s remaining strategies Student 2’s

Q3

Q4 i) Q4 ii)

Problem 2 Q1 i) Player 1’s remaining strategies Player 2’s Player 3’s Q1 ii)

Q2

Q3

Q4 i) Q4 ii)

Deleted

strategy

Dominating

strategy

Order of

knowledge Example:

1’s S 1’s T 2

Deleted

strategy

Dominating

strategy

Order of

knowledge

Q1 ii) Example: player 1’s strategy S is deleted in IDSDS, because 1’s own strategy T is strictly dominates S

under 2nd order knowledge of rationality

(i.e., knowing other’s rationality).

HW 1/ Econ 3519

Your name: (Example: NASH, John F.)

Problem 3 Q1. 1) [Player 1’s exp payoff of x]= [Player 1’s exp payoff of y]=

[Player 2’s exp payoff of x]= [Player 2’s exp payoff of y]=

2) [p in Player 1’s best response]=

[q in Player 2’s best response]=

Q2. (p,q)=

Homework for Part 1: Strategic form analysis Instructor: Dr.Dai ZUSAI Econ 3519, Temple University Due: February 14, 2018, 11:59 pm

Try each problem/question right after finishing the referred topic’s lectures.

Do not procrastinate until the due date!

For exam prep, do also the end of chapter problems.

Instructions If your HW does not meet any of these, points may be deducted or your HW may not be accepted.

 Please submit the HW on Canvas. Merge all the pages into a single pdf file. Unreadable handwriting may be returned with zero grade and no comments.

 Use the answer sheet(s), if provided. Attach paper as much as needed for additional explanation.

 If you submit a scan of your handwriting, please make a black-white scan with appropriate contrast, not full color or gray scale. 300-dpi is a recommended resolu- tion. You should check the scanner setting and also the scan result. A camera is not recommended, even if you use a camera scan app of iPhone or Android. Use a

scanner at Paley library or computer labs, if you don’t have one: http://guides.temple.edu/c.php?g=78259. If your file size is bigger than 1MB, it is too big (and an- noys me quite a lot).

 Never copy your classmates’ answers or an Internet website. Plagiarism will be severely punished.

 The absolute score, not the percentage, will be counted into the grading formula. Bonus points will be just added to your score of this HW.

 You must submit the HW on Canvas. If and only if the Canvas system is down around (an hour before) the deadline, you can send me the HW by e-mail. (Other- wise, please not. Remember I can check the system status history; so don’t tell a lie!) Although still I will ask you to upload it to Canvas later by the next day (unless

the system down persists so long), I will take it as an evidence that you’ve finished the HW by the deadline and will not impose the penalty. But, whether on Canvas

or by e-mail, you must submit it by the deadline. No late homework will be accepted.

Problem 1: Ch.3 Exercise 10 (Topics 1−3) 50 points By answering the following questions, solve Exercise 10 in the end of Chapter 3 (Harrington, Ed.2).

Q0 (Topic 1). Write the payoff table of the game in this exercise (as a complete visualize its strategic form; the payoff table must

show all the components of the strategic form). Attach a paper to explain the process to get the payoff table.

Q1 (Topic 2). Do the iterative deletion of strictly dominated strategies (IDSDS) and answer the following questions.

1) List the strategies that finally survive IDSDS 2) For each of the deleted strategy, state the strategy that dominates the deleted strategy (dominating strategy) and the order of

knowledge of rationality that is needed for this deletion. (On the answer sheet, each row corresponds with deletion of one strat-

egy: i.e., you can write only one strategy in the left column. So one “round” can be expressed by several rows. Of course, you

may not use all rows.)

Q2 (Topic 3). Completely identify the best responses of each player. (Circle corresponding payoffs.) State the reason briefly or show

the process to find equilibria. If you want to visualize best responses on the payoff table, circle the corresponding payoffs in the

payoff table that you wrote for Q1, as the textbook does.

Q3 (Topic 3). State all the Nash equilibria. Caution: An equilibrium must be a strategy profile.

Q4 (Topic 3). If there are multiple Nash equilibria, answer the following questions; briefly explain the procedure to derive your an-

swers. If there is only one Nash equilibrium, answer “NA”. Ignore the strategies that are deleted in IDSDS: think as if they did not

exist at all.

1) Which equilibria are eliminated and which remain by equilibrium Pareto (equilibrium payoff-dominance) principle? 2) Which equilibria are eliminated and which remain by weak-dominance principle.

Problem 2: Ch.4 Exercise 12 (Topics 1−3) 25 points Solve Exercise 12 in the end of Chapter 4 (Harrington, Ed.2) by answering the same questions Q1–4 for Problem 2.

Problem 3: Ch.17 Exercise 1 (Topic 4) 25 points We consider the game in Exercise 1 in the end of Chapter 17 (Harrington, Ed.2). (But we answer different questions and you don’t have to read Ch.17.)

Q1. Denote by p the probability that the player 1 chooses action x and by q the probability that the player 2 chooses action x. Using

this notation, answer the following questions. Briefly explain the procedure to derive your answers.

1) Express each player’s expected payoff from choosing x and the one from choosing y. 2) State each player’s best response mixed strategy in each possible case of the opponent’s mixed strategy. Plot the graph of each

player’s best response on the (p,q)-plane.

Q2. Find all the mixed-strategy Nash equilibria; state them by the values of the pair (p,q). (e.g. (p.q)=(0.2,0.5).) (Recall that a pure

strategy is just a special class of mixed strategies. So the answer to this question includes pure-strategy Nash equilibria, if exist.)

Problem 4: Ch.3 Exercise 15 (Topic 2) 20 points (bonus) Solve Exercise 15 in the end of Chapter 3 (Harrington, Ed.2). (Do this on your own paper.)

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