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Applied Finite Mathematics

Rupinder sekhon, game theory: homework - all with video answers.

Chapter Questions

Determine whet her the games are strictly determined. If the games are strictly determined, find the optimal strategies for each player and the value of the game. a. $\left[\begin{array}{cc}1 & 2 \\ -2 & 3\end{array}\right]$ b. $\left[\begin{array}{ll}6 & 3 \\ 2 & 1\end{array}\right]$ c. $\left[\begin{array}{ccc}-1 & -3 & 2 \\ 0 & 3 & -1 \\ 1 & -2 & 4\end{array}\right]$ d. $\left[\begin{array}{ccc}2 & 0 & -4 \\ 3 & 4 & 2 \\ 0 & -2 & -3\end{array}\right]$ e. $\left[\begin{array}{cc}0 & 2 \\ -1 & -1 \\ -1 & 1 \\ 3 & 2\end{array}\right]$ f. $\left[\begin{array}{rrr}5 & -3 & 2 \\ 3 & -1 & 4\end{array}\right]$

Two players play a game which involves holding out one or two fingers simultaneously. If the sum of the fingers is more than 2, Player II pays Player I the sum of the fingers; otherwise, Player I pays Player II the sum of the fingers. a. Write a payoff matrix for Player $\mathrm{I}$. b. Find the optimal strategies for each player and the value of the game.

A mayor of a large city is thinking of running for re-election, but does not know who his opponent is going to be. It is now time for him to take a stand for or against abortion. If he comes out against abortion rights and his opponent is for abortion, he will increase his chances of winning by $10 \% .$ But if he is against abortion and so is his opponent, he gains only $5 \% .$ On the other hand, if he is for abortion and his opponent against, he decreases his chance by $8 \%$, and if he is for abortion and so is his opponent, he decreases his chance by $12 \%$. a. Write a payoff matrix for the mayor. b. Find the optimal strategies for the mayor and his opponent.

A man accused of a crime is not sure whether anybody saw him do it. He needs to make a choice of pleading innocent or pleading guilty to a lesser charge. If he pleads innocent and nobody comes forth, he goes free. However, if a witness comes forth, the man will be sentenced to 10 years in prison. On the other hand, if he pleads guilty to a lesser charge and nobody comes forth, he gets a sentence of one year and if a witness comes forth, he gets a sentence of 3 years. a. Write a payoff matrix for the accused. b. If you were his attorney, what strategy would you advise?

Determine the optimal strategies for both the row player and the column player, and find the value of the game. a. $\left[\begin{array}{cc}-1 & 1 \\ 1 & -1\end{array}\right]$ b. $\left[\begin{array}{cc}1 & -1 \\ -4 & 0\end{array}\right]$ $=\left[\begin{array}{cc}3 & -2 \\ 2 & 4\end{array}\right]$ $$\text { d. }\left[\begin{array}{cc}-3 & 2 \\1 & -4\end{array}\right]$$

Find the expected payoff for the given game matrix $G$ if the row player plays strategy $R,$ and column player plays strategy $C$. $$\text { a. } G=\left[\begin{array}{cc}-3 & 2 \\1 & -4\end{array}\right] R=\left[\begin{array}{cc}2 / 3 & 1 / 3\end{array}\right] C=\left[\begin{array}{c}1 / 4 \\3 / 4\end{array}\right]$$ b. $G=\left[\begin{array}{cc}-1 & 1 \\ 1 & -1\end{array}\right] R=\left[\begin{array}{ll}1 / 3 & 2 / 3\end{array}\right] C=\left[\begin{array}{c}2 / 3 \\ 1 / 3\end{array}\right]$

Two players play a game which involves holding out one or two fingers simultaneously. If the sum of the fingers is even, Player II pays Player I the sum of the fingers. If the sum of the fingers is odd, Player I pays Player II the sum of the fingers. a. Write a payoff matrix for Player $\mathrm{L}$. b. Find the optimal strategies for both the row player and the column player, and the value of the game.

In December 1995 , President Clinton ordered the first of $20,000 \mathrm{U}$. S. troops to be sent into BosniaHerzegovina as a peace keeping force. Unfortunately, the heavy fog made visibility very poor at the Tuzla airfield, and at the same time increased the threat of sniper attacks from the Serbian forces. U. S. Air Force Col. Neal Patton, and Lt. Col. Sid Kooyman, the advance specialists, had two choices: either to send in the troops by air with the difficulties already described or by road thus exposing the troops to ambush by the Serbian forces. The Serbian army, with its limited resources, had a choice of deploying its forces near the airport or along the road route. If the U. S. lands its troops on the airfield in the fog while the Serbs are concentrating on the road route, the payoff for $U . S$, is 20 points. But if the $U .$ S. lands its troops on the airfield, and Serbians are there hiding in the fog, U. S. wins only 5 points. On the other hand, if U. S. transports its troops by road and avoids Serbs its payoff is 35 points, but if $\mathrm{U} . \mathrm{S} .$ meets Serb resistance on the road route, it loses 50 points. a. Write a payoff matrix for the game. b. If you were Air Force Col. Neal Patton's advisor, what advice would you give him?

Find the optimal strategy for each player and the value of the game. $\left[\begin{array}{cc}2 & -1 \\ 0 & 3 \\ -2 & 0\end{array}\right]$

Find the optimal strategy for each player and the value of the game. $\left[\begin{array}{ll}2 & 3 \\ 4 & 5 \\ 5 & 4\end{array}\right]$

Find the optimal strategy for each player and the value of the game. $\left[\begin{array}{ccc}1 & 3 & -2 \\ -2 & 9 & 4 \\ -5 & 0 & 1\end{array}\right]$

Find the optimal strategy for each player and the value of the game. $\left[\begin{array}{lll}0 & 1 & 1 \\ 0 & 1 & 2 \\ 1 & 2 & 4 \\ 3 & 1 & 3\end{array}\right]$

Find the optimal strategy for each player and the value of the game. $\left[\begin{array}{cccc}2 & 0 & -4 & 8 \\ 0 & -6 & -6 & 2 \\ 2 & -2 & 4 & 6 \\ -2 & -4 & -8 & 0\end{array}\right]$

Find the optimal strategy for each player and the value of the game. $\left[\begin{array}{cccc}-1 & 3 & 2 & 4 \\ 1 & 2 & 0 & 5\end{array}\right]$

Find the optimal strategy for each player and the value of the game. $\left[\begin{array}{cccc}-5 & -1 & -1 & 3 \\ -10 & 1 & 2 & -8 \\ 4 & 0 & 1 & 5 \\ 3 & -8 & 0 & 5\end{array}\right]$

Find the optimal strategy for each player and the value of the game. $\left[\begin{array}{cccc}1 & -3 & -4 & 1 \\ 1 & -4 & -1 & 3 \\ 1 & 1 & -1 & 1 \\ 0 & -1 & 1 & 1\end{array}\right]$

Determine whether the games are strictly determined. If the games are strictly determined, find the optimal strategies for each player and the value of the game. a. $\left[\begin{array}{ll}2 & 3 \\ 3 & 4\end{array}\right]$ b. $\left[\begin{array}{ccc}0 & 3 & -1 \\ 1 & 3 & -2 \\ -1 & 2 & -5\end{array}\right]$ c. $\left[\begin{array}{ccc}3 & 2 & -1 \\ 5 & 3 & 4\end{array}\right]$ d. $\left[\begin{array}{cc}4 & 2 \\ -1 & 3 \\ 4 & 3 \\ 1 & -3\end{array}\right]$

Two players play a game which involves holding out a nickel or a dime simultaneously. If the sum of the coins is more than 10 cents, Player I gets both the coins; otherwise, Player II gets both the coins. a. Write a payoff matrix for Player $\mathrm{I}$ b. Find the optimal strategies for each player and the value of the game.

Lacy's department store is thinking of having a major sale in the month of February, but does not know if its competitor store Hordstrom's is also planning one. If Lacy's has a sale and Hordstrom's does not, Lacy's sales go up by $30 \%,$ but if both stores have a sale simultaneously, Lacy's sales go up by only $5 \%$. On the other hand, if Lacy's does not have a sale and Hordstrom's does, Lacy's loses $5 \%$ of its sales to Hordstrom's, and if neither of the stores has a sale, Lacy's experiences no gain in sales. a. Write a payoff matrix for Lacy's. b. Find the optimal strategies for both stores.

Mr. Halsey has a choice of three investments: Investment A, Investment B, and Investment C. If the economy booms, then Investment A yields $14 \%$ return, Investment B returns $8 \%,$ and Investment C $11 \% .$ If the economy grows moderately, then Investment A yields $12 \%$ return, Investment $\mathrm{B}$ returns $11 \%$, and Investment C $11 \%$. If the economy experiences a recession, then Investment A yields a $6 \%$ return, Investment $\mathrm{B}$ returns $9 \%,$ and Investment $\mathrm{C} 10 \%$. a. Write a payoff matrix for Mr. Halsey. b. What would you advise him?

Mr. Thaggert is trying to decide whether to invest in stocks or in CD's(Certificate of deposit). If he invests in stocks and the interest rates go up, his stock investments go down by $2 \%,$ but he gains $1 \%$ in his CD's. On the other hand if the interest rates go down, he gains $3 \%$ in his stock investments, but he loses $1 \%$ in his CD's. a. Write a payoff matrix for Mr. Thaggert. b. If you were his investment advisor, what strategy would you advise?

Determine the optimal strategies for both the row player and the column player, and find the value of the game. a. $\left[\begin{array}{cc}2 & -2 \\ -2 & 2\end{array}\right]$ b. $\left[\begin{array}{cc}-2 & 2 \\ 5 & 0\end{array}\right]$ c. $\left[\begin{array}{cc}3 & 5 \\ 4 & -1\end{array}\right]$ $$\text { d. }\left[\begin{array}{cc}-2 & 5 \\4 & -3\end{array}\right]$$

Find the expected payoff for the given game matrix $\mathrm{G}$ if the row player plays strategy $\mathrm{R},$ and the column player plays strategy $\mathrm{C}$. $$\begin{array}{l}\text { a. } G=\left[\begin{array}{cc}3 & 5 \\4 & -1 \end{array}\right] R=\left[\begin{array}{cc}1 / 2 & 1 / 2\end{array}\right] C=\left[\begin{array}{c} 1 / 4 \\3 / 4\end{array}\right] \\\text { b. } G=\left[\begin{array}{cc}-2 & 5 \\4 & -3\end{array}\right] R=\left[\begin{array}{ll}2 / 3 & 1 / 3\end{array}\right] C=\left[\begin{array}{c} 1 / 3 \\2 / 3\end{array}\right]\end{array}$$

A group of thieves are planning to burglarize either Warehouse A or Warehouse B. The owner of the warehouses has the manpower to secure only one of them. If Warehouse $\mathrm{A}$ is burglarized the owner will lose $$\$ 20,000$$, and if Warehouse $\mathrm{B}$ is burglarized the owner will lose $$\$ 30,000$$. There is a $40 \%$ chance that the thieves will burglarize Warehouse $\mathrm{A}$ and $60 \%$ chance they will burglarize Warehouse B. There is a $30 \%$ chance that the owner will secure Warehouse A and $70 \%$ chance he will secure Warehouse B. What is the owner's expected loss?

Two players play a game which involves holding out a nickel or a dime. If the sum of the coins is odd, Player I gets both the coins, and if the sum of the coins is even, Player II gets both the coins. Determine the optimal strategies for both the row player and the column player, and find the expected payoff.

A football quarterback has to choose between a pass play or a run play depending on how the defending team is going to react. If he chooses a pass play and the defending team is expecting a pass, he expects to gain 4 yards, but if the defending team is expecting a run, he gains 20 yards. On the other hand, if he calls a run play and the defending team expects a pass, he gains 7 yards, and if he calls a run play and the defending team expects a run, he loses 2 yards. If you were the quarterback, what would your strategy be?

The Watermans go fishing every weekend either at Eel River or at Snake River. Unfortunately, so do the Nelsons. If both families show up at Eel River, the Watermans can hope to catch only 3 fish, but if the Watermans fish at Eel River and the Nelsons at Snake River, the Watermans can catch as many as 12 fish. On the other hand, if both families fish at Snake river, the Watermans can catch about 5 fish, and if Watermans fish at Snake river while the Nelsons fish at Eel river, the Watermans can catch up to 15 fish. Determine a mixed strategy for the Watermans, and the expected payoff.

Terry knows there is a quiz tomorrow, but does not remember whether it is in his math class or in his biology class. He has time to study for only one subject. If he studies math and there is a quiz in it, he gains 10 points and even if there is no quiz he gains two points for acquiring the extra knowledge which he will apply towards the final exam. If he studies biology and there is a quiz in it, he gains ten points but there is no gain if there is no quiz. Determine a mixed strategy for Terry, and the expected payoff.

Reduce the payoff matrix by dominance. Find the optimal strategy for each player and the value of the game. a. $\left[\begin{array}{ccc}-3 & 1 & 2 \\ -3 & 5 & 3 \\ 2 & 4 & -1\end{array}\right]$ b. $\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 1 & 4 \\ 2 & 3 & 4 \\ 1 & 2 & 2\end{array}\right]$ c. $\left[\begin{array}{cccc}4 & 3 & 9 & 7 \\ -7 & -5 & -3 & 5 \\ -1 & 4 & 5 & 8 \\ -3 & -5 & 1 & -1\end{array}\right]$ d. $\left[\begin{array}{cccc}2 & 3 & 1 & 5 \\ -2 & 2 & 1 & 3\end{array}\right]$ e. $\left[\begin{array}{cccc}0 & 3 & 2 & 1 \\ 0 & 2 & 1 & -7 \\ -4 & -9 & 5 & 4 \\ 4 & -7 & 6 & 6\end{array}\right]$ f. $\left[\begin{array}{cccc}1 & 0 & 2 & 2 \\ 2 & 2 & 0 & 2 \\ 2 & -3 & 0 & 4 \\ 2 & -2 & -3 & 2\end{array}\right]$

Game Theory

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Game theory is a multi-disciplinary approach to the study of human behavior, used in such fields as mathematics, economics, and the social sciences. By "games" we actually mean human interactions governed by opposing strategies of the people involved. As an example, in a patent system, the first research lab to invent a device gets the patent. Game theorists will apply the principles of game theory to find out how many research labs need to be involved to establish a Nash equilibrium , and compare that to the most efficient number of labs. Everything else follows from there. A good course in non-cooperative game theory will most likely deal with these areas:

• Games in strategic form and Nash equilibrium
• Iterated strict dominance, rationalizability, and correlated equilibrium
• Extensive-form games
• Applications of multi-stage games with observed actions
• Repeated games
• Bayesian games and Bayesian equilibrium
• Bayesian games and mechanical design
• Equilibrium refinements
• Reputation effects
• Sequential bargaining with incomplete information
• Payoff-relevant strategies and Markov equilibrium
• Common knowledge and games

The International Journal of Game Theory is a good place to stay up to date in this field, and game theory tutorials are not hard to find.

A Brief History of Game Theory

The work of Von Neumann and Morgenstern (1944) marked an official start of game theory. Then came the second milestone contribution of John Nash (1951), which basically becomes chapter 1 of any modern game theory textbook. In the 1950s, in the intense atmosphere of war, there were many brilliant breakthroughs in fundamental fields such as computer science, mathematics and physics that set the foundation and landmarks for today's standard. Some said it was a time when “giants roamed the Earth”.

Game theory considers interaction among individuals in a narrow context. A context is narrow when the action of one player can have influence over the course of the whole game, hence each player has to take into account the reasoning of the other in order to make their action. There are two parallel lines in game theory: cooperative game theory and non-cooperative game theory. The cooperative strand studies coalition and contract formation from a normative point of view. It answers questions such as: What should be the society we would like to enter? What kind of contract or coalition could we form? The non-cooperative approach (which becomes more prominent in modern game theory) starts from the individual's perspective. Starting from building blocks of decision theory such as preference and utility function, it investigates positive questions such as: What would really happen after we have all agreed to an agreement? Would we still comply with it? The rationale is that we would comply to an agreement if and only if it aligns with our own interest. In this non-cooperative realm, a Nash equilibrium is defined as a profile of players' actions that maximize the utility of everyone given their correct expectation on everyone else's behavior. There are two parts of this definition. First, the player maximizes his own expected utility based on some belief (or prediction) over the other's behavior. Second, that expectation turns out to be correct. In this way, a Nash equilibrium becomes a situation that is self-enforced (i.e. no one wants to deviate from her chosen action).

This is a broad concept because any self-fulfilling prophecy therefore can be a Nash equilibrium, making the number of Nash equilibria to be more than one. And some people may suggest that some Nash equilibria seem to be better (more rational or more equitable) than others; hence there have been many attempts to refine the Nash equilibrium concept which results in a vast amount of literature on refinements in the 1980s. Even now the problem of equilibrium selection remains open for active contribution. Game theory has had extensive applications in the study of cooperation and competition in society. Different situations can be captured in different hypothetical situation such as the Prisoner's Dilemma for the cooperative decision, the bargaining game for negotiating decision, and the trust game for investment decision. Further information on classical game theory can be found at Standford.edu .

References :

Von Neumann, J.; Morgenstern, O. 1944. Princeton University Press. Theory of games and economic behavior . Princeton, NJ, US.

Nash, Jr. J. F., 1951. Noncooperative games. Annals Math ., 54, 289-95.

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Browse Course Material

Course info.

• Prof. Muhamet Yildiz

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• Game Theory

Learning Resource Types

Economic applications of game theory, assignments.

PROBLEM SETS	SOLUTIONS

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Topic	Professor	Format
	Mike Shor
	Mike Shor
Denis Peguin
	Bernhard von Stengel
John Hillas
	Stefan Waner
	Mike Shor
	Mike Shor
	Levent Koçkesen
	Eric Rasmusen

• Q U I C K L I N K S:

Prof. Olga Radko, MS5356	TA: Steve Chan, MS 2925
Office hours: M 1:30-2:30, W 1-2, F 11-12	Office hours: T 2-3, Th (12-1) & 1-2

Textbook: K. Benmore , Fun and Games

Prerequisite: math 115a or equivalent (first part of undergraduate Linear Algebra).  

Class Calendar

Homework due date	Assigned Homework	Material	Remarks
Oct. 8th	1.10: 1a)-e), 2, 5, 7, 17	0.1, 0.1.3, 0.2, 0.3, 0.4.1, 1.1-1.5	Problems 1f) and  20 from 1.10 are moved to the next homework
Oct. 15th	1.10: 1f), 18-21, 25	1.7-1.9, 2.1-2.3	Quiz # 1 on Tuesday
Oct. 23nd	1.10: 25 (moved from hw 2); 2.6: 5, 7, 8, 11, 23.	2.4-2.5, 3.1-3.2, 3.4
Oct. 29th	2.6: 26; 3.7: 1--7	3.1-3.2, 3.4	Quiz # 2 on Tuesday
Nov. 5th	3.7: 10 -- 12, 15, 18; 4.8: 1, 2	3.4, 3.5, start Ch. 4	Review: Thursday, at noon in 5252 Boelter.
Nov. 12th	4.8: 14, 18, 29; 5.9: 2, 3, 5.	Ch. 4 and begin Ch. 5	Midterm on Monday, Nov. 8th in
Nov. 19th	5.9: 7, 8, 11, 12, 17, 19, 20	Finish Ch. 5, start Ch. 6	Quiz # 3 on Tuesday
Nov. 26th	5.9: 21, 27; 6.10: 2, 5	6.1--6.4	Happy Thanksgiving!
Dec. 3rd	6.10: 15, 18, 19, 27b, 28a	6.5--6.7
Dec. 10th	7..9: 1, 6, 13	7.1, 7.2, 7.7	Quiz #4 on Tuesday.
Final Exam:	Dec. 17th, 8-11am, room TBA

Midterm: There will be one midterm, on November 8, during regular class hours.  Please, make sure that you have no time conflict on this date. Unless you have a documented medical illness, there will be no make-up exam.

Quizzes: There will be several quizzes throughout the quarter. Quiz topics and dates will be announced in advance. One lowest quiz score will be dropped in the computation of your grade.

Grading policy:   Y our grade will be computed as the best of the  following:

Homework (20%)+Quizzes(20%)+Midterm(25%)+Final(35%)     or: Homework (20%)+Quizzes(20%)+Midterm(30%)+Final(30%)

Open Yale Courses

You are here, game theory.

This course is an introduction to game theory and strategic thinking. Ideas such as dominance, backward induction, Nash equilibrium, evolutionary stability, commitment, credibility, asymmetric information, adverse selection, and signaling are discussed and applied to games played in class and to examples drawn from economics, politics, the movies, and elsewhere.

This Yale College course, taught on campus twice per week for 75 minutes, was recorded for Open Yale Courses in Fall 2007.

A. Dixit and B. Nalebuff. Thinking Strategically , Norton 1991

J. Watson. Strategy: An Introduction to Game Theory , Norton 2002

P.K. Dutta. Strategies and Games: Theory And Practice , MIT 1999

Who should take this course? This course is an introduction to game theory. Introductory microeconomics (115 or equivalent) is required. Intermediate micro (150/2) is not required, but it is recommended. We will use calculus (mostly one variable) in this course. We will also refer to ideas like probability and expectation. Some may prefer to take the course next academic year once they have more background. Students who have already taken Econ 156b should not enroll in this class.

Course Aims and Methods. Game theory is a way of thinking about strategic situations. One aim of the course is to teach you some strategic considerations to take into account making your choices. A second aim is to predict how other people or organizations behave when they are in strategic settings. We will see that these aims are closely related. We will learn new concepts, methods and terminology. A third aim is to apply these tools to settings from economics and from elsewhere. The course will emphasize examples. We will also play several games in class.

Outline and Reading. Most of the reading for this course comes from the first ten chapters of Dutta or from the first two parts of Watson. There will be a reading packet for weeks 6-7. The readings are not compulsory, but they will help back up the class material.

Problem sets: 30% Midterm examination: 30% Final examination: 40%

Lecture 1
Lecture 2
Lecture 3
Lecture 4
Lecture 5
Lecture 6
Lecture 7
Lecture 8
Lecture 9
Lecture 10
Lecture 11
Lecture 12
Exam 1
Lecture 13
Lecture 14
Lecture 15
Lecture 16
Lecture 17
Lecture 18
Lecture 19
Lecture 20
Lecture 21
Lecture 22
Lecture 23
Lecture 24
Exam 2