How to play games, mathematically
Autumn Term 2005
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Instructor: Jianqiang
Zhao |
Index |
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Time and Locations:
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CLASSES |
Mon.-Fri. |
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LOCATION: |
SHA 103. |
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Course Description:
Game theory studies optimal strategies when more than one player interact, or
when one player has more than one choice to deal with a problem. It has found
many applications in a variety of subjects such as economics, computer science,
psychology, philosophy, political science, and military conflicts. In this
course we will play a dozen or so different games and analyze the mathematics
behind them. We will consider the real world applications of the game theory
too.
Here is the outline of this course (not necessarily in this order):
1. Nim and binary system
2. Scissor-Rock-Paper and probability
3. Bomb-Sorties
4. A Poker Game
5. Game Tree
6. Zero Sum Games
7. Application to Economics
8. Application to Philosophy
9. Application to Social Psychology
10. Non-zero Sum Game
11. Prisoner's Dilemma
12. Application to Political Science
13. Application to Arbitration
Homework:
We will have daily homework. I encourage you to work in groups but prepare your writing in your own words. No late homework will be accepted.
Examinations:
There will be three exams during the autumn term.
Examination Schedule |
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Date of Exam |
Material Covered |
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Thursday, Aug. 18 |
First Week's Material |
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Thursday, Aug. 25 |
Second Week's material |
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Thursday, Sept. 1 |
Third Week's material |
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It is expected that you will adhere to the Shared Commitment and Student Honor Pledge so that all work on examinations will be your own: On my honor, as an Eckerd College student, I pledge not to lie, cheat, or steal, nor to tolerate these behaviors in others.
Grading Policy:
Your final grade in the course will be based on the following:
class participation 20%, homework 35%, first midterm: 15%, second midterm 15%, third mid term: 15%.
Writing Center
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Sunday, Aug 14 |
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Monday, , Aug 15 |
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Tuesday, Aug 16 |
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Wednesday, Aug 17 |
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Thursday, Aug 18 |
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Sunday, Aug 21 |
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Monday, Aug 22 |
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Tuesday, Aug 23 |
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Wednesday, Aug 24 |
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Thursday, Aug 25 |
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Sunday, Aug 28 |
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Monday, Aug 29 |
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Tuesday, Aug 30 |
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Wednesday, Aug 31 |
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Lecture Notes:
08/13 (Nim),
08/15 (Scissors-Stone-Paper),
08/16 (A matrix Game),
08/17 (Bomb-Sorties),
08/17 (General 2x2 Matrix Game),
08/18
(Application to Anthropology),
08/22
(Application to Military: NMD),
08/23
(Application to Philosophy: Free Will),
08/23 (A
Poker Game),
08/23
(Another Poker Game),
08/24
(Non-Zero Sum Games),
08/25 (Movie:
A Beautiful Mind),
08/26
(Application to Business),
08/29 (Application
to Social Psychology),
08/29 (Chicken
- Strategic Moves),
08/29 (Wholesale
vs. Retail),
08/30 (Voting
Strategies Evolutionary),
08/31 (Stable
Strategy),
09/01 (Third
Midterm Exam),
Assignment
08/15
1.
Write 90 in
binary form.
2.
Write (1010010)2 in decimal
form.
3.
Determine whether
you want to move first in the following ordinary Nim
games:
(3,1,2,2,1,1),
(2,3,5,17,21),
(4,11,15).
If yes, provide
the first move.
Choose ONE
of the following two writing assignments: two-page essay, 12 pt, double spaced.
4. Tell a real story that you’ve experienced concerning the application of mathematics.
5. Tell a real story about one of your favorite science teachers.
08/16
Problem 1.
What’s the probability to draw 5 cards and get a royal flush (A,K,Q,J,10
of the same suit)?
Problem 2.
What’s the probability to throw 2 dice and get a sum less than 4?
Problem 3.
What’s the probability to flip three coins and get three heads at the
same time?
Problem 4.
What’s the probability to draw 4 hearts in a row from a deck of 52 poker
cards?
Problem 5. The
weather forecast says that the chances of rain for the next two days are 20%
and 60% respectively. What’s the probability that neither of the two days
rains?
Problem 6. Expand
the binomial polynomial (x+1)5.
Problem
7. Find the value and saddle
points (if there are any) of the following matrix games:
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A
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K
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Q
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J
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A
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4
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2
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5
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1
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K
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2
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1
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–1
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–8
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Q
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3
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2
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4
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2
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J
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–9
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0
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9
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1
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A
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K
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Q
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J
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A
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3
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1
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4
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4
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K
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2
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1
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3
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0
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Q
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2
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3
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2
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3
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08/17
Problem 1.
Rose Strategy III. Pace bomb on bomber 70% of the
time. Colin’s counter strategy III: Attack bomber 60% of the time.
Find out the percentage of successful missions of Rose.
Problem 2.
Let R denote a strategy for Rose
and C denote a strategy for
Colin for the given game G. Find the expected payoff of the following game: R=[.6,.2,.2], C=[.1,.4,.3,.2], G =
Problem 3. The Huckster. At a temporary
booth on a busy street Sarah found that she can sell 50 umbrellas when it rains
and about 10 umbrellas and 100 sunglasses when it shines. Umbrellas cost her
$2.5 and sell for $5; glasses cost $1 and sell for $2.5. She is will to invest
$500 in the project. Describe her profit by a table. Then find the best
strategy for Sarah for The Huckster problem in the
last assignment
Problem 4. “Fair” Game. One
day Colin says to his friend Rose, “Let’s play a coin game. We each
throw a coin at the same time.
I’ll give you $1 if I get a head and you get a tail. I’ll
give $3 if it’s the other way around. And ― just make it fair
― you give me $2 when we match.” Describe the game by a table. Then
find the best strategies for Colin and Rose in the “Fair” Game in
the last assignment. Is the game really fair?
Writing Assignment. Revise the essay on your story. Revision is due next
Monday 08/22.
08/17
The following is the schedule of your mentor meetings on this Friday afternoon.
I will discuss your registration with each of you. Before you come, try your
best to choose a language course and other two courses. Then bring the course
catalog with you when you come. The registration forms with your registration
codes are with me. You can NOT register without your codes.
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Student Name |
Meeting
Time |
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Cynthia and Samantha |
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Delia and Alyssa |
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Danielle G. and Alice |
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Katie and James |
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Jason |
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Peter |
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Kat |
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Breck |
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Daniel |
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Matthew K |
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Matt L. |
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T.J. |
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Anne |
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Hank |
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Geoffrey |
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Mike |
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Danielle and Janielle |
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Current
Run Not Rrun
08/18
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Current |
1.
Find a current percentage
for which In-Out would be the best fishing strategy.
2.
Suppose the
payoff for In-Out with no current were 15.0 instead of 17.0. What would be the
effect on the game-theoretic solution? On the value of the
game?
3.
Writing
Assignment. Revise the essay on
your story. Revision is due next Monday 08/22.
08/22
1. Solve the NMD game where Purple has three dummies and one warhead while Green has only one anti-missile of the same type as in the class.
2. Read the article http://www.gametheory.net/News/Items/087.html. I would like to know your opinion. Please write an essay of 150-200 words on this subject. EMAIL your essay to me as an attachment of Word file.
3. Here are your scores: http://home.eckerd.edu/~zhaoj/teach/courses/studentgame.php
Your password is your student id. If anything is wrong, please let me know as soon as possible.
08/23
1. Levi considered a modified situation in which if you take both boxes and BB happens to contain $1,000,000 (because Being made a mistake), you have to pay a "greediness fee" of $1500.
(a). Write out the matrix game and show that the Dominance Principle no longer applies.
(b). Can you think of reasons which might still lead you to take both boxes?
2. If Rose is dealt one card from two cards A and K. She can bet or drop. If she drops then she loses two dollars. If she bets then Colin can either call or fold. If Colin folds then he loses four dollars. If he calls then he wins or loses eight dollars depending whether Rose has a K or an Ace. Draw the game tree and find the best strategies for Rose and Colin. What’s the expected payoff to Rose in this game?
3. Let's replay the Cuban missile crisis. In 1963, the Soviet
Union (SU) starts the game by deciding whether or not to place
intermediate range ballistic missiles in Cuba Cuba
(a). Please draw the game tree for the Cuban missile crisis. You can assign the payoffs to (US, SU) at the end nodes as much as you think appropriate. Note this is not a zero-sum game.
(b). How many strategies does US have?
(c) How many strategies does SU have?
(d) (Extra credit) Write down the matrix game corresponding to the Cuban missile crisis, based on all the strategies to both players.
08/24
1. Do the test on F Scale and record your score on a paper. Bring it to class tomorrow but you don’t need show me.
2. For each of the following games, draw the movement diagram and the payoff polygon, and identify all (pure-strategy) equilibria and Pareto optimal outcomes. Determine whether the game is solvable in the strict sense (there is at one Pareto optimal equilibrium, or if there are more than one Pareto optimal equilibrium then they are all the same and exchangeable). If so, give the solution. If not, why not.
(c)
(a)
(b)
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(c) |
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(a) |
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(b) |
08/25 (Thursday)
Prepare for tomorrow’s exam.
08/26
1. Suppose that the companies move simultaneously, but before they do, Zeus
conducts a market survey. Athena knows the existence of the
survey, but not its results.
a) What are the information sets for this game?
b) Write and solve the resulting 4X2 matrix game.
c) What effect would it have if Athena did not know of the existence of the
survey?
2. Suppose that Athena must move first, but that neither side knows
Chance's move. Further, Zeus knows Athena's move.
a) Draw the game tree and show the information sets.
b) Write the resulting 4x2 matrix game (don't solve).
08/29
The games on the right all have outcomes which do not give Rose her maximal
payoff. For each one, say whether Rose could benefit by any of the following
strategic moves: i) seizing (or committing to) the
first move, ii) forcing Colin to move first, iii) making a threat, iv) making a
promise, or v) making both a threat and a promise. For each game, it is
possible that more than one of these could work, or that none of them will. For
each commitment, threat or promise, show how Rose could implement it by lowering
some of her payoffs.
Rose threatens “If A then K” or lowers AA from 3 to 1.
Nothing works for Rose in this case.
Rose threatens “If A then K” and Promises “If K then A,” or lowers AA from 2 to 0 and KK from 4 to 2.
Rose moves first, or commits to K by lowering AA from 2 to 0 and AB from 4 to 2. Alternatively Rose can promise “If K then K,” lowering only AK.
Rose makes Colin move first.
08/30
1. Compute the Shapley-Shubik power indices for [3;2,1,1]. Here the dean has 2 votes, the faculty has 1 and the student has 1. Then compute the Banzhaf indices for them.
2. Find the information about the legislative system of the US
Hint. Here is a simplified version. Consider a legislative scheme consisting of a president, three senators S, and five representatives R. A bill had to be approved by the president and a majority of both the senators and the representatives. Assume senators and representatives don’t have veto power. The player in this game are: PSSSRRRRR. So there are 9!/1!3!5!=504 different orderings. For an R to pivot it must be preceded by P, exactly two other R’s and either two or three S’s. Here is the total number of orderings:
(PSSRR)R(SRR): (5!/1!2!2!)(3!/1!2!)=(120/4)(3)=90
(PSSSRR)R(RR): (6!/1!3!2!)(2!/2!) =(720/12) = 60
Total: 150.
Similarly, for an S to pivot it must be preceded by P, exactly on other S, and either three, or four, or five R’s:
(PSRRR)S(SRR): (5!/1!1!3!)(3!/1!2!)=(120/6)(3)=60
(PSRRRR)S(SR): (6!/1!1!4!)(2!/1!1!) =(720/24)(2) = 60
(PSRRRRR)S(S): (7!/1!1!5!)(1!/1!) =(7x720/120) = 42
Total: 162.
In the remaining 504-150-162=192 orderings, P pivots. The division of power is as follows:
President: 192/504=.381
Total senators: 162/504=.321 Each senator: .107
Total representatives: 150/504=.298 Each representative: .060
The ratio of power of P:S:R=192:54:30=6.4:1.8:1
3. A corporation has only six stock holders. The CEO has 40% and the remaining 60% shares are divided among the other five board members equally. Now one board member suggest to introduce an additional board member while increase the share of the CEO to 41%. She further suggests that the remaining 59% shares are divided among the other six board members equally except CEO. Should the CEO agree to this? Explain why.