EXAM paper on "Game of Theory"
moneyboy632
selected-downloads11.zipT-8.pdf
GET1018 / GEK 1544 The Mathematics of Games
Tutorial 8
-: This is the last tutorial . No Webcast discussion. :-
1. Consider a zero - sum game between players A and B , with the payoffs for A shown in the following diagram .
B1 B2 B3
A1 −4 6 − 6 A2 −1 4 3 A3 −2 −3 −1
(a) Argue by eliminating dominated (or weaker) strategies that there is a pure pair of strategies (each for each player) for this game. Explain what should happen with rational play, and what payoff will result?
(b) Now change the A3 − B3 payoff from −1 to 4 :
B1 B2 B3
A1 −4 6 − 6 A2 −1 4 3 A3 −2 −3 4
Show that there are no dominated strategies. Then use the Maxi - mini method to find a saddle point solution. That is, the situation where
Maxi - mini (A) = −Maxi - mini (B) .
2. Consider the payoff diagram of a zero - sum game between A and B.
B1 B2 B3
A1 3 − 2 2 A2 −1 0 4 A3 −4 −3 1
(a) Argue by eliminating dominated (or weaker) strategies that the only meaningful strategies are A1 , A2 , B1 and B2 .
(b) Show that the Maxi - mini method does not yield a saddle point.
(c) Using the probability method, show that A can avoid having to make an expected payment of more than 1/3 by choosing A1 with probability 1/6 and A2 with probability 5/6, no matter what B chooses. Likewise, B can ensure an expected gain of at least 1/3 by using a randomized strategy of choosing B1 with probability 1/3 and B2 with probability 2/3 , no matter what A chooses.
3. An airline loses two suitcases belonging to two different travelers. Both suitcases happen to be identical and contain identical antiques. An airline manager tasked to settle the claims of both travelers explains that the airline is liable for a maximum of $ 100 per suitcase, and in order to determine an honest appraised value of the antiques the manager separates both travelers so they can’t confer, and asks them to write down the amount of their value at no less than $ 2 and no larger than $ 100 . He also tells them that if both write down the same number, he will treat that number as the true dollar value of both suitcases and reimburse both travelers that amount. However, if one writes down a smaller number than the other, this smaller number will be taken as the true dollar value, and both travelers will receive that amount along with a bonus/malus: $ 2 extra will be paid to the traveler who wrote down the lower value and a $ 2 deduction will be taken from the person who wrote down the higher amount. Note that in this theoretical game, the amount quoted can be any real number between 2 and 100 ( inclusive ) .
What is (are) the Nash equilibrium ?
T-7.pdf
GEK 1544 / GET1018 The Mathematics of Games
Tutorial 7
1. Consider a Tic Tac Toe analysis. We first ignore symmetry and assume that games are continued until all nine boxes are filled.
(a) Show that the number of different games that can be played is 9 ! = 362, 880 .
(b) Argue that there are 9 × 7 8 different strategies for the first two moves of the “ X ” (first) player.
(c) Prove that there are 9 · 7 8 · 58×6 · 38×6×4 ( ≈ 1.89 · 1041 � 9 ! = 362, 880 ) different strategies for the “ X ” player.
2. The Pirate Game. There are five rational pirates, say A, B, C, D and E. They find 100 gold coins. They must decide how to distribute them.
The pirates have a strict order of seniority: A is superior to B, who is superior to C, who is superior to D, who is superior to E.
The Pirate world’s rules of distribution are this: that the most senior pirate should propose a distribution of coins. The pirates should then vote on whether to accept this distribution; the proposer is able to vote, and has the casting vote in the event of a tie. If the proposed allocation is approved by vote, it happens. If not, the proposer is thrown overboard from the pirate ship and dies, and the next most senior pirate makes a new proposal to begin the system again.
Pirates base their decisions on three factors. First of all, each pirate wants to survive. Secondly, each pirate wants to maximize the amount of gold coins he receives. Thirdly, each pirate would prefer to throw another overboard, if all other results would otherwise be equal.
What is your suggested strategy to pirate A ?
T-6.pdf
GE T1018 / GEK 1544 The Mathematics of Games
Tutorial 6
1. A person decides to bet $ 1 on the (fixed) number 8 in Las Vegas roulette, and continue to do so for a total of 500 times. The pay-off on betting on a number is 35 : 1 . Using the normal distribution table, estimate (±0.1 ) the person’s probability on winning $ 40 or more (accurate up to 3 decimal places).
2. Let p and q be positive numbers with p + q = 1 . Consider the binomial expansion
1 = (p + q)n = pn + C(n, 1) pn−1q + · · · + C(n, r) pn−rqr + · · · + qn
= pn [
1 + C(n, 1)
( q
p
) + · · · + C(n, r)
( q
p
)r + · · · +
( q
p
)n ] .
Using the Stirling formula
k ! ≈ √
2π ·k 1 2 · ( k
e
)k for k � 1 ,
estimate the value of r ( in terms of n when it is large ) so that the expression
C(n, r)
( q
p
)r =
n !
(n − r) ! r !
( q
p
)r (2.1)
is at maximum. [ Suggestion. Express (2.1) using the Stirling formula, and treat it as a function of r . Then differentiate the expression with respect to r , and let n →∞ . ]
Compare in the roulette example when we use the normal distribution, we take
Z = r −
( nq + 1
2
) √ npq
.
T-5.pdf
GET1018 / GEK1544 The Mathematics of Games
Tutorial 5
The standard 52 poker cards consist of the thirteen kinds :
Ace = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q , K .
Each kind has four suits : ♠ , ♣ , ♥ and ♦ .
1. In a poker hand of getting 5 consequent cards out of the standard 52 poker cards, find the following.
P ( three of a kind , no better) & P ( two pairs , no better) .
Conclude that
P ( three of a kind , no better) < P ( two pairs , no better) .
2. Consider drawing 7 cards out of the 52 standard poker cards ( well shuffled ) . Find the probability of obtaining a full house out of the 7 cards drawn. Note that a full house consists of 3 cards from a kind and a pair from another kind. For example,
A A A J J ( order not important ) .
Note that the 7 cards being drawn can be of the following ;
A A A A J J J .
Q. 3 begins in the next page.
3. Let {x1, x2, · · · , xn } be a collection of n numbers, among which k of them are the same, and the remaining n − k numbers are all distinct. (E.g, for the numbers {4, 4 , 3, 2} , n = 4 and k = 2 .) Show that there are
P (n, n−k) = n !
k !
ways to arrange the n-numbers in order . E.g. P (4, 2) = 4 · 3 = 12 :
4, 4 , 3, 2 2, 3 , 4, 4
4, 4 , 2, 3 3, 2 , 4, 4
4, 3 , 4, 2 2, 4 , 3, 4
4, 3 , 2, 4 4, 2 , 3, 4
3, 4 , 4, 2 2, 4 , 4, 3
3, 4 , 2, 4 4, 2 , 4, 3
P.S. Due to the e - Test , there will be no tutorial discussion on the week 8 – 13 , March. This is to give you more time to study. Tutorial 5 will be discussed online on the week 15 – 20 , March .
T-4.pdf
GET1018 / GEK 1544 The Mathematics of Games
Tutorial 4
1. In Craps, a field bet is a bet that a total of 2, 3, 4, 9, 10, 11, or 12 will come up on the roll ( refer to the craps table in the lecture note ). Double 2 and double 6 pay 2 : 1, all others paying “even” odds ( 1:1 ). Compute the bettor’s expectation per unit bet on field .
2. A casino in which the “ no result ” craps roll for Don’t Pass bettors is “total of 3” ( instead of the usual “ total of 12 ” ). That is, when the initial roll has total equal to 3 , it results a loss to a Pass bettor, and neither win nor loss for a Don’t Pass bettor. Compute the new expectation per unit bet for Don’t Pass bettors. Would you prefer to bet Pass or Don’t Pass in Craps at this casino ?
3. Consider the binomial expansions :
( a + b )2 = a2 + 2 ab + b2 ,
( a + b )3 = a3 + 3 a2b + 3 ab2 + b3 ,
( a + b )4 = a4 + 4 a3b + 6 a2b2 + 4 ab3 + b4 ,
( a + b )5 = a5 + 5 a4b + 10 a3b2 + 10 a2b3 + 5 ab4 + b5 ,
· · ·
( a + b )n = an + C( n, 1 ) · an−1 · b + C( n, 2 ) · an−2 · b2 + · · · + C( n, r ) · an−r · br + · · · + bn .
Here
C( n, r ) = n !
r ! ( n − r ) ! for n ≥ 1 , r ≥ 1 ( integers ; n ≥ r ; 0 ! = 1 ) .
Show that when n is even,
C( n, 1 ) < C( n, 2 ) < · · · < C ( n,
n
2
) > · · · > C( n, n − 1 ) ;
and when n is odd,
C( n, 1 ) < C( n, 2 ) < · · · < C ( n ,
n − 1 2
) = C
( n ,
n + 1
2
) > · · · > C( n, n − 1 ).
P.S. Due to the Chinese New Year holidays, the online tutorial discussion for this tutorial is set on the week 1 - 6 March, 2021.
T-3.pdf
GET1018 / GEK 1544 The Mathematics of Games
Tutorial 3
1. Consider a Las Vegas roulette wheel with a bet of $ 5 on black (payoff = 1 : 1) and a bet of $ 2 on the specific group of 4 (e.g. 13, 14, 16, 17 ; payoff = 8 : 1). What is the bettor’s expectation on this combined bet?
2. A mythical slot machine has three wheels, each containing ten symbols. On each wheel there is 1 JACKPOT symbol and 9 other non-paying symbols. You put in 1 silver dollar ( 1 unit ) in the slot and the payoffs are as follows (the silver coin you put in the slot is not returned to you) :
3 JACKPOT symbols – $ 487 in silver is returned.
2 JACKPOT symbols – $ 10 in silver is returned.
1 JACKPOT symbols – $ 1 in silver is returned.
Define what it would mean to say that this slot machine is fair and then show that it is indeed a fair (that is , the expectation is zero ) “one-armed bandit” !
3. The martingale betting strategy had the gambler double his bet after every loss, so that the first win would recover all previous losses plus win a profit equal to the original stake. Since a gambler with infinite wealth will with probability 1 eventually flip heads, the Martingale betting strategy was seen as a sure thing by those who practised it.
Of course, none of these practitioners in fact possessed infinite wealth, and the exponential growth of the bets would eventually bankrupt those who choose to use the martingale betting strategy. Moreover, it has become impossible to implement in modern casinos, due to the betting limit at the tables. Because the betting limits reduce the casino’s short term variance, the Martingale system itself does not pose a threat to the casino, and many will encourage its use, knowing that they have the house advantage no matter when or how much is wagered.
Let one ‘round ’ be defined as a sequence of consecutive losses followed by a win, or consecutive losses resulting in bankruptcy of the gambler.
After a win, the gambler “resets” and is considered to have started a new round. A continuous sequence of martingale bets can thus be partitioned into a sequence of independent rounds. We will analyze the expected value of one round.
Let ℓ be the probability of losing ( e.g. betting on “ even ” in the Las Vegas roulette has 20/38 chance of losing ) . Let the commencing bet be equal to 1 unit , and n the finite number of bets you can afford to lose. Show that the expected profit (a loss if the number is negative) per round is given by
1 − ( 2 ℓ)n .
T-2.pdf
GET1018 / GEK1544 The Mathematics of Games
Tutorial 2
1. Using the property on probability :
( 1 ) P ( A∪B ) = P ( A ) + P ( B ) − P ( A∩B ) .
show that
(2)
P ( A∪B ∪C ) = P ( A ) + P ( B ) + P ( C )
−P ( A∩B ) − P ( A∩C ) − P ( B ∩C ) +
+ P ( A∩B ∩C ) .
Guess the formula for ( you are not required to prove it )
( 3 ) P ( A ∪ B ∪ C ∪ D ) .
Using the formula ( 3 ), argue in the positive direction that the probability on obtaining
at least one six in four rolls of a single die is 671
1, 296 , exactly the same result as Pascal’s
argument in the negative direction.
2. Suppose you are dealt 3, 4, 5, as the first three cards of a poker hand, and there are two subsequent cards to be served to you ( out of the remaining 52 − 3 = 49 cards ) . What is the probability of getting a straight ( i.e., getting “ A , 2 , 3 , 4 , 5 ,” ; “ 2 , 3 , 4 , 5 , 6 ” or “ 3 , 4 , 5 , 6 , 7 ” ) ?
3. After Chevalier de Mere found out the probability on obtaining one or more double sixes in 24 rolls of a pair of dice, and saw that it is less than half, he stopped playing the game ( refer to Q. 2 in T. 1 ) . Define a new pay - off R : 1 so that de Mere would think it is a fair game and consider playing it again.
T-1.pdf
GET 1018 / GEK 1544 The Mathematics of Games
Tutorial 1
1. In the movie “ 21 ”, Professor Rosa asked the student Ben the following question.
“There are three doors, and behind one of them is a car, while behind the other two are goats. If you choose the door with the car behind it, you win the car. Now, say you choose Door 1. The host then opens either Door 2 or Door 3, behind which is a goat. (The host knows what is behind each door, and never opens the door with the car behind it. ) The host now gives you the choice: do you want to stick with Door 1, or switch to the other door. What should you do? Or does it matter ? ”
Without hesitation Ben answered this correctly, which convinced Professor Rosa that Ben would be a good addition to their “card counting team”. Explain why Ben’s answer is correct, that is, by switching the choice rather than sticking with the original one, the probability increases to 2/3 .
2. Solve Chevalier de Mere’s problem on determining the probability of obtaining one or more double sixes in 24 rolls of a pair of honest dice.
3. Consider the basic property on probability :
P( C ∪ D) = P(C) + P(D) when C ∩ D = ∅
( ↑ i.e., no common events) .
Show that P(A ∪ B) = P(A) + P(B) − P(A ∩ B) when A ∩ B ̸= ∅ .
4. Compute the probability on getting at least 1 six in 3 rolls of an honest die. Then expose the flaw in the following argument, which attempts the same problem in the “ positive direction” :
The probability of a six in any one roll is 1 6 .
Since we have 3 rolls, P (at least 1 six ) = 1 6 + 1
6 + 1
6 = 1
2 .
Tutorials start on the third week. You will receive information on tutorial registration in the second week. No use to e-mail me regarding tutorial classes, unless you cannot settle it online. Thank you !
Course Homepage: LumiNUS. E-mail: [email protected]
