NATIONAL UNIVERSITY OF SINGAPORE

FACULTY OF SCIENCE

SEMESTER 2 EXAMINATION 2009 2010

GEK1544 The Mathematics of Games

May 2010 Time allowed : 2 hours

INSTRUCTIONS TO CANDIDATES

1. This examination paper contains a total of FOUR (4) questions andcomprises FIVE (5) printed pages (including this page).

2. Answer ALL questions.

3. Candidates are each allowed to bring in ONE (1) hand-written, double-sided help-sheet no larger than A 4 size.

4. Candidates may use calculators. However, they should lay out systemati-cally the various steps in the calculations.

PAGE 2

Answer ALL the four questions.

Question 1 [ 25 marks]

In a collection of 52 poker cards, consider the thirteen kinds

Ace = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q , K .

(i) Explain why there are

(4 13)(4 12)(4 11)(4 10)(4 9)P ( 5 , 5)

different arrangements (order not important) of the 5 cards, taken out of the 52cards, so that all the five cards are of different kinds.

(ii) In a 5 cards draw from the 52 poker cards, find the following.

(ii)a Probability on obtaining straight flush.

(ii)b Probability on obtaining flush but not straight flush.

(ii)c Probability on obtaining straight but not straight flush ,

[ It is given that C( 52, 5) = 2, 598, 960 . ]

(iii) Using (i) and (ii), find

P (worse than a pair)

in a 5 cards draw from the 52 poker cards.

In (ii) and (iii), the answers should be written at the end of your answers toQuestion 1, and accurate up to 7 decimal places. That is

P = 0. ??? ???? .

Provide the details of the calculations leading to the answers.

Question two starts on page 3 .

PAGE 3

Question 2 [ 25 marks]

Consider the Fibonacci numbers

c1 = 1 , c2 = 1 , c3 = 2 , c4 = 3 , c5 = 5 , c6 = 8 , , cn+1 = cn + cn1 , ,

where n 2 is an integer. In the Fibonacci betting strategy, one fixes a resultin a game and always bets on the same result ( payoff is 1 : 1 ), with starting betequal to $ c1 = $ 1 , then follows the procedure described below.

(a) If one wins the opening game, one stops and makes a profit of $ 1 .

(b) If one loses the bet with $ c1 = $ 1 , one bets on with the amount equal to thenext Fibonacci number, that is, $ c2 = $ 1 .

(c) In case one wins on the bet of $ c2 = $ 1 , one goes back to c1 and bets with theamount equal to $ c1 = $ 1 . The process is stopped if one wins the bet $ c1 = $ 1 .Otherwise, one follows step (b) .

(d) In case one loses on the bet of $ c2 = $ 1 , one continues the process, each timebets with the amounts equal to the subsequent Fibonacci numbers c3 , c4 , ,until a future win appear, say at the bet of $ cn+1 (n 2 is an integer) .(e) Suppose one wins on the bet of $ cn+1 , then one crosses out the numberscn+1 and cn in the Fibonacci sequence. Next, one bets with the amount equal to$ cn1 .

(f) If one loses the next game with bet $ cn1 , one bets with the next Fibonaccinumber, that is, $ cn , and continues with step (d) until one wins again, then backto the instruction in step (e) .

(g) If one wins after betting $ cn1 , one crosses out the numbers cn1 and cn2in the Fibonacci sequence, and next bets with $ cn 3 .

The process continues until either one comes back to the number c1 = 1 again, andbets with $ c1 = $ 1, and wins ; or one comes back to the number c2 , and wins thenext two rounds with bets $ c2 = $ 1 and $ c1 = $ 1 , respectively. In both cases,after the last winning, one stops the process and finishes one round.

For example, after 5 consecutive loses [ losing $ (1 + 1 + 2 + 3 + 5) = $ 12 ] , onewins on the next bet of $ c6 = $ 8 . Next, one bets with $ c4 = $ 3 , and loses.Accordingly, one bets with $ c5 = $ 5 . Suppose one wins, then one bets with$ c3 = $ 2 . Again one wins, one bets with $ c1 = $ 1 . If one wins again, onefinishes a round and stops. As the payoff equals to 1 : 1 , we count the profit

$ (8 + 5 + 2 + 1 12 3) = $ 16 $ 15 = $ 1 .

Question two continues on page 4 .

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[ Question 2 continues ... ]

(i) On applying the Fibonacci betting strategy, one finds herself/himself bettingwith the amount $ cN+1 , where N 1 is an integer. Show that one has lost

$ (c1 + c2 + c3 + + cN) .

(ii) Using mathematical induction, or otherwise, show that

c1 + c2 + c3 + + cN = cN+2 1 for N 1 .

(iii) For any one round in the Fibonacci betting strategy, show that the profit isexactly one dollar.

Question 3 [ 25 marks]

Consider a zero-sum game with players A and B . Player A has strategies A1 andA2 , while player B has strategies B1 and B2 . The payoffs are shown in the tablebelow .

B1 B2

A1 a2 b2

A2 c2 d 2

Here a, b, c and d are positive numbers. (The payoffs for B are negative of thenumbers shown.)

(i) Show that there are no dominating strategies.

(ii) Show that the Maxi-Mini method does not yield a saddle point.

(iii) Let p be the probability on A playing strategy A1 , and q the probability onB playing strategy B1 . Using the 2 2 table :

B1 B2

A1 p q p (1 q)A2 (1 p) q (1 p)(1 q)

find the optimal value of p for A to play the mixed strategies. Express youranswers to part (iii) in terms of a, b, c and d .

Question four is on page 5 .

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Question 4 [ 25 marks]

In Game Theory, Guess 23of the Average is a game where n people guess

what two third of the average of their guesses will be ( here n 2) . The numbersare restricted between 0 and 100 (including 0 and 100). Let ai be the guess of thei-th player ( 1 i n) . The payoff for the i-th player is given by

100 ai 23 a1 + + ai + + ann

.Here | | represents the absolute value of the number inside, i.e.

| a| = a if a 0 ; | a| = a if a < 0 .

For example , | 1| = 1 , whereas | 1 | = (1) = 1 .

(i) Suppose n = 10 and

a2 = a3 = = a10 = 23 100 .

Find a1 so that the payoff for the first player is 100 . Write your answers in theform ?? . ?? , that is, accurate up to 2 decimal places.

(ii) Show that

a1 = a2 = = an = 0 (i.e., zero for all ai)

is a Nash equilibrium for the game.

(iii) Show that for any strategy

a1, , ai, , anwith at least one number not equal to zero (say, ai 6= 0), then (at least) one ofthe players can gain from the original payoff by either increasing or decreasing theoriginal guess a little bit, while other players guesses remain unchanged. (Caution :the player who can gain from making small changes may or may not be the i -th

player.)

END OF THE EXAMINATION PAPER.