Game Description
 Player 1: Decides on integer x > 0
 Player 2 : Has to find a number yn so that yn  x
 Rules:


Player 2: y1< y2< …< yn
y1< y2< …< yn-1 < x and yn  x
 On a guess yj, player 1 either says:


smaller than x, please provide a next guess
larger or equal x, and reveals x stopping the game
Optimization Criteria
 Let the guesses be y1, y2, ….yn, so that
yn  x
yj < x for all j  n – 1
 The optimization criteria is the
performance ratio:
y
j
x
j
Programming Examples
1. Write a C program that randomly generates the number to
guess and ask user to provide the guesses. The program
will calculate and print the performance ratio for one
game
2. Write a C program that randomly generates the number to
guess and ask user to provide the guesses. The program
will calculate and print the performance ratio for 10
games and also calculates the average of performance
ratios for 10 games.
3. Modification of example 2: let user decide on amount of
games that user would like to play and output the average
performance ratio over all games.
EXAMPLE
Player 1
Player 2
x is chosen, please provide a guess
Smaller than x, next guess
Smaller than x, next guess
Smaller than x, next guess
Smaller than x, next guess
STOP!
x = 37
1
3
10
28
76
Performance Ratio
118 / 37 = 3.189189
EXAMPLE
Player 1
Player 2
x is chosen, please provide a guess
Smaller than x, next guess
Smaller than x, next guess
Smaller than x, next guess
Smaller than x, next guess
STOP!
x=5
1
2
3
4
23
Performance Ratio
(1+2+3+4+23) / 5 = 6.6
The Powers of 2 Strategy for the
Second Player
 It turns out that the simple strategy that selects
powers of 2: y0 = 1, y1 = 2, y2 = 4, … yi = 2i …
is an optimal deterministic strategy for this game
 The worst case for the strategy is when the number
selected by the first player is x = 2j + 1
 In this case the game is played until the second
player suggests 2j+1
EXAMPLE
 x = 13



guesses: 1, 2, 4, 8, 16
sum: 1+2+4+8+16 = 31
performance ratio is 2.384615
 x = 65



guesses: 1, 2, 4, 8, 16, 32, 64, 128
sum: 1+2+4+8+16+32+64+128 = 255
performance ratio is 3.923
A Randomized Strategy
The following simple randomized strategy gives an
improved expected value
 Let  R [0, 1) – randomly and uniformly chosen
from interval [0, 1)
 Define yj = exp( j +  )
 Let i be so that exp( i - 1 +  ) < x  exp( i + 
)
 The expected performance ratio is:
e x 1
e
x
EXAMPLE
Player 1
Player 2
x is chosen, please provide a guess
Smaller than x, next guess
Smaller than x, next guess
Smaller than x, next guess
Smaller than x, next guess
STOP!
x = 48
 = 0.419
exp() = 1
exp(+1) = 4
exp(+2) = 11
exp(+3) =30
exp(+4) = 83
Performance Ratio
129 / 48 = 2.6875
EXAMPLE
Player 1
Player 2
x is chosen, please provide a guess
Smaller than x, next guess
Smaller than x, next guess
Smaller than x, next guess
Smaller than x, next guess
STOP!
x = 63
 = 0.866
exp() = 2
exp(+1) = 6
exp(+2) = 17
exp(+3) =47
exp(+4) = 129
Performance Ratio
201 / 63 = 3.190476
EXAMPLE
Player 1
Player 2
 = 0.195
x is chosen, please provide a guess
Smaller than x, next guess
Smaller than x, next guess
STOP!
x=6
exp() = 1
exp(+1) = 3
exp(+2) = 8
Performance Ratio
12 / 6 = 2.0000
Discussion Questions
 Discussion of the Optimization Criteria selection:
Why not to choose yi/x, where yi  x?
 Answer: simple strategy 1, 2, 3, …is optimal
 Discussion of possible strategies:
 Why not to start with some “large” number?
 Why do we not benefit from increasing the next
guess only a little compared to the previous
guess?
 What is the disadvantage of making the next
guess, say, 100 times larger than previous guess?
