1. 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

2. 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:

3. 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.

4. EXAMPLE 76 Performance Ratio 118 / 37 = 3.189189 1 3 10 28
Player 1
Player 2
x is chosen, please provide a guess
Smaller than x, next guess 1
Smaller than x, next guess 3
Smaller than x, next guess 10
Smaller than x, next guess 28
STOP! x = 37 76
Performance Ratio
118 / 37 =

5. EXAMPLE 23 Performance Ratio (1+2+3+4+23) / 5 = 6.6 1 2 3 4
Player 1
Player 2
x is chosen, please provide a guess
Smaller than x, next guess 1
Smaller than x, next guess 2
Smaller than x, next guess 3
Smaller than x, next guess 4
STOP! x = 5 23
Performance Ratio
( ) / 5 = 6.6

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

7. EXAMPLE x = 13 x = 65 guesses: 1, 2, 4, 8, 16 sum: 1+2+4+8+16 = 31
performance ratio is
x = 65
guesses: 1, 2, 4, 8, 16, 32, 64, 128
sum: = 255
performance ratio is 3.923

8. 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  ) < x  exp( i +  )
The expected performance ratio is:

9. EXAMPLE Performance Ratio 129 / 48 = 2.6875  = 0.419 STOP! x = 48
Player 1
Player 2
x is chosen, please provide a guess
 = 0.419
Smaller than x, next guess
exp() = 1
Smaller than x, next guess
exp(+1) = 4
Smaller than x, next guess
exp(+2) = 11
Smaller than x, next guess
exp(+3) =30
STOP! x = 48
exp(+4) = 83
Performance Ratio
129 / 48 =

10. EXAMPLE Performance Ratio 201 / 63 = 3.190476  = 0.866 STOP! x = 63
Player 1
Player 2
x is chosen, please provide a guess
 = 0.866
Smaller than x, next guess
exp() = 2
Smaller than x, next guess
exp(+1) = 6
Smaller than x, next guess
exp(+2) = 17
Smaller than x, next guess
exp(+3) =47
STOP! x = 63
exp(+4) = 129
Performance Ratio
201 / 63 =

11. Smaller than x, next guess Smaller than x, next guess
EXAMPLE
Player 1
Player 2
x is chosen, please provide a guess
 = 0.195
Smaller than x, next guess
exp() = 1
Smaller than x, next guess
exp(+1) = 3
STOP! x = 6
exp(+2) = 8
Performance Ratio
12 / 6 =

12. 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?