Game Description

1
Game Description

• Player 1 Decides on integer x gt 0
• Player 2 Has to find a number yn so that yn ?
  x
• Rules
• Player 2 y1lt y2lt lt yn
• y1lt y2lt lt yn-1 lt 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 lt 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
Player 1
Player 2
x is chosen, please provide a guess
1
Smaller than x, next guess
3
Smaller than x, next guess
10
Smaller than x, next guess
28
Smaller than x, next guess
76
STOP! x 37
Performance Ratio 118 / 37 3.189189
5
EXAMPLE
Player 1
Player 2
x is chosen, please provide a guess
1
Smaller than x, next guess
2
Smaller than x, next guess
3
Smaller than x, next guess
4
Smaller than x, next guess
23
STOP! x 5
Performance Ratio (123423) / 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 2j1

7
EXAMPLE

• x 13
• guesses 1, 2, 4, 8, 16
• sum 124816 31
• performance ratio is 2.384615
• x 65
• guesses 1, 2, 4, 8, 16, 32, 64, 128
• sum 1248163264128 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 - 1 ? )? lt x ? ?exp( i
  ? )?
• The expected performance ratio is

9
EXAMPLE
Player 1
Player 2
x is chosen, please provide a guess
? 0.419
?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
Smaller than x, next guess
?exp(?4)? 83
STOP! x 48
Performance Ratio 129 / 48 2.6875
10
EXAMPLE
Player 1
Player 2
x is chosen, please provide a guess
? 0.866
?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
Smaller than x, next guess
?exp(?4)? 129
STOP! x 63
Performance Ratio 201 / 63 3.190476
11
EXAMPLE
Player 1
Player 2
x is chosen, please provide a guess
? 0.195
?exp(?)? 1
Smaller than x, next guess
?exp(?1)? 3
Smaller than x, next guess
STOP! x 6
?exp(?2)? 8
Performance Ratio 12 / 6 2.0000
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?