All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Games
Quit as a winner (Posted on 2015-09-24) Difficulty: 4 of 5
You have a normal deck of 52 playing cards You draw cards one by one (Cards drawn are not returned to the deck).
A red card pays you a dollar. A black one fines you a dollar.
You can stop any time you want.

a. What is the optimal stopping rule in terms of maximizing expected payoff?
b. What is the expected payoff following this optimal rule?
c. What amount in dollars (integer values only ) are you willing to pay for one session (i.e. playing as long as you wish, not exceeding the deck), using your strategy?

Source will be disclosed after the solution is published.

No Solution Yet Submitted by Ady TZIDON    
Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Some Thoughts computer exploration | Comment 3 of 8 |
The below table shows the results of 10,000 trials each of setting an amount by which one's being ahead would cause one to stop playing.

stop           prob of  expected
point  wins     win      value
  1    9654    0.9654     .9654
  2    8569    0.8569    1.7138
  3    7181    0.7181    2.1543
  4    5489    0.5489    2.1956
  5    3873    0.3873    1.9365
  6    2472    0.2472    1.4832
  7    1531    0.1531    1.0717
  8     862    0.0862     .6896
  9     433    0.0433     .3897
 10     220    0.022      .22

So it looks as if, when considering a constant value to be ahead, regardless of the stage of the game, you should stop when you're ahead by 4, giving an expected gain of about $2.20.

It might (likely) be possible to improve the expected value by changing the stop point as the game progresses.


DefDbl A-Z
Dim crlf$


Private Sub Form_Load()
 Form1.Visible = True
 
 Text1.Text = ""
 crlf = Chr$(13) + Chr$(10)
 
 For v = 1 To 10
 Randomize Timer
 wins = 0
 For tr = 1 To 10000
   DoEvents
   red = 26: black = 26
   posn = 0
   For draw = 1 To 52
     r = Rnd(1)
     If r < black / (black + red) Then
       black = black - 1
       posn = posn - 1
     Else
       red = red - 1
       posn = posn + 1
       If posn >= v Then wins = wins + 1: Exit For
     End If
   Next
   If tr = 10000 Then
     Text1.Text = Text1.Text & Str(v) & "    " & wins & "    " & wins / tr & Str(wins / tr * v) & crlf
   End If
 Next tr
 Next v

 Text1.Text = Text1.Text & crlf & " done"
  
End Sub



  Posted by Charlie on 2015-09-24 15:05:47
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (6)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information