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Guessing Game (Posted on 2014-05-16) Difficulty: 3 of 5
We decide to play the following game: An integer N will be randomly selected from the interval 0 - 100, inclusive. You try to guess N. After each guess, I tell you whether N is higher or lower than your guess.

If you successfully guess the integer, you win N dollars. Each guess costs you K dollars.

For each of the variants (a) and (b) below, what is the maximum value of K for which you'd be willing to play this game? Which strategy would you use to try to maximize your winnings in the long run?

(a) Once you start a round, you must continue until you guess N exactly.

(b) You may stop playing a round if you determine that N is too small to keep paying more money to guess N exactly. The money you've already spent on guesses is lost, but you may then start a new round with a new N.

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

Comments: ( Back to comment list | You must be logged in to post comments.)
re(5): Additional thoughts Comment 18 of 18 |
(In reply to re(4): Additional thoughts by tomarken)

Tomarken:

Well, if you want to pursue this, see my first part(b) post, where I give

Expected winnings = P(X) = (X+100)*(101 - X)/202.
Expected Additional cost =   K*(1+E(100-X)*(100-X)/101). 

After after determining that the maximum winnings is 68, for instance, the game changes to
  
  Expected winnings = P(X) = (X+68)*(69 - X)/138.
  Expected Additional cost =   K*(1+E(68-X)*(68-X)/69). 
  
You and I both expect that  (X+68)*(69 - X)/138 - K*(1+E(68-X)*(68-X)/69)
is negative for all X <= 69, where 
K = ((X+100)*(101 - X)/202) / (1+E(100-X)*(100-X)/101)

  Posted by Steve Herman on 2014-05-21 16:43:31
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