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 Guessing Game (Posted on 2014-05-16)
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)

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 Repaired VB program agrees with Steve's answer to part b | Comment 11 of 18 |
DefDbl A-Z
Dim crlf\$
Dim guess(100, 100)
Dim expVal(100, 100), k

Function mform\$(x, t\$)
a\$ = Format\$(x, t\$)
If Len(a\$) < Len(t\$) Then a\$ = Space\$(Len(t\$) - Len(a\$)) & a\$
mform\$ = a\$
End Function

Text1.Text = ""
crlf\$ = Chr(13) + Chr(10)
Form1.Visible = True
DoEvents
For k = 11 To 12 Step 0.01
For span = 1 To 101
For start = 0 To 100
fin = start + span - 1
If fin <= 100 Then
If span = 1 Then
expVal(start, fin) = start - k
guess(start, fin) = start
Else
bestev = -999999
For g = start To start + span - 1
ev = g / span  ' hit it this guess
If g > start Then ev = ev + ((g - start) / span) * expVal(start, g - 1)
If g < start + span - 1 Then ev = ev + (((start + span - 1) - g) / span) * expVal(g + 1, start + span - 1)
ev = ev - k
If ev > bestev Then
bestev = ev: bestg = g
End If
Next g
If bestev > 0 Then
guess(start, fin) = bestg
expVal(start, fin) = bestev
Else
guess(start, fin) = -1
expVal(start, fin) = 0
End If
End If

End If
Next start
Next span
Text1.Text = Text1.Text & mform(k, "###.00  ") & mform(expVal(0, 100), "##0.00000") & mform(guess(0, 100), "##0") & crlf
DoEvents
Next k
End Sub
`		     best		     init   K    exp. gain    guess   4.00   27.80198     48  5.00   22.87129     45  6.00   18.33663     37  7.00   14.30693     42  8.00   10.63366     48  9.00    7.31683     54 10.00    4.35644     60 11.00    1.75248     66 12.00    0.00000     -1 13.00    0.00000     -1 14.00    0.00000     -1 15.00    0.00000     -1 16.00    0.00000     -1 17.00    0.00000     -1 18.00    0.00000     -1 19.00    0.00000     -1 20.00    0.00000     -1 21.00    0.00000     -1 22.00    0.00000     -1`

where -1 indicates not to bet, which is the only reason the expected gain is zero rather than negative.

Finer distinctions:

` 11.00    1.75248 66 11.01    1.72851 67 11.02    1.70455 67 11.03    1.68059 67 11.04    1.65663 67 11.05    1.63267 67 11.06    1.60871 67 11.07    1.58475 67 11.08    1.56079 67 11.09    1.53683 67 11.10    1.51287 67 11.11    1.48891 67 11.12    1.46495 67 11.13    1.44099 67 11.14    1.41703 67 11.15    1.39307 67 11.16    1.36911 67 11.17    1.34535 68 11.18    1.32198 68 11.19    1.29861 68 11.20    1.27525 68 11.21    1.25188 68 11.22    1.22851 68 11.23    1.20515 68 11.24    1.18178 68 11.25    1.15842 68 11.26    1.13505 68 11.27    1.11168 68 11.28    1.08832 68 11.29    1.06495 68 11.30    1.04158 68 11.31    1.01822 68 11.32    0.99485 68 11.33    0.97149 68 11.34    0.94851 69 11.35    0.92574 69 11.36    0.90297 69 11.37    0.88020 69 11.38    0.85743 69 11.39    0.83465 69 11.40    0.81188 69 11.41    0.78911 69 11.42    0.76634 69 11.43    0.74356 69 11.44    0.72079 69 11.45    0.69802 69 11.46    0.67525 69 11.47    0.65248 69 11.48    0.62970 69 11.49    0.60693 69 11.50    0.58416 69 11.51    0.56139 69 11.52    0.53861 69 11.53    0.51584 69 11.54    0.49307 69 11.55    0.47030 69 11.56    0.44752 69 11.57    0.42475 69 11.58    0.40198 69 11.59    0.37921 69 11.60    0.35644 69 11.61    0.33366 69 11.62    0.31089 69 11.63    0.28812 69 11.64    0.26535 69 11.65    0.24257 69 11.66    0.21980 69 11.67    0.19703 69 11.68    0.17426 69 11.69    0.15149 69 11.70    0.12871 69 11.71    0.10594 69 11.72    0.08317 69 11.73    0.06040 69 11.74    0.03762 69 11.75    0.01485 69 11.76    0.00000 -1 11.77    0.00000 -1 11.78    0.00000 -1 11.79    0.00000 -1 11.80    0.00000 -1 11.81    0.00000 -1 11.82    0.00000 -1 11.83    0.00000 -1 11.84    0.00000 -1 11.85    0.00000 -1 11.86    0.00000 -1 11.87    0.00000 -1 11.88    0.00000 -1 11.89    0.00000 -1 11.90    0.00000 -1 11.91    0.00000 -1 11.92    0.00000 -1 11.93    0.00000 -1 11.94    0.00000 -1 11.95    0.00000 -1 11.96    0.00000 -1 11.97    0.00000 -1 11.98    0.00000 -1 11.99    0.00000 -1 12.00    0.00000 -1`

 Posted by Charlie on 2014-05-18 12:06:44

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