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.
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
Private Sub Form_Load()
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
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Posted by Charlie
on 2014-05-18 12:06:44 |
Repaired VB program agrees with Steve's answer to part b
