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.
Below is a table of what guesses would be made, using optimal strategy, when the uniformly-distributed random N is the various possibilities, and the expected winnings when that happens, when K is the highest integer that allows one to play rationally in part b, which is K = 11:
0 66 quit -11
1 66 quit -11
2 66 quit -11
3 66 quit -11
4 66 quit -11
5 66 quit -11
6 66 quit -11
7 66 quit -11
8 66 quit -11
9 66 quit -11
10 66 quit -11
11 66 quit -11
12 66 quit -11
13 66 quit -11
14 66 quit -11
15 66 quit -11
16 66 quit -11
17 66 quit -11
18 66 quit -11
19 66 quit -11
20 66 quit -11
21 66 quit -11
22 66 quit -11
23 66 quit -11
24 66 quit -11
25 66 quit -11
26 66 quit -11
27 66 quit -11
28 66 quit -11
29 66 quit -11
30 66 quit -11
31 66 quit -11
32 66 quit -11
33 66 quit -11
34 66 quit -11
35 66 quit -11
36 66 quit -11
37 66 quit -11
38 66 quit -11
39 66 quit -11
40 66 quit -11
41 66 quit -11
42 66 quit -11
43 66 quit -11
44 66 quit -11
45 66 quit -11
46 66 quit -11
47 66 quit -11
48 66 quit -11
49 66 quit -11
50 66 quit -11
51 66 quit -11
52 66 quit -11
53 66 quit -11
54 66 quit -11
55 66 quit -11
56 66 quit -11
57 66 quit -11
58 66 quit -11
59 66 quit -11
60 66 quit -11
61 66 quit -11
62 66 quit -11
63 66 quit -11
64 66 quit -11
65 66 quit -11
66 66 55
67 66 82 74 70 68 67 1
68 66 82 74 70 68 13
69 66 82 74 70 68 69 3
70 66 82 74 70 26
71 66 82 74 70 72 71 5
72 66 82 74 70 72 17
73 66 82 74 70 72 73 7
74 66 82 74 41
75 66 82 74 78 76 75 9
76 66 82 74 78 76 21
77 66 82 74 78 76 77 11
78 66 82 74 78 34
79 66 82 74 78 80 79 13
80 66 82 74 78 80 25
81 66 82 74 78 80 81 15
82 66 82 60
83 66 82 91 86 84 83 17
84 66 82 91 86 84 29
85 66 82 91 86 84 85 19
86 66 82 91 86 42
87 66 82 91 86 88 87 21
88 66 82 91 86 88 33
89 66 82 91 86 88 89 23
90 66 82 91 86 88 89 90 13
91 66 82 91 58
92 66 82 91 97 93 92 26
93 66 82 91 97 93 38
94 66 82 91 97 93 95 94 17
95 66 82 91 97 93 95 29
96 66 82 91 97 93 95 96 19
97 66 82 91 97 53
98 66 82 91 97 99 98 32
99 66 82 91 97 99 44
100 66 82 91 97 99 100 34
The average of these expected winnings are the previously listed 1.75247524752475.
The strategy used in the above is the following optimal list:
possibilities rational
remaining guess
0 - 100 66
67 - 67 67
67 - 69 68
67 - 73 70
67 - 81 74
67 - 100 82
69 - 69 69
71 - 71 71
71 - 73 72
73 - 73 73
75 - 75 75
75 - 77 76
75 - 81 78
77 - 77 77
79 - 79 79
79 - 81 80
81 - 81 81
83 - 83 83
83 - 85 84
83 - 90 86
83 - 100 91
85 - 85 85
87 - 87 87
87 - 90 88
89 - 90 89
90 - 90 90
92 - 92 92
92 - 96 93
92 - 100 97
94 - 94 94
94 - 96 95
96 - 96 96
98 - 98 98
98 - 100 99
100 - 100 100
no other ranges come up in the optimal play with K = 11.
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Posted by Charlie
on 2014-05-18 13:16:04 |
rational play for the highest integral K allowing play in part b
