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PETERSBURG PARADOX
A game of tossing a coin, at which the stake is doubled in each toss, is known in the probability theory as the Petersburg paradox from 1783. It was the fact that the experienced gamblers did not want to stake more than 3 till 40 roubles on the win, despite that Daniel Bernoulli believed and proved that the mean gain should be infinite great. The autority of the Swis mathematician was also great inough, that his reasoning about the full probability at this game was accepted for so long.
Feller pointed that if the game stops at some moment without the win, the probability is not full and tried to determine some limit of the stake according to the length n of the game S /nlog n. But nobody analyzed the reasoning of gamblers.
Suppose that a good coin is tossed and that the string of tosses is normal. Therefore, in the long run there is equal number of heads and eagles (better 0 and 1, 1 representing the win and simultaneously the end of the game). A game ends with the first win and a new game starts from the lowest stake. This limits the mean wins considerably.
Moreover, each game must end. The reasons can be different. The gambler can not continue, since he is too drunken (remember that is was Petersburg), he lost all his money, he must fall in or the night club closes.
It can be supposed that the remaining stake was lost.
We write all possible results of three games. Gains are differences between the stakes and the wins
Sequence | Gains | ||||
1 | 1 | 1 | 1 | 1 | 1 |
1 | 1 | 0 | 1 | 1 | -1 |
1 | 0 | 1 | 1 | 0 | 2 |
0 | 1 | 1 | 0 | 2 | 1 |
1 | 0 | 0 | 1 | 0 | -2 |
0 | 1 | 0 | 0 | 2 | -1 |
0 | 0 | 1 | 0 | 0 | 4 |
0 | 0 | 0 | 0 | 0 | -4 |
We calculate all possible results for sequences of growing lengths and the number of games played
Length n | Number of tosses | Number of games | Gain | Total win | Mean win | ||||||
0 | 1 | 2 | 4 | 8 | 16 | 32 | |||||
0 | 1 | 1 | 1 | 0,5 | |||||||
1 | 2 | 2 | 1 | 1 | 1 | 0,833 | |||||
2 | 8 | 6 | 2 | 3 | 1 | 5 | 1,125 | ||||
3 | 24 | 16 | 4 | 8 | 3 | 1 | 18 | 1,400 | |||
4 | 64 | 40 | 8 | 20 | 8 | 3 | 1 | 56 | 1,666 | ||
6 | 384 | 224 | 16 | 48 | 20 | 8 | 3 | 1 | 432 | 1,928 | |
7 | 996 | 612 | 32 | 112 | 48 | 20 | 8 | 3 | 1 | 1120 | 2,187 |
Zero gain means that the game does not end, the last toss is 0.
The number of tosses is n x 2n. The number of games is the difference of the consecutive numbers of tosses, thus (n+1) x 2n-2. Otherwise, in the block of (n-1) tosses (n-1) x 2n-2 ones are, representing the end of the game. To them 2n-1 ones are added on the last places of sequences. In the longer sequences the wins can be only doubled, their numbers remain the same.
The increased number of games ends in games ending after the first toss, with the gain one. It is (n+2) x 2n-3. The ratio of these games is greater than 0.5.
Now, a question appears, what the pace of the game was? How many tosses per minute were usual, how fastly stakes were made, bets were payed after the end of each run? Supposing that one session lasted mostly only one night, and that it was drunken at, it could not be more than some thousands of tosses. The experienced gamblers did not see in their lifes the full probability, the stake 3 corresponds to the length of the all replayed sequences of length 5, and about 50 games. Did not the banker and the player change their roles after each play?
The win of short plays is much lower than the win from the full probability. Gamblers knew that the game ends too often too early and that then a new game begins from the lowest stake. Bernoulli did not calculated that when one gains, the game starts again from the lowest stake. In real life it is not possible to count on the full probability, but each serie ends at latest in the half of the infinity.
After adding two dummy rows to the matrix above, an interesting inverse matrix exists which elements are
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
-1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
-1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 |
-1 | -1 | -3 | 1 | 0 | 0 | 0 | 0 |
-1 | -1 | 1 | -3 | 1 | 0 | 0 | 0 |
-1 | -1 | 1 | 1 | -3 | 1 | 0 | 0 |
-1 | -1 | 1 | 1 | 1 | -3 | 1 | 0 |
-1 | -1 | 1 | 1 | 1 | 1 | -3 | 1 |
The columns Number of tosses and Number of plays are just sums of following columns.
Literature
William Feller An Introduction to Probability Theory and its Applications, J.Willey, New York, 1970, Chapter 10.4.