File:Petersburg-Paradox.png

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Summary
Graph of average value per play over approximately 20,000 game plays of the St. Petersburg Paradox game. The average value rises from about one to five over the first 500 plays and then slowly rises to an average value of eight between 500 and 20,000 plays.

The graph illustrates the characteristics of the St. Petersburg Paradox game:


 * as game play progresses there is a typical slow decline in average value punctuated occasionally be a very large jump in average value


 * as game play progresses there is a very, very slow overall rise of the average winning over many game plays because the occasional large jumps are on average slightly larger than the long, slow declines between them.


 * although the average winning rises overall, it rises more and more slowly as gameplay progresses.

The graph illustrates the characteristics that make the game seem paradoxical:


 * Since, on the whole, the average value continually rises, the average value of a gameplay does indeed go to infinity as the number of gameplays goes to infinity.


 * Since the average value rises towards infinity at a very, very slow rate that continues to slow further as gameplay progresses, typical values of a few gameplays, or even a few hundred or a few thousand gameplays, are very small. In order to reach even modestly high average returns, tremendously huge amounts of gameplays are typically required.

The combination of infinitely high average winnings with very low typical winnings from a few gameplays seems paradoxical, but the graph shows how both of these can coexist.

Graph created by Brent Hugh.

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{| border="1" ! date/time || username || edit summary
 * 03:05, 30 September 2005 || |
 * 02:44, 30 September 2005 || |
 * 02:20, 30 September 2005 || |}
 * 02:20, 30 September 2005 || |}

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