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		1.5.3— Example Where the Average Does Not Exist: The St. Petersburg Game

		1— The Average Winnings Is 50¢ for a Non-Fractal, Ordinary Coin Toss Game

		Play the following game. Toss a coin. If it lands Heads, you win $1; if it lands Tails, you win nothing. The average payback to you is the sum of the probability of each outcome multiplied by the winnings associated with it, namely (1/2) x 0 + (1/2) x 1 = $0.50. The more times N that you play this game, the closer the payback to you, averaged over all the N games, approaches $0.50. The house should be ready to pay out $0.50, on average, and you should be willing to bet $0.50 to play each game.

		2— There Is No Average Winnings for the Fractal, St. Petersburg Game

		Now play a game formulated about 250 years ago by Niklaus Bernoulli and analyzed and published by his cousin Daniel Bernoulli. You toss a coin and continue to do so until it lands Heads. You get $2 from the house if it lands Heads on the first toss, $4 if on the second toss, $8 if on the third toss, $16, if on the fourth toss, and so on, so that with each additional toss the number of dollars that the house must pay is doubled. The average payback to you is (1/2) x 2 + (1/4) x 4 + (1/8) x 8 + (1/16) x 16 . . . = 1 + 1 + 1 + 1 . . . = ¥. There is NO number that we can identify as the average winnings!

		The more times N that you play this game, the larger the payback to you averaged over all the N games. The average winnings after N games continues to increase as N increases. It does not reach a finite, limiting value that we can identify as the average winnings. The average winnings for playing this game does not exist. This game is fractal because the distribution of winnings (how much you win for each probability of winning) has a power law scaling.

		This game is called the St. Petersburg paradox. Since there is a 50/50 chance that you will $2 on each game, you would be willing to put up $2 to play. But since the average winnings is infinite, the house would want you to put up more than all the money in the world to play. This type of game and the statistical properties associated with it were studied by mathematicians since it was first proposed. However, these ideas became separated from the main thread of probability theory that became popular among natural scientists.

 

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