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# The St Petersburg game: you have an advantage at Russian Roulette

27 August 2007

The 18th Century mathematician, Nicholas Bernoulli, wondered about guidelines for making risk-reward decisions. His first guess was that a good criterion would involve "expected value," a way of relating odds and payoffs, comparable to using "edge."

To see how this applies, picture a \$10 bet with a \$30 payoff a chance for a fourfold increase from \$10 to \$40. The bet would be "fair," no edge either way, were prospects of winning one out of four (25 percent). You'd have an edge if the shot was better, say one out of three (33-1/3 percent). You'd be an underdog if your chances were less, such as one out of five (20 percent). Expected value or edge says make at out of three, don't bet at one out of five, and take your pick at one out of four.

But, Bernoulli noted that rational individuals often don't make decisions this way. To investigate why not, he hypothesized a game. It later became known as the St Petersburg Paradox because it was described in scholarly form in the prestigious Russian Commentaries of the Imperial Academy of Science of St Petersburg.

To play, you buy-in for a specified fee. The house keeps this money regardless of what happens. You then flip a coin. On heads, the chance of which is one out of two, you're paid \$2 and the game ends. If the coin shows tails, you flip again. Heads this time pays \$4, given that the chance of tails followed by heads is one out of four. Tails on the second round leads to a third flip; now, heads pays \$8 the chance of tails-tails-heads being one out of eight. Flips continue, paying \$16, \$32, \$64, and so forth with chances qual to one out of 16, 32, and 64, etc, respectively, until heads finally shows and the game ends.

On the average, half of all players should be paid \$2 by winning on the first flip, a quarter will receive \$4 with heads on the second, an eighth will get \$8 on the third, and so on. Fractions such as one out of 1,024 will get \$1,024 with on the 10th flip, one out of 1,048,576 will receive \$1,048,576 on the 20th, and one out of 33,554,432 will pick up \$33,554,432 on the 25th.

Bernoulli wanted to see how much people would pay for the gamble. The expected value, and therefore the players' edge, is infinite. On this basis alone, any solid citizen should play no matter what the fee. Infinite expected return, an unending amount of money, would always be a smart move. But, Bernoulli found and contemporary decision support experts agree that few people would buy a chance for more than the equivalent of \$20 or \$25.

An intuitive explanation of the hesitancy to go whole hog is the far greater chance of loss than profit. Say the fee is \$100. Half of all players could anticipate being paid \$2 and finishing \$98 behind. A quarter would lose \$96 with a \$4 payoff. Only those lasting more than six flips would pick up over \$100 and come out ahead; the chance of reaching this stage is one out of 64.

Casinos won't offer a game precisely like St Petersburg. One reason is that bettors with an infinite edge, as a group, will eventually break the bank. Another reason is that casinos would need infinite cash reserves to back the action. With extensive but ultimately limited funds, an operator could conceivably offer a version with a cut-off point. For instance, after 19 tails, the game might be aborted and the player get paid at the 20-flip level \$1,048,576. However, this constraint changes the math significantly. A game funded with \$50 million would have an expected value of only \$13.29. Players would only have an edge paying less than this amount to participate.

Of course, 18th Century math mavens weren't satisfied with such informal logic. They wanted to find a rule to explain and help predict how people actually decide what they're willing to risk for various rewards under alternate conditions. The result, the "utility" principle, is about when and why a bird in the hand is worth two in the bush. We'll explore it in a later exposition. For now, just remember this rhyme of the poet, Sumner A Ingmark:

Though rewards are attractive, it's wise to be leery,
Of expensive investments that seem great in theory.

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Best of Alan Krigman
Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.