What Is the St. Petersburg Paradox?

a man getting ready to flip a coin
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You’re on the streets of St. Petersburg, Russia, and an old man proposes the following game. He flips a coin (and will borrow one of yours if you don’t trust that his is a fair one). If it lands tails up then you lose and the game is over. If the coin lands heads up then you win one ruble and the game continues. The coin is tossed again. If it is tails, then the game ends. If it is heads, then you win an additional two rubles.

The game continues in this fashion. For each successive head we double our winnings from the previous round, but at the sign of the first tail, the game is done.

How much would you pay to play this game? When we consider the expected value of this game, you should jump at the chance, no matter what the cost is to play. However, from the description above, you probably wouldn’t be willing to pay much. After all, there is a 50% probability of winning nothing. This is what is known as the St. Petersburg Paradox, named due to the 1738 publication of Daniel Bernoulli Commentaries of the Imperial Academy of Science of Saint Petersburg.

Some Probabilities

Let's begin by calculating probabilities associated with this game. The probability that a fair coin lands heads up is 1/2. Each coin toss is an independent event and so we multiply probabilities possibly with the use of a tree diagram.

  • The probability of two heads in a row is (1/2)) x (1/2) = 1/4.
  • The probability of three heads in a row is (1/2) x (1/2) x (1/2) = 1/8.
  • To express the probability of n heads in a row, where n is a positive whole number we use exponents to write 1/2n.

Some Payouts

Now let's move on and see if we can generalize what the winnings would be in each round.

  • If you have a head in the first round you win one ruble for that round.
  • If there is a head in the second round you win two rubles in that round.
  • If there is a head in the third round, then you win four rubles in that round.
  • If you have been lucky enough to make it all the way to the nth round, then you will win 2n-1 rubles in that round.

Expected Value of the Game

The expected value of a game tells us what the winnings would average out to be if you played the game many, many times. To calculate the expected value, we multiply the value of the winnings from each round with the probability of getting to this round, and then add all of these products together.

  • From the first round, you have probability 1/2 and winnings of 1 rubles: 1/2 x 1 = 1/2
  • From the second round, you have probability 1/4 and winnings of 2 rubles: 1/4 x 2 = 1/2
  • From the first round, you have probability 1/8 and winnings of 4 rubles: 1/8 x 4 = 1/2
  • From the first round, you have probability 1/16 and winnings of 8 rubles: 1/16 x 8 = 1/2
  • From the first round, you have probability 1/2n and winnings of 2n-1 rubles: 1/2n x 2n-1 = 1/2

The value from each round is 1/2, and adding the results from the first n rounds together gives us an expected value of n/2 rubles. Since n can be any positive whole number, the expected value is limitless.

The Paradox

So what should you pay to play? A ruble, a thousand rubles or even a billion rubles would all, in the long run, be less than the expected value. Despite the above calculation promising untold riches, we would all still be reluctant to pay very much to play.

There are numerous ways to resolve the paradox. One of the simpler ways is that no one would offer a game such as the one described above. No one has the infinite resources that it would take to pay someone who continued to flip heads.

Another way to resolve the paradox involves pointing out how improbable it is to get something like 20 heads in a row. The odds of this happening are better than winning most state lotteries. People routinely play such lotteries for five dollars or less. So the price to play the St. Petersburg game should probably not exceed a few dollars.

If the man in St. Petersburg says that it will cost anything more than a few rubles to play his game, you should politely refuse and walk away. Rubles aren’t worth much anyway.

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Taylor, Courtney. "What Is the St. Petersburg Paradox?" ThoughtCo, Oct. 27, 2017, thoughtco.com/what-is-the-st-petersburg-paradox-3126175. Taylor, Courtney. (2017, October 27). What Is the St. Petersburg Paradox? Retrieved from https://www.thoughtco.com/what-is-the-st-petersburg-paradox-3126175 Taylor, Courtney. "What Is the St. Petersburg Paradox?" ThoughtCo. https://www.thoughtco.com/what-is-the-st-petersburg-paradox-3126175 (accessed January 21, 2018).