We’re on the streets of St. Petersburg, Russia, and an old man proposes the following game to us. He flips a coin (and will borrow one of ours if we don’t trust that his is a fair one). If it lands tails up then we lose and the game is over. If the coin lands heads up then we 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 we 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 we pay to play this game? When we consider the expected value of this game, we should jump at the chance, no matter what the cost is to play. However from the description above, we probably wouldn’t be willing to pay much. After all, there is a 50% probability that we will win 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.
We 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 that we have 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.
We now move on and see if we can generalize what the winnings would be in each round.
- If we have a head in the first round we win one ruble for that round.
- If there is a head in the second round we win two rubles in that round.
- If there is a head in the third round, then we will four rubles in that round.
- If we have been lucky enough to make it all the way to the nth round, then we will win 2n-1 rubles in that round.
Expected Value of the Game
The expected value of a game tells us what our winnings would average out to be if we 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, we have probability 1/2 and winnings of 1 rubles: 1/2 x 1 = 1/2
- From the second round, we have probability 1/4 and winnings of 2 rubles: 1/4 x 2 = 1/2
- From the first round, we have probability 1/8 and winnings of 4 rubles: 1/8 x 4 = 1/2
- From the first round, we have probability 1/16 and winnings of 8 rubles: 1/16 x 8 = 1/2
- From the first round, we 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.
So what should we play to pay? 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 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 for 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, we will politely refuse and walk away. Rubles aren’t worth much anyway.