Expected value, martingales, and the St. Petersburg paradox

A martingale is a technical term for a type of betting strategy, the simplest of which involves doubling your bet size until you win. It is more or less the mathematical name for the “double or nothing” strategy, provided you can keep doubling it as many times as you’d like.

In reality, this course of action quickly becomes problematic. To see how rapidly a doubling bet can grow, consider the following example. Place 2 pennies on the first square of a standard chessboard with 64 squares. For each subsequent square, imagine doubling the number of pennies. The second square will have 4, the next will have 8, then 16, and so on until you reach the last square. How much money is piled on the last square?

The correct answer is “a ridiculous amount”, in excess of one hundred trillion dollars. The precise number of pennies is 2 multiplied together 64 times or

\(2^{64} = 18\text{,}446\text{,}744\text{,}073\text{,}709\text{,}551\text{,}616\)

Considering this example, it’s easy to see how you can quickly run into trouble using a martingale betting strategy. When you hit a string of losses, which is bound to happen if you keep pushing your luck, the amount of money you’ll eventually need to put up will meet or exceed your entire wealth. If you lose that bet, you’re wiped out.

Note that if you repeatedly apply a betting system like this at a casino, they will almost certainly come out ahead since their bankroll will outlast yours. You might get lucky and come out ahead if you try it a few times, but I wouldn’t recommend it.

The martingale betting system leads to many strange situations. One of the best-known is the St. Petersburg paradox. It was first described in writing by Nicolas Bernoulli in the early 1700s. He was a member of the Bernoulli family, who famously produced many talented academics.

Before we dig into the paradox, we need to discuss expected values. If you’ve had a statistics or probability course, you’re likely already familiar with the term expected value. In the context of gambling, it considers the probability of each event occurring and its corresponding payout, providing an estimate of what to expect in the long term.

The expected value of 3.5 for the die-rolling example in the video tells us that if we were to play this game many times, record our payout each turn, and then compute the average of all the payouts, it would be close to 3.5. In this way, the expected value gives the “fair price” of the game. If we paid $3.5 dollars for each roll, our long-term winnings would be close to $0 (as would the long-term profit of the person operating the game).

Here is an even simpler example. Suppose we win 2 dollars if a coin flip lands on heads and 1 dollar if it lands on tails. Then the expected value of the game is:

\(\text{heads payout} \times \text{probability of heads} + \text{tails payout} \times \text{probability of tails} \)

which is equal to:

\(2 \times \frac{1}{2} + 1 \times \frac{1}{2} = 1.5\)

So if we paid $1.5 to play the game, our long-term winnings would hover around $0.

Applying the expected value computation to the specific coin-flipping game in the video yields infinity. This happens because the minute probability of a long string of tails is offset by the possibility of an exceedingly large payout.

The “paradox” is that an infinite expected value suggests that we should be willing to pay any amount of money to play this game. Yet most people wouldn’t pay more than 20 bucks to play it, even if you could play more than once.

There have been many proposed solutions to the paradox. The simplest of which is that in the real world, we tend to ignore very small probabilities as if they were 0.

My favorite practical solution was offered by Buffon (the same fellow behind Buffon’s Needle Problem). He argued that a 56-year-old man ignores the possibility of dying within the next 24 hours, which has a probability of about 1 in 10,000 according to mortality tables, so events with a probability of less than 1 in 10,000 should be ignored. I think overall mortality rates have improved since the 1700s, and I don’t know why he chose age 56, but let’s use his numbers for the sake of demonstration.

In the context of the coin-flipping game, this means that long strings of tails beyond 13 flips or so should simply be ignored (i.e. assume that this will never happen and set the probability to 0). If we do this, then the expected value is much more reasonable at about $7.

However, suppose we had an infinite line of credit and an infinite amount of time to play the game as many times as we’d like. Then it’s true that in the long run we could always come out ahead regardless of the fixed price of the game. Even if it was 1 million dollars. Of course, we’d also have to assume that the person running the game had access to an infinite line of credit so that we would get paid when that long sequence of tails finally occurs.

The St. Petersburg paradox is a veridical paradox, meaning that the method to arrive at the result is valid, but the result seems absurd. This often happens in mathematics when you introduce the concept of infinity. In this example, it is present in the game itself, since a string of tails beyond any finite number you can imagine could theoretically occur.

In any case, the St. Petersburg paradox is an interesting thought experiment and has led to many improved theories of how probabilities and expectations should be treated in the real world.