In the world of probability and decision theory, few paradoxes are as perplexing and thought-provoking as the St. Petersburg Paradox. Named after the Russian city where it was first introduced by Swiss mathematician Daniel Bernoulli in 1738, this paradox challenges our intuitive understanding of rational decision-making and the value we place on uncertain outcomes. In this blog post, we will explore the St. Petersburg Paradox, its implications, and the proposed solutions that have arisen over the centuries.
To understand the St. Petersburg Paradox, let’s begin with a simple hypothetical gambling game. In this game, you’re offered the opportunity to participate by paying a fixed amount, which we say is $10. A fair coin is then flipped and the game continues as follows:
- If the coin lands heads (a 50% chance), you receive double your initial bet, making it $20.
- If the coin lands tails, you flip the coin again. If it lands heads on the second flip, you receive four times your initial bet, making it $40. If it lands tails again, you flip once more and so on.
- This process continues until the coin lands heads, at which point your winnings are calculated by doubling the previous prize money. This doubling continues indefinitely until a heads is obtained.
Now, the critical question is: How much would you be willing to pay to play this game?
At first glance, it seems pretty profitable. After all, the potential winnings seem astronomical, with each flip of the coin doubling the prize money and potential rewards. However, the paradox emerges when we consider the expected value of this game.
The Expected Value (EV) of a game is calculated by multiplying each possible outcome by its probability and summing them up. In this case, the EV can be expressed as follows:
EV = (0.5 * $20) + (0.25 * $40) + (0.125 * $80) + …
As you can see, the EV of this game is an infinite series that keeps doubling with each flip of the coin. Mathematically, this series sums up to infinity, indicating that the expected value of the game is infinitely high.
Herein lies the St. Petersburg Paradox. According to what the math says, people should be willing to pay any finite amount to play a game with an infinite expected value. In reality, most people aren’t willing to pay a large amount to participate in such a game. This contradiction between rationality and human behavior is at the heart of the paradox.
While the paradox challenges our conventional usage of expected value, it also highlights the importance of incorporating psychological and behavioral factors into decision theory. It is worth noting that one common flaw in the initial set-up is that money is unlimited, because in reality, there is a finite amount of money. Using this as a cap for the potential gains, there is a finite expected value to the game. However, understanding the paradox encourages us to delve deeper into the complexities of human choices and the dissonance between mind and logic.