What is the St. Petersburg Paradox?

Okay, let's talk about this interesting "St. Petersburg Paradox."

What is the St. Petersburg Paradox?

You can think of it as a coin game where, in theory, you're guaranteed to profit immensely, but in reality, anyone who plays is a fool.

Let's first look at the rules of the game

Suppose we play a game with very simple rules:

You pay me an "entry fee" first.
I start flipping a perfectly fair coin.
If the first flip is heads, the game ends, and I pay you 2 yuan.
If the first flip is tails, I flip again. If the second flip is heads, the game ends, and I pay you 4 yuan.
If the second flip is still tails, I flip a third time. If the third flip is heads, the game ends, and I pay you 8 yuan.
...And so on. Each time tails appears, the prize money doubles until you flip heads.

Now the question is: What is the maximum "entry fee" you would be willing to pay to play this game?

Think about it intuitively first.

Most people's first reaction might be: "Hmm... there's a 50% chance of winning on the first flip and getting 2 yuan. A 25% chance of winning on the second flip and getting 4 yuan. It feels like... I'd pay 5 or 10 yuan to play, at most, right?"

This thought is very normal and aligns well with the intuition of ordinary people.

Where is the Paradox (Contradiction)?

The contradiction lies in the "Expected Value" calculated by mathematicians.

What is 'Expected Value'? Simply put, it's the "average return" of this gamble. If you could play this game an infinite number of times, add up all the money won, and divide by the number of times played, that average would be the expected value. In rational decision-making, the maximum price you should be willing to pay is this expected value.

Alright, let's do the math:

There's a 1/2 probability that you flip heads on the 1st try, winning 2 yuan. The contribution to the expected value is: (1/2) * 2 = 1 yuan.
There's a 1/4 probability that you flip heads on the 2nd try, winning 4 yuan. The contribution to the expected value is: (1/4) * 4 = 1 yuan.
There's a 1/8 probability that you flip heads on the 3rd try, winning 8 yuan. The contribution to the expected value is: (1/8) * 8 = 1 yuan.
There's a 1/16 probability that you flip heads on the 4th try, winning 16 yuan. The contribution to the expected value is: (1/16) * 16 = 1 yuan.
...And this can go on indefinitely.

So, the total expected value of this game is the sum of the contributions from all possible scenarios:

Expected Value = 1 + 1 + 1 + 1 + 1 + ... = ∞ (infinity)

The paradox is thus born:

Mathematical calculation tells you: The expected return of this game is infinite, so theoretically, no matter how expensive the entry fee is (even a hundred million yuan), you should participate, because "in the long run" you are guaranteed to profit.
Your intuition tells you: Are you crazy? I wouldn't pay a large sum to play this game! Most of the time, I'd probably just walk away with 2 or 4 yuan. Risking my entire fortune for a super prize that has an extremely small probability of occurring? No way.

This is the St. Petersburg Paradox: A decision that is mathematically "absolutely worthwhile," yet appears utterly absurd in reality.

Why does this happen? How can we explain this paradox?

This paradox has troubled many people. Later, economists and mathematicians provided several very practical explanations, which also gave rise to crucial concepts in decision theory.

1. Diminishing Marginal Utility

This is the most classic and important explanation.

It means: The value of money to you is not constant.

For example:

Suppose you're starving. If I give you 100 yuan at that moment, it's life-saving money, and the "happiness" or "utility" of those 100 yuan is extremely high.
Suppose you're already a billionaire. If I give you another 100 yuan, you might not even bother to bat an eyelid. For you, those 100 yuan would bring almost no additional happiness.

Back to our game:

Winning from 0 to 2 yuan is very happy.
Winning from 2 to 4 yuan is also quite good.
However, the joy of winning from 1 million to 2 million yuan is definitely not as intense as winning from 0 to 1 million yuan.
When you've won a certain amount of money (e.g., tens of millions), doubling it again no longer brings a substantial change to your life; its "utility" grows very slowly.

Therefore, when we make decisions, what we're truly measuring is not the "numerical value" of money, but the "utility" (satisfaction) it can bring us. If we calculate expected value using "utility," the sum would not be infinite, but a finite, relatively small number. This number would be very close to the few yuan you intuitively would be willing to pay.

2. Real-world limitations

The paradox's calculation is based on an ideal model, but the real world is not that ideal.

The house is not infinitely wealthy: Who would play this game with you? If tails were flipped dozens of times consecutively, the prize money would become an astronomical figure (e.g., exceeding the global GDP), which the house simply couldn't afford to pay. Since the house's payout capacity has a limit, the expected value of this game is also not infinite, but finite.
Your money is also limited: You cannot have unlimited money to play this game repeatedly until you hit the big prize.

3. Risk Aversion

Most people are risk-averse.

Certainty: 10 yuan in your hand is concretely 10 yuan.
Uncertainty: Although this game has an infinite expected value, you're likely to only win 2 or 4 yuan. Taking on such immense risk for a minuscule, almost negligible probability (e.g., 20 consecutive tails) to win a huge sum is too much.

People tend to prefer a certain, albeit small but decent, return, rather than gambling on a "dream" with extremely low probability and extremely high reward.

To summarize

The St. Petersburg Paradox doesn't actually mean there's a problem with mathematics itself; rather, it reveals a profound truth:

A purely mathematical expected value model cannot fully describe complex human decision-making behavior.

When people make decisions, they consider the actual utility of money, various real-world limitations, and their degree of risk aversion. This paradox has greatly advanced the fields of economics and decision theory, helping us understand that "value" is a more subjective and complex concept than "price."