St. Petersburg Paradox

What is the St. Petersburg Paradox?

Let’s say you’re at a funfair and you come across a game where the cash prize can come as quite a surprise. Here’s how the game works: a coin is flipped again and again until it shows heads. If it shows heads on the first try, you win $2. Sounds simple? Well, if it’s tails first and then heads on your second try, your prize increases to $4. If it’s tails again and heads on the third flip, now you win $8. Every time you flip tails first, the amount you could win doubles right up until heads appear. The big question is, how much money is fair to pay to play?

The St. Petersburg Paradox is a famous problem in the study of economics and chance that looks at ideas like odds and making smart choices. It reveals a strange situation where what we feel is a fair price to play this coin game is way less than what the game might actually pay out on average, also known as the ‘expected value’.

Simple Definitions

A paradox is something that goes against what we would normally expect or what seems like common sense. In this case, even though math tells us that the average win from the game could be huge, even without limit, most people wouldn’t dream of paying a huge amount to participate. It’s weird because, well, usually we think more money equals better, right?

Expected value is a math concept where you look at all the possible results of a game or situation and figure out what outcome you might expect on average if you could repeat the game over and over. Imagine if you could play a game a zillion times; the expected value is what you think you’d typically make from each play in the long run.

Examples of the St. Petersburg Paradox

If someone offers you a chance to play a game where the prize can go incredibly high, but the chances of winning those high amounts are super slim, like the coin game we’ve discussed, that’s the paradox in action. The game’s potential winnings fly off the charts, but our choice in the real world doesn’t follow that same sky-high pattern.
Imagine you have a lottery where the jackpot is higher than any jackpot in history. According to the expected value, each ticket should be worth a lot because of the possible enormous payout. However, lots of people wouldn’t be willing to empty their bank accounts for a ticket, because they understand the likelihood of winning is tiny. This is another example of the paradox where our thinking doesn’t match the data.
Let’s think about an auction where an item that could potentially be very valuable but also could be worth very little is up for sale. Even though it could be worth a lot, people tend to bid much less than that highest potential value. This is because they weigh the risk of it being worth less more heavily than the possible high reward.
Suppose an insurance company offers a policy covering an extremely rare event. The expected value of what you’d need to claim is high, but most people wouldn’t pay premiums that are anywhere near as high. This mismatch is the paradox at work in the real world.

Answer or Resolution

The St. Petersburg Paradox makes us think twice about what we consider smart money decisions. The riddle was eventually solved by adding the idea of utility to the mix. It’s not just about pure value; it’s about the usefulness or enjoyment that money gives, which doesn’t just keep going up forever the more money you have. Bernoulli figured out that if you take this principle of diminishing returns into account, the expected utility doesn’t keep getting bigger, and that lines up more closely with what people would actually pay to play the game.

Major Criticism

People have taken shots at the St. Petersburg Paradox because of the assumptions it makes. Critics say that it forgets that most people don’t have limitless wealth to play with or that you can’t really have a game that offers unlimited money. Others say that it doesn’t take into account the personal side of things—like how some people are just more willing to take risks than others.

Practical Applications

This whole paradox thing isn’t just theory—it ties into the real world too:

Economics: It’s helped us get a clearer idea of why people make the money choices they do, especially when those choices seem a bit risky. It’s been a big deal for the part of economics that looks at how our brains and feelings impact our financial decisions.
Insurance: The kind of thinking behind the St. Petersburg Paradox is also behind the way insurance companies work out the price for insuring something. People prefer a small, certain cost over facing a huge bill if something bad happens unexpectedly.
Lotteries and Gambling: These businesses know how the paradox works and use it. They offer big jackpots for a small chance to win. They’re betting on the fact that we’re okay with paying a little for the slim chance of striking it rich.
Decision Theory: The paradox has led to smarter, more complex ways to understand the risks we’re willing to take, which has made a splash in the world of figuring out how people make choices under uncertainty.

Why is it Important

The St. Petersburg Paradox matters because it’s made us rethink some of our ideas about money and choices. It has changed economics by showing that there’s more to our decisions than the cold hard numbers might suggest. This is important for the average person because every time we make choices about spending or saving, we’re juggling similar ideas of risk and reward. Understanding why we don’t always ‘go for the gold’ can help us make better financial decisions in life.

Conclusion

The St. Petersburg Paradox isn’t just a head-scratcher; it’s a real game-changer in how we understand the economy, decisions, and our own behavior. It has shown us that we can’t just look at the probable financial gain; we have to consider how much we value each dollar. By recognizing that most of us aren’t willing to risk it all based on a math equation, the paradox has paved the way for new ways to think about money, business, and the decisions we make every day.