If you’ve fallen down even one foot of Bitcoin’s burrow, you’ve probably seen references to the idea of ”everything there is, divided by 21 million.” Originally postulated by Knut Svanholm, Ioni Appelberg, and Guy Swann in a Youtube video with the same title, the concept reflects the seemingly infinite potential value of Bitcoin. As it continues to outperform various asset classes such as gold and spreads into new markets, the total market value of Bitcoin appears to be only limited by the absolute maximum of 21 million bitcoins in circulation.
When a financial endeavor promises, in theory, to produce infinite returns (divided by 21 million), it’s easy to get caught up in the hype. People even mortgages taken out to buy bitcoins. Faced with infinitely high returns, this is not irrational. Why wouldn’t you pay anything and everything for the opportunity to make your life infinitely better? However the Saint Petersburg game, a thought experiment in economics that can trace its roots back to 1713, may offer some insights into the hype surrounding the potential value of bitcoin.
The Saint Petersburg party means much more than the Bitcoin bowl in St. Petersburg, Florida. Originally posted on Papers of the Imperial Petersburg Academy of Sciences By Daniel Bernoulli, The St. Petersburg Paradox is a thought experiment that tests traditional behavioral economics. Imagine that you are asked to pay a certain amount of money to participate in a bet. In the bet, a fair coin is tossed until it is heads. When it finally shows its head, you get paid two dollars to the power of North the number of times it was launched. If you land heads on the first toss, you are paid $ 2. If you land tails on the first toss and heads on the second toss, you are paid $ 4. If you land heads on the third toss, $ 8, and so on. How much should you be willing to pay to play this game?
The theory of pure economic decision would tell you to pay any finite amount. You need to sell your car, take out a third mortgage on your home, and contact all the lenders in your address book just for a chance to play the St. Petersburg game. Let’s see why a ideally A rational person could end up in such a seemingly irrational conclusion.
For a person seeking to maximize expected value, the traditional approach to decision theory is to take the amount of value in a payment and multiply it by the probability that that payment will occur. The probability that the coin will land heads on the first flip is 50 percent and the payoff is $ 2. Multiplying them together gives us an expected value of $ 1. The probability that the coin will land heads on the second flip is 25 percent and the payment is $ 4, which gives us an expected value of $ 1 again. You can calculate the expected value for any number of tosses North and it will always give $ 1.
What are the probabilities that the coin will land tails a million times before it finally lands heads? Really very low (around 10−301030), but the amount you can win is $ 2 million. To find the maximum amount of money you should be willing to play this game, you can add up the sum of the expected values for all the potential outcomes. As the number of cross throws in a row increases, the probability becomes incredibly small, but it is never zero. As such, we should take the sum of an infinite set.
As you can see, this gives us a total expected value of infinite for the game. With a potential payout approaching infinity like this, you could pay some finite amount of money to play the game and still maximize its expected value.
This recommendation has confused economists for centuries. How could it be reasonable to pay millions or even billions of dollars for an infinitesimally small chance of a positive net reward? So far there is no conclusive answer, but decreasing marginal utility is a long-standing suggestion why should not pay a ridiculous amount. Diminishing marginal returns is the idea that at some point, once all your needs are met, the extra dollars are worth less to you than before. Vitalik Buterin explained in this article that a gallon of water a day is worth much more when you have nothing than when you are already 99. Similarly, in the St. Petersburg game, it may seem justified to risk losing a billion dollars to win three billion, but if you already have a billion dollars, how much can an extra billion really do for you? If you lose, the lifestyle difference between zero dollars and one billion dollars is much more extreme than the difference in lifestyle between one billion and three billion dollars. This is also true for more realistic sums of money.
Now what does all this have to do with Bitcoin? If Bitcoin can truly represent the full value of the entire economy (past, present, and future), then it’s the closest we’ve ever seen to a St. Petersburg game in real life. Like the St. Petersburg game, the payout itself would be finite (bitcoin will likely always have a quantifiable value attached to it), but its potential approaches infinity.
In a sense, this highlights the asymmetric risk-reward tradeoff of investing in bitcoin. You can only lose what you invest, but the potential payoff is much, much greater. In another sense, this thought experiment recommends putting everything you can into bitcoin, paying no attention to your potential losses. The only thing you need to consider when deciding whether or not to put an additional $ 100 of your fiat currency into bitcoin is how you could use that $ 100 right now. If money is the difference between paying for utilities or having your water shut off, it’s probably smarter to keep it. After all, even the best-case scenario for bitcoin won’t add as much happiness to your life as giving up showers for a month would. But if you’ve paid off all of your credit cards and the rent check is in the mail, then know that the extra dollars you invest in bitcoin are backed by the economic theory of the last few hundred years.
This is a guest post from Will Henson. The opinions expressed are entirely my own and do not necessarily reflect those of BTC, Inc. or Bitcoin Magazine.