The Crazy Russian Merchant
At the St. Petersburg casino there's a game: starting with a prize of 2\$, a coin is tossed and each time the coin comes up tail the prize money doubles. When it comes up head you win the prize and the game ends.
The question is: how much money is fair to spend for taking part in such a game?
This seems like a paradox if we consider as “fair” estimating the expected value of the game and spending such amount. We can do so by multiplying each possible outcome by its probability of occurring
$$ 2 \cdot \frac{1}{2} + 4 \cdot \frac{1}{4} + 8 \cdot \frac{1}{8} + ... $$
Which is equals to evaluating the series
$$ \sum_{n=1}^{\infty} 2^n \cdot \frac{1}{2^n} $$
This is an infinite sum of 1s so it would seem reasonable to spend any finite amount of money to take part in such a game, a thing that many consider paradoxical given that the probability of winning back that same amount of money is very low.
Many proposed solutions involve not using the monetary value but rather a function of it, called utility function. A first attempt was made by Daniel Bernoulli [1] using a logarithmic function of monetary value, the reason being that the more we have money, the less valuable added money becomes. With this function the series converges, but as Menger [2] points out, it is sufficient to offer a reward that grows in the order of $ 2^{2^n} $ for the paradox to reappear.
Other solutions [3] aren't really satisfactory to me: keeping as a problem constraint that we use the expected value and consider all possible outcomes, there is a game of coin tossing for which it is reasonable to spend all my money as an entry fee.
Let's consider the following problem instead, which I call the Crazy Russian Merchant problem.
At the St. Petersburg market there's a crazy merchant who is the only one selling wheat due to global shortage. To decide wheat cost he tosses a coin until it comes up head, and the wheat price is given by $2^n$ where $n$ is the number of times it came up tail plus one. How much is rational to bring in order to be able to buy wheat?
For this problem to be somewhat realistic we assume that we are broke and so all money must be loaned. We can say at least two things:
- if I have to buy wheat once, and I want it with 100% certainty, that's not possible because of the random nature of the coin tosses;
- if I have to buy wheat multiple times I can specify a percentage of times I can tolerate to not buy wheat.
In this perspective it's useful to think of probabilities as embedded in the time domain rather than in possible universes, as [4] points out.
Our problem can be modeled as a random variable $Y = 2^X$ where $X$ is a geometrically distributed random variable with parameter $p = 0.5$.
That is, if I require only 50% as a threshold, it's fine to just loan 2\$: that's because $ P(Y \leq 2) = P(X \leq 1) = 0.5 $. So if I buy multiple times with 2\$ I may be able to get wheat half the times. If I require that I need wheat 75% of the times, than I should bring 4\$, because $ P(Y \leq 4) = 0.75 $.
If we don't have a success rate requirement over time though, we don't have any rational reason to bring more than 2\$, that is, the minimum we need to be able to buy wheat with a probability $p$ with $p > 0$. We will be fine being successful 50% of the times. If we bring less we won't ever “win” at this game, because $ P(Y \leq 1) = P(X \leq 0) = 0 $.
Back to the original game, where you win instead of losing, you can't win all the times: what win ratio is rational? This seems dependent entirely on an individual, but let's assume I don't have a time requirement: 2\$ should be the fee.
In both example it's probably 2\$ + $\epsilon$, because in the wheat problem we assumed there's added value from having wheat (either eating it or selling it for higher profit – storing it would be meaningless) and in the original game you always win something, so the casino would be losing money whatever the results.
I don't pretend to have found a particularly important insight mathematically speaking for this problem. Rather reasoning on it has been fun and helpful to learn new things.
Fortunately casinos have already set their fees.
[1] https://www.jstor.org/stable/1909829
[2] Menger (1934) in the english translation by Wolfgang Schoellkopf
[3] https://plato.stanford.edu/archives/sum2022/entries/paradox-stpetersburg/