Paradox of Infinite Rich Gamblers

In a recent stream, I talked to philosopher Djalma Beorne about a philosopher named Charles Sanders Peirce. We talked about how you can have a betting game where 50% of people will win and 50% of people will lose in each round, but you will have a probability of 100% that you can play it to become richer. 

So we spend a bit of time trying to sort out this paradox. In the stream we unravel this problem. I will work out the solution here, but if you want to watch the whole stream, the link can be found here. 

You can see the statement of the problem in Pierce’s lecture on The Maxim Pragmatism found in Chapter 10 of his collected works.

There are many problems connected with probabilities which are subject to doubt. One of them, for example, is this: Suppose an infinitely large company of infinitely rich men sit down to play against an infinitely rich bank at a game of chance, at which neither side has any advantage, each one betting a franc against a franc at each bet. Suppose that each player continues to play until he has netted a gain of one franc and then retires, surrendering his place to a new player.

In order to solve this problem you want to figure out the probability you will eventually reach a gain of one franc. He sets up a basic principle in probability, independent events. The point here is simple, let us say that you have a chance of ⅙ of rolling a 1 on a 6 sided dice. What is the probability of it happening twice or in other words getting two 1s. Here it is 1/6 * 1/6 = 1/36. If the events don’t affect each other they are independent and thus if you want to see the probability of them both happening you simply multiply them. 

So Peirce points out this nugget of wisdom. Each player has an infinite quantity of money. This means no matter how much money you win or lose it will not affect your probability of being able to win or lose; In other words you can’t go broke. This means that the probability of winning on net 2 francs is just the probability of winning on net 1 franc twice. Each event is independent.

He then wants to try to calculate the probability of being able to win 1 franc and thus walk away a winner. The probability of gaining 1 franc is the probability of winning plus the probability of losing and then gaining on net 2 francs from the position of losing one. You either win 50% of the time and get to walk away instantly; If you lose the other 50% of the time, then you will need to win twice to come back.

We need to remember that since we have an infinite quantity of francs, that being up or down any quantity of francs does not change the probability of being able to go up or down any quantity of francs. This means that we can substitute the probability of gaining 2 francs for gaining 1 franc squared. 

Now that we have only 1 variable to solve for we can turn this into a quadratic equation. This is something you learn how to solve in algebra.

Here the result of these equations tells us that there is only one solution, the probability you will walk away 1 franc richer is 1 or in order words 100%. This is assuming you are infinitely rich and playing against an infinitely rich bank in the way Peirce described. However under this scheme you have a 50% chance of winning or losing each bet. There is always someone who will in theory lose all the games they play, especially if they are among an infinite quantity of rich men. So how does this make sense that the probability you will win is 100%. 

Let us imagine here that you have an apples orchard that is full of trees. Now say that so far you have yielded 1000 apples. Now before the next season you plant your last tree that will yield 10 apples per year, this brings the total yield to 100 apples per year. What is the probability that out of all the apples you have gained from the orchard they came from the final tree. Here it would depend on the year.

After 10 years the probability will be 5%. But the question I put to you is this, what would the probability be after an infinite amount of time has passed. Understanding the logic of limits, we observe that this probability will tend towards 10% as the years progress. What is significant about this relationship is that both the final tree and the whole orchard will have produced an infinite quantity of trees, but we can see that just because you are comparing infinities does not mean they are equal.

Now let us think about a bank. You have an account in the bank that starts with 100 dollars and it grows at 10% per year. Now the whole bank starts with 1000 dollars and grows as a whole at 21% per year. What is the probability at a given time a dollar in the bank is yours?

We can simplify this equation.

So here about 10 years in, the probability is about 3.8%. What about an infinite number of years? Well we can observe the size of the bank grows exponentially faster compared to your bank account. This means that the ratio of your bank account to the size of the bank will forever shrink and approach zero. Just because the probability of selecting one of your dollars in the bank is zero doesn’t mean you don’t have any money in the bank. Here in fact, it doesn’t mean you don’t have an infinite quantity of money in the bank. The relationship as we have determined can still be zero.

Pierce shows that we think about probability as the relationship of occurrences in an infinite quantity of instances. What we observe is that over time an infinite quantity of infinitely rich men who sit down to play against the infinitely large bank are down money. But in comparison to the total lot of the infinite gamblers this tends towards zero as more games are played. This explains how an infinite quantity of people can lose, but still the probability of losing is zero.