I have come into one problem that I don't know the solution of.
Let's suppose there is a casino, where you can bet on toin coss, and you get the double if you bet right. Let's also suppose that the casino always has money to pay to everyone.
It is a fair coin, so the gamblers have 50% chance to be right.
Now, imagine that all the gamblers always bet everything they have and what they have won already. In every bet. New gamblers enter the casino time from time. Gamblers do have finite amount of money.
The gamblers have high probability of losing everything, very soon. So, most of the gamblers will come out empty-handed after a while.
However, from the casino's point of view, they have zero expected income, because you do have zero expected income at this type of game.
So, the players lose everything, while the casino shouldn't have any income.
How is that possible?
edit:
There was a question if casino has infinite amount of money. I am not sure about that, actually. Does it change the problem? If it does, how?
Best Answer
The casino only has zero expected income at every finite time step, because at every stage there is a small chance that one of the gamblers has won an exponentially large amount of money.
However, if there are a finite number of gamblers, then after some finite time they will all have run out of money, and the casino will be in profit. But a priori you don't know when this time is, so there is always a zero expectation for the casino.
A nice way to see this is by looking at some random realizations. I wrote a quick Matlab script to simulate this situation. I took 1024 gamblers who each came into the casino with a dollar, and ran the simulation for 20 time steps (so that there's only a very small probability that any of the gamblers have any money remaining by the 20th step). I then repeated this 100 times, and plotted the casino's net gain/loss after each time step for each of the 100 runs.
As you can see, although all the casinos end up in profit, some of them suffer extreme losses before they enter profitability. It is this possibility for extreme loss that means that they always have zero expectation of profit.
Note that I had to truncate the bottom of the plot for this to be anything other than a mess of blue lines - the worst-performing casino in my sample incurred losses of $65,000 before returning to profitability!
Here's another way to look at it. After $t$ timesteps, the total player winnings are
$$\sum_{i=1}^n X_i$$
where each $X_i$ is a random variable taking the values $2^t$ with probability $2^{-t}$ or $0$ with probability $1-2^{-t}$.
For each player, their expected winnings at time $t$ is 1, but the variance of their winnings is $2^t-1$. Therefore the variance of the total winnings for $n$ players is $n(2^t-1)$. And the variance of casino winnings is the same, by symmetry.
This explains why the result is so unintuitive - for most problems, the variance stabilises as the number of timesteps increases. For this problem the variance keeps increasing, so talking about the limit of infinitely many timesteps doesn't make a lot of sense.