I’m not much of a mathematician but I do like to go to Atlantic City, so this is my take on the paradox. I suspect this approach was previously taken by someone else and I would appreciate any feedback.

It appears to me that whatever fee one decides to pay, the odds of losing that fee(investment) is 50%.

What are the odds of breaking even?

If your fee is $2, the odds are 50%

If your fee is $4, the odds are 25%

If your fee is $8, the odds are 12.5%

If your fee is $16, the odds are 6.25%

If your fee is $32, the odds are 3.125%

If your fee is $64, the odds are 1.5% (rounded)

If your fee is $128, the odds are .75%.

Therefore the odds of losing $128 is approx 99%, very bad bet.

Since the odds of losing one’s fee is never in one’s favor a reasonable person would not play the game at $2. If you don’t play there is a 100% chance of keeping $2, if you play there is only a 50% chance of keeping $2.

Playing at a higher fee is more unreasonable….

…. on the other hand if you pay only $1 you still have a 50% chance of losing that dollar but you also have a 50% chance of DOUBLING your investment. The less you pay still keeps you at 50/50 but you can triple, quadruple… your investment.

So I think a reasonable person could play for less than $2.

I think the utility theory comes into play only when you change the starting amount of the game. As a middle class cautious gambler I would be comfortable paying a $20 fee for a game that started at $20 and a $50 fee for a game that started at $100.

I would not be comfortable paying a $100 fee for a game that started at $100 or $200.