You are at a casino. You have $1. A table offers you a game: you have to bet all your money; a fair coin will be tossed; if it lands heads, you triple your money; if it lands tails, you lose everything.
In the first round, it is rational to take the bet since the expected value of winning is $1.50, which is greater than what you started out with.
If you win the first round, you'll have $3. In the next round, it is rational to take the bet again, since the expected value is $4.50 which is larger than $3.
If you win the second round, you'll have $9. In the next round, it is rational to take the bet again, since the expected value is $13.50 which is larger than $9.
You get the idea. At every round, if you won the previous round, it is rational to take the next bet.
But if you follow this strategy, it is guaranteed that you will eventually lose everything. You will go home with nothing. And that seems irrational.
Intuitively, it feels that the rational thing to do is to quit while you are ahead, but how do you get that prediction out of the maximization of expected utility? Or does the above analysis only feel irrational because humans are loss-averse? Or is loss-aversion somehow optimal here?
Anyway, please dissolve my confusion.
Money doesn't work in a way that makes this interesting.
If I have one dollar, I basically have nothing. What can I buy for one dollar? Well, a shot at this guys casino. Why not?
Now I'm broke, ah well, but I was mostly here before. OR Now I've got 3 dollars. Still can't buy anything,except for this crazy casino.
Now I'm broke, ah well, but I was mostly here before, OR Now I've got 9 dollars. That's a meal if I'm near a fast food joint, or another spin at the casino which seems to love me
Now I'm broke, ah well, but I was mostly here before, OR Now I've got 27 dollars. That's a meal at a casino's prices. Do I want a burger or another spin?
And so on. At each level, compare the thrill of gambling with whatever you could buy for that money. You will eventually be able to get something that you value more than the chance to go back up to the table.
Always do the thing with this money that you will enjoy the most. Initially that is gonna be the casino, because one dollar. Before it gets to something interesting you will lose, because odds.