Suppose your friend offers you the opportunity to play this game for free: Your friend will flip a biased coin 1000 times, which has a 60% chance of coming up heads each time, and a 40% chance of coming up tails. Every time it comes up heads, you are paid $100, and every time it comes up tails, you pay $100. You should definitely play this game, unless you are ridiculously risk-averse, because the chances of getting more than 500 heads are almost 100%, so you are virtually guaranteed to make money.
Suppose the game is changed slightly: Instead of agreeing upfront that the coin will be flipped 1000 times, you are given the option after each coin flip whether to continue flipping, to a maximum of 1000 flips, or to settle all payments and stop flipping (so this game is the same as the first one, with an additional option to terminate early). In order to decide whether to play this game, you think it through in your head, and you realize that, after 999 flips, you would probably end up terminating (i.e. not flipping the 1000th time), because at that time, you’d have a 40% chance of losing $100, and you are too risk-averse for that. But then you realize that, after 998 flips, you would effectively have only 1 flip left (because you’d know you would terminate after the 999th), so you’d apply the same logic and terminate at that time. You realize that this same logic can be applied repeatedly, backwards, all the way to the 1st flip, so you decide not to play the game at all.
But that doesn’t make any sense – the 2nd game is the same as the 1st, except for an option to terminate early (which makes the 2nd game superior), so it can’t make sense for you to agree to play the 1st game but not the 2nd.
So where is the logic wrong?
————
Note: I thought of this game in response to hearing the financial advice which I complain about in my last post.
The logic is wrong the instant you don’t want to make a bet with 1:1 payout but which has 3:2 odds in your favor. It’s a bet with a positive expectation. If you don’t want to make it the 1000th time, you’re dumb enough that you don’t even want to make it once.
“because at that time, you’d have a 40% chance of losing $100, and you are too risk-averse for that.”
That was true back on the first flip. That was true on every individual flip. How did someone with this much risk aversion proceed past the first flip?
“you’re dumb enough…”
It is not true that only dumb people refuse bets with a positive expectation. Many smart people would refuse a bet with a 51% chance of winning $100,000 and a 49% chance of losing $100,000.
Whether you are too risk averse to play the game with only single flip may be a different issue from whether you are too risk averse to play the game with many flips, since you have a *much* higher chance of winning the game with many flips (you have a 99.999999993% chance of winning the game with many flips).
Yes, I agree that bets with positive expectations aren’t uniformly smart, because money has diminishing returns in terms of how much happiness it gets you.
I’m going to guess that the error is a failure to appreciate or apply conditional probability. I’m not sure that’s the right way to describe it, but the likelihood of failure obviously gets compounded because of the many trials, but this person is behaving as though his foresight doesn’t extend beyond the next bet. I don’t think that numbers tend to regress to the mean, but I am pretty sure that as the number of trials rises, the distribution gets smoother and more likely to accord with the P of the event (right?). But this person is ignoring that fact.
Interesting puzzle. My only “solution” is that your risk aversion should depend fairly strongly on your existing wealth. If you are reasonably sure to win a large sum of money by playing the first 999 rounds, you won’t be as risk averse to the 1000th round.
Playing the game once might make me broke for the month; playing the game one additional time after 999 games almost certainly won’t. So I’m not sure the backward induction is appropriate.
Julian – yes you’re right, as the # of events increases, the ratio of heads/tails will converge to 60/40.
that fact, of course, does by itself solve the puzzle though.
GregS – suppose you found someone who said that he could be certain that, no matter his wealth, he would just barely be too risk averse to play the 60/40 coin flip game once.
Although I agree this person may not be typical (people are more likely to behave like the person you describe, who becomes less risk-averse as he becomes more wealthy), we should be allowed to consider this type of person as well.
How does your analysis change if we are considering such a person?
I’ve been exploring in this vicinity for quite a while but had to comment. This is quite concise, do another like this one!
Couldn’t have explained it much better myself
Your mode of telling everything in this post is in fact pleasant,
all be capable of easily understand it, Thanks a lot.