How much would you pay for a 50/50 chance to win $1B?
This post (https://old.reddit.com/r/wallstreetbets/comments/vkncj5/would_you_rather_get_a_guaranteed_million_dollars/) showed up on wallstreetbets and the more interesting question to me that came up was: how much is this opportunity worth? There must be a way to calculate how much someone would pay for a 50/50 chance to win $1 billion, so I’m curious how that would be calculated and what that dollar amount is?
The Kelly criterion (https://en.wikipedia.org/wiki/Kelly_criterion) is one possible way to decide on the amount to bet, in order to "maximiz[e] the expected value of the logarithm of wealth".
For a fixed probability of winning a fixed proportion of the bet, the amount turns out to depend on the size of your bankroll. However, as your example fixes the payoff but allows a variable buy-in, I don't see how to directly apply the formula from that article.
But, long story short, it is almost sure to depend on the size of your current bankroll, if you accept a Kelly-like criterion.
I'd say it depends on your "demand" to change lifestyle, from 0% to 100%, then the max bet should be: demand * min($500M, fun_money). Of course, the hard part is figuring out one's personal demand for change.
Exactly 1 cent. I maximize the gain and minimize the risk.
If I was already a billionaire, I would pay up to 500 million dollars. At 500 million dollars it's a EV neutral move, but there is also entertainment value, because I get to see if I win or lose. If it's less than 500 million dollars, it's a EV positive move, so I'll happily take it.
Since I'm not a billionaire, I would put up as much as I could comfortably lose, and still meet all of my existing financial obligations.
EDIT -- Instead of paying 500 million, I would pay up to 1 billion dollars, because paying 1 billion for a 50/50 chance to win 1 billion is an EV neutral move.
It's important to realize that if you pay for a chance to win, if you lose you don't get your money back. That makes the risk calculation dependent on how much you can afford to lose.
There's a trick here. I don't really want $1B. $2M is probably the max I'd need in my life. So I'd pay about max $0.99M.
If I were not as lazy (and with that much money I should be), I'd calculate the point where it's easier to earn more money. Having just 6 years runway would let me do a lot of experimental business stuff, but would those still have a 50% chance of earning a billion?
Anything less than $500m, especially if you could redo this 50/50 chance multiple times.
I'd give about 1000 hours of effort to it. I'm old, so I don't have a lot of time left, but that seems reasonable to me.
It depends; how much money do I have in the bank?