First, if you think that $8400 is not a life-changing amount of money, consider giving it to a charity in which it will go to help people who think otherwise, whether they are American or Malawian.
Here's another angle to look at re EV: Consider this scenario -- you can bet $8400 at a roulette table that's weighted 60/40 in your favor. Additionally, if you win, you can play double-or-nothing as much as you want, but each new spin must be a full double-or-nothing -- no going south (i.e., no taking money off the table) until you walk away. What do you do? Strict EV maximization says you play forever, since every bet has a positive EV, but obviously no human player would do that.
Let's say you have $8400 and can bet any or all of it on the first spin, after which future bets are locked based on the first one. How much do you stake on that first bet? The EV maximizing answer is to put on $8400, but each person's personal answer will be different. Same holds true for regular roulette, or any wager -- with real roulette I might bet $25 but I wouldn't be like that one guy some years ago who liquidated everything he owned and bet $250,000. (I'd also never play roulette since it's a sucker's bet with some of the worst odds of any casino game, but that's beside the point. :p)
1. With the first scenario, I would do it once, and then stop, because I would have a >50% chance of turning up a profit.
2. I don't really understand. If you bet $8400 on each bet, so you could theoretically go indefinitely into the red by continuing to lose money even when you are below zero, I would bet $8400, because if the odds are >50% of winning, I could eventually go to the big bucks. There is a mathematical proof that if you indefinitely bet $1, with a 50% chance of winning each time, you will at some point have every integer number of dollars, including extremely high and low ones. So, eventually, I'd hit the billions.
The proof is as follows:
Say you have, at the moment, $-1. Call the probablility that you will eventually advance to $0, and not have to lose money, p
. Call the probability of the same thing happening if you arrive at $-2 q
. You have a 1/2 probability of getting to $0, and a 1/2 probability of getting to $-2, so that means that the probability of getting to $0 from $-1 is equal to (1+q)/2. p=(1+q)/2. Now, we can look at what q
equals. Assuming we call the probability of getting to $0 from $-3 r
, we get q=(p+r)/2. Here, we see that the probabilities of getting to $0 from $0, $-1, $-2, $-3, etc. form an arithmetic sequence (constant difference between consecutive terms), and the first term in that sequence is 1. If the difference between consecutive terms is anything other than zero, we either go above 1 or below 0 eventually, both of which is impossible. This proves that all of the terms of the sequence must be equal to 1, meaning that you are guaranteed to eventually get back to $0 from any negative value. We can easily transfer the probabilities a googolplex over and get that if you have a negative googolplex of dollars and are guaranteed to get to zero, you are guaranteed to be able to get to a googleplex of dollars from $0. Similar results happen if you have a higher chance of advancing past $1.
However, if I could not bet money if I were in the red, then I would most likely bet a small amount. I wouldn't really want to lose a lot.
Golf wrote:I still believe the wager that maximizes winning chances is either all or nothing. Now, if you know for certain you opponent is betting everything, it would probably change the frequency of times you should bet everything/nothing.
I'm curious how you would answer the following question.
Say that right now, you get to bet a certain amount of your net worth on a single FJ-style clue in a wheelhouse category of yours.
How much of your net worth do you bet? Are the only correct answers zero or everything?
If someone asked me that question today, I would not bet literally every dollar I owned. But I would bet something.
After thinking about that for a scenario, I figured out that the solution to this paradox is that things are strange because every dollar does not affect one's quality of life equally. A dollar does much more for a poor person than a wealthy one, so betting your entire net worth would require an incredibly high percent chance of winning for it to be a favorable bet, since the last dollars taken away from you would lower your quality of life profoundly.
For the original question, I said that I would bet all but $1, because it was a wheelhouse. Also, because it is extra money I am wagering rather than what I already have, the odds don't have to be quite as high for it to be a good move. Even being a $1 champion would be an interesting distinction. If it was U.S. GEOGRAPHY or MATH, I would want to bet all but $1, but if it were OPERA or THE OSCARS, or even CLASSICAL MUSIC, I would bet $0.