Our protagonist, Euclid, has devised a scheme to beat the house in Math World Las Vegas:
He bets a dollar, double-or-nothing with even odds, on red in roulette. If he wins, great, he's done. If he loses, he doubles his previous bet and repeats until he wins, which with probability 1 he eventually will.
According to the Wikipedia page on Martingale betting, even though each bet has zero expected value, Euclid is certain to make money with this strategy. But what about his expected winnings? Are those still zero?
Argument for NO: He either makes $1 off the bat or he loses some number of times first. If he loses $1, loses $2, loses $4, then finally wins $8 -- lo and behold, he's netted $1. That's true for any possible losing streak (except an infinite losing streak, which has zero probability), so he always makes $1.
Argument for YES: Euclid is making a sequence of bets, each of which has zero expected value. The sum of any number of zeros is zero. (Also he has an infinitesimal chance of losing infinite money so that has to factor in.)
FAQ
1. How much money does Euclid start with?
Infinite money! Or, rather, we'll say he has unlimited credit. Any finite amount of money he wants to bet, he borrows it as needed. This is magical math world.
2. What's the house's edge?
None! Again: math world. Euclid wins each spin with exactly 50% probability.
3. What if the expected value (EV) is infinitesimally positive?
I think the EV is either zero or one but just in case, we'll count infinitesimal as zero. So that would be a YES here.
4. What if the EV is undefined or the answer is "it depends"?
We'll add clarifications until we get a meaningful EV and thus a definitive YES or NO.
5. What happens if Euclid has an infinite string of losses?
He gets a -$∞ payout! Being math world, all the infinitely many roulette spins happen instantaneously.
Resolution Criteria
Consensus in the comments. Failing that, expert consensus. Bet with caution until we've hashed out in the comments what that should mean! And be sure to ask about anything missing from the FAQ before betting.
Related Markets
The original Snake Eyes market (lots more related markets linked there as well)
1,000
3.00Let the change in bankroll on the nth round be denoted by W_n. That is, it will equal 2^n for a win, -2^n for a loss, and 0 if W_m =2^m for m < n.
Then, for a player with finite bankroll - say, enough funds to play at most N rounds, their wealth after playing will be equal to W_1 + W_2 + … + W_N. The expectation of each W_i is equal to 0, so by linearity of expectation, the expectation of the game is zero.
Now, in the infinite case the result of the game is equal to W_1 + W_2 + …, so by countable additivity the expectation of the game is 0 + 0 + … and so overall equal to 0.
To me this seems like the obvious way to set up the problem, and the answer follows immediately from it. Is there anything wrong with this?
@dreev could you help me cash out so I can donate? I made a limit order to sell all my NO shares at the current market price.
@dreev if you intended to fill my order you got the amount wrong. I have 7691 remaining NO shares to sell, which would cost around 7k mana (but since you have a YES position the shares would immediately be redeemed).
@jskf Sorry to be a tease there! I was just like, "sure, trivial amount of money, no problem" without noticing that that was only a fraction of your limit order. In the meantime, it looks like there's a better way for people to donate the expected value of their positions:
If you're looking to donate over M10k, apply for a loan from Manifold so that you can donate the value of your shares immediately: https://forms.gle/XjegTMHf7oZVdLZF7
@Primer Is there a way to see the most anyone has tied up in a market? I expected everyone's mana to be loaned back by now.
@ShitakiIntaki Thanks! And nice work! Though I'm skeptical of the numbers. It shows me as having only an eighth of what I've spent in this market loaned back to me? I collect my mana loan almost every day and it only takes something like 90 days to get over 90% loaned back, I thought...
@dreev mmm it is what ever the API is reporting as loaned.
for the moment https://api.manifold.markets/v0/bets?contractId=tBwjJKw0FDngm9guTrE6 returns 464 records, so I dont think there are issues with a truncated api response.
@dreev You're loaned back a percentage of the shares' worth, not a percentage of your initial spending.
@ShitakiIntaki Now I'm thinking there's just a problem in how we're interpreting "tied up". I think we want to think of that in expectation, using the current market price. Like it doesn't matter anymore what you spent; that's sunk cost. If you've been loaned back the current EV of your shares, you should have no preference for the market to resolve early. If you think your shares are worth more than the market's EV, that means you also don't want the market to resolve yet, because you want to trade more.
The key thing here is unlimited funds. The problem is normally that eventually you will have a losing streak that takes so long, you run out of money. And casinos also limit max bet size because of this. You can only double your bet this much times until you have a nasty loss. But if you can bet with unlimited money, then in theory it should be a good strategy.
Suggest the following to resolve to make things as simple as possible. You can post more about your case if you want. Then we choose a judge and each side stakes 10k mana. The judge reads the comments and decides (we can point them to the posts we think are most important, but they can skim the rest or skip some stuff if they don’t feel they need to read everything). The judge makes a post explaining the decision but it can be relatively short. 5k pays the judge, the other 5k goes to the winning side. I can contribute at least 6k that I have free for NO, willing to do more if no one else wants in.
Current status is that the case for YES is based on something like this:
(a) To avoid zero times infinity, we define this question in terms of the limit as the starting capital goes to infinity.
The case for NO is based on something like this:
(b) In measure theory it's standard to take zero times infinity as zero.
Per FAQ4, something like either (a) or (b) needs to be added to the FAQ and I'd like to defer to an expert on which.
@dreev As I’ve explained before b) is a fact that is not an assumption of NO and is no more necessary for this problem than any other probability problem that is not finite and discrete and therefore needs an integral. If you really want to add a “assumption” the appropriate one is what I’ve suggested before.
“Expected value is defined as the integral of a random variable with respect to probability measure”
Even this is a little misleading since you’d get the same thing with high school calculus using Riemann integral and the probability density function definition of EV.
@WilliamEhlhardt Or perhaps we could resolve it on the authority of notes and/or textbooks written by such professors that include this problem 🤔.
Realized today that at some point @dreev added:
5. What happens if Euclid has an infinite string of losses?
He gets a -$∞ payout! Being math world, all the infinitely many roulette spins happen instantaneously.
which makes me less sure what a prof would say. I was referring to the problem as written in the original description of this market, which I do think is the natural interpretation of "Martingale with infinite credit" and the one that best models a game one could imagine playing in the real world.
@WilliamEhlhardt even if you explicitly define the payout for that sequence of events to be infinite, it will be the only such sequence, and it has measure/probability 0, so it does not affect the expected value (which is an integral). See my earlier comment here, and a bunch of comments from Nate.