Just been thinking about this some more and I wonder if it's really just a question of your own personal "utility of money" function rather than a question of it being or not being an "infinitely repeatable game", in which case there really is no correct answer; just a continuum of reasonable/sensible answers. eg:

- Somebody from a western country ins't going to be that bothered about the chance of winning a big chunk occasionally (but most of the time getting nothing) and a linear utility function is probably quite appropriate...

- Whereas somebody from a country where the daily wage is $1/day may well be better off trying to get something out of the game most times and hence a logarithmic utility function (or even a square root utility function) may well be most appropriate for them...

I guess this just reiterates what JoeC2012 mentioned above, but if so it does mean there is no real point in trying to answer the question without reference to a specific utility function.

Juk

IMO the key here is that while it seems like you don't want to bet much early, on the occasions the coin is significantly biased you want to be smashing the all in button from round 2 to get that $10k from compounding. It makes no difference what you do when the coin isn't significantly biased.

Very easy all in all the way for probably almost anyone reading this. For me the real question is how much would the initial stake have to be before I go to a different strategy.

Anyone who has spent $20 on lottery tickets in big jackpots because they think it's +ev should probably be betting it all on this game for stakes up to at least $1000 or so, probably a lot more if they're at all logically consistent.

Yup. Obviously took me a while to get to the right answer but I see it now.

I was surprised to run the math the other night and see that you cash out about 1/40 times by smashing the all in. Initially assumed it was well under 1% which is why I thought Kelly might still be relevant.

You might even go all in on round 1 without losing value.

The assumption is we don't want to gamble with zero edge but if you value your time highly that could be optimal :P

The 20 bucks belongs to us, as does all money in our balance sheet. We should invest it in a sequence that has maximal return vs the risks offered. Losing 5k on a final flip might be appropriate or not, and there should be math to justify choosing a path.

With only 10 flips, can not all outcomes be enumerated, both for a successful Bayes analysis, and for an unsuccessful (yet still appropriate) analysis?

By the last flip, should we not have the most information about the coin possible, and be able to make a precise wager on the information acquired?

We have 5k on the final flip because we've already seen 9 heads or 9 tails and the math justifies it quite nicely.

Ahhhh. I see that now.

What about the more likely scenarios where it is less clearly defined? Do we still bet all that we have?

Thanks.

MLE is completely irrelevant, disregard that.

A little bird has told me that besides uniform, one might instead use Jefferys prior. @stinkypete, is that what you had in mind, or something else?

I don't know what that means (and a quick google didn't clear it up with minimal effort), but I stand by what I said regarding it being wrong to just assume a uniform distribution.

I'm not a mathematician and I've always hated formal mathematics so maybe this has been addressed somewhere in mathematical theory, but I don't believe there's a strict mathematical way to use Bayes here without making some sort of judgement call on defining the prior.

So, when someone presents us a coin that will be proven biased, and we are to determine what the bias is, we should decide to start at a biased point or start at even.

This is so complicated that my sentences are run-on failures, lol.

Before the first flip, the probability of Heads can be any real number from 0 to 1 exclusive, each with equal probability. That's as uniform as it gets. I believe that's why RR said it's not an assumption.

How do you avoid judgment? I'm pretty certain that if you don't use Bayes, you are making a judgment call. If the first flip is Heads, what would you say is the objective probability on the 2nd flip? Or do you think there is no single true probability?

You have to start at even because you don't know which side it's biased for. There's a 50% chance it's biased for Heads and 50% chance it's biased for Tails.

It's like, if there's a sports game where one team is heavily favored (say 80% to win), but you pick the winner by flipping a (fair) coin, your chance of being right is 50% even though the game is a mismatch. There's a 50% chance you'll pick the 80% team and a 50% chance you'll pick the 20% team, which averages out to 50%.

You are awesome, and I really enjoy learning from your posts. However, if we started with .75 bias, would our potential profit be higher, given that we only get 10 trials at this?

We know the bias is between .5 and either 1 or 0.

It's not only an assumption, but it's clearly a wrong assumption.

There's a huge difference between having no information and knowing something is a uniform distribution and treating them equally is mathematically incorrect, even if it produces a reasonable result for the infinitely repeating version of this game

Does it matter if there are two ranges:

(0,.5) and (.5,1)

Also, have we inspected it to see if it does not have identical sides?

Just spitballin’ here.

It looks like the definitive paper was written on this the same year I got my master's.

edit to add: Obviously my knowledge of Bayesian stats is limited to what is taught in a section of a class. I never talk to the Bayesian guys in my department.

If it's clearly not uniform, then what is it? Which bias is more likely than others?

I fail to see a difference and this is my intuition speaking, not my limited math background. When I say it's uniform I'm saying I have literally no clue what the % is and therefore, for all I know, each one is equally likely. To claim it's anything other than uniform is to claim that you have information. You wouldn't say that X% bias is more likely than Y% bias unless you had info not provided in the OP.

Care to link or cite? Or summarize?

Beta(1/2, 1/2) distribution, whereas Uniform is Beta(1,1). I don't get why one would be allowed to use anything other than Uniform for this, but said bird is much more knowledgeable than me. Perhaps the existence of more than one choice supports your argument that it's subjective. However, I still contend that not using Bayes would not avoid subjectivity.

You mean start with a pure guess? Your potential profit would be higher (by virtue of making a real bet on the 1st flip) but your EV wouldn't, and definitely not your EG.

Nifty, nice work

Ralph Vince has written about exactly this. I don't know of any links or free material that provide it, so some time I'll show what his formula would yield for this problem. He called it EACG (expected avg compound growth?) because he understandably didn't want to name it "Vince Criterion". When n=1, Vince = 100%; as n→∞, Vince → Kelly. I forget whether you decrement n after each completed bet, and that's an important detail, so hopefully it's in my notes because I don't have his book with me in my travels. I'm guessing you don't decrement n, otherwise the formula would imply that your utility function becomes more linear after each completed bet, which to me doesn't make sense. If I'm 70 years old and about to make what I know is the last bet of my life, and my bankroll is 1M, I'm probably not risking 100% on a 55-45 coinflip. Losing that 1M probably impacts me more than doubling it up, unless there's something specific I want to buy costing closer to 2M.

In reality, since we're gamblers on a gambling forum, it's not like these 10 bets will be the only gambling/investing we do for the rest of our lives. It wouldn't be good to Vince-bet as though n=10 if in fact n=1000.

Also like a few people ITT have said, $20 is a small fraction of our true bankroll IRL, so if we're limited to a $20 stake for this game then the growth-optimal strat is identical to the max-EV strat until maybe the later bets.

Agree

We don't know what it is. It's nothing. Not knowing the prior distribution is not the same as knowing that it's uniform.

If we're playing any non-max-EV (all in) strat we definitely want to reduce bet sizes on earlier bets rather than later bets.

The bankroll is known, but how much is the minimum bet?
If I have to roll 10 times a biased coin I know that some of the 1024 different WL patterns are more likely to happen.

Min bet is zero, some assumed 1c. Makes no practical difference which you use.

Since I do not know which side is favorite and by how much, I'd let it go the first nine hands without betting, then betting the entire $20 on the most frequent side right on the last roll.

if the first 8 are heads you wouldn't wanna gamble?

There's no such thing as nothing. The first flip has a probability: 50%. It doesn't have "no probability". And if the first flip is heads, you know the 2nd flip is >50% to be heads. You wouldn't know that if the prior distribution were somehow "nothing". What you're proposing is some kind of nihilism in which there's no such thing as probability and everything is just a guess with no percentages attached. If that's an unfair description, then tell me, after flipping heads on the 1st flip, what is P(heads on 2nd flip)?