If both players have infinite bankrolls, but only one of them is allowed to stop the game once they are ahead, the one with the option of stopping has an advantage. They can play until they are in the lead, then stop. The reason this doesn't work in real life is that real bankrolls aren't infinite.
this post was submitted on 01 Sep 2024
40 points (95.5% liked)
No Stupid Questions
36154 readers
554 users here now
No such thing. Ask away!
!nostupidquestions is a community dedicated to being helpful and answering each others' questions on various topics.
The rules for posting and commenting, besides the rules defined here for lemmy.world, are as follows:
Rules (interactive)
Rule 1- All posts must be legitimate questions. All post titles must include a question.
All posts must be legitimate questions, and all post titles must include a question. Questions that are joke or trolling questions, memes, song lyrics as title, etc. are not allowed here. See Rule 6 for all exceptions.
Rule 2- Your question subject cannot be illegal or NSFW material.
Your question subject cannot be illegal or NSFW material. You will be warned first, banned second.
Rule 3- Do not seek mental, medical and professional help here.
Do not seek mental, medical and professional help here. Breaking this rule will not get you or your post removed, but it will put you at risk, and possibly in danger.
Rule 4- No self promotion or upvote-farming of any kind.
That's it.
Rule 5- No baiting or sealioning or promoting an agenda.
Questions which, instead of being of an innocuous nature, are specifically intended (based on reports and in the opinion of our crack moderation team) to bait users into ideological wars on charged political topics will be removed and the authors warned - or banned - depending on severity.
Rule 6- Regarding META posts and joke questions.
Provided it is about the community itself, you may post non-question posts using the [META] tag on your post title.
On fridays, you are allowed to post meme and troll questions, on the condition that it's in text format only, and conforms with our other rules. These posts MUST include the [NSQ Friday] tag in their title.
If you post a serious question on friday and are looking only for legitimate answers, then please include the [Serious] tag on your post. Irrelevant replies will then be removed by moderators.
Rule 7- You can't intentionally annoy, mock, or harass other members.
If you intentionally annoy, mock, harass, or discriminate against any individual member, you will be removed.
Likewise, if you are a member, sympathiser or a resemblant of a movement that is known to largely hate, mock, discriminate against, and/or want to take lives of a group of people, and you were provably vocal about your hate, then you will be banned on sight.
Rule 8- All comments should try to stay relevant to their parent content.
Rule 9- Reposts from other platforms are not allowed.
Let everyone have their own content.
Rule 10- Majority of bots aren't allowed to participate here.
Credits
Our breathtaking icon was bestowed upon us by @Cevilia!
The greatest banner of all time: by @TheOneWithTheHair!
founded 2 years ago
MODERATORS
And, in real life, the house has a much larger bankroll.
I don't know if that applies to this scenario. In this game, the player is always in the lead until they aren't, but I don't see how that works in their favor.
Oh wait you mean the player has to stop if they lose? That's different.
Well, they have to start over with a $1 bet.
Once the player loses, the chain ends, and the house wins. So as long as the house can afford to keep pushing the player in to trying again, they're going to create more opportunities for the player to return their winnings to the house.
Right, and as the chain continues, the probability of the player maintaining their streak becomes infinitesimal. But the potential payout scales at the same rate.
If the player goes for 3 rounds, they only have a 1/8 chance of winning... but they'll get 8 times their initial bet. So it's technically a fair game, right?
If everyone has the same amount of starting capital it is a fair game assuming both can opt out at any time.
That said, the house appears to not be able to opt out (they definitely can, you just don't think about that part), and the house has more capital. For them each time someone plays a round there are only 3 possible outcomes. Half are the player loses, then a quarter are the player wins and plays another round, and lastly a quarter are the player wins and ends the game. The only case where the player wins is option 3, in all other cases, so 75%, the house wins because the next round has another chance to make the player lose directly at a 50/50 chance or play another round.
They have a 50/50 chance each round. Doesn't matter how many rounds they've won.
I'm looking at the game as a whole. The player has a 1 in 8 chance of winning 3 rounds overall.
the house can only make $1 per play, and the bettor can make a functionally unlimited amount.
see the martingale strategy. you are basically sticking the house with a martingale strategy in which you get to decide when they bet.
But the odds of the player managing to do so are proportionate. In theory, if 8 players each decide to go for three rounds, one of them will win, but the losings from the other 7 will pay for that player's winnings.
You're right that the house is performing a Martingale strategy. That's a good insight. That may actually be the source of the house advantage. The scenario is ideal for a Martingale strategy to work.
That looks like the St. Petersburg Paradox. Much ink has been spilled over it.
The expected payout is infinite. At any point, the "rational" (profit-maximizing) decision is to keep flipping, since you wager a finite sum of money to win an infinite sum. It's very counter-intuitive, hence called a paradox.
In reality, a casino has finite money. You can work out how many coin flips it takes to bankrupt it. So you can work out how likely it is to reach that point with a given, finite sum of money. Martingale strategies have already been mentioned.
Not quite the same, since in my scenario the player loses everything after a loss while in the St. Petersburg Paradox it seems they keep their winnings. But it does seem relevant in explaining that expected value isn't everything.
If you have 100$, and you bet 1$ at a time, infinitely, you will lose.
More generally (simplified to assume you're always betting the same amount):
P(ruin after X bets) = (edit: I removed my formula because it was wrong..but I'm sure you could mathematically prove a formula)
You're saying that the player pays a dollar each time they decide to "double-or-nothing"? I was thinking they'd only be risking the dollar they bet to start the game.
That change in the ruleset would definitely tilt the odds in the house's favor.