Gambling with the magical genie

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Cradarc
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Gambling with the magical genie

Postby Cradarc » Fri Jan 23, 2015 7:42 am UTC

A magical genie offers you a single chance to play his gambling game. The rules are as follows:
1. You must pay your entire financial worth.
2. The genie will flip a perfectly probabilistic coin.
3. If it lands on heads, he will give you ten fold what you paid.
4. If it lands on tails, you will get nothing in return.

Unfortunately you know nothing about the coin he will flip other than the fact that it is perfectly probabilistic and has a fixed chance of landing on heads. However, the genie also made this offer to many other people. You are allowed to observe the outcomes of other people's games.
You observe the first person: He won! You observe the second person: She won, too! The third, fourth, fifth, all won.
In fact, everyone you observe wins game!

After observing how many other people would you decide to play the game yourself?
(Remember each person can only play the game once. However you can assume there are an unlimited number of other players.)

Also pretend economics doesn't matter so if everyone is rich, 10 fold your worth is still desirable.

Edit:
The probability of heads is a random variable with an unknown distribution. However, there is a non-zero chance that P can be any value in the range [0,1]
Last edited by Cradarc on Fri Jan 23, 2015 8:11 pm UTC, edited 3 times in total.
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twinsen
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Re: Gambling with the magical genie

Postby twinsen » Fri Jan 23, 2015 1:09 pm UTC

I have personal request,
Can you explain what measn:
"perfectly probabilistic" (does it mean that it does not change in time?)
and also what is:
"10 fold" (is it 1/2^10 or something like you are folding something, and again and again and so 10 times).

Im really sorry, my english is not good,
so can you please tell me this simple, like to a kid, so I can understand?

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Sizik
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Re: Gambling with the magical genie

Postby Sizik » Fri Jan 23, 2015 4:11 pm UTC

twinsen wrote:I have personal request,
Can you explain what measn:
"perfectly probabilistic" (does it mean that it does not change in time?)
and also what is:
"10 fold" (is it 1/2^10 or something like you are folding something, and again and again and so 10 times).

Im really sorry, my english is not good,
so can you please tell me this simple, like to a kid, so I can understand?


Tenfold means ten times as much. So the possible outcomes are -X on tails (where X is "your entire financial worth"), or +9X on heads (10X for winning, minus X for the bet).
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Re: Gambling with the magical genie

Postby Cauchy » Fri Jan 23, 2015 4:50 pm UTC

This depends on your prior distribution for the coin's weighting. Depending on this prior, after some number of flips you should be convinced that the chance of a heads is greater than 10% (the expected value break-even point), or 50%, or whatever bar you've set for yourself. The exact number of flips depends on your prior: for uniform, it's 0, and different priors can require arbitrarily high numbers of flips before you'll be convinced.
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Re: Gambling with the magical genie

Postby SDK » Fri Jan 23, 2015 6:17 pm UTC

So let's say I'm willing to do this if I'm 90% sure that I'll win. How do we go about calculating how many people I need to see?
The biggest number (63 quintillion googols in debt)

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Re: Gambling with the magical genie

Postby Cradarc » Fri Jan 23, 2015 8:00 pm UTC

I think this problem is interesting because you have to consider the gambler's fallacy.
Suppose someone else had been observing the players long before you began. Would having access to their data change your conclusions? All you know is that everyone YOU observed won the game.


twinsen wrote:I have personal request,
Can you explain what measn:
"perfectly probabilistic" (does it mean that it does not change in time?)
and also what is:
"10 fold" (is it 1/2^10 or something like you are folding something, and again and again and so 10 times).

Im really sorry, my english is not good,
so can you please tell me this simple, like to a kid, so I can understand?


The coin has a probability of landing on heads with chance P, and probability of landing tails with chance 1-P. P is a random variable, but it constant through all games. P is independent of everything.
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Re: Gambling with the magical genie

Postby HonoreDB » Fri Jan 23, 2015 10:30 pm UTC

Aside from calculating your personal "risk tolerance", the math here is surprisingly simple:

Spoiler:
We can invoke Laplace's Rule of Succession, which gives the probability of personally winning, given that you've observed n wins and no losses, as (n+1)/(n+2). So given a minimum probability p at which we'd take the bet, we can calculate the minimum number of games to observe as

c <= n+1/n+2
cn + 2c <= n + 1
n(c-1) + 2c <= 1
n(c-1) <= 1 - 2c
n >= (1 - 2c)/(c-1)
n >= (2c - 1)/(1-c)

So, for example, if we want a 90% or better probability, we need to observe (2*0.9-1)/(1-0.9) = 8 people winning.


As for calculating your risk tolerance, it depends for me on just how inclusive "all your financial worth" is. If I lost all my liquid assets and my physical possessions, but still had my job and friends and family to borrow money from, it wouldn't really be that disastrous.

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Re: Gambling with the magical genie

Postby twinsen » Sat Jan 24, 2015 9:02 am UTC

Well, now when I know that 10 fold = 10x (thanks to the guys for telling me this, and for the probabilistic termin)

I can put my strange (probably not so true) logic:
Spoiler:
First part, to determine what chance of falling on head I want.
If considered what is economically rational, the risk I am taking, should be equeal or less than the profit im getting.
In other words, if the game is fair for both me and the dwarf, It must be so, that:
"I win one game, every 10 games". this way, 9 times I will lose X = 9X lost, and one time I will win 9X."
But I dont really care for the game to be fair to the dwarf, so I just want:
"To win at least once every 10 games."

And to continue, with my answer I need more information.


Another question, for the probabilistic coin. There is a chance P, to fall on head, that does not change.
But if there is a bag, full of similar coins, and we get array of P[COINS_COUNT], are chance:
P[i] between 0 and 0,5 , and the chance: P[i] between 0,5 and 1 equal? Are thouse all P distributed evenly?

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Re: Gambling with the magical genie

Postby twinsen » Sat Jan 24, 2015 1:00 pm UTC

HonoreDB wrote:Aside from calculating your personal "risk tolerance", the math here is surprisingly simple:

Spoiler:
We can invoke Laplace's Rule of Succession, which gives the probability of personally winning, given that you've observed n wins and no losses, as (n+1)/(n+2). So given a minimum probability p at which we'd take the bet, we can calculate the minimum number of games to observe as

c <= n+1/n+2
cn + 2c <= n + 1
n(c-1) + 2c <= 1
n(c-1) <= 1 - 2c
n >= (1 - 2c)/(c-1)
n >= (2c - 1)/(1-c)

So, for example, if we want a 90% or better probability, we need to observe (2*0.9-1)/(1-0.9) = 8 people winning.


As for calculating your risk tolerance, it depends for me on just how inclusive "all your financial worth" is. If I lost all my liquid assets and my physical possessions, but still had my job and friends and family to borrow money from, it wouldn't really be that disastrous.


Do you know how stupid was the person who wrote this "Rule".

You can easy say what "was" the success rate, by looking at results of number of events. Like Rate of success = times you done it well / times you done it.
He decided that you can say what "will be" the success rate for event. And this is one of the most stupid ideas anyone on this earth come with:
Just before the observation is done, there is one sucess and one failure, since there is a chance, that this happen.

So instead of S observations successful, we have S+1(ONE SUCCESS).
And instead of N tries to do something, we have N +2 (ONE SUCCESS + ONE FAILURE)

And from how many years? Few hundred? There is noone there to tell him: " God damn , you have to be retarded ".

First of all, I dont know, why people even call this a RULE. It is so god damn estimate.
Second the whole use of this is:
1) For small N values(when you have really hard time giving estimates), he tries to average the error you can make, for example: Instead of having errors: 0% or 50%, you have errors : 17% or 33% (People think its cool, but I know im wrong anyway)
2) For larger N values, to not interfere with the result.(People think its cool, but I still know there is a slight error)

I would rather start believing in god, than use this stuff.

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Re: Gambling with the magical genie

Postby HonoreDB » Sat Jan 24, 2015 2:31 pm UTC

To me, that read like a nice, intuitive discussion of ways to look at the Rule, interspersed with unsupported hostility toward it. It's an unusual combination.

The intuitive ways of thinking about it aren't the only ways, though; the article I link in my spoiler block gives the derivation from Bayes Theorem. The fact that it's derivable from that means that it's grounded firmly in decision theory, which to me makes it worthy of being called a Rule, or even a Law. You could convert it into a statement about the natural world by saying something like

"Suppose we ran a simulation in which agents are presented with a series of gambling games, where the probability of winning is some unknown (but constant for a given game) p in [0,1] and the payout ratio is known and constant for each game. Agents are presented with a game and can choose whether to bet $1 or not. If they choose to play, all agents see the result. Some agents will always play, others use algorithms to decide. We iterate this process many times, using the same set of games over and over and allowing agents to take on as much debt as they want. In such a simulation, an agent that uses the Rule to decide whether or not to play a given game will eventually end up with more money than an agent using any other method."

Now this is a probabilistic Law, but so, explicitly, are the Laws of Thermodynamics, and so, implicitly, are all other laws describing reality.

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Re: Gambling with the magical genie

Postby twinsen » Sat Jan 24, 2015 4:07 pm UTC

HonoreDB wrote:To me, that read like a nice, intuitive discussion of ways to look at the Rule, interspersed with unsupported hostility toward it. It's an unusual combination.

The intuitive ways of thinking about it aren't the only ways, though; the article I link in my spoiler block gives the derivation from Bayes Theorem. The fact that it's derivable from that means that it's grounded firmly in decision theory, which to me makes it worthy of being called a Rule, or even a Law. You could convert it into a statement about the natural world by saying something like

"Suppose we ran a simulation in which agents are presented with a series of gambling games, where the probability of winning is some unknown (but constant for a given game) p in [0,1] and the payout ratio is known and constant for each game. Agents are presented with a game and can choose whether to bet $1 or not. If they choose to play, all agents see the result. Some agents will always play, others use algorithms to decide. We iterate this process many times, using the same set of games over and over and allowing agents to take on as much debt as they want. In such a simulation, an agent that uses the Rule to decide whether or not to play a given game will eventually end up with more money than an agent using any other method."

Now this is a probabilistic Law, but so, explicitly, are the Laws of Thermodynamics, and so, implicitly, are all other laws describing reality.


Ok, your arguments convinced me it. It is nice method for guessing.
Now it is not more likely to start to believe in god, than use this, but its about the same.

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Re: Gambling with the magical genie

Postby Lopsidation » Sun Jan 25, 2015 2:17 pm UTC

> inb4 religious debate in Logic Puzzles

Will the genie let me communicate with other players? If I ally with 10 other players, then we all make a big profit even if a few of us lose. In that case, I would take this bet after seeing maybe six other players win.

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Re: Gambling with the magical genie

Postby Cradarc » Sun Jan 25, 2015 11:26 pm UTC

Lopsidation wrote:>Will the genie let me communicate with other players? If I ally with 10 other players, then we all make a big profit even if a few of us lose. In that case, I would take this bet after seeing maybe six other players win.


Nope. You and only you get one bet. The other players exist in separate dimensions if you will. Their sole purpose is to provide you with information. It's equivalent to asking the genie to flip the coin without making a bet.
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Re: Gambling with the magical genie

Postby twinsen » Tue Jan 27, 2015 9:04 am UTC

I dont get it, I waited to hear something about P distribution, just to see the edited post, that its unknown.

Spoiler:
So, as I said, we need to win at least one of ten games, to be rational in our choice to play.
(Or lets just say, that one person of ten should win, becouse we can play only once)

So, now I have to watch N people, to make sure that P is in [10%,100%], and not in [0%,10%].
But I cant be sure. If I watch really big number of people, however... I can be almost sure that P is in the wanted diapason,
leaving somewhat just theoretical chance of beeing wrong. In other words - I will be sure, that the game is fair.


So next question, how sure I want to be? If its absolutely sure, then I cant say...

If I have to be 100%:
Spoiler:
No solution. You can roll 9999 times head, and roll tails till the end of your time.


If its up to me to decide:
Spoiler:
One person if any at all.
Two will be optimal.
Three persons is overkill.

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Re: Gambling with the magical genie

Postby douglasm » Tue Jan 27, 2015 10:09 am UTC

I'll assume that the genie can't cherry pick the people you observe to skew the distribution - such as showing you the 100 people that win while not mentioning 1000 that lose - and that I know this fact.

If there's an unlimited supply of other people to observe playing the game and no negative consequence to observing more, I think I'd keep going to 100 before deciding to play, just to be certain. If the genie cuts me off earlier and demands a decision, I'm not sure where the tipping point would be for me. Definitely higher than 2 or 3, instant poverty is a really big penalty, and I picked 100 in part because there's very little reason to not go at least that high if you can, but where in the range between those would take some serious thought.

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Re: Gambling with the magical genie

Postby jaap » Tue Jan 27, 2015 10:53 am UTC

douglasm wrote:I'll assume that the genie can't cherry pick the people you observe to skew the distribution - such as showing you the 100 people that win while not mentioning 1000 that lose - and that I know this fact.

If there's an unlimited supply of other people to observe playing the game and no negative consequence to observing more, I think I'd keep going to 100 before deciding to play, just to be certain. If the genie cuts me off earlier and demands a decision, I'm not sure where the tipping point would be for me. Definitely higher than 2 or 3, instant poverty is a really big penalty, and I picked 100 in part because there's very little reason to not go at least that high if you can, but where in the range between those would take some serious thought.

It's a question of utility. The perceived value of gaining 9X is not simply 9 times the perceived value of losing X; a dollar is worth more to a person who has nothing than to a millionaire. This assessment will be different for different people, and as HonoreDB points out it depends for example on whether you have a job or friends to help you out. Once you have worked out your personal utility function, then you can determine how sure you need to be for the game to result in an expected gain in utility.

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Re: Gambling with the magical genie

Postby Cradarc » Tue Jan 27, 2015 11:44 pm UTC

jaap wrote:Once you have worked out your personal utility function, then you can determine how sure you need to be for the game to result in an expected gain in utility.

In this context, what does the "expected gain" mean? You get one shot at the bet. You either win or you lose. You may argue that the average over infinite hypothetical universes is the expected gain, but that has no impact whatsoever on the one universe you exist in.
Unless p = 1, there's always a chance you will lose all your wealth. So does it devolve into psychology? The threshold p value is inversely related to how greedy/daring someone is?

We digress. I still haven't seen an explanation for how one can deduce the general behavior of the coin simply by looking at the result of other players' games.
In our everyday lives, we simply assume coin flipping is 50-50 and then there's this paper.. Does that mean coins are not 50-50? Or does it only mean the results of that experiment yielded 51% of the flips being heads?
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Re: Gambling with the magical genie

Postby Xias » Wed Jan 28, 2015 6:53 am UTC

I find the mathematical answer ("this is how many you need to see to have an expected return > 0") to be far more compelling than the psychological answer ("this is how many I would need to see before I would be comfortable.") Humans are generally really bad at intuiting about risk and probability. To take care of the question of utility, I think we could speak in general terms where the payout is nX instead of just 10X.

Cradarc wrote:
jaap wrote:Once you have worked out your personal utility function, then you can determine how sure you need to be for the game to result in an expected gain in utility.

In this context, what does the "expected gain" mean? You get one shot at the bet. You either win or you lose. You may argue that the average over infinite hypothetical universes is the expected gain, but that has no impact whatsoever on the one universe you exist in.
Unless p = 1, there's always a chance you will lose all your wealth. So does it devolve into psychology? The threshold p value is inversely related to how greedy/daring someone is?


It really shouldn't matter that you're only doing it once, since risk aversion would be calculated into the personal utility in the first place. If you could pay a dollar to flip a fair coin and win $1000000 on heads, the only reason that the uncertainty should drive you against playing is if you really really really value that dollar.

We digress. I still haven't seen an explanation for how one can deduce the general behavior of the coin simply by looking at the result of other players' games.
In our everyday lives, we simply assume coin flipping is 50-50 and then there's this paper.. Does that mean coins are not 50-50? Or does it only mean the results of that experiment yielded 51% of the flips being heads?


You can't deduce the general behavior, nor do you need to. You end up with a probability distribution. Of course it's possible that the coin is biased towards tails but still (un)luckily turned up heads a dozen times. But if your friend flipped a coin 10 times and hit 10 heads, would you think the coin was biased? What if he did it 100 times? A million? As you can see, the more flips you see turn up heads, the more likely it is that the coin is biased because the odds of such a result if it were not biased is so incredibly low.

I'm not sure what to think of the paper you shared. It seems like a lot of maths to support a claim, and then a relatively small sample to demonstrate it. However, given enough flips, and enough of a gap between heads and tails, and a high enough probability that the coin is representative of the norm, then yes: such a demonstration would imply that coins are not 50/50.

If you're still not convinced:

Spoiler:
Let's consider a more discrete case where either A) the coin is perfectly fair, or B) the coin is 100% biased in your favor, with equal probability of either. The odds of you losing blind are 1/4, so you would take the bet if the utility of the payout is greater than 1/3 of the utility of your current wealth (note that the payout could be much greater than 1/3 of your wealth, since it's the utility that matters.) If you saw one winner (call this C) then you have a set of equations:

P(A|C) = P(C|A)P(A)/P(C) #This is the probability of the coin being fair given that someone won the last flip.
P(B|C) = P(C|B)(P(B)/P(C) #This is the probability of the coin being biased given that someone won the last flip.

Since P(C|A) is 1/2 (the odds of C given A) and P(A) is 1/2, that means that P(A|C) = (1/4)/P(C). Likewise, P(B|C)=(1/2)/P(C). Then P(B|C) = 2P(A|C), and since P(B|C)+P(A|C) = 1, that means that there is now a 2/3 chance that the coin is biased in your favor, and your chance of losing is 1/6. So now you can take the bet if the payout utility is only 1/5 the utility of your current wealth.

If C is seeing two wins in a row, then your chance of losing is reduced to 1/10. This keeps getting smaller as you see more wins come up.


The actually problem is more complex because you aren't limited to two discrete cases with equal probability; rather, you have a continuous set of probabilities, so you end up with a curve.

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Re: Gambling with the magical genie

Postby quantropy » Wed Jan 28, 2015 11:50 am UTC

I'm not sure that I get what this thread is supposed to be about, but here are some thoughts

It looks like we assume that the coin comes up heads with probability P (this is an objective probability). We don't know what P is but it seems reasonable to assume a uniform distribution on [0,1] (this is a subjective probability). This means that without seeing any other games you would assign a probability of half to it coming up heads, so from an expected gain point of view it is worth playing the game.

Then comes the fact that it is your entire wealth that you are playing with, so as others have said you have to modify the expected gain or loss via a utility function. However, the 'Also pretend economics doesn't matter so if everyone is rich, 10 fold your worth is still desirable.' doesn't seem to fit in with this.

But supposing you assign +1 utility to getting 10xyour present wealth and -100 utility to losing it all. Then I think you can use Bayesian analysis to use the outcome of other games to modify your prior distribution for P. You can thus find out how many wins you need to observe to get positive expected utility.

What some people seem to want is to have a high subjective probability that the objective probability P lies in a given range. I don't see that this is necessary, you just want to know the expected utility based on your modified subjective probability distribution.

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Re: Gambling with the magical genie

Postby Yablo » Wed Jan 28, 2015 7:49 pm UTC

I'm generally a very logical and reasonable person, and I'm also generally suspicious of any sort of gambling of this sort (I get the same impression I would from a shell game). I'm also very suspicious of anything at all involving genies, good or bad. That's why I'm a little surprised at my answer, and my reasoning:

The number of players before I decide wouldn't make a difference for me. I could see one win; I could see a million win. I'd probably just assume they were all in on it. And still, I think I'd probably flip a normal coin of my own to decide whether I'd let the genie flip his. Of course, even if I bet and won, my wife would probably leave me, so I'd have to hide my winnings from her.

And the IRS. I'm not sure genies are legally authorized to operate a gambling operation of any sort, and even if this one was, a profit of nine times my current financial worth would be of great interest to the government.
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Re: Gambling with the magical genie

Postby Sizik » Wed Jan 28, 2015 10:29 pm UTC

What if a have a mortgage? Does my "entire financial worth" take debts into account, or or does it only include assets?
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Re: Gambling with the magical genie

Postby Xias » Thu Jan 29, 2015 1:29 am UTC

quantropy wrote:I'm not sure that I get what this thread is supposed to be about, but here are some thoughts

It looks like we assume that the coin comes up heads with probability P (this is an objective probability). We don't know what P is but it seems reasonable to assume a uniform distribution on [0,1] (this is a subjective probability). This means that without seeing any other games you would assign a probability of half to it coming up heads, so from an expected gain point of view it is worth playing the game.


Good point. How much more interesting does the problem get if we don't assume a uniform distribution on the outset? Like, a more cynical view that if the genie is willing to make the deal in the first place then it's more likely that the coin is biased toward a loss. Then we have to view some number of other results.

Then comes the fact that it is your entire wealth that you are playing with, so as others have said you have to modify the expected gain or loss via a utility function. However, the 'Also pretend economics doesn't matter so if everyone is rich, 10 fold your worth is still desirable.' doesn't seem to fit in with this.


I think that statement was to negate any argument about hyperinflation as the number of previous games you view increases. Your utility function won't change based on how many people have played the game. Could have been phrased better.

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Re: Gambling with the magical genie

Postby Cradarc » Thu Jan 29, 2015 7:17 am UTC

I think lots of people are getting caught up on the money issue. What if I changed it to this?

1. There are two groups of people, A and B. Group B has an infinite number of people. Group A has a large number of people, on the order of millions.
2. Everyone in group A starts with 1 point. Everyone in group A wants to maximize the number of people in the group that have strictly fewer points than he/she. The only way to change one's point value is via a single bet with the genie.
3. You competing in group A. Your rivals in group A are equally intelligent as you and have access to any resource you do.
4. You get to choose when to make your bet (if ever). In the mean time you can spectate bets being made by people in group B.
5. However, you have absolutely no access to information about bets made by people in group A. You would not even know if any such bets occurred. You also have no access to information known by any other person involved with the game.
6. If you win the bet, the genie will change your point value to 10+1/(N+1). If you lose, your point value becomes -1+1/(N+1). N is the number of bets you observed.
7. The genie uses the same coin for group B and group A. The coin has a fixed probability p of landing on heads (win). You have no information about p other than it is some value in [0,1].
8. Knowing all of the above conditions, you decided to observe some bets. Every bet you observed yielded a win. How many straight wins would you observe until you're confident enough to make your own bet?
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Re: Gambling with the magical genie

Postby SDK » Thu Jan 29, 2015 6:52 pm UTC

Cradarc wrote:I think lots of people are getting caught up on the money issue. What if I changed it to this?

1. There are two groups of people, A and B. Group B has an infinite number of people. Group A has a large number of people, on the order of millions.
2. Everyone in group A starts with 1 point. Everyone in group A wants to maximize the number of people in the group that have strictly fewer points than he/she. The only way to change one's point value is via a single bet with the genie.
3. You competing in group A. Your rivals in group A are equally intelligent as you and have access to any resource you do.
4. You get to choose when to make your bet (if ever). In the mean time you can spectate bets being made by people in group B.
5. However, you have absolutely no access to information about bets made by people in group A. You would not even know if any such bets occurred. You also have no access to information known by any other person involved with the game.
6. If you win the bet, the genie will change your point value to 10+1/(N+1). If you lose, your point value becomes -1+1/(N+1). N is the number of bets you observed.
7. The genie uses the same coin for group B and group A. The coin has a fixed probability p of landing on heads (win). You have no information about p other than it is some value in [0,1].
8. Knowing all of the above conditions, you decided to observe some bets. Every bet you observed yielded a win. How many straight wins would you observe until you're confident enough to make your own bet?

Based on the parameters of this game, I'm going to assume that there is in fact some logical way for me to gain points. Since the only way to do so appears to be by flipping this coin, and the number of observations apparently matters, the expected value must be positive, otherwise observing bets by group B would cause the players in group A to decline to bet. Therefore, I will bet before seeing even a single win. :wink:
The biggest number (63 quintillion googols in debt)

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Adam H
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Re: Gambling with the magical genie

Postby Adam H » Thu Jan 29, 2015 8:14 pm UTC

Rule 2 makes the game interesting. I disagree with SDK that the rules imply the coin is likely to be biased towards success. Because you are trying to maximize the number of players with fewer points than you, if 100% of the other players are declining to flip the coin, you should want to flip the coin if there's greater than 0% chance of success, because you wouldn't lose anything if you failed. It's tricky since you don't know if other players are flipping coins, but it wouldn't be trivial so the game would still work if the coin was biased towards failures.

I would flip my coin either immediately, or after 1 observed success, or never if the first observation was failure.
-Adam

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Qaanol
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Re: Gambling with the magical genie

Postby Qaanol » Fri Jan 30, 2015 2:31 am UTC

How about this variant:

The genie creates a sequence {ai} of real numbers, with 0<ai<1 for all i∈ℕ. You have no other information about this sequence. For each i∈ℕ, the genie creates a coin with probability ai of landing on heads.

The genie flips each coin a countably-infinite number of times and records the results. The sequence {bi,j} represents the j’th flip of the i’th coin with a 1 if it was heads and a 0 if it was tails.

You are part of a game with a total of n players, for some possibly-infinite cardinality n. Each player starts with 1 point and plays by the same rules.

You begin on (0, 0), meaning the first flip of the first coin. Every turn, you may bet all your points, or bet nothing. The genie then reveals the value of bi,j for the (i, j) where you currently are, and pays you 10× your bet if the value is 1 (meaning the i’th coin came up heads on the j’th flip when the genie flipped it before the game started) or nothing if the value is 0 (at which point if you made a bet then you lose everything).

After the genie reveals the result, you may move to the next flip of the same coin, meaning to increment j, or you may move to the first flip of the next coin, meaning to increment i and reset j to zero. If you placed a bet, then you must move on to the next coin. If you did not bet, then you may choose which move to make.

In other words, you can watch as many flips of each coin as you like, then either bet on the next flip, or move on to the next coin.

You do not know how the other players are doing, but you do know that everyone makes their moves at the same time, so when you have seen, for example, 100 results, then so has everybody else. Furthermore, the sequences of flips are exactly the same for each player, because the genie preflipped them.

The game will continue for m moves, for some m which is known ahead of time. What strategy do you use—a. if the player with the most points wins, or b. to maximize your expected number of points at the end?
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marzis
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Re: Gambling with the magical genie

Postby marzis » Fri Jan 30, 2015 2:48 pm UTC

I don't believe the later modification of including a 1/n+1 modifier to your win/loss value changes things, but if so then this is written with the original setup in mind:

I observe 10 games, I observe 10 wins.
I observe 100 games, I observe 100 wins.
I observe 1000 games, I observe 1000 wins.
So, if I observe X games, then I will observe X wins.

Okay, for a second, let's think about electrons. While I can calculate the energy of 1/2 of an electron, our physical reality eliminates the possibility of my ever observing the energy of 1/2 of an electron. It cannot be done because it cannot exist.

Okay, so the genie has a coin. I'm going to assume it is a physical coin existing under the same rules of the universe in which this forum exists, and therefore has the same physical quantum that we are familiar with (existence of genies aside, obv.)

If I observe enough games, I believe that my confidence in the probability of the coin landing on the 'loss' side will be smaller than the physical limitations on the existence of a 'loss' side large enough to ever be revealed existing on the coin, for the limitations of my universe. Instead of a coin, consider it being a die (since it is easier to consider a die with 1/6 chances of revealing a 'win' than it is to consider a coin that somehow has a 1/6 chance of revealing a 'win', but for our gambling purposes, they are the same).

The die has X sides for 'win' and Y sides for 'lose'. As we observe a win after a win after a win, we can confidently assume that, in the case of 10 wins in a row, that the chances are good the coin has more than 11 sides, of which more than 10 are 'win' (and at least 1 side for 'lose'. As we continue in this way, we will run into a limitation of how many sides can exist on this die such that it is reasonable for there still to exist a side on which we would lose, should that side turn up.

Eventually we'll observe so many games that while the 'lose' side might exist, it now has to be so small (proportionally to the die) that it exists between our minimum measurable distance, and therefore the die will actually never reveal it, not because the probability is zero (it isn't, btw, it is still > 0) but because our universe doesn't allow for the measurement of distances that small.

So, I'd observe a crap ton of games until the minimum physical size of the 'lose' side of the coin is smaller than some relevant quantum distance, and therefore won't happen in my universe since we can't measure things less than 1 of that quantum.

Just a fun thought, no clue if it is reasonable at all, but I feel like someone with a better understanding of this stuff might be able to make it work.

quantropy
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Re: Gambling with the magical genie

Postby quantropy » Sat Jan 31, 2015 11:21 am UTC

Cradarc wrote: Everyone in group A wants to maximize the number of people in the group that have strictly fewer points than he/she.

You competing in group A. Your rivals in group A are equally intelligent as you and have access to any resource you do.

If you win the bet, the genie will change your point value to 10+1/(N+1). If you lose, your point value becomes -1+1/(N+1). N is the number of bets you observed.

So you have a sequence W0>W1>...>Wn...>O>L0>L1...>Ln... of winning scores and losing scores. You get score Wn if you win after observing n tosses, and Ln if you lose. (O is what you have if you do nothing) Assigning numerical values to the scores is irrelevant - your real score is the number of people with scores strictly less than you.

It would still seem reasonable to assign a uniform distribution on [0,1] as your prior distribution for P and update this using your observations of tosses.
(A bit of calculation says that the distribution will be (n+1)xn after observing n wins with a probability (n+1)/(n+2) of winning)

Note however that since you are competing with everyone else in group A, who are equally intelligent as you, you need to apply game theory (this is despite the fact that you have no interaction with them). I would therefore expect that the optimum strategy would be probabilistic - bet immediately with probability a, otherwise observe a toss and choose between betting, stopping and observing another toss with different probabilities.

I think working it all out might be a bit of difficult though


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