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Old 2012-07-06, 10:47   #1
jinydu
 
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Default Statics Question, in the form of a storey

--- Part 1 ---

Alice: Hello Bob. So glad you could spare a few minutes out of your busy day.
Bob: No problem Alice. It's on my way back home anyway. Now what did you ask me to come for again? Some kind of new online game you found?
A: That's right. It's a very simple game really. All you do is click this button, and the game runs an animation that ends with either Victory or Defeat.
B: Very simple indeed. How did you get interested in something so trivial?
A: You know, that's the same thing I thought until I started playing; and all of a sudden I was hooked. I can't explain it. It's as if I'm a character in some story someone wrote just to make a point.
B: Weird... Anyway, what seems to the problem?
A: Something's wrong with the game. I keep losing.
B: Excuse me?
A: You heard me. I keep losing. Every single time I try. Not once have I gotten the Victory message.
B: But you said this is a game of chance, right? These things happen. Sometimes you win, sometimes you lose.
A: Right, sometimes. Not always. If you always lose then something's wrong.
B: Look, we've had this conversation before. What would you like me to do?
A: Can you try this game for me please?
B: I don't see how that's going to... Ok... What do I have to do?
A: First create an account by registering here
B: Alright, giving them my spam-collecting email address... Ready to start now. Trying once... Defeat. Again... Defeat. Yet again... Victory. There we go! See, there's nothing wrong with the game.
A: Guess you're right, although it feels odd... How you managed to get a win on the 3rd try whereas I haven't been able to get one in scores upon scores of attempts.
B: Like I said, all luck of the draw. Well... I need to get going soon; I have an appointment in an hour. How about we play 10 times each, and call it a day.
A: Sure. Thanks.
[Alice and Bob alternate playing the game, each on his/her account. After playing 10 rounds each, Bob has 3 victories, Alice none.]
A: Oh dear.
B: Oh wow... Sorry Alice. Today's just not your lucky day.
A: It's ok...
B: I can see you're not happy. Let me try to make it up to you. I'm free for a few hours tomorrow. How about I drop by and we play to 100 each?
A: Really? You don't have to... Thanks a lot!
B: No problem.

--- Part 2 ---

A: Thanks so much for coming.
B: Hey, happy to help. Let's get started shall we?
A: Sure.
B: Like I said last time, we play 100 times each alternating. I stick with my account; you stick with yours.
A: Got it.
[After 100 rounds each, Bob gets 17 wins, Alice 0.]
A: Noo. Nooooo...
B: Oh wow... I'm sorry. Really, I am...
A: How could this be?
B: Today's just not your lucky day. These things happen from time to time with random events. Sometimes, you just get long dry streaks...
A: You're sure that's all it is? Dumb luck? What if there's something else?
B: Like what?
A: Account-specific luck, maybe?
B: No. Why would the game developers do that?
A: I don't know.
B: It doesn't sound likely... How about we check the game's official forums?
A: Sure.
B: Searching... Searching... Here we are, a post from the head admin on exactly this topic: "There is no account-specific luck in this game. Every round has an equal probability of ending in a win. This probability is independent of what account is playing, how many turns have been played, the outcomes of previous runs, everything. It is a universal constant."
A: I see... Does it say what that constant is?
B: Let me check... Nope. Looks like the game devs are keeping that to themselves. But that's beside the point. As you can see, this settles it. There is no account-specific luck. Everyone has the same chance of winning.
A: You're sure? What if that admin just isn't telling the truth, or there's some bug in the programming? I mean... my losing streak...
B: Alice, we've been over this. This is the way randomness and life in general is. Sometimes, you just go on really long streaks. You have to learn to accept it. Ok?
A: I... guess...
B: I see that look in your eye. You don't. Sigh... Look... I haven't got much else to do this weekend. How about I come by again, and we play the game up to a thousand?
A: Really? Thanks so much!
B: My pleasure. I'm certain that with this many runs, you'll not only get a win, but roughly as many as me.
A: You're certain?
B: Hehehe. Well, almost certain... Just one thing: Don't jump to any conclusions until we've both played 1000 each, ok? Because this is a test for 1000 runs and not one less.
A: Alright

--- Part 3 ---

A: Good morning Bob.
B: Good morning Alice. Let's get right to it. I brought my laptop this time to save time. You play on your own computer on your account, I'll play on my computer on my account.
A: Sounds good.
B: Let's play to 500 today, and then tomorrow we'll play from there to 1000.
A: Alright.
[8 hours later]
B: Hello again Alice. I got 90 wins in 500 tries. How about you?
A: Zero.
B: ... ... You remember what we agreed about not jumping to conclusions before this little experiment is over, right?
A: Yes.
B: Ok, see you tomorrow.
A: Good night, and thank you.
[At the end of the next day, Bob has won 191 times out of 1000. Alice? Still 0.]
B: Alice, I'm sorry to say this, but you are well and truly the unluckiest gamer I have ever seen. I mean it. I truly am sorry for you...
A: [Eyes closed, wincing in pain and frustration]...
B: Really, I've never seen this before. Such an incredibly long streak of bad luck. Even now, I can hardly believe a mere mortal can be so atrociously unlucky.
A: Is that what you think this is? Nothing but just bad luck? You still don't think account-specific luck is to blame?
B: Well, no. The head admin was very clear. This game has no account-specific luck.
A: But what if he's wrong? What if there's a flaw in the coding? It's not like we have the code in front of us and can verify it for ourselves.
B: Alice, random number generators are not exactly a new thing. It's implausible that the game devs could mess up something so simple. And what reason would the head admin have for lying about something like this... Wait... scratch that last point...
A: ... And my thousand-strong losing streak?
B: I've said this many times before and I'll say it again. A 1-in-10 chance of success, for example, is not a guarantee that the event will happen if you make 10 attempts. There are no guarantees. A 1-in-10 probability event could fail to show itself in a hundred, a thousand, even a million runs. Nothing is for certain.
A: Yes, yes... But aren't such long streaks very improbable.
B: Improbable, yes. But not impossible.
A: ....
B: I see you still don't see things my way. After such a shocking, horrific run of bad luck, I can hardly blame you.
A: ....
B: I'll tell you what. Let's settle this once and for all. Let's play to 10,000. I'll come here every weekend until we get there.
A: Thanks... Thanks a lot.
B: I bet that after we've both reached 10,000, this nightmarish fluke of yours will be just a distant memory, and the difference between our win counts will be well within the expected margin of statistical error.
A: And I bet that my streak will be alive and stronger than ever.
B: Let's bet $100, ok?
A: Agreed

--- Part 4 ---

[Days turn to weeks. Weeks turn to months. As promised, Bob visits every weekend to play, until at last, one day, the experiment finally concludes. At the end of it all, Bob has 2012 wins in 10,000 tries. Alice has still won none.]

B: ... I never... imagined... I'd have to do this... [Reaches into his wallet, pulls out a hundred dollar bill, gives it to Alice.]
A: Thanks... But Bob... There's something I want more...
B: What is that?
A: An admission from you, that I was right, that account-specific luck is real, that in this game some accounts have a better chance of winning than others.
B: ... I'm sorry Alice, but I don't believe that.
A: After all this, after ten thousand runs and this large a difference, you still believe this is all just dumb luck? That my inability to get even a single win is just a super-improbable fluke?
B: I'm afraid that's exactly what I think.
A: ...
A: What would it take for you to change your mind? What If we played up to 100,000, you got 20,086 wins and I got 0, or something close to that?
B: That wouldn't happen.
A: And why not?
B: Or rather, it almost almost almost almost almost certainly wouldn't happen. The odds against running ten thousand independent, identically distributed random variables twice getting a result like that... It's like one of those "not in a trillion trillion times the size and lifetime of the Universe" sort of things.
A: But if it did happen... Just hypothetically... You would still believe what the head admin said, that there is no account-specific luck?
B: Err... Well... Yes. As astronomically low as the probability is, it still wouldn't be zero. So it would still be consistent with what the admin said, that every account has the same chance of winning every time.
A: ....
B: But again, what you're talking about is an extremely improbable scenario with a practically negligible chance.
A: My streak though...
B: ... is nothing but another ultra-improbable, ultra-unlucky and I understand ultra-frustrating fluke. Still, a fluke is a fluke; it can end anytime.
A: ...
B: You could get your first win 100 tries from now, 50 tries, 20 tries... Heck, maybe even on your next try.
A: ...
B: In fact, judging by our combined record of 2012 victories in 20,000 tries, I think there's a very decent chance you'll get your first win within your next 10 tries.
A: Would you like to bet on that?
B: Sorry, but I'm a little short on cash right now from a certain bet I just lost. Haha.
A: ...
B: ...
A: Sorry Bob, but it looks like we're going to have to agree to disagree.
B: Just as I thought... Good day.
A: Good day.
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Old 2012-07-06, 10:55   #2
axn
 
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Is there a question in there somewhere?
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Old 2012-07-06, 11:00   #3
jinydu
 
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Well, that was fun, and a bit tiring. I didn't know about the 10,000 character per post limit, which prevented me from asking my question in the original post. Anyway, my question was this: Is there a statistical test that could vindicate Alice in that argument, to some prescribed degree of confidence? I heard once of something called the t-test, or something like that, but I don't think it fits this situation exactly...

And a more complicated version of the question: Imagine that 10,000,000 people hear this story, and decide to form a "distributed gaming" project to test out for themselves whether that game has account-specific luck or not. Each person runs the game 10,000 times, records the number of wins, and reports it. When everyone has finished, a plot is made of (number of wins out of 10,000) vs. (number of people with this number of wins). If account-specific luck is not real, we'd expect the plot to look something like a bell curve with a peak at the (a priori unknown) mean. If account-specific luck is real, the plot will deviate from that; for instance, it could have an \cap\cap shape. Is there a statistical test that given a plot, says whether or not to reject the null hypothesis, with some fixed degree of confidence?

(Obviously I could have asked in a much smaller number of words. But what would be the fun in that? )

Thanks

Last fiddled with by jinydu on 2012-07-06 at 11:31
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Old 2012-07-06, 11:26   #4
ET_
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It looks like a good start of a SF novel... :-) I liked it!

Will Alice and Bob fall in love with each other?
Is there anything not working on Alice's computer?
Does Alice live in a "psychic negative zone" that influences reality?
If so, how much do psychic powers influence statistics as unnoticed dependent variables?

(walks out, gazing at the sky while still blabbering...)
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Old 2012-07-06, 11:57   #5
retina
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I think Alice is the site admin (or perhaps in good with the admin) and is scamming Bob for the $100. Is Alice from Nigeria by any chance?
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Old 2012-07-06, 12:37   #6
LaurV
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What the hack? Switch those dammit accounts, I play for you, you play for me. We stop if more then 20 games were played and Alice has two wins (on Bob account) and Bob still none or any other version (like Bob making points on Alice account) would settle it. I suspect Alice hold the mouse click too lung
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Old 2012-07-06, 13:25   #7
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Can't she just create a new account and play with that one, after all, Bob managed to play the game immediately...
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Old 2012-07-06, 18:34   #8
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You didn't specify how long each game takes, but wouldn't having Alice
play and lose an infinite number of games (in a finite amount of time, say
by a diminishing geometric ratio) provide a positive confidence level that
the game is "rigged""?

I would say no finite number of strictly losing games would provide
a statistically "certain" result.
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Old 2012-07-06, 18:54   #9
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Possibly relevant?

http://www.mersenneforum.org/showthread.php?t=3371
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Old 2012-07-07, 00:36   #10
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Quote:
Originally Posted by jinydu View Post
Well, that was fun, and a bit tiring. I didn't know about the 10,000 character per post limit, which prevented me from asking my question in the original post. Anyway, my question was this: Is there a statistical test that could vindicate Alice in that argument, to some prescribed degree of confidence? I heard once of something called the t-test, or something like that, but I don't think it fits this situation exactly...
It's been many years since I used any of this stuff, but anyway. Yes, there are tests. Alice's number of wins should follow a binomial distribution (assuming each of her games is independent of her other games & her chance of winning is constant), and the same for Bob. Binomial probabilities have a simple formula, but for a largish number of games you can approximate the distribution by a normal one, mean np & variance np(1-p). So if you're prepared to assume Bob's games are independent of Alice's - but might have a different chance of winning - you can test the difference between their win rates using the normal distribution.
Quote:
And a more complicated version of the question: Imagine that 10,000,000 people hear this story, and decide to form a "distributed gaming" project to test out for themselves whether that game has account-specific luck or not. Each person runs the game 10,000 times, records the number of wins, and reports it. When everyone has finished, a plot is made of (number of wins out of 10,000) vs. (number of people with this number of wins). If account-specific luck is not real, we'd expect the plot to look something like a bell curve with a peak at the (a priori unknown) mean. If account-specific luck is real, the plot will deviate from that; for instance, it could have an \cap\cap shape. Is there a statistical test that given a plot, says whether or not to reject the null hypothesis, with some fixed degree of confidence?

(Obviously I could have asked in a much smaller number of words. But what would be the fun in that? )

Thanks
You're right - it's more complicated (which actually means I can't remember enough to answer this part properly - it was over forty years ago & I didn't use this bit again). There are tests for whether the distribution you see is different from your null hypothesis, but about all I recall is that getting the 'right' sensitivity in the tails seemed tricky. In a real life situation like this I'd investigate a bit first. (Probably before 100 games unless they were quick, and definitely well before 1000 games let alone 10000!!) As LaurV & sonjohan already said, swap accounts and/or computers and try new ones. Try forums & the game makers. Do accounts have just two different win probabilities or a range, is each constant? This really affects the kind of difference you expect from the null hypothesis and therefore how you test. Testing some property of the overall distribution, like its standard deviation or some percentile, might be an alternative to a shape-of-the-whole-distribution test.

Last fiddled with by markr on 2012-07-07 at 00:47
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Old 2012-07-12, 09:47   #11
jinydu
 
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Just checking in to this thread again...

Perhaps I should have been even more explicit. I'm looking for the name of a specific test that allows one to accept or reject the null hypothesis, which is that Alice and Bob have the same probability of winning.

For example, had the game admin said "Everyone always has a 20% probability of winning", this null hypothesis could have been tested with the chi-squared test. Using Alice's data over 10,000 tries, this would give a chi-squared value of (0-2000)^2/2000 + (10000-8000)^2/8000 = 2500, which means one can reject the null hypothesis with an uncertainty of less than 1 part in 4.6 x 10^544.

What I'm looking for is a test that can allow me to perform an analogous calculation for my question.

Quote:
Originally Posted by markr View Post
It's been many years since I used any of this stuff, but anyway. Yes, there are tests. Alice's number of wins should follow a binomial distribution (assuming each of her games is independent of her other games & her chance of winning is constant), and the same for Bob. Binomial probabilities have a simple formula, but for a largish number of games you can approximate the distribution by a normal one, mean np & variance np(1-p). So if you're prepared to assume Bob's games are independent of Alice's - but might have a different chance of winning - you can test the difference between their win rates using the normal distribution.
Thanks for the reply. But wasn't the Binomial \approx Normal approximation only valid when p = 0.5? Because for any other value of p, the distribution is skewed, which doesn't happen in the normal distribution.
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