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What total do zero score players get when they lose if they play forever?
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skrumgaer wrote
at 3:12 PM, Sunday September 2, 2007 EDT
The thread "I am in math heaven" seems to be irretrievable so I will give the answer to the math homework I assigned there in this thread.
The question was, if there are only 7 players, and they play each other only, and they play forever, what will their average scores be? Will it be infinite? Answer: Infinity, yes. I don't have the math skills to work it out directly, but I set up a simpler problem on a spreadsheet. Suppose you flip a fair coin over and over. For each head, you gain one point, for each tail, you lose one point, but if your total score falls below zero, it is restored to zero. I ran this for 12,000 iterations on a spreadsheet and found that you have a positive score about 98 percent of the time. So every 50th toss, on average, you will go into negative territory and be given one point. So, your score will roughly be 200 points for every 10,000 interations. In other words, unbounded. In kdice, will the average player have a positive score 98 percent of the time? It remains to be seen. |
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cheesewiz wrote
at 3:24 PM, Sunday September 2, 2007 EDT Genius.
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Elithrion wrote
at 3:24 PM, Sunday September 2, 2007 EDT Nice. You have the incorrect answer to your own problem. In your coin example, you pointed out that every 50th toss you will go into negative territory. If this is the case, you can't have an infinite score, since every 50th toss you have 0. If this is not the case, given infinite time you can have any score you want, but your score will never be steadily increasing, or decreasing. It will just be random.
Ergo, if the 7 players play infinitely and have equal probabilities of winning a given game, their scores will be random and arbitrarily large. If, conversely, some have a higher or lower probability of winning games, *then* you'll actually get infinite scores. |
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skrumgaer wrote
at 3:29 PM, Sunday September 2, 2007 EDT I said, on every 50th toss.....you go into negative territory...and GIVEN one point. Try running the iteration and see if you get the incorrect answer to your own post.
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Elithrion wrote
at 3:34 PM, Sunday September 2, 2007 EDT But you aren't given 1 point. You're still at 0. You skip losing 1 point, but you're no better off than you were at the very beginning. Your result with the coin is just going to be a binomial distribution with the negative half cut off.
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skrumgaer wrote
at 3:57 PM, Sunday September 2, 2007 EDT The binomial distribution with the left half cut off does not have a mean of zero.
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Elithrion wrote
at 4:14 PM, Sunday September 2, 2007 EDT I'm not saying it has a mean of 0, but it sure as hell doesn't have a mean of infinity.
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skrumgaer wrote
at 4:26 PM, Sunday September 2, 2007 EDT Why not?
In an infinite number of games: the number of heads will be infinite the number of tails will be infinite the longest run of consecutive heads will be infinite the longest run of consecutive tails will be infinite the longest number of turns in a row where your score remains at zero will be infinite the longest number of turns in a row where your score is greater than zero is infinite |
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Elithrion wrote
at 4:39 PM, Sunday September 2, 2007 EDT Let me fix that for you:
"the number of heads will be infinite the number of tails will be infinite the longest run of consecutive heads will be arbitrarily large the longest run of consecutive tails will be arbitrarily large the longest number of turns in a row where your score remains at zero will be arbitrarily large the longest number of turns in a row where your score is greater than zero is arbitrarily large" Those things are not infinite because giving infinite time, they'll stop. Hence, arbitrarily large. Your current argument is akin to the whole "enough monkes with typewriters with time=Shakespeare" thing. Yeah, it's true, but it doesn't really mean very much. See my first reply for correct answer (actually, I mentioned your wonderful argument there as well, briefly). |
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skrumgaer wrote
at 4:53 PM, Sunday September 2, 2007 EDT It's true, you said it was true, that's all it needs to be.
I win. |
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Tommen wrote
at 5:40 PM, Sunday September 2, 2007 EDT if two players (or 7) keep betting a dollar back and forth on coin flips, and the bank gives them handouts whenever one of them goes negative, two things are true:
the bank will eventually pay out an infinite amount of money each player will go bankrupt an infinite number of times This is a direct result of the mathematical theorem which states that, on a one or two dimensional (but not 3 or more dimensional) lattice, anyone who does a random walk where he is 50% likely to choose left or right each time (or in 2-D 25% likely to choose north/south/east/west), will find his way back to the origin an infinite number of times. |
