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Nontransitive dice

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Re: Nontransitive dice

#41

Post by frumiousirc »

tgb wrote: post #37
allenz wrote: post #33

Earlier quoted context omitted.

You might find the result less surprising after you solve a riddle by Martin Gardner: > A young man lives in Manhattan near a subway express station. He has two girlfriends, one in Brooklyn, one in the Bronx. To visit the girl in Brooklyn, he takes a train on the downtown side of the platform; to visit the girl in the Bronx, he takes a train on the uptown side of the same platform. Since he likes both girls equally w…

I see an immediate solution to that riddle and it matches the idea of the Wikipedia page you link. But I don't see any connection to Penney's game. Can you explain?

The operative words in the rules of both riddles are "appears first".

Re: Nontransitive dice

#43

Post by [deleted] »

frumiousirc wrote: post #41
tgb wrote: post #37

Earlier quoted context omitted.

I see an immediate solution to that riddle and it matches the idea of the Wikipedia page you link. But I don't see any connection to Penney's game. Can you explain?

The operative words in the rules of both riddles are "appears first".

[deleted]

Re: Nontransitive dice

#44

Post by [deleted] »

tgb wrote: post #37
allenz wrote: post #33

Earlier quoted context omitted.

You might find the result less surprising after you solve a riddle by Martin Gardner: > A young man lives in Manhattan near a subway express station. He has two girlfriends, one in Brooklyn, one in the Bronx. To visit the girl in Brooklyn, he takes a train on the downtown side of the platform; to visit the girl in the Bronx, he takes a train on the uptown side of the same platform. Since he likes both girls equally w…

I see an immediate solution to that riddle and it matches the idea of the Wikipedia page you link. But I don't see any connection to Penney's game. Can you explain?

[deleted]

Re: Nontransitive dice

#45

Post by allenz »

frumiousirc wrote: post #41
tgb wrote: post #37

Earlier quoted context omitted.

I see an immediate solution to that riddle and it matches the idea of the Wikipedia page you link. But I don't see any connection to Penney's game. Can you explain?

The operative words in the rules of both riddles are "appears first".

Yup, Penney's game is surprising because at first glance the probabilities of two sequences of the same length are unrelated. But if more than half of the sequences overlap, then one sequence will tend to arrive before the other. As the proportion of overlap tends to 100%, Player B has a 2:1 advantage over Player A: https://i.imgur.com/eKujwrK.png

Re: Nontransitive dice

#46

Post by hammock »

hawkice wrote: post #28
hammock wrote: post #25

Earlier quoted context omitted.

Mean EV doesn't work because the dice have different variances? Trying to wrap my head around the rationale here

Variance isn't the best lens here, better to look at the distributions. It's also a discrete problem, so EV isn't as precise as looking at all the possible combinations.

Variance describes distribution...

Discreteness is another layer. What are the exact defining characteristics of this problem though? You could take two dice 1-2-3-4-5-6 and 2-3-4-5-6-7 and the typical mean EV calculation would work fine...

1-2-3-4-5-7 and 2-3-4-5-6-8 have different variances but are probably not intransitive (haven't checked the math) since they are translations of each other

I'm just thinking out loud here and trying to narrow it down...maybe someone reading wants to help me :)

Re: Nontransitive dice

#47

Post by hawkice »

hammock wrote: post #46
hawkice wrote: post #28

Earlier quoted context omitted.

Variance isn't the best lens here, better to look at the distributions. It's also a discrete problem, so EV isn't as precise as looking at all the possible combinations.

Variance describes distribution... Discreteness is another layer. What are the exact defining characteristics of this problem though? You could take two dice 1-2-3-4-5-6 and 2-3-4-5-6-7 and the typical mean EV calculation would work fine... 1-2-3-4-5-7 and 2-3-4-5-6-8 have different variances but are probably not intransitive (haven't checked the math) since they are translations of each other I'm just thinking out l…

Variance does describe the distribution, but not well enough, in this case. It reduces the vector of possibilities to a scalar, and hides the most important aspect, the non-transitivity. Numbers have transitive comparisons, so there's essentially no way a scalar value _could_ capture that in any straightforward way. Basically, abandon all single-value summaries. They'll do you no good in understanding this phenomenon. But the detailed analysis isn't too complicated, so it's okay.

Re: Nontransitive dice

#49

Post by tripzilch »

amalcon wrote: post #22
CydeWeys wrote: post #9

Earlier quoted context omitted.

There are many sets of non-transitive dice where the average is not the same though (I should have used one of those examples). Here's one: A: 4, 4, 4, 4, 0, 0 (avg: 8/3) B: 3, 3, 3, 3, 3, 3 (avg: 9/3) C: 6, 6, 2, 2, 2, 2 (avg: 10/3) D: 5, 5, 5, 1, 1, 1 (avg: 9/3) There's no reason that the average values need to be the same to have non-transitive dice. To modify the original three to have the same winning properties…

It doesn't play into this game because of the rounding . Again, you seem to be attempting to disagree with me, by echoing the exact same things I just said.

That's because the second person that disagrees with the same thing is expected to be more right on average.
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