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Nontransitive dice

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Re: Nontransitive dice

#11

Post by CydeWeys »

TTPrograms wrote: post #5
CydeWeys wrote: post #4

Earlier quoted context omitted.

Negative. Non-transitive dice work even when the expected value of each die is the same. The first example in the Wikipedia article exhibits this: Die A has sides 2, 2, 4, 4, 9, 9. Die B has sides 1, 1, 6, 6, 8, 8. Die C has sides 3, 3, 5, 5, 7, 7. The expected value of each die is 5, yet A beats B, B beats C, and C beats A. Other examples in the article have sets of transitive dice where the expected value is not th…

The "rounding" he's describing is the "beating" you're describing. If the game you play is just long term sums of values then expected value is all you need, and in your example they're all even.

See my other example. Most sets of non-transitive dice do not have the same expected value, and the expected value doesn't tell you which one will win. To use an extreme example, take two dice, (2 2 2 2 2 2) and (1 1 1 1 1 999). The average value of the first die is much smaller than that of the second, yet it wins 5/6ths of the time.

Re: Nontransitive dice

#12

Post by [deleted] »

dws wrote: post #10

The non-transitivity of Rock-Scissors-Paper is easy to understand, partly because it's so simple, but mostly because you're likely never played outside the usual rules, even if adding Lizard-Spock. Non-transitive dice screw with the 'nature' of dice that most of us expect. To get to the mathematical intuition, one may have to get past a deeply-ingrained feeling that something about these dice just isn't right. That's…

[deleted]

Re: Nontransitive dice

#13

Post by latkin »

You can purchase a set of non-transitive dice here: https://mathsgear.co.uk/products/non-transitive-grime-dice (I am not affiliated in any way with this store)

These particular ones are fun because the winning order reverses if you double the count of dice. i.e. A beats B beats C beats A, but 2 As lose to 2 Bs lose to 2 Cs lose to 2 As.

I wrote up a blog a while back that explores the various possibilities and chains using Mathematica: http://latkin.org/blog/2015/01/16/non-transitive-grime-dice-...

Re: Nontransitive dice

#14

Post by kybernetikos »

dws wrote: post #10

The non-transitivity of Rock-Scissors-Paper is easy to understand, partly because it's so simple, but mostly because you're likely never played outside the usual rules, even if adding Lizard-Spock. Non-transitive dice screw with the 'nature' of dice that most of us expect. To get to the mathematical intuition, one may have to get past a deeply-ingrained feeling that something about these dice just isn't right. That's…

I think the numeric and gambling aspect helps too. If you're used to probability, you probably start thinking about expected value automatically in these situations, but it isn't the case that the die with the highest expected value wins most of the time against a die with lower expected value. (e.g. 2,2,2,2,2,2s vs 1,1,1,1,1,100).

Re: Nontransitive dice

#15

Post by Buge »

CydeWeys wrote: post #9
amalcon wrote: post #7

Earlier quoted context omitted.

I'm confused, because this reads to me like exactly what I was trying to say, but you seem to be disagreeing with me. Could you clarify? In the game with rounding, A beats B, B beats C, C beats A. In the game without rounding, where totals are summed, they are evenly matched and it's down to chance. That's exactly the effect I was referring to.

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…

In normal dice play expected value doesn't come into play. But if we were to do a long term sum of results (like amalcon proposed) then it expected value would come into play, and would determine the winner.

This illustrates the difference between premature rounding (normal dice play) and non premature rounding (long term summing of results).

Re: Nontransitive dice

#17

Post by CydeWeys »

Buge wrote: post #15
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…

In normal dice play expected value doesn't come into play. But if we were to do a long term sum of results (like amalcon proposed) then it expected value would come into play, and would determine the winner. This illustrates the difference between premature rounding (normal dice play) and non premature rounding (long term summing of results).

But a long term sum of results is a completely different game. It doesn't matter what's on the face of the dice at all; all that matters is the average value per roll. It's not dice anymore! And you've defined the winning condition completely differently!

I also don't understand how normal dice play counts as "premature rounding". It's just how playing dice works -- you compare the numbers versus each other.

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