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Nontransitive dice

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Re: Nontransitive dice

#21

Post by jameshart »

dragonwriter wrote: post #8
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.

You seem to be using "rounding" to mean "comparing the exact, unmodified results of individual rolls", which is probably the source of the confusion because that doesn't really correspond to any common definition of the word "rounding".

One way to interpret OP's 'rounding' concept is that in a round of a dice game, the winner's score is rounded up to 1, and the loser's score rounded down to 0. If, on the other hand, each turn each player got a fractional score between 0 and 1 (without 'rounding'), then that would represent a 'truer' evaluation of the dice.

The best way I can think of to allocate a fractional score to each die player is to sum all the dice rolled, and give each player their roll divided by the sum. So if players rolled 2 and 5, one player would get 2/7 and the other 5/7 - instead of rounding the scores to 0 and 1.

Note that to calculate the EV for a die under this scheme, you have to calculate it versus a particular opposing die, so it would produce a different EV for the transitive A/B/C dice in different combinations.

Perhaps that's what the OP was trying to suggest?

Re: Nontransitive dice

#22

Post by amalcon »

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…

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.

Re: Nontransitive dice

#23

Post by amalcon »

jameshart wrote: post #21
dragonwriter wrote: post #8

Earlier quoted context omitted.

You seem to be using "rounding" to mean "comparing the exact, unmodified results of individual rolls", which is probably the source of the confusion because that doesn't really correspond to any common definition of the word "rounding".

One way to interpret OP's 'rounding' concept is that in a round of a dice game, the winner's score is rounded up to 1, and the loser's score rounded down to 0. If, on the other hand, each turn each player got a fractional score between 0 and 1 (without 'rounding'), then that would represent a 'truer' evaluation of the dice. The best way I can think of to allocate a fractional score to each die player is to sum all th…

More or less this, except with a simpler way of allocating fractional scores: simply divide the value rolled on that die by the largest such value on any of the dice in the set (or, equivalently, round to either 0 or largest-such-value and skip the division step).

Re: Nontransitive dice

#24

Post by barrkel »

CydeWeys wrote: post #17
Buge wrote: post #15

Earlier quoted context omitted.

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.

"But a long term sum of results is a completely different game."

That's exactly the point! A naive intuition about dice is that they have a single probabilistic long-term score. E.g. 6-sided die is 3.5. So if one dice beats another, it should be transitive (the intuition goes). It's the summation (implied by averaging over time) that leads to the naive intuition.

Re: Nontransitive dice

#25

Post by hammock »

kybernetikos wrote: post #14
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).

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

Re: Nontransitive dice

#28

Post by hawkice »

hammock wrote: post #25
kybernetikos wrote: post #14

Earlier quoted context omitted.

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).

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.

Re: Nontransitive dice

#29

Post by [deleted] »

amalcon wrote: post #7
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…

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.

[deleted]

Re: Nontransitive dice

#30

Post by Double_Cast »

CydeWeys wrote: post #4
amalcon wrote: post #3

I find nontransitive dice to be a clear demonstration of the effects of premature rounding. The nontransitivity is only possible because, after each iteration, the result is rounded to a victory for one die. If the totals were summed over time, they could clearly be ranked by expected value. You can see this result in other places, also. It's especially visible in sports, for example, or in the stock market.

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…

You two are speaking past each other. CydeWes is using EV in the "first moment" sense. Amalcon is using EV in the "expected utility" sense. E.g. consider rock-paper-scissors. The expected utility of rock is 1/2. But since rock is represented non-numerically, each event has no first moment.
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