Nontransitive dice
en.wikipedia.org
Nontransitive dice
1–10 of 49 posts
Re: Nontransitive dice
#2Brought to my attention from Timothy Gowers' blog
https://gowers.wordpress.com/2017/04/28/a-potential-new-poly...
Re: Nontransitive dice
#3I 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.
Re: Nontransitive dice
#4I 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.
Re: Nontransitive dice
#5I 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…
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.
Re: Nontransitive dice
#6This reminded me of Penney's Game (https://en.wikipedia.org/wiki/Penney's_game).
Re: Nontransitive dice
#7I 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…
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.
Re: Nontransitive dice
#8Earlier 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.
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".
Re: Nontransitive dice
#9Earlier 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.
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)
Die A has sides 2, 2, 4, 4, 99, 99.
Die B has sides 1, 1, 6, 6, 8, 8.
Die C has sides 3, 3, 5, 5, 7, 7.
Re: Nontransitive dice
#10The 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 a big part of the fun.