Nontransitive dice

Re:Nontransitive dice by CydeWeys on Saturday April 29, 2017 @05:55PM (#14228985)

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, but radically different averages, just imagine doing this:

   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.

The expected value of an individual die roll doesn't play into it at all.

[ Parent | on HN ]

Re:Nontransitive dice by amalcon on Saturday April 29, 2017 @07:58PM (#14229465)

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.

[ Parent | on HN ]

Re:Nontransitive dice by tripzilch on Tuesday May 09, 2017 @08:49AM (#14299522)

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

[ Parent | on HN ]

Re:Nontransitive dice by Buge on Saturday April 29, 2017 @06:53PM (#14229210)

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).

[ Parent | on HN ]

Re:Nontransitive dice by CydeWeys on Saturday April 29, 2017 @07:03PM (#14229251)

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.

[ Parent | on HN ]

Re:Nontransitive dice by barrkel on Saturday April 29, 2017 @08:24PM (#14229553)

"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.

[ Parent | on HN ]

Re:Nontransitive dice by zodiac on Sunday April 30, 2017 @08:18AM (#14231202)

Maybe this description of the two games might help clarify the link.

Game 1: Player 1 chooses a dice, then player 2 chooses a dice. They both roll numbers, say a and b. Then player 2 gives player 1 (a - b) dollars.

Game 2: Player 1 chooses a dice, then player 2 chooses a dice. They both roll numbers, say a and b. Then player 2 gives player 1 (a - b > 0 ? 1 : -1) dollars.

You can kind of squint and see that game 2 is the same as game 1 just with (a - b) "rounded" to either 1 or -1.

[ Parent | on HN ]