4 ms·

Non-transitive Grime Dice, via Mathematica

28 points by latkin 12y ago

superobserver 12y ago
Not non-intuitive at all. Reminds me of Rock-paper-scissors-lizard-Spock but in dice form.
- dj-wonk 12y ago
  Yes, the pairwise comparisons form a similar pattern as shown in the top-right pentagram in http://latkin.org/blog/wp-content/uploads/2015/01/cycles12.png http://latkin.org/blog/wp-content/uploads/2015/01/cycles12.p... (each die beats two others and loses to two others, in pairwise comparisons)
  - latkin 12y ago
    The fact that rolling doubles causes some relationships to reverse (but some to stay the same) is very interesting and non-obvious, at least to me.
    - dj-wonk 12y ago
      Good point. As an example: olive tends to win "blue vs olive" rolls. However, blue+blue tends to win "blue+blue vs olive+olive" rolls. I uploaded blue vs olive histograms at http://imgur.com/a/p4zK8 http://imgur.com/a/p4zK8 -- note, for the comparison ones (labeled with "vs"), -1 means that the leftmost (first) roll won, 0 means a tie, and +1 means that the rightmost (last) roll won. P.S. Blue vs Magenta here: http://imgur.com/a/9e6ne http://imgur.com/a/9e6ne
dj-wonk 12y ago
Here is what may surprise some people. Here is an expectation table (i.e. just averages): | :olive | :magenta | :blue | :yellow | :red | |--------+----------+-------+---------+-------| | 4.167 | 4.333 | 4.5 | 4.667 | 4.833 | Generated by this Clojure code (`exp` means expectation): (print-table (order-keys-by exp dice) [(exp dice)]) I'm surprised that the two (very good!) articles ([1] and [2]) I've read did not point that the non-transitive property [3] holds on the dice even though the expectation are transitive: E(olive) < E(magenta) < E(blue) < E(yellow) < E(red) Of course the expectations have to be transitive; they are scalars. When you apply a function to pairs (e.g. compare one die against another), you can get non-transitive behavior. This is not earth-shattering, but it is interesting. Put another way: this is yet another reason to not trust a single summary statistic (e.g. the average in this case) when you really should look at the distribution. My code is here: https://gist.github.com/xpe/30ae93b107c91ec2ccf5 https://gist.github.com/xpe/30ae93b107c91ec2ccf5 (Edited at 12:57 PM EST.) [1] OP: http://latkin.org/blog/2015/01/16/non-transitive-grime-dice-via-mathematica/ http://latkin.org/blog/2015/01/16/non-transitive-grime-dice-... [2] http://www.singingbanana.com/dice/article.htm http://www.singingbanana.com/dice/article.htm [3] Actually, there are multiple cycles; the 'secondary' cycles are not as 'strong'.

28 points by latkin 12y ago

superobserver 12y ago

Not non-intuitive at all. Reminds me of Rock-paper-scissors-lizard-Spock but in dice form.

dj-wonk 12y ago
Yes, the pairwise comparisons form a similar pattern as shown in the top-right pentagram in http://latkin.org/blog/wp-content/uploads/2015/01/cycles12.png http://latkin.org/blog/wp-content/uploads/2015/01/cycles12.p... (each die beats two others and loses to two others, in pairwise comparisons)
- latkin 12y ago
  The fact that rolling doubles causes some relationships to reverse (but some to stay the same) is very interesting and non-obvious, at least to me.
  - dj-wonk 12y ago
    Good point. As an example: olive tends to win "blue vs olive" rolls. However, blue+blue tends to win "blue+blue vs olive+olive" rolls. I uploaded blue vs olive histograms at http://imgur.com/a/p4zK8 http://imgur.com/a/p4zK8 -- note, for the comparison ones (labeled with "vs"), -1 means that the leftmost (first) roll won, 0 means a tie, and +1 means that the rightmost (last) roll won. P.S. Blue vs Magenta here: http://imgur.com/a/9e6ne http://imgur.com/a/9e6ne

dj-wonk 12y ago

Here is what may surprise some people. Here is an expectation table (i.e. just averages): | :olive | :magenta | :blue | :yellow | :red | |--------+----------+-------+---------+-------| | 4.167 | 4.333 | 4.5 | 4.667 | 4.833 | Generated by this Clojure code (`exp` means expectation): (print-table (order-keys-by exp dice) [(exp dice)]) I'm surprised that the two (very good!) articles ([1] and [2]) I've read did not point that the non-transitive property [3] holds on the dice even though the expectation are transitive: E(olive) < E(magenta) < E(blue) < E(yellow) < E(red) Of course the expectations have to be transitive; they are scalars. When you apply a function to pairs (e.g. compare one die against another), you can get non-transitive behavior. This is not earth-shattering, but it is interesting. Put another way: this is yet another reason to not trust a single summary statistic (e.g. the average in this case) when you really should look at the distribution. My code is here: https://gist.github.com/xpe/30ae93b107c91ec2ccf5 https://gist.github.com/xpe/30ae93b107c91ec2ccf5 (Edited at 12:57 PM EST.) [1] OP: http://latkin.org/blog/2015/01/16/non-transitive-grime-dice-via-mathematica/ http://latkin.org/blog/2015/01/16/non-transitive-grime-dice-... [2] http://www.singingbanana.com/dice/article.htm http://www.singingbanana.com/dice/article.htm [3] Actually, there are multiple cycles; the 'secondary' cycles are not as 'strong'.