

Non-transitive Grime Dice, via Mathematica - latkin
http://latkin.org/blog/2015/01/16/non-transitive-grime-dice-via-mathematica/

======
dj-wonk
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-...](http://latkin.org/blog/2015/01/16/non-transitive-grime-dice-via-
mathematica/)

[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'.

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

~~~
dj-wonk
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.p...](http://latkin.org/blog/wp-
content/uploads/2015/01/cycles12.png) (each die beats two others and loses to
two others, in pairwise comparisons)

~~~
latkin
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
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)

