Here is what may surprise some people. Here is an expectation table (i.e. just averages):<p><pre><code> | :olive | :magenta | :blue | :yellow | :red |
|--------+----------+-------+---------+-------|
| 4.167 | 4.333 | 4.5 | 4.667 | 4.833 |
</code></pre>
Generated by this Clojure code (`exp` means expectation):<p><pre><code> (print-table (order-keys-by exp dice) [(exp dice)])
</code></pre>
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:<p><pre><code> E(olive) < E(magenta) < E(blue) < E(yellow) < E(red)
</code></pre>
Of course the expectations have to be transitive; they are scalars.<p>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.<p>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.<p>My code is here: <a href="https://gist.github.com/xpe/30ae93b107c91ec2ccf5" rel="nofollow">https://gist.github.com/xpe/30ae93b107c91ec2ccf5</a><p>(Edited at 12:57 PM EST.)<p>[1] OP: <a href="http://latkin.org/blog/2015/01/16/non-transitive-grime-dice-via-mathematica/" rel="nofollow">http://latkin.org/blog/2015/01/16/non-transitive-grime-dice-...</a><p>[2] <a href="http://www.singingbanana.com/dice/article.htm" rel="nofollow">http://www.singingbanana.com/dice/article.htm</a><p>[3] Actually, there are multiple cycles; the 'secondary' cycles are not as 'strong'.