# Re: Nontransitive dice (Gates/Buffett)

RDAWSON@HUSKY1.STMARYS.CA
Tue, 5 Mar 1996 08:49:26 -0500

jcnash@aix1.uottawa.ca (John C. Nash) writes:
>In Fortune magazine (Feb 5 1996, pp. 102-104), republished from the
>Harvard Business Review of Jan/Feb 1996, Bill Gates relates how
>Warren Buffett offered him 4 dice (call them A B C D) with numbers
>from the set [0..12] on the faces. The game was for Gates to select
>a dice, then Buffett, and two are set aside. Then play for highest
>roll or rolls. Gates claims that the dice were nontransitive in that
>you could have A beating B on average, B > C, C > D yet D > A, for
>example.

>Does anyone have a description and analysis of such a set of dice
>(or some equivalent problem)? It could be a useful didactic tool.
>Or perhaps it is simply a hoax.

No, not a hoax... In fact, it can be done with as few as three dice.
Consider:

A: 333333 B: 222266 C: 115555

Now, A clearly beats B iff B rolls 2 : probability 2/3
C beats A iff C rolls 5: probability 2/3
And B beats C if B rolls 6 [P=1/3] or B rolls 2 and C rolls 1
[P = 2/9]. Thus P(B beats C) = 5/9.

Of course, for the purposes of gambling^H^H^H^H^H^H^H^H informal
lectures in probability theory, you would use more values and mix them
up a little, to make the principle less obvious...

As for why it works, the real question is why people [myself
included] find it surprising. We seem to think that there "ought" to
be a one-parameter ranking of "fitness" - and there isn't.

-Robert