A brief note on non-transitive dice

I've talked before about non-transitive dice. We say that given a pair of dice X and Y, X beats Y if more than half the time when the pair is rolled X has a larger number face up than Y. It turns out one can construct dice A, B and C such that A beats B, B beats C, but C in fact beats A. This is a neat and weird property.

During a recent discussion I used non-transitive dice as an example of a counter-intuitive aspect of mathematics, I was pointed to an even weirder variant. Consider the following set of dice: A has sides (5,5,5,2,2,2), B has sides (4,4,4,4,4,1) and C has sides (6,3,3,3,3,3).

Here A beats B, B beats C and C beats A. But here's the really cool part: Let's say I roll two copies of A, two copies of B or two copies of C. Now things actually reverse! That is, a pair of Bs beats a pair of As and a pair of As beats a pair of Cs and a pair of Cs beats a pair of Bs.

This is a much more sensitive property than just non-transitive dice. Most sets of non-transitive dice will not have this property. We can also describe this sensitivity in a more rigorous fashion. Suppose we have a strictly increasing function f(x). That is, a function such that f(x) is greater than f(y) whenever x is greater than y. Now suppose we take a set of non-transitive dice and relable each value x with f(x). Then they will still be non-transitive. But, given a set of non-transitive, reversable dice, reversibility is not necessarily preserved by the f mapping. This reflects the much more sensitive nature of the reversible dice.

Here's a question I have so far been unable to answer: Is it possible to make a set of die which do an additional reversal? That is, is there a set of dices such rolling three copies the dice results in another reversal direction?

Voluntary Taxation and Gratuitous Promotion of Family Members

It looks like the last of my siblings has no entered the blogosphere. My sister Jacoba has a piece up at the Huffington Post discussing the reaction to Warren Buffet's statements that the rich are not being taxed enough. She points out that if people think that they aren't being taxed enough then they can always just right a check to the US Treasury.

Her point is an interesting one but I think it is misguided: Very few rich people think like Buffet does. Almost everyone who is in favor of higher taxes thinks that the group who should be paying higher taxes are the people with income slightly above their own income. Moreover, even if a large number of rich people agreed with Buffet, if those people gave much more of their money to the federal government while others in their income bracket did not those volunteers would suffer a relative loss of income. There's a fair bit of evidence that people's sense of status and wealth is a function of the people around them. So if most of the rich aren't paying as much it is quite understandable that the other rich would not want to. It is thus reasonable for some high income people to call for higher taxes even as they don't make voluntarily payments. In any event, the idea is an interesting one and the piece is worth reading.