--- "M. Oskar van Deventer" > There are no comparable graphs where any three points can be chosen. > Is this your conjecture or a mathematical statement? Are you telling me that > I have been looking for a four-player version in vain, because it cannot > exist? I did say that, but now that I understand the problem better, I think I'm wrong. :) First off, you need a graph where all possible triplets can be dominated. I think that is possible with a K15 graph. We extend the game to three players, Eric, you and me. We have a larger set of dice. Eric has the first pick, you have the second pick and I pick last. Again surprisingly, I'll statistically beat both Eric and you. That is because I designed a three-player non-transitive set of dice. My mathpuzzle to you and your readers: How many dice are there in my set and what are the numbers on their faces? And how about the four-player version? For quite some time, I have been looking in vain for a four-player non-transitive set of dice. Can you or your readers find such a set, or prove that it does not exist?