Mathematicians roll the dice and get rock-paper-scissors

In their paper, which was posted online in late November 2022, an important piece of evidence is showing that talking about whether a single cube is strong or weak makes no sense for the most part. Buffett’s dice, none of which are the strongest of the pack, aren’t that unusual: If you pick one dice at random, the Polymath project has shown, it’s likely to beat about half the other dice and lose to the other half . “Almost every cube is pretty average,” Gowers said.

The project deviated from the AIM team’s original model in one respect: To simplify some technical details, the project explained that the order of the numbers on a cube matters – for example, 122556 and 152562 would be considered two different cubes. But the Polymath result, combined with the experimental evidence from the AIM team, creates a strong conjecture that the conjecture is also true in the original model, Gowers said.

“I was absolutely delighted that they provided this evidence,” said Conrey.

For clusters of four or more cubes, the AIM team had predicted behavior similar to that for three cubes: For example, if A beats B, B beats CAnd C beats Dthen there should be about a 50-50 chance D beats Aapproaches exactly 50-50 as the number of faces on the dice approaches infinity.

To test the conjecture, the researchers simulated head-to-head tournaments for sets of four dice of 50, 100, 150, and 200 sides. The simulations didn’t follow their predictions quite as closely as in the case of three dice, but were still close enough to reinforce their belief in the conjecture. But although the researchers didn’t notice, these small discrepancies carried a different message: For sets of four or more dice, their guess is wrong.

“We really wanted to [the conjecture] to be true because that would be cool,” Conrey said.

In the case of four cubes, Elisabetta Cornacchia from the Swiss Federal Institute of Technology in Lausanne and Jan Hązła from the African Institute for Mathematical Sciences in Kigali, Rwanda, showed in an article published online at the end of 2020 that if A beats B, B beats CAnd C beats DThen D has a slightly better chance of beating 50 percent A– probably somewhere around 52 percent, said Hązła. (As in the Polymath work, Cornacchia and Hązła used a slightly different model than in the AIM work.)

Cornacchia and Hązła’s insight stems from the fact that while a single die is typically neither strong nor weak, a pair of dice can sometimes share common areas of strength. Cornacchia and Hązła have shown that if you choose two dice at random, there is a good chance that the dice will correlate: you tend to hit or lose against the same dice. “If I ask you to create two cubes that are close to each other, it turns out that it can be done,” Hązła said. These little pockets of correlation push tournament results away from symmetry once there are at least four dice in the picture.

The recent papers are not the end of the story. The work of Cornacchia and Hązła only begins to reveal exactly how correlations between dice throw the symmetry of tournaments out of balance. In the meantime, however, we know that there are many sets of intransitive dice out there – maybe even one subtle enough to get Bill Gates to choose first.

Original story Reprinted with permission from quanta magazine, an editorially independent publication Simons Foundation whose mission is to improve public understanding of science by covering research developments and trends in mathematics and the natural and life sciences.

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