In their paper, to be published online in late November 2022, a key part of the proof involves showing that, in most cases, it doesn’t make sense to talk about individual chips being strong or weak. None of Buffett’s dice is the strongest, but that’s not uncommon: If you pick one at random, the Polymath project shows, it’s likely to beat about half the other dice and lose to the other half. “Almost every die is pretty average,” Gowers said.
The project differs from the AIM team’s original model in one respect: To simplify some technical details, the project states that the order of the numbers on the dice matters — so, for example, 122556 and 152562 would be considered two different dice. But Polymath’s results, combined with the AIM team’s experimental evidence, created a strong hypothesis that the conjecture was also true in the original model, Gowers said.
“I’m very pleased that they came up with this proof,” Conley said.
The AIM team predicted similar behavior to three dice when sets of four or more dice were involved: for example, if A the beat Second, Second the beat Cand C the beat Manthen there should be about 50-50 probability Man the beat Awhich happens to be close to 50-50 as the number of sides of the dice approaches infinity.
To test this conjecture, the researchers simulated a head-to-head match with a set of four dice with sides of 50, 100, 150 and 200. The simulation didn’t quite match their predictions like in the case of three dice, but it was still close enough to support their belief in the conjecture. But, though the researchers didn’t realize it, these tiny differences conveyed a different message: For sets of four or more dice, their guesses were wrong.
“We really want to [the conjecture] Really, because that would be cool,” Conley said.
For the case of four dice, Elisabetta Cornacchia of EPFL in Switzerland and Jan Hązla of the African Institute of Mathematical Sciences in Kigali, Rwanda, showed in a paper published online in late 2020 that if A the beat Second, Second the beat Cand C the beat ManThen Man Slightly more than 50% chance of defeating A– Probably around 52%, says Hązła. (As in the Polymath paper, Cornacchia and Hązła use a slightly different model than the one in the AIM paper.)
Cornacchia and Hązła’s discovery arose from the fact that, although usually a single dice is neither strong nor weak, a pair of dice can sometimes have a common area of strength. Cornacchia and Hązła showed that if you choose two dice at random, there is a good chance that the two dice are related: they tend to beat or lose to the same die. “If I ask you to create two dice that are close to each other, it turns out that’s possible,” Hązła said. Once there are at least four dice in the picture, these little pockets of correlation can throw tournament outcomes out of symmetry.
The recent paper is not the end of the story. Cornacchia and Hązła’s paper is only beginning to reveal exactly how correlations between dice can throw tournament symmetry out of balance. In the meantime, though, we now know that there are quite a few sets of intransitive dice out there—maybe even one subtle enough to trick Bill Gates into choosing it in the first place.
ability Reprinted with permission from Quanta Magazine, Edit stand-alone publications Simons Foundation Its mission is to enhance public understanding of science by covering research developments and trends in the mathematical, physical and life sciences.