University of Michigan researchers have added a new dimension to the mathematics used to predict the outcomes of all manner of competitions, including sports, games and social hierarchies in both humans and animals.

The research is published in the journal Science Advances.

This dimension, which they call “depth of competition,” can be integrated into a variety of important and lucrative fields. It could, for instance, help project winners of match-ups in sports, forecast consumer preferences, rank universities and evaluate hiring practices.

But it also provides a single framework to compare and glean insights from a wide spectrum of competitive interactions. As long as the researchers have data, be it from board games or baboon fights, their model can calculate the depth of any competition with a winner and loser.

“The model doesn’t know if it’s getting a sports data set or an animal data set,” said Max Jerdee, a doctoral student in physics at U-M and an author of the new study. “What we’re trying to do is build a general way of measuring inequality in a bunch of these different settings.”

In this framework, the more unequal a competition, the deeper it is. In deeper competitions, then, the competitors are more stratified by their skill and status. So it might be surprising to learn that, in this context, human games and sports end up at the shallow end of the spectrum.

But this is by design, said Mark Newman, U-M professor of physics and complex systems.

“A game like basketball, for example, is not actually a shallow sport,” Newman said. “There’s a huge range of abilities and you can play at many different levels. But people don’t because there wouldn’t be any point having the average high-schooler going against an NBA professional.”

Even within the NBA, worse teams have better odds to draft better young players onto their teams as a pathway to improvement, Jerdee pointed out.

“Saying something is shallow may have a negative connotation, but you could also say it’s more competitive, less predictable or more exciting,” Jerdee said. “They’re all describing the same thing.”

While humans have norms and structures to promote parity and exciting competitions, other animals do not.

Consider chickens, the animal responsible for the expression “pecking order.” Within a flock, there is a clear hierarchy where stronger birds peck those lower in the order with very little risk of “losing” the competition or being bested in a display of dominance.

So, using Jerdee and Newman’s assessment, basketball has a depth of less than one layer while chickens’ hierarchy is closer to 20 layers. The social hierarchy of hyenas is even more rigid and predictable, with a depth of more than 100 layers.

Competitive interactions in human society, including university rankings and social hierarchies within high school friend groups, land between sports and animal competitions in terms of depth.

Beyond assessing the depth of a competition, the new model can also predict the “winner” of certain competitions. The method could thus be used to assess university rankings, to project consumer preference, or to predict the outcome of sporting events, even between competitors who have never met before.

To demonstrate this predictive capability, the team showed that the 2022 U-M football team would have had an 89% chance of beating the University of Wisconsin had they played one another.