The hand game “rock-paper-scissors” is a classic way to settle playground disputes, with rock smashing scissors, scissors cutting paper, and paper covering rock. But it turns out that nature plays its own versions of the game, and mathematicians and biologists have used it to study everything from human societies to bacteria in a petri dish. Now, researchers have found that when players change their strategies on the fly, a stable pattern arises in which each of the three weapons gains and loses popularity in turn. The discovery could shed light on how living creatures maintain competing strategies in the struggle for existence.

When applied to biology, rock-paper-scissors blossoms from a two-person children’s game into a complex dance among multiple players. Certain lizards, for example, use three competing strategies—aggression, cooperation, and deception—to win mates, with each tactic beating one and losing to another—just like rock, paper, and scissors. For the lizards, winning the game equates to making babies.

In the biologically inspired version of the game, random pairs in a large population continually square off, and each player maintains the same strategy throughout—consistently throwing rock, paper, or scissors against every opponent. After each encounter, the winner adds a new copy of itself that uses the same strategy, like a parent producing offspring; the loser disappears. A well-studied set of mathematical equations governs how the relative numbers of rock-, paper-, and scissors-wielders fluctuate over time. Depending on the initial prevalence of each strategy, the game can fall into different long-term behaviors—such as a stable state in which a third of players use each strategy, or wild fluctuations in which one strategy nearly disappears and then rebounds at the expense of the others.

Inspired by computer simulations of a related game, two mathematicians—Steven Strogatz and Danielle Toupo of Cornell University—decided to get to the root of what happens when players switch strategies midgame. “I thought it was fascinating, and I wanted to find a mathematical model that would describe this in its simplest form,” Strogatz says. They went back to basics, studying the pure equations instead of complicated computer simulations.

Strogatz and Toupo modified the rock-paper-scissors equations to allow some “mutant” offspring to play different strategies than their parents. Previous researchers had also studied mutations, but they had assumed that the changes were symmetric—that is, that each strategy mutated to the others at the same rate. Strogatz and Toupo considered many other patterns, such as one in which rock players can spawn paper players but not vice versa.

Every type of mutation they examined led to cyclic patterns, with the fractions of rock, paper, and scissors rocking gently up and down forever. More surprisingly, they also proved that the games fell into these recurring orbits even for mutation rates very close to zero, the two report in this month’s issue of *Physical Review E*. A little bit of mutation kept the game from spiraling into a state in which either all three strategies appear in equal numbers or the ratios fluctuate wildly.

“I think the fascination is that you get these kinds of games in nature,” says Barry Sinervo, an ecologist at the University of California, Santa Cruz, who was not involved in the new study. “You don't have to be a mathematician to appreciate that.”

Sinervo has studied how side-blotched lizards in California get locked into a similarly oscillating game of rock-paper-scissors. By observing lizards in the field, Sinervo and a colleague showed that the number of lizards employing the strategies of aggression, cooperation, and deception fluctuated over the course of 6 years, the dominant strategy changing as new lizards were born. The new research provides a mathematical model that accounts for such swings. “That's what makes this paper really interesting to me,” Sinervo says.

Mathematical challenges kept the Cornell researchers from proving their result for all possible mutation patterns, but Strogatz says they expect it to hold. Even so, with their treatment of a wider class of mutations, the result could provide a mathematical foundation for understanding the ebb and flow of many different strategies in the games nature plays.