Ivars Peterson's MathTrek

April 15, 1996

## Mating Games and Lizards

Scissors-paper-rock is a game that children play, mathematicians analyze, and a certain species of lizard takes very seriously.

In the playground version of the game, each of two players holds a hand behind his back. On the count of three (or by chanting some ritual phrase), both players bring their hidden hands forward in one of three configurations. Two fingers in a "V" represent scissors, the whole hand spread out and slightly curved means paper, and a clenched fist signifies rock. The winner is determined by the following sequence of rules: Scissors cut paper, paper wraps rock, and rock breaks scissors. If both players present the same configuration, the game is a draw.

Is there a winning strategy for this game? It certainly doesn't make sense to show the same configuration each time. An alert opponent would quickly learn to anticipate your move, make the appropriate response, and always win. A similar danger lies in following any kind of pattern. Thus, unless you can find a flaw in your opponent's play, your best bet is to mix the three choices in a random manner.

Of course, this isn't a completely satisfying result. If you stick to a strategy of random choices, your opponent can't profit. But then, you can't profit from your opponent's mistakes either.

Curiously, the scissors-paper-rock game has a counterpart in the mating rituals of a certain species of lizard native to California. Instead of just one mating strategy, these lizards have three, distinct types of behavior that constantly compete with one another in a perpetual cycle of dominance.

In the side-blotched lizard (Uta stansburiana), males have one of three throat colors, each one declaring a particular strategy. Dominant, orange-throated males establish large territories within which live several females. But these territories are vulnerable to infiltration by males with yellow-striped throats -- known as sneakers -- who mimic the markings and behavior of receptive females. The orange males can't successfully defend all their females against these disguised interlopers, who cluster on the fringes of the territories held by the orange lizards.

However, a large population of sneakers, which have no territory of their own to defend, can be quickly overrun by blue-throated males, who defend territories large enough to hold just one female. Sneakers have no chance against a vigilant, blue-throated guard. But once the sneakers become rare, powerful orange males flourish, grabbing territory and females from the blue lizards. Now, the blue males lose out.

As in the scissors-paper-rock game, the wide-ranging, ultradominant strategy of orange males is defeated by the sneaker strategy of the yellow males, which is in turn defeated by the mate-guarding strategy of blue males. The orange strategy defeats the blue strategy to complete the cycle.

Reporting in a recent issue of Nature, biologists Barry Sinervo and Curt M. Lively of Indiana University discuss field data showing that the populations of each of these three types, or morphs, of male lizard oscillate over a six-year period. They found that when a morph population hits a low, this particular type of lizard produces the most offspring in the following year, helping to perpetuate the cycle. This arrangement somehow succeeds in maintaining substantial genetic diversity while keeping the overall population reasonably stable.

The mathematical side of the scissors-paper-rock game gets a little more interesting when a scoring system is introduced. Suppose, for instance, that scissors scores one point against paper, paper scores two against rock, and rock scores three against scissors. In this situation, would you automatically form a rock and hope to score three, or would you expect your opponent to form a rock, which you could beat by forming paper?

As in the basic game, making the same choice every time doesn't work. What seems to make sense, again, is to mix the three choices randomly, forming each of scissors, paper, and rock with a certain probability. The scoring system determines what these probabilities ought to be to achieve an optimal result. In the example given, you can calculate that the probability should be 1/3 for scissors, 1/2 for paper, and 1/6 for rock. Other scoring schemes give different probabilities.

Any deviation from a random mixing strategy gives your opponent an opportunity to profit from your actions. At the same time, by sticking strictly to these probabilities, you forgo any chance of taking advantage of bad play on the part of your opponent. When both players adopt exactly the same strategy, no one wins -- or loses -- in the long run.

What side-blotched lizards have figured out quite naturally, mathematicians can emulate with their reasoning and their proofs.