Random rewards enrich classic game-theory contests

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In real life, though, we are driven by external factors that are not under our control. The rabbit does not control the rain that floods its burrow or the drought that kills its food. The game changes each round because the risks and rewards of each strategy change over time. The researchers recognized that incorporating these dynamics into a mathematical model might reveal new behaviors. And they were not wrong.

Τhe prisoner’s dilemma as described above has only a single stable point (everyone loses). The model quickly converges to that point, where all players adopt the same strategy within a few rounds. But the new work found that if the rewards change with time (even by a small amount), then a second stable point emerges from the model, allowing both cooperators and defectors to coexist. Under even greater variation, the defector point becomes unstable, meaning that only cooperators exist.

In chicken, the model’s results are a bit scarier: With no change in rewards, the stable point is that everyone swerves and we all survive. Add in just a bit of variation, and a population that does not swerve emerges. Add in more noise, and a bistable flipping between survival and crashing emerges. (Given that chicken was basically the Cold War strategy, I’m even more amazed we all survived to face our next existential challenge.)

For rock-paper-scissors, an even more complex dynamic emerges. In terms of stable and unstable points, regular rock-paper-scissors has no stable points—the strategy never settles into everyone choosing rock, for instance. Instead, everyone keeps flipping continuously among the three options. If the rewards change randomly per round, however, new stable and unstable points emerge. Depending on the case, this can cause the game to more quickly evolve to the flipping strategy or develop a limit cycle. Limit cycles emerge if the rewards are uneven (for instance, rock versus scissors gives the winner a greater payoff than paper versus rock). In this case, the population continuously cycles, with the probability of choosing rock versus paper versus scissors evolving predictably and stably over time.