// ARS TECHNICA — MOBILE & WEB
Random rewards enrich classic game-theory contests
Simple games gain rich strategies in the face of noise.
Games may be life with all the hard bits removed, but they provide a way to study why people make the choices they make. Traditional games are usually played against a static background: the rewards per outcome are constant. That limits their relevance to behavior because, in real life, the rewards and consequences of strategic choices are ever changing. Now, researchers have used a mathematical model to study a series of games that include evolving strategies and randomly varying returns.
Perhaps the most famous game-theory contest is the prisoner’s dilemma. In the prisoner’s dilemma, a pair of thieves have been captured and are being separately interrogated by the police. If both clam up, they will be punished for a lesser crime. If one prisoner makes a deal (defects) then that prisoner gets to go free and the other gets a heavier sentence. If both make a deal, they both get an in-between punishment.
The person running the game can start it with different rewards for cooperating and defecting to explore how the optimum strategy varies with reward and risk, which the players can figure out by varying the strategies across multiple rounds. Depending on the balance between the reward for staying silent (cooperating) and betrayal, the game stabilizes with everyone betraying everyone. In this simple situation, everyone loses.
Similar dynamics can be found in games of chicken, rock-paper-scissors, and more. The evolution of strategies can lead to stable populations, bistable populations (where the population flips between two stable strategies), or limit cycles, where the population shifts continuously among multiple strategies.
There is also a rich history of changing a game as it is played. Usually, these are within-game variations. For instance, you can set a limit on the amount of reward available, so strategies evolve to take into account increasingly limited resources as the number of rounds goes up. In other words, most of this older work studied situations where the player’s behavior in the current round changed the resources or rewards available for the next round.
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 deve