Random rewards enrich classic game-theory insights

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.

A bit of history

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.

Your influence is limited

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.

The conclusion from all this game theorizing is that, even though the behavioral tendencies of the players may influence the game, a varying game environment can have a huge effect on the optimal strategy.

Life revealed through simple rules

If we take the prisoner’s dilemma as the classic game theory tool, it yields a depressing conclusion: Cooperation is a losing strategy. Yet even under the conditions set by the prisoner’s dilemma, we observe cooperation. Why? Because there are external influences on the reward—the prisoner who defects and is released may well be retaliated against under slightly different circumstances, and not under others. Therefore, a more complex mix of strategies is likely to emerge.

However, it was still surprising to see that a relatively small amount of variation in the reward structure could lead to such strong changes in behavior.

Game-theory models are also often used to try and understand economic behavior. I’ve always been skeptical (perhaps overly skeptical) of the insights drawn from these games, and I think this paper makes it explicit why I distrust these models. But the results here also point a way forward, where the richer dynamics that we observe in real life are replicated in games that are still quite simple.

Physical Review Letters, 2026, DOI: 10.1103/3yby-qq2n

Chris writes for Ars Technica's science section. A physicist by day and science writer by night, he specializes in quantum physics and optics. He Lives and works in Eindhoven, the Netherlands.