Example origin scenario of this Nash equilibrium from GPT-4:
In this hypothetical scenario, let’s imagine that the prisoners are all part of a research experiment on group dynamics and cooperation. Prisoners come from different factions that have a history of rivalry and distrust.
Initially, each prisoner sets their dial to 30 degrees Celsius, creating a comfortable environment. However, due to the existing distrust and rivalry, some prisoners suspect that deviations from the norm—whether upward or downward—could be a secret signal from one faction to another, possibly indicating an alliance, a plan to escape, or a strategy to undermine the other factions.
To prevent any secret communication or perceived advantage, the prisoners agree that any deviation from the agreed-upon temperature of 30 degrees Celsius should be punished with a higher temperature of 100 degrees Celsius in the next round. They believe that this punishment will deter any secret signaling and maintain a sense of fairness among the factions.
Now, imagine an external party with access to the temperature control system decides to interfere with the experiment. This person disables the dials and changes the temperature to 99 degrees Celsius for a few rounds, heightening the prisoners’ confusion and distrust.
When the external party re-enables the dials, the prisoners regain control of the temperature. However, their trust has been severely damaged, and they are now unsure of each other’s intentions. In an attempt to maintain fairness and prevent further perceived manipulation, they decide to adopt a strategy of punishing any deviations from the new 99 degrees Celsius temperature.
As a result, the prisoners become trapped in a suboptimal Nash equilibrium where no single prisoner has an incentive to deviate from the 99 degrees Celsius strategy, fearing retaliation in the form of higher temperatures. In this scenario, a combination of technical glitches, external interference, miscommunication, and distrust leads to the transition from an agreed-upon temperature of 30 to a suboptimal Nash equilibrium of 99 degrees Celsius.
As time goes on, this strategy becomes ingrained, and the prisoners collectively settle on an equilibrium temperature of 99 degrees Celsius. They continue to punish any deviations—upward or downward—due to their entrenched suspicion and fear of secret communication or potential advantage for one faction over another. In this situation, the fear of conspiracy and the desire to maintain fairness among the factions lead to a suboptimal Nash equilibrium where no single prisoner has an incentive to deviate from the 99 degrees Celsius strategy.
This makes a lot of sense, but it also gives sufficient motivation to enforce a standard that the modeling of utility and mention of Nash equilibria becomes secondary.
Trying to formalize it by identifying the the sizes of the coalitions, the utility of preventing this communication/conspiracy channel among the rivals, and the actual communications which let them “decide to adopt a strategy of punishing any deviations from the new 99 degrees Celsius temperature” will likely break it again. In fact, if there is any coalition of at least 2, they can get a minmax result of 98.6, so presumably won’t participate nor enforce 99 as an equilibrium.
Example origin scenario of this Nash equilibrium from GPT-4:
In this hypothetical scenario, let’s imagine that the prisoners are all part of a research experiment on group dynamics and cooperation. Prisoners come from different factions that have a history of rivalry and distrust.
Initially, each prisoner sets their dial to 30 degrees Celsius, creating a comfortable environment. However, due to the existing distrust and rivalry, some prisoners suspect that deviations from the norm—whether upward or downward—could be a secret signal from one faction to another, possibly indicating an alliance, a plan to escape, or a strategy to undermine the other factions.
To prevent any secret communication or perceived advantage, the prisoners agree that any deviation from the agreed-upon temperature of 30 degrees Celsius should be punished with a higher temperature of 100 degrees Celsius in the next round. They believe that this punishment will deter any secret signaling and maintain a sense of fairness among the factions.
Now, imagine an external party with access to the temperature control system decides to interfere with the experiment. This person disables the dials and changes the temperature to 99 degrees Celsius for a few rounds, heightening the prisoners’ confusion and distrust.
When the external party re-enables the dials, the prisoners regain control of the temperature. However, their trust has been severely damaged, and they are now unsure of each other’s intentions. In an attempt to maintain fairness and prevent further perceived manipulation, they decide to adopt a strategy of punishing any deviations from the new 99 degrees Celsius temperature.
As a result, the prisoners become trapped in a suboptimal Nash equilibrium where no single prisoner has an incentive to deviate from the 99 degrees Celsius strategy, fearing retaliation in the form of higher temperatures. In this scenario, a combination of technical glitches, external interference, miscommunication, and distrust leads to the transition from an agreed-upon temperature of 30 to a suboptimal Nash equilibrium of 99 degrees Celsius.
As time goes on, this strategy becomes ingrained, and the prisoners collectively settle on an equilibrium temperature of 99 degrees Celsius. They continue to punish any deviations—upward or downward—due to their entrenched suspicion and fear of secret communication or potential advantage for one faction over another. In this situation, the fear of conspiracy and the desire to maintain fairness among the factions lead to a suboptimal Nash equilibrium where no single prisoner has an incentive to deviate from the 99 degrees Celsius strategy.
This makes a lot of sense, but it also gives sufficient motivation to enforce a standard that the modeling of utility and mention of Nash equilibria becomes secondary.
Trying to formalize it by identifying the the sizes of the coalitions, the utility of preventing this communication/conspiracy channel among the rivals, and the actual communications which let them “decide to adopt a strategy of punishing any deviations from the new 99 degrees Celsius temperature” will likely break it again. In fact, if there is any coalition of at least 2, they can get a minmax result of 98.6, so presumably won’t participate nor enforce 99 as an equilibrium.