The Prisoner's Dilemma and Nash Equilibrium
A standalone explanation, written for someone with no prior knowledge, of why two rational players betray each other even when silence would serve them both.
To understand the Prisoner’s Dilemma, imagine a situation in which two prisoners, prisoner A and prisoner B, are held in separate prison cells. Each prisoner is questioned and presented with two choices: stay silent or betray. If both prisoners stay silent, both are sentenced to only one year in prison. If prisoner A stays silent but prisoner B betrays, then prisoner A gets a ten year sentence while prisoner B goes free. The opposite is true as well — prisoner B gets ten years for staying silent while prisoner A goes free for betrayal. If both prisoners decide to betray each other, they each get a five year prison sentence.
The best option for both prisoners is for each of them to stay silent, receiving a one year sentence. However, from each prisoner’s individual point of view, betrayal appears to be the better option. Unaware of the decision that prisoner B will make, prisoner A must consider each scenario. If prisoner B stays silent, then prisoner A can either stay silent or betray. If they betray, they go free. If they stay silent, they get a one year sentence. They are therefore incentivized to betray. Conversely, in the case that prisoner B betrays, they can also decide whether to betray or stay silent. Betrayal will get them a five year sentence, while silence will get them a ten year sentence. Once again, prisoner A is incentivized to betray.
Thus, prisoner A is always incentivized to betray irrespective of the decision made by prisoner B. Using the same logic, prisoner B is incentivized to always betray as well. This situation is known as a Nash Equilibrium — neither player can improve their outcome by changing their choice alone. The prisoner’s dilemma illustrates that rational individual choices can lead to a worse collective outcome.
A repeated prisoner’s dilemma is the same game played more than once by the same players. A repeated game is different to a singular game since players have to think about future rounds. In the one-time prisoner’s dilemma, players do not face any future consequences for their actions. In a repeated prisoner’s dilemma, each player must consider how their actions will influence the decisions of the other player. Players must consider not only the immediate payoffs of their actions but also the potential future payoffs.