The Prisoner's Dilemma is a fundamental concept in game theory that illustrates why two rational individuals might not cooperate, even when it's in their best interest to do so. It is represented mathematically as a payoff matrix showing the outcomes for two players, each choosing between two strategies: Cooperate or Defect. In this matrix, each cell contains a pair of payoffs, one for each player. These payoffs are typically labeled as R for the reward for mutual cooperation, T for the temptation to defect, S for the sucker's payoff, and P for the punishment for mutual defection.
The Prisoner's Dilemma is defined by specific relationships between the payoff values. R represents the reward for mutual cooperation. T is the temptation payoff when a player defects while the other cooperates. S is the sucker's payoff received when cooperating while the other defects. And P is the punishment for mutual defection. For a scenario to be a true Prisoner's Dilemma, these payoffs must satisfy the inequality: T greater than R greater than P greater than S. This mathematical condition ensures that defection is always the dominant strategy for each player, despite mutual cooperation yielding a better collective outcome. In our numerical example, with R equals 3, T equals 5, S equals 0, and P equals 1, we can verify that 5 is greater than 3, which is greater than 1, which is greater than 0, satisfying the required condition.
A Nash Equilibrium is a situation where no player can benefit by changing only their own strategy while the other players keep theirs unchanged. In the Prisoner's Dilemma, the only Nash Equilibrium occurs when both players defect, represented by the cell (D,D) with payoffs (1,1). This is because defection is always the best response regardless of what the other player does. The blue arrows show Player 1's best responses: if Player 2 cooperates, Player 1 is better off defecting (moving from (3,3) to (5,0)); if Player 2 defects, Player 1 is still better off defecting (moving from (0,5) to (1,1)). Similarly, the green arrows show Player 2's best responses. This creates the famous dilemma: the rational pursuit of self-interest leads both players to defect, resulting in a worse outcome (1,1) than if they had both cooperated (3,3).
When the Prisoner's Dilemma is played repeatedly, it becomes an Iterated Prisoner's Dilemma. In this version, players face the same dilemma over multiple rounds and can remember and respond to each other's previous choices. This iteration allows for the emergence of cooperation through conditional strategies. Mathematically, the total payoff in an iterated game is represented as the sum of discounted payoffs across all rounds, where delta is a discount factor between 0 and 1 that represents how much players value future payoffs compared to immediate ones. One famous strategy for the iterated game is 'Tit-for-Tat,' where a player begins by cooperating and then simply copies the opponent's previous move. As shown in our example, Player 1 starts with cooperation, continues cooperating until Player 2 defects in round 3, then retaliates with defection in round 4, before returning to cooperation. This strategy has proven remarkably effective in promoting cooperation, despite the temptation to defect in any single round.
To summarize what we've learned about the Prisoner's Dilemma: First, it's mathematically represented by a payoff matrix where the payoffs must satisfy the inequality T greater than R greater than P greater than S. This means the temptation to defect exceeds the reward for cooperation, which exceeds the punishment for mutual defection, which exceeds the sucker's payoff. Second, in a single game, defection is always the dominant strategy regardless of what the other player does, leading to a Nash Equilibrium at mutual defection with payoffs (P,P). Third, when the game is played repeatedly, cooperation can emerge through conditional strategies like Tit-for-Tat, which starts with cooperation and then mirrors the opponent's previous move. Finally, this mathematical model helps explain cooperation in various social, economic, and biological systems, showing how rational self-interest can sometimes lead to suboptimal outcomes for all participants. The Prisoner's Dilemma remains one of the most powerful and widely applied concepts in game theory.