Question

Suppose players 1 and 2 will play the following prisoners' dilemma. Prior to interacting in the prisoners' dilemma, simultaneously each player \(i\) announces a binding penalty \(p_{i}\) that this player commits to pay the other player \(j\) in the event that player \(i\) defects and player \(j\) cooperates. Assume that these commitments are binding. Thus, after the announcements, the players effectively play the following induced game. \({ }^{6}\) (a) What values of \(p_{1}\) and \(p_{2}\) are needed to make \((\mathrm{C}, \mathrm{C})\) a Nash equilibrium of the induced game? (b) What values of \(p_{1}\) and \(p_{2}\) will induce play of \((\mathrm{C}, \mathrm{C})\) and would arise in a subgame perfect equilibrium of the entire game (penalty announcements followed by the prisoners' dilemma)? Explain. (c) Compare the unilateral commitments described here with contracts (as developed in Chapter 13).

Short answer

Penalties \( p_1, p_2 \geq T - R \) make (C, C) a Nash equilibrium; set \( p = T - R \) for subgame perfection. Unilateral commitments resemble contracts.

Step-by-step solution

Examine the Payoff Matrix

Consider the basic payoff matrix for a typical Prisoner's Dilemma. Each player can either Cooperate (C) or Defect (D). The payoffs, generally, are such that mutual cooperation (C, C) gives both players a payoff higher than mutual defection (D, D), but the temptation to defect when the other cooperates offers the highest individual payoff.

Define the Penalty Terms

Based on the scenario described, if a player chooses to defect while the other player cooperates, the defecting player pays a penalty to the cooperating player. This modifies the payoff outcomes depending on the penalties chosen, \( p_1 \) for Player 1 and \( p_2 \) for Player 2.

Identify the Nash Equilibrium Condition

For (C, C) to be a Nash Equilibrium, neither player should have an incentive to unilaterally change their strategy from C to D. The payoff from choosing C must be at least as great as the payoff from choosing D, considering the penalties imposed on defection.

Set Up Inequalities for Nash Equilibrium

For Player 1, the inequality is \( T - p_1 \leq R \), where \( T \) is the payoff for defecting while the other cooperates and \( R \) is the payoff for mutual cooperation. For Player 2, \( T - p_2 \leq R \).

Determine Values for Penalties

From the inequalities, we determine that \( p_1 \geq T - R \) and \( p_2 \geq T - R \). These are the minimum penalties needed to sustain cooperation as a Nash equilibrium.

Subgame Perfect Equilibrium Analysis

In the subgame perfect equilibrium, penalties are set such that any deviation after penalty announcements leads to a lower payoff. Therefore, each player chooses the minimum penalty \( p = T - R \), which sustains cooperation plays (C, C) as the outcome in both the penalty subgame and the entire game.

Compare with Contracts

The unilateral commitments here function similarly to contracts by creating credible penalties for defection, aligning individual incentives with cooperative outcomes. This mirrors contract enforcement which also aligns private incentives with collective agreements.

Key concepts

Nash Equilibrium

The Nash Equilibrium is a crucial concept in game theory where no player has an incentive to unilaterally change their strategy, given the other player's action. In the context of the Prisoner's Dilemma, for (C, C) to be a Nash Equilibrium, both players need to find cooperating more beneficial or equally beneficial than defecting.
This means that each player's payoff from cooperating (C) should be at least as high as their payoff if they decide to defect (D), once penalty commitments are factored into the scenario.
In technical terms, for Player 1, the condition is given by the inequality: \( T - p_1 \leq R \). Here, \( T \) represents the temptation payoff for defecting when the other cooperates, and \( R \) is the reward for mutual cooperation. The penalty \( p_1 \) acts as a deterrent, making defection less attractive.
Similarly for Player 2, the inequality \( T - p_2 \leq R \) holds. These conditions ensure that the threat of penalties aligns incentives, making cooperation a stable strategy at Nash Equilibrium.

Subgame Perfect Equilibrium

The concept of Subgame Perfect Equilibrium extends Nash Equilibrium into a framework where players consider all possible future scenarios and their outcomes when making decisions. In the context of the prisoner's dilemma with penalties, it ensures cooperation across both stages of the game: the penalty announcement and the subsequent decision to cooperate or defect.
For a strategy to be subgame perfect, it must not only be the best response given past actions but must also lead to beneficial outcomes in future stages.
In our exercise, this means that after penalties \( p = T - R \) are agreed upon in the first stage, both players mutually cooperate (C, C) in the second stage. Deviating from this by defecting will result in incurring penalties greater than the immediate gain from defection.
The subgame perfect equilibrium thus holds because any deviation, considering the penalty, results in a lesser payoff, prompting both players to stick to cooperation as the rational outcome.

Game Theory

Game Theory provides the tools to analyze strategic interactions between players, each with their own preferences and available strategies. In our scenario, game theory helps model the conflict and potential cooperation in the Prisoner's Dilemma.
Each player has two strategies, cooperate (C) or defect (D), and the game is structured in such a way that mutual cooperation yields better payoffs than mutual defection, albeit each player is tempted to defect for an individual advantage.
The introduction of penalty commitments is a strategic maneuver analyzed using game theory. By adjusting the payoffs through penalties, players realign incentives to make cooperation more attractive than defection, thus overcoming the dilemma.
Understanding how these strategic choices interact and lead to equilibria, be it a Nash or Subgame Perfect Equilibrium, helps us appreciate how penalty commitments can influence outcomes in strategic interactions.

Penalty Commitments

Penalty Commitments act as binding promises that deter undesirable actions by imposing costs. In our problem, they encourage cooperation by altering the traditional payoff matrix of the Prisoner's Dilemma.
When players are allowed to set penalties \( p_1 \) and \( p_2 \), they commit to compensating the other if they defect and the other cooperates. These commitments add a layer of strategy where the choice of penalty significantly impacts the game's equilibrium.
Having a sufficient penalty ensures that defection becomes costly, thus nudging players towards mutual cooperation, which is a favorable outcome in social settings.
This mechanism is akin to contracts in economic theory, where penalties enforce cooperation by ensuring that deviation leads to higher costs than the immediate gains from such actions, aligning incentives effectively.