Introduction
The Prisoner’s Dilemma is one of the most famous examples in game theory that demonstrates how two individuals may not cooperate, even if it is in their best interest to do so. In this question, we are given a payoff matrix with the number of years in prison that individuals A and B might receive based on whether they choose to confess or not confess. Using this matrix, we will determine their optimal strategies and analyze whether it represents a Prisoner’s Dilemma.
Payoff Matrix
Payoffs are in terms of years of imprisonment (i.e., lower numbers are better):
| Individual A / B | Confess | Don’t Confess |
|---|---|---|
| Confess | (-5, -5) | (-1, -10) |
| Don’t Confess | (-10, -1) | (-2, -2) |
Each ordered pair in the matrix shows (A’s prison years, B’s prison years) for each possible combination of decisions.
Part (i): Optimal Strategy for Each Individual
To find the optimal strategy, we analyze each individual’s choices and determine which one gives the best outcome regardless of the other’s decision. This approach is called dominant strategy analysis.
Strategy for Individual A
- If B confesses:
- If A confesses → A gets -5 years
- If A doesn’t confess → A gets -10 years
⇒ Confessing is better (-5 is better than -10)
- If B doesn’t confess:
- If A confesses → A gets -1 year
- If A doesn’t confess → A gets -2 years
⇒ Confessing is better (-1 is better than -2)
Conclusion: No matter what B does, A is better off confessing. So, confessing is a dominant strategy for A.
Strategy for Individual B
- If A confesses:
- If B confesses → B gets -5 years
- If B doesn’t confess → B gets -10 years
⇒ Confessing is better (-5 is better than -10)
- If A doesn’t confess:
- If B confesses → B gets -1 year
- If B doesn’t confess → B gets -2 years
⇒ Confessing is better (-1 is better than -2)
Conclusion: No matter what A does, B is also better off confessing. So, confessing is a dominant strategy for B.
Resulting Strategy Combination
Both individuals will choose to confess because it is the dominant strategy for both. The outcome will be (-5, -5).
Part (ii): Is This a Prisoner’s Dilemma?
Definition: A Prisoner’s Dilemma is a game in which:
- Each player has a dominant strategy (confess)
- The outcome when both follow their dominant strategies is worse for both than if they had cooperated
Check Conditions
- Dominant strategies? Yes, both A and B will confess.
- Outcome from mutual confession = (-5, -5)
- Outcome from mutual cooperation (don’t confess) = (-2, -2)
So, if both didn’t confess, they would get only 2 years in prison each, which is better than 5 years. However, fear of betrayal and individual incentives lead them to confess.
This is a classic Prisoner’s Dilemma because the rational decision leads to a worse outcome for both players.
Conclusion
From the given payoff matrix, both individuals have a dominant strategy to confess, which leads to the Nash equilibrium of (-5, -5). Although mutual cooperation (not confessing) would result in a better outcome (-2, -2), the lack of trust and fear of being betrayed drives both to confess. This makes the scenario a clear example of the Prisoner’s Dilemma, highlighting the conflict between individual rationality and collective benefit.