Question

Bob and Tom are two criminals who have been arrested for burglary. The police put Bob and Tom in separate cells. They offer to let Bob go free if he confesses to the crime and testifies against Tom. Bob also is told that he will serve a 15 -year prison sentence if he remains silent while Tom confesses. If Bob confesses and Tom also confesses, they will each serve a 10-year sentence. Separately, the police make the same offer to Tom. Assume that Bob and Tom know that if they both remain silent, the police have only enough evidence to convict them of a lesser crime, and they will both serve 3 -year sentences. a. Use the information provided to write a payoff matrix for Bob and Tom. b. Does Bob have a dominant strategy? If so, what is it? c. Does Tom have a dominant strategy? If so, what is it? d. What prison sentences do Bob and Tom serve? How might they have avoided this outcome?

Short answer

The payoff matrix for Bob and Tom is \[ \begin{bmatrix} (10,10) & (0,15) \ (15,0) & (3,3) \end{bmatrix} \]. Both Bob and Tom have a dominant strategy to confess. Therefore, they will each serve a 10-year sentence. They could have avoided this outcome and only served a 3-year sentence if they both remained silent.

Step-by-step solution

Construct the Payoff Matrix

First, create a 2x2 matrix with Bob’s strategic choices, confess or remain silent, along one dimension, and Tom's strategic choices along the other. The matrix cells contain the corresponding payoffs: \[ \begin{bmatrix} (10,10) & (0,15) \ (15,0) & (3,3) \end{bmatrix} \]. The first number in each pair represents Bob’s sentence while the second represents Tom's.

Identify Bob's Dominant Strategy

Now compare Bob’s sentence for each of his possible actions (Confess or Remain Silent) given each of Tom’s actions. Bob prefers to confess whether Tom confesses or remains silent because in each case, confessing leads to less jail time (10 years compared to 15 years if Tom confesses, and 0 years compared to 3 years if Tom remains silent). Thus, Bob's dominant strategy is to confess.

Identify Tom's Dominant Strategy

Perform a similar analysis for Tom by comparing his sentences for each of his possible actions given each of Bob’s actions. Tom also prefers to confess whether Bob confesses or remains silent (10 years instead of 15 years if Bob confesses, and no jail time instead of 3 years if Bob remains silent). Thus, Tom's dominant strategy is to confess.

Determine Their Prison Sentences

Since both possess a dominant strategy to confess, in equilibrium, both Bob and Tom will confess. Using the payoff matrix, we find that when both confess, they each will serve 10 years.

Discuss the Possible Different Outcome

They each could have avoided a 10-year sentence if they had both remained silent, in which case they would have each served 3 years. However, because they couldn't communicate and coordinate their strategies, they were unable to achieve this outcome.

Key concepts

Payoff Matrix

Understanding the payoff matrix is crucial in analyzing strategic decisions in a game. In this case, Bob and Tom face decisions of whether to confess or remain silent. Here's how the matrix works:

• The matrix is a 2x2 grid, with actions for Bob (rows) and actions for Tom (columns).
• Each cell in the matrix shows the outcomes based on the pair of decisions made. For example, if both confess, the outcome is (10,10), meaning both face 10-year sentences.
• The numbers are arranged in pairs, where the first number is the outcome for Bob and the second number is for Tom.

This matrix is a simple yet powerful tool that helps in visualizing potential outcomes based on different actions, allowing players to assess the consequences of their strategies easily.

Dominant Strategy

A dominant strategy is an action that yields a better outcome for a player, no matter what the other player does. For Bob and Tom:

• Bob examines his options: whether Tom confesses or not, confessing always gets him a reduced sentence (10 or 0 years).
• Similarly, Tom sees confessing is better regardless of Bob's action, as he ends up with a shorter sentence (10 or 0 years).

Since confessing always leads to less jail time for both Bob and Tom, it becomes their dominant strategy. This outcome is independent of the other person's choice, making it a key aspect of strategic decision-making in competitive scenarios.

Game Theory

Game theory delves into the strategic interactions between decision-makers. In the Prisoner’s Dilemma, Bob and Tom serve as classic examples:

• Each player tries to minimize their prison sentence, operating in their own best interest.
• Theory suggests that rational players, like Bob and Tom, will gravitate towards strategies that best serve them, even if mutual cooperation (both staying silent) would lead to a better collective outcome.

Through game theory, we see how individuals navigate complex decisions, often resulting in less-than-optimal results for all parties involved when cooperation is sacrificed for individual gain.

Nash Equilibrium

The Nash Equilibrium is a key concept in game theory, where each player's strategy is optimal given the other player's strategy. For Bob and Tom:

• Both confessing leads to a Nash Equilibrium. At this point, neither Bob nor Tom can reduce their sentence by changing their decision alone.
• In this equilibrium, they both serve 10-year sentences, even though a different strategy (both staying silent) would have been better for both.

The dilemma lies in their inability to communicate and trust each other to remain silent, highlighting the challenges of achieving cooperation in competitive environments.