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?

Step by step solution

01

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.

02

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.

03

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.

04

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.

05

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.

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