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Chapter 10: Problem 21

Jane and Bill are apprehended for a bank robbery. They are taken into separate rooms and questioned by the police about their involvement in the crime. The police tell them each that if they confess and turn the other person in, they will receive a lighter sentence. If they both confess, they will be each be sentenced to 30 years. If neither confesses, they will each receive a 20-year sentence. If only one confesses, the confessor will receive 15 years and the one who stayed silent will receive 35 years. Table 10.7 below represents the choices available to Jane and Bill. If Jane trusts Bill to stay silent, what should she do? If Jane thinks that Bill will confess, what should she do? Does Jane have a dominant strategy? Does Bill have a dominant strategy? \(\mathrm{A}=\) Confess; \(\mathrm{B}=\) Stay Silent. (Each results entry lists Jane's sentence first (in years), and Bill's sentence second.)

Short Answer

Expert verified
Jane's dominant strategy is to confess, as it leads to the best outcome for her in both scenarios (assuming Bill will stay silent or confess). Likewise, Bill's dominant strategy is also to confess. Therefore, both Jane and Bill should confess, regardless of the other's actions.

Step by step solution

01

Understand the problem context and scope

This problem is a classic Prisoner's Dilemma problem, where two individuals need to decide whether to trust each other or betray each other. The main goal is to find the best strategy for both parties (i.e., Jane and Bill).
02

Examine the given table

The table contains four possible outcomes, with the sentences for Jane and Bill represented as (J, B): 1. Both confess: (30, 30) 2. Jane confesses, Bill stays silent: (15, 35) 3. Jane stays silent, Bill confesses: (35, 15) 4. Both stay silent: (20, 20)
03

Determine Jane's best action assuming Bill will stay silent

If Jane thinks Bill will stay silent, she can choose between confessing and staying silent herself. Let's compare her sentences based on her choices: - If Jane confesses: (15, 35) - If Jane stays silent: (20, 20) Jane's best action is to confess since she will receive a 15-year sentence instead of a 20-year sentence.
04

Determine Jane's best action assuming Bill will confess

If Jane thinks Bill will confess, she can again choose between confessing and staying silent. Let's compare her sentences based on her choices: - If Jane confesses: (30, 30) - If Jane stays silent: (35, 15) Jane's best action is to confess since she will receive a 30-year sentence instead of a 35-year sentence.
05

Determine if Jane has a dominant strategy

A dominant strategy is a strategy that always leads to the best outcome, regardless of the other players' actions. Since Jane's best action is to confess in both scenarios (assuming Bill will stay silent or confess), her dominant strategy is to confess.
06

Determine if Bill has a dominant strategy

Following a similar reasoning for Bill: - If Bill thinks Jane will stay silent: he should confess (15, 35) instead of staying silent (20, 20). - If Bill thinks Jane will confess: he should confess (30, 30) instead of staying silent (35, 15). Since Bill's best action is to confess in both scenarios, he also has a dominant strategy: to confess.
07

Conclusion

Jane's dominant strategy is to confess. Bill's dominant strategy is also to confess. Therefore, regardless of what Bill does, Jane should confess, and regardless of what Jane does, Bill should confess.

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Most popular questions from this chapter

When OPEC raised the price of oil dramatically in the mid-1970s, experts said it was unlikely that the cartel could stay together over the long term-that the incentives for individual members to cheat would become too strong. More than forty years later, OPEC still exists. Why do you think OPEC has been able to beat the odds and continue to collude? Hint: You may wish to consider non- economic reasons.

Would you expect the kinked demand curve to be more extreme (like a right angle) or less extreme (like a normal demand curve) if each firm in the cartel produces a near-identical product like OPEC and petroleum? What if each firm produces a somewhat different product? Explain your reasoning.

How does a monopolistic competitor choose its profit-maximizing quantity of output and price?

How is the perceived demand curve for a monopolistically competitive firm different from the perceived demand curve for a monopoly or a perfectly competitive firm?

Mary and Raj are the only two growers who provide organically grown corn to a local grocery store. They know that if they cooperated and produced less corn, they could raise the price of the corn. If they work independently, they will each earn \(100\)dollar. If they decide to work together and both lower their output, they can each earn \(150\). If one person lowers output and the other does not, the person who lowers output will earn \(0\)dollar and the other person will capture the entire market and will earn \(200\)dollar Table 10.6 represents the choices available to Mary and Raj. What is the best choice for Raj if he is sure that Mary will cooperate? If Mary thinks Raj will cheat, what should Mary do and why? What is the prisoner's dilemma result? What is the preferred choice if they could ensure cooperation? \(A=\) Work independently; \(\mathrm{B}=\) Cooperate and Lower Output. (Each results entry lists Raj's earnings first, and Mary's earnings second.)
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