Question

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 data-custom-editor="chemistry" $15$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? A = Confess; B = Stay Silent. (Each results entry lists Jane’s sentence first (in years), and Bill's sentence second.)

Short answer

Game theory is a mathematical method for distinct players compete with one other to make decisions.

Step-by-step solution

Definition

Dominant strategy refers to a situation in which one player picks a specific strategy regardless of what the other player decides.

Explanation

• Jane will receive a $15$-year sentence if she confesses and a $20$-year term if she keeps silent if she trusts Bill to remain silent, thus she will choose to confess because it is the best alternative open to her given Bill's choice.
• If Jane believes Bill will confess, she will face a $30$-year sentence if she confesses and a $35$-year sentence if she remains silent; therefore, Jane will choose to confess because it is the best alternative open to her given Bill's choice
• If Bill chooses to confess, Jane's best option is to confess and receive a $30$-year sentence; if Bill chooses to remain silent, Jane's best option is to confess and receive a $15$-year sentence. This means that no matter what Bill chooses, Jane's best strategy is to confess, so Jane has the dominant strategy of confessing.
• If Jane chooses to confess, the best option for Bill is to confess and receive a $30$-year sentence; if Jane chooses to remain silent, the best option for Bill is to confess and receive a $15$-year sentence, indicating that no matter what Jane chooses, the best strategy for Bill is to confess, indicating that Bill has the dominant strategy of confessing.