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 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

In conclusion, if Jane trusts Bill to stay silent, she should confess to minimize her own sentence (15 years instead of 20 years). If she thinks Bill will confess, she should also confess to minimize her sentence (30 years instead of 35 years). Both Jane and Bill have a dominant strategy, which is to confess.

Step-by-step solution

Create a payoff matrix

: Create a 2x2 payoff matrix to represent the outcomes based on the choices available to Jane and Bill. Label the rows for Jane's choices (Confess or Stay Silent) and the columns for Bill's choices (Confess or Stay Silent). Fill in the matrix with the respective outcome based on the problem statement. $$ \begin{array}{c|c|c} & \textbf{Confess} & \textbf{Stay Silent} \\ \hline \textbf{Confess} & (30, 30) & (15, 35) \\ \hline \textbf{Stay Silent} & (35, 15) & (20, 20) \\ \end{array} $$

Determine Jane's decision if she trusts Bill to stay silent

: If Jane trusts that Bill will stay silent, we should examine the outcomes of her choice in the column of Bill's "Stay Silent" decision. Compare the two possible outcomes for Jane: (15,35) if she confesses and (20,20) if she stays silent: - If Jane confesses, her sentence will be 15 years, and Bill's will be 35 years. - If Jane stays silent, her sentence will be 20 years, and Bill's will be 20 years. Therefore, if Jane trusts Bill to stay silent, she should confess to minimize her own sentence (15 years sentence instead of 20 years).

Determine Jane's decision if she thinks Bill will confess

: Now, if Jane believes that Bill will confess, we should examine the outcomes of her choice in the "Confess" column. Compare the two possible outcomes for Jane: (30,30) if she confesses and (35,15) if she stays silent: - If Jane confesses, her sentence will be 30 years, and Bill's will be 30 years. - If Jane stays silent, her sentence will be 35 years, and Bill's will be 15 years. In this situation, if Jane thinks Bill will confess, she should confess to minimize her sentence (30 years sentence instead of 35 years).

Determine if either player has a dominant strategy

: A dominant strategy is a strategy that provides the best outcome for a player, regardless of what the other player chooses. To determine this, we must compare the potential outcomes of each strategy. - For Jane, confessing has better outcomes in both cases where Bill confesses (30 years vs. 35 years if stay silent) and when Bill stays silent (15 years vs. 20 years if stay silent). Therefore, Jane's dominant strategy is to confess. - Similarly, for Bill, confessing has better outcomes in both cases where Jane confesses (30 years vs. 35 years if stay silent) and when Jane stays silent (15 years vs. 20 years if stay silent). Thus, Bill's dominant strategy is also to confess. In conclusion, Jane should confess whether she trusts Bill to stay silent or if she thinks he will confess to minimize her sentence. Additionally, both Jane and Bill have a dominant strategy, which is to confess.