Imagine that a zealous prosecutor (P) has accused a defendant (D) of committing a crime. Suppose that the trial involves evidence production by both parties and that by producing evidence, a litigant increases the probability of winning the trial.
Specifically, suppose that the probability that the defendant wins is given by eD>(eD + eP), where eD is the expenditure on evidence production by the defendant and eP is the expenditure on evidence production by the prosecutor. Assume that eD and eP are greater than or equal to 0. The defendant must pay 8 if he is found guilty, whereas he pays 0 if he is found innocent. The prosecutor receives 8 if she wins and 0 if she loses the case.
(a) Represent this game in normal form.
(b) Write the first-order condition and derive the best-response function for each player.
(c) Find the Nash equilibrium of this game. What is the probability that the defendant wins in equilibrium.
(d) Is this outcome efficient? Why?