Consider the following simplified view of a jury weighing evidence (treat the jury as a single decision maker). A defendant in a court case is accused of committing a crime. Independent of guilt (G) or innocence (I), with probability 3/4, the defendant possesses evidence. This evidence is in the set {d1, d2, d3}. When he possesses evidence, the probability of which document it is depends on whether he is guilty or innocent. We denote the probabilities with which di is realized, conditional on I or G, respectively, (and conditional on evidence being realized) by pIi and pGi. Suppose these are given by pI3 = 3/8, pI2 = 3/8, pI1 = 1/4, pG3 = 1/8, pG2 = 1/8, and pG1 = 3/4.
Assume the prior belief that the defendant is innocent is 3/8. Suppose that the jury believes that both guilty and innocent types of the defendant do not disclose d1, but both do disclose d2 and d3. What is the jury’s posterior belief that the defendant is innocent when it observes no document disclosed?
A particular model of used car comes in four categories: perfect, good, bad, and hopeless. There are equal numbers of cars in each category. The market consists of at least two potential buyers and sellers of the car. Assume that there are equal numbers of buyers and sellers. All agents are interested in maximizing the expected surplus they obtain from having a car net of any payments they give or receive. They obtain surplus of zero if they do not own a car. The table below gives both the buyer and the seller’s valuation of each type of car.
Category Buyer Seller
Perfect 48 46
Good 44 42
Bad 40 38
Hopeless 36 34
So, for example, if a buyer pays the price 44 for a perfect car, he obtains
net surplus of 48 − 44 = 4. If a seller receives the price 48 for her
hopeless car, then she obtains surplus of 48 − 34 = 14.
The game works as follows: Simultaneously each buyer announces a price that he is willing to pay for a car. Each seller decides whether or not to sell her car (and to whom to sell it). If more than one seller accepts the price of the same buyer, then one of the sellers is randomly selected to sell the car.
Find the equilibrium price (or prices) assuming that the sellers know the value of the car that they own, and buyers do not. Describe which cars are sold.