Q. The owner of a new restaurant is planning to advertise to attract customers. In the Bayesian game, Nature determines the restaurant's quality, which is either high or low. Assume that each quality occurs with equal probability. After the owner learns about quality, he decides how much to advertise. Let A denote the amount of advertising expenditure. For simplicity, assume that there is a single consumer. The consumer observes how much advertising is conducted, updates her beliefs about the quality of the restaurant, and then decides whether or not to go to the restaurant. (One can imagine that A is observed by noticing how many commercial spots are on local television and how many ads are in the newspaper and on billboards.) Assume that the price of a meal is fixed at $50. The value of a high-quality meal to a consumer is $85 and of a low-quality meal is $30. A consumer who goes to the restaurant and finds out that the food is of low quality ends up with a payoff of 20 which is the value of a low-quality meal, 30, less the price paid, 50. If the food is of high quality, then the consumer receives a value of 35 Furthermore, upon learning of the high quality, a consumer anticipates going to the restaurant a second time. Thus, the payoff to a consumer from visiting a high-quality restaurant is actually 70
For the restaurant owner, assume that the cost of providing a meal is 35 whether it is of low or high quality. If the restaurant is of high quality, the consumer goes to the restaurant, and the restaurant spends A in advertising, then its profit (and payoff) is If the restaurant is of low quality, the consumer goes to the restaurant, and the restaurant spends A in advertising, then its profit is These payoffs are summarized in the following table. If the consumer does not go to the restaurant, then her payoff is zero and the owner's payoff is A.
a. Find a separating perfect Bayes-Nash equilibrium.
b. At a separating perfect Bayes-Nash equilibrium, what is the maximum amount of advertising that a restaurant conducts? What is the minimum amount?