Consider the following game. There is a Ski Resort who wants to sell a season pass to a Skier for price p. The Skier values the pass at v. But the Ski Resort does not observe v. The Ski Resort only knows that v is distributed N(mu, sigma^2). The Skier observes their own value of v and the price p. The skier chooses to Buy if v > p and Not Buy if v < p. Assume that there is no cost to the Ski Resort of having the Skier buy the pass. Write down the game, including players, strategies, payoffs and beliefs. Hint: the Skier's strategy is a function of their value v, which is not observed by the Ski Resort.
Let mu = 2 and sigma = 2. Find the profit maximizing price for a firm that can only charge one price to all skiers.
Now allow the Ski Resort to offer two types of ski passes. The passes can have different prices. One price must allow access to the Ski Resort every day, while the other is allowed to have restricted access to the resort. Hint we can have a paramater alpha that discounts v for the second pass. That is if alpha = 0.5, the second pass is valued at half the first pass prior if they were both offered for free. Determine the optimal prices for the two passes and the size of alpha for the discounted pass. Remember that you need to make sure that people who prefer the undiscounted pass don't want to buy the discounted pass.
The next question is based on the Netflix documentary Made You Look. There is an episode of Reveal on the scandal: https://revealnews.org/podcast/fancy-galleries-fake-art-update-2023/. There is also a podcast https://www.iheart.com/podcast/1119-art-fraud-91404660/ as well as other articles about the Knoedler Gallery fakes of American modern masters such as Mark Rothko.
Consider a model of an art market with a Seller, a Buyer and an Expert. The Seller knows whether their painting is Fake or Real, and values the real painting at V. The Buyer values the real painting at V + delta. Everyone values the fake at 0. The Buyer pays the Expert a commission rate (k) to authenticate the painting. That is, the Expert earns k p if the Seller Accepts the Buyer's offer to pay p for the painting. The Expert chooses how much effort (e) to put into the authentication, where e = Pr(DetermimneReal | ActuallyReal) = Pr(DetermineFake | ActuallyFake), this is the accuracy of the authentication, and the cost of authentication is c/(1 - e). The Buyer does not observe the effort of the Expert (e). Everyone knows that prior to authentication, the probability that the painting is Fake is q. Let p denote the amount of money the Buyer will offer to the Seller for the painting. The Seller can Accept or Reject the offer. Let q_new denote the Buyer's belief about the probability the painting is Fake given the Seller Accepts the offer.
a. Initially consider the case without the Expert. Write down the game.
b. Assume that q and delta are such that the following inequality holds: (1 - q) delta - q V < 0. What is the Bayes Nash equilibrium of the initial game?
c. Consider the following model of an expert Art Gallery acting as an intermediary in the art world. What is the optimal commission rate (k)?
V <- 10
c <- 0.00001
q <- 0.01
delta <- 0.1
U <- function(k, e) { q*(1 - e)*k*V + (1 - q)*e*k*V - c/(1 - e) }
e_star <- function(k) { optimize(U, c(0, 1), k=k, maximum=TRUE)$maximum }
q_e <- function(e) { q*(1 - e)/(q*(1 - e) + (1 - q)*e) }
B <- function(k) { (1 - q_e(e_star(k)))*(V + delta) - k*V - V }