In the first part of the problem set, we will delve more deeply into auction theory, which Sara introduced in lecture. We will demonstrate some auction theory properties by performing simulations of data. In these simulations, we will compare different schemes for auctions by varying the number of bidders and valuations. At an auction, bidders make offers to buy the goods, and a bidder's valuation is how much the bidder offers to pay for the good.

To start, try to understand the following R code. Run the code to test your understanding. We will start with the assumption that there are 2 bidders. We will simulate the auction 1000 times, resulting in 1000 valuations for these 2 bidders. Imagine that you are the person trying to sell a particular good, and that you are using R to figure out the perfect pricing and allocation scheme.

#Preliminaries rm(list = ls()) setwd("---------------------") #Uniform Valuations number_of_bidders <- 2 number_of_simulations <- 1000 set.seed(1) valuations1 <- matrix(runif( number_of_bidders*number_of_simulations, min=0, max=1), nrow = number_of_simulations)

QUESTION NO.1

Try to figure out what the set.seed() command is doing, and then answer the following true or false question:

True or False: Even though we are simulating random numbers, the use of the set.seed() function allows us to have the same valuations each time we re-run this code.

QUESTION NO. 2

Let us consider the posted price model. In lecture, we saw that the expected revenue when there are posted prices is given by: pPr(Vi>_p)for at least one i,where p is the posted price, and vi is individual 's valuation of the good. Then, the expected revenue is equal to: p(1-Pr(vN

What is the optimal price in the case of two bidders, and a U[0,1] distributions for valuations?

QUESTION NO.3

Using the same scenario from Question 2, what is the expected revenue for the seller?

QUESTION NO. 4

Now, we will use the R code above to test whether these predictions hold.

First, let's find the maximum valuation among all 1000 simulations among the two bidders. Name the function in R that allows you to get this value. The function you use should return the maximum valuation when you run funcName(valuations1).

Please enter only the function name (what you typed for funcName -- no arguments or parentheses!)

QUESTION NO. 5

Take a look at the following R code that calculates the analytic solution to the expected revenue, and compare it with the one coming from the simulation.

#Uniform Valuations number_of_bidders <- 2 N <- number_of_bidders V <- 10000

set.seed(5) valuations <- matrix(runif( N*V, min = 0, max = 1), nrow = V)

maximum_valuation <- apply(valuations, 1, max) optimal_price <- 1/((N+1)^(1/N)) expected_revenue <- (N/(N+1)) * 1/((N+1)^(1/N))

revenue <- optimal_price*(maximum_valuation >= optimal_price) mean(revenue) expected_revenue

What variable captures the number of simulations we are using in the code?

Please enter ONLY the name of the variable without any additional text. Make sure that the capitalization matches the code!

QUESTION NO. 6

Now, perform this exercise for different numbers of simulations: 10, 100, 1000, and 10000. As you increase the number of simulations, does the mean of the numeric revenue vector coincide more or less with the analytic solution?

QUESTION NO. 7

Now, we will compare the results we just computed, which hold for the posted price model, with the results we would get from an auction . Let's consider an English auction, where the buyers' optimal strategy is to stay in the bidding until p=Vi and then leave once p>Vi

. As shown in lecture, the equilibrium price in this case is the second highest valuation.

What is the expected revenue when there are two bidders [N=2] Again, assume that bidders' valuations follow a uniform [0,1] distribution

QUESTION NO. 8

What is the minimum number of bidders such that a buyer prefers to sell the good in an English Auction rather than at a posted price auction?

Note: You can do this in two different ways: one is to solve the question mathematically (difficult!), and the other one is to use the simulation in R to answer the question. To use the simulation in R, you will need to write code that computes the expected revenue in an English Auction and the expected revenue in a posted price auction given some number of bidders. You can then compare the two expected revenues for different numbers of bidders.