2- Simple auctions II

Question 1

What is a first-price sealed-bid auction?

Answer 1

Bidders simultaneously submit a secret bid, whichever is highest wins paying their own bid

Question 2

Why is there no incentive to bid one’s valuation (or above) in a first-price auction?

Answer 2

Since you pay your own bid, any price at your valuation or above yields no utility

Question 3

What are the 3 main assumptions on bidders’ functions?

Answer 3

-Bidders’ strategies are symmetric: If two bidders have the same valuation, v, they bid the same s(v)
-Bidders’ strategies are strictly increasing in valuation
-For each bidder i, s(vi) ≤ vi

Question 4

What is the fundamental trade off for bidders in a first-price sealed auction?

Answer 4

Higher bids increase probability of winning but reduce payoff if they do

Question 5

What is the strategy of a bidder [s(v)]?

Answer 5

The strategy maps a valuation to a bid e.g. bid half my valuation s(v) = v/2

Question 6

What is a simple way of expressing a bidding strategy?

Answer 6

b = αv
where 0≤α≤1 scales down bidder’s valuation

Question 7

What is the optimal bid in first-price sealed with n i.i.d bidders with private valuations?

Answer 7

(n-1)v/n
i.e. second highest valuation, if we win our valuation v is the highest

Question 8

Outline how a Dutch/descending price auction works

Answer 8

Price clock showing a decreasing price, auction stops as soon as someone says ‘mine’

Question 9

Why are Dutch auctions strategically equivalent to first-price sealed-bid?

Answer 9

Because bidding one’s true valuation is not a dominant strategy

Question 10

What is the main difference between Dutch and first-price sealed-bid auctions?

Answer 10

First-price is a strategic form game, Dutch is an extensive form game

Question 11

What is the revenue equivalence theorem?

Answer 11

Expected seller revenue is the same for first-price & second-price sealed-bid auctions

Question 12

What are the 4 conditions for the revenue equivalence theorem?

Answer 12

-Private and i.i.d valuations
-Strictly increasing CDF F(v) = v/r
-Winner is bidder with highest valuation
-Bidder with lowest valuation has expected payoff 0

Question 13

What is a reserve price?

Answer 13

Only bids above the reserve price are considered

Question 14

What is a main pro and con of having a reserve price?

Answer 14

-Pro: guarantees a minimum revenue for seller if sold
-Con: if no bid meets the reserve price then the seller will make nothing

Question 15

What is the fundamental trade off when setting a reserve price?

Answer 15

Lower probability of selling but higher revenue if you do

Question 16

What is the optimal auction reserve price theorem?

Answer 16

If bidders’ valuations are drawn independently from the distribution F then a seller’s optimal auction consists of choosing and running a Vickrey auction with the reserve price r that satisfies ψ(r) = 0, where
ψ(r) = r - (1-F(r))/f(r)

Question 17

What is the intuition behind the optimal auction reserve price theorem?

Answer 17

Seller revenue is maximised when Marginal Revenue is zero

Question 18

What is the virtual valuation of a bidder ψ(v)?

Answer 18

The marginal revenue brought by the bidder

Question 19

What is the probability a bidder will win a first-price sealed-bid auction?

Answer 19

P(b>αv̅) = (b/αr)ⁿ⁻¹
Probability their bid is greater than the next highest valuation scaled down as per strategy

Question 20

What is the expected payoff of a bid EU(b)?

Answer 20

(b/αr)ⁿ⁻¹ x (v-b)
Probability of winning times payoff

Question 21

What is the expected revenue for the seller in first & second price sealed-bid auctions?

Answer 21

r(n-1/n+1)
Where r is max valuation

Question 22

What is the probability density function (PDF) of valuations f(v)?

Answer 22

f(v) = 1/r

Question 23

What is the cumulative distribution function (CDF) of valuations F(v)?

Answer 23

F(v) = v/r