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Auctions. Strategic Situation. You are bidding for an object in an auction. The object has a value to you of $20. How much should you bid? Depends on auction rules presumably. Review: Second Price Auctions. Suppose that the auction is a second-price auction High bidder wins

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strategic situation
Strategic Situation
  • You are bidding for an object in an auction.
  • The object has a value to you of $20.
  • How much should you bid?
    • Depends on auction rules presumably
review second price auctions
Review: Second Price Auctions
  • Suppose that the auction is a second-price auction
    • High bidder wins
    • Pays second highest bid
    • Sealed bids
  • We showed (using dominance) that the best strategy was to bid your value.
  • So bid $20 in this auction.
review english auctions
Review: English Auctions
  • An English (or open outcry) auction is one where bidders shout bids publicly.
  • Auction ends when there are no higher bids.
  • Implemented as a “button auction” in Japan
  • Implemented on eBay through proxy bidding.
what to bid
What to Bid
  • Again, suppose you value the object at $20.
  • Dominance says to drop out when bid = value.
  • The fact that bidding strategies are the same in the two auction forms means that they are strategically equivalent.
revenues
Revenues
  • How much does the seller earn on the auction?
  • Depends on the distribution of values.
  • Suppose that there are 2 bidders and values are equally likely to be from $0 to $100.
  • The seller earns an amount equal to the expected losing bid.
order statistics
Order Statistics
  • The seller is interested in the expected value of the lower of two draws from 0-100.
    • This is called the second order statistic of the distribution.
    • We will sometimes write this as E[Vk(n)] where the k denotes the order (highest, 2nd highest, etc.) of the draw and (n) denotes the number of draws.
    • So we’re interested in E[V2(2)]
order statistics of uniform distributions
Order Statistics of Uniform Distributions
  • There order statistics have simple regularity properties
  • The mean of a uniform draw from 0-100 is 50.
    • Note the mean could be written as E[V1(1)].

100

0

50

two draws
Two Draws
  • Now suppose there are two draws.
  • What are the first and second order statistics?

100

0

66

33

key observation
Key Observation
  • With uniform distributions, the order statistics evenly divide the number line into n + 1 equal segments.
  • Let’s try 3 draws:

1st

3rd

2nd

50

0

75

100

25

generalizing
Generalizing
  • So in general,
    • E[Vk(n)] = 100* (n – k + 1)/(n + 1)
  • So revenues in a second price or English auction in this setting are:
    • E[V2(n)] = 100 * (n – 1)/(n + 1)
  • As the number of bidders grows large, the seller’s revenues increase
  • As the number of bidders grows unbounded, the seller earns all the surplus, i.e. 100!
first price auctions
First Price Auctions
  • Now suppose you have a value of $20 and are competing with one other bidder in a first-price auction
  • You don’t know the exact valuation of the other bidder.
  • But you do know that it is randomly drawn from 0 to 100.
  • How should you bid?
setting up the problem
Setting Up the Problem
  • As usual, you want to bid to maximize your expected payoff
  • But now you need to make a projection about the strategy of the other bidder
  • Presumably this strategy depends on the particular valuation the bidder has.
  • Let b(v) be your projection for the bid of the other bidder when his valuation is v.
bidder s problem
Bidder’s Problem
  • Choose a bid, B, to maximize expected profits.
    • E[Profit] = (20 – B) x Pr(B is the highest bid)
  • What is Pr(B is the highest bid)?
    • It is Pr(B > b(v))
what is pr b b v
What is Pr(B > b(v))?

b(v)

B

v

I lose

I win

b-1(B)

conjectures about b v
Conjectures about b(v)
  • Suppose that I believe that my rival’s strategy is to bid a constant fraction of his value
    • Then b(v) = av
    • Where a is some fraction
  • I win whenever
    • B >= av
  • Or, equivalently
    • v <= B/a
  • So Pr(B > b(v)) becomes:
    • Pr( v <= B/a) = B/100a
bidder s problem revisited
Bidder’s Problem Revisited
  • So now I need to choose B to maximize
    • E[Profit] = (20 – B)(B/100a)
  • Optimize in the usual way:
    • (1/100a) x (20 – 2B) = 0
    • Or B = 10
  • So I should bid 10 when my value is 20.
other values
Other Values
  • Suppose my value is V?
    • E[Profit] = (V – B)(B/100a)
  • Optimize in the usual way:
    • (1/100a) x (V – 2B) = 0
    • Or B = V/2
  • So I should always bid half my value.
equilibrium
Equilibrium
  • My rival is doing the same calculation as me.
    • If he conjectures that I’m bidding ½ my value
    • He should bid ½ his value (for the same reasons)
  • Therefore, an equilibrium is where we each bid half our value.
uncertainty about my rival
Uncertainty about my Rival
  • This equilibrium we calculated is a slight variation on our usual equilibrium notion
  • Since I did not exactly know my rival’s payoffs in this game
    • I best responded to my expectation of his strategy
    • He did likewise
bayes nash equilibrium
Bayes-Nash Equilibrium
  • Mutual best responses in this setting are called Bayes-Nash Equilibrium.
    • The Bayes part comes from the fact that I’m using Bayes rule to figure out my expectation of his strategy.
comments
Comments
  • In this setting, dominant strategies were not enough
  • What to bid in a first-price auction depends on conjectures about how many rivals I have and how much they bid.
  • Rationality requirements are correspondingly stronger.
revenues1
Revenues
  • How much does the seller make in this auction?
    • Since the high bidder wins, the relevant order statistic is E[V1(2)] = 66.
    • But since each bidder only bids half his value, my revenues are
      • ½ x E[V1(2)] = 33
  • Notice that these revenues are exactly the same as in the second price or English auctions.
revenue equivalence
Revenue Equivalence
  • Two auction forms which yield the same expected revenues to the seller are said to be revenue equivalent
  • Operationally, this means that the seller’s choice of auction forms was irrelevant.
more rivals
More Rivals
  • Suppose that I am bidding against n – 1 others, all of whom have valuations equally likely to be 0 to 100.
  • Now what should I bid?
    • Should I shade my bid more or less or the same?
  • In the case of second-price and English auctions, it didn’t matter how many rivals I had, I always bid my value
  • What about in the first-price auction?
optimal bidding
Optimal Bidding
  • Again, I conjecture that the others are bidding a fraction a of their value.
    • E[Profit] = (V – B) x Pr(B is the high bid)
  • To be the high bid means that I have to beat bidder 2.
    • Pr( B >= b(v2)) = B/100a
  • But I also have to now beat bidders 3 through n.
probability of winning
Probability of Winning
  • So now my chance of winning is
    • B/100a x B/100a x …B/100a
      • For n – 1 times.
    • Or equivalently
      • Pr(B is the highest) = [B/100a]n-1
bidder 1 s optimization
Bidder 1’s optimization
  • Choose B to maximize expected profits
    • E[Profit] = (V – B) x Pr(B is highest)
    • E[Profit] = (V – B) x [B/100a]n-1
    • E[Profit] = (1/100a)n-1 x (V – B) x [B]n-1
  • Optimizing in the usual way:
    • (1/100a)n-1 x ((n-1)V – nB) [B]n-2 = 0
  • So the optimal bid is
    • B = V x (n-1)/n
equilibrium1
Equilibrium
  • I bid a proportion of my value
  • But that proportion is (n-1)/n
    • As I’m competing against more rivals, I shade my bid less.
  • Since all my rivals are making the same calculation, in equilibrium everyone bids a fraction (n-1)/n of their value.
revenues2
Revenues
  • How much does the seller make in this auction?
    • The relevant order statistic is E[V1(n)] = 100* n/(n + 1)
    • But eveyone shades by (n-1)/n so
    • Revenues = (n-1)/n x E[V1(n)]
    • Revenues = 100 x (n-1)/(n+1)
comments1
Comments
  • Revenues are increasing in the number of bidders
  • As that number grows arbitrarily large, the seller gets all the surplus, i.e. 100!
  • How does this compare to the English or Second-Price auction?
comparing revenues
Comparing Revenues
  • First-price:
    • R = (n-1)/n x E[V1(n)]
    • R = 100 x (n-1)/(n+1)
  • Second-price:
    • R = E[V2(n)]
    • R = 100 x (n-1)/(n+1)
  • The auctions still yield the same expected revenues.
revenue equivalence theorem
Revenue Equivalence Theorem
  • In fact, revenue equivalence holds quite generally
    • Consider any auction which:
      • Allocates the object to the highest bidder
      • Gives any bidder the option of paying zero
    • Then if bidders know their values
    • Values are uncorrelated
    • Values are drawn from the same distribution
  • Then all such auctions are revenue equivalent!
implications
Implications
  • This means that we can determine the revenues quickly and easily for all sorts of auctions
  • Consider an all-pay auction
      • Bidders submit cash payments to the seller (bribes)
      • The bidder submitting the highest bribe gets the object
      • The seller keeps all the bribe money
  • This auction auction yields the same revenues as an English auction.
other strange auction forms
Other Strange Auction forms
  • Suppose that all bidders submit bribes to the auctioneer
  • The object is awarded to the person paying the highest bribe
  • And the seller gives back the bribe of the winner, but keeps all the others
  • This is also revenue equivalent.
optimal auctions
Optimal Auctions
  • Revenue equivalence says that the form of the auction does not affect how much money the seller makes.
  • But there are other tools the seller has to make money.
one bidder auctions
One Bidder Auctions
  • Suppose that the seller is running an auction that attracts only one bidder.
  • What should he do?
  • If he goes with the usual auction forms, he’ll make nothing since the second highest valuation for the object is zero.
monopoly
Monopoly
  • Since the seller is a monopoly provider of the good, maybe some tricks from monopoly theory might help.
  • Suppose a monopolist faced a linear demand curve and could only charge a single price
  • What price should he charge?
monopoly problem
Monopoly Problem

P

Demand curve

Q

monopoly problem1
Monopoly Problem
  • The monopolist should choose p to maximize profits
    • Profits = P x Q(P) – C(Q(P))
  • Or equivalently, the monopolist could choose Q to maximize profits
    • Profits = P(Q) x Q – C(Q)
    • P(Q) is the inverse demand function
  • Optimizing in the usual way, we have:
    • MR = MC
monopoly problem2
Monopoly Problem

P

Marginal Revenue

P*

MC

Q

Q*

back to auctions
Back to Auctions
  • What is the demand curve faced by a seller in a one bidder auction?
    • One can think of the “quantity” as the probability of making a sale at a given price.
      • So if the seller asks for $100, he will make no sales.
      • If he asks for $0, he will sell with probability = 1
      • If he asks $50, he will sell with probability .5
auction monopoly problem
Auction/Monopoly Problem

P

100

50

0

1

Q = Pr of sale

1/2

auction monopoly problem1
Auction/Monopoly Problem

P

Q = 1 – F(p)

100

50

0

1

Q = Pr of sale

1/2

demand curve
Demand Curve
  • So the demand curve is just the probability of making a sale
    • Pr(V > P)
  • If we denote by F(p) the probability that V <=p, then
    • Q = 1 – F(p)
  • But we need the inverse demand curve to do the monopoly problem the usual way.
    • P = F-1(1 – Q)
auction monopoly problem2
Auction/Monopoly Problem
  • Now we’re in a position to do the optimization.
  • The seller should choose a reserve price to maximize his expected profits
    • E[Profits] = p x (1 – F)
  • Equivalently, the auctioneer chooses a quantity to maximize
    • E[Profits] = F-1(1 – Q) x Q
optimization
Optimization
  • As usual the optimal quantity is where MR = MC
    • But MC is zero in this case
    • So the optimal quantity is where MR = 0
auction monopoly problem3
Auction/Monopoly Problem

P

Marginal Revenue

100

P*

0

1

Q = Pr of sale

Q*

so what is marginal revenue
So what is Marginal Revenue?
  • Revenue = F-1(1 – Q) x Q
  • Marginal Revenue = F-1(1 – Q) – Q/f(F-1(1 – Q))
    • where f(p) is (approximately) the probability that v = p
  • Now substitute back:
    • P – (1 – F(p))/f(p) = 0
uniform case
Uniform Case
  • In the case where valuations are evenly distributed from 0 to 100
    • F(p) = p/100
    • f(p) = 1/100
  • So
    • P – (1 – P) = 0
  • Or
    • P = 50!
recipe for optimal auctions
Recipe for Optimal Auctions
  • The seller maximizes his revenue in an auction by:
    • Step 1: Choosing any auction form satisfying the revenue equivalence principle
    • Step 2: Placing a reserve price equal to the optimal reserve in a one bidder auction
  • Key point 1: The optimal reserve price is independent of the number of bidders.
  • Key point 2: The optimal reserve price is NEVER zero.
conclusions
Conclusions
  • Optimal bidding depends on the rules of the auction
    • In English and second price auctions, bid your value
    • In first-price auctions, shade your bid below your value
      • The amount to shade depends on the competition
        • More competition = less shading
more conclusions
More Conclusions
  • As an auctioneer, the rules of the auction do not affect revenues much
  • However reserve prices do matter
  • The optimal reserve solves the monopoly problem for a one bidder auction
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