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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