 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

 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.

 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.

 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.

 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.

 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[V k
(n)
] where the k denotes the order (highest, 2 nd highest, etc.) of the draw and (n) denotes the number of draws.

So we’re interested in E[V 2
(2)
]

 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[V 1
(1)
].
100
0
50

 Now suppose there are two draws.
 What are the first and second order statistics?
0 33 66
100

 With uniform distributions, the order statistics evenly divide the number line into n + 1 equal segments.
 Let’s try 3 draws:
3rd
2nd
1st
50
0
25
75 100

 So in general,
 E[V k
(n)
] = 100* (n – k + 1)/(n + 1)
 So revenues in a second price or English auction in this setting are:
 E[V 2
(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!

 Now suppose you have a value of $20 and are competing with one other bidder in a firstprice 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?

 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.

 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))

b(v)
B
I win b -1 (B)
I lose 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

 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.

 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.

 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.

 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

 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.

 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.

 How much does the seller make in this auction?
 Since the high bidder wins, the relevant order statistic is E[V 1
(2)
] = 66.
 But since each bidder only bids half his value, my revenues are

½ x E[V 1
(2)
] = 33
 Notice that these revenues are exactly the same as in the second price or English auctions.

 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.

 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?

 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(v
2
)) = B/100a
 But I also have to now beat bidders 3 through n.

 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

 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

 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.

 How much does the seller make in this auction?
 The relevant order statistic is E[V 1
(n)
] = 100* n/(n + 1)
 But eveyone shades by (n-1)/n so
 Revenues = (n-1)/n x E[V 1
(n)
]
 Revenues = 100 x (n-1)/(n+1)

 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?

 First-price:
 R = (n-1)/n x E[V 1
(n)
]
 R = 100 x (n-1)/(n+1)
 Second-price:
 R = E[V 2
(n)
]
 R = 100 x (n-1)/(n+1)
 The auctions still yield the same expected revenues.

 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!

 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.

 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.

 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.

 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.

 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?

P
Demand curve
Q

 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

P
Marginal Revenue
P*
MC
Q
Q*

 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

P
100
50
0
1 Q = Pr of sale
1/2

P
Q = 1 – F(p)
100
50
0
1 Q = Pr of sale
1/2

 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)

 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

 As usual the optimal quantity is where MR =
MC
 But MC is zero in this case
 So the optimal quantity is where MR = 0

P
Marginal Revenue
100
P*
0
1 Q = Pr of sale
Q*

 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

 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!

 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.

 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

 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