1. Econ 805 Advanced Micro Theory 1
Dan Quint
Fall 2009
Lecture 2

2. Today
Common auction formats
The Independent Private Values model

3. Common Auction Formats and Strategic Equivalences

4. Dutch Auction
Auctioneer begins at a high price and lowers it until a buyer claims the object at the current price
A slightly abstracted view: the price falls continuously (on a clock) instead of in increments
In a literal sense, a bidder’s strategy can be thought of as a choice of whether or not to buy at each price; but for practical purposes, it can be reduced to a decision of at what price to shout “mine!” if the item’s still available

5. “Sealed Tender” or “First-Price” Auction
Each bidder submits a sealed bid
The object goes to the bidder with the highest bid, at that price
A strategy is simply a choice of how much to bid

6. Dutch Auction = First-Price Auction
If all the matters is who wins the object and how much they pay, then the Dutch Auction and the First-Price Auction are equivalent
In each, a bidder’s strategy is reduced to picking a number; the highest number wins, and pays that much

7. English (Ascending) Auction
Think of art auctions
Price begins low; auctioneer solicits bids at the next price, keeps naming higher prices until no one is willing to raise their bid
Or, bidders name their own prices until no one is willing to outbid the high bidder – think of online auctions without proxy bidding
High bidder pays what he bid

8. Simplified English Auction (or Button Auction)
Price begins low, rises continuously
At each price, bidders can remain active (hold down a button) or drop out permanently
Bidders only know the current price (not who has dropped out and at what price)
When the second-to-last bidder drops out, the last man standing pays the current price
A bidder’s strategy can be reduced to choosing a price at which to drop out if he hasn’t won

9. Second-Price (or Vickrey) Auction
Each bidder submits a sealed bid
The object goes to the highest bidder, but the price they pay is the second-highest bid

10. Simplified English = Second-Price
Again, if we reduce the game to the question of who wins and how much they pay, the Simplified English Auction and Second Price Auction are equivalent
Strategies are reduced to picking a number
Highest number wins; payment is second-highest number
But the Simplified English Auction changes if bidders can see who is still active at each price
If I’m unsure of the exact value of the object, I may revise my estimate depending on how other bidders bid
Then strategies can no longer be reduced to picking a single number, and the equivalence breaks down

11. All-Pay Auctions and Wars of Attrition
In an All-Pay Auction, bidders submit sealed bids, the high bid wins the object, but everyone pays what they bid
All-Pay Auctions are sometimes used to model lobbying, attempts to buy political influence, and patent races – the losers already made their contributions or incurred their costs
War of Attrition is the same, but dynamic – like an all-pay button auction where bidders can see who’s still active
Great game for an undergrad game theory class – auction off a $20 bill, highest bid wins, highest two bids both pay what they bid

12. Multi-Unit Auctions with Unit Demand
Suppose there are k > 1 identical items for sale, but each bidder can only have one
“Pay-as-bid” auction is like a first-price auction – the k highest bidders win and pay their bids
Analog to the second-price auction is the “k+1st-price” auction
Button auction works similarly, ends when the k+1st bidder left drops out

13. The Independent Private Values Model

14. Baseline model of an auction as a Bayesian Game: Symmetric Independent Private Values
N > 1 bidders in an auction for a single object
Nature moves first, assigning each bidder a private valuation vi for the object
Each bidder’s value vi is an independent draw from a common probability distribution F
Each bidder knows his own value vi but not that of his opponents
F is common knowledge
Bidder i’s payoff is vi – p if he wins, 0 if he loses, where p is the price he pays for the object
Like the Cournot game, i’s payoff depends on j’s type only through j’s action – this is what’s meant by “private values”

15. Note all the implicit assumptions we’re making
The number of bidders is fixed – there is no decision over whether or not to participate
Each bidder knows his own valuation perfectly, does not care what the other bidders think of the object
The bidders are symmetric ex-ante – valuations are drawn from the same distribution, which is common knowledge
Valuations are statistically independent
Bidders are risk-neutral

16. Auctions to sell versus auctions to buy
Suppose the government holds an auction for a contract to provide some service
Bids are now offers to provide the service at a given price, and the lowest bid wins
Where buyers were distinguished by their valuations for winning their object, firms can be thought of as distinguished by their cost of providing the service
So firm i’s payoffs would be p – ci, where p is the price received, and all the same analysis goes through

17. Solving for Equilibrium in the First- and Second-Price Auctions

18. Second-price (Vickrey) auctions in the IPV world
Claim. In a second-price sealed-bid auction, submitting a bid equal to your value is a weakly dominant strategy
Proof. Let B be the highest of your opponents’ bids.
When B > v, you could only win the object at price B, for a payoff of v – B < 0; bidding b = v gives you 0, which is as good as you can do
When B < v, any bid b > B gives the same payoff, v – B > 0, which is payoff from bidding b = v and the best you can do
When B = v, any bid gives the same payoff, 0
Corollary. Every bidder playing the strategy bi(vi) = vi is a Bayesian Nash Equilibrium of the second-price auction

19. Similarly…
In a button auction, it’s a dominant strategy to drop out when the price reaches your private value vi
Doesn’t matter if you can observe who’s already dropped out or not
In an open-outcry ascending auction…
Equilibrium strategies are not clear
But it is a dominant strategy to never bid above your private value vi, nor to let the auction end at price below vi – d (where d is the minimum bid increment)
So any equilibrium will involve the highest-value bidder winning (unless the highest two are within d of each other), and paying within d of the second-highest value
So with private values, as d gets small, second-price or button auctions give approximately the same outcome as ascending auctions
Also similar is a first-price auction with proxy bidding, a la eBay
Bidders can name a maximum, then the computer raises their bid to the minimum required to win until that maximum is reached

20. Sadly, “everyone bids their value” is not the only equilibrium of the second-price auction
Suppose bidder values were drawn from a distribution with support [0,10]
The following is an equilibrium of the second-price auction:
Bidder 1 bids 15 regardless of his type
All other bidders bid 0 regardless of their type
bi(vi)=vi is “nearly” the only symmetric equilibrium; and it involves bidders playing a strict best-response at nearly every type; and it’s the equilibrium we’ll focus on

21. First-price auctions in the symmetric IPV world
We’ll look for “nice” equilibria:
Symmetric (bidders all play the same strategy)
Bids are increasing in valuations
Tomorrow, we’ll learn a trick that makes finding this type of equilibrium much easier
Suppose such an equilibrium exists, and let b : [0,V]  R+ be the common bid function; then at a given type v, b(v) must be a solution to
max x Î R+ (v – x) Pr(win | bid x, opponents bid b(-))
= max x Î R+ (v – x) Pr(b(vj) < x " j ¹ i)
= max x Î R+ (v – x) Pr(vj < b-1(x) " j ¹ i)
= max x Î R+ (v – x) FN-1(b-1(x))

22. If there is a symmetric, increasing equilibrium in a first-price auction…
b(v) must solve
First-order condition
(b-1)’ = 1/b’ x = b(v) in equilibrium
so integrating from 0 to v,

23. So if there is a “nice” equilibrium, it must be b(v) = ò0v sd(FN-1(s)) / FN-1(v)
What is this?
Well, if a random variable y has cumulative distribution G with positive support, then
ò0v s dG(s) / G(v) = E(y | y < v)
And FN-1(v) is the cumulative distribution function of the highest of N-1 independent draws from F
So if we let v1 and v2 refer to the highest and second-highest valuations in a symmetric IPV model, then
b(v) = E(v2 | v1 = v)

24. Now here’s where it gets cool…
In the symmetric equilibrium of the second-price auction, the price paid is v2, so the seller’s expected revenue is simply E(v2)
In the symmetric, increasing equilibrium in the first-price auction (if it exists),
The bidder with the highest value wins
If the highest value is v, the winner pays E(v2 | v1 = v)
So the seller’s expected revenue is
E v1 E(v2 | v1) = E(v2)
So the seller’s expected revenue is the same in both auctions!

25. And similarly… In the first-price auction…
A bidder with type v expects to win with probability FN-1(v), and to pay b(v) = E(v2 | v1 = v) when he wins
So his expected payoff is FN-1(v) [ v – E(v2 | v1 = v) ]
In the second-price auction…
A bidder with type v expects to win whenever he has the highest value (v1 = v), and to pay v2 when he wins
So his expected payment, conditional on winning, is E(v2 | v1 = v)
And so his expected payoff is FN-1(v) [ v – E(v2 | v1 = v) ]
So each type of bidder gets the same expected payoff in the two auctions

26. This turns out not to be a fluke
This is exactly what we’ll prove more generally next class:
With independent private values, any two auctions in which, in equilibrium,
the player with the highest value wins the object, and
any player with the lowest possible type gets expected payoff of 0
will give the same expected payoff to each type of each player, and the same expected revenue to the seller
So, (a) this is pretty interesting, and (b) once we’ve proven this, we can use it to calculate equilibrium strategies much more easily

27. In Case We Have Time, A Few Slides on Second-Order Stochastic Dominance

28. When is one probability distribution less risky than another?
Two random variables X and Y with the same mean, with distributions F and G
Three conditions to consider:
1. “Every risk-averse utility maximizer prefers X to Y”, i.e., E u(X) ³ E u(Y) for every nondecreasing, concave u, or ò-¥¥ u(s) dF(s) ³ ò-¥¥ u(s) dG(s) (also called SOSD)
2. “Y is a mean-preserving spread of X”, or “Y = X + noise”: $ r.v. Z s.t. Y =d X + Z, with E(Z|X) = 0 for every value of X
3. For every x, ò-¥x F(s) ds £ ò-¥x G(s) ds
Rothschild-Stiglitz (1970): 1 « 2 « 3

29. What does this tell us?
Risk-averse buyers greatly impact auction design – changes equilibrium strategies – we’ll get to that in a few lectures (Maskin and Riley)
Risk-averse sellers have less impact – equilibrium strategies are the same, all that changes is seller’s valuation of different distributions of revenue
Claim. With symmetric IPV, a risk-averse seller prefers a first-price to a second-price auction

30. Proof: we’ll show revenue in second-price auction is MPS of revenue in first-price
Recall that revenue in a second-price auction is v2, and revenue in a first-price auction is E(v2 | v1)
Let X, Y, and Z be random variables derived from bidders’ valuations, as follows:
X = g(v1)
Z = v2 – g(v1)
Y = v2
where g(t) = ò0t s dFN-1(s) / FN-1(t) = E(v2 | v1 = t)
Note that Y = X + Z, and E(Z | X=g(t)) = E(v2 | v1 = t) – E(v2 | v1 = t) = 0
So Y is a mean-preserving spread of X, so any risk-averse utility maximizer prefers X to Y
But X is the revenue in the first-price auction, and Y is the revenue in the second-price auction – Q.E.D.

31. A cool proof SOSD º “ò-¥x F(s) ds £ ò-¥x G(s) ds everywhere”
We’ll use the “extremal method” or “basis function method”
We’ll rewrite our generic (increasing concave) function u(s) as a positive sum of basis functions
u(s) = ò-¥¥ w(q) h(s,q) dq
with w(q) ³ 0, where these basis functions are themselves increasing and concave
Then we’ll show that X SOSD Y if and only if
ò-¥¥ h(x,q) dF(x) ³ ò-¥¥ h(y,q) dG(y)
for all the basis functions
(“Only if” is trivial, since h(s,q) is increasing and concave; “if” just involves multiplying this inequality by w(q) and integrating over q)

32. A cool proof SOSD º “ò-¥x F(s) ds £ ò-¥x G(s) ds everywhere”
We’ll do the special case of u twice differentiable. Our basis functions will be a constant, a linear term, and the functions h(x,q) = min(x,q)
Claim is that u(x) = a + bx + ò0¥ (-u’’(q)) h(x,q) dq
Note that -u’’(q) is nonnegative, since u is concave
To see the equality, integrate by parts, with db = -u’’ dq, a = h: ò a db = a b – ò b da = –h(x,q)u’(q)|q=-¥¥ – ò-¥¥ –u’(q) 1q<x dq = –xu’(¥) + constant + ò-¥x u’(q) dq
Since X and Y have the same mean,
ò-¥¥ (a+bx) dF(x) = ò-¥¥ (a+by) dG(y)

33. A cool proof SOSD º “ò-¥x F(s) ds £ ò-¥x G(s) ds everywhere”
So all that’s left is to determine when
ò-¥¥ h(s,q) dF(s) ³ ò-¥¥ h(s,q) dG(s)
Integrate by parts: u = h(s,q), dv = dF(s), LHS becomes h(¥,q) F(¥) – h(-¥,q) F(-¥) – ò-¥¥ F(s) hs(s,q) ds = q – 0 – ò-¥¥ F(s) 1s<q ds = q – ò-¥ q F(s) ds
Similarly, the right-hand side becomes q – ò-¥ q G(s) ds
So Es~F h(s,q) ³ Es~G h(s,q) « ò-¥ q F(s) ds £ ò-¥ q G(s) ds
So X SOSD Y if and only if this holds for every q