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Auction Theory
Jonathan Levin
October 2004
Our next topic is auctions. Our objective will be to cover a few of the main ideas and highlights. Auction theory can be approached from different angles – from the perspective of game theory (auctions are bayesian games of incomplete information), contract or mechanism design theory (auctions are allocation mechanisms), market microstructure (auctions are models of price formation), as well as in the context of different applications (procure- ment, patent licensing, public finance, etc.). We’re going to take a relatively game-theoretic approach, but some of this richness should be evident.
1 The Independent Private Value (IPV) Model
1.1 A Model
The basic auction environment consists of:
- Bidders i = 1, ..., n
- One object to be sold
- Bidder i observes a “signal” Si ∼ F (·), with typical realization si ∈ [s, s], and assume F is continuous.
- Bidders’ signals S 1 , ..., Sn are independent.
- Bidder i’s value vi(si) = si.
Given this basic set-up, specifying a set of auction rules will give rise to a game between the bidders. Before going on, observe two features of the model that turn out to be important. First, bidder i’s information (her signal) is independent of bidder j’s information. Second, bidder i’s value is independent of bidder j’s information – so bidder j’s information is private in the sense that it doesn’t affect anyone else’s valuation.
1.2 Vickrey (Second-Price) Auction
In a Vickrey, or second price, auction, bidders are asked to submit sealed bids b 1 , ..., bn. The bidder who submits the highest bid is awarded the object, and pays the amount of the second highest bid.
Proposition 1 In a second price auction, it is a weakly dominant strategy to bid one’s value, bi(si) = si.
Proof. Suppose i’s value is si, and she considers bidding bi > si. Let ˆb denote the highest bid of the other bidders j 6 = i (from i’s perspective this is
a random variable). There are three possible outcomes from i’s perspective: (i) ˆb > bi, si; (ii) bi > ˆb > si; or (iii) bi, si > ˆb. In the event of the first or third outcome, i would have done equally well to bid si rather than bi > si. In (i) she won’t win regardless, and in (ii) she will win, and will pay ˆb regardless. However, in case (ii), i will win and pay more than her value if she bids ˆb, something that won’t happen if she bids si. Thus, i does better to bid si than bi > si. A similar argument shows that i also does better to bid si than to bid bi < si. Q.E.D.
Since each bidder will bid their value, the seller’s revenue (the amount paid in equilibrium) will be equal to the second highest value. Let Si:n denote the ith highest of n draws from distribution F (so Si:n^ is a random variable with typical realization si:n). Then the seller’s expected revenue is E
S2:n
The truthful equilibrium described in Proposition 1 is the unique sym- metric Bayesian Nash equilibrium of the second price auction. There are also asymmetric equilibria that involve players using weakly dominated strate- gies. One such equilibrium is for some player i to bid bi(si) = v and all the other players to bid bj (sj ) = 0. While Vickrey auctions are not used very often in practice, open as- cending (or English) auctions are used frequently. One way to model such auctions is to assume that the price rises continuously from zero and players each can push a button to “drop out” of the bidding. In an independent pri- vate values setting, the Nash equilibria of the English auction are the same as the Nash equilibria of the Vickrey auction. In particular, the unique sym- metric equilibrium (or unique sequential equilibrium) of the English auction has each bidder drop out when the price reaches his value. In equilibrium, the auction ends when the bidder with the second-highest value drops out, so the winner pays an amount equal to the second highest value.
It is easy to check that b(s) is increasing and differentiable. So any symmetric equilibrium with these properties must involve bidders using the strategy b(s).
B. The “Envelope Theorem” Approach
A closely related, and often convenient, approach to identify necessary conditions for a symmetric equilibrium is to exploit the envelope theorem. To this end, suppose b(s) is a symmetric equilibrium in increasing dif- ferentiable strategies. Then i’s equilibrium payoff given signal si is
U (si) = (si − b(si)) F n−^1 (si). (1)
Alternatively, because i is playing a best-response in equilibrium:
U (si) = max bi (si − bi) F n−^1 (b−^1 (bi)).
Applying the envelope theorem (Milgrom and Segal, 2002), we have:
d ds
U (s)
s=si
= F n−^1 (b−^1 (b(si)) = F n−^1 (si)
and also,
U (si) = U (s) +
Z (^) si
s
F n−^1 (˜s)ds˜. (2)
As b(s) is increasing, a bidder with signal s will never win the auction – therefore, U (s) = 0. Combining (1) and (2), we solve for the equilibrium strategy (again drop- ping the i subscript):
b(s) = s −
R (^) si s F^
n− (^1) (˜s)ds˜ F n−^1 (s)
Again, we have showed necessary conditions for an equilibrium (i.e. any increasing differentiable symmetric equilibrium must involve the strategy b(s)). To check sufficiency (that b(s) actually is an equilibrium), we can exploit the fact that b(s) is increasing and satisfies the envelope formula to show that it must be a selection from i’s best response given the other bidder’s use the strategy b(s). (For details, see Milgrom 2004, Theorems 4. amd 4.6).
Remark 1 In most auction models, both the first order conditions and the envelope approach can be used to characterize an equilibrium. The trick is to figure out which is more convenient.
What is the revenue from the first price auction? It is the expected winning bid, or the expected bid of the bidder with the highest signal, E
b(S1:n)
. To sharpen this, define G(s) = F n−^1 (s). Then G is the proba- bility that if you take n − 1 draws from F , all will be below s (i.e. it is the cdf of S1:n−^1 ). Then,
b(s) = s−
R (^) s s F^
n− (^1) (˜s)ds˜
F n−^1 (s)
F n−^1 (s)
Z (^) s
s
sdF ˜ n−^1 (˜s) = E
S1:n−^1 |S1:n−^1 ≤ s
That is, if a bidder has signal s, he sets his bid equal to the expectation of the highest of the other n − 1 values, conditional on all those values being less than his own. Using this fact, the expected revenue is:
E
b
S1:n
= E
S1:n−^1 |S1:n−^1 ≤ S1:n
= E
S2:n
equal to the expectation of the second highest value. We have shown:
Proposition 2 The first and second price auction yield the same revenue in expectation.
1.4 Revenue Equivalence
The result above is a special case of the celebrated “revenue equivalence the- orem” due to Vickrey (1961), Myerson (1981), Riley and Samuelson (1981) and Harris and Raviv (1981).
Theorem 1 (Revenue Equivalence) Suppose n bidders have values s 1 , ..., sn identically and independently distributed with cdf F (·). Then all auction mechanisms that (i) always award the object to the bidder with highest value in equilibrium, and (ii) give a bidder with valuation s zero profits, generates the same revenue in expectation.
Proof. We consider the general class of auctions where bidders submit bids b 1 , ..., bn. An auction rule specifies for all i,
xi : B 1 × ... × Bn → [0, 1] ti : B 1 × ... × Bn → R,
a constant. Q.E.D.
The revenue equivalence theorem has many applications. One useful trick is that it allows us to solve for the equilibrium of different auctions, so long as we know that the auction will satisfiy (i) and (ii). Here’s an example.
Application: The All-Pay Auction. Consider the same set-up – bidders 1 , .., n, with values s 1 , ..., sn, identically and independently distributed with cdf F – and consider the following rules. Bidders submit bids b 1 , ..., bn and the bidder who submits the highest bid gets the object. However, bidders must pay their bid regardless of whether they win the auction. (These rules might seem a little strange – the all-pay auction is commonly used as a model of lobbying or political influence). Suppose this auction has a symmetric equilibrium with an increasing strategy bA(s) used by all players. Then, bidder i’s expected payoff given value si will be (if everyone plays the equilibrium strategies):
U (si) = siF n−^1 (si) − bA(si) =
Z (^) si
s
F n−^1 (˜s)ds˜
So
bA(s) = sF n−^1 (s) −
Z (^) s
s
F n−^1 (˜s)d˜s
In addition to the all-pay auction, many other auction rules also satisfy the revenue equivalence assumptions when bidder values are independently and identically distributed. Two examples are:
- The English (oral ascending) auction. All bidders start in the auction with a price of zero. The price rises continuously, and bidders may drop out at any point in time. Once they drop out, they cannot re- enter. The auction ends when only one bidder is left, and this bidder pays the price at which the second-to-last bidder dropped out.
- The Dutch (descending price) auction. The price starts at a very high level and drops continuously. At any point in time, a bidder can stop the auction, and pay the current price. Then the auction ends.
2 Common Value Auctions
We would now like to generalize the model to allow for the possibility that (i) learning bidder j’s information could cause bidder i to re-assess his estimate of how much he values the object, and (ii) the information of i and j is not independent (when j’s estimate is high, i’s is also likely to be high). These features are natural to incorporate in many situations. For in- stance, consider an auction for a natural resource like a tract of timber. In such a setting, bidders are likely to have different costs of harvesting or processing the timber. These costs may be independent across bidders and private, much like in the above model. But at the same time, bidders are likely to be unsure exactly how much merchanteable timber is on the tract, and use some sort of statistical sampling to estimate the quantity. Because these estimates will be based on limited sampling, they will be imperfect – so if i learned that j had sampled a different area and got a low estimate, she would likely revise her opinion of the tract’s value. In addition, if the areas sampled overlap, the estimates are unlikely to be independent.
2.1 A General Model
- Bidders i = 1, ..., n
- Signals S 1 , ..., Sn with joint density f (·)
- Signals are (i) exchangeable, and (ii) affiliated.
— Signals are exchangeable if s^0 is a permutation of s ⇒ f (s) = f(s^0 ). — Signals are affiliated if f(s ∧ s^0 )f(s ∨ s^0 ) ≥ f(s^0 )f(s) (i.e. sj |si has monotone likelihood ratio property).
- Value to bidder i is v(si, s−i).
Example 1 The independent private value model above is a special case: just let v(si, s−i) = si, and suppose that S 1 , ..., Sn are independent.
Example 2 Another common special case is the pure common value model with conditionally independent signals. In this model, all bidders have the same value, given by some random variable V. The signals S 1 , .., Sn are each correlated with V , but independent conditional on it (so for instance, Si = V + εi, where ε 1 , .., εn are independent. Then v(si, s−i) = E[V |s 1 , ..., sn].
Remark 2 While we will not purse it here, in this more general environ- ment the revenue equivalence theorem fails. Milgrom and Weber (1982) prove a very general result called the “linkage principle” which basically states that in this general symmetric setting, the more information on which the winner’s payment is based, the higher will be the expected revenue. Thus, the first price auction will have lower expected revenue than the second price auction because the winner’s payment in the first price auction is based only on her own signal, while in the second price auction it is based on her own signal and the second-highest signal.
3 Large Auctions & Information Aggregation
An interesting question that has been studied in the auction literature arises if we think about auction models as a story about how prices are determined in Walrasian markets. In this context, we might then ask to what extent prices will aggregate the information of market participants. We now consider a series of results along these lines. For each result, the basic set-up is the same. There are a lot of bidders, and the auction has pure common values with conditionally independent signals. We consider second price auctions (or more generally, with k objects, k + 1 price auctions, where all winning bidders pay the k + 1st highest bid). The basic question is: as the number of bidders gets large, will the auction price converge to the true value of the object(s) for sale?
3.1 Wilson (1977, RES)
Wilson considers information aggregation in a setting with a special infor- mation structure. In his setting, bidders learn a lower bound on the object’s value. The model has:
- Bidder’s 1, ..., n
- A single object with common value V ∼ U [0, 1]
- Bidder’s signals S 1 , ..., Sn are iid with Si ∼ U [0, v] (so signal si ⇒ V ≥ si).
Wilson shows that as n → ∞, the expected price converges to v. This is easy to see: with lots of bidders, someone will have a signal close to v, and you have to bid very close to your signal to have any chance of winning.
3.2 Milgrom (1979, EMA)
Milgrom goes on to identify a necessary and sufficient condition for infor- mation aggregation with many bidders and a fixed number of objects. Mil- grom’s basic requirement is, in the limit as n → ∞, that for all v^0 ∈ Ω and all v < v^0 , and M > 0, there exists some s^0 ∈ S such that
P (s^0 |v^0 ) P (s^0 |v)
> M.
That is, for any possible value v^0 , there must be arbitrarily strong signals that effectively rule out any value v < v^0. This condition builds on Wilson, and the intuition is again quite easy. As n → ∞, there is a very strong winner’s curse. In a second price auction, the high bid is:
EV
V | s1:n, max j 6 =i Sj = s1:n
While it is clear that s1:n^ is likely to be very high as n → ∞, the only way anyone would ever bid v^ would be to have a signal so strong that conditioning on millions of other signals being lower than your own would not push down your estimate too much.
3.3 Pesendorfer and Swinkels (1997, EMA)
Pesendorfer and Swinkels consider a model that is quite similar to Milgrom.
- n bidders, k objects
- Each bidder has value V ∼ F (·) on [0, 1]
- Signals Si ∼ G(·|v) on [0, 1], where g(·|·) has the MLRP.
- Signals S 1 , ..., Sn are independent conditional on V = v.
Remark 3 Clearly with a large number of independent signals, there is enough information to consistently estimate v. The question then is whether the bids will accurately aggregate this information – i.e. will the price be a consistent estimator?
Pesendorfer and Swinkel’s big idea is the following. While the winner’s curse will make a bidder with s = 1 shade her bid as n → ∞ with a fixed number of objects k, as k → ∞, there is also a loser’s curse: if lowering your bid ε matters there must be k people with higher valuations. This operates against the winner’s curse, counterbalancing it.
[5] Milgrom, Paul and Robert Weber (1982) “A Theory of Auctions and Competitive Bidding,” Econometrica, 50,.
[6] Milgrom, Paul and Ilya Segal (2002) “Envelope Theorems for Arbitrary Choice Sets,” Econometrica, 70, 583-601.
[7] Myerson, Roger (1981) “Optimal Auction Design,” Math. Op. Res, 6, 58—73.
[8] Pesendorfer, Wolfgang and Jeroen Swinkels (1997) “The Loser’s Curse and Information Aggregation in Auctions,” Econometrica, 65,.
[9] Riley, John and William Samuelson (1981) “Optimal Auction,” Amer- ican Economic Review, 71, 381—392.
[10] Vickrey, William (1961) “Counterspeculation, Auctions and Competi- tive Sealed Tenders,” Journal of Finance, 16, 8—39.
[11] Wilson, Robert (1977) “A Bidding Model of Perfect Competition,” Re- view of Economic Studies.
[12] Wilson, Robert (1992) “Strategic Analysis of Auctions,” Handbook of Game Theory.
