Econ 805 – Advanced Micro Theory I Dan Quint Fall 2008 Lecture 13 – Oct 21 2008

One of the points always stressed when teaching about : “the game is always larger than you think.” The first several weeks of this course focused on models of auctions where the bidders have already shown up, knowing what they know, and play noncooperative equilibrium strategies. But we can also step back and think about the bigger game – these bidders have to come from somewhere and decide to show up (endogenous participation/entry), there’s a reason they know what they know (endogenous information), and under some circumstances, they may try to collude which each other or otherwise discourage participation. Changes to the itself may have a direct impact on revenue, even keeping this other factors fixed; but they may also affect the number of bidders who choose to participate, the amount of information they choose to acquire, how easy it is for them to collude with each other, and so on. In his book, Klemperer emphasizes that these problems – ensuring adequate genuine competition – often dwarf whatever differences there would otherwise be between different types of auctions. Last Tuesday, we introduced models of auctions with endogenous entry. Last Thursday, we discussed a couple of papers on endogenous information acquisition. Today, I want to talk a bit more about endogenous information, and also a bit about collusion and other deviations from noncooperative equilibrium play. The two papers we saw on Thursday made polar opposite assumptions about whether “well- informed” bidders were known to be well-informed:

• The Persico paper assumed that information acquisition was covert. Bidders simultaneously choose how accurate a signal to observe (or how much to invest in information acquisition), learn their signal, but do not observe the choices made by their competitors. Persico finds that first-price auctions are more risk-sensitive than second-price auctions – basically, better information is more valuable in a first-price auction, so a first-price auction leads to greater information acquisition. (This is in a slightly-modified Milgrom-Weber affiliated world.) Per- sico also points out that this can lead to a reversal of the – with exogenous information and affiliation, second-price auctions revenue-dominate first-price auctions; but with endogenous information, this can be flipped.

• Hernando-Veciana, on the other hand, assumed that information acquisition was public – bid- ders observe each others’ decisions about how accurate a signal to invest in. He uses a more rigidly structured model – the winner’s value is the sum of a private-value component and a common-value component, and each bidder’s signal is one-dimensional but contains informa- tion about both components. He finds that which type of auction induces greater information

1 acquisition depends on which component of value (the PV or CV bit) the incremental infor- mation is about. If a better signal is more informative about the common-value component, then sealed-bid auctions induce more information acquisition than ascending auctions. If a better signal is more informative about the private-value component, then (roughly) the op- posite: ascending auctions induce more information acquisition. He also shows that in both cases, efficiency (roughly) favors ascending auctions.

So that’s one paper that considers covert information acquisition, and one that considers observ- able (open) information acquisition. My own paper, “Common Value Auctions with Two Bidders – When To Brag About What You Know,” compares the two. That is, I ask the question, if you had a choice between gaining better information covertly or openly, which would a bidder prefer? In other words, aside from the obvious benefit of having better information, is there an additional gain (or a loss) from appearing to have better information. The answer turns out to depend on the nature of this incremental information – whether it is independent or highly correlated with the other bidder’s signal. I restrict attention to the two-bidder case, with pure common values. Rather than signal accuracy being a continuous variable, I make it discrete – specifically, I look at the value of observing an additional signal in addition to the one you started with. Most of my results are generalizations of those from the drainage tract auction (common-value auctions with just one bidder having private information) to both bidders have private information. I assume the common value V is additively separable into two independent components,

V = V1 + V2

I assume there are two signals, X and Y , which are correlated with V1, and two signals, Z and W , which are correlated with V2; and that (V1,X,Y ) are independent of (V2,Z,W ). The “base case” will be bidder 1 observing X (but not Y ), and bidder 2 observing both Z and W . This way, we can look at the incremental effect of bidder 1 learning Y (information which is independent of his opponent’s, which I’ll refer to as novel information) or learning Z (information which is a subset of what his opponent knows, which I’ll refer to as redundant information). I focus on first-price auctions, in part because in this setting, the second-price auction has multiple equilibria. (However, my results all generalize to convex combinations between first- and second-price auctions, called “hybrid auctions,” where the winner pays

²b1 + (1 − ²)b2 for any ² ∈ (0, 1]. For ² > 0, this auction has a unique equilibrium. If we take the limit of the equilibrium strategies as ² → 0, this gives us an equilibrium of the second-price auction; if we assume that’s the equilibrium that’s played, then the results generalize to second-price auctions as well.)

Let Si be the set of signals bidder i observes. First result:

2 Lemma 1. Let S1 and S2 be common knowledge (the bidders know which signals their opponents know), and let S1 ⊂ {X,Y } and S2 ⊂ {Z,W }. Let

0 ti = E(Vi|i s information)

and Fti its probability distribution. Then

• Like in the drainage tract model, the probability distributions of b1 and b2 are identical

• Given type ti, bidder i wins with probability Fti (ti), and gets expected payoff Z ti

Fti (s)ds 0

−1 To see why, let Gj be the distribution of bidder j’s bids, and tj(b) = bj (b) the type of bidder j who bids b. Then given type ti and bid b, bidder i gets

Z b ui(ti, b) = (ti + tj(s) − b) dGj(s) 0 Differentiating with respect to b gives Z ∂ b ui(ti, b) = (ti + tj(b) − b) gj(b) − dGj(s) ∂b 0

Plugging in ti = ti(b) and setting this equal to 0 (equilibrium conditions) imply the first-order condition 1 g (b) = j ti(b) + tj(b) − b Gj(b) Since the left-hand side does not depend on i, we get g (b) g (b) j = i Gj(b) Gi(b) which implies Gi(b) = Gj(b), so the bid distributions match.

Once Gi(b) = Gj(b), we know that

Pr(i wins|ti) = Pr(bj(tj) < bi(ti)) = Gj(bi(ti)) = Gi(bi(ti)) = Pr(bi < bi(ti)) = Fti (ti)

More importantly, since Z b ui(ti, b) = (ti + tj(s) − b) dGj(s) 0 we know that Z ∂ b ui(ti, b) = dGj(s) = Gj(b) ∂ti 0 so, plugging in b = bi(ti) and applying the envelope theorem, Z ti

ui(ti, bi(ti)) = Fti (s)ds 0 This lemma leads pretty directly to the two main results of the paper:

3 Theorem 1. Let S1 = {X} and S2 = {Z,W }.

1. Bidder 1 gains more from learning Y openly than secretly; and this holds for every realization (x, y) of (X,Y )

2. Bidder 1 gains more from learning Z secretly; this holds in expectation over (X,Z), but not necessarily at every realization

We’ll prove the second part first. Let T = E(V |X), F its distribution, and t its realization. X R 1 X X tX Knowing only X, bidder 1’s expected payoff is 0 FX (s)ds. Now suppose bidder 1 learns Z openly. Then the realization of Z is common knowledge, X is bidder 1’s private information, and W is bidder 2’s private information. But then Lemma 1 still R tX holds, and bidder 1’s expected payoff is 0 FX (s)ds – the same as before he knew Z! It’s easy to show that under very general conditions, knowing Z secretly is strictly better than not knowing it; so in expectation, bidder 1 strictly prefers to learn Z secretly. However, this is an ex-ante result; in the paper, I give a simple example where for some real- izations (x, z), bidder 1 would prefer to have learned z publicly.

On to the first result. Let TXY = E(V1|X,Y ), FXY its distribution, and tXY its realization. Now,

tX = EY |X=xE(V1|X = x, Y = y) = EY |X=xtXY so for a given X, tXY − tX is a random variable with zero mean; which means that tXY is a mean- preserving spread around tX , or FX second-order stochastically dominates FXY . Which means that for any k, Z k Z k FX (s)ds ≤ FXY (s)ds 0 0 Now, when 1 learns Y publicly, it’s common knowledge that S = {X,Y } and S = {Z,W }, so R 1 2 tXY lemma 1 holds, with t1 = tXY ; so bidder 1’s expected payoff is 0 FXY (s)ds. What if 1 learns Y secretly? When we figured out 1’s equilibrium strategy when he only knew

X, he was maximizing his expected payoff given his estimate of E(V1) and bidder 2’s equilibrium bid strategy. If he learns Y secretly, he gets a better estimate of V1, but bidder 2 still behaves the same way; so his maximization problem is the same as before, just with a sharper estimate of V1.

So 1’s expected payoff, after learning Y secretly and calculating tXY , is exactly what it would have been if he hadn’t learned Y , but had learned tX = tXY ; which is Z tXY FX (s)ds 0 And second-order stochastic dominance already proved that this is less than what he got from learning Y publicly.

And both these payoffs were calculated for a particular realization of tXY , that is, for a particular realization of both x and y.

So bidder 1 prefers to learn novel information publicly, but redundant information secretly.

4 In the paper, I then go on to ask the next question: what if bidder 1 learned something secretly, but had the option to announce that he had learned it? That is, what if bidder 1 had a credible way to show he was better-informed than expected, but could choose to do this after learning his private information? Turns out, like in many signaling games, there are lots of equilibria. But the comparative statics results do establish this: in the case of novel information, it is an equilibrium to always announce it; and in the case of redundant information, it is not an equilibrium to always announce it, but it is an equilibrium to never announce it. Like I said, though, there are lots of other equilibria as well. I also give results in the paper about the seller’s incentive to reveal his information. These are similar to what we saw in Milgrom and Weber. If S1 = {X} and S2 = {Z,W }, then under fairly general conditions, the seller gains ex-ante by revealing both Y and Z. If he’s expected to know one of them for sure, and can reveal it credibly, then there’s a unique equilibrium where he always reveals whatever he knows.

5 Collusion

So that’s endogenous information. Next, a few things about collusion among bidders.

• Obviously, collusion is more important in some settings than in others. We don’t worry about two bidders colluding in an eBay auction – if two friends decide not to bid against each other in an auction for a digital camera they both want, it doesn’t really matter, because there are lots of other bidders. Anonymity will make it almost impossible to collude with all the bidders interested in a particular auction, so the problem pretty much goes away.

• On the other hand, there are some settings where the universe of potential bidders is small, they all know each other, and they expect to face each other over and over in a large number of auctions. This is the case with timber auctions – there are a large number of auctions for logging rights held every year, with the same competitors facing each other over and over. Similarly with government procurement auctions – reverse-auctions to provide a service, with the lowest price winning.

• In these settings, we should take seriously the possibility that bidders collude, either explicitly or tacitly

• Hendricks and Porter (1989), “Collusion in Auctions,” point out that from 1979 to 1988, over four-fifths of the cases filed by the U.S. DOJ’s under the section 1 of the Sherman Antitrust Act were in auction markets

Collusion in auctions could occur in one of several ways. For example:

• The colluding bidders could agree to all bid the exact same amount, say, the reserve price, and rely on the seller to randomly give the object to one of them

• (McAfee and McMillan (1992) show this is the optimal collusion mechanism for “weak cartels,” cartels which are unable to make cash transfers among themselves)

• Or the colluding bidders could agree to have one of them submit a low winning bid and the others bid still lower

• With common values, any rule for picking the designated winner will work; with private values, the bidders must decide among themselves who values the object most, which can be done by holding a second, private auction either before or after the “real” auction. (Such an auction is referred to as a “knockout” auction.)

Of course, since colluding bidders submit bids that are lower than their static best-responses to each others’ bids, they also need some means of punishing each other for deviating from the collusive arrangement. This could be outside of the auction framework (the organized crime approach), or it could be via a “grim trigger”-type punishment in a repeated-games framework. A simple example will help...

6 Example

I’m guessing you all remember the idea of in a repeated game, but let’s do a simple example to see how it would work in an auction. Consider a simple common-value auction with P n bidders: ti are independently U[0, 1], and V = i ti. Suppose the seller is holding a first-price auction with no reserve price, and that the same auction is to be repeated infinitely many times, with independent draws of ti each time, and a discount factor of δ.

The equilibrium in the first price auction is symmetric, so a bidder with type ti wins with n−1 probability ti , so by the envelope theorem, Z ti n−1 1 n Vi(ti) = s ds = ti 0 n

Taking the expectation over ti,

Z 1 1 n 1 πi = ti dti = 0 n n(n + 1) Suppose that all bidders’ bids were to be announced publicly after the auction, and that in the event of a tie, the winner would be chosen at random. Suppose the bidders decided to collude by all 0, letting the seller pick the winner randomly. If anyone ever deviated from this, they would revert to playing the Nash equilibrium thereafter. n 1 Since the object is worth an average of 2 , colluding like this leads to an expected payoff of 2 1 in every period, since each bidder wins with probability n and gets an object for free which is on n average worth 2 . The most either bidder would ever be tempted to deviate was when he got the highest possible signal, 1, and considered outbidding his opponent by a minimal amount, ², to win with probability 1 n−1 n+1 1 instead of probability n . Given ti = 1, the object is worth, in expectation, 1 + 2 = 2 . So colluding is an equilibrium as long as 1 n + 1 1 ¡ ¢ n + 1 1 ¡ ¢ + δ + δ2 + ... ≥ + δ + δ2 + ... n 2 2 2 n(n + 1) Multiplying through by 2n gives ¡ ¢ 2 ¡ ¢ n + 1 + n δ + δ2 + ... ≥ n(n + 1) + δ + δ2 + ... n + 1 or µ ¶ 2 δ n − ≥ (n + 1)(n − 1) n + 1 1 − δ µ ¶ n2 + n − 2 δ ≥ (n + 1)(n − 1) n + 1 1 − δ µ ¶ (n − 1)(n + 2) δ ≥ (n + 1)(n − 1) n + 1 1 − δ µ ¶ n + 2 δ ≥ (n + 1) n + 1 1 − δ

7 δ(n + 2) ≥ (1 − δ)(n + 1)2 ¡ ¢ δ (n + 1)2 + (n + 2) ≥ (n + 1)2 (n + 1)2 δ ≥ (n + 1)2 + (n + 2) So for δ sufficiently big, tacit collusion in the repeated auction is sustainable as an equilibrium; but not surprisingly, collusion is harder to sustain for higher n, since the short-term gains from deviating are so much larger.

Some Easy Conclusions

A number of simple conclusions fall out of this example will a bit of thought.

• Sealed-bid auctions are more robust to collusion than ascending auctions. This is because in ascending auctions, a deviating bidder is noticed immediately, and can be punished immedi- ately, so his gains from deviating can be made very small. In a sealed-bid auction, a deviating bidder could at least win that one auction at a low price, so the temptation to deviate is much greater; or, to put it another way, punishment is delayed, and therefore less effective.

• If the seller releases less information after the auction, it makes it harder for the bidders to sustain collusion. Announcing the winner and the winning bid allows the colluding bidders to verify that the winner did what he was supposed to do. Announcing only the identity of the winner (keeping the winning bid secret) is disruptive when bidders were colluding via the first strategy, since it is unclear whether the winner was chosen at random or deviated to a higher bid. (Of course, in many government auctions, the seller is required by law to disclose the winning bid, to reduce the possibility of collusion between the seller and one of the bidders.)

• Similarly, choosing a tiebreaking rule other than randomization makes an auction more robust to that type of collusion. (Not quite on topic, but I believe this is how Rockefeller broke the railroad cartel. He was shipping a lot of steel, and the various railroads agreed among themselves to all charge him the same high rates. Rockefeller picked one of them at random and used them exclusively; the other railroads assumed that one had cut its rates, and the cartel crumbled.)

• Similarly, if the bidders are colluding by pre-selecting the winner and allowing the others to knowingly submit losing bids, it may be optimal for the seller to sometimes award the object to one of the losing bidders.

• (Again, in both these last two cases, when it is a , certain details in how it’s run may be mandated by law. However, a private seller is free to do what they want. Not randomizing among tied bids costs nothing; awarding the object to the second-highest bidder costs little if the bids are close together, and might only have to be done once in a while to disrupt collusion.)

8 • Also, if collusion is suspected, a higher reserve price may be optimal, since it is much more likely the sale will conclude at the reserve price.

One of these observations was that ascending auctions are worse (from a collusion standpoint) than sealed-bid auctions. There’s another way in which this turns out to be true: in settings where bidders are free to name their own bids, bids can be used to communicate with other bidders, enabling collusion. Two cool examples of this come from spectrum auctions, and are discussed in the Klemperer book:

• In 1999, Germany used a simultaneous ascending auction to auction off 10 blocks of spectrum, with the rule that a new bid had to exceed the previous high bid by 10%. Mannesman opened the bidding by bidding 18.18 million deutschmarks on blocks 1-5 and 20 million on blocks 6-10. The only other credible bidder – T-Mobile – correctly took this as a signal that if they bid 10% more (20 million) on blocks 1-5 and didn’t bid on 6-10, the two would divide the market in that way and capture everything at the same low price; which was exactly the result.

• In a 1996-97 US , U.S. West was competing with McLeod for lot number 378 – a license in Rochester MN. Most bids were in thousands of dollars, but at one point, US West bid $313,378 and $62,378 for licenses in Iowa where McLeod had been the only bidder. McLeod got the message that these bids in Iowa were “punishment” for competing in Rochester, outbid US West for the Iowa licenses and stopped bidding in Rochester.

In both cases, there was no explicit agreement to collude, but the auction structure allowed bidders to communicate with each other and reach an understanding in the middle of the auction. Collusion using the strategy of all bidding the same amount is, of course, easy to detect. (In fact, a large number of sealed-bid government procurement auctions have indeed resulted in all bidders bidding the same amount to the dollar, which is very suspicious.) However, a more clever collusive arrangement – say, all bidders shading their bids downwards relative to equilibrium levels – is not necessarily detectable. A paper by Pat Bajari and Lixin Ye, “Detecting Collusion in Procurement Auctions,” shows that collusion among bidders is impossible to detect if information is not available about the competing firms’ costs. However, if data about the firms’ costs is available, they give an empirical method to test for collusion. (In fact, McAfee and McMillan claim that most of the DOJ’s convictions for bid-rigging have come after one of the cartel members, unhappy with his share of the profits, turns in his co- conspirators.) McAfee and McMillan (1992, “Bidding Rings”) point out that a successful cartel among bidders must overcome four challenges:

• They need a way to divide the profits – just like truthful revelation must be incentive- compatible in the auction itself, bidders will still have private information that must be shared within the cartel

9 • Contracts to fix prices are generally illegal, and therefore some other way to enforce the agreement must be found

• Collusion leads to higher profits, which makes an industry look more attractive to new en- trants, which may destroy the collusive arrangement

• And of course, the seller can take actions to disrupt the cartel

(McAfee and McMillan focus on the first problem – coming up with the optimal incentive-compatible mechanism for the bidders to use to determine who gets the object.)

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