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Oct 21 2008 One of the Points Paul Milgrom Always Stress

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Econ 805 { Advanced Micro Theory I Dan Quint Fall 2008 Lecture 13 { Oct 21 2008 One of the points Paul Milgrom always stressed when teaching about auctions: \the game is always larger than you think." The ¯rst 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 auction itself may have a direct impact on revenue, even keeping this other factors ¯xed; but they may also a®ect 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 di®erences there would otherwise be between di®erent 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: ² 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 ¯nds that ¯rst-price auctions are more risk-sensitive than second-price auctions { basically, better information is more valuable in a ¯rst-price auction, so a ¯rst-price auction leads to greater information acquisition. (This is in a slightly-modi¯ed Milgrom-Weber a±liated world.) Per- sico also points out that this can lead to a reversal of the linkage principle { with exogenous information and a±liation, second-price auctions revenue-dominate ¯rst-price auctions; but with endogenous information, this can be flipped. ² 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 ¯nds 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, e±ciency (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 bene¯t 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 { speci¯cally, 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 e®ect 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 ¯rst-price auctions, in part because in this setting, the second-price auction has multiple equilibria. (However, my results all generalize to convex combinations between ¯rst- and second-price auctions, called \hybrid auctions," where the winner pays ²b1 + (1 ¡ ²)b2 for any ² 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 ½ fX; Y g and S2 ½ fZ; W g. Let 0 ti = E(Viji s information) and Fti its probability distribution. Then ² Like in the drainage tract model, the probability distributions of b1 and b2 are identical ² Given type ti, bidder i wins with probability Fti (ti), and gets expected payo® 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 Di®erentiating 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 ¯rst-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 winsjti) = 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 = fXg and S2 = fZ; W g. 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 ¯rst. Let T = E(V jX), F its distribution, and t its realization. X R 1 X X tX Knowing only X, bidder 1's expected payo® 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 payo® 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 ¯rst result. Let TXY = E(V1jX; Y ), FXY its distribution, and tXY its realization. Now, tX = EY jX=xE(V1jX = x; Y = y) = EY jX=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 = fX; Y g and S = fZ; W g, so R 1 2 tXY lemma 1 holds, with t1 = tXY ; so bidder 1's expected payo® is 0 FXY (s)ds. What if 1 learns Y secretly? When we ¯gured out 1's equilibrium strategy when he only knew X, he was maximizing his expected payo® given his estimate of E(V1) and bidder 2's equilibrium bid strategy.
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