Weak bidders prefer first-price (sealed-bid) auctions. (This holds both ex-ante, and once the bidders have learned their types)

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1 Econ 805 Advanced Micro Theory I Dan Quint Fall 2007 Lecture 9 Oct Last week, we egan relaxing the assumptions of the symmetric independent private values model. We examined private-value auctions with risk-averse idders, and showed that the first-price (sealed-id) auction led to higher expected revenue than the secondprice (or ascending) auction. Today, we return to risk-neutral idders, ut remove the assumption of symmetry that is, we allow for auctions where the idders values are drawn from different proaility distriutions. For simplicity, we will focus on the case of two idders, one strong and one weak, which we will define. Most of today s results come from the Maskin and Riley paper, Asymmetric Auctions. However, their paper kind of drowns you in notation. One of the nice things aout the Milgrom ook is that he simplifies the proofs of some of the results, so we ll e orrowing some proofs from there. Today s main results: trong idders (idders whose F i favor higher values) prefer second-price (or ascending) auctions Weak idders prefer first-price (sealed-id) auctions (This holds oth ex-ante, and once the idders have learned their types) The seller s preferences can go either way The general setup: two risk-neutral idders, one named trong and one named Weak, with independent private values drawn from F and F W. We will generally assume that F first-order stochastically dominates F W, which we showed earlier was equivalent to F (t) F W (t) for every t. Actually, we ll assume that this holds strictly at all interior values of t, plus something a it stronger. (Like efore, what we re really trying to do is to compare first-price sealed-id auctions to ascending (English) auctions; ut in this setting, that s the same as comparing them to second-price auctions.) In a second-price auction with asymmetric idders, it is still a dominant strategy to id your value; so the initial focus is on understanding equilirium id strategies in the asymmetric first-price auction. Maskin and Riley cite an earlier paper estalishing that in general, an asymmetric twoidder first-price auction has a unique equilirium, which is a solution to the differential equations defined y the idders first-order conditions, suject to the appropriate oundary

2 conditions. If we let and W e the equilirium id functions, then the strong idder s prolem is or with first-order condition max F W ( W ())(t ) max log F W ( W ()) + log(t ) f W ( W ()) ( ) F W ( W ()) W () t and, setting this equal to 0 at = (t ), or t = (), imilarly, for the weak type, f W ( W ()) ( F W ( W ()) W f ( ()) ( F ( ()) ) () = ) () = () W () In a first-price auction, it is never a est-response to id strictly higher than your opponent s highest id, so if we let the supports of the type distriutions e [t, t ] and [t W, t W ], then W (t W ) = (t ). As for the oundary condition at the ottom, first-order stochastic dominance implies that t W t. When these are equal, the other oundary condition is W (t W ) = (t ) = t W = t. (It s easy to show that these are necessary conditions for equilirium. Along with the differential equations, they turn out to e sufficient as well.) When t W < t, the oundary condition gets a little more complicated, as there may e a mass of weak idders who knowingly sumit losing ids. Later on, we ll also consider auctions with reserve prices, in which case the relevant oundary condition is W (r) = (r) = r if r is within the support of oth idders type distriutions. We can use the ranking lemma from last week to show that the proaility distriution of (t ) first-order stochastically dominates the distriution of W (t W ). Recall the ranking lemma: if g(x ) h(x ), and for x x, g(x) = h(x) g (x) > h (x), then g > h aove x. In this case, the oundary condition at the top is cleaner, so we ll rephrase the ranking lemma to start from the top. Lemma. Consider two continuous, differentiale functions g and h. uppose g(x ) h(x ), and for x x, g(x) = h(x) g (x) < h (x). Then g > h elow x. (Proof: see g and h as functions of x x, not x. Then the signs of the derivatives flip and satisfy the conditions of our original ranking lemma.) Lemma 2. If the strong idder s type distriution strictly FOD the weak idder s, then the strong idder s id distriution strictly FOD the weak idder s. 2

3 For i either or W, Pr( i (t i ) < ) = Pr(t i < i ()) = F i ( i ()), so what we want to show is F ( ()) < F W ( W ()) for in the interior of the range of the id functions. As we argued, the id functions must have the same upper ound, which we will call ; so F ( ( )) = F W ( W ( )) =. For <, suppose F ( ()) = F W ( W (). By assumption, F (t) < F W (t) (the type distriution of the strong idder first-order stochastically dominates); and F and F W are increasing, so this requires () > W (). Then from the two idders first-order conditions, f W ( W ()) ( ) F W ( W () = so we just showed that F W ( W ()) = F ( ()) implies W ()) () < F ( ()) W () = f ( ()) ( ) () d d F W ( W ()) < d d F ( ()) o the conditions for the ranking lemma are met, and F W ( W ()) > F ( ()); or the strong idder s id distriution strictly FOD the weak idder s. Maskin and Riley make a stronger assumption than FOD aout how the distriutions of the two idders values differ. pecifically, they require a condition they call Conditional tochastic Dominance: Assumption. For any t and t with t < t, Pr(t < t t < t ) < Pr(t W < t t W < t ). (Usual FOD is simply that this holds for t = t only; so CD implies FOD, ut is stronger.) This is equivalent to F (t) F (t ) < F W (t) F W (t ) for any t < t, which (rearranging) is equivalent to for any t < t, or F F W F (t) F W (t) < F (t ) F W (t ) strictly increasing, that is, which (taking the derivative) is equivalent to d F (t) dt F W (t) > 0 (F W (t)) 2 ( FW (t)f (t) F (t)f W (t) ) > 0 or, dropping the denominator and rearranging, F (t) F (t) > F W (t) F W (t) 3

4 (Milgrom recasts the type distriutions as F (t s), where s takes the value of 0 for the weak idder and for the strong idder. And since F /F is the derivative of log F, he recasts the assumption of Conditional tochastic Dominance as the requirement that d dt log F e increasing in s, or log F is supermodular i.e., has increasing differences in t and s.) Lemma 3. If F CD F W, then (t) < W (t) for interior t. That is, at the same type, a strong idder shades his id further elow his actual value. (Note that this immediately implies that asymmetric first-price auctions are not efficient the weak idder will get the oject when his type is slightly elow the strong idder s.) First-order stochastic dominance implies t W t (ecause F W (t ) F (t ) = ). ince the id functions have the same range and is increasing, W (t W ) = (t ) (t W ) Now pick t < t W, and suppose (t) = W (t). The two idders first-order conditions are f W (t) F W (t) W (t) = t W (t) = t (t) = f (t) F (t) (t) But Conditional tochastic Dominance implies f W (t) F W (t) < f (t) F (t), so for equality, W (t) < (t). Which implies (y the ranking lemma) that W (t) > (t) for t elow t W. o under the assumption that F conditionally first-order stochastically dominates F W, we now know that at the same value t, the weak idder ids more aggressively ut the strong idder s id distriution still first-order stochastically dominates Now we come to the interesting part: Theorem. uppose F conditionally first-order stochastically dominates F W. Then comparing a first-price and second-price auction, lowest. Every type of strong idder prefers the second-price auction Every type of weak idder prefers the first-price auction (For auctions without reserve prices, the preferences are strict for all types aove the For auctions with reserve prices, the preferences are strict for all types aove the reserve price. We prove it elow for a first- and second-price auctions with the same (inding) reserve price; for auctions without a reserve price, just set r = 0.) Proof. First of all, we know that (t) and W (t) have the same range, so define m(t) W ( (t)) as the type of weak idder who ids the same as a strong idder with type t in the first-price auction. (This function is sometimes referred to as a matching function or tying function.) ince (t) < W (t) for interior t, we know that m(t) < t for interior t. 4

5 We will calculate the strong idder s expected payoff at a type t using the envelope theorem. By the usual argument, Taking the derivative with respect to t gives and evaluating this at x = (t) gives f (x, t) = Pr( W (t W ) < x)(t x) f t (x, t) = Pr( W (t W ) < x) f t ( (t), t) = Pr( W (t W ) < (t)) = Pr(t W < m(t)) = F W (m(t)) ince given reserve price r, V (r) = 0, y the envelope theorem, V F P (t) = t r F W (m(s))ds in a first-price auction with a reserve price r. On the other hand, in a second-price auction, oth idders id their types, so and so f t ( (t), t) = Pr(t W < t) = F W (t) V P (t) = t r F W (s)ds ince m(s) < s and F W is strictly increasing, the strong idder strictly prefers the secondprice auction for t > r. By the same exact logic, the weak idder s expected payoff is V F P W (t) = t r F (m (s))ds in the first-price auction, and VW P (t) = t r F (s)ds in the second-price auction; since m (s) > s, the first is higher. o we have the nice, fairly general result that strong idders prefer open auctions and weak idders prefer sealed-id auctions. (My intuition for this is that strong idders are very likely to value the oject more, and therefore very likely to try to outid whatever the weak idder does. In an ascending auction, he can see the weak idder s id, so he can just arely outid him; in a sealed-id first-price auction, he has to guess, and since he likely values the good so highly, he has to aim near the top of the weak idder s id distriution. Maskin and Riley show the extreme example where F W = U[0, ] and F = U[2, 3]. In the second-price auction, the strong idder always wins and pays t W, which on average is 2. In the first-price auction, the equilirium is for the weak idder to id his type and the strong idder to always id, regardless of type; so the strong idder still always wins, and pays. In this case, the 5

6 results aove don t hold equilirium isn t characterized y the first-order conditions ut the intuition goes the same way.) Maskin and Riley point out that empirically, this seems to have een understood y idders, at least in some instances. In timer auctions in the Pacific Northwest, local insiders have loied hard for open auctions. imilarly, uyers who perceived themselves to e strong loied the FCC to use some form of open auction in the early spectrum auctions. This is also something Klemperer points to as one of the failings of the ill-fated 3G auction in witzerland, which raised one-thirtieth of the per-capita revenue of the British and German auctions held the same year: witzerland ran an ascending auction, causing potential entrants who were expected to e weaker than the few incument firms to give up. (At least one company hired idding consultants and then chose not to other idding.) Expected Revenue Now we come to the seller s prolem. Consider auctions without reserve prices. We know from efore that the expected revenue of any auction is the expected value of the marginal revenue of the winner. We know that the second-price auction is efficient gives the oject to the idder with the highest value which without symmetry need not e the idder with the highest marginal revenue. We just showed that the asymmetric first-price auction is not efficient the strong idder ids lower relative to his true valuation, so when the two idders types are close together, the first-price auction is iased toward giving the oject to the weak idder. Which is etter from the seller s point of view depends on whether the two distortions go in the same direction or different directions. To come up with a general principle... If, conditional on their types eing in the same allpark, the weak idder is likely to have significantly higher marginal revenue, then firstprice auction is proaly revenue-superior. If, conditional on their types eing in the same allpark, the weak idder is likely to have the same or lower marginal revenue, the secondprice auction is proaly revenue-superior. Two examples to ear out this intuition. First, suppose the weak idder s types are U[0, ], and the strong idder s types are U[0.5,.5]. Marginal revenue for each is { MR i (t) = t F i(t) t ( t) = 2t for weak idder = f i (t) t (.5 t) = 2t.5 for strong idder o for types that are reasonaly close together, the weak idder has higher marginal revenue than the strong idder; so the first-price auction, which is skewed towards giving the oject to the weak idder, turns out to e revenue-superior. Now suppose instead that the strong idder has types that are U[0, ], and the weak idder has types which with proaility 2 are U[0, ], and with proaility 2 are 0. The marginal revenue for a strong idder with type t is 2t ; the marginal revenue for a weak idder is MR W (t) = t F i(t) f i (t) 6 = t ( t)/2 /2 = 2t

7 o in this case, even though idders are asymmetric, the idder with the higher type also has the higher marginal revenue. In that case, the second-price auction, which is efficient, is also higher-revenue. Maskin and Riley give three results on settings where one auction is revenue-superior to another. Unfortunately, in each case, they need to e fairly restrictive aout the conditions they impose on the distriutions; so interpreting these as roust positive results is a little tricky. On the other hand, it s easy to interpret them as negative results: that is, that sometimes the first-price auction revenue-dominates, and sometimes the second-price auction revenue-dominates, and that it doesn t take a particularly pathological example for either one to happen. Theorem 2. uppose that F is just F W right-shifted y a constant. If F W is weakly convex and strictly log-concave, and MR W (t) 0 for t < t, then the sealed-id auction gives higher expected revenue than the ascending auction. The requirement that F W e convex rules out many familiar-looking distriutions; ut these conditions do allow the uniform distriution, the exponential distriution F (t) e t, and the distriution F (t) t. The other condition is asically that the right-shift not e too severe. (Our example with U[0, ] and U[.5,.5] fit these requirements. The example with U[0, ] and U[2, 3] does not, since the equilirium characterization y differential equations reaks down. In that particular example, the first-price auction still turns out to e revenuesuperior, ut the same proof doesn t work and the result may not generalize.) Theorem 3. uppose that F W is a truncation of F, that is, there is some t such that F W (t) = F (t)/f (t ). Under some regularity conditions on F, the sealed-id auction gives higher expected revenue than the ascending. Theorem 4. uppose F W is just F, with some of the weight taken from the distriution and moved to the lower end. If F has an increasing hazard rate, and with an assumption on how the mass is taken, the ascending auction gives higher expected revenue than the sealed-id auction. 7