Optimal Auctions Continued

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1 Lecture 6 Optimal Auctions Continued 1 Recap Last week, we... Set up the Myerson auction environment: n risk-neutral bidders independent types t i F i with support [, b i ] residual valuation of t 0 for the seller Named our goal: maximize expected seller payoff over all conceivable auction/sales formats, subject to two constraints: bidders participate voluntarily and play equilibrium strategies Defined mechanisms, and showed the Revelation Principle that without loss of generality, we can restrict attention to direct revelation mechanisms Showed that feasibility of a mechanism is equivalent to four conditions holding: 4. Q i is weakly increasing in t i 5. U i t i ) = U i ) + t i Q i p, s)ds for all i, all t i 6. U i ) 0 for all i, and 7. j p jt) 1, p i 0 So we defined our goal as solving { max E t x i t) + t 0 1 )} p i t) direct revelation mechanisms s.t. 4), 5), 6), 7) So... onward! 59

2 2 Rewriting our Objective Function By adding and subtracting E i i p it)t i, we can rewrite the seller s objective function as { U 0 p, x) = E t x i t) + t 0 1 )} p i t) = t 0 + E t p i t)t i t 0 ) E t p i t)t i x i t)) Note that... the first term is a constant the payoff to doing nothing) the second term is the total surplus created by selling the object the third term, which is being subtracted, is the expected payoff going to the bidders Next, we do some work expressing the last term, expected bidder surplus, in a different form: E t p i t)t i x i t)) = b i U i t i )dt i = b i U i ) + t i Q i s i )ds i ) dt i = U i ) + b i Q i s i )ds i dt i = U i ) + b i bi s i Q i s i )dt i ds i = U i ) + b i 1 F i s i ))Q i s i )ds i = U i ) + b i ) 1 F i s i )) p t i is i, t i )f i t i )dt i ds i = U i ) + b i 1 F i s i ) ) f i s i ) p t i is i, t i )f i t i )dt i f i s i )ds i = U i ) + T 1 F i s i ) f i s i ) p i t i, s i )ft i, s i )dt i ds i = U i ) + T 1 F i t i ) p i t)ft)dt = U i ) + E t p i t) 1 F it i ) 60

3 So we can rewrite the objective function as U 0 p, x) = t 0 + E t p i t)t i t 0 ) U i ) E t p i t) 1 F it i ) = t 0 + E t p i t) t i 1 F ) it i ) t 0 U i ) So the seller s problem amounts to choosing an allocation rule p and expected payoffs for the low types U i ) to maximize t 0 + E t p i t) t i 1 F ) } it i ) t 0 U i ) subject to feasibility which just means p plausible, Q i increasing in type, and U i ) 0 The envelope formula for bidder payoffs is no longer a constraint we ve already imposed it.) Once we find a mechanism we like, each U i is uniquely determined by the envelope formula, and so the rest of the transfers x i are set to satisfy those required payoffs. Once we ve phrased the problem in this way, Myerson points out, revenue equivalence becomes a straightforward corollary: Corollary 1. For a given environment, the expected revenue of an auction depends only on the equilibrium allocation rule and the expected payoffs of the lowest possible type of each bidder. The Revenue Equivalence Theorem is usually stated in this way: Corollary 2. Suppose bidders have symmetric independent private values and are risk-neutral. Define a standard auction as an auction where the following two properties hold: 1. In equilibrium, the bidder with the highest valuation always wins the object 2. The expected payment from a bidder with the lowest possible type is 0 Any two standard auctions give the same expected revenue. Two standard auctions also give the same expected surplus to each type of each bidder U i t i ). So this means with symmetry, independence, and risk-neutrality, any auction with a symmetric, strictly-monotone equilibrium gives the same expected revenue. Examples.) 61

4 Now, back to maximizing expected revenue We ve redefined the problem as choosing p and U i ) to maximize t 0 + E t p i t) t i 1 F ) } it i ) t 0 U i ) Clearly, to maximize this, we should set U i ) = 0 for each i This leaves Myerson s Lemma 3: Lemma 1. If p maximizes t 0 + E t p i t) t i 1 F ) } it i ) t 0 subject to Q i increasing in t i and p possible, and then p, x) is an optimal auction. x i t) = t i p i t) p i t i, s i )ds i The transfers are set to make U i ) = 0 and give payoffs required by the envelope theorem To see this, fix t i and take the expectation over t i, and we find or E t i x i t) = t i Qt i ) Q i s i )ds i Q i s i )ds i = t i Q i t i ) E t i x i t i, t i ) = U i t i ) which is exactly the envelope theorem combined with U i ) = 0 The exact transfers x i t) are not uniquely determined by incentive compatibility and the allocation rule p; what is uniquely pinned down is E t i x i t i, t i ), because this is what s payoff-relevant to bidder i. The transfers above are just one rule that works.) Next, we consider what the solution looks like for various cases. 62

5 3 The Regular Case With one additional assumption, things fall into place very nicely. Define a distribution F i to be regular if is strictly increasing in t i This is not that crazy an assumption t i 1 F it i ) Most familiar distributions have increasing hazard rates that is, which would imply 1 F f decreasing This is a weaker condition, since f 1 F f 1 F is increasing, is allowed to decrease, just not too quickly. When the bid distributions are all regular, the optimal auction becomes this: Calculate c i t i ) = t i 1 F it i ) for each player If max i c i t i ) < t 0, keep the good; if max i c i t i ) t 0, award the good to the bidder with the highest value of c i t i ) Charge the transfers determined by incentive compatibility and this allocation rule Note that this rule makes Q i monotonic Q i t i ) = 0 for t i < c 1 i t 0 ), and ) j i F j c 1 j c i t i )) above it So the rule satisfies our two constraints, and it s obvious that it maximizes the seller s objective function There s even a nice interpretation of the x i we defined above. Fixing everyone else s type, p i is 0 when c i t i ) < max{t 0, max{c j t j )}} and 1 when c i t i ) > max{t 0, max{c j t j )}}, so this is just x i t) = t i ds i = t i t i t i ) = t i t i where t i is the lowest type that i could have reported given everyone elses reports) and still won the object. This payment rule makes incentive-compatibility obvious: for each combination of my opponents bids, I face some cutoff t i such that if I report t i > t i, I win and pay t i ; and if I report less than that, I lose and pay nothing. Since I want to win whenever t i > t i, just like in a second-price auction, my best-response is to bid my type. 63

6 3.1 Symmetric IPV In the case of symmetric IPV, each bidder s c function is the same as a function of his type, that is, c i t i ) = ct i ) = t i 1 F t i) ft i ) which is strictly increasing in t i. This means the bidder with the highest type has the highest c i, and therefore gets the object; and so his payment is the reported type of the next-highest bidder, since this is the lowest type at which he would have won the object. Which brings us to our first claim: Theorem 1. With symmetric independent private values, the optimal auction is a second-price auction with a reserve price of c 1 t 0 ). Note, though, that even when t 0 = 0, this reserve price will be positive. The optimal auction is not efficient since ct i ) < t i, the seller will sometimes keep the object even though the highest bidder values it more than him but he never allocates it to the wrong bidder. Also interesting is that the optimal reserve price under symmetric IPV does not depend on the number of bidders it s just c 1 t 0 ), regardless of N. 3.2 Asymmetric IPV When the bidders are not symmetric, things are different. With different F i, it will not always be true that the bidder with the highest c i also has the highest type; so sometimes the winning bidder will not be the bidder with the highest value. As we ll see later, efficiency is not standard in auctions with asymmetric bidders: even a standard first-price auction is sometimes not be won by the bidder with the highest value.) One special case that s easy to analyze: suppose every bidder s bid is drawn from a uniform distribution, but uniform over different intervals. That is, suppose each F i is the uniform distribution over a potentially) different interval [, b i ]. Then c i t i ) = t i 1 F it i ) = t i b i t i )/b i ) 1/b i ) = t i b i t i ) = 2t i b i So the optimal auction, in a sense, penalizes bidders who have high maximum valuations. This is to force them to bid more aggressively when they have high values, in order to extract more revenue; but the price of this is that sometimes the object goes to the wrong bidder. 3.3 What About The Not-Regular Case? Myerson does solve for the optimal auction in the case where c i is not increasing in t i, that is, where the auction above would not be feasible. Read the paper if you re interested. 64

7 4 Bulow and Klemperer, Auctions versus Negotiations We just learned that with symmetric IPV and risk-neutral bidders, the best you can possibly do is to choose the perfect reserve price and run a second-price auction. This might suggest that choosing the perfect reserve price is important. There s a paper by Bulow and Klemperer, Auctions versus Negotiations, that basically says: nah, it s not that important. Actually, what they say is, it s better to attract one more bidder than to run the perfect auction. Suppose we re in a symmetric IPV world where bidders values are drawn from some distribution F on [a, b], and the seller values the object at t 0. Bulow and Klemperer show the following: as long as a t 0 all bidders are serious ), the optimal auction with N bidders gives lower revenue than a second-price auction with no reserve price and N + 1 bidders. To see this, recall that we wrote the auctioneer s expected revenue as t 0 + E t p i t) t i 1 F ) } it i ) t 0 U i ) Consider mechanisms where U i ) = 0, and define the marginal revenue of bidder i as MR i = t i 1 F it i ) so expected revenue is ) t 0 } E t p i t)mr i t) + 1 p i t) So if we think of the seller as being another possible buyer, with marginal revenue of t 0, then the expected revenue is simply the expected value of the marginal revenue of the winner. Jump back to the symmetric case, so F i = F. Continue to assume regularity. In an ordinary second-price or ascending auction, with no reserve price, the object sells to the bidder with the highest type, which is also the bidder with the highest marginal revenue; so the expected revenue in this type of auction what Bulow and Klemperer call an absolute English auction ) is Expected Revenue = E t max{mrt 1 ), MRt 2 ),..., MRt N )} This is Bulow and Klemperer Lemma 1.) The fact that expected revenue = expected marginal revenue of winner also makes it clear why the optimal reserve price is MR 1 t 0 ) this replaces bidders with marginal revenue less than t 0 with t 0. So counting the seller s value from keeping the unsold object) an English auction with an optimal reserve price has expected revenue Expected Revenue = E t max{mrt 1 ), MRt 2 ),..., MRt N ), t 0 } So here s the gist of Bulow and Klemperer, Auctions Versus Negotiations. They compare the simple ascending auction with N + 1 bidders, to the optimal auction with N bidders. We 65

8 discovered last week that with symmetric independent private values, the optimal auction is an ascending auction with a reserve price of MR 1 t 0 ).) The gist of Bulow and Klemperer is that the former is higher, that is, that E max{mrt 1 ), MRt 2 ),..., MRt N ), MRt N+1 )} E max{mrt 1 ), MRt 2 ),..., MRt N ), t 0 } so the seller gains more by attracting one more bidder than by holding the perfect auction. They normalize t 0 to 0, but this doesn t change anything.) Let s prove this. The proof has a few steps. First of all, note that the expected value of MRt i ) is a, the lower bound of the support. This is because ) E ti MRt i ) = E ti t i 1 F t i) ft i ) = ) b a t i 1 F t i) ft i ) ft i )dt i = b a t ift i ) 1 + F t i )) dt i Now, tft) + F t) has integral tf t), so this integrates to t i F t i ) b t i =a b a) = b 0 b a) = a which by assumption is at least t 0. So EMRt i )) t 0. Next, note that for fixed x, the function gy) = max{x, y} is convex, so by Jensen s inequality, If we take an expectation over x, this gives us E y max{x, y} max{x, Ey)} E x {E y max{x, y}} E x max{x, Ey)} or E max{x, y} E max{x, Ey)} Now let x = max{mrt 1 ), MRt 2 ),..., MRt N )} and y = MRt N+1 ); E max{mrt 1 ), MRt 2 ),..., MRt N ), MRt N+1 )} E max{mrt 1 ), MRt 2 ),..., MRt N ), EMRt N+1 ))} E max{mrt 1 ), MRt 2 ),..., MRt N ), t 0 } and that s the proof. Finally and leading to the title of the paper), Bulow and Klemperer point out that negotiations really, any process for allocating the object and determining the price cannot outperform the optimal mechanism, and therefore leads to lower expected revenue than a simple ascending auction with one more bidder. They therefore argue that a seller should never agree to an early take-it-or-leave-it offer from one buyer when the alternative is an ascending auction with at least one more buyer, etc. 66

9 References Roger Myerson 1981), Optimal Auctions, Mathematics of Operations Research 6 Jeremy Bulow and Paul Klemperer 1996), Auctions Versus Negotiations, American Economic Review 86 67

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