Day 3. Myerson: What s Optimal

Day 3. Myerson: What s Optimal 1 Recap Last time, we... Set up the Myerson auction environment: n risk-neutral bidders independent types t i F i with support [, b i ] and density f i residual valuation of t 0 for the seller Everything works with very very slight modification) when t 0 > 0, but for simplicity today, we ll consider t 0 = 0, i.e., the seller has no valuation for the object and the objective is just to maximize expected revenue Named our goal: maximize expected revenue 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 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 can redefine our goal as solving { So... onward! max direct revelation mechanisms E t } x i t) i N s.t. 4), 5), 6), 7) 29

2 Rewriting our Objective Function By adding and subtracting E t i p it)t i, we can rewrite the seller s objective function as } U 0 p, x) = E t { i N x i t) = i N E t p i t)t i i N E t p i t)t i x i t)) Note that... the first term is the total surplus created by selling the object the second 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)) = E ti Et i p i t)t i x i t)) } = b i U i t i )f i t i )dt i = b i U i ) + t i Q i s i )ds i ) f i t i )dt i = U i ) + b i ti Q i s i )f i t i )ds i dt i = U i ) + b i bi s i f i t 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 s i, t i )fs i, t i )dt i ds i = U i ) + T 1 F i t i ) f i t i ) p i t)ft)dt = U i ) + E t p i t) 1 F it i ) f i t i ) 30

So we can rewrite the objective function as U 0 p, x) = i N E t p i t)t i i N U i ) i N E t p i t) 1 F it i ) f i t i ) = E t p i t) t i 1 F ) it i ) U i ) f i t i ) i N i N So the seller s problem amounts to choosing an allocation rule p and expected payoffs for the low types U i ) to maximize { E t p i t) t i 1 F ) } it i ) U i ) f i t i ) i N i N 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.) 31

Now, back to maximizing expected revenue We ve redefined the problem as choosing p and U i ) to maximize { E t p i t) t i 1 F ) } it i ) U i ) f i t i ) i N i N Clearly, to maximize this, we should set U i ) = 0 for each i This leaves Myerson s Lemma 3: Lemma 1. If p maximizes { E t p i t) t i 1 F ) } it i ) f i t i ) i N subject to Q i increasing in t i and j p jt) 1 for all t, and then p, x) is an optimal auction. ti x i t) = t i p i t) p i s i, t 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 ti ti 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. Also note that these are the transfers we found last time would make the outcome implementable in dominant strategies, not just BNE, as long as p i, not just Q i, are monotonic.) Next, we consider what the solution looks like for various cases. 32

3 The Regular Case With one additional assumption, things fall into place very nicely. Define a distribution F i to be regular if t i 1 F it i ) f i t i ) is strictly increasing in t i This is not that crazy an assumption Many 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 ) f i t i ) for each player Note that this will sometimes be negative) If max i c i t i ) < 0, keep the good; if max i c i t i ) 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 This rule makes Q i monotonic Q i t i ) = 0 for t i < c 1 i 0), and ) j i F j c 1 j c i t i )) above In fact, it makes each p i monotonic too, so we can implement in dominant strategies!) So the rule satisfies our constraints, and it s clear 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{0, max{c j t j )}} and 1 when c i t i ) > max{0, max{c j t j )}}, so this is just where ti x i t) = t i ds i = t i t i t i ) = t i t i ) t i = c 1 i max{0, max c jt j )} j 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. 33

3.1 Special Case: Symmetric Bidders In the case of symmetric IPV, each bidder s c function is the same as a function of his type, that is, which is strictly increasing in t i c i t i ) = ct i ) = t i 1 F t i) ft 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 a cool result: Theorem 1. With symmetric independent private values, the optimal auction is a second-price auction with a reserve price of c 1 0). 1 F r) Note that this reserve price will be strictly positive it s the solution to r fr) = 0, which will often be above the bottom of the bidders type distribution This means the optimal auction is not efficient the seller will sometimes withhold the object even though the highest bidder values it more than 0 but he never allocates it to the wrong bidder If t 0 > 0, everything goes through, and the profit-maximizing mechanism is a second-price auction with reserve c 1 t 0 ). Also interesting is that the optimal reserve price under symmetric IPV does not depend on the number of bidders it s just c 1 0) or c 1 t 0 )), regardless of N. 34

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 in the optimal mechanism, the winning bidder will not always 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 ) f i t 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. 35

4 The Marginal Revenue interpretation We defined a function c i t i ) = t i 1 F it i ) f i t i ) which told us which buyer we wanted to allocate the good to in the optimal mechanism. Myerson defined this as the buyer s virtual surplus or virtual value. become more common to refer to it as the buyer s marginal revenue. Since then, it s To understand why, consider a monopolist who faces a continuum of buyers whose valuations take a distribution F. By setting a price p, the monopolist sells to 1 F p)) of the buyers and earns revenue Rp) = p1 F p)) Now consider the question, how much more revenue will I get from selling to one more buyer? First, how much will we have to lower our price? Well, demand is 1 F p), so dq/dp = fp), so to sell to one more buyer, we need to lower the price by 1/fp) So the effect on revenue of selling to one more customer will be 1 d Rev fp) dp = 1 d p1 F p))) fp) dp = 1 1 F p)) pfp)) fp) = p 1 F p) fp) Of course, the indifferent buyer that we re now selling to is the one whose willingness-to-pay is exactly p So we can think of t valuation t 1 F t) ft) as the marginal revenue from selling to the consumer with Jumping back to the auction setting, we found that expected revenue is E t p i t) t i 1 F ) it i ) f i t i ) i So the revenue from any auction is the expected value of the winner s marginal revenue 36

5 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] Bulow and Klemperer show the following: 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 as long as U i ) = 0, expected revenue is { E t p i t) t i 1 F ) } { } it i ) = E t p i t)mr i t) + p 0 t)0 f i t i ) i N i N where p 0 t) = 1 i p it) is the probability the seller keeps the object So we can think of the seller as being another possible buyer, with marginal revenue of 0, and expected revenue remains the expected value of the marginal revenue of the winner In the symmetric case, if F is regular, then in an an ordinary second-price 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 ) with N + 1 bidders is E t max{mr 1, MR 2,..., MR N, MR N+1 } where MR i = t i 1 F t i) t i. 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 0) this just prevents selling to bidders with marginal revenue less than 0, replacing them with revenue 0. So an English auction with N bidders and optimal reserve price has expected revenue Expected Revenue = E t max{mr 1, MR 2,..., MR N, 0} You sell to the bidder with highest MR as long as MR is above 0.) 37

So the gist of Bulow and Klemperer, Auctions Versus Negotiations is to show that with symmetric IPV, E max{mr 1, MR 2,..., MR N, MR N+1 } E max{mr 1, MR 2,..., MR N, 0} That is, the revenue from a simple auction no reserve price) with N + 1 bidders, is higher than the revenue from the optimal auction with N bidders That is, the seller gains more by attracting one more bidder than by holding the perfect auction This also holds when t 0 > 0, as long as t 0 < a, so all bidders have valuations higher than the seller; the optimal reserve is then MR 1 t 0 ), and the profit from the optimal mechanism with N bidders is E max{mr 1, MR 2,..., MR N, t 0 }.) 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 MR i = E ti t i 1 F t ) i) ft i ) = b a t i 1 F t ) i) ft i )dt i ft i ) = b a t i ft i ) 1 + F t i )) dt i The integrand integrates to t i 1 F t i )), so so the integral evaluates to t i 1 F t i )) b t i =a = 0 a) = a so EMR i ) = a. This makes some sense. Suppose there s only one buyer, so expected revenue is the expected value of the marginal revenue when we sell to him. Suppose we want to always sell to him. The only way to do that is to charge a, because if we charge more, he won t always buy; so the expected value of a player s marginal revenue must be the bottom of the support.) 38

Next, note that for fixed x, the function gy) = max{x, y} is convex draw it) So by Jensen s inequality, E y gy) gey)) E y max{x, y} max{x, Ey)} If this holds for any value of x, it also holds in expectation over x if x is a random variable: E x {E y max{x, y}} E x max{x, Ey)} or E max{x, y} E max{x, Ey)} So now let let x = max{mr 1, MR 2,..., MR N } and y = MR N+1 ; then RevN + 1 bidders, r = 0) = E max{mr 1, MR 2,..., MR N, MR N+1 } = E max{max{mr 1, MR 2,..., MR N }, MR N+1 } E max{max{mr 1, MR 2,..., MR N }, E{MR N+1 }} = E max{max{mr 1, MR 2,..., MR N }, a}} E max{max{mr 1, MR 2,..., MR N }, 0}} = RevN bidders, optimal reserve) and that s the proof. Finally leading to the title of the paper), Bulow and Klemperer point out that negotiations really, any other 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. 39

6 Asymmetric Bidders and Other Mechanisms Myerson gave us the optimal mechanism even with asymmetric bidders bidders whose private values come from different probability distributions But the Myerson mechanism is pretty unusual-looking with asymmetric bidders and doesn t occur much in the wild There s an additional literature out there on how the two most common auction formats first-price and second-price auctions compare to each other when bidders are asymmetric, since revenue equivalence doesn t hold The main results are these, taken from Maskin and Riley 2000), which assumes just two bidders, one strong and one weak That is, t 1 F S and t 2 F W, where F S F OSD F W In fact, they assume something a little stronger conditional stochastic dominance read the paper for details The main results they find: In a second-price auction, bidding your value is still a dominant strategy, so the equilibrium is efficient the bidder with the higher value wins) In a first-price auction, the weak bidder bids more aggressively than the strong bidder: a weak bidder with a particular valuation bids higher than a strong bidder would with that same valuation This means the first-price auction is not efficient sometimes the weak bidder will win even though the strong bidder had a higher valuation Strong bidders prefer second-price auctions Weak bidders prefer first-price auctions But the revenue ranking is ambiguous there are some environments favoring FP and some favoring SP They give examples of both, and some partial characterization) I m not going to prove the results read the Maskin and Riley paper if you want to see how it all works But I will mention a more recent paper I like, which used the Myerson framework to give sufficient conditions for revenue superiority of one format or the other 40

Kirkegaard 2012 ECMA) takes the Myerson approach to understanding when one type of auction revenue-dominates the other In the second-price auction, whichever bidder has the higher valuation, wins; while in the first-price auction, the weak bidder sometimes wins even when the strong bidder s valuation is higher t w W wins in either auction format 45 line W wins in FP auction, S wins in SP auction S wins in either auction format t s Recall what we learned: expected revenue is the expected value of the marginal revenue of the person you give the thing to So in this case, we can answer which auction gives more revenue by asking... which bidder has higher marginal revenue in the cases where the strong bidder s valuation is above the weak bidder s, but the weak bidder would win the first-price auction? And he finds sufficient conditions which guarantee the weak bidder s MR is higher in those cases, and therefore that the first-price auction gives higher revenue For example, if t w U[0, 1] and t s U[0, 2], we saw earlier that at the same type, the strong bidder has lower marginal revenue; so for cases close to the 45 degree line, the weak bidder would have higher marginal revenue. This is one of the known cases where the first-price auction gives higher revenue.) 41

7 A cool application of Bulow/Klemperer Let s go back to Bulow and Klemperer that a simple second-price auction with N +1 bidders is better than the optimal auction with N This is generally interpreted as, don t worry so much about optimizing the mechanism, participation is more important But there s one case we can t ignore the mechanism: when there s just one bidder With one bidder, you can t just do nothing and run a second-price auction, of course The optimal mechanism is a posted price of c 1 0) But what if you don t know F? There s a cute result relating to this, and the proof follows directly from Bulow and Klemperer The result is from the approximately optimal mechanism design literature at the intersection of mechanism design and computer science I found it in a survey by Roughgarden 2014) Some of the focus in that literature is on revenue guarantees basically, if you have limited information or constraints on what mechanisms you can use, what fraction of the revenue of the optimal mechanism can you guarantee yourself across all environments)? With one bidder, if you know nothing about F, there s literally no fraction you can guarantee For example, suppose the distribution of values puts probability 1 M M, and the rest of the weight on 0 on the bidder having value The optimal mechanism gets revenue of 1 But if you don t know M, there s no price that guarantees you revenue above 0, since if you set the price r above M, revenue is 0 and if you set it below M, revenue is just r M, which is close to 0 for M large) Even if you got a finite number of samples from the distribution, if you make M large enough, all your samples would likely be 0, so you wouldn t learn anything So with one bidder, without prior knowledge of F and without putting any restriction on it, you re pretty screwed 42

But suppose we know that F is regular Then, things are much better Even if you start out knowing nothing about F except regularity, it turns out that if you get to see one sample draw from F, there s a mechanism that guarantees you at least half the revenue of the actual optimal mechanism That is, we get one random draw from F to use as data, and then we run a one-bidder auction with an independent) value drawn from F, and we re guaranteed we can get at least half the revenue we would get if we knew F perfectly and ran the optimal mechanism The simple rule that achieves this: just demand the price equal to the sample draw And the proof is straight from Bulow/Klemperer Let t 0 be the sample draw, and t 1 the value of the actual buyer we re trying to sell to And let MR 0 and MR 1 be t 0 1 F t 0) ft 0 ) and t 1 1 F t 1) ft 1 ) Once we pick a price r for the actual mechanism, expected revenue is E t1 MR 1 1 t1 r = E t1 MR 1 1 MR1 MRr) If we decide ahead of time to set r = t 0, then in expectation over t 0 as well as t 1, revenue is E t0,t 1 MR 1 1 t1 t 0 = E t0,t 1 MR 1 1 MR1 MR 0 By symmetry since t 0 and t 1 are drawn from the same distribution, so MR 0 and MR 1 are drawn from the same distribution), this is equal to 1 2 E t 0,t 1 max{mr 0, MR 1 } But E t0,t 1 max{mr 0, MR 1 } is the revenue from a second-price auction with two bidders and no reserve price And by Bulow and Klemperer, that s at least the revenue from the optimal mechanism with one bidder So with a single sample from F, we can get half the revenue of the optimal mechanism for every regular F ) 43

8 References and Further Reading Myerson 1981), Optimal Auction Design, Mathetmatics of Operations Research 6.1 Bulow and Klemperer 1996), Auctions Versus Negotiations, AER 86.1 Maskin and Riley 2000), Asymmetric Auctions, REStud 67 Kirkegaard 2012), A Mechanism Design Approach to Ranking Asymmetric Auctions, ECMA 80 Roughgarden 2014), Approximately Optimal Mechanism Design: Motivation, Examples, and Lessons Learned, ACM SIGEcom Exchanges 10.2 44