# Optimal Auctions. Jonathan Levin 1. Winter Economics 285 Market Design. 1 These slides are based on Paul Milgrom s.

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1 Optimal Auctions Jonathan Levin 1 Economics 285 Market Design Winter 29 1 These slides are based on Paul Milgrom s. onathan Levin Optimal Auctions Winter 29 1 / 25

2 Optimal Auctions What auction rules lead to the highest average prices? Di culty: the set of auction rules is enormous, e.g.: The auction is run in two stages. In stage one, bidders submit bids and each pays the average of its own bid and all lower bids. The highest and lowest rst stage bid are excluded. All other bidders are asked to bid again. At the second stage, the item is awarded to a bidder with a probability proportional to the square of its second-stage bid. The winning bidder pays the average of its bid and the lowest second-stage bid. Can we really optimize revenue over all mechanisms? Jonathan Levin Optimal Auctions Winter 29 2 / 25

3 Revelation Principle For any augmented Bayesian mechanism (Σ, ω, σ), there is a corresponding direct mechanism (T, ω ) for which truthful reporting is a Bayes-Nash equilibrium. The otcome function for the corresponding direction mechanism is given by: ω (t) = ω (σ 1 (t 1 ),..., σ N (t N )). Jonathan Levin Optimal Auctions Winter 29 3 / 25

4 Auction Revenues Suppose that De nition value of good to bidder i is v i (t i ) each v i is increasing and di erentiable. types are idependent, drawn from U[, 1]. An augmented mechanism (Σ, ω, σ) is voluntary if the maximal payo V i (t i ) is non-negative everywhere. The expected revenue from an augmented mechanism is the expected sum of payments: " N # R (Σ, ω, σ), E pi ω (σ 1 (t 1 ),..., σ N (t N )). i=1 onathan Levin Optimal Auctions Winter 29 4 / 25

5 Revenue maximization problem Allow randomized mechanisms, and let x i (t) denote the probability that i is awarded the good at equilibrium for the augmented mechanism. Then, we have the following constraints on feasible mechanisms: Consider the problem x i (t) for all i 6= N x i (t) 1 i= max R (Σ, ω, σ). (Σ,ω),σ2NE Performance: x i (t 1,..., t N ), x ω i (σ 1 (t 1 ),..., σ N (t N )). Jonathan Levin Optimal Auctions Winter 29 5 / 25

6 Marginal revenue Simple case... assume there is a single bidder with value v(t), where v is increasing and t is drawn from U[, 1]. If the seller xes a price v(s) it sells whenever t s, or with probability 1 s. Can view 1 s as the quantity sold Expected total revenue is (1 s)v(s). Marginal revenue is drev/dq: MR(s) = v(s) (1 s)v (s) = d ((1 s)v(s)) d(1 s) Jonathan Levin Optimal Auctions Winter 29 6 / 25

7 Revenue formula Theorem Consider any augmented mechanism (N, Σ, ω, σ ) for which σ is a BNE, x j (t) is the probability that j is allocated the good at type pro le t, and V j () is his expected payo with type equal to zero. Then: R(Σ, ω, σ ) = N x i (s 1,..., s N )MR i (s)ds 1...ds N i=1 N V i (). i=1 Note that: Expected revenue is expressed as a function of the allocation x but not the payment function p. The expression is linear in the x i s and depends critically on the marginal revenues. onathan Levin Optimal Auctions Winter 29 7 / 25

8 Derivation of revenue formula, I Consider mechanism (N, Σ, ω) with equilibrium pro tle σ. Consider a given bidder i. Its equilibrium payo is: V i (t i ) = max E t i [x i (σ i, σ i (t i ))v i (t i ) σ i 2Σ i p i (σ i, σ i (t i ))] = max E t i [x i (σ i, σ i (t i ))] v i (t i ) σ i 2Σ i E t i [p(σ i, σ i (t i ))] The derivative of this objective wrt t i is: E t i [x i (σ i, σ i (t i ))] v i (t i ) By the envelope theorem this expression gives V i (t)... Jonathan Levin Optimal Auctions Winter 29 8 / 25

9 Derivation of revenue formula, II Applying Myerson s Lemma, we have for any i, V i (t i ) V i () = = Z ti E s i [x i (s i, s i )] vi (s i )ds i Z ti x i (s 1,..., s N )ds i vi (s i )ds i So the bidder s average payo across all types is: E ti [V i (t i )] V i () = [V i (t i ) V i ()] dt i Jonathan Levin Optimal Auctions Winter 29 9 / 25

10 Derivation of revenue formula, III Using integration by parts we obtain E ti [V i (t i )] V i () Z ti = E s i [x i (s i, s i )] vi (s i )ds i dt i = = = E s i [x i (s i, s i )] v i (s i )ds i (1 s i ) E s i [x i (s i, s i )] v i (s i )ds i (1 s i ) x i (s 1,..., s N )vi (s i )ds 1...ds N t i E s i [x i (t i, s i )] v i (t i )dt i Jonathan Levin Optimal Auctions Winter 29 1 / 25

11 Derivation of revenue formula, IV Total realized surplus is N x i (t 1,..., t N )v i (t i ) = x(t) v(t) i=1 Total expected revenue is therefore R(Σ, ω, σ) N = E t [x(t) v(t)] E ti [V i (t i )] i=1 = x i (s 1,..., s N ) v i (1 s i ) vi (s i ) ds 1...ds N = x i (s 1,..., s N )MR i (s i )ds 1...ds N N V i () i=1 N V i () i=1 Jonathan Levin Optimal Auctions Winter / 25

12 Revenue maximization theorem Theorem Suppose that MR i is increasing for all i. Then and augmented voluntary mechanism is expected revenue maximizing if and only if (i) V i () = for all i, (ii) bidder i is allocated the good exactly when MR i (t i ) > max, max j6=i MR j (t j ). Furthermore at least one such augmented mechanism exists, with expected revenue of: E t [max f, MR 1 (t 1 ),..., MR N (t N )g]. onathan Levin Optimal Auctions Winter / 25

13 Proof, I From the prior theorem we have R(Σ, ω, σ) = x i (s 1,..., s N )MR i (s i )ds 1...ds N N V i () i=1 max f, MR 1 (s 1 ),..., MR N (s N )g ds 1...ds N The in the max expression can be viewed as a reserve price, that is a condition under which the item is not sold. We show on the next slide that the revenue bound can be achieved using a dominant strategy mechanism. Jonathan Levin Optimal Auctions Winter / 25

14 Proof, II: an optimal auction Ask bidders to report their types. Assign the item to the bidder with the highest marginal revenue, provided it exceeds zero. Payments are p i (t) = x i (t) v i MR 1 i max, max j6=i MR j (t j ). So bidder i pays only if she gets the item, and in that event she pays the value corresponding to the lowest type she could have reported and still been awarded the item. It s dominant to report one s true type and V i () = for all i. Jonathan Levin Optimal Auctions Winter / 25

15 Example An optimal auction may distort the allocation away from e ciency to increase revenue. Bidder 1 s value v 1 (t 1 ) = t 1, so value is U[, 1]. Bidder 2 s value v 2 (t 2 ) = 2t 2, so value is U[, 2]. Marginal revenues: MR 1 (t 1 ) = t 1 (1 t 1 ) = 2t 1 1 = 2v 1 1 MR 2 (t 2 ) = 2t 2 2(1 t 2 ) = 4t 2 2 = 2v 2 2. Bidder 1 may win despite having lower value, and auction may not be awarded despite both bidders having positive value. Jonathan Levin Optimal Auctions Winter / 25

16 Example, cont. Allocation in the optimal auction Jonathan Levin Optimal Auctions Winter / 25

17 Relation to Monopoly Theory (Bulow-Roberts) The problem is analogous to standard multi-market monopoly problem. Each bidder is a separate market in which the good can be sold. Quantity is analogous to probability of winning probabilities must sum to one like a quantity constraint. Monopolist can price discriminate, setting di erent prices in di erent markets. Allocating the probability of winning to individual markets is analogous to allocating quantities across markets. Solution in both cases: sell marginal unit where marginal revenue is highest, provided it is positive! Jonathan Levin Optimal Auctions Winter / 25

18 Optimal Reserve Prices Corollary Suppose valuation functions are identical v 1 =... = v N = v and v(s), MR(s) = v(s) (1 s)v (s) are increasing and take positive and negative values. A voluntary auction maximizes expected revenue i at equilibrium, V () =, and the auction is assigned to the high value bidder provided its value exceeds v(r), where r satis es MR(r) =. onathan Levin Optimal Auctions Winter / 25

19 More optimal auctions Corollary Suppose the valuation functions are identical and v(s), MR(s) are increasing. Then the following auctions are optimal: A second price auction with minimum bid v(r). A rst price auction with minimum bid v(r). An ascending auction with minium bid v(r). An all-pay auction with minimum bid v(r)r N 1. onathan Levin Optimal Auctions Winter / 25

20 Reserve Price Example Suppose N bidders and v(t) = t for all bidders. From above, MR(t) = 2t 1. The optimal reserve price is v(r) = 1/2, i.e. from MR(r) =. In a second price auction, each bidder bids its value, but doesn t bid if t < 1/2. In a rst price auction, each bidder places no bid if t < 1/2 and t. otherwise bids β(t) = N 1 N + (2t) N N Jonathan Levin Optimal Auctions Winter 29 2 / 25

21 Bulow-Klemperer Theorem Theorem Suppose that the marginal revenue function is increasing and that v i () = for all i. Then, adding a single buyer to an otherwise optimal auction but with zero reserve yields more expected revenue than setting the reserve optimally, that is, E [max fmr 1 (t 1 ),..., MR N (t N ), MR N +1 (t N +1 )g] E [max fmr 1 (t 1 ),..., MR N (t N ), g]. onathan Levin Optimal Auctions Winter / 25

22 Bulow-Klemperer Math Lemma E[MR i (t i )] = v i () Proof. so therefore E [MR i (s)] = TR i (s) = (1 s) v i (s) d MR i (s) = d (1 s) TR i (s) = d ds TR i (s) MR i (s)ds = [TR i (1) TR i ()] = v i (). onathan Levin Optimal Auctions Winter / 25

23 More Bulow-Klemperer Math Lemma Given any random variable X and any real number α, E [maxfx, αg] E [X ] and E [maxfx, αg] α, so E [maxfx, αg] max fe [X ], αg. Proof. This follows from Jensen s inequality. onathan Levin Optimal Auctions Winter / 25

24 Bulow-Klemperer Proof Applying the two lemmata, with X = MR N +1 (t N +1 ), E [Revenue, N + 1 bidders, no reserve] = E t1,...,t N,t N +1 [max fmr 1 (t 1 ),..., MR N (t N ), MR N +1 (t N +1 )g] = E t1,...,t N [E tn +1 [max fmr 1 (t 1 ),..., MR N (t N ), MR N +1 (t N +1 )g E t [max fmr 1 (t 1 ),..., MR N (t N ), g] = E [Revenue, N bidders, optimal auction]. Jonathan Levin Optimal Auctions Winter / 25

25 Using the RET... Bulow-Klemperer theorem is one example of how to us the RET to derive new results in auction theory. Many other results also rely on RET arguments McAfee-McMillan (1992) weak cartels theorem Weber (1983) analysis of sequential auctions Bulow-Klemperer (1994) analysis of dutch auctions Bulow-Klemperer (1999) analysis of war of attrition. Milgrom s chapter ve contains many examples. Jonathan Levin Optimal Auctions Winter / 25

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