1 Notable features of

Ancient “market” mechanisms. Widespread in • use. A lot of varieties.

Simple and transparent games (mechanisms). Uni- Auctions 1: • versal rules (does not depend on the object for sale), anonymous (all bidders are treated equally). Common auctions & Operate well in the incomplete information envi- • Revenue equivalence & ronments. Seller (and sometimes bidders as well) Optimal mechanisms does not know how the others value the object.

Optimality and efficiency in broad range of set- • tings.

Probably the most active area of research in eco- • nomics. 2Notation(S ymmetricIP V) 3 Common auctions

Independent private values setting with symmetric risk- SEALED-BID Auctions. neutral buyers, no budget constraints. First price sealed-bid : • Single indivisible object for sale. Each bidder submits a bid b (sealed, or un- • i R observed by the others). The ∈winner is the buyer with the highest bid, the winner pays her bid. N potential buyers, indexed by i. N commonly • known to all bidders. Second price sealed-bid auction: • As above, the winner pays second highest bid – X – valuation of buyer i – maximum willing- • i highest of the bids of the others. ness to pay for the object.

Kth price auction: • Xi ∼ F [0,ω]withcontinuousf = F and full • 0 support. The winner pays the Kth highest price.

All-pay auction: X is private value (signal); all X are iid,which • • i i is common knowledge. All bidders pay their bids. : • OPEN (DYNAMIC) Auctions. The price of the object starts at zero and in- creases. Bidders start active – willing to buy the : object at a price of zero. At a given price, each • bidder is either willing to buy the object at that The price of the object starts at some high level, price (active) or not (inactive). While the price is when no bidder is willing to pay for it. It is de- increasing, bidders reduce(*) their demands. The creased until some bidder announces his willing- auction stops when only one bidder remains ac- ness to buy. He obtains the object at this price. tive. She is the winner, pays the price at which Note: Dutch and First-price auctions are equiva- the last of the others stopped . lent in strong sense. Note: English auction is in a weak sense equiva- lent to the second-price auction. 4 First-price auction Expected payoff from bidding b when receiving xi is 1 G (β− (b)) (xi b). Payoffs Y1 × − FOC: xi bi, if bi > maxj=i bj, Πi = − 6 g(β 1(b)) ( 0, otherwise. − 1 1 (x b) G(β− (b)) = 0. β0(β− (b)) − − Proposition: Symmetric equilibrium strategies in a first-price auction are given by In symmetric equilibrium, b(x)=β (x), so FOC ⇒ I β (x)=E [Y1 Y1

1. Suppose values are uniformly distributed on [0, 1]. Proposition: In a second-price sealed-bid auction, it N 1 F (x)=x, then G(x)=x − and is aweaklydominantstrategytobid N 1 βII(x)=x. βI(x)= − x. N

2. Suppose values are exponentially distributed on In the second price auction expected payment of the [0, ). winner with value x is the expected value of the second ∞ highest bid given x, which is the expectation of the F (x)=1 e λx,fors omeλ> 0a ndN =2, − − second-highest value given x. then x F (y) βI(x)=x dy Thus, expected payment in the second-price auction − Z0 F (x) is 1 xe λx = − . I λ − 1 e λx m (x)=Pr[Win] E [Y1 Y1

Independent private values setting with risk-neutral buyers, no budget constraints. Not necessarily sym- A selling mechanism ( ,π,µ): metric. B – a set of messages (or bids) for player i. •Bi Single indivisible object for sale. π : ∆ – allocation rule; here ∆ is the set • • B → of probability distributions over N.

N potential buyers, indexed by i. N commonly n • µ : R –payment rule. known to all bidders. • B →

X – private valuation of buyer i – maximum Example: First- and second-price auctions. • i willingness to pay for the object.

Every mechanism defines an incomplete information

Xi ∼ Fi[0,ωi]withcontinuousf i = F and full • i0 game: support, independent across buyers. β :[0,ω ] is a strategy; • i i → Bi = N , = , f(x) is joint den- Equilibrium is defined accordingly. •X ×i=1Xi X i ×j=iXj sity. − 6 • 9 Revelation principle 10

Direct mechanism (Q, M): Define qi(zi)andm i(zi)tobeaprobabilitythat i gets the object and her expected payment from reporting i = i; •B X zi while every other bidder reports truthfully: Q : ∆,whereQ (x) is the probability that • X → i i gets the object. qi(zi)= Qi(zi, x i)f i(x i)dx i, i − − − − n ZX − M : R ,whereM i(x) is the expected pay- • X → mi(zi)= Mi(zi, x i)f i(x i)dx i. ment by i. i − − − − ZX−

Expected payoff of the buyer i with value xi and re- Proposition: (Revelation principle) Given a mecha- porting zi is nism and an equilibrium for that mechanism, there q (z )x m (z ). exist a direct mechanism in which: i i i − i i 1. it is an equilibrium for each buyer to report truth- fully, and 2. the outcomes are the same. Direct mechanism (Q, M)is incentive compatible (IC) if i, x ,z , equilibrium payoff function U (x )satisfies ∀ i i i i U (x ) q (x )x m (x ) q (z )x m (z ). Proof: Define Q(x)=π (β(x)) and M(x)=µ (β(x)). i i ≡ i i i − i i ≥ i i i − i i Verify. U convex i →

IC implies that U is absolutely continuous i → Ui(xi)= max qi(zi)xi mi(zi) zi i { − } ∈X Ui is differentiable almost everywhere (Ui0(xi)=q i(xi) – maximum of a family of affine functions, thus, and so q (x ) is non-decreasing) i i → Ui(xi) is convex.

Ui is the integral of its derivative: x By comparing expected payoffs of buyer i with zi of i Ui(xi)=U i(0) + qi(ti)dti. reporting truthfully (zi) and of reporting xi,weo b- Z0 tain: Conclusion: The expected payoff to a buyer in an in- Ui(zi) Ui(xi)+ qi(xi)(zi xi), ≥ − centive compatible direct mechanism (Q, M) depends so qi(xi)isthes lopeofthel inethat“supports”U i(x) (up to a constant) only on the allocation rule Q. at xi.

Note: IC q (x) is non-decreasing. ⇐⇒ i 11.1 An application of Revenue Equiva- lence 11 Revenue Equivalence Consider symmetric (iid) environment.

In the second-price auction Proposition: (Revenue Equivalence) If the direct mech- anism (Q, M) is incentive compatible, then i, x the βII(x)=x. ∀ i expected payment is and

xi II m (x )=m (0) + q (x )x q (t )dt . m (x)=G (x) E [Y1 Y1

Consider direct mechanism (Q, M). 12 Individual rationality The expected revenue to the seller is

E[R]= E[mi(Xi)], where Direct mechanism (Q, M)is individually rational (IR) i X∈ ωN if i, x , i i E[mi(Xi)] = mi(xi)fi(xi)dxi ∀ 0 Z ω Ui(xi) 0. i ≥ = mi(0) + qi(xi)xifi(xi)dxi 0 ωi xZi Corollary: If mechanism (Q, M)is IC then it is IR qi(ti)fi(xi)dtidxi. − Z0 Z0 if for all buyers Ui(0) 0(orm i(0) 0). The last term is equal to (with changing variables of ≥ ≤ integration)

ωi ωi ωi qi(ti)fi(xi)dxidti = (1 Fi(ti)) qi(ti)dti. Z0 Zti Z0 − Substituting back 14 Solution

Define the virtual valuation of a buyer with value xi as 1 Fi(xi) ψi(xi)=x i − . E[mi(Xi)] − fi(xi)

ωi 1 Fi(xi) = mi(0) + xi − qi(xi)fi(xi)dxi Z0 Ã − fi(xi) ! Then seller should choose (Q, M) to maximize 1 Fi(xi) = mi(0) + xi − Qi(x)f(x)dx. Ã − fi(xi) ! ZX mi(0) + ψi(xi)Qi(x) f(x)dx. ⎛ ⎞ iX ZX iX ∈N ⎝ ∈N ⎠ Look at i ψi(xi)Qi(x). It is best to give the ∈N Optimal mechanism maximizes E[R] subject to: IC highest wPeights Qi(x) to the maximal ψi(xi). and IR.

Design problem is regular if for i, ψ ( )isanincreas- ∀ i · ing function of xi.

Regularity would imply incentive compatibility of the optimal mechanism. Define

yi(x i )= inf zi : ψi(zi) 0andψ i(zi) max ψj(xj) − ( ≥ ≥ j=i ) The following is the optimal mechanism (Q, M): 6 – the smallest value for i that “wins” against x i . − Allocation rule Q: • Thus,

Qi(x) > 0 ψi(xi)=maxψj(xj) 0. 1, if zi >yi(x i ), ⇐⇒ j ≥ Qi(zi, x i )= − ∈N − ( 0, if zi yi(x i ), Qi(zi, x i )= − − − 0 − ( 0 , if zi