2 Risk-averse bidders 1 IPV and Revenue Equivalence:

Key assumptions Each bidder has u : R+ R with u(0) = 0, • → u0 > 0, and u00 < 0. Independence of values. • Proposition: With risk-averse symmetric bidders the Risk-neutrality. expected revenue in a first-price is greater • than in a second-price auction.

No budget constraints. bidder in the first-price auction. • Intuition: Consider a

By reducing current bid b by some ∆, a bidder gains Symmetry (Same allocation rule!). • ∆ when wins, but increases a probability of losing, which has a greater effect on expected utility. Other considerations: • Result: more aggressive in the first-price auc­ — Collusion tion.

— Resale possibilities No change in strategies in the second-price auction. Formally, suppose γ :[0,w] R+ is an equilibrium strategy (incr. diff.). → 3 Budget-constrained bidders max EU =maxG (z)u(x γ(z)). z z − Every bidder obtains value (signal) X [0, 1] FOC: • i ∈ and absolute budget Wi [0, 1]. g(z)u(x γ(z)) G(z)u (x γ(z))γ (z)=0. ∈ − − 0 − 0 In symmetric eqm: (Xi,Wi) are iid across bidders. (Xi and Wi need u(x γ(x)) g(x) • γ0(x)= − , not be independent.) u0(x γ(x)) G(x) − g(x) β0(x)=(x β(x)) . − G(x) Proposition: With budget-constrained bidders the ex­ pected revenue in a first-price auction is greater than u(y) in a second-price auction. (provided symmetric equi­ Note that for all y>0, u (y) >y.Therefore, 0 librium exists.) g(x) γ (x) > (x γ(x)) . 0 − G(x) Intuition: The bids in second-price auction are higher on average and so are more often constrained. Now β(x) >γ(x) γ0(x) >β0(x). Together with β(0) = γ(0) = 0 we⇒ obtain (Not enough: players will reduce bids in the first-price β(x) <γ(x). auction). Proof: In the second-price auction: II β (x, w)=min x, w . I { } Define x ∼ (x, w)asthes olutiont o Define (effective type) xII (x, w)asthet ypet hat ∼ I I I I I is effectively unconstrained and submits the same bid β (x, w)=β (x , 1) = β(x ) E[RII]. We have

II II(N) E[R ]=E Y2 . ∙ ¸

In the first-price auction: Suppose a symmetric in­ All-pay dominate first-price auctions in terms creasing equilibrium exists with of revenue generated to the seller. βI(x, w)=min β(x),w . { } 5Resale(a ndeffi ciency)

Intuition: If resale is possible, low-value bidders • will bid more aggressively: revenue to the seller 4 Asymmetric bidders should be higher. Counter-argument: High-value bidders will bid Revenue equivalence theorem applies only to mech­ • less, and possibly will not reveal their values via anisms (equilibria) with the same allocation rule. bidding in the first period.

Second price auction is efficient. If outcome is efficient after resale (no additional • • information is exogenously revealed) revenue equiv­ alence holds. First price auction generally is not. • Second-price auction with resale: efficient, no re­ — Weaker bidder will bid higher. • sale happens. In general: Any efficient mechanism followed by No general revenue ranking. resale would have an equilibrium like that. • First-price auction with resale: inefficient in gen­ • eral (asymmetry), values are not revealed in the first period. A simple illustration: Two bidders, F1[0,w] = F2[0,w], 6 E[X1] = E[X2]. 6 6 Collusion The winner can make a take it or leave it offer to the loser. Very brief:

Claim: There is no efficient equilibrium in the first- price auction followed by resale that reveals valuations Typically modelled as bidding rings. • of the bidders in the first stage. (Why then efficiency would not be possible to obtain in general?) A bidding ring is a collection of bidders who exchange information, decide on the participation in the auction Suppose β1 and β2 are increasing strategies with in­ 1 (who and how bids), decide on transfers. verses φi = βi− . Analysis: Stability of a ring (coalition), ¯ Step 1. β1(w)=β 2(w)=b . Effects on the other bidders, and the seller. Step 2. Use revenue equivalence. Expected payments of a bidder with X have to be the same here and in i Counter-measures by the seller. the second price auction. Thus, β (w)=E [X2] = 1 6 E[X1]=β 2(w). Second-price auction. First-price auction. • •

A group of bidders exchange information (conduct an Thesamestructure roughly. auction among themselves), the winner goes to the main auction and bids her value, others do not go or Now, however, a “representative” bidder can send a bid 0. “friend” who will just overbid him, and thus capture all the spoils without sharing them among the mem­ The ring obtains (in case of win) bers of the ring.

i max Y1N\ ,r max Y1N\ I,r , ½ ¾ − ½ ¾ Other types of collusion: where i is the winner and is the ring. I Seller can cheat by inserting “fake” bids – has an Relatively easy to support. No bidder from the ring effect in the second-price auction. can go (incognito) to the main auction and benefit (need to overbid). In case of multiple units, by specific bidding patterns buyers can signal their intentions and support collu­ Seller might respond by setting a higher reserve price. sion. 7 Multi-unit auctions

M units of the same object are offered for sale. Practical auction design: entry and collusion. • Each bidder has a set of (marginal values) V i = i i i i Better to attract another bidder and have no re­ (V1 ,V2 ,...VM), the objects are substitutes, Vk i ≥ serve than to set an optimal reserve price. (Sym­ Vk+1. metry is crucial). Extreme cases: unit-demand, the same value for all vs sealed-bid auctions: discour­ objects. ages entry, more susceptible to collusion. Open­ ness and information revelation (feedback) maybe crucial. Types of auctions: • Multiple units for sale: other issues, parallel si­ multaneous or sequential ascending price auctions The discriminatory (“pay-your-bid”); • are particularly susceptible.

Uniform-price; •

Vickrey; • 8 Interdependent (common) values.

Each bidder receives private signal X [0,w ]. • i ∈ i (wi = is possible) Multi-unit English; ∞ • (X1,X2,...,Xn) are jointly distributed accord­ • Ausubel; ingtocommonlyknownF (f>0). •

Dutch, descending uniform-price, • • Vi = vi(X1,X2,...,Xn).

vi(x1,x2,...,xn) E Vi Xj = xj for all j . ... ≡ | • h Ni Typically assumed that functional forms vi i=1 are commonly known. { } Issues: Existence and description of equilibria, price series if sequential, efficiency, optimality, non-homogenous vi(0, 0,...,0) = 0 and E[Vi] < . goods, complementarities,... • ∞

Symmetric case: • vi(xi, x i)=v (xi, x i)=v (xi,π(x i)). − − − 9 Brief analysis 10 Second-price auction Common values / Private values / Affiliated val­ • ues / Interdependent values. Define

Winner’s curse. v(x, y)=E [V1 X1 = x, Y1 = y] . • |

Second-price auction: Pivotal bidding–I bid what Equilibrium strategy • I get if i just marginally win. βII(x)=v (x, x).

First-price auction: “Usual” analysis–differential • Indeed, equation, .... 1 β− (b) Π(b, x)= (v(x, y) β(y)) g(y x)dy 0 − | English auction: See below. Z 1 • β− (b) = (v(x, y) v(y, y)) g(y x)dy. Z0 − | Revenue ranking: English > SPA > FPA. Π is maximized by choosing β 1(b)=x ,thatis,b = • − β(x). (!) Interdependency and affiliation are important for the first part. 12 11 Example Define 1. Suppose S1,S2,andT are uniformly and inde- A W (z, x)=E [P (z) X1 = x, Y1