1 IPV and Revenue Equivalence: Key assumptions • Independence of values. • Risk-neutrality. • No budget constraints. • Symmetry (Same allocation rule!). • Other considerations: — Collusion — Resale possibilities 2 Risk-averse bidders • Each bidder has u : R+ → R with u(0) = 0, u0 > 0, and u00 < 0. Proposition: With risk-averse symmetric bidders the expected revenue in a first-price auction is greater than in a second-price auction. Intuition: Consider a bidder in the first-p rice auction. By reducing current bid b by some ∆, a bidder gains ∆ when wins, but increases a probability of losing, which has a greater effect on expected utility. Result: more aggressive bidding in the first-price auc-tion. No change in strategies in the second-price auction.Formally, suppose γ :[0,w] → R+ is an equilibrium strategy (incr. diff.). max z EU =max z G(z)u(x − γ(z)). FOC: g(z)u(x − γ(z)) − G(z)u 0(x − γ(z))γ0(z)=0. In symmetric eqm: γ0(x)= u(x − γ(x)) u0(x − γ(x)) g(x) G(x), β0(x)=(x − β(x)) g(x) G(x) . Note that for all y>0, u(y) u0(y) >y.Therefore, γ0(x) > (x − γ(x)) g(x) G(x) . Now β(x) >γ(x) ⇒ γ0(x) >β0(x). Together with β(0) = γ(0) = 0 we obtain β(x) <γ(x). 3 Budget-constrained bidders • Every bidder obtains value (signal) Xi ∈ [0, 1] and absolute budget Wi∈ [0, 1]. • (Xi,Wi) are iid across bidders. (Xi and Wi need not be independent.) Proposition: With budget-constrained bidders the ex-pected revenue in a first-price auction is greater than in a second-price auction. (provided symmetric equi-librium exists.) Intuition: The bids in second-p rice auction are higher on average and so are mo re often constrained. (Not enough: players will reduce bids in the first-price auction).Proof: In the second-price auction: βII(x, w)=min{x, w}. Define (effective type) xII ∼ (x, w)asthe type that is effectively unconstrained and submits the same bid as (x, w). Can be found as a solution to βII(x, w)= βII(x II , 1) = x II . Let Y II(N) 2 be the second highest of the equivalent values, xII i ,among N bidders. Its distribution is GII(z)= ³ F II(z) ´ N−1 , where F II(z) is the probability that βII(x, w)= βII(xII , 1) = xII <z = βII(z, 1). We have E[RII]= E ∙ Y II(N) 2 ¸ . In the first-price auction: Suppose a symmetric in-creasing equilibrium exists with βI(x, w)=min{β(x),w}. Define xI ∼ (x, w)asthe solution to βI(x, w)= βI(x I , 1) = β(x I) <xI . Let Y I(N) 2 be the second highest of the equivalent values, xI i,among N bidders. Its distribution is GI(z)= ³ F I(z) ´ N−1 . We have E[RI]= E ∙ Y I(N) 2 ¸ . Note that F I(z) <F II(z), and thus E[RI] >E[RII]. All-pay auctions dominate first-price auctions in terms of revenue generated to the seller.4 Asymmetric bidders • Revenue equivalence theorem a pplies only to mech-anisms (equilibria) with the same allocation rule. • Second price auction is efficient. • First price auction generally is not. — Weaker bidder will bid higher. • No general revenue ranking. 5Resale (and efficiency) • Intuition: If resale is possible, low-value bidders will bid more aggressively: revenue to the seller should be higher. Counter-argument: High-value bidders will bid less, and possibly will not reveal their values via bidding in the first period. • If outcome is efficient after resale (no additional information is exogenously revealed) revenue equiv-alence holds. • Second-price auction with resale: efficient, no re-sale happens. In general: Any efficient mechanism followed by 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 sim ple illustration: Two bidders, F1[0,w] 6= F2[0,w], E[X1] 6= E[X2]. The winner can make a take it or leave it offer to the loser. Claim: There is no efficient equilibrium in the first-price auction follo wed by resale that reveals valuations of the bidders in the first stage. (Why then efficiency would not be possible to obtain in general?) Suppose β1 and β2 are increasing strategies with in-verses φi = β−1 i . Step 1. β1(w)= β2(w)= ¯b. Step 2. Use revenue equivalence. Expected payments of a bidder with Xi have to be the same here and in the second price auction. Thus, β1(w)= E[X2] 6= E[X1]= β2(w). 6 Collusion Very brief: • Typically modelled as bidding rings. A bidding ring is a collection of bidders who exchange information, decide on the participation in the auction (who and how bids), decide on transfers. Analysis: Stability of a ring (coalition), Effects on the other bidders, and the seller. Counter-measures by the seller.• Second-price auction. A group of bidders exchange information (conduct an auction among themselves), the winner goes to the main auction and bids her value, others do not go o r bid 0. The ring obtains (in case of win) max ½ YN\i 1 ,r ¾ − max ½ YN\I 1 ,r ¾ , where i is the winner and I is the ring. Relatively easy to support. No bidder from the ring can go (incognito) to the main auction and benefit (need to overbid). Seller might respond by setting a higher reserve price. • First-price auction. Thesamestructure roughly. Now, however, a “representative” bidder can send a “friend” who will just overbid him, and thus capture all the spoils without sharing them among the mem-bers of the ring. Other types of collusion: Seller can cheat by inserting “fake” bids – has an effect in the second-price auction. In case of multiple units, by specific bidding patterns buyers can signal their intentions and support collu-sion.• Practical auction design: entry and collusion. Better to attract another bidder and have no re-serve than to set an optimal reserve price. (Sym-metry is crucial). English auction vs sealed-bid auctions: discour-ages entry, more susceptible to collusion. Open-ness and information revelation (feedback) maybe crucial. Multiple units for sale: other issues, parallel si-multaneous or sequential ascending price auctions are particularly susceptible. 7 Multi-unit auctions M units of the same object are offered for sale. Each bidder has a set of (marginal values) Vi = (Vi 1 ,Vi 2 ,...Vi M), the objects are substitutes, Vi k ≥ Vi k+1. Extreme cases: