Examples of auctioning similar items Auctioning Many • Treasury bills Similar Items • Stock repurchases and IPOs • Telecommunications spectrum • Electric power Lawrence Ausubel and Peter Cramton • Emissions permits Department of Economics University of Maryland 1 2 Pay-as-bid Auction: Ways to auction many similar items All bids above P 0 win and pay bid • Sealed-bid: bidders submit demand schedules – Pay-as-bid auction (traditional Treasury practice) Price – Uniform-price auction (Milton Friedman 1959) Supply – Vickrey auction (William Vickrey 1961) Aggregate Bidder 1 Bidder 2 P P P 0 P Demand (stop-out) + = Demand (Bids) Q S Quantity Q 1 Q 2 Q 3 4 Vickrey Auction: Uniform-Price Auction: All bids above P 0 win and pay opportunity cost All bids above P 0 win and pay P 0 Price Residual Supply Q S − ∑ j ≠ i Q j (p) Price Supply p 0 P 0 Demand Q i (p) (stop-out) Demand (Bids) Q S Q i (p 0 ) Quantity Quantity 5 6
Payment rule affects behavior More ways to auction many similar items Price Residual Supply • Ascending-bid: Clock indicates price; Q S − ∑ j ≠ i Q j (p) Pay-as-bid bidders submit quantity demanded at each price until no excess demand Uniform-Price – Standard ascending-bid p 0 – Ausubel ascending-bid (Ausubel 1997) Demand Vickrey Q i (p) Q i (p 0 ) Quantity 7 8 Ausubel Ascending-Bid: Standard Ascending-Bid Auction: All bids at P 0 win and pay price at which clinched All bids at P 0 win and pay P 0 Price Residual Supply Q S − ∑ j ≠ i Q j (p) Price Supply p 0 P 0 Demand Excess Demand Q i (p) Demand Excess Demand Clock Clock Q S Q i (p 0 ) Quantity Quantity 9 10 Research Program More ways to auction many similar items How do standard auctions compare? • Ascending-bid – Simultaneous ascending auction (FCC spectrum) • Efficiency • Sequential – FCC: those with highest values win – Sequence of English auctions (auction house) • Revenue maximization – Sequence of Dutch auctions (fish, flowers) – Treasury: sell debt at least cost • Optimal auction – Maskin & Riley 1989 11 12
Efficiency Inefficiency Theorem (not pure common value; capacities differ) In any equilibrium of uniform-price auction, • Uniform-price and standard ascending-bid with positive probability objects are won by – Inefficient due to demand reduction bidders other than those with highest values. • Pay-as-bid – Inefficient due to different shading • Vickrey • Winning bidder influences price with positive probability – Efficient in private value setting • Creates incentive to shade bid – Strategically simple: dominant strategy to bid true demand • Incentive to shade increases with additional units – Inefficient with affiliated information • Differential shading implies inefficiency • Ausubel ascending-bid – Same as Vickrey with private values – Efficient with affiliated information 13 14 Inefficiency from differential shading Vickrey inefficient with affiliation Large Bidder Small Bidder • Winner’s Curse in single-item auctions – Winning is bad news about value mv 1 • Winner’s Curse in multi-unit auctions – Winning more is worse news about value mv 2 P 0 – Must bid less for larger quantity D 1 D 2 – Differential shading creates inefficiency in b 1 b 2 Vickrey Q 2 Q 1 Large bidder makes room for smaller rival 15 16 What about seller revenues? Uniform price may perform poorly • Independent private values uniform on [0,1] Price Residual Supply Q S − ∑ j ≠ i Q j (p) • 2 bidders, 2 units; L wants 2; S wants 1 Pay-as-bid • Uniform-price: unique equilibrium – S bids value Uniform-Price p 0 – L bids value for first and 0 for second Demand – Zero revenue; poor efficiency Vickrey Q i (p) • Vickrey – price = v (2) on one unit, zero on other Q i (p 0 ) Quantity 17 18
Standard ascending-bid may be worse Efficient auctions tend to yield high revenues • 2 bidders, 2 units; L wants 2; S wants 2 Theorem. With flat demands drawn independently • Uniform-price: two equilibria from the same regular distribution, seller’s revenue is maximized by awarding good to those – Poor equilibrium: both L and S bid value for 1 with highest values. • Zero revenue; poor efficiency Generalizes to non-private-value model with – Good equilibrium: both L and S bid value for 2 independent signals: • Get v (2) for each (max revenue) and efficient v i = u(s i ,s -i ) • Standard ascending-bid: unique equilibrium Award good to those with highest signals if – Both L and S bid value for 1 downward sloping MR and symmetry. • S’s demand reduction forces L to reduce demand • Zero revenue; poor efficiency 19 20 Downward-sloping demand: p i (q i ) = v i − g i (q i ) But uniform price has advantages Theorem. If intercept drawn independently from the • Participation same distribution, seller’s revenue is maximized – Encourages participation by small bidders by (since quantity is shifted toward them) – awarding good to those with highest values if constant hazard rate – May stimulate competition – shifting quantity toward high demanders if increasing • Post-bid competition hazard rate – More diverse set of winners may stimulate • Note: uniform-price shifts quantity toward low competition in post-auction market demanders 21 22 Auctioning Securities Models A pure common-value model with affiliation • Common uncertainty • n risk-neutral symmetric bidders – Bidders have no private information • Each bidder has pure common value V for • Affiliated private signals security and can purchase any quantity – Bidder i gets signal S i (flat demand curve w/o capacity) – Random variables V, S 1 , …, S n are affiliated 23 24
Results: Common Uncertainty Results: Common Uncertainty Proposition. (Wilson ‘79; Maxwell ‘83; Back & Zender ‘93) Theorem. • Wide range of prices can be supported as equilibrium • Vickrey auction has a unique equilibrium that in uniform-price auction, even if supply is stochastic; survives elimination of weakly-dominated highest yields EV strategies Proposition. (Wang & Zender ‘96) • Vickrey auction has a unique symmetric equilibrium consistent with stochastic supply • Many equilibria in pay-as-bid auction, even if supply is stochastic; highest yields EV • This equilibrium revenue-dominates all equilibria of all auction formats consistent with voluntary • Indeterminacy avoided if set reserve price (even 0) bidder participation 25 26 Results: Affiliated Private Signals Results: Affiliated Private Signals Vickrey and Ausubel ascending-bid eliminate • With affiliated signals, each auction format bottom end of revenue indeterminacy: has a “simple equilibrium” where bidders Revenues submit flat demand curves • Conjecture: These simple equilibria provide upper bounds on revenues from each format • Alt. ascending-bid > Vickrey > Pay-as-bid • Std. ascending-bid > Uniform > Pay-as-bid Ausubel Pay-as- Standard Uniform Vickrey Ascending Bid Ascending Price Bid Bid 27 28 Conclusion • Efficient auctions should be favored • Treasury should try Ausubel ascending-bid • IPOs should be auctioned 29
Recommend
More recommend
