Optimal auction design

Optimal auction design Roger Myerson Mathematics of Operations research 1981

Auctions • What is an auction? • Agreement between seller and bidders • Who gets the item? • How much does everyone pay?

Optimal auction design problem • The seller has a single item to sale • She doesn’t know how bidders value the item • She wants to make as much money as possible

The setting • A seller has 1 item for sale, which she values at 0 • A set of bidders: bidder i’svaluation (type) ti towards the item is private info • Others view ti as a random variable in [ai, bi] drawn from Fi(ti) • An outcome: a probability pi of allocation and a payment xi,for each i • Who gets the item at what price • Bidder’s utility: ui = pi ti-xi • Seller’s goal: maximizes her expected utility/revenue thru a mechanism • Bidders maximize their expected utility

Auction mechanisms • A mechanism • Specifies a set Ai of actions for each bidder i • Outcome function: a1×…an  outcome • A bidder i’sstrategy si(): [ai, bi]  Ai • Bidders’ strategies forms a (Bayes) Nash equilibrium • Infinite space: action can be anything!

Revelation principle • Direct revelation mechanisms • Everyone’s action is to report a valuation (Ai=[ai,bi]) • Being truthful is an equilibrium (incentive compatible) • Revelation principle • It is WLOG to focus on direct revelation mechanism • In other words, anything outcome implemented by a mechanism can also be implemented by a direct revelation mechanism

Proof

The seller’s problem • Design direct revelation mechanism (p(t),x(t)), so as to maximize Et(∑i xi(t)) where (t=(t1,…tn)) • Subject to • Incentive compatibility (IC) • truthful is NE • Individual rationality (IR) • participation • Resource feasibility (RF) • Seller should never

Analysis: constraints simplification • Interim allocation probability • Lemma: Constraints simplification • IC, IR and RF iff

Proof

Analysis: objective simplification lemma • Subject to RF and Q being increasing and

Proof

Optimal auction: the regular case • Virtual value: ti-(1-Fi(ti))/fi(ti) • Regularity: ti-(1-Fi(ti))/fi(ti) is increasing in ti • So that Q is increasing (last constraint satisfied) • Allocation rule: give the item to the highest non-negative virtual value • Payment rule: max {0, Inverse of 2nd highest VV }

Summary • Upon receiving bids ti from each bidder i • The seller calculates VV: ti-(1-Fi(ti))/fi(ti) • The seller gives the item to j who has the highest non-negative VV • The seller charges j the amount that would tie him to the 2nd highest VV • If all VV are negative, the seller keeps the item

Discussion: bidders may have different virtual valuation functions Actual valuation ranking t3 t1 t2 Virtual valuation ranking ~t1 ~t3 ~t2 winner is 2

Discussion: symmetric bidders • Assume Fi=Fj (symmetric bidders) • Every bidder has the same virtual valuation function • Myerson auction is 2nd-price auction with a reserve price

Discussion: Fi=Fj Actual valuation ranking reserve t1 t2 t3 Virtual valuation ranking 0 ~t1 ~t2 ~t3 winner is 3

Exercise: envelope theorem [Milgron, <Putting auction theory to work>]

Recent progresses • Optimal auction • Single item setting (Myerson) • Multiple identical item (Maskin&Riley) • Combinatorial items with single parameter (Levin) • Two items with discrete distribution (Armstrong) • Approximate optimal auction • 2nd-price auction 2-approximates Myerson (Hartline and Roughgarden, EC-09) • VCG 2-approximates Levin (Tang and Sandholm, IJCAI-11) • One bidder, two item: Separate Myerson 2-approximates optimal (Hart-Nisan, EC-12) • Unfortunately: even for 1 bidder, 2 item case, the optimal auction is unknown! • Two sellers?

Optimal auction design