Exchanges markets with many buyers and many sellers

Exchanges = markets with many buyers and many sellers Let’s consider a 1 -item 1 -unit exchange first

Exchange game in class • 1 buyer, 1 seller, 1 good • The agents’ valuations for the good are drawn uniformly from [0, 100]. This is common knowledge • The agents don’t know each others’ valuations

Does a good exchange mechanism exist ? • E. g: Keith is selling a car to Tuomas – Both have quasilinear utility functions – Each party knows his valuation, but not the other’s valuation – Probability distributions of valuations are common knowledge • Want a mechanism that is – Budget balanced: Keith gets what Tuomas pays – Pareto efficient: Car changes hands if and only if vbuyer > vseller – Individually rational: Both Keith and Tuomas get higher expected utility by participating than not • Thrm. Such a mechanism does not exist (even if randomized mechanisms are allowed) [Myerson-Satterthwaite] – This impossibility is at the heart of more general exchange settings (NYSE, NASDAQ, combinatorial exchanges, …) !

Multi-unit auctions & exchanges (multiple indistinguishable units of one item for sale) Tuomas Sandholm Carnegie Mellon University

Markets with multiple indistinguishable units for sale • Application examples – IBM stocks – Barrels of oil – Pork bellies – Trans-Atlantic backbone bandwidth from NYC to Paris –…

Multi-unit auctions: pricing rules • Auctioning multiple indistinguishable units of an item • Naive generalization of the Vickrey auction: uniform price auction – If there are k units for sale, the highest k bids win, and each bid pays the k+1 st highest price – Demand reduction lie [Crampton&Ausubel 96]: • k=5 • Agent 1 values getting her first unit at $9, and getting a second unit is worth $7 to her • Others have placed bids $2, $6, $8, $10, and $14 • If agent 1 submits one bid at $9 and one at $7, she gets both items, and pays 2 x $6 = $12. Her utility is $9 + $7 - $12 = $4 • If agent 1 only submits one bid for $9, she will get one item, and pay $2. Her utility is $9 -$2=$7 • Incentive compatible mechanism that is Pareto efficient and ex post individually rational – Clarke tax. Agent i pays a-b • b is the others’ sum of winning bids • a is the others’ sum of winning bids had i not participated

Multi-unit exchanges • Multiple buyers, multiple sellers, multiple units for sale • By Myerson-Satterthwaite thrm, even in 1 unit case cannot obtain all of • Pareto efficiency • Budget balance • Individual rationality (participation)

Multi-unit auctions & exchanges: Clearing complexity [Sandholm & Suri IJCAI-01 & new draft]

Screenshot from e. Mediator [Sandholm AGENTS-00]

Supply/demand curve bids Quantity Aggregate supply Aggregate demand profit psell pbuy Unit price profit = amounts paid by bidders – amounts paid to sellers Can be divided between buyers, sellers & market maker One price for everyone (“classic partial equilibrium”): profit = 0 One price for sellers, one for buyers ( nondiscriminatory pricing ): profit > 0

Nondiscriminatory vs. discriminatory pricing Quantity Aggregate demand Supply of agent 1 Supply of agent 2 psell pbuy Unit price One price for sellers, one for buyers ( nondiscriminatory pricing ): profit > 0 p 2 sell pbuy p 1 sell One price for each agent ( discriminatory pricing ): greater profit

Shape of supply/demand curves • Piecewise linear curve can approximate any curve • Assume – Each buyer’s demand curve is downward sloping – Each seller’s supply curve is upward sloping – Otherwise absurd result can occur • Aggregate curves might not be monotonic • Even individuals’ curves might not be continuous

Pricing scheme has implications on time complexity of clearing • Piecewise linear curves (not necessarily continuous) can approximate any curve • Clearing objective: maximize profit • Thrm. Nondiscriminatory clearing with piecewise linear supply/demand: O(p log p) – p = total number of pieces in the curves • Thrm. Discriminatory clearing with piecewise linear supply/demand: NP-complete • Thrm. Discriminatory clearing with linear supply/demand: O(a log a) – a = number of agents • These results apply to auctions, reverse auctions, and exchanges • So, there is an inherent tradeoff between profit and computational complexity