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Multi-unit auctions & exchanges (multiple indistinguishable units of one item for sale). Tuomas Sandholm Computer Science Department Carnegie Mellon University. Auctions with multiple indistinguishable units for sale. Examples IBM stocks Barrels of oil Pork bellies

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Multi unit auctions exchanges multiple indistinguishable units of one item for sale

Multi-unit auctions & exchanges (multiple indistinguishable units of one item for sale)

Tuomas Sandholm

Computer Science Department

Carnegie Mellon University


Auctions with multiple indistinguishable units for sale
Auctions with multiple indistinguishable units for sale

  • Examples

    • IBM stocks

    • Barrels of oil

    • Pork bellies

    • Trans-Atlantic backbone bandwidth from NYC to Paris


Bidding languages and expressiveness
Bidding languages and expressiveness

  • These bidding languages were introduced for combinatorial auctions, but also apply to multi-unit auctions

    • OR [default; Sandholm 99]

    • XOR [Sandholm 99]

    • OR-of-XORs [Sandholm 99]

    • XOR-of-ORs [Nisan 00]

    • OR* [Fujishima et al. 99, Nisan 00]

    • Recursive logical bidding languages [Boutilier & Hoos 01]

  • In multi-unit setting, can also use price-quantity curve bids


Multi unit auctions exchanges multiple indistinguishable units of one item for sale

Screenshot from

eMediator

[Sandholm AGENTS-00,

Computational Intelligence 02]


Multi unit auctions pricing rules
Multi-unit auctions: pricing rules

  • Auctioning multiple indistinguishable units of an item

  • Naive generalization of the Vickrey auction: uniform price auction

    • If there are m units for sale, the highest m bids win, and each bid pays the m+1st highest price

    • Downside with multi-unit demand: Demand reduction lie [Crampton&Ausubel 96]:

      • m=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

    • I.e., if i wins n items, he pays the prices of the n highest losing bids

    • What about revenue (if market is competitive)?


General case of efficiency under diminishing values
General case of efficiency under diminishing values

  • VCG has efficient equilibrium. What about other mechanisms?

  • Model: xik is i’s signal (i.e., value) for his k’th unit.

    • Signals are drawn iid and support has no gaps

    • Assume diminishing values

  • Prop. [13.3 in Krishna book]. An equilibrium of a multi-unit auction where the highest m bids win is efficient iff the bidding strategies are separable across units and bidders, i.e., βik(xi)= β(xik)

    • Reasoning: efficiency requires xik >xir iff βik(xi) > βir(xi)

      • So, i’s bid on some unit cannot depend on i’s signal on another unit

      • And symmetry across bidders needed for same reason as in 1-object case


Revenue equivalence theorem which we proved before applies to multi unit auctions
Revenue equivalence theorem (which we proved before) applies to multi-unit auctions

  • Again assumes that

    • payoffs are same at some zero type, and

    • the allocation rule is the same

  • Here it becomes a powerful tool for comparing expected revenues


Multi unit auctions clearing complexity sandholm suri ijcai 01
Multi- to multi-unit auctionsunit auctions: Clearing complexity[Sandholm & Suri IJCAI-01]


Multi unit auctions exchanges multiple indistinguishable units of one item for sale

In all of the curves together to multi-unit auctions


Multi unit auctions exchanges multiple indistinguishable units of one item for sale

to multi-unit auctions


Multi unit reverse auctions with supply curves
Multi-unit reverse auctions with supply curves to multi-unit auctions

  • Same complexity results apply as in auctions

    • O(#pieces log #pieces) in nondiscriminatory case with piecewise linear supply curves

    • NP-complete in discriminatory case with piecewise linear supply curves

    • O(#agents log #agents) in discriminatory case with linear supply curves


Multi unit exchanges
Multi-unit to multi-unit auctionsexchanges

  • 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)


Supply demand curve bids

profit to multi-unit auctions

psell

pbuy

  • One price for everyone (“classic partial equilibrium”):

  • profit = 0

  • One price for sellers, one for buyers ( nondiscriminatory pricing ): profit > 0

Supply/demand curve bids

profit = amounts paid by bidders – amounts paid to sellers

Can be divided between buyers, sellers & market maker

Quantity

Aggregate supply

Aggregate demand

Unit price


Nondiscriminatory vs discriminatory pricing

p to multi-unit auctionsbuy

p2sell

p1sell

One price for each agent

( discriminatory pricing ): greater profit

Nondiscriminatory vs. discriminatory pricing

Aggregate demand

Quantity

Supply of agent 1

Supply of agent 2

Unit price

psell

pbuy

One price for sellers, one for buyers

( nondiscriminatory pricing ):

profit > 0


Shape of supply demand curves
Shape of supply/demand curves to multi-unit auctions

  • 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
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 – even without worrying about incentives