Multi-unit auctions exchanges multiple indistinguishable units of

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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 – Trans-Atlantic backbone bandwidth from NYC to Paris –…

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

Screenshot from e. Mediator [Sandholm AGENTS-00, Computational Intelligence 02]

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+1 st 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 • 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 • 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

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