Auction Theory, Lecture Notes - Computer Science, Study notes of Game Theory

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Auction Theory

David C. Parkes

Division of Engineering and Applied Science, Harvard University

CS 286r–Spring 2002

Auctions: A Special Case of Mech.

Design

• Allocation problems
  • finite set G of items to allocate
  • variations possible (e.g. information goods, configurable items)
  • 1:N settings typical, N:M possible.
• Agent models
  • private values vs. common values
  • “no externalities”
  • quasi-linear, i.e. ui(S, p) = vi(S) − p for item(s)

S ⊆ G at price p; i.e. risk-neutral

• Mechanism properties
  • Budget-balanced (“trading mechanisms”)
  • efficiency (maximize total value ), or revenue (maximize the utility of a single agent)

Private Values

• Single-item variations
• Reverse auctions
• Iterative vs. sealed-bid
• Collusion, trust, privacy
• Variations: double auctions, multi-unit auctions, combinatorial auctions, multiattribute auctions, etc.

Single-item: Efficient [Vickrey 61]

• English/Vickrey (second-price)
• Dutch/FPSB (first-price) All efficient. (Why does Vickrey not break Green-Laffont imposs?) Also, all revenue-equivalent (if IID, quasi-linear, symmetric).

Let v(k) denote the k-th order statistic.

• First-price Sealed-bid/Dutch
  • best-response, B(v) = E[v(2)|v(1) = v]; expected

revenue, E[B(v(1))] = E[v(2)]

• Vickrey/English
  • revenue E[V(2)] Thm. [Rev. Equiv.] In any efficient auction, the expected payoff to every bidder, and the seller is the same.

As a Constrained Optimization Problem...

Consider a single-item allocation problem with N agents, let

pi(v 1 ,... , vN ) denote the expected payment from agent i

to the mechanism and, xi(v 1 ,... , vN ), denote the

probability with which agent i is allocated the item. Let v 0

denote the value of the seller.

{p^ max i ,xi }

∑^ N

i=

E [pi(v 1 ,... , vN )] −

∑^ N

i=

E [xi(v 1 ,... , vN )] v 0

s.t.

∑^ N

i=

xi(v 1 ,... , vN ) ≤ 1 , ∀v (feas) Eui(vi) ≥ Eui(ˆvi), ∀vˆi 6 = vi, ∀vi, ∀i (IC) Eui(vi) ≥ 0 , ∀vi (IR) where Eui(ˆvi) = Ev−i [xi(v 1 ,... , ˆvi,... , vN )vi]−E [pi(v 1 ,... , vˆi,... , vN )]

Effect of an Aftermarket [Ausubel & Cramton 99]

• Optimal auction designs makes two assumptions:
  • seller can prevent resale
  • seller can commit not to sell goods withheld after the auction
• Assume “perfect resale” (all gains from trade exhausted in resale)
  • seller’s incentives to misassign goods now destroyed
  • optimal to be efficient In addition: efficient marketplaces often will be the only markets to survive in long-term competition with other marketplaces [“larger pie to share”].

Closing Rules [Roth & Ockenfels 01]; eBay vs. Amazon (now dead).

• eBay [hard closing rule]
  • industry in “sniping”, favors bidders with better technology
  • empirically, limits information revelation during the auction, many bidders do not use proxy agents [esp. experienced bidders]
  • bidders can implicitly collude; avoid price wars; “strategic demand reduction” in a long-term game
  • at the end there is a probability that bids will fail, helps commitment issue.
• Amazon [soft closing rule]
  • removes this “arms race” for bidding technology
  • empirically, encourages bidding earlier in the auction
  • now it is hard to enforce implicit collusion

Multi-period Auctions (e.g. Priceline, eBay, etc.)

You want a single item, and can participate in a sequence of Vickrey auctions. What should you do?

• The strategyproofness of Vickrey is quite brittle. [design of s’proof seq. auctions is an interesting open problem]

Iterative vs. Sealed-bid

• Cost of communication
• Cost of delay
• Cost of information revelation
• Common vs. Private values
• Cost of valuation
• Ability to manipulate
• Cost of participation
• Transparency

Collusion

• FCC auction. Simultaneous ascending-price auction for multiple licenses. Collusion, “strategic-demand reduction” via trailing digits.
• Bidder rings. [Robinson 85] Group of bidders get together beforehand, and decide that only one will participate in the auction. Share gains afterwards.
  • problems in reaching an agreement, sharing rewards
  • first-price [Dutch, FPSB], this collusion is not self-enforcing because the selected bidder must submit a very small bid
  • second-price [Vickrey, English], this collusion is self-enforcing, because deviators are punished.
  • shills , “pulling bids off chandelier”, are a tool for sellers to fight collusion

Information Revelation [Rothkopf et al. 90]

• In a contracting example, the Vickrey auction awards a contract to the lowest bidder, but makes payment equal to the second-lowest bid.
  • Political problems?
• Repeated auctions. In the context of repeated auctions, whenever I reveal my true value for an item, that can be used against me in the future.
  • Business implications, within a supply-chain context? perhaps English auctions have more desirable properties? computational remedies?

Double Auctions Multiple buyers, multiple sellers, each with private

information. Suppose bids, b 1 ≥ b 2 ≥... ≥ bm, and asks,

s 1 ≤ s 2 ≤... sn. Compute l∗, s.t. bids i ≤ l∗^ and asks

j ≤ l∗^ trade; and determine payments.

• strategyproof, efficient and budget-balanced impossible
• McAfee-Double auction
  • compute a payment based on the bids not quite accepted, use this when IR; otherwise, implement one less trade.
  • strategy-proof, BB, not EFF.
• k-DA
  • clear double auction to maximize reported surplus
  • set a price equal to sl∗^ + k(bl∗^ − sl∗^ ), for some

k ∈ [0, 1].

• not strategyproof or EFF, but BB and “good” EFF in practice, in particular for large markets.

Multi-unit Auctions

Single bid, (ki, bi), for ki units, from each agent. Let

xi ∈ { 0 , 1 } define whether bid i is accepted, and pi denote

payment by agent i.

(1) compute x∗^ to solve (weighted knapsack) problem:

V ∗^ = max x

i

xipi

s.t.

i

xiki ≤ N

(2) compute payments, pi = bi − (V ∗^ − V −i) if xi = 1,

with pi = 0 otherwise; where V −i^ is maximal value over

subproblem induced by removing bid from agent i.

Note. exclusive-or bid generalizations easy to define.

Multi-unit Auctions: Approx. Use greedy method to select the winning bids: (1) sort in decreasing per-unit bid price (2) greedily accept, with highest per-unit bid price first. Then (a) compute price as per-unit price of first rejected bid; or (b) use VCG rule to compute price.

Prop. Payment rule (a) is strategy-proof; but VCG is no longer strategy-proof.

...good example of problems with introducing approximations into the VCG mechanism.