Combinatorial Auctions and Multicast Cost-Sharing: CS 286r by David C. Parkes, Study notes of Interface between Computer Science and Economics

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Combinatorial Auctions

David C. Parkes

[email protected] Spring, 2002

Basics

Bidsget B (”. Auction mechanism determines thep, S) for set S ⊆ G of items, “I only want allocation A if I also, and agent payments. Variants:

• Bidding languages:– OR-of-XORS, XOR-of-ORs, restricted bundles, etc.
• Iterative vs. Sealed-bid vs. Multi-round
• Linear, Non-linear, Non-anonymous ask prices
• First-price vs. Second-price (GVA)
• Reserve price

FCC Spectrum Auction

• 51 Major Trading Areas (MTAs), 30 MHz spectrum perMTA. 492 Basic Trading Areas (BTAs), each with one 30 Mhz and four 10 MHz blocks.– 51 × 1 + 492 × (1 + 4) = 2511 items
• Clear efficiencies to aggregating licenses.– fixed cost of infrastructure, marketing, roaming synergies, etc.
• Simultaneous ascending-price auction.– prices on items
  • complex activity , and stopping rules, click-box bidding.
• 1994–2001, more than $40 billion.

Auction 31: Combinatorial

• 700 MHz auction, 12 licenses (6 regions, 10 MHz and20 MHz in each)
• Diverse preferences (30 MHz for high-speed dataservice; “fill holes”; build a “national footprint”).
• Limited number of bundles, XOR-across rounds, ORwithin round, stopping rules, activity rules.
• Still debated, and still not happened.

Multicast Cost-Sharing

[FPS01,JV01] Network per-node, ( N, Ei ∈ N), cost. Source node, ce ≥ 0 per-edge, αS ∈ N e. ∈ E, value vi Task: T ∗ (^) ⊆ (^) Select receiver-setE, to maximize social welfare: R∗^ ⊆ N , and multi-cast tree W (v) = max R⊆N^ [∑ i∈R vi − (^) T ∈minT (R) e^ ∑∈T ce^ ] [EFF] where T (R) is the set of all trees that “touch” R. Self-interested receivers, private information about values.Must collect payments to balance total cost to network. Feigenbaum et al. decentralized algorithms to implement mechanisms.: assume a Universal tree, propose Jain & Vazirani propose a centralized: assume a general biconnected network, approximation mechanism.

Example: Universal Tree Multicast

Welfare=40+15-(20+10)=

Mechanism Design Problem

Values,announce vi , of receivers are private information. Agentsvˆ = (ˆv 1 ,... , vˆ|N |). Propose a mechanism to compute receiversand tree T (ˆv). Let rwi(ˆiv(ˆv) (^) ) =∈ { 0 q,i (^) (ˆ (^1) v}), paymentsvi − xi(ˆv) x.i(ˆv) ≥ 0 , Desirable properties to achieve in equilibrium:[BB] ∑ [EFF] implement eff. outcome^ i∈N^ xi(ˆv) =^ ∑^ e∈T^ (ˆ (vR)^ c∗e, T ∗) [VP] [NPT] x ix(ˆiv(ˆv) (^) )≤ ≥ q i 0 (ˆv)ˆvi [CS] ri(ˆv) = 1, if vˆi large enough What about the solution concepts?

Solution Concepts

• Strategy-proof w [SP] i(vi,^ v−i)^ ≥^ wi(ˆvi,^ v−i),^ ∀v−i,^ ∀vˆi
• Group Strategy-proof for all coalitions S ⊆ [GSP] N , we have: ∀v−S , ∀vˆS wi(vS , v−S ) = wi(ˆvS , v−S ), ∀i ∈ S or ∃i ∈ S s.t. wi(vS , v−S ) > wi(ˆvS , v−S )

Distributed Implementations

[Feigenbaum et al. 01]

• VCG, or “marginal cost”. (SP) and (EFF).– pgva,i = ri(ˆv)ˆvi − (W (ˆv) − W (ˆv−i))
  • propose a mechanism to compute with 2 messagesper-link (tree-traversal).
• Shapley. (GSP) and (BB).– cost-sharing function ξ(R, i) to share cost of multicast to receivers – select maximal number of receivers that can stand the R, with ∑ i ξ(R, i) = C(R) cost-share– any implementation has worst-case linear number of messages per-link Network complexity: total # msgs, max # msgs. on any link,max size msg, local comput. burden on agents.

Budget-Balanced Mechanisms for

General Multicast Tree Problem

Moulin&Shenker, 01 Class of (GSP), (BB) mechanisms defined by cross-monotone cost-sharing methods, ξ(Q, i), with properties: ∑^ ξ(Q, i) = 0,∀i^6 =^ Q i ξ^ ξ(Q, i(Q, i) ) =≥ ξ^ C(R, i(Q))∀Q ⊆ R M (ξ) mechanism:

• Agents report values, vˆ; initialize R ← N.
• Select an agentdrop i from R. i ∈ R at random, if vˆi < ξ(R, i) then
• Continue until vˆi ≥ ξ(R, i) for all i ∈ R.