Bandwidth auctions and their parallels to power Fernando Paganini . Universidad ORT, Uruguay . Outline: 1. Intro: resource allocation and pricing in comm networks. 2. Background on auctions. 3. Auctions for Internet bandwidth. 4. Connections to auctions in the power grid. Lund, May 2011
1. Intro : Rate allocation in the Internet Input rate x Link rate y r l C apacity c l x r Netw ork of links, indexed by w ith capacity (e.g. M bps). l , c l End-to-end flow s, indexed by rate r , x r if route uses link 1 r l Link rate aggregation: , w here y R x R l lr r lr 0 otherw ise r Netw ork Utility M axim ization (Kelly ´98) Rx c subject to max U ( x ) , x r r r LINK CAPACITY SO URCE CO NSTRAINTS UTILITY
Solution through duality (Low- Lapsley ’ 99) Lagrangian L x p ( , ) [ U ( x ) q x ] p c . r r r r l l r l " Lagrange m ultiplier p : congestion price" of link l . l T q R p ( total price of route q R p : i ). i li l l Route price q i Source rate x Link rate y i l Link price p l Dual algorithm Sources so lve argm ax [ U ( x ) q x ], x : r r r r r x r Links update prices as prices sent back. : [ y c ] , l l l l l
Uses for this “virtual economy” of bandwidth • Interpretation of the equilibrium and dynamic properties of current TCP congestion control protocols. • Guides in design of new protocols with: – Better dynamic properties (convergence, etc.) – Use other utility functions, achieving other notions of fairness . • Extension to cross-layer optimization, including other layers of the protocol stack, for both wired and wireless networks: – Routing – Medium access control (Scheduling, random access). – Physical layer control (power control, modulation,…) • However, the “real” economy of bandwidth doesn’t work this way. Why? – Bandwidth has been abundant, not crucial to optimally allocate it. – This control is faster than the human time-scale.
1. Auctions • Popular trading mechanism – Fast, reliable and transparent way of setting market price. – Various mechanisms exist, single and multi-unit auctions. • Open auctions for sale of a single unit – English: ascending bids, open-outcry, terminates when one bidder is left. – Dutch: descending bids, open-outcry, terminates when one bidder shouts “mine”. • Closed, sealed-bid auctions: – First price: highest bidder wins, pays his/her bid. – Second price (Vickrey): highest bidder wins, pays 2 nd bid.
Vickrey Auctions and truth revelation $5 Bidder of $5 bid wins the auction, but pays $3.7 for the item. $3.7 $2.5 • In a second-price auction, it is rational for participants to bid their true valuation for the item: – They gain no reduction in payment by bidding below their valuation. – They might lose the auction if they do so.
Multiple unit auctions (1) b ( 2 ) b (3) b ( 4 ) b 1 2 3 4 Exam ple: sell 3 units, choose 3 highest bidders. Alternatives: Charge bid am ount: gives incentives for bidding below valuation. Vickrey-Clarke-G roves (VCG ) principle: charge users the loss of valuation im posed to others by their presenc e. In this case: presence of 2nd bidder changes others' total valuation (1) (3) ( 4 ) (1) (3) ( 4 ) . from to Should charge b b b b b b It is shown VCG m akes it rational to reveal the true valuations.
Is there a cost for (1) b ( 2 ) b truth revelation in (3) b loss of revenue for ( 4 ) b the seller? 1 2 3 4 Revenue Equivalence Theorem (Vickrey, M yerson, Riley-Sam uelson) Assum e: Risk neutral buyers, valuations draw n from a know n distribution. M echanism assigns objects to bidders w ith highest valuation. At Bayesian Nash equilibrium , all auction m echanism s yield the sam e expected revenue for the auctioneer. However, equivalence does not hold if buyers are risk averse, do not know the distribution, or do not have unbounded rationality. In such cases a first-price auction m ay give higher revenue than VCG .
Procurement auctions ( 4 ) a Auctioneer buys one or m ore (3) a item s from low est offers (asks). ( 2 ) a e.g., buy 2 item s from offers 1, 2. (1) a (3) VCG : pay to b oth. . a 2 1 3 4 Double or two-sided auctions (1) b ( 4 ) a Bids for buyin g an d selli ng ( 2 ) b (3) a ( k ) ( k ) Sell m ax Here, 2 item s. k : b a . ( 3 ) b ( 2 ) a E quilibrium pri ce: crossin g point. ( 4 ) b (1) a 1 2 3 4
2. Internet Bandwidth Auctions Based on 2011 paper in Computer Networks. Joint work with: • Pablo Belzarena (Universidad de la República, Uruguay) • Andrés Ferragut (Universidad ORT, Uruguay) Scenario: a netw ork periodically auctions capacity. Users subm it bids for am ounts of end-to-end bandw idth. A distributed algorithm m ust optim ally assign capacity. M otivation: overlay for prem ium services over the Internet. : ’00, ’0 , Related w ork Lazar Sem ret Shu Varaiya 3 ’0 , ’0 , . ’0 Reichl W rzaczek 5 M aillé Tuffin 6 Courcoubetis et al 7.
Three issues and solution features 1. Auction allocation/payment mechanism: – Optimize the value of accepted bids. – Charge 1 st price, VCG would have high complexity (Maillé-Tuffin ´07). – Revenue equivalence argument. 2. Distributed auction over a general network topology. – Bids submitted to “bandwidth brokers” distributed across the network. – Bidders need not know network topology, capacity, etc. – Brokers run a distributed algorithm to allocate the auction. 3. Inter-temporal constraints. – Auctions are held periodically, for currently available capacity. – Service may be longer than the auction period, and reservations are in place: a connection, once assigned, cannot be displaced by future bids. – So the seller optimize over the risk of future bids: selling capacity now with a low bid can cause the rejection of a better bid in the future.
Notation: auction for a single service (1) ( 2 ) (3) ( N ) bids N b b b b for units of bandwidth. Revenue from allocating U ( ) a b bandw idth a N : (1) ( 2 ) b b (first-price auction) a ( ) i (1) U ( ) a b . b b i 1 Interpolate to piecew ise linear, concave function. 1 2 3 4 a
Auction over a network Let represent a service, characterized by a r reserved bandw idth betw een 2 endpoints. r Single-path case: service has a fixed route, defined by a routing m atrix : R iff route us es link R 1 r l . lr M ul tipath generalizations are available. ( N ) (1) ( 2 ) (3) Broker collects bids for this service: . b b b b r a r r r r r r ( ) i If w e adm it bids for a total rate a , the total revenue is U ( a ) b . b r r r r i 1 Network optim al revenue auction Integer program . r m ax U ( a ) r b Relaxation is a concave r utility m axim ization as a subject to r , R c a Z introduced in Kelly '98. lr r l r r
Distributed allocation algorithm r Lagrangian L U ( a ) ( , a ) [ c R a ] c r l l lr r l l b r l s l Dual algo rithm (from Low -La psley '99 ) Brokers solv e argm ax [ U ( a ) q a ], w ith q a : R a r r r r r b rl l r r l route price. Am ounts to selecting bids better than q . r Com m unicate to links, w ho update pri ce s as : [ y c ] , a l l l l l r w ith link bandw idth dem and. Prices sent back to brokers. y R a r l lr s Can be im plem ented in the control plane, variant of RSVP protocol . Difficulty: U ( a ) not strictly concave, algorithm will ``chatter" around optim um . r b r Solution: prox im a l optim ization with extra variable d r 2 r subject to max U ( a ) ( a d ) , R c , a b r r r lr r l r 2 r s r (see Li n-Shroff '06, also useful for m ultipath case).
Periodic auctions for one link Service of bandw idth , single link of capacity . 1 C k Collect bids for tim e of length allocate bandw idth units at tim e T , a kT Allocated users have a reservation for service duration, assum e exp( ) . T : probability that a connection is still active at the next auct ion. p e C k k x a Occupied k 1 k k x ~ Bin x ( a , p ). bandwidth k x k 1 x ( k 1) T ( k 1) T kT time k k M yopic policy : , sell all currently available capacity. a C x M ay m iss higher bids in the fu ture. W hat is the optimal policy?
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