Combinatorial Auctions Practical Applications, Generators, and Equilibria

Transcription

1 Combinatorial Auctions Practical Applications, Generators, and Equilibria Benjamin Lubin Boston University Questrom School of Management Joint work with: Sven Seuken (UZH) Michael Weis (UZH) David Parkes (Harvard) Jeff Kephart (IBM) FOR CLASS USE ONLY NOT FOR REDISTRIBUTION

2 Combinatorial Auctions 2

3 What is a Combinatorial Auction? A mechanism that mediates the purchase or sale of bundles of goods Multiple buyers (or sellers in a reverse auction) Allows expression of complements and substitutes in value over goods Example: Delta values a landing slot at LGA at $3,000 But it values 10 landing slots 1 hour apart for more then $30,000 because this allows a shuttle service FAA wants a market What mechanism should we use for allocation? 3

4 Combinatorial Auctions Are Everywhere 4 Internet Advertising Markets Supply Chain / Procurement CombineNet: More than $35B in CAs Saved $4B Walmart, Target, Proctor & Gamble Bandwidth Auctions Computational Resource Allocation Benjamin Lubin - NOT FOR REDISTRIBUTION 4

5 Sealed-Bid Combinatorial Auctions Bidder 1 Bidder 2 Center Allocation & Payments Bidder n Bidders state their values for multiple bundles of items, e.g.: {(A,3), (B,5), (C,2), (AB, 9), (ABC, 15)} Center determines: winners and their allocation payments 5

6 Value-Queries & Bidding Languages 2014 Canadian Spectrum Auction 98 different licenses in 14 regions, 10 bidders Bidder 1 Bidder 2 Bidder 10 Center Allocation & Payments Partial Problem Information Solved? Inefficiency 6

7 Types of Bidding Languages XOR (fully expressive, but verbose), e.g. {(A,3), (B,5), (C,2), (AB, 9), (BC,10),(AC,6), (ABC, 15)} OR (not fully expressive, but concise), e.g. A3 + B2 + C1 OR* (fully expressive, possibly concise), e.g: A3 + BD3 + CD1 + BE2 + CE3 (Include phantom goods) L GB /TBBL Logical Languages (fully expressive, concise), e.g.: A3 XOR B2 AND C3 Domain Specific Languages (fully expressive, highly concise) Specific to the domain at hand 7

8 Iterated Combinatorial Auctions Many examples from the literature (e.g ibundle, Clock Auction) Basic Structure: Prices start at zero In each round: o Center quotes prices to bidders o Bidders state their preferred bundle at these prices (demand query instead of value query) o Center updates prices o Auction stops when bidders no longer change their requests Final payments are then determined, which may or may not be different than the interim prices 8

9 Interim Prices for Iterative CAs (like ibundle) Prices can be quoted on: Individual items (linear prices, like you are used to) Non-linear prices some higher order price terms included o E.g. polynomial prices include pairs, triples, etc Full Bundles every bundle gets its own price Prices can be: Anonymous one set of prices for every player Non-Anonymous each player gets their own prices To clear an iterative CA with fully general bidder valuations, you may need non-anonymous bundle prices. 9

10 Activity Rules and Incentives In Mechanism Design, we assume bidders are strategic They may not reveal information unless required to do so Mechanism typically include activity rules that force bidders to reveal information or incur penalties Weak rules will lead to bad equilibria Strong rules may force to too much revelation What is wrong with too much revelation? 10

11 Payments and Their Properties The final payments have a big impact on agent behavior We may want payments that have: Budget ballance: mechanism doesn t need to be subsidized (problem in exchanges vs auctions) Individually rationality: bidder gets non-negative payoff Yield High Efficiency: equilibrium outcome given strategic players is welfare maximizing Yield High Revenue: equilibrium outcome given strategic players is good for seller Fairness by any of several measures (relative, absolute etc) 11

12 Strategyproofness If the payment rule (in a sealed-bid auction) incentivizes every player to bid truthfully regardless of other players bids (dominant strategy incentive compatibility), it is called strategyproof The famed Vickrey-Clark-Groves (VCG) rule is strategyproof ( folk theorem -- the only such mechanism for general CAs) Idea in VCG: You pay your marginal impact on the economy You pay: Social welfare to all agents but ego of allocation excluding ego less the social welfare to all agents but ego of the allocation including ego For CAs seller is held out; for CEs its not (leading to BB problems) 12

13 Core Pricing VCG can easily lead to very small (zero) revenue A group of bidders (together with the seller) may wish to defect from the mechanism: their total gain from trade may be greater than that realized in the mechanism Core prices are precisely those that never have this problem: no coalition of players can make more by defecting from the mechanism VCG can be outside the core in CAs; core may be empty in CEs. Core will generally have higher revenue than VCG Most CCAs in practice use core pricing (complex to compute) 13

14 The Combinatorial Clock Auction (CCA) Milgrom/Ausubel/Cramton 2006 defined a hybrid format: Run a iterative auction with clock prices that rise where there is over-demand round on round A revealed preference activity rule forces bidders to reveal which bundle they prefer most at the current prices in each round Alternatively eligibility points may be required to submit a bid Or a hybrid of these At the end a last and final sealed bid auction is run, with all the data from clock phase included as bids, plus any new bids (subject to additional activity rules) This then determines final allocation and payments. 14

15 Domains 15

16 Ad Auctions (in brief) RightMedia, DoubleClick, AdECN Combinatorial preferences on the part of advertisers o e.g. specific demographics & multiple slots on a page (c.f. NYTimes.com) 16

17 Procurement (in brief) Reverse auctions May be massive in scale Typically combinatorics come from volume discounts Complementarities in shipping (don t send empty trucks) UX for elicitation is important: must allow bidders to place bids in a way that is natural for them, in order to get high-quality outcomes Large scale makes solving the WD efficiently important 17

18 Spectrum Auctions (in brief): Timeline 20 th Century July 1994 FCC 2011 UK 4G Grant Simultaneous Ascending Auction Combinatorial Clock-Proxy Auction Two-side Combinatorial Auction Clock Proxy combinatorial auction [Ausubel, Cramton & Milgrom 06] mitigates exposure; can use core prices Current FCC incentive auction is two-sided Run reverse auction to buy back spectrum for low-use TV stations Sell this bandwidth to cell companies in a forward auction Phases interleaved Highly complex anti-interference constraints NOT a full Combinatorial Exchange (with swaps) 18

19 Recent CCA Spectrum Auctions Ausubel & Baranov

20 Data Center Domain 20

21 Data Centers Power Use 2006 Data Centers: 61 billion kwh (1.5% of total) Equiv. 5.8M US Households (5%) 37 billion metric tons of CO 2 (.6%) Cost $4.5B, load of 7GW (15 plants) Double 2000 level Double again by 2011 to 12GW 10 new plants 21

22 The Data Center Allocation Problem 3 Tier Architecture Queuing Application processing (us) Database storage Middle layer: Thousands of heterogeneous servers Dozens of applications Problem: Map applications to servers Obey constraints Set server power/performance levels 22

23 Related Work Long History back to markets to timeshare PDP-1 Chase et. al. 01 data center market, we add: Market bids based on SLAs / modeling language Modern power management Sophisticated winner determination Utility Computing (e.g. Byde 06): sell computational resources like gas or electricity Assumes very high level of fungibility We want an incremental step from existing practice Related areas Markets in Grid computing Markets in High Performance Computing 23

24 Data Center Market Overview Buyers specify SLAs (Service Level Agreements) for the applications they wish to run. Sellers specify a model for the costs of provisioning those applications at various performance levels 24

25 Main Results A Data Center allocated by a Combinatorial Auction can Reduce power consumption Maintain performance objectives Triage client load Market as optimization method Finds maximally advantageous tradeoff Consistent with Existing business model (SLAs) Various allocation constraints 25

26 Value per Month Buyers Provide Generalization of standard SLA: Response Time per Transaction in Seconds Measured at e.g. 95 th percentile 26

27 Dollars / Period Value per Month MCycles / Period Buyer Model Value of Supply Given Load/Performance Prediction: Application B Demand Response Time per Transaction in Seconds U(r) R(s,d) Proxy Maintains Period Sample Demand prediction Actual Predicted U(s,d) = U(R(s,d)) Value Curve (For a Expected Short-Term Load) Supply in Mcycles / Period 27

28 Defining the Goods Partition data center into groups of identically configured machines Goods: cores of each machine in each group in a given power state (and transitioning from a possibly identical previous power state). Simplify WD by tracking only counts of such cores (across machines) Requires a post process outside the market to reify these counts (c.f. Cramton UK bandwidth auctions) 28

29 Efficiency (MCycles/Watt-hour) Seller Model Seller Ask is specified as a Model: Specifies the power / performance DVFS hardware (most efficient at < full power) All cores/machine same power level Costs: Hardware lease cost Operating Power Transition Power Transition Failure Rate costs Overhead (cooling, network etc) Energy Efficiency Active Cores Low Power High Power 29

30 Languages for Combinatorial Markets Common Method, Agents: Find preferred bundles Value these bundles Convey in a (concise) language Here, Agents: Model valuation problem directly Optimize in-line with WD No expensive conversion required Constraints Domain specific constraints (e.g. anti-colocation) 30

31 Constraints The market must also have supply constraints: Only sell what you have Can t buy more then available If bought, then sold Purchases must start from state at the end of the last period These stipulate the hard constraints A system administrator could also supply soft constraints over: Min/Max cores/machines for a given application Min/Max energy in a given group, or for a given application Number of cores in a given machine group that can be allocated to a given application, if a certain number are already allocated to specific alternative application (anti-colocation) etc. etc. 31

32 Winner Determination Market solved as a Mixed Integer Program via ILog CPLEX Scales near polynomially in the number of machines in equivalence classes Scales exponentially with the complexity of the bipartite graph between applications and machine equivalence classes 32

33 MCycles / Period Experimental Setup Application B Demand Actual Predicted Response Time for Given Supply & Demand Experiment uses M/M/1 model for response time: RT=1/(S-D) Can use any prediction method Buyer Demand Prediction Experiment uses moving average / Std. Dev. to define a Gaussian Can use any prediction method Experimental Demand Noisy double sinusoid pattern (e.g. days/weeks) Period Benjamin Lubin - NOT FOR REDISTRIBUTION 33

34 Response Time per Transaction Response Timeline Even with noisy prediction, good response times: Period Actual Predicted 34

35 Power & Response Time Cheap power Buy lots low RT And Converse: 35

36 Revenue Net Costs Our market produces more revenue-net-costs: 36

37 Experimental Results: Application and State When constrained shift to load with higher value/cycle And shift machines towards their most efficient state (i.e. pay transition costs to closely track demand) 37

38 Experimental Results: Complexity Complexity of WD rises with bipartite graph of Applications Machine Groups 38

39 Scaling & Overhead Allocation problem with 5 groups each with 200 quad core machines supporting 10 applications Solvable on 1 machine in 10 minutes Thus.1% overhead Feasible 39

40 Data Center Summary A data center allocated by a combinatorial auction can: Reduce power consumption Maintain performance objectives Triage client load Market as optimization method Finds maximally advantageous tradeoff Consistent with Existing business model (SLAs) Various allocation constraints 40

41 Generators CATS, SATS and others 41

42 Combinatorial Auction Test Suite (CATS) Generator Widely used, CATS was developed in 2000 with the primary purpose of exercising Winner Determination algorithms It uses OR* for its language, but the dummy goods are used to effectively create XOR (no conciseness) Includes legacy distributions and the following stylized models: Arbitrary relationships (e.g. random) Paths through a planer graph (e.g. railroad routes) Regions in planar space (e.g. real estate, closest to spectrum) Scheduling in time (e.g. job shop scheduling) Not explicitly calibrated to real world problems or data 42

43 Motivation for SATS: Improve Spectrum Auction Simulations CA research often uses simulations, e.g., Day and Raghavan, 2007 (evaluation of core-pricing algorithms) Lubin et al., 2009, 2016 (evaluation of payment rules) How to create auction instances for simulations? CATS, the current standard, not designed to model spectrum Community needs a new generator 43

44 Contributions 1. Multi Region Value Model (MRVM) Geographical division + multiple frequency bands 2. The Spectrum Auction Test Suite (SATS) Web Interface, Command Line Tool, Java API 3. Evaluation of the MRVM Fit to Canadian 700 MHz Auction Data 44

45 The Value of Spectrum Licenses allow use of frequencies Can be complements and substitutes Licenses are heterogeneous, e.g., Paired licenses and unpaired licenses Number of MHz Covered frequencies We call a set of nearly-equivalent licenses a band Regional Division Value depends on population Not all Mobile Network Operators operate nationwide 45

46 Example: Canadian 700 MHz Auction Service Area A B C C1 C2 D E Eastern Quebec Rogers Rogers TELUS Vidéotron Bell TELUS TELUS Southern Quebec Rogers Rogers TELUS Vidéotron Bell TELUS TELUS Eastern Ontario & Outaouais Rogers Rogers TELUS Vidéotron Bell TELUS TELUS Northern Quebec Bell Bell Rogers Vidéotron TELUS Bell Bell Southern Ontario Rogers Rogers Bell Vidéotron TELUS Bell Bell Northern Ontario Bell Bell Rogers Bragg TELUS Bell Bell Manitoba TELUS TELUS Rogers MTS Bell TELUS TELUS Partial allocation of the 2014 Canadian 700 MHz auction Three bidder types: Global, Regional, Local 46

47 Spectrum Value Models from the Literature 47

48 MRVM: The Multi Region Value Model 1. MRVM captures geographical division and multiple bands 2. Succinct enough to be represented as a Mixed Integer Program (MIP) In auction simulations, want to evaluate efficiency need efficient allocation o Requires full value functions (not a sample of XOR bids) Tractability of winner determination is essential concise MIPs o Need 1-1 mapping between model s value function and MIP objective MIP formulation uses model parameters o Can be used a domain specific bidding language in mechanisms that support it 3. Precise enough to capture the 2014 Canadian 700 MHz auction 48

49 MRVM: The Multi Region Value Model Physical Band Properties Bandwidth and Quality Function Business Case Geographical Properties Regional Value Regional Discount 3 Nonlinearities Total Value 49

50 Modelling of Intra-Band Synergies Nonlinear change of bandwidth for having multiple licenses of the same band and same region x b,r c b syn b x b,r x b,r is the number of licenses of band b in region r c b is the base capacity of band b syn b is the nonlinear synergy function of band b Similar to the Multi Band Value Model (Bichler et al., 2013) 50

51 Subscriber value Subscriber Value Value depends on the quality of service Quality depends on: Bandwidth provisioned Population in region and firm market share Firm strength -- value of fully provisioned customer bandwidth 51

52 Bidder Types Global Discount Factor Dependent on number of uncovered regions Regional Values are multiplied with a discount factor [0,1] based on the bidder type Regional Discount Factor for region r = λ i distance(h i,r) λ i 0,1 is a bidder specific discount factor h i is the bidders center of interest Local Only interested in small set of regions 52

53 SATS: Accessing Spectrum Value Models Web Interface Command Line Tool Java Library 53

54 The features of SATS Six value models 3 bidding languages Complete preferences of bidders Atomic bid orderings WDP solvers Different file formats Complete Atomic WDP File Bidding Value Formats Solvers Models Bid Languages Preferences Orderings (VM) Base VM In Value Random XOR (Bichler CATS concise et al., defined bidding 2013) for language every set of Size-Based licenses XOR Multi JSON Band with VM Quantities (Bichler et al., 2013) Suitable Domain Local Synergy Specific for Multi-Round VM Auctions (Goeree and Holt, 2008) Global Synergy VM (Scheffel et al., 2012) Single Region VM (UK 4G VM) (Kroemer et al., 2016) Multi Region VM (this paper) 54

55 Evaluation: Fitting the MRVM Goal: Parameterize MRVM so that. it generates random auction instances, which well-fit the Canadian 700 MHz Auction in expectation. Setup: Set number and type of bidders, licenses, regions and population as in the Canadian Auction Find reasonable distributions for other parameters by gradient ascent Evaluate resulting model/parameterization on 1000 Instances by solving WD MIPs and comparing result with the real-world data Evaluation Metrics: Average Number of licenses won per bidder Average Number of regions covered by bidder Average Winning Bid/MHzPop 55

56 Evaluation: Fitting the MRVM Number of Licenses per bidder Number of Regions per bidder Bidder Type MRVM (using SATS) Canadian Auction Global (0.247) (2.848) Regional (0.047) (1.436) Local (0.013) (0.333) Global (0.005) (0.333) Regional (0.047) (1.436) Local (0.013) (0.333) Global (0.031) (0.832) Bid/MhzPop Regional (0.016) (0.466) Local (0.005) (0.137) Mean (Std Error) 56

57 Summary of SATS MRVM: New spectrum value model, capturing both geographical division and multiple bands of spectrum SATS: Simple yet powerful way to generate auction instances MRVM can capture complex domains such as the 2014 Canadian 700 MHz auction SATS will be open source released in May when the corresponding paper is published, but you can early access 57

58 Final Thoughts Combinatorial Auctions are fascinating and have found important applications; but it is the spectrum domain that is probably still the most important We have seen how to construct various mechanisms, and to use various generators to drive them. While there has been a huge amount of research into them in the last 20 years, from the business, computer science and economic sides, there are still many open questions 58