Simultaneous First-Price Auctions with Preferences Over Combinations: Identification, Estimation and Application

Transcription

1 Simultaneous First-Price Auctions with Preferences Over Combinations: Identification, Estimation and Application Matthew Gentry, Tatiana Komarova, Pasquale Schiraldi London School of Economics September , Interactions conference

2 How does cross-auction interaction affect market performance? This question arises in a variety of contexts (e.g. online), but we focus on simultaneous first-price auctions: I.e. settings where many first-price auctions are run at once, without the possibility of combination bids Common pattern in procurement: E.g. highway contracting, OCS drilling, recycling and waste collection Simultaneity clearly may induce cross-auction interaction: Particularly if preferences over objects are not additively separable: e.g. due to capacity effects, project types, locations Parallels exposure problem in simultaneous ascending auctions (Milgrom 1999, among many others) but (if anything) more acute But little known about economic consequences of cross-auction interaction, particularly in first-price markets

3 Motivating Application: Michigan Department of Transportation (MDOT) highway procurement letting rounds per year, with many projects let in each round (average of 45 per round in our data) Bidders compete for multiple projects simultaneously: about 2.7 bids per round on average Heterogeneous projects with plausibly non-additive payoffs: Capacity constraints: known to matter in highway applications Project characteristics: complementary locations or work types But bids on combinations explicitly prohibited potential exposure problems and / or cross-auction interaction

4 Multiple bidding in MDOT auctions

5 So how do we think about an application like MDOT? Relatively typical of other procurement settings But not much in the literature on this kind of simultaneity RQ1: What can we learn about combinatorial preferences from auction-by-auction bids? RQ2: How important are cross-auction effects in practice? And in which direction do they operate? RQ3: How might switching to a known efficient mechanism (e.g. combinatorial Vickery-Clarke Groves) affect overall performance? Key tradeoff in multi-object mechanism design: complexity of the mechanism versus expressiveness of the message space But to evaluate this we need to know magnitude and direction of combinatorial preferences, i.e. what we re trying to measure

6 Our contributions Introduce a novel preference decomposition leading to new theoretical results on equilibria and to a viable empirical model for simultaneous 1P auctions Establish identification of this model under exclusion restrictions Either set of rivals i faces does not affect i s primitives... Or observe at least one factor shifting complementarities but not standalone valuations Structural estimation on MDOT highway contracting data Substantial reduced-form evidence of cross-auction interaction Structural estimates (preliminary!) suggest small projects are complements while large projects are substitutes Counterfactual: combinatorial VCG may substantially reduce total costs for both MDOT and bidders

7 Closely related literature Simultaneous auctions with combination bids Cantillon-Pesendorfer (2006): Combinatorial 1P auctions Kim, Oliveres, Weintraub (2014): Large-scale combinatorial 1P Lunander and Lundberg (2012, others): reduced-form comparisons of simultaneous 1P and combinatorial 1P formats Other forms of strategic dependence Milgrom (2002): Simultaneous ascending with some combo bids Ghosh (2012): Simultaneous 1P auctions for two identical objects with budget-constrained bidders Szentes-Rosenthal (2003): Three-object two-bidder auctions Computer science work on efficiency bounds (Bayesian Price of Anarchy): e.g. Feldman et al (2012)

8 Plan of the presentation General framework for simultaneous 1P auctions Setup and notation Theory highlights Empirical model and identification Identifying restrictions Identification argument Application to MDOT highway contracting Data and descriptive regressions Preliminary structural estimates Counterfactual: VCG vs FPA

9 The Auction Game N risk-neutral bidders compete for (subsets of) L prizes allocated via simultaneous first-price auctions: High bidder in auction l wins prize l Bids are binding, bidders may not submit combination bids, and bidders may not make bids contingent across auctions For now: ties in auction l broken randomly, i.i.d across auctions Allocations and outcomes: An allocation a is an assignment of prizes to bidders An outcome ω from the perspective of bidder i is an L 1 vector of zeros and ones with a one in the lth place indicating that prize l is allocated to bidder i

10 Preference structure: Combinatorial valuations Bids submitted by auction, values defined over combinations Yi ω : combinatorial valuation bidder i assigns to outcome ω Y i : 2 L 1 vector of combinatorial valuations i assigns to each possible outcome (normalize Yi 0 = 0); i s private type A1 (Independent Private Values): For all i, Y i distributed on compact space Y i R 2L according to continuous CDF F i Y i private information, F i common knowledge Types independent across bidders: Y i Y j for all i j

11 Preference structure: Final payoffs Bidders risk-neutral and pay bids in each auction won. Thus i s net payoff from winning outcome ω at bid vector b is Y ω i ω b Let Ω be the 2 L L matrix whose rows contain (transposes of) each outcome ω: e.g. if L = 2 [ ] Ω T = Then i s payoffs over possible outcomes described by 2 L 1 vector Y i Ωb.

12 Standalone valuations and complementarities Standalone valuations: Let V il be value i assigns to winning object l alone Let V i = (V i1,..., V il ) be i s vector of standalone valuations Complementarities: Let Ki ω be difference between i s combinatorial valuation for ω and the sum of i s standalone valuations for objects won in ω: K ω i = Y ω i V i ω. Let K i describe i s complementarities over all possible ω: K i = Y i ΩV i (2 L 1). Note that K i = 0 if and only if valuations are additive

13 The bidding problem Consider bidder i with type y i facing rival strategy profile σ i. Let p(b) be the distribution over outcomes arising when i submits bid b, with p ω (b) the probability of outcome ω. Let Γ(b) Ω T p(b) be the L 1 vector of marginal win probabilities implied by distribution p(b) (so Γ k (b) is probability of winning auction k) Bidder i then chooses b B i to maximize π i (b) = (y i Ωb) T p(b) = (v i b) T Ω T p(b) + K i p(b) = (v i b) T Γ(b) + K i p(b). Note problem is separable if K i = 0 standard theory applies.

14 Theory highlights Existence of equilibrium: Approach 1: based on communication extension of Jackson, Simon, Swinkels and Zame (2002) (JSSZ): bidders send cheap talk type signals which auctioneer uses to break ties. Approach 2: discrete bid space (Milgrom-Weber 1985) Best response bidding: b need not be monotone in y, but... Holding K i constant, b il is increasing in v il. For cross-auction monotonicity need supermodular K: K ω 1 ω 2 i + K ω 1 ω 2 i K ω 1 i + K ω 2 i ω 1, ω 2.

15 Theory highlights: Our results (joint with Wiroy Shin) Applying Athey (2001), McAdams (2003) and Reny (2011), establish existence of an equilibrium in pure monotone strategies when Complementarities can be stochastic Complementarities are supermodular Partial order on the space of types defined in a non-trivial way: it is not enough to require y y in the component-wise sense also need additional inequalities on (differences of differences of differences... ) marginal valuations

16 The identification problem What can we learn about combinatorial preferences when we observe only standalone bids? Suppose we observe a market-level sample with T auction rounds, L t auctions per round, N t bidders in market per round. For each bidder, observe: Bidder characteristics Z it (common knowledge) Bids submitted b it (L it 1) for each round For each auction round, observe: Generic auction-level characteristics X t Optional: combination characteristics W t (assumed relevant for K it but excludable from V it )

17 Identifying assumptions Obviously, without further assumptions, we won t be able to make much progress: L i bids for up to 2 L i unobservables. We thus introduce two key identifying assumptions: A2 (Deterministic K i ): V i F ( Z, X, W ), K i = κ(z, X, W ) A3 (Exclusion): F ( Z, X, W ) = F ( Z i, X ), κ(z, X, W ) = κ(z i, X, W ). Can generalize A2 and A3 to affine K i : K i = κ(z, X, W ) + M(Z, X, W )V i Formally, model is static interpret κ i ( ) as expectation over combination shock realized after a multiple win But we also think of κ as capturing reduced form dynamics Should be a good approximation so long as firm i s future profit is determined mainly by its own state

18 Identification with deterministic complementarities Identification now involves recovery of F ( Z it, X t ) and κ(z it, X t, W t ). We consider this problem under the following regularity conditions: A4: Conditional on covariates, observed outcomes are generated by repeated play of a single JSSZ-style equilibrium A5: In this equilibrium, the distribution of bids submitted by each bidder admits an absolutely continuous CDF Two notes on A5: A5 is implied by K = 0 formally nest null of additivity A5 is verifiable: no mass points in observed data

19 Overview of the identification argument Main idea: How do bids in Auction 1 respond to characteristics of Auction 2? Number of competitors in Auction 2 (Z i ) Size of project in Auction 2 (W ) Cross auction patterns + identifying assumptions pin down κ i More formally: Three-step identification argument An aside: For the moment, we take participation as exogenous

20 Step 1: Inverse bid system identified up to K i Bidder i s problem at (Z, W ) is max{(v i b) ˆΓ i (b Z, W ) + ˆp i (b Z, W ) T K i }. b B i An interior solution b must satisfy the first order necessary conditions bˆγ i (b Z, W ) (v i b ) = ˆΓ i (b Z, W ) b ˆp i (b Z, W ) T K i. (1) Proposition: For almost every b i G i ( Z, W ) and every vector K i R 2L i, there exists a unique vector v i R L i solving (1): [ ] 1 ] v i = b i + bˆγ i (b i Z, W ) [ˆΓ i (b i Z, W ) b ˆp i (b i Z, W ) T K i Note: Reduces to GPV (2000) auction by auction if K i = 0.

21 Step 2: From inverse bid system to identification criterion First enforce A2: K i = κ(z, W ). Then [ 1 v i = b i + bˆγ i (b i Z, W )] [ˆΓ i (b i Z, W ) b ˆp i (b i Z, W ) T κ(z, W )] a.e.g i ( Z, W ). Taking conditional mean then gives E[V i Z i, Z i, W ] = Υ i (Z i, Z i, W ) + Ψ i (Z i, Z i, W )κ(z, W ), where Υ(Z i, Z i, W ) (B i + b Γ i (B i Z, W ) 1 Γ i (B i Z, W ))G i (db i Z, W ) B i Ψ(Z i, Z i, W ) b Γ i (B i Z, W ) 1 b P i (B i Z, W ) T G i (db i Z, W ). B i

22 From invariance criterion to identification of K i Next enforce A3: κ(z, W ) = κ(z i, W ) E[V i Z i, Z i, W ] = Υ i (Z i, Z i, W ) + Ψ i (Z i, Z i, W )κ(z i, W ) (2) Then holding (Z i, W) fixed, we must have for any Z i : E[V i Z i, Z i, W ] = Υ i (Z i, Z i, W ) + Ψ i (Z i, Z i, W )κ(z i, W ), (3) Enforce the other part of A3: F ( Z i, Z i, W ) = F ( Z i ) to obtain that ( Υ(Zi, Z i, W ) Υ(Z i, Z i, W ) ) + ( Ψ(Zi, Z i, W ) Ψ(Z i, Z i, W ) ) κ(z i, W ) = 0.

23 From invariance criterion to identification of K i Only L i 1 system in 2 L i L i 1 unknowns in K(Z i, W ), but... Proposition: Suppose there exist Z i,0, Z i,1,..., Z i,j in the support of Z i Z i, W such that the final 2 L i L i 1 columns of Ψ(Z i, Z i,1, W ) Ψ(Z i, Z i,0, W ) M Ψ. Ψ(Z i, Z i,j, W ) Ψ(Z i, Z i,0, W ) have full column rank. Then κ(z i, W ) is identified. Step 3: After κ(z i, W ) is identified, the identification of F ( Z i ) is straightworfward

24 More on identification of κ How restrictive is full rank of M Ψ? Key thought experiment is varying n in other auctions Under (weak) assumption that competition varies distinctly in each auction, cardinality of Z i will be at least 2 L i With variation in Z i inducing nonlinear variation in Ψ(Z i ) What about variation in factors W shifting κ but not V i? E.g. distance between projects, size of other projects Doesn t help nonparametrically; κ(w ) arbitrary But parametrically (e.g. κ(w ) = W θ 0 ) can ID κ( )

25 Application: MDOT highway procurement 8187 auctions from , where for each auction observe Contract characteristics: Contract location, contract description, engineer s estimate Letting outcomes: planholders (by firm), bids submitted (by firm), award decision, winning bid 725 unique bidders, classified as follows: Regular bidders (35 firms): participate in more than 100 auctions in sample period, subclassified as follows: Large bidders (9 firms): at least 6 plants in Michigan Medium bidders (26 firms): fewer than 6 plants in Michigan Fringe bidders: all other firms

26 Summary statistics, MDOT lettings VARIABLES mean std max min Number of auctions per round Number of bidders per round Number of bids per round Engineer s estimate in Project completion time (days) ,838 Number of bidders per auction Number of regular bidders Number of bids per bidder Frac bids by regular bidder Frac bids by large reg bidder Backlog in

27 Cross-auction effects in the MDOT market y = ln(bid) (FE) Log Engineer s 0.983*** ( ) Log N Bidders *** ( ) Log Distance ** ( ) Log N Own Bids *** (0.0107) Log Total Size *** ( ) Log Distance Nearest *** ( ) Share Days Overlapping *** ( ) Same Type *** ( ) Standardized Backlog *** ( )

28 Estimation Overview Two-step algorithm building on Athey, Levin and Siera (2011) and Cantillon and Pesendorfer (2006) Step 1: Estimate parametric approximation to bid dist n G i : log(b it ) MVN( µ(d t ), Σ(D t )), where D t is a subset of observables (Z t, W t, X t ) and µ(d t ) = βd t σ 2 l (D t) = exp(αd t ) ρ lk (W t ) = T (γ [W lt W kt ]), with T (v) = exp(v 1)/ exp(v + 1) Step 2: Substitute estimated Ĝ i, ˆΓ i, ˆp i in expectations criterion and estimate κ( )

29 Step 1: Estimating G i Mean parameters MLE SEs 95% CI Constant (mean) Number of rivals Multiple auction Num rivals (other) ln_sum_eng ln_eng Covariance parameters MLE SEs 95% CI Constant (covariance) Same county Same type Product of ln_eng in pair Product of rivals in pair Variance parameters MLE SEs 95% CI Constant (variance) Number of rivals ln_sum_eng ln_eng ln_tnb

30 Step 2: Estimating κ Parametrize κ 0 (W ) = W θ 0 to obtain ID criterion linear in θ 0 : ( Υ(Zi, Z i, W i ) Υ(Z i, Z i, W i ) ) = ( Ψ(Zi, Z i, W i )W i Ψ(Z i, Z i, W i )W i ) θ0. For each observation i, approximate criterion above via simulation: Draw comparison (Z i, W i ) from empirical distribution Draw {bi0 r }R r=1, {br i1 }R r=1 from G i( Z, W, X ), G i ( Z, W, X ) Compute ˆΓ i, ˆΓ i and ˆp i for each bid drawn Average across draws to approximate Υ i, Υ i, Ψ i, Ψ i

31 Estimates of κ Parametrizing κ(w ) = W θ 0 and applying the algorithm above: Value SE Constant Sum EngEst Days overlap Distance among projects Projects of same type Log(Number of auctions -1) FE yes yes Units are thousands of dollars, higher κ means larger cost synergies, [10, 50, 90]th percentiles of Sum EngEst are [364, 1200, 5000] Point estimates suggest joint win leads to cost savings at median, transitioning to cost increases at the 90th percentile

32 Counterfactual: Combinatorial VCG mechanism How would procurement costs and allocative efficiency change if, instead of the current simultaneous FPA format, MDOT awarded contracts via a combinatorial Vickery-Clarke-Groves auction? Can t (yet) answer this question fully: preliminary estimates yield preferences only for bidders competing in one or two auctions First pass: run counterfactual on subsample of auctions where all bidders in each auction compete in no more than 2 auctions Not a representative sample... But yields an internally valid thought experiment And results underscore why the problem is important!

33 Combinatorial VCG mechanism Bidders report vectors y i Ω T c i + κ i describing private costs over all combinations of projects for which they bid Final allocation a chosen to minimize total completion costs: min a A N y j (a), j=1 where y j (a) denotes bidder i s total cost at allocation a A Payments from MDOT to bidders determined by usual VCG rule: p i (y i, y i ) = min y j (a) y j (a ). a A\i j i As usual, weakly dominant for bidders to report costs truthfully. j i

34 Results: combinatorial VCG versus simultaneous FPA Our first-take counterfactual results are very favorable for VCG: relative to FPA, VCG reduces MDOT procurement costs by 7.1 percent and total project completion costs by 14.7 percent Magnitude of efficiency gains driven partly by counterfactual sample definition: no more than two auctions tends to select smaller projects with relatively large estimated synergies MDOT cost reduction is net of two competing effects: Increased social efficiency tend to reduce MDOT costs Exposure problem in simultaneous FPA leads bidders to bid above valuations for projects with positive synergies, an incentive which disappears in combinatorial VCG

35 What we ve done thus far: Conclusion Propose a framework for analysis of simultaneous first-price auctions with potentially non-additive preferences Establish identification under exclusion restrictions in the special case of stochastic V i but stable K i First take on MDOT application (U-shaped costs) First take on VCG counterfactual (potential ) What we re working on now: More flexible specifications for G i, e.g. Gaussian copula with nonparametric marginals Richer specification for κ (backlog, work dates) Unobserved auction-level heterogeneity

36 Thank you!