Game theory and empirical economics: The case of auction data J-J. Laffont European Economic Review, 1997

October 14, 2008

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Introduction

• Bidder faces 3 types of uncertainty: – Uncertainty about the value of the object auctioned – Uncertainty (strategic) about the strategies of other players – Uncertainty about the characteristics of the other players (opponents) • Assumption: The characteristics of the other players are drawn by Nature from probability distributions which are common knowledge to all players

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Introduction

• Fact: These probability distributions governing the characteristics of the other players are unknown, yet important in explaining bidding behavior • The issue: How does the empirical analysis of auctions deal with the fact that players’ characteristics are unknown? – Reviews the major issues – Proposes a simple framework with few restrictions on the probability distributions

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Competitive bidding

Friedman’s model: Expected profit of bid b in a first price sealed bid auction E(b) = (v − b)P (b) where: v is the expected valuation, and P (b) is the probability that bid b wins the auction. Qn • P (b) = i=1 Gi (b), where Gi (·): estimated distribution of the bidding behavior of bidder i for all past auctions • “The probability of being the highest bidder with a bid of b, when the competitors are known, is simply the product of the probabilities of defeating each of the known competitors.” 4

Friedman’s model Expected profit of bid b in a first price sealed bid auction E(b) = (v − b)P (b) • Extended to the following cases: – Unknown number of bidders – Simultaneous auctions with financial constraints • Competitive bidding literature approximates for P (b) – P (b) is assumed stable, given enough data – P (b) changes through time, because other firms have adaptive behavior • Is valid only for the IPV specification 5

Winner’s curse Capen et al (1971): Firms bidding in oil drilling right auctions in the Gulf of Mexico have lower returns than the local credit union • Friedman’s optimal bid: max(v(σi ) − b)P (b), ∀σi R – Expected value v(σi ) = vh(v/σi )dv – Signal σi in [σ, σ] is correlated with true value v – h(·/σi ) is the pdf of the value conditional on σi • Capen et al. (1971): expected value conditional on winning R – Expected value: Eyi vh(v/σi and yi < bi )dv R bi R – Expected utility: v [ v vh(v/σi and yi < bi )dv − bi ]dP (yi ) – P (·) is the cdf of the highest bid yi of other firms – Let bi = kσi and search for ks providing fair returns 6

The competitive bidding lit • Takes the perspective of the bidding firm and searches for optimal bidding procedures • Aim: Identify the relevant P (·) function – P (b) is the probability that bid b wins the auction – Statistical analysis of the distribution of bids • History of this statistical analysis – Lognormality of bids: Arps (1965), Pelto (1971), Dougherty and Lorenz (1976), Lorenz and Peterson (1979) etc. – Cdf highest bid: product of estimated value of current bid and empirical distribution of the ratio of the highest bid to the bidder’s estimate of the value for past auctions (Hansmann and Rivett (1959)) • Lederer (1995)’s criticism: do not acknowledge that variance of predictive distribution depends on auction’s characteristics

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The competitive bidding lit - Limitations • Limited to optimal bidding strategies within a given environment. Ignores: – Alternative auctions – Optimality of auctions – Collusion • Ignores the complexity of firm’s optimization problem – Approach holds for IPV auctions, but IPV not the correct setting in most cases – Fails to properly specify objective function of firms when signals are dependent – Uses improper statistical methods • Fails to give insights on how to behave in different, new environments 8

Implications • Shift away from estimating bid function to estimating underlying stochastic distributions • Allows to address more questions than just bid optimization • Focus of this paper: 3 environments – Symmetric common values – Independent private values – Asymmetric common values

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Symmetric CV - Bayes Nash equilibrium • n bidders bid for good of unknown value v˜ – The cdf of v˜ is G(·) and is common knowledge • Each bidder i receives a signal σi – Signals are i.i.d with cdf F (·|v) and support [v, v] v , σ˜i , i = 1, ..., n) are affiliated random variables – (˜ – High σi signal high v • Each bidder i chooses his bid b by maximizing a Ev,yi {(v − b) |σi } [b∗ (yi )
–

` [b∗ (yi )
is 1 of bidder i wins and 0 otherwise

– yi is the 1st order statistic of the other bidders’ signals 10

Symmetric CV - Optimal bid Z ∗

b (σi ) = v(σi , σi ) −

σi

L(α|σi )dv(α, α)

σ

• L(α|σi ) = exp(

Rα

h(s|s) ) α H(s|s)

• H(yi |σi ) is the cdf of yi conditional on σi Prediction 3.1: Under Nash behavior, expected profits are non-negative b∗ (σi ) ≤ Ev,yi (v|σ ≤ yi ≤ σi , σi ) ≤ Ev,yi (v|σ ≤ yi ≤ σ, σi ) = Ev (v|σi ) • Ev,yi (v|σ ≤ yi ≤ σi , σi ) is the expectation of v˜ conditional on winning • Ev (v|σi ) is the interim expected value

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Symmetric CV - Optimal bid Z ∗

b (σi ) = v(σi , σi ) −

σi

L(α|σi )dv(α, α)

σ

Prediction 3.2: Optimal bids decrease eventually with number of bidders • As n increases, bid closer to the willingness to pay in order to win • Willingness to pay, Ev,yi (v|σ ≤ yi ≤ σi , σi ), decreases with n • Fear of winner’s curse dominates the competitive effect Prediction 3.3: The expected winning bid converges almost surely to true value in n1 • from Wilson (1977) and Milgrom (1979) • Reece (1978): this result does not hold for n small Caveat: ignores endogeneity in and uncertainty about n, collusion, post-auction markets, joint bidding, sequential auctions 12

Symmetric CV - Empirical validation • Bids as a function of n – Decreasing (Mead, 1967; Smith, 1977; Gaver and Zimmerman, 1977) ∗ accept CV model, or reject PV model? – Increasing (Brannman et al. (1987); Giliberto and Varaiya (1989)) ∗ reject CV model, or accept winner’s curse? • Hendricks et al. (1987): auctions of leases for wildcat tracts from 1954-1969 – Heterogeneity in bidding activity and returns: is the symmetric CV model appropriate? – Winning bids and money left on the table % with n: ∗ interpreted as consistent with CV ∗ but: consistent with most game theory models – Net profits & with n, and are < 0 with n ≥ 7: ∗ reflects Winner’s curse ∗ or: uncertainty about n

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Symmetric CV - Structural approach What is the best structural specification for bid functions? • Paarsch (1992) compares: – specification where bids are linear in signals ∗ uniform distribution for v ∗ Gumbel, Log-Normal or Weibull distribution for signals – to specification where bid functions combine linear and exponential terms ∗ Diffuse prior on v ∗ i.i.d and normal signals around the true value – Prefers linear specification, since the linear-exponential specification predicts < 0 cost estimates • Big picture question: Is it possible to identify all the distributions involved?

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Independent PV - Optimal bid Z ˆb∗ (vi ) = vi − n

vi v

Fˆ (u)n−1 Fˆ (vi )n−1

• Fˆ (·) is the cdf of vi = V (σi ) • equilibrium bid from Vickrey (1961) Prediction 4.2: The expected winning bid is an increasing function of the number of bidders R Eˆb∗ (vmax ) = vv ˆb∗n (u)dFˆ (u)n Prediction 4.3: Bid functions increase when valuations increase in the strong first order stochastic dominance case Prediction 4.4: First- and second-price, sealed bid auctions are revenue equivalent (Vickrey, 1961)

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IPV - Tests of Revenue Equivalence (RE) • Competitive bidding literature – Sealed bid auctions yield 10% more revenue than EA (Mead et al. (1981), Mead (1967)) – Caused by selection bias (Hansen, 1985) • Results depend on whether or not there is symmetry of bidders (Johnson, 1979) – Symmetric case: no difference between sealed bid and EA – Asymmetric case: Seal bid auctions yield higher revenue 16

IPV - Tests of Revenue Equivalence (RE) • In general, RE may not hold for various reasons: risk aversion, affiliation vs independence, symmetry vs aymmetry (Maskin and Riley (1983) – Test first whether there is independence of bids – If independence holds, test for RE • Multi-unit auctions: – Uniform price auction yield higher revenue than discriminatory auction (Weber, 1983; Tenorio, 1993) ∗ Depends on selection of equilibria (Back and Zender, 1993) ∗ Affected by changes in participation – In general: lack of theory for CV models w/multiple bids, units, asy info and collusion

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IPV - Structural approach • Estimation of bidding behavior – Simulated moment method (Laffont et al., 1995) – Maximum likelihood (Donald and Paarsch, 1993) – Non-parametric estimation (Elyakime et al., 1994) – Numerical integration (Elyakime et al., 1995) • ∃ more than 1 distribution of bidder’ characteristics compatible with the observed bid? – No: distributions are identifiable, both parametrically (Donald and Paarsch, 1995) and non-parametrically (Guerre et al., 1995)

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Asymmetric CV - Bayes Nash equilibrium • Wilson (1967) • One informed bidder who completely knows the true value v • One uninformed bidder who does not know the true value v˜ – The cdf of v˜ is G(·), with density g(·) on [v, v] • Equilibrium: uninformed bidder has random strategy, with cdf M (·) and pdf m(·) • Expected utility: – Informed bidder: (v − b)M (b) R v(p) – Uninformed bidder: v (v − p)dG(v) • Optimal bid b of informed bidder: Rv b(v) = v −

v

G(u)du G(v)

– where M (·) is defined by: v = b +

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M (b) m(b)

Asymmetric CV - Predictions • Prediction 5.1: The mixed strategy of the uninformed bidder is identical to the distribution of bids of the informed bidder –

b0 (v)m(b(v)) M (b(v))

=

0 b R v(v)G(v) G(u)du v

• Prediction 5.2: The expected profit of the uninformed bidder is zero • Prediction 5.3: The strategy of the informed bidder is independent of the number of uninformed bidders

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Asymmetric CV - Empirical validation Hendricks and Porter: drainage oil track data set • Hendricks et al. (1987): firms having neighbor tracks to the one being auctioned had much better info • Hendricks and Porter (1988) extend Wilson (1967) model: reservation price – Random reservation price (Hendricks et al. (1989), (1994)) – Main predictions in all models: ∗ Uninformed bidders’ expected profits = 0 ∗ Relation between distribution of bids of informed and uninformed bidders – Implications of theory: supported by data

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Extensions of the empirics 1. Endogeneity of number of bidders and of reservation price 2. Sequential auctions and multi-unit valuations 3. Auctions and bargaining 4. Collusion

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1. Endogeneity of number of bidders and of reservation price • Important when players must consider the uncertainty that arises from an unobserved number of participants • Existing work: – Secret reservation price rule (Elyakime et al. (1994, 1995)) – Free entry with participation costs in common value model (Mc Afee and Vincent, 1992) • Issues: – Modeling may be difficult – Additional insights not guaranteed

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2. Sequential auctions and multi-unit valuations • Empirical regularities: – Prices decline (Ashenfelter, 1989) – Prices are peaked shaped (Laffont et al., 1994) • Theory: buyers interested in 1 item – Winning bids follow a martingale in PV auction (Weber, 1983) – Winning bids follow a submartingale in symmetric affiliated auction (Milgrom and Weber, 1982) – Winning bids decreasing with nondecreasing ARA (Vincent, 1993) • Theory: buyers interested in more than 1 item – In a 2-object, 2-bidder, sequential first price auction ∗ Independent valuations conditional on a state variable whose distribution is common knowledge (Ortega-Reichert, 1976) ∗ Private signals are received before auctions occur (Hausch, 1986) • Effect of externalities (e.g., Caillaud and Jehiel, 1994)

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3. Auctions and bargaining • Auctions: just one stage of a more complex bargaining problem • Existing work: – T-bills: preceded by forward trading, and followed by markets with strategic behavior – Auctions where bids < reservation price: bidders may want to hide some private info because of the next stage in game (Elyakime et al. (1995) – Condominium markets: auction prices > than prices from face-to-face negociations (Ashenfelter and Genesove (1992)) • Recommendation: – Combine auction model and bargaining framework (as in Elyakime et al., 1995) with dynamic structure (as in Laffont et al. (1994))

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4. Collusion • Bid-rigging: concern of anti-trust policy • Existing theoretical work: Is collusion possible? – Yes: sealed bid and English auctions (Graham and Marshall, 1987) – Yes: all bidders are included in the ring (McAfee and McMillan, 1992) – Yes: in a SPA even if only some bidders are included (Mailath and Zemsky, 1992) – No: not if ∃ externalities (Caillaud and Jehiel, 1994) • Existing empirical work: very limited – Losing bids of colluders 6= losing bids on non-cooperative bidders (Porter and Zona, 1993) • Future work: different goals can be pursued – Analyze collusive behavior once it has been identified – Identify collusive behavior econometrically – Study relevant collusive behavior for different judicial environments

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Recommendation Develop a general theory of identification • Theory: cannot tell us in which particular auctions model we are • Empirics: identify PV and CV elements Proposed framework • Symmetric model where: – Utility of bidder i: increasing in (σi , v) – (˜ v , σ˜1 , ..., σ˜n ): affiliated random variables • Identification possible through: σ ˆi = bi +

GB i |bi (bi |bi ) gB i |bi (bi |bi )

= ξ(bi )

– where σ ˆi = t(σi ), t(α) = E[u(˜ σi , v˜)|˜ σi = α, y˜i = α], y˜i = maxj6=i σ ˜j – and GB i |bi : cdf of the maximal bid of bidders j, j 6= i conditional on bidder i’s bid 27

Recommendation Identification possible through: GB i |bi (bi |bi ) i σ ˆ i = bi + = ξ(b ) gB i |bi (bi |bi ) • GB i |bi is identifiable from observation of bids ˆi is identifiable • Hence: distribution of transformed signals σ • But: function u(·) and distributions of v, σ1 , .., σn not identifiable, since different functions and distribution yield the same σ ˆi • Select a flexible family of distributions for (σˆ1 , ..., σˆn ): clean test of IPV under the maintained hypothesis of the symmetric AV

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