Vincent P. Crawford & Nagore Iriberri Universitat Pompeu Fabra CERGE-EI, April 19th

Level-k Auctions:Can a Non-Equilibrium Model of Strategic Thinking Explain the Winner's Curse and Overbidding in Private-Value Auctions?

Vincent P. Crawford & Nagore Iriberri Universitat Pompeu Fabra

CERGE-EI, April 19th

Motivation• Sealed-Bid Auctions are theoretically well

understood. Standard solution: Risk Neutral Bayesian Nash equilibrium.

• Experimental anomalies:– Overbidding in private-value auctions (value

of object is known when bidders bid and different for each bidder)

– Winner’s curse in common-value auctions (value of object is unknown when bidders bid but the same for all bidders)

• Private-Value Auctions: (preferences)

– Risk Aversion: Cox, Smith and Walker (1983,1988), Holt and Sherman (2000)– Joy of winning: Cox, Smith and Walker (1992), Holt and Sherman (1994)

• Common-Value Auctions: (not conditioning on winning)

– Naïve bidding: Kagel and Levin (1986), Holt and Sherman (1994)– Cursed Equilibrium: Eyster and Rabin (2005)

Explanations

An alternative approach: A structural non-equilibrium model of

initial responses to auctions based on "level-k" thinking

• Level-k models have been useful in explaining subjects' initial responses in experiments with complete-information games.

• A suitable generalization from complete to incomplete-information games might yield a unified explanation of – the winner's curse in common-value auctions – overbidding in independent-private-value auctions – non-equilibrium behavior in other incomplete-information games

Level-k Models

• Players are drawn from a common distribution (estimated or translated from other settings) over a hierarchy of decision rules or "types”: – Level-0 are non-strategic and naïve anchoring level– Level-1 best responds to Level-0 type– Level-2 best responds to Level-1 type and so on...

• Level-k agents are rational and maximize expected payoffs as equilibrium players but they have simpler models of other individuals’ behavior.

• Extend level-k analysis to incomplete-information games sealed-bid auctions.

• Explore the robustness of the conclusions of equilibrium auction theory to failures of the equilibrium assumption.

• Provide a unified explanation for systematic patterns of non-equilibrium bidding behavior in private and common-value auctions.

• Explore how to model initial responses to games (strategic thinking): link between empirical auction studies and non-auction experiments.

Contributions

Outline1. Set up: Sealed-Bid Auctions

2. Alternative theories:1. Equilibrium Theory2. Cursed Equilibrium Theory3. Level-k Auction Theory

1. Specifying Level-02. Bidding Behavior

3. Estimate the models and compare their ability to explain the experimental data

4. Conclusions





4. Conclusions

• Signals: where

• Y: highest signal among (n-1)– Affiliated-signals: – Independent-signals:

• Values:– Private-Value (PV):– Common-Value (CV):

• Price Rules: – First-Price: winning bidder pays his own bid– Second-Price: winning bidder pays the second highest bid

),...,,( 21 NXXXX

1. Set up: Sealed Bid Auctions (Milgrom and Weber 82)

)|( xyfY

)(yfY

),( SXuVi ii XV

iV

],[~ xxX i

• Bidder’s problem:

– Probability of winning: Assume others bid according to a monotonic bidding function , then I win the auction if I bid higher than the bidder with the highestsignal among the rest of the bidders

– Different value functions:

• Private-Value (PV) :

• Common-Value (CV) conditional on winning:

• Common-Value (CV) not conditional on winning:

]|[max winningpriceVE ib

)(1

)|(),(bb

xY

i

dyxyfpriceyxv

)( xb i

)( ybb i ybb i

)(1

xyxv ),(

yYxXVEyxv ii ,|),(

xXVExr ii |)(




3. Calibrate the models and compare their ability to explain the experimental data

4. Conclusions

2.1.a Symmetric Equilibrium: First-Price

• First-Price:

First-Order Conditions:

– CV:

– IPV:

)(1

*

)|(),(maxbb

xYb dyxyfbyxv

0)|()(

1)|())(),(( '*

* xxFxb

xxfxbxxv YY

0)()(

1)())(( '*

* xFxb

xfxbx YY

Bidding trade-offValue Adjustment

2.1.b Symmetric Equilibrium: Second-Price

• Second-Price:


– CV:

– IPV:

)(

*

1*

)|()(),(maxbb

xYb dyxyfybyxv

0)(),( * xbxxv

0)(* xbx

Value Adjustment





4. Conclusions

2.2. Cursed Equilibrium:Eyster and Rabin 2005

• Cursed bidders believe that with probability χ (level of cursedness) each other bidder bids the average of others' bids over all signals rather than the bid her strategy specifies for her own signal.

• Cursedness, χє[0,1], only affects the value function:– χ = 0 Bayesian Nash Equilibrium ( )– χ = 1 Fully-cursed equilibrium or naïve bidding ( )– χє(0,1) levels of cursedness ( )

• PV Auctions: cursedness has no effect.

),( xxv)(xr)(),()1( xrxxv

2.2.a.Cursed Equilibrium: First-Price

• First-Price:


– CV:

– IPV:

)(1

)|()(),()1(maxbb

xYb dyxyfbxryxv

1(1 ) ( , ) ( )) ( ) ( | ) ( | ) 0'( )Y Yv x x r x b x f x x F x x

b x

'

1( ) ( ) ( ) 0( )Y Yx b f x F x

b x

Same as equilibrium

Different Value Adjustment

2.2.b.Cursed Equilibrium: Second-Price

• Second-Price:


– CV:

– IPV:

)(1

)|()()(),()1(maxbb

xYb dyxyfybxryxv

0)()(),()1( xbxrxxv

0)( xbx Same as equilibrium

Different Value Adjustment





4. Conclusions

2.3. Level-k Auctions: Specifying Level-0

• Level-k bidders believe opponents behave as level-(k-1) and best respond to those beliefs.

• What are plausible specifications of the non-strategic, anchoring type Level-0, which is the starting point for players’ thinking about others’ likely bids?

• Two leading possibilities :

RANDOM L0: TRUTHFUL L0:

??)(0 xb0 ( ) ~ [ , ]b x U v v

xXVExb ii |)(0

RANDOM LEVEL-K(1)RANDOM L0:

(2)RANDOM L1: best responds to RL0– Random L0s do not condition on their own signal no

information revealed by winning: r(x).– Uniform: Highest bid among (N-1) uniform bids ( )

the actual distribution of the value and signal is ignored.

(3)RANDOM L2: best responds to RL1– Random L1s’ bidding function is monotonic in signal

information revealed by winning: v(x,y).– The actual distribution of the value and signal is

incorporated.

0 ( ) ~ [ , ]b x U v v

1Z

TRUTHFUL LEVEL-K

(1)TRUTHFUL L0:

(2)TRUTHFUL L1: best responds to TL0– Truthful L0s’ bidding function is monotonic in signal

information revealed by winning: v(x,y).– The actual distribution of the value and signal is incorporated.

(3)TRUTHFUL L2: best responds to TL1– Truthful L1s’ bidding function is monotonic in signal

information revealed by winning: v(x,y). – The actual distribution of the value and signal is incorporated.

xXVExb ii |)(0

RANDOM L1: Bidding Behavior

• First-Price:

– CV:

– IPV:

• Second-Price:

– CV :

– IPV:

b

zZb dzzfbxr )()(max

1

b

zZb dzzfzxr )()(max

1

0)()( 1 xbxr r

0)(1 xbx r

Same as Fully-cursed

Same as equilibrium

0)()())(( 111 11 r

Zr

Zr bFbfbxr

0)()()( 111 11 r

Zr

Zr bFbfbx

Different value adjustment

Different bidding trade-off

RANDOM L2,TRUTHFUL L1 AND TRUTHFUL L2: Bidding Behavior

• These decision rules are similar to equilibrium:– Other bidders are assumed to bid monotonically in their signal:

information revealed by winning is taken into account: v(x,y).

– The original distribution of signals is also taken into account.

• BUT they differ in not having equilibrium beliefs. How do First-Order Conditions change?– Different Value Adjustment: expected value of the item

conditional on winning. Isolated in second-price CV auctions.

– Different Bidding Trade-Off: change in the optimization problem when increasing the bid. Isolated in first-price PV auction.

Value Adjustment: actions are strategic substitutes

• Level-k bids according to the expected value given its own signal, conditional on just winning. A level-k bidder believes it wins when it bids at least and not when it has the highest signal, as a symmetric equilibrium bidder does.

• Value adjustment tends to make bidders' bids strategic substitutes.

)(

1

11

)|()(),(maxbb

xYkb

k

kdyxyfybyxv

)(1 Ybk

0))(,( 11

kkk bbbxv

Assume others

overbid w.r.t equilibrium

Winning means others’ signals are lower than it would mean in equilibrium

v(x,y) increasing in y, so lower

value

Reduce optimal bid

Bidding trade-off: no clear direction

• When – upward shifts in the slope of others' bidding strategy , γ, make

bidders' bids strategic complements (respectively substitutes) iff is convex (concave) in y

– upward shifts in the level, δ, make bidders' bids strategic complements.

)(1

1

)(maxbb

xYb

k

kdyyfbx

xxbk )(1

)(/)( yfyF YY

k

kk

kkY

kkY

k

bqbbqbbfqbbF

bx

),()),(()),((

)( 11

11

11

The numerator is decreasing in q

The denominator is also

decreasing in q

No clear direction in general

q: parameter that shifts

others’ bids

Random L1(RL1)

Random L2

(RL2)

Truthful L1 (TL1)

Truthful L2(TL2)

CV First-Price

CV Second-Price

IPV First-Price

IPV Second-Price

Summary: do level-k bidders overbid or underbid w.r.t. equilibrium?

],[~ vvUb i

],[~ vvUb i

Value Adjustment: -If r(x)>v(x,x): overbidding -If r(x)<v(x,x): underbidding

Bidding trade-off:

Value Adjustment: strategic substitutes -If level-(k-1) overbids then level-k underbids -If level-(k-1) underbids then level-k overbids

Bidding trade-off: Complements or Substitutes

Value Adjustment only: -If r(x)>v(x,x): overbidding -If r(x)<v(x,x): underbidding

Value Adjustment only: strategic substitutes -If level-(k-1) overbids then level-k underbids -If level-(k-1) underbids then level-k overbids

Bidding trade-off only:

If uniform values~Equilibrium

Bidding trade-off only: Complements or Substitutes

If uniform values~Equilibrium

~Equilibrium ~Equilibrium





4. Conclusions

3. Experimental Designs in the literature

• Kagel and Levin (1989, 1994): CV First and Second-Price:

– CV Function: and Affiliated Signals:

– Variation in the number of players (4,6 and 7 bidders) and the precision of the signals (a=12,18,24).

– 51 individuals in first-price and 28 in second-price.

• Avery and Kagel (1997): CV Second-Price

– Independent Uniform Signals: and CV Function:– Two bidders. – 23 individuals.

• Goeree, Holt and Palfrey (2002): discrete uniform IPV:– Independent Uniform Signals and PV Functions:

• Low Value Treatment:• High Value Treatment:

– Two bidders.– 80 individuals, 40 for each treatment.

]4,1[~ UX i

N

iixsxu

1

),(

ssxu ),( ]2

,2

[~| asasiidUSX

]12,9,7,5,3,0[~ UX i

]11,8,6,4,2,0[~ UX i

ixsxu ),(

Estimating Models

• All models (except equilibrium) are based on behavioral parameters.

• Two alternative models: – Mixture of types model where all separated Level-k

decision rules and equilibrium are included.– Mixture of types model of cursed equilibrium where

types represent different cursedness levels ( ).• Which model explains better the behavior in the

experiments? • What is the type distribution?

Identifying initial responses

• Initial responses: first 5 initial periods for inexperienced individuals.

• Data:– Editing of individual “crazy” bids.– Payoffs adjusted by CPI when comparing different

experimental designs.• Logit decision rules: deviations from optimal

decision rules will be proportional to the cost of such deviations in terms of payoffs. Precision of the decision rules given by λ: – λ0 random– λ∞ optimal decision with probability 1

Mixture of types model: 3 specifications

Parameters to estimate:

– Type proportions:

• Fixed chi (cursed equilibrium only):

• Fix K types (cursed equilibrium only):

– Precision:• Individual-specific precisions:

• Type-specific precisions:

• Same precision over types and individuals:

( , | ) ( | )g

g gk k i

ki N

L b L b

( , , | ) ( , | )g

g gk k i

ki N

L b L b

1 2( , ,..., )K

1 2( , ,..., )gN

1 2( , ,..., )K

(0,0.1,0.2,...,1)

1 2( , ,..., )K

Level-k+Equilibrium Cursed equilibrium

k̂k̂

Table 3a. Models and Estimates for Kagel and Levin First-PriceSubject Specific Precision

Model Level-k plus equilibrium Cursed equilibriumTypes Types

Random L0 0.04 Random L0 -- 0.06

Random L1 0.61 Type 1 1 0.47Random L2 0.04 Type 2 0.9 0.02

Truthful L1 0.16 Type 3 0.8 0.08Truthful L2 ~Eq. Type 4 0.7 0.06

Equilibrium 0.16 Type 5 0.6 0Type 6 0.5 0

Type 7 0.4 0.04Type 8 0.3 0.04

Type 9 0.2 0.04Type 10 0.1 0

Type 11 0 0.20Log-likelihood -1658.39 Log-likelihood -1640.5

BIC -1724.57 BIC -1715.1

Distribution of chi, Kagel and Levin, first price

0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Chi

Freq

uenc

y

k̂k̂

Table 3b. Models and Estimates for Kagel and Levin Second-PriceSubject Specific Precision

Model Level-k plus equilibrium Cursed equilibrium

Types Types

Random L0 0 Random L0 -- 0.18

Random L1 0.25 Type 1 1 0.18

Random L2 0.14 Type 2 0.9 0.11

Truthful L1 ~R.L2 Type 3 0.8 0.04

Truthful L2 0.32 Type 4 0.7 0

Equilibrium 0.29 Type 5 0.6 0.07

Type 6 0.5 0.04

Type 7 0.4 0.04

Type 8 0.3 0

Type 9 0.2 0.11

Type 10 0.1 0.07

Type 11 0 0.18

Log-likelihood -920.68 Log-likelihood -950.91

BIC -955.01 BIC -992.76

k̂ k̂

Table 3c. Models and Estimates for Avery and Kagel Second-PriceSubject Specific Precision


Types Types

Random L0 0 Random L0 -- 0.13

Random L1 0.65 Type 1 1 0.43

Random L2 0.09 Type 2 0.9 0

Truthful L1 ~RL2 Type 3 0.8 0

Truthful L2 0.22 Type 4 0.7 0.13

Equilibrium 0.04 Type 5 0.6 0.04

Type 6 0.5 0.09

Type 7 0.4 0.04

Type 8 0.3 0

Type 9 0.2 0.04

Type 10 0.1 0.04

Type 11 0 0.04


BIC -696.05 BIC -714.13

k̂k̂

ˆ 0

Table 3d. Models and Estimates for Goeree, Holt, and Palfrey FPSubject Specific Precision

Model Level-k plus equilibrium QRE

Types Types

Random L0 0 Random L0 0

Random L1 0.62 1

Random L2 0.04

Truthful L1 0.14

Truthful L2 0.01

Equilibrium 0.19


BIC -678.12 BIC -728.36

Summary: Empirical Findings

• Significant advantage of Level-k model over cursed equilibrium in 3 out of 4 experimental designs.

• Random specification shows higher ability to explain data than Truthful specification.

• Significant individual heterogeneity regarding precision.

• Type estimates are similar to those found in other experimental designs (except KL Second): – L0 exist only in the minds of other levels, – L1 is the most frequent type, – then equilibrium and then higher levels.





4. Conclusions

4. Conclusions

• Extend level-k analysis to incomplete-information games sealed-bid auctions.

• Explore the robustness of the conclusions of equilibrium auction theory to failures of the equilibrium assumption.

• Provide a unified explanation for systematic patterns of non-equilibrium bidding behavior in PV and CV auctions (except when uniform private value auction).

• Find support for level-k thinking in the experimental data. Establish a link between empirical auction studies and non-auction experiments.

• Most CV auctions are better explained with level-k model than with the mixture of cursed types.

• IPV designs are especially useful to separate the cursed equilibrium model and level-k and. GHP experimental design level-k shows significant advantage over cursed/equilibrium model.

Table with different decision rules

Table 1. Types' Bidding Strategies

Auction/Type Equilibrium χ-cursedEquilibrium Random L1 Random L2 Truthful L1 Truthful L2

KL’s: First-Price CV

KL’s: Second-Price

CV

AK’s: Second-Price

CV

GHP’s:First-Price IPV

2ax

Nax

2ax

2ax

2ax

Na

Naa

xx )2

)(1(

Naax

2 NNax2

2)1( x

12

2 NNax

12

2 NNax xxbt )(2

xx 2)1(25

25

x

543210

543210

1186420

1297530

54,33,22,11,0

0

1186420

1297530

6,55,4

4,3,1110

43

3,22,1

10

1186420

222210

1297530

2x3.5 if x≤2.5

6.5 if x>2.5

Low Value

Low Value

Low Value

High Value

High Value

High Value

3.5 if x≤2.5

6.5 if x>2.5

b v

543210

1186420

1297530

643210

High Value

Low Value

Low Value

1297530

443210

b v b vb v b v b v b v b v b v

High Value

3.5 if x<1.753.5<b<6.5 if x<3.256.5 if x>3.25

~Eq

k̂k̂ k̂ ̂ kk̂ k̂kk̂ ̂

Table 3a. Models and Estimates for Kagel and Levin First-Price

Model Level-k plus equilibrium Cursed equilibriumSpecification Type-specific

precision Constant precision

Type-specific precision

Constant precision

Random L0 0 -- 0 -- -- 0 -- -- 0 --

Random L1 0.35 1 0.49 1.62 0.99 0.83 0.6 1 0.5 0.68

Random L2 0.03 280.9 0 1.62 0.78 0.06 46.20 0 0.5 0.68

Truthful L1 0.54 1.21 0.29 1.62 0 0.11 14.74

Truthful L2 ~Eq. ~Eq. ~Eq. ~Eq.

Equilibrium 0.08 11.09 0.22 1.62

Log-likelihood -1739.6 -1753.54 -1736.62 -1762.24

BIC -1749.23 -1759.56 -1747.45 -1768.26

k̂ k̂k̂ ̂ k k̂k̂ kk̂ ̂

Table 3b. Models and Estimates for Kagel and Levin Second-Price


Specification Type-specific precision

Constant precision

Type-specific precision

Constant precision

Random L0 0 -- 0 -- -- 0.43 0 -- 0 --

Random L1 0.10 95.84 0.62 8.91 0.86 0.27 8.89 0.79 0.43 2.95

Random L2 0.27 2.50 0.11 8.91 0.18 0.30 5.35 0.33 0.15 2.95

Truthful L1 ~RL2 ~RL2 ~RL2 ~RL2 0 0.42 2.95

Truthful L2 0.33 6.10 0.27 8.91

Equilibrium 0.30 49.76 0 8.91

Log-likelihood -967.80 -973.81 -987.48 -995.59

BIC -976.39 -979.17 -997.14 -1003.1

k̂k̂ k̂ ̂ k k̂k̂kk̂ ̂

Table 3c. Models and Estimates for Avery and Kagel Second-Price

Model Level-k plus equilibrium Cursed equilibriumSpecification Type-specific

precisionConstant precision

Type-specific precision Constant precision

Random L0 0 -- 0 -- -- 0 -- -- 0 --

Random L1 0.56 12.77 0.94 4.3 1 0.37 9.67 0.8 1 2.77

Random L2 0 -- 0.06 4.3 0.73 0.08 161.45

Truthful L1 ~RL2 ~RL2 ~RL2 ~RL2 0.63 0.55 1.33

Truthful L2 0.05 1000 0 4.3

Equilibrium 0.39 0.63 0 4.3

Log-likelihood -702.34 -710.53 -706.00 -715.77

BIC -710.58 -715.68 -715.27 -719.89

k̂k̂ k̂̂k̂k̂ ̂k̂

Table 3d. Models and Estimates for Goeree, Holt, and Palfrey First-Price

Model Level-k plus equilibrium QRE

Specification Type-SpecificPrecision

Constant Precision

Type-Specific Precision

ConstantPrecision

Random L0 0 -- 0 -- -- 0 -- 0

Random L1 0.98 8.54 0.99 8.71 2.74 0.80 3.14 1

Random L2 0 -- 0 8.71 9.63 0.20

Truthful L1 0 -- 0 8.71

Truthful L2 0 -- 0 8.71

Equilibrium 0.02 29.84 0.01 8.71

Log-likelihood -642.91 -644.12 -684.81 -688.44

BIC -655.92 -651.93 -688.71 -689.74

Documents

Vincent P. Crawford & Nagore Iriberri Universitat Pompeu Fabra CERGE-EI, April 19th