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)

Explanations • 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)

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.

Contributions • 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.

Outline • Set up: Sealed-Bid Auctions • Alternative theories: • Equilibrium Theory • Cursed Equilibrium Theory • Level-k Auction Theory • Specifying Level-0 • Bidding Behavior • Estimate the models and compare their ability to explain the experimental data • Conclusions

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

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 highest signal 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:

Outline • Set up: Sealed-Bid Auctions • Alternative theories: • Equilibrium Theory • Cursed Equilibrium Theory • Level-k Auction Theory • Specifying Level-0 • Bidding Behavior • Calibrate the models and compare their ability to explain the experimental data • Conclusions

2.1.a Symmetric Equilibrium: First-Price • First-Price: First-Order Conditions: • CV: • IPV: Value Adjustment Bidding trade-off

2.1.b Symmetric Equilibrium: Second-Price • Second-Price: First-Order Conditions: • CV: • IPV: Value Adjustment

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.

2.2.a.Cursed Equilibrium: First-Price • First-Price: First-Order Conditions: • CV: • IPV: Different Value Adjustment Same as equilibrium

2.2.b.Cursed Equilibrium: Second-Price • Second-Price: First-Order Conditions: • CV: • IPV: Different Value Adjustment Same as equilibrium

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 : RANDOML0: TRUTHFULL0:

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.

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.

RANDOM L1: Bidding Behavior • First-Price: • CV: • IPV: • Second-Price: • CV : • IPV: Different value adjustment Different bidding trade-off Same as Fully-cursed Same as equilibrium

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. Winning means others’ signals are lower than it would mean in equilibrium Assume others overbid w.r.t equilibrium v(x,y) increasing in y, so lower value Reduce optimal bid

Bidding trade-off: no clear direction • When • upward shifts in the slopeof 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. q: parameter that shifts others’ bids The numerator is decreasing in q The denominator is also decreasing in q No clear direction in general

Summary: do level-k bidders overbid or underbid w.r.t. equilibrium? 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: Complements or Substitutes If uniform values~Equilibrium Bidding trade-off only: If uniform values ~Equilibrium ~Equilibrium ~Equilibrium

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.

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: • λ0 random • λ∞ optimal decision with probability 1

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: Mixture of types model: 3 specifications Level-k+Equilibrium Cursed equilibrium

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 • 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 3.5 if x≤2.5 6.5 if x>2.5 3.5 if x≤2.5 6.5 if x>2.5 3.5 if x<1.75 3.5<b<6.5 if x<3.25 6.5 if x>3.25 2x Low Value High Value Low Value High Value Low Value High Value Low Value High Value Low Value High Value b v b v b v b v b v b v b v b v b v ~Eq

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