1 Motivation

• Game theory is fundamental in industrial organi-zation

• Agents are not anonymous, they interact often insmall numbers

• Game theory starts with a specification of strate-gies, payoffs and information

• From these objects, predictions of behavior are

derived

• Economic theory: Model Parameters=⇒Predictions

• Some problems with using game theoretic modelsfor ”real” economic problems:

1. Predictions can depend delicately on the specifi-

cation of the model

2. Little evidence to guide us on how to make these

choices

3. Difficult to derive the predictions for all but the

simplest games

4. Multiplicity of outcomes

5. Unwillingness of theorists to confront dynamics

• Next, we will study how to use a game as an

econometric model

• Econometrics: Data=⇒Payoffs and other modelparameters

• The flexibility of game theory is an advantageeconometrically

• It allows the researcher to fit a broad class ofbehaviors

• Models that are trivially rejected are not very use-ful empirically

• Estimation involves two steps:

1. Flexibly estimate reduced form decision rules

• ”Solution” to the multiple equilibrium problem is

to condition on the one that is in the data

2. Find structural parameters that make these deci-

sion rules optimal

• Computationally simple and accurate methods

• Emphasis on implimentation, substantive empiri-cal applications.

2 Static Games

• A game is a generalization of a standard discretechoice model (e.g. logit or probit)

• Payoffs depend on exogenous covariates and pref-erence shocks

• In a game, payoffs also depend on actions of otherplayers

• Observed behavior is assumed to be an equilib-rium to the game

• Objective: Estimate agents utilities and equilib-rium selection mechanism

• Early work:Vuong and Bjorn (1984), Bresnahanand Reis (1990,1991) and Berry (1992).

• More recent work: Mazzeo (2002), Tamer (2003),Sweeting (2004), Ackerberg and Gowrisankaran

(2006), Aradillas-Lopez (2005), Ryan and Tucker

(2006), Pakes, Porter, Ho and Ishii (2005), Ho(2005),

Ishii (2005), Ciliberto and Tamer (2007)

• Dynamic Games: Aguirregabiria and Mira (2007),Pesendorfer and Schmidt-Dengler (2007),Pakes,

Ovtrovsky and Berry (2007) and Bajari, Benkard

and Levin (2007).

3 Outline

1. Simple entry example

2. Static Games

3. Nonparametric identification

4. Nonparametric/semiparametric estimator

3.1 Entry Example.

• Data on a cross section of markets.

• Entry by Walmart and/or Target.

• Markets = 1 and firms = 1 2

• Let = 1 denote entry and = 0 denote noentry.

• Economic theory suggests that profits depends ondemand, costs and number of competitors

• is population of market (demand)

• is distance from headquarters (costs)

• − indicates entry by competitors

• The profit of firm is:

= ·+ ·+ ·−+ if = 1

= 0 if = 0

• is private information

• ( = 1) is probability that enters market .

• In a Bayes-Nash equilibrium, agent makes bestresponse to −(− = 1)

• Therefore, ’s decision rule is:

= 1⇐⇒ · + · + · −(−= 1) + 0

• If error terms are extreme value, then

(= 1) =exp(·+·+·−(−=1))1+exp(·+·+·−(−=1))

• Equilibrium- two equations in two unknowns (1(1 =1) and 2(2 = 1)):

1(1= 1) =exp(·+·1+·2(2=1))1+exp(·+·1+·2(2=1))

2(2= 1) =exp(·+·2+·1(1=1))1+exp(·+·2+·1(1=1))

Two-Step Estimator

• First, estimate b( = 1|12)

using a “flexible” method.

• This is the frequency that entry is observed em-pirically.

• Standard problem.

• We are recovering an agent’s equilibrium beliefs

by using the sample analogue.

• Given this first stage estimate, agent ’s decisionrule is estimated as :

= 1⇐⇒ · + · +·b−(−= 1) + 0

• The probability that choose to enter is

(= 1) =exp(·+·+·b−(−=1))1+exp(·+·+·b−(−=1))

• This is the familiar conditional logit model!

• In second step, let ( ) denote the pseudolikelihood function defined as:

( ) =

Y=1

2Y=1

³exp(·+·+b−(−=1)·)1+exp(·+·+b−(−=1)·)´1{=1}

³1− exp(·+·+b−(−=1)·)

1+exp(·+·+b−(−=1)·)´1{=0}

• Maximize psuedo-likelihood to estimate .

• Bottom line: this is the logit model with b( =1|12) as an additional in-

dependent variable.

• Simple generalization of widely used model.

• Computationally simple and accurate

• Easy to include unobserved heterogeneity.

• Generalize to richer models, including dynamicmodels.

4 General Model

• Players, = 1 and actions ∈ {0 1}.

• Let = {0 1} and = (1 ).

• Let ∈ denote a vector of state variables.

• Two strategy assumption can be generalized.

• State is common knowledge and observed by

econometrician.

• Preference shocks private information, extremevalue.

• Period utility for :

( ) = Π( − ) +

• Π( − ) mean utility

• Utility similar to standard discrete choice model(e.g. conditional logit).

• Normalize Π( = 0 − ) = 0

• E.g. profit from not entering is zero.

• “Outside good” assumption

• Define the choice specific value function as

Π( = 1 ) =X−

−(−|)Π( = 1 − )

• This is the expected utility from choosing = 1

(excluding the preference shock)

• The optimal decision rule satisfies:

= 1⇐⇒ Π( = 1 ) + 0

• Since is extreme value

( = 1|) = exp(Π(=1))1+exp(Π(=1))

• Analogous to our simple example.

5 Identification

• The model is identified if we can reverse engineerΠ( − ) that uniquely rationalize (|)

• Π( − ) is a nonparametric function of ( − )

• The error terms () are distributed i.i.d. ex-treme value across actions and agents .

• We cannot nonparametrically identify both errorterms and Π( − )

• This is true in single agent models as well.

• We first do the “Hotz-Miller” inversion, i.e.

( = 1|) = exp(Π( = 1 ))

1 + exp(Π( = 1 ))⇒

Π( = 1 ) = log(( = 1|))− log(( = 0|))

• We can reverse engineer the choice specific valuefunction Π( = 1 ) from choice probabilities

( = 1|)

• Identification requires inversion of the followingsystem:

Π( = 1 ) =P− −(−|)Π( = 1 − )∀ =

1

• For a fixed , there are × 2−1 unknowns cor-responding to the Π( = 1 − )

• However, there are only equations

• In general, this system cannot be inverted and themodel is underidentified!

• One way to identify the system is to impose ex-

clusion restrictions.

• Partition = ( −), and suppose Π( − ) =Π( − ) depends only on .

• Ex. profit of firm excludes distance of other

agents.

• Then

Π( − ) =P− −(−|− )Π( − )

• By varying − we increase the number of equa-tions but not the number of unknowns

• Sufficient condition: ∃ 2−1 points in the supportof the conditional distribution of − given .

6 Nonparametric Estimation

• Nonparametric estimation in 3 steps.

• Empirical analogue of identification argument

Step 1: Estimation of Choice Probabilities.

• There are = 1 repetitions of the game

with actions and states ( ), = 1 .

• In the first step we form an estimate b(|) of(|) using flexible method.

• E.g. sieve logit (see Newey (1990) and Ai andChen (2003))

• Let () denote the vector of terms in a orderpolynomial

• ( = 1| ) = exp(()0)

1+exp(()0)

• Let → ∞ as sample size → ∞ but not too

fast, i.e. → 0

• Other basis functions (e.g. splines, orthogonal

polynomials) are also possible

Second Step: Inversion

• Perform the empirical analogue of the Hotz-Millerinversion.

Π̂( = 1 ) = log (b( = 1|))−log (b( = 0|))• This gives an estimate of the choice specific valuefunction.

Third Step: Recovering The Structural Parameters

• In the third step, we “invert” our system of equa-

tions to estimate Π( − ).

• Choose Π( − ) to solve the following weightedleast squares problem:

1

P=1

ÃΠ̂( )−P

− b−(−|− )Π( − )

!2( )

• Local linear regression.

• Note that Π̂( ) and b−(−|− ) arefrom first two steps.

• The weights ( ) are kernel weights:

µ −

¶

• The kernel measures the distance between and

• As we send the bandwidth to zero, we get a

consistent estimate of Π( − )

6.1 A Semiparametric Estimator

• Nonparametric estimators suffer from a curse of

dimensionality.

• Wemight wish to use a parametric model of Π( − )

• Approximate Π( − ) by a set of basis func-tions Φ (e.g. linear index, polynomial, etc...)

Π( − ) = Φ( − )0

• Replace Π( − ) in the above withΦ( − )0

P=1

ÃΠ̂( )−P

− b−(−|− )Φ( − )0

!2

• The estimator b is least squares!

• Alternatively, we can do a psuedo-mle procedureas in our simple example:

( = 1|) = exp(Π( = 1 ))

1 + exp(Π( = 1 ))

=exp(

P− b−(−|)Π( = 1 − ))

1 + exp(P− b−(−|)Π( = 1 − ))

=exp(

P− b−(−|)Φ( − )0)

1 + exp(P− b−(−|)Φ( − )0)

• The log-likelihood function is:

()= 1

P

Ãexp(

P− b−(−|)Φ(−)0)

1+exp(P

− b−(−|)Φ(−)0)

!1n =

Ã1

1+exp(P

− b−(−|)Φ(−)0)

!1n = 0

o

• This is just the logit model with a generated re-gressor b−(−|)

• The hard work in Bajari, Hong, Krainer and Nekipelov(2007) is the asymptotic theory.

• We show that b converges at a 12 rate and hasnormal asymptotics.

• The asymptotic distribution is invariant to thechoice of method used to estimate the first stage.

• The bootstrap can be used to compute standarderrors

7 Examples

• Seim

- location choices in video retailing

- Estimate demand and markups using BLP

- Discrete choices are locations

• Andrew Sweeting

- Radio stations benefit from have the commercials

at the same time

- Discrete choice is when to have a commercial dur-

ing drive time

- Spillover from havinng commerials at the same

time

• Ackerberg and Gowrisankaran

- Discrete Choice is choice of payment system

- Spillover from network effect

- Frequency of choices made by

• Bajari, Hong, Krainer and Nekipelov

- stock analyst ratings during tech bubble

- strong buy, buy, hold, sell are ratings choices (or-

dered logit)

- conflicts of interest, publicly available information

and ratings of other analysts determine ratings

8 Conclusions

• Simple model of strategic interaction

• Nonparametric identification

• Simple Nonparametric/Semiparametric Estimators

• Identification and estimation of dynamic gamesfollows similar principals