Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Estimation of Discrete Games
with Weak Assumptions on Information
L. Magnolfi
C. Roncoroni
Yale University
February 3, 2016
1 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Introduction
• Empirical game-theoretic models are an important tool in industrial
organization
• Useful to endogenize firms’ strategic decisions in contexts such as
I
I
I
I
Entry (Bresnahan and Reiss 1991, Berry 1992, Ciliberto and Tamer 2009)
Product or location choice (Mazzeo 2002, Seim 2006)
Advertising (Sweeting 2009)
Technology adoption (Ackerberg and Gowrisankaran 2006)
• Require a priori assumptions on primitives unknown to the researcher
• We focus on assumptions on private information that players have on
each others’ payoffs
2 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Introduction
• Empirical game-theoretic models are an important tool in industrial
organization
• Useful to endogenize firms’ strategic decisions in contexts such as
I
I
I
I
Entry (Bresnahan and Reiss 1991, Berry 1992, Ciliberto and Tamer 2009)
Product or location choice (Mazzeo 2002, Seim 2006)
Advertising (Sweeting 2009)
Technology adoption (Ackerberg and Gowrisankaran 2006)
• Require a priori assumptions on primitives unknown to the researcher
• We focus on assumptions on private information that players have on
each others’ payoffs
2 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Our Contribution
• Current models impose strong assumptions on nature of information
available to players
• We propose a method that leaves this aspect of the model incomplete
I
I
Nest all assumptions in previous literature
Allow for information structures relevant in applications
• We bring to the data a solution concept with weaker restrictions on
behavior
• The method provides informative estimates and counterfactuals
3 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Do Assumptions on Information Matter?
• In general, restrictive assumptions on information structure can result
in biased parameter estimates and misleading conclusions
• In an application we look at effect of large malls on entry by local
supermarkets in Italy
I
Heterogeneity of information across firms and markets cannot be
accommodated by existing models
• Counterfactual policy analysis depends crucially on assumptions
maintained on information
I
Standard assumption of complete information does not fit
features of the industry, and drives counterfactual results
4 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Do Assumptions on Information Matter?
• In general, restrictive assumptions on information structure can result
in biased parameter estimates and misleading conclusions
• In an application we look at effect of large malls on entry by local
supermarkets in Italy
I
Heterogeneity of information across firms and markets cannot be
accommodated by existing models
• Counterfactual policy analysis depends crucially on assumptions
maintained on information
I
Standard assumption of complete information does not fit
features of the industry, and drives counterfactual results
4 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Outline of the Talk
• Model: class of discrete games
• Robust identification results
• Empirical application to effect of large malls on local supermarkets in
Italy
5 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Literature
• Estimation of static discrete games: Bresnahan and Reiss (1991), Berry
(1992), Tamer (2003), Ciliberto and Tamer (2009), de Paula and Tang (2012)
• Recent work on weak assumptions on information:
I
I
Dickstein and Morales (2015) in single agent models
Grieco (2014) in empirical games with flexible information structure
We also build on:
• Theory literature on robust prediction: Bergemann and Morris (2013,
2015)
6 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Model and Empirical Setup
• We observe firm-level outcomes y = (y1 , ..., yn ) in markets indexed by
exogenous market-level covariates x
• Data are conditional distributions Py|x
• Example: data on entry in markets
• Assume data are generated by play of static games
• Players in set N = {1, ..., n} , discrete set of actions ×i∈N Yi = Y 3 y
• Object of inference is parameter vector θ that pins down players’
preferences
7 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Model and Empirical Setup
• We observe firm-level outcomes y = (y1 , ..., yn ) in markets indexed by
exogenous market-level covariates x
• Data are conditional distributions Py|x
• Example: data on entry in markets
• Assume data are generated by play of static games
• Players in set N = {1, ..., n} , discrete set of actions ×i∈N Yi = Y 3 y
• Object of inference is parameter vector θ that pins down players’
preferences
7 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Model and Empirical Setup
• We observe firm-level outcomes y = (y1 , ..., yn ) in markets indexed by
exogenous market-level covariates x
• Data are conditional distributions Py|x
• Example: data on entry in markets
• Assume data are generated by play of static games
• Players in set N = {1, ..., n} , discrete set of actions ×i∈N Yi = Y 3 y
• Object of inference is parameter vector θ that pins down players’
preferences
7 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Payoff Environment
• Players have payoff types ε i ∈ Ei , ε ∼ F (·; θε ) common prior over types
• Payoff functions:
πix,θπ : Y × Ei 7−→ R
• Parameter vector θπ shifts payoff function, θε shifts distribution of
types; θ = (θπ , θε )
8 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Payoff Environment
• Players have payoff types ε i ∈ Ei , ε ∼ F (·; θε ) common prior over types
• Payoff functions:
πix,θπ : Y × Ei 7−→ R
• Parameter vector θπ shifts payoff function, θε shifts distribution of
types; θ = (θπ , θε )
8 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Payoff Environment
• Players have payoff types ε i ∈ Ei , ε ∼ F (·; θε ) common prior over types
• Payoff functions:
πix,θπ : Y × Ei 7−→ R
• Parameter vector θπ shifts payoff function, θε shifts distribution of
types; θ = (θπ , θε )
8 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Entry Game Example
Two-player entry game:
• Players in set N = {1, 2}, discrete set of actions Yi = {0, 1}
• Payoff functions:
πix,θπ (y, ε) = yi xi0 β i + y−i ∆i + ε i
Player 2:
Player 1:
Out
Enter
Out
Enter
0
0
1
1
0,
(0, 0)
x10 β 1
+ ε1, 0
x20 β 2
+ ε2
x10 β 1 + ∆1 + ε 1 ,
x20 β 2 + ∆2 + ε 2
!
• Payoff types ε ∼ F (·; θε ) , payoff parameters θπ = ( βi , ∆i )i=1,2
• We observe, for each x and every outcome y, the probability Py|x
9 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Informational Environment
• Each player i observes her payoff type ε i
• and gets private signals τix (ε) , distributed over (Tx )x∈X according to
n
Pτ x | ε : ε ∈ E , x ∈ X
o
• Information structure
n
o
S = (T x )x∈X , Pτ x | ε : ε ∈ E , x ∈ X
belongs to nonparametric class S .
10 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Informational Environment
• Each player i observes her payoff type ε i
• and gets private signals τix (ε) , distributed over (Tx )x∈X according to
n
Pτ x | ε : ε ∈ E , x ∈ X
o
• Information structure
n
o
S = (T x )x∈X , Pτ x | ε : ε ∈ E , x ∈ X
belongs to nonparametric class S .
10 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Examples of Information Structures
• Complete Information, SCI (Bresnahan and Reiss 1991, Berry 1992):
Tix = E and Pτx |ε {ε} = 1, ∀ε, x, i
i
• Minimal Information, SMIN (Parsch 1992 in auctions, Brock and
Durlauf 2001 in Social Interaction, Seim 2006 in Entry):
Pτx |ε = Pτx , ∀ε, x, i
i
i
• Privileged Information, SP (similar to Proprietary Information of
Engelbrecht-Wiggans, Milgrom and Weber (1983) in common value
auctions):
N = {1, 2} , Pτx |ε {ε} = 1 ∀ε, Pτx |ε = Pτ2x
1
2
11 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Examples of Information Structures
• Complete Information, SCI (Bresnahan and Reiss 1991, Berry 1992):
Tix = E and Pτx |ε {ε} = 1, ∀ε, x, i
i
• Minimal Information, SMIN (Parsch 1992 in auctions, Brock and
Durlauf 2001 in Social Interaction, Seim 2006 in Entry):
Pτx |ε = Pτx , ∀ε, x, i
i
i
• Privileged Information, SP (similar to Proprietary Information of
Engelbrecht-Wiggans, Milgrom and Weber (1983) in common value
auctions):
N = {1, 2} , Pτx |ε {ε} = 1 ∀ε, Pτx |ε = Pτ2x
1
2
11 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Examples of Information Structures
• Complete Information, SCI (Bresnahan and Reiss 1991, Berry 1992):
Tix = E and Pτx |ε {ε} = 1, ∀ε, x, i
i
• Minimal Information, SMIN (Parsch 1992 in auctions, Brock and
Durlauf 2001 in Social Interaction, Seim 2006 in Entry):
Pτx |ε = Pτx , ∀ε, x, i
i
i
• Privileged Information, SP (similar to Proprietary Information of
Engelbrecht-Wiggans, Milgrom and Weber (1983) in common value
auctions):
N = {1, 2} , Pτx |ε {ε} = 1 ∀ε, Pτx |ε = Pτ2x
1
2
11 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Information Matters for Behavior
Consider an entry game with payoff environment:
1
iid
πi (y, ε) = yi − yj + ε i for i = 1, 2 and ε i ∼ U [−1, 1] .
2
Complete Information
ε2
1
(0,1)
(0,1)
(1,1)
(0,1)
(1,0) or
(0,1)
(1,0)
(0,0)
(1,0)
(1,0)
1/2
0
-1
-1
0
1/2
1 ε1
Player i knows ε 1 , ε 2
12 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Information Matters for Behavior
Consider an entry game with payoff environment:
1
iid
πi (y, ε) = yi − yj + ε i for i = 1, 2 and ε i ∼ U [−1, 1] .
2
Complete Information
Minimal Information
ε2
1
ε2
1
(1,1)
(0,1)
(0,1)
1/2
1/2
(0,1)
(1,0) or
(0,1)
(1,0)
(0,0)
(1,0)
(1,0)
(0,1)
(1,1)
1/5
0
0
(1,0)
(0,0)
-1
-1
-1
0
1/2
1 ε1
-1
0 1/5
1/2
1 ε1
Player i only knows ε i
13 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Information Matters for Behavior
Consider an entry game with payoff environment:
1
iid
πi (y, ε) = yi − yj + ε i for i = 1, 2 and ε i ∼ U [−1, 1] .
2
Complete Information
Minimal Information
ε2
1
ε2
1
(1,1)
(0,1)
(0,1)
(0,1)
(1,0) or
(0,1)
(1,0)
(0,0)
(1,0)
(1,0)
1/2
1/2
1/2
(0,1)
(1,1)
ε *2
є [1/8,1/4]
0
0
(1,0)
(0,0)
0
1/2
(1,0)
(0,0)
-1
-1
1 ε1
(1,1)
(0,1)
1/5
0
-1
Privileged Information
ε2
1
-1
-1
0 1/5
1/2
1 ε1
-1
0
1/2
1
ε1
Player 1 knows ε 1 , ε 2 ,
player 2 only knows ε 2
14 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Assumptions on Information and Identified Sets
• Py|x are generated by equilibrium behavior in game with θ0 , S0
Current approach in the literature:
• For a subset S 0 ⊂ S , we denote the identified set ΘBNE
(S 0 )
I
• Assumption on information S 0 ⊂ S seldom motivated by application
• Restrictions on information ΘBNE
(S 0 ) might lead to misspecification
I
Our approach:
• We want to identify the robust set
ΘBNE
(S)
I
15 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Assumptions on Information and Identified Sets
• Py|x are generated by equilibrium behavior in game with θ0 , S0
Current approach in the literature:
• For a subset S 0 ⊂ S , we denote the identified set ΘBNE
(S 0 )
I
• Assumption on information S 0 ⊂ S seldom motivated by application
• Restrictions on information ΘBNE
(S 0 ) might lead to misspecification
I
Our approach:
• We want to identify the robust set
ΘBNE
(S)
I
15 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Equilibrium and Information
• Direct computation of ΘBNE
(S) is intractable
I
• Would require to specify every possible S, and find corresponding
BNE strategy profiles
• To recover object of interest we adopt alternative solution concept –
Bayes Correlated Equilibrium (Bergemann and Morris, 2015)
• Equilibrium as joint distribution of actions and payoff types
16 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Equilibrium and Information
• Direct computation of ΘBNE
(S) is intractable
I
• Would require to specify every possible S, and find corresponding
BNE strategy profiles
• To recover object of interest we adopt alternative solution concept –
Bayes Correlated Equilibrium (Bergemann and Morris, 2015)
• Equilibrium as joint distribution of actions and payoff types
16 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Bayes Correlated Equilibrium
A Bayes Correlated Equilibrium (BCE) for the game (SMIN ,θ ) is a probability
measure ν over (Y × E ) that satisfies
(i) consistency with the prior:
∑
Z
y∈Y
{ε̃≤ε}
dν {y, ε̃} = F (ε; θε ) ,
for all ε ∈ E , and
(ii) best response:
∀i, ε i , yi , yi0 such that if ν {yi |ε i } > 0,
Z
x,θπ
π
y,
ε
dν
y
,
ε
|
y
,
ε
≥
(
)
{
}
∑
i
−i −i i i
i
E −i
y−i ∈ Y−i
∑
y−i ∈ Y−i
Z
E −i
πix,θπ
yi0 , y−i , ε i dν {y−i , ε −i |yi , ε i }
17 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Predictions for Entry Game
BCE Predictions
Minimal Information Predictions
Complete Information
Privileged Information
P11
P01
P10
18 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Predictions for Entry Game
Minimal Information Predictions
Privileged Information
BCE Predictions
Complete Information
P11
P01
P10
19 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Bayes Correlated Equilibrium Identification
•
The BCE distribution ν induces an equilibrium prediction
pν (y) =
•
Z
ε∈E
dν {y, ε}
Set of predictions
n
o
QBCE
(x) = p ∈ ∆|Y|−1 : ∃νx,θ ∈ BCE (θ ) such that p = pνx
θ
•
Sharp identified set is
ΘBCE
I
=
n
θ ∈ Θ such that Py|x ∈ QBCE
(x) x − a.s.
θ
o
20 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Bayes Correlated Equilibrium Identification
•
The BCE distribution ν induces an equilibrium prediction
pν (y) =
•
Z
ε∈E
dν {y, ε}
Set of predictions
n
o
QBCE
(x) = p ∈ ∆|Y|−1 : ∃νx,θ ∈ BCE (θ ) such that p = pνx
θ
•
Sharp identified set is
ΘBCE
I
=
n
θ ∈ Θ such that Py|x ∈ QBCE
(x) x − a.s.
θ
o
20 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Bayes Correlated Equilibrium Identification
•
The BCE distribution ν induces an equilibrium prediction
pν (y) =
•
Z
ε∈E
dν {y, ε}
Set of predictions
n
o
QBCE
(x) = p ∈ ∆|Y|−1 : ∃νx,θ ∈ BCE (θ ) such that p = pνx
θ
•
Sharp identified set is
ΘBCE
I
=
n
θ ∈ Θ such that Py|x ∈ QBCE
(x) x − a.s.
θ
o
20 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Robust Identification Result
Proposition
The identified set under BCE behavior coincides with the identified set
under BNE and robust assumptions on information:
ΘBCE
= ΘBNE
(S) .
I
I
• Identifying ΘBCE
as a way of obtaining ΘBNE
(S)
I
I
• The result builds on the robust prediction property of BCE
21 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Robust Identification Result
Proposition
The identified set under BCE behavior coincides with the identified set
under BNE and robust assumptions on information:
ΘBCE
= ΘBNE
(S) .
I
I
• Identifying ΘBCE
as a way of obtaining ΘBNE
(S)
I
I
• The result builds on the robust prediction property of BCE
21 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Characterization of the Identified Set
Sharp Identified set is characterized through the support function of the
convex set of predictions
n
o
BCE
ΘBCE
=
θ
∈
Θ
such
that
P
∈
Q
x
x
−
a.s.
(
)
y|x
I
θ






=
θ ∈ Θ| sup Py0 |x b − sup p0 b ≤ 0 , x − a.s.


b∈B
p∈QBCE (x)
θ
where B is the |Y| − 1 dimensional unit ball.
Since the inner maximization is linear (in p), the program can be simplified
exploiting duality.
22 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Characterization of the Identified Set
Sharp Identified set is characterized through the support function of the
convex set of predictions
n
o
BCE
ΘBCE
=
θ
∈
Θ
such
that
P
∈
Q
x
x
−
a.s.
(
)
y|x
I
θ






=
θ ∈ Θ| sup Py0 |x b − sup p0 b ≤ 0 , x − a.s.


b∈B
p∈QBCE (x)
θ
where B is the |Y| − 1 dimensional unit ball.
Since the inner maximization is linear (in p), the program can be simplified
exploiting duality.
22 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Inference
• Inference is based on level sets of sample criterion


1 n
Cn (θ ) = ∑ sup P̂y0 |x b − sup p0 b
j
n j=1 b∈B
p∈QBCE
( xj )
θ
+
where P̂y|xj is empirical frequency of strategy profile y in observations
with covariates x = xj
• We discretize covariates, and use subsampling as in Chernozhukov
Hong and Tamer (2007) to recover cutoff ĉ such that Cˆ = {θ | Cn (θ ) ≤ ĉ}
is .95 confidence set for ΘI
23 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Inference
• Inference is based on level sets of sample criterion


1 n
Cn (θ ) = ∑ sup P̂y0 |x b − sup p0 b
j
n j=1 b∈B
p∈QBCE
( xj )
θ
+
where P̂y|xj is empirical frequency of strategy profile y in observations
with covariates x = xj
• We discretize covariates, and use subsampling as in Chernozhukov
Hong and Tamer (2007) to recover cutoff ĉ such that Cˆ = {θ | Cn (θ ) ≤ ĉ}
is .95 confidence set for ΘI
23 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Identifying Power of BCE in Simple Games
We investigate identification power of BCE in 2-players entry games with
payoffs linear in covariates and parametric distribution of payoff types
• Point identification can be achieved under assumptions on the
distribution of covariates x
• We discuss identification of distribution of payoff types
• We look at the effect of strong assumptions on information and
falsification
24 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Identifying Power of BCE in Simple Games
We investigate identification power of BCE in 2-players entry games with
payoffs linear in covariates and parametric distribution of payoff types
• Point identification can be achieved under assumptions on the
distribution of covariates x
• We discuss identification of distribution of payoff types
• We look at the effect of strong assumptions on information and
falsification
24 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Point Identification
Proposition
Consider a two-player entry game with payoffs linear in covariates x:
πix,θπ (y, ε) = yi xi0 β i + y−i ∆i + ε i ,
Assume
(i) ε i has positive everywhere marginal density and is independent of x
(ii) for each i = 1, 2 there exists xik ∈
/ x−i , such that β ik 6= 0 and xik |x−ik has
positive everywhere density
Then ( β i , ∆i )i=1,2 are point identified
25 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Identification of Joint Distribution of Payoff Types
• Variation in excluded covariates point-identifies marginal distribution
of payoff types
• We partially identify the joint distribution of payoff types
Consider the bounds implied by dominance:
Py|x (y) ≥
Z
1 {ε : y is dominant given θ π , x} dF (ε) , ∀y ∈ Y
26 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
A Computed Example of ΘBCE
I
• Entry game with payoffs linear in covariates (xc , x1 , x2 ), with x−i
excluded from i’s payoff
iid
• Payoff shocks ε i ∼Logit
• Data generating processes: BNE in the games with information
structure SCI , SMIN , SP
0
• Covariates iid uniformly over a finite support: X = {−1, 0, 1}, or
00
X = {−3, 0, 3}
• In next table we report projections of ΘBCE
for all the combinations of
I
DGP and support of covariates
27 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Variation in Excluded Covariates at work
Projection of ΘBCE
under different S0 , X
I
DGP:πix,θπ (y, ε) = yi x0 βc + xi0 β i + y−i ∆i + ε i
θ0
0
S0 = SCI
S0 = SMIN
S0 = SP
[.90,1.02]
[.81,1.11]
[.76,1.04]
βC
1
β1
1
[.92,1.09]
[.90,1.21]
[.91,1.17]
β2
1
[.92,1.09]
[.90,1.21]
[.83,1.17]
∆1
-1
[-2.36,-.82]
[-1.48,-.78]
[-1.99,-.84]
∆2
-1
[-2.36,-.82]
[-1.48,-.78]
[-2.13,-.83]
βC
1
[.97,1.08]
[.87,1.08]
[.86,1.08]
β1
1
[.96,1.05]
[.96,1.09]
[.96,1.06]
X
X
00
β2
1
[.96,1.05]
[.96,1.09]
[.95,1.08]
∆1
-1
[-1.30,-.90]
[-1.23,-.90]
[-1.26,-.90]
∆2
-1
[-1.30,-.90]
[-1.23,-.90]
[-1.29,-.89]
28 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Projection of ΘBCE
under different S0 , X
I
DGP:πix,θπ (y, ε) = yi x0 βc + xi0 β i + y−i ∆i + ε i
θ0
βC
1
β1
1
X
0
S0 = SCI
S0 = SMIN
S0 = SP
[.90,1.02]
[.81,1.11]
[.76,1.04]
[.92,1.09]
[.90,1.21]
[.91,1.17]
β2
1
[.92,1.09]
[.90,1.21]
[.83,1.17]
∆1
-1
[-2.36,-.82]
[-1.48,-.78]
[-1.99,-.84]
∆2
-1
[-2.36,-.82]
[-1.48,-.78]
[-2.13,-.83]
βC
1
[.97,1.08]
[.87,1.08]
[.86,1.08]
β1
1
[.96,1.05]
[.96,1.09]
[.96,1.06]
β2
1
[.96,1.05]
[.96,1.09]
[.95,1.08]
∆1
-1
[-1.30,-.90]
[-1.23,-.90]
[-1.26,-.90]
∆2
-1
[-1.30,-.90]
[-1.23,-.90]
[-1.29,-.89]
X
00
• Identified sets shrink with variation in covariates
29 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Identification Under Restrictions on Information
•
We may also restrict the class of information structures to S 0 ⊆ S , identify
ΘBNE
(S 0 )
I
0
For the entry game model as above with X , the set ΘBNE
(S 0 ) identified under
I
correct restrictions on information:
•
•
If S 0 = {SMIN } = {S0 } the model is point identified
If S 0 = {SCI } = {S0 }
θ0
Proj. of ΘBNE
(S 0 )
I
Proj. of ΘBCE
I
βC
1
[0.96,1.02]
[.90,1.02]
β1
1
[0.96,1.04]
[.92,1.09]
β2
1
[0.96,1.04]
[.92,1.09]
∆1
-1
[-1.11,-0.97]
[-2.36,-.82]
∆2
-1
[-1.11,-0.97]
[-2.36,-.82]
30 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Identification Under Restrictions on Information
•
We may also restrict the class of information structures to S 0 ⊆ S , identify
ΘBNE
(S 0 )
I
0
For the entry game model as above with X , the set ΘBNE
(S 0 ) identified under
I
correct restrictions on information:
•
•
If S 0 = {SMIN } = {S0 } the model is point identified
If S 0 = {SCI } = {S0 }
θ0
Proj. of ΘBNE
(S 0 )
I
Proj. of ΘBCE
I
βC
1
[0.96,1.02]
[.90,1.02]
β1
1
[0.96,1.04]
[.92,1.09]
β2
1
[0.96,1.04]
[.92,1.09]
∆1
-1
[-1.11,-0.97]
[-2.36,-.82]
∆2
-1
[-1.11,-0.97]
[-2.36,-.82]
30 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Identification Under Restrictions on Information
•
We may also restrict the class of information structures to S 0 ⊆ S , identify
ΘBNE
(S 0 )
I
0
For the entry game model as above with X , the set ΘBNE
(S 0 ) identified under
I
correct restrictions on information:
•
•
If S 0 = {SMIN } = {S0 } the model is point identified
If S 0 = {SCI } = {S0 }
θ0
Proj. of ΘBNE
(S 0 )
I
Proj. of ΘBCE
I
βC
1
[0.96,1.02]
[.90,1.02]
β1
1
[0.96,1.04]
[.92,1.09]
β2
1
[0.96,1.04]
[.92,1.09]
∆1
-1
[-1.11,-0.97]
[-2.36,-.82]
∆2
-1
[-1.11,-0.97]
[-2.36,-.82]
30 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Identification Under Restrictions on Information
However, restrictive assumptions
• may lead to misspecification:
θ0 ∈
/ ΘBNE
S 0 6= ∅
I
• or falsification of the model
ΘBNE
S0 = ∅
I
In our example
• If S 0 = {SCI } but S0 = {SMIN } (and viceversa), then set ΘBNE
(S 0 ) = ∅
I
31 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Identification Under Restrictions on Information
However, restrictive assumptions
• may lead to misspecification:
θ0 ∈
/ ΘBNE
S 0 6= ∅
I
• or falsification of the model
ΘBNE
S0 = ∅
I
In our example
• If S 0 = {SCI } but S0 = {SMIN } (and viceversa), then set ΘBNE
(S 0 ) = ∅
I
31 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Bias from Assumptions on Information
Consider model with:
1
iid
πi (y, ε) = yi ∆yj + ε i for i = 1, 2, ∆0 = − and ε i ∼ U [−1, 1] .
2
If we correctly assume SMIN , point identification is guaranteed by:
Py (1, 1) = (si (yi = 1|∆))2
Similarly, if we correctly assume SCI , then
Py (1, 1) = (1 − Fi (−∆))2
ˆ = −0.2
If S0 = SMIN , but we assume SCI , we recover ∆
32 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Bias from Assumptions on Information
Consider model with:
1
iid
πi (y, ε) = yi ∆yj + ε i for i = 1, 2, ∆0 = − and ε i ∼ U [−1, 1] .
2
If we correctly assume SMIN , point identification is guaranteed by:
Py (1, 1) = (si (yi = 1|∆))2
Similarly, if we correctly assume SCI , then
Py (1, 1) = (1 − Fi (−∆))2
ˆ = −0.2
If S0 = SMIN , but we assume SCI , we recover ∆
32 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Bias from Assumptions on Information
Consider model with:
1
iid
πi (y, ε) = yi ∆yj + ε i for i = 1, 2, ∆0 = − and ε i ∼ U [−1, 1] .
2
If we correctly assume SMIN , point identification is guaranteed by:
Py (1, 1) = (si (yi = 1|∆))2
Similarly, if we correctly assume SCI , then
Py (1, 1) = (1 − Fi (−∆))2
ˆ = −0.2
If S0 = SMIN , but we assume SCI , we recover ∆
32 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
The Method
Adopting BCE:
• We obtain robust identified set
• Estimation of discrete games with weak assumptions on info structure
is feasible, with computationally manageable inference
• In standard models of entry, with variation in excluded covariates, the
identified sets are informative
33 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Application
• Large grocery-anchored malls are relatively recent phenomenon in
Italy
• Debate on their impact on local retail
• Similar to Walmart debate in the US
• Do large malls create underserved markets by preventing presence of
local supermarkets?
• Does the answer change when relying on models without strong
assumptions on information?
I
Institutional details do not support standard assumptions on
information
34 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Application
• Large grocery-anchored malls are relatively recent phenomenon in
Italy
• Debate on their impact on local retail
• Similar to Walmart debate in the US
• Do large malls create underserved markets by preventing presence of
local supermarkets?
• Does the answer change when relying on models without strong
assumptions on information?
I
Institutional details do not support standard assumptions on
information
34 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Data: Grocery Markets and Industry Players
• Define local grocery markets, based on commuting areas, excluding
big cities
• Collect data on market level demographics, presence of supermarkets
and large malls in Central and Northern Italy in 2013
• Focus on supermarkets with floor-space > 1,500m2 (~16,000 ft2 )
• Three types of supermarket groups:
1. Networks of cooperatives, under same umbrella-organization
2. Italian independent groups (Esselunga, Finiper, Bennet, ...)
3. French groups Carrefour and Auchan
• Malls have≥ 50 shops and large grocery anchor (~50,000-100,000 ft2 )
35 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Data: Grocery Markets and Industry Players
• Define local grocery markets, based on commuting areas, excluding
big cities
• Collect data on market level demographics, presence of supermarkets
and large malls in Central and Northern Italy in 2013
• Focus on supermarkets with floor-space > 1,500m2 (~16,000 ft2 )
• Three types of supermarket groups:
1. Networks of cooperatives, under same umbrella-organization
2. Italian independent groups (Esselunga, Finiper, Bennet, ...)
3. French groups Carrefour and Auchan
• Malls have≥ 50 shops and large grocery anchor (~50,000-100,000 ft2 )
35 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Data: Grocery Markets and Industry Players
• Define local grocery markets, based on commuting areas, excluding
big cities
• Collect data on market level demographics, presence of supermarkets
and large malls in Central and Northern Italy in 2013
• Focus on supermarkets with floor-space > 1,500m2 (~16,000 ft2 )
• Three types of supermarket groups:
1. Networks of cooperatives, under same umbrella-organization
2. Italian independent groups (Esselunga, Finiper, Bennet, ...)
3. French groups Carrefour and Auchan
• Malls have≥ 50 shops and large grocery anchor (~50,000-100,000 ft2 )
35 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Malls and the Supermarket Industry
• 2000-2013 is a period of expansion of the supermarket industry,
following regulatory overhaul
• Most large malls are either built or planned by early 2000s
• We consider as exogenous the presence of malls
I
I
Larger “catchment area”, driven by non-grocery shopping
Subject to idiosyncratic constraints on location due to zoning and
land availability
36 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Presence of Large Malls and Market Structure
Variable
# of Supermarkets
Large Mall in Market
-0.437
(0.278)
(0.165)
(0.145)
(0.175)
Market Size
3.764∗∗∗
2.658∗∗∗
1.213∗∗∗
1.766∗∗∗
(0.236)
(0.158)
(0.109)
(0.143)
Constant
-0.222
# of Players in Market
0.167
-0.150
-0.242
0.022
(0.378)
(0.230)
R2
0.677
0.255
0.434
0.225
Model
Linear Regression
Ordered Probit
Linear Regression
Ordered Probit
37 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Presence of Large Malls and Market Structure
Variable
Large Mall in Market
Market Size
Constant
•
# of Supermarkets
# of Players in Market
-0.437
-0.222
-0.150
-0.242
(0.278)
(0.165)
(0.145)
(0.175)
3.764∗∗∗
2.658∗∗∗
1.213∗∗∗
1.766∗∗∗
(0.236)
(0.158)
(0.109)
(0.143)
0.167
0.022
(0.378)
(0.230)
R2
0.677
0.255
0.434
0.225
Model
Linear Regression
Ordered Probit
Linear Regression
Ordered Probit
Negative but statistically weak effect of large malls on number of
supermarkets/industry players in a market
38 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Probit Regressions
Variable
Large Mall in Market
Market Size
COOP Entry
.
.
1.640∗∗∗
(0.225)
IT Entry
.
.
1.644∗∗∗
(0.268)
FR Entry
.
.
0.882∗∗∗
(0.198)
-0 722∗∗∗
-0 182
(0 243)
(0 248)
(0 220)
.
(0.172)
.
(0.186)
0.252
(0.192)
0 098
Competitive Effects ∆ji
0 452∗∗∗
COOP Present in 2013
IT Present in 2013
.
.
0.132
(0.196)
0 569∗∗∗
(0 170)
FR Present in 2013
N
481
0 234
.
(0.206)
0 043
447
447
39 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Probit Regressions
Variable
Large Mall in Market
Market Size
COOP Entry
.
(0.243)
1.640∗∗∗
(0.225)
-0 722∗∗∗
IT Entry
FR Entry
.
(0.248)
1.644∗∗∗
(0.268)
.
(0.220)
0.882∗∗∗
(0.198)
.
(0.172)
.
(0.186)
0.252
(0.192)
-0 182
0 098
Competitive Effects ∆ji
0 452∗∗∗
COOP Present in 2013
IT Present in 2013
.
.
0.132
(0.196)
0 569∗∗∗
(0 170)
FR Present in 2013
•
0 234
.
(0.206)
0 043
Heterogeneous effect of large malls on different industry players
40 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
The Game-Theoretic Model
• We model market structure as static game between main supermarket
industry players
I
No data on timing of entry decisions
• Features of the industry suggest complex information structure
I
I
I
I
Straightforward business model, complicated entry regulation
Information about competitors’ costs and profits can vary by
group, market
Political connections of cooperatives suggest informational
advantage
No regret argument used to justify complete information: not
obvious in this industry
41 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
The Game-Theoretic Model
• We model market structure as static game between main supermarket
industry players
I
No data on timing of entry decisions
• Features of the industry suggest complex information structure
I
I
I
I
Straightforward business model, complicated entry regulation
Information about competitors’ costs and profits can vary by
group, market
Political connections of cooperatives suggest informational
advantage
No regret argument used to justify complete information: not
obvious in this industry
41 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
The Game-Theoretic Model
•
Payoff for player i from entry yi = 1 in market m with observable
characteristics xm is:
πix,θπ (yi , y−i ; ε im ) = xim β i + ∑ yj ∆j + ε im ,
j6 =i
where θ π = ( β, ∆) is parameter vector, and ε im is payoff type
•
xim are market-level observables including:
I
I
I
•
market size
distance from headquarters/home region
presence of large malls in local market
Mkt-level correlation among payoff types allowed
42 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
The Game-Theoretic Model
•
Payoff for player i from entry yi = 1 in market m with observable
characteristics xm is:
πix,θπ (yi , y−i ; ε im ) = xim β i + ∑ yj ∆j + ε im ,
j6 =i
where θ π = ( β, ∆) is parameter vector, and ε im is payoff type
•
xim are market-level observables including:
I
I
I
•
market size
distance from headquarters/home region
presence of large malls in local market
Mkt-level correlation among payoff types allowed
42 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Inference
•
Compute confidence sets for both ΘBCE
and ΘI (SCI )
I
•
Inference on ΘBCE
based on empirical criterion function
I


1 n
0
0
Cn (θ ) = ∑ sup P̂y|xj b − sup p b
n j=1 b∈B
p∈QBCE
( xj )
θ
+
•
Inference on sharp ΘI (SCI ) based on empirical criterion


1 n
CI
0
0
Cn (θ ) = ∑ sup P̂y|xj b −
sup
p b
n j=1 b∈Dir
p∈QPSNE
( xj )
θ
+
where Dir contains vectors corresponding to core-determining class (Galichon
and Henry 2011)
•
Cutoffs found by subsampling as in Ciliberto and Tamer (2009)
43 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Inference
•
Compute confidence sets for both ΘBCE
and ΘI (SCI )
I
•
Inference on ΘBCE
based on empirical criterion function
I


1 n
0
0
Cn (θ ) = ∑ sup P̂y|xj b − sup p b
n j=1 b∈B
p∈QBCE
( xj )
θ
+
•
Inference on sharp ΘI (SCI ) based on empirical criterion


1 n
CI
0
0
Cn (θ ) = ∑ sup P̂y|xj b −
sup
p b
n j=1 b∈Dir
p∈QPSNE
( xj )
θ
+
where Dir contains vectors corresponding to core-determining class (Galichon
and Henry 2011)
•
Cutoffs found by subsampling as in Ciliberto and Tamer (2009)
43 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Inference
•
Compute confidence sets for both ΘBCE
and ΘI (SCI )
I
•
Inference on ΘBCE
based on empirical criterion function
I


1 n
0
0
Cn (θ ) = ∑ sup P̂y|xj b − sup p b
n j=1 b∈B
p∈QBCE
( xj )
θ
+
•
Inference on sharp ΘI (SCI ) based on empirical criterion


1 n
CI
0
0
Cn (θ ) = ∑ sup P̂y|xj b −
sup
p b
n j=1 b∈Dir
p∈QPSNE
( xj )
θ
+
where Dir contains vectors corresponding to core-determining class (Galichon
and Henry 2011)
•
Cutoffs found by subsampling as in Ciliberto and Tamer (2009)
43 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Preview of Results
• If S0 = SCI , then ΘBNE
. In this case, we expect the
({SCI }) ⊆ ΘBCE
I
I
corresponding confidence sets also to be nested, up to sampling
variation
• Under misspecification, the identified set ΘBNE
({SCI }) would be
I
empty. Not clear what to expect from confidence sets in that case
• We find the sets non-nested, pointing in the direction of SCI not being
supported by the data
• Difference in confidence sets important when looking at effect of malls,
competition effects
44 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Preview of Results
• If S0 = SCI , then ΘBNE
. In this case, we expect the
({SCI }) ⊆ ΘBCE
I
I
corresponding confidence sets also to be nested, up to sampling
variation
• Under misspecification, the identified set ΘBNE
({SCI }) would be
I
empty. Not clear what to expect from confidence sets in that case
• We find the sets non-nested, pointing in the direction of SCI not being
supported by the data
• Difference in confidence sets important when looking at effect of malls,
competition effects
44 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Estimation Results I
Parameter
_____ Weak Assumptions on Info - BCE, ____ Complete Information - Pure Nash
Constant
Market Size
Home Region:
Cooperatives
Italian Supermarket Groups
French Supermarket Groups
45 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Estimation Results II
Parameter
_____ Weak Assumptions on Info - BCE, ____ Complete Information - Pure Nas
Presence of Large Malls:
Cooperatives
Italian Supermarket Groups
French Supermarket Groups
Correlation Of Unobs. Profitability
•
With weaker assumptions on information, not reject larger positive effect of
malls
46 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Estimation Results III
Parameter
_____ Weak Assumptions on Info - BCE, ____ Complete Information - Pure Nash
Competitive Effects:
Cooperatives
Italian Supermarket Groups
French Supermarket Groups
•
Systematically larger (in absolute value) competitive effects with weak
assumptions
47 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Counterfactual: Large Malls in Underserved Markets
• Concern: large malls with grocery anchor result in no other
supermarkets in small geographical markets, hurting local consumers
• Look at the effect of removing malls on market structure
• Focus on markets with a large mall and no local supermarkets
Report the change in average (across market) upper bound on probability of
events
48 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Results: Predicted Probabilities
ub
Change in average upper bounds in probabilities P (θ ) :
h
i
ub
ub
infθ ∈Θ̂ P (θ ) , supθ ∈Θ̂ P (θ )
Outcome
Weak Assumptions on Info - BCE
Complete Information - Nash
(1)
(2)
No Entry
[-0.44, 0.06]
[-0.42, 0.06 ]
Entry by Cooperatives
[-0.54, 0.40 ]
[-0.03, 0.53 ]
Entry by Italian Groups
[-0.21, 0.11 ]
[-0.14, 0.45 ]
Entry by French Groups
[-0.08, 0.17 ]
[-0.22, 0.69 ]
Entry by at least 1 Player
[-0.03, 0.32 ]
[0.05, 0.43 ]
Entry by at least 2 Players
[-0.50, 0.33 ]
[0.16, 0.44 ]
49 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Results: Predicted Probabilities
ub
Change in average upper bounds in probabilities P (θ ) :
i
h
ub
ub
infθ ∈Θ̂ P (θ ) , supθ ∈Θ̂ P (θ )
Outcome
Weak Assumptions on Info - BCE
Complete Information - Nash
(1)
(2)
No Entry
[-0.44, 0.06]
[-0.42, 0.06 ]
Entry by at least 1 Player
[-0.03, 0.32 ]
[0.05, 0.43 ]
Entry by at least 2 Players
[-0.50, 0.33 ]
[0.16, 0.44 ]
•
Complete Information model predicts increase in upper bound probability of
entry by at least 1, 2 players
50 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Results: Predicted Probabilities
ub
Change in average upper bounds in probabilities P (θ ) :
h
i
ub
ub
infθ ∈Θ̂ P (θ ) , supθ ∈Θ̂ P (θ )
Outcome
Weak Assumptions on Info - BCE
Complete Information - Nash
(1)
(2)
No Entry
[-0.44, 0.06]
[-0.42, 0.06 ]
Entry by at least 1 Player
[-0.03, 0.32 ]
[0.05, 0.43 ]
Entry by at least 2 Players
[-0.50, 0.33 ]
[0.16, 0.44 ]
•
This result does not hold any longer with weak assumptions on information
51 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Conclusions
• We propose a method to estimate models of discrete games under
weak assumptions on information
• The method yields informative bounds in standard models of entry
• In an application to the Italian supermarket industry, we find that
evaluation of counterfactual policy changes when strong assumptions
on information are removed
• Future research: beyond discrete games
52 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Conclusions
• We propose a method to estimate models of discrete games under
weak assumptions on information
• The method yields informative bounds in standard models of entry
• In an application to the Italian supermarket industry, we find that
evaluation of counterfactual policy changes when strong assumptions
on information are removed
• Future research: beyond discrete games
52 / 52
Introduction
Model
Robust Identification
Identification in Simple Entry Games
Application
Conclusions
• We propose a method to estimate models of discrete games under
weak assumptions on information
• The method yields informative bounds in standard models of entry
• In an application to the Italian supermarket industry, we find that
evaluation of counterfactual policy changes when strong assumptions
on information are removed
• Future research: beyond discrete games
52 / 52