A revealed preference theory of monotone choice and
strategic complementarity
Natalia Lazzati
UC Santa Cruz
joint work with John Quah, Johns Hopkins U, and Koji Shirai, Kwansei Gakuin U
September, 2016
(joint work with John Quah, Johns Hopkins A
U,revealed
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U ) choice and strategic complementarity
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Introduction
What is the empirical content of the supermodular games?
We provide a revealed preference test and out-of-sample predictions for
Nash equilibrium behavior.
We extend the ideas to cross-sectional data.
We apply our results to study smoking decisions among married couples:
results are easy to implement
conditions are quite informative
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U ) choice and strategic complementarity
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Motivation
Revealed preference theory is often used for single agent models.
One of the most well known results is Afriat’s Theorem.
We apply a similar approach to the analysis of a multi-agent model:
we focus on supermodular games
we have variation of set constraints and payo¤-relevant parameters
we provide an econometric implementation of our results
don’t assume groups are randomly formed
agnostic about eq selection
random treatment assignment
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U ) choice and strategic complementarity
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Closest Work
Revealed preference analysis of eq behavior with monotone best-replies:
Topkis (1998) and Carvajal (2004)
Econometric implementation:
Manski (2007)
Echenique and Komunjer (2009)
Aradillas-Lopez (2011), Kline and Tamer (2012), Molinari and Rosen
(2008), Uetake and Watanabe (2013) and Lazzati (2014)
Application to smoking decisions:
Cutler and Glaeser (2010)
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U,revealed
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U ) choice and strategic complementarity
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Smoking Decisions among Married Couples
Example
Each member of a married couple decides whether to smoke.
Payo¤s depend on his/her partner’s choice and the smoking policy at work
Ui (ai , a i , yi ) : f0, 1g
f0, 1g
fA, R g ! R
where
ai = 1 if the person smokes (and 0 otherwise)
= 1 if his/her partner smokes (and 0 otherwise)
yi = A if it is fully allowed to smoke at work (and R if restricted)
a
i
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U ) choice and strategic complementarity
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Smoking Decisions among Married Couples
Example
Let
4i (a i , yi ) = Ui (ai = 1, a i , yi )
Ui (ai = 0, a i , yi )
We believe there are strategic complementarities (spm game):
between own action and other players’actions
4i (a
i
= 1, yi ) > 4i (a
i
= 0, yi )
between own action and the parameter (if we let A > R)
4i (a i , yi = A) > 4i (a i , yi = R )
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Smoking Decisions among Married Couples
Example
Strategic complementarities guarantee that:
Best-replies increase in ξ i = (a i , yi )
BRi (ξ i , f0, 1g) = arg max fUi (ai , ξ i ) : ai 2 f0, 1gg
Results involve simple extensions of a single-agent model.
The game has at least one pure strategy Nash equilibrium
Assuming Nash behavior is internally consistent.
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U ) choice and strategic complementarity
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Smoking Decisions among Married Couples
Example
We observe the smoking choices of couples for di¤erent smoking policies.
For instance, we might observe (0, 1, A, A), (1, 1, A, R), etc.
We are interested in two questions:
Are these choices consistent with the Nash eq of a spm game?
Can we predict smoking choices for speci…c policies at work?
Remark. The analysis in the paper is more complicated as we also have
variation of constraint sets.
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Smoking Decisions among Married Couples
Example
We extend the idea to cross-sectional data.
We might observe µ1 =
1 1 1 1
2, 8, 8, 4
, A, A , µ2 =
1 1 1
4, 2, 4,0
, A, R , etc.
[µt = (Pr (0, 0) , Pr (1, 0) , Pr (0, 1) , Pr (1, 1)) , y1t , y2t ]
(We are also interested in consistency and extrapolation.)
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The Game
N = f1, 2, ..., n g is the set of players
R is the action space of player i
Xi
The feasible action sets of player i are intervals
Ai = a i , a i = a i 2 Xi
a
i
= ( aj : j
R : ai
ai
ai
n, j 6= i ) is a vector of other players’actions
yi is a payo¤-relevant parameter for player i
ξ i = (a i , yi ) 2 Ξi , where (Ξi ,
) is a POSET
for a …xed ξ i , %i is a preference relation on Xi that induces
BRi (ξ i , Ai ) = ai0 2 Ai : ai0 , ξ i %i (ai , ξ i ) for all ai 2 Ai
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The Game
A pro…le (%i )i 2N , y = (yi )i 2N , and A = (Ai )i 2N de…ne a game
G(y , A) = [(%i )i 2N , y , A] .
If we vary y , A we get a family of games
G = fG(y , A)g(y ,A )2Y
A.
G has strategic complementarities if all agents have monotone best
response correspondences, i.e., 8ξ i00 > ξ i0 ,
a00 2 BRi ξ i00 , Ai
and a0 2 BRi ξ i0 , Ai ) a00
a0 .
This notion of monotonicity is stronger than the strong set order.
Similar to non-satiation in the standard revealed preference analysis.
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Properties of this Class of Games
Theorem (Milgrom and Shannon (1994))
Suppose %i is a regular preference on Xi Ξi .
Then, agent i has a monotone BR correspondence i¤ %i obeys SSCD.
Theorem (Vives (1990), Milgrom and Roberts (1990), Zhou (1994))
Suppose that G has strategic complementarities.
Then, each G(y , A) 2 G has a pure strategy Nash equilibrium.
(In fact, each game has extremal equilibria that increase with y .)
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Conditions on Preferences
%i is regular if BRi (ξ i , Ai ) is non-empty and compact
%i has Strict Single Crossing Di¤ (SSCD), 8ai00 > ai0 and 8ξ i00 > ξ i0 ,
(ai00 , ξ i0 ) %i (ai0 , ξ i0 ) ) (ai00 , ξ i00 )
i
(ai0 , ξ i00 )
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U ) choice and strategic complementarity
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Data and Consistency
From a collection of games G = fG(y , A)g(y ,A )2Y A , we observe
io
nh
O = at = ait i 2N , y t = yit i 2N , At = Ati i 2N
t 2T
De…nition
O is consistent with strategic complementarities if there exists a pro…le of
regular preferences (%i )i 2N each of which derives a monotone best
response correspondence such that each observation constitutes a Nash
equilibrium, i.e., 8t 2 T and 8i 2 N,
(ait , ξ ti ) %i (ai , ξ ti ) for all ai 2 Ati .
(Recall that ξ ti = at i , yit .)
State minimal conditions on O so that the data set is consistent
with strategic complementarities.
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U ) choice and strategic complementarity
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Direct and Indirect Revealed Preference
The direct revealed preference %Ri is given by
(ai00 , ξ i ) %Ri (ai0 , ξ i ) if (ai00 , ξ i ) = (ait , ξ ti ) and ai0 2 Ati for some t 2 T .
The indirect revealed preference %RT
is the transitive closure of %Ri .
i
That is, (ai00 , ξ i ) %RT
(ai0 , ξ i ) if there exists zi1 , zi2 , ..., zik in Xi such that
i
(ai00 , ξ i ) %Ri (zi1 , ξ i ) %Ri (zi2 , ξ i ) %Ri ... %Ri (zik , ξ i ) %Ri (ai0 , ξ i ).
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U ) choice and strategic complementarity
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Axiom of Revealed Complementarity (ARC)
Data set O = f(at , y t , At )gt 2T obeys ARC if for each i 2 N, 8s, t 2 T ,
ξ ti > ξ si , ait < ais , and (ais , ξ si ) %RT
(ait , ξ si ) =) (ait , ξ ti ) 6%RT
(ais , ξ ti ).
i
i
(Recall that ξ ti = at i , yit .)
ARC can be checked in a …nite number of steps (as data is …nite)
ARC is clearly refutable
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U ) choice and strategic complementarity
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Revealed Complementarity
Theorem
The following statements on O = f(at , y t , At )gt 2T are equivalent
(a) O is consistent with strategic complementarities
(b) O obeys ARC
(c) O is rationalizable by a pro…le of regular and SSCD preferences
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U ) choice and strategic complementarity
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Revealed Complementarity
Proof.
a =) b and c =) a follow by simple arguments.
b =) c involves many steps.
construct the incomplete preference that must be satis…ed by any
SSCD preference that explains the data
construct a complete SSCD pref (%i )i 2N that extends the latter
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U ) choice and strategic complementarity
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Utility Representation
Theorem
Suppose O obeys ARC and Xi is a closed interval of R (8i 2 N).
Then the preference %i admits a utility representation.
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U ) choice and strategic complementarity
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Out-of-Sample Predictions
Suppose the data set obeys ARC.
What can we say about the equilibrium set for a given y 0 , A0 ?
The question is interesting even if y 0 , A0 = (y t , At ) for some t 2 T !
eq selection might depend on time index t
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U ) choice and strategic complementarity
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Out-of-Sample Predictions
If the data obeys ARC, then there is a non-empty set of consistent
preferences pro…les that explain the data as Nash equilibria.
We show the set of all consistent preferences induces a possible best reply
correspondence PR(a, y 0 , A0 ) = A0
A0 where
PR(a, y 0 , A0 ) = PR1 a
0
0
1 , y1 , A1
, ..., PRn a
0
0
n , yn , An
.
Its …xed points are the set of all possible Nash equilibria ε y 0 , A0 .
(These elements can be easily calculated.)
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U ) choice and strategic complementarity
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Out-of-Sample Predictions
PRi a i , yi0 , A0i = [%i 2Pi BRi a i , yi0 , A0i , %i .
PRi a i , yi0 , A0i = e
ai 2 A0i : O i = Oi [
e
ai , a i , yi0 , A0i
obeys ARC .
We o¤er a third characterization that facilitates the computation of PRi .
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U,revealed
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U ) choice and strategic complementarity
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Out-of-Sample Predictions
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U ) choice and strategic complementarity
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Out-of-Sample Predictions
Theorem
Suppose O = f(at , y t , At )gt 2T obeys ARC and let (y 0 , A0 ) 2 Y
A.
Then, ε y 0 , A0 is a non-empty …nite union of hyper-rectangles in A0 .
Its closure has a largest and a smallest element that increase in y 0 .
We show how to construct ε y 0 , A0 .
The procedure involves a …nite number of steps.
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U ) choice and strategic complementarity
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Cross-Sectional Data
We extend our results to cross-sectional data.
The population involves many groups with di¤erent preferences.
In this case, we observe
nh
O = µt , y t = yit
i 2N
, At = Ati
i 2N
io
t 2T
where µt : At ! [0, 1] is a distribution probability on action pro…les.
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U ) choice and strategic complementarity
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Cross-Sectional Data
Are choice distributions consistent with the hypothesis that each group
behaves as the Nash eq of a spm game?
Group type is a set of choices consistent with Nash eq of spm game.
The population can be decomposed in a distribution on group types
We assume treatments are randomly assigned
Similar to "unconfounded" in treatment e¤ects
Can be relaxed
Check whether there is a distribution of group types that explains the data.
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U ) choice and strategic complementarity
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Stochastic Revealed Complementarity
Let E = f(y t , At )gt 2T be the set of all treatments we observe.
We know that a = a1 , a2 , ..., aT 2
t 2T
At is a SC-rationalizable path i¤
f at , y t , At gt 2T obeys ARC.
A SC-rationalizable path de…nes a group type (pref + eq selection).
Let A be the set of all consistent group types on E = f(y t , At )gt 2T .
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U ) choice and strategic complementarity
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Smoking Decisions among Married Couples
Example
In the smoking model there are 256 types.
Only 64 are consistent!
type 1
type 2
...
R R R A A R A A
(0, 0) (0, 1) (0, 1) (0, 1) consistent
(0, 0) (1, 0) (0, 1) (1, 1) inconsistent
...
...
...
...
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U ) choice and strategic complementarity
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Stochastic Revealed Complementarity
Theorem
A data set O = f(µt , y t , At )gt 2T is stochastically SC-rationalizable i¤
there exists a probability distribution Q on A such that
µt (x ) =
∑
Q (a)1(at = x )
a2A
8t 2 T and 8x 2 At .
We assume the distribution of types is the same across treatments.
This is true if the treatments y t , At are randomly assigned.
We might need to condition on covariates.
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Stochastic Revealed Complementarity
Consistency requires checking whether a system of linear equations
Ax = b
has a positive solution in x.
A is a matrix of 0’s and 1’s that depends on the group types in A
Each column captures the behavior of a given type under all treatments
b builds on the observed distribution of choices
x is a possible distribution of group types
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Out-of-Sample Predictions
The predictions for y 0 , A0 are the choice probabilities µ0 for which
O[ µ0 , y 0 , A0 passes our test of stochastic SC.
Let A
be the set of all group types on f(y t , At )gt 2T [0 .
Predictions are all µ0 for which exists a distribution Q on A
µt (x ) =
∑
such that
Q (a)1(at = x )
a2A
8t 2 T [0 and 8x 2 At .
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U ) choice and strategic complementarity
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Out-of-Sample Predictions
The set of all possible choice probabilities solve a linear system.
They constitute a convex set.
The fraction of type i players choosing ai at y 0 , A0 is an interval.
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U ) choice and strategic complementarity
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Application: Smoking Choices among Married Couples
We study smoking decisions among married couples.
Each married couple is a group whose members decide whether to smoke.
The smoking decision of each person depends on:
the smoking decision of his/her partner
the smoking policy at his/her workplace
Hypothesis is couples play a game of strategic complementarities.
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U ) choice and strategic complementarity
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Application: Smoking Choices among Married Couples
We use the Tobacco Use Supplement of the Current Population Survey.
It is part of the US Census Bureau’s Current Population Survey.
We focus on indoor workers for the period 1992-1993.
We have data on 5,363 married couples across the US.
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U ) choice and strategic complementarity
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Application: Smoking Choices among Married Couples
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Application: Smoking Choices among Married Couples
There are 44 = 256 possible choices across treatments.
64 satisfy ARC.
The data is not consistent with stochastic SC.
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Application: Smoking Choices among Married Couples
At least one violation is given by
µ (N, S j 1, 0) = 9.1% > 8.6% = µ (N, S j 0, 1) .
Any couple type that selects (N, S ) at (1, 0) must select (N, S ) at (0, 1) .
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Application: Smoking Choices among Married Couples
The violation is not severe.
The closest compatible distribution is as follows.
(Compatible distribution that requires minimal changes on choices.)
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Application: Smoking Choices among Married Couples
Kitamura and Stoye (2013)
The Null-Hypothesis is
(H) minx 2R64
(b
+
Ax )0 (b
Ax ) = 0.
The sample counterpart is
JN = N min[x
τ N 1 64 /64 ]2R64
+
b
b
Ax
0
b
b
We get p-value 37% with τ N = 0 and 16% with τ N =
Ax
= 0.
p
ln(N )/N.
Thus, we don’t reject the Null-Hypothesis at the 5% con…dence level.
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Application: Smoking Choices among Married Couples
Is exogeneity assumption credible? What about omitted variables bias?
A natural candiate to control for is education!
Willigness to smoke might di¤er across education levels
Distribution of smoking policies at work di¤ers across education levels
HE Pr(R,R) = 0.79 Pr(R,A) = 0.07 Pr(A,R) = 0.11 Pr(A,A) = 0.03
LE Pr(R,R) = 0.59 Pr(R,A) = 0.13 Pr(A,R) = 0.20 Pr(A,A) = 0.08
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Application: Smoking Choices among Married Couples
High Educated couples — at least some colleague— pass the test directly.
For Low Educated couples we don’t reject the Null-Hypothesis at the 5%.
(The results hold for all tuning parameters we tried.)
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Application: Smoking Choices among Married Couples
What if we assume Pareto optimality instead of Nash?
Unfortunately, Pareto optimality is non-refutable in our application!
Consider the next preferences irrespective of treatment
Husband
Wife
(S, N )
(N, S )
(S, S )
(N, N )
(N, N )
(S, S )
(N, S )
(S, N )
These two preference relations obey strict single-crossing di¤erences.
At each treatment, every strategy pro…le is Pareto optimal.
Thus, they can generate any data set!!!
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Application: Smoking Choices among Married Couples
What do we learn from the application?
The requirements for consistency are stricter than what they seem.
The test can be easily implemented if the action space is not too large.
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Concluding Remarks
We provide a revealed preference test for strategic complementarity
standard time-series data
cross-sectional data
We provide out-of-sample predictions of equilibrium points
We show our results can be easily implemented
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Thanks!!!
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