Nonparametric analysis of monotone choice Natalia Lazzati John Quah Koji Shirai November, 2018 Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 1 / 27
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 games: we focus on games with monotone best-replies variation of set constraints and payoff-relevant parameters tight econometric implementation of our theoretical results Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 2 / 27
Main results What is the empirical content of games with monotone best-replies? We provide a revealed preference test for Nash equilibrium behavior and monotone shape restrictions (RM axiom): we also study Bayesian Nash equilibrium behavior We extend the idea to cross-sectional data. (Manski (2007)) We apply our results to an IO model of entry decisions: method is easy to implement method delivers meaningful restrictions on data Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 3 / 27
Application: IO entry model of airlines Berry (1992), Ciliberto and Tamer (2009), Kline and Tamer (2016) We observe many markets defined as trips between two airports In each market, two airline firms: i ∈ { 1 , 2 } each firm decides to enter it ( y i = E ) or not ( y i = N ) we observe covariates ( x 1 , x 2 ) : Market Presence (P 1 , P 2 ∈ { 0 , 1 } ) and Market Size (S ∈ { 0 , 1 } ) We test a specific IO model of entry decisions Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 4 / 27
Standard parametric specification α � 1 x 1 + δ 1 1 ( y 2 = E ) + ε 1 if y 1 = E Π 1 ( y 1 , y 2 , x 1 , ε 1 ) = 0 if y 1 = N α � 2 x 2 + δ 2 1 ( y 1 = E ) + ε 2 if y 2 = E Π 2 ( y 1 , y 2 , x 2 , ε 2 ) = 0 if y 2 = N Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 5 / 27
Cross-sectional data Econometrician observes distribution of choices for different ( x 1 , x 2 ) . Firm 2 N E Firm 1 N P ( N,N | x 1 , x 2 ) P ( N,E | x 1 , x 2 ) P ( E,N | x 1 , x 2 ) P ( E,E | x 1 , x 2 ) E Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 6 / 27
Standard analysis Estimate α � 1 , δ 1 and α � 2 , δ 2 Some common assumptions: firms play Nash equilibrium distribution of ( ε 1 , ε 2 ) belongs to a known family ( ε 1 , ε 2 ) is independent of ( x 1 , x 2 ) interaction effects, δ 1 and δ 2 , are negative Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 7 / 27
Our approach We offer a joint test of eq behavior and signed effect restrictions. We can find payoffs Π 1 ( y 1 , y 2 , x 1 , ε 1 ) and Π 2 ( y 1 , y 2 , x 2 , ε 2 ) such that have single-crossing property in ( y 1 ; ( − y 2 , x 1 )) and ( y 2 ; ( − y 1 , x 2 )) � � � � − y �� j , x �� − y � j , x � For all > i i � � > Π i � � = � � > Π i � � E, y � j , x � N, y � j , x � E, y �� j , x �� N, y �� j , x �� ⇒ Π i Π i i , ε i i , ε i i , ε i i , ε i observed distribution at each covariate arises from a distribution of pure-strategy Nash eq (We also offer bound estimates for the distribution of firms’ payoffs.) Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 8 / 27
Advantages of our approach Nonparametric No assumption on eq selection mechanism Very general approach to model heterogeneity no assumption on the joint distribution of ( ε 1 , ε 2 ) no assumption on group formation We do assume ( ε 1 , ε 2 ) is independent of ( x 1 , x 2 ) Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 9 / 27
How does the idea work? Suppose we have the following data (for some fixed x 1 ) x 2 = ( 0 , 0 ) x 2 = ( 0 , 1 ) x 2 = ( 1 , 0 ) Firm 2 Firm 2 Firm 2 N E N E N E Firm 1 N 3/12 3/12 Firm 1 N 1/12 5/12 Firm 1 N 2/12 4/12 E 4/12 2/12 E 3/12 3/12 E 2/12 4/12 Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 10 / 27
How does the idea work? We test joint hypothesis of Nash eq behavior and signed effect restrictions. Let a (behavioral) type be a sequence of joint choices y 1 , y 2 across covariates x 1 , x 2 . We say a (behavioral) type is consistent if it can be generated by eq behavior with payoffs that satisfy the signed restrictions . As possible joint choices and covariate values are finite, we can enumerate all consistent types. The data is consistent with our model if we can decompose the population into a distribution of consistent types that can explain the observations. Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 11 / 27
How does the idea work? (A bit more on consistent types...) A realization of ε 1 , ε 2 leads to Π 1 ( y 1 , y 2 , x 1 , ε 1 ) and Π 2 ( y 1 , y 2 , x 2 , ε 2 ) . Suppose these payoffs satisfy the signed restrictions. Combined with an eq selection rule, these payoffs induce a sequence of Nash eq choices across different covariates x 1 , x 2 . This sequence of joint choices across covariates corresponds to one consistent type. Varying ε 1 , ε 2 and the eq selection we generate all consistent types. Notice that many different ε 1 , ε 2 could lead to the same consistent type. Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 12 / 27
How does the idea work? Data can be rationalized by the following consistent types. x 2 = ( 0 , 0 ) x 2 = ( 0 , 1 ) x 2 = ( 1 , 0 ) Type Weight Action profiles Action profiles Action profiles N,N N,E E,N E,E N,N N,E E,N E,E N,N N,E E,N E,E • • • 1 • • • 2 • • • 3 • • • 4 • • • 5 • • • 6 • • • 7 × ⊗ ⊗ ⊗ Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 13 / 27
How does the idea work? Data can be rationalized as follows. x 2 = ( 0 , 0 ) x 2 = ( 0 , 1 ) x 2 = ( 1 , 0 ) Type Weight Action profiles Action profiles Action profiles N,N N,E E,N E,E N,N N,E E,N E,E N,N N,E E,N E,E 1 1/12 1/12 1/12 1/12 2 2/12 2/12 2/12 2/12 3 2/12 2/12 2/12 2/12 4 1/12 1/12 1/12 1/12 5 1/12 1/12 1/12 1/12 6 2/12 2/12 2/12 2/12 7 3/12 3/12 3/12 3/12 Sum 1 3/12 3/12 4/12 2/12 1/12 5/12 3/12 3/12 2/12 4/12 2/12 4/12 Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 14 / 27
Can we explain the data with a linear model? Notice that Π 2 ( y 1 , E , x 21 , x 21 , ε 2 ) = α 21 x 21 + α 22 x 22 + δ 2 1 ( y 1 = E ) + ε 2 with α 21 > 0, α 22 > 0, and δ 2 < 0 . Then ∆Π 2 > ∆Π 2 ⇐ ⇒ α 21 > α 22 . ∆ x 21 ∆ x 22 Whether > or < in the left side, does not depend on realizations of ε 2 ! Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 15 / 27
Can we explain the data with a linear model? In the data, we have that (I avoid x 1 as it is fixed) P ( E , E | ( 1 , 0 )) − P ( E , E | ( 0 , 0 )) > P ( E , E | ( 0 , 1 )) − P ( E , E | ( 0 , 0 )) = ⇒ α 22 < α 21 P ( N , N | ( 0 , 0 )) − P ( N , N | ( 1 , 0 )) < P ( N , N | ( 0 , 0 )) − P ( N , N | ( 0 , 1 )) = ⇒ α 22 > α 21 This is not possible! Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 16 / 27
In Summary... Revealed Monotonicity (RM) axiom: a type is a sequence of joint choices across all covariates in the data not every type is consistent with our model! we offer an axiom that checks consistency of types Test with cross-sectional data: we need to find a distribution on consistent types that explains data Estimation with cross-sectional data: we can get bounds on different subsets of consistent types Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 17 / 27
Distribution of entry choices in the data Data are from Kline and Tamer (2016). Covariates = ( 0 , 0 , 0 ) Covariates = ( 0 , 1 , 0 ) N,N N,E E,N E,E N,N N,E E,N E,E 30 . 37 68 . 21 0 . 55 0 . 87 19 78 . 51 0 . 26 2 . 23 Covariates = ( 1 , 0 , 0 ) Covariates = ( 1 , 1 , 0 ) N,N N,E E,N E,E N,N N,E E,N E,E 19 . 38 36 . 71 25 . 33 18 . 58 12 . 15 54 . 22 4 . 99 28 . 64 Covariates = ( 0 , 0 , 1 ) Covariates = ( 0 , 1 , 1 ) N,N N,E E,N E,E N,N N,E E,N E,E 15 . 88 82 . 28 0 . 12 1 . 73 7 . 80 88 . 93 0 3 . 27 Covariates = ( 1 , 0 , 1 ) Covariates = ( 1 , 1 , 1 ) N,N N,E E,N E,E N,N N,E E,N E,E 10 . 64 32 . 64 30 . 58 26 . 14 5 . 53 50 . 07 2 . 14 42 . 26 Data looks very consistent with our hypothesis! Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 18 / 27
Testing hypothesis We can find payoffs Π 1 ( y 1 , y 2 , x 1 , ε 1 ) and Π 2 ( y 1 , y 2 , x 2 , ε 2 ) such that have single-crossing property in ( y 1 ; ( − y 2 , x 1 )) and ( y 2 ; ( − y 1 , x 2 )) can explain the data as pure-strategy NE Natalia Lazzati, John Quah, Koji Shirai () Nonparametric analysis of monotone choice 11/18 19 / 27
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