Identification and estimation of incomplete information games with multiple equilibria

Accepted Manuscript Identification and estimation of incomplete information games with multiple equilibria Ruli Xiao

PII: DOI: Reference:

S0304-4076(17)30245-2 https://doi.org/10.1016/j.jeconom.2017.12.005 ECONOM 4460

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Journal of Econometrics

Received date : 23 March 2015 Revised date : 5 December 2017 Accepted date : 5 December 2017 Please cite this article as: Xiao R., Identification and estimation of incomplete information games with multiple equilibria. Journal of Econometrics (2018), https://doi.org/10.1016/j.jeconom.2017.12.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Identification and Estimation of Incomplete Information Games with Multiple Equilibria∗ Ruli Xiao† Indiana University December 20, 2017

Abstract In games, the multiplicity of equilibria poses a challenge for identification and estimation. The existing literature typically abstracts from this multiplicity by assuming that the data are generated from a single equilibrium. Instead of imposing such restrictions, this paper provides sufficient conditions to non-parametrically identify payoff primitives in finite action games with incomplete information, while allowing for multiple equilibria. I then propose a two-step estimator and illustrate its finite-sample performances via Monte Carlo simulations. Furthermore, I study the strategic interaction among radio stations when choosing different time slots to air commercials. I indeed find evidence to support the existence of multiple equilibria. JEL Classification: C14 ,C57 Keywords: Multiple equilibria, discrete games, measurement error models, non-parametric identification, semi-parametric estimation ∗

I am deeply indebted to Yingyao Hu for his generous support and guidance. I also benefited greatly from the comments of Yuya Sasaki and Richard Spady. I thank the co-editor, two anonymous referees, Andrew Sweeting, Victor Aguirregabiria, Matt Shum, Juan Carlos Escanciano, and seminar participants at JHU and IU for their helpful comments. The usual disclaimer applies. † Department of Economics, Indiana University, 100 S Woodlawn, Bloomington, IN 47405. Email: [email protected]

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1

Introduction

Games generally admit multiple equilibria, which greatly complicates the identification and estimation of the underlying game structures. This multiplicity in games does not necessarily preclude estimation, but ignoring it may result in mis-specification and lead to inconsistent estimates (Jovanovic (1989) and Tamer (2003)). Indeed, the two-step estimators pioneered by Hotz and Miller (1993) enable consistent estimation of the payoff primitives in the second step if the conditional choice probabilities (CCPs) can be consistently estimated in the first step. The existing literature obtains consistent CCPs by imposing the assumption that the data are generated from a single equilibrium. This assumption, however, has rarely been supported by empirical studies. Meanwhile, in empirical studies, researchers have suggested that the multiplicity of equilibria is important in explaining various aspects of economic data.1 Thus, it is important to understand under what conditions we can uniquely identify and consistently estimate the payoff primitives without imposing the assumption of a single equilibrium. Allowing for multiple equilibria, this paper provides sufficient conditions to identify the underlying structure of the game in two steps. The first step involves identifying the equilibrium-specific components for equilibria being selected with positive probabilities by nature (referred to active equilibria). These components include the number of active equilibria, the equilibrium selection probabilities, and the individual players’ equilibrium strategies associated with all active equilibria. The second step is to identify payoff primitives with exclusion restrictions, as in Bajari et al. (2010c). Based on these identification results, I propose a two-step estimator and illustrate its finite-sample performances via Monte Carlo simulations. Furthermore, I study the strategic interaction among radio stations when choosing different time slots to air commercials. I indeed find evidence to support the existence of multiple equilibria. The identification in the first step adopts results developed in the measurement error literature by considering the equilibrium index as a latent variable. The identification relies on the usual conditional independence assumption, with which players’ actions are independent in a single equilibrium. When the data are generated by multiple equilibria, the outcome distribution represents a mixture of the underlying equilibrium outcome distributions, resembling the structure in the measurement error literature. Using individual players’ actions as measurements of the latent equilibrium index, the correlation among actions identifies the equilibrium CCPs via matrix decomposition. 1

Such as the different clustering of commercial timings in different cities in Sweeting (2009).

2

The proposed identification method differs from that of the measurement error literature in several aspects. First, in the measurement error literature, the latent variable is often assumed to be exogeneous, and so the cardinality is typically inferred by the economics context of the variable or by assumption. In contrast, the number of equilibria in games is endogenous by nature and must be identified from the data. To address this issue, this paper first shows that the number of equilibria can be bounded from below. This lower bound enables me to construct a test with a null hypothesis that the data are generated by a single equilibrium. This null hypothesis should be rejected if the lower bound on the number of equilibria is greater than one. This test on the existence of multiple equilibria is important in its own right because the conventional two-step estimators are only valid if the test is not rejected. Moreover, I identify the number of equilibria as the rank of the joint distribution matrix with a full rank condition. More importantly, this paper relaxes the assumptions of distinctive eigenvalues and monotonicity imposed in the measurement error literature for unique identification. These assumptions are very restrictive in games because the equilibrium is endogenous. Without imposing these assumptions, this paper does not identify the ordering of equilibria. This paper instead is agnostic on the exact ordering of the equilibrium since its index does not convey an economic meaning. Consequently, I identify the equilibrium CCPs up to a permutation of the equilibrium ordering. I also show unique identification of the payoff primitives without knowing the exact ordering of the equilibrium. Note that payoff primitives are identified via plugging the equilibrium CCPs into the equilibrium conditions. The key is to make sure that all players’ CCPs plugged in come from the same equilibrium, which is guaranteed via exploring the identification structure. Specifically, I determine the equilibrium index for one player first. Then, for the rest of the players, the equilibrium index is structured to be the same as that of the first player. This paper examines the strategic interactions of radio stations who choose air commercials between two time slots. Since listeners generally dislike commercials and tend to switch stations during commercials, advertising firms prefer stations to air commercials at the same time to avoid switching behaviors by listeners. However, because a station’s listenership is averaged over both commercial and non-commercial time slots, radio stations do not necessarily prefer commercial clustering. The interaction among radio stations in the same broadcasting domain is captured by a game with incomplete information, which fits the structure of the proposed methodology. In summary, I find that data of the overall markets admit two equilibria. Furthermore, to control for market heterogeneity, I divide the markets into large and small, based on the market populations. I find that multiple equilibria exist in

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small market, but not in the large.2 I also divide markets into ‘drivetime’ and ‘non-drivetime’ based on the hour of the timing decisions to make and find the market of ‘drivetime’ admit two equilibria. Besides multiple equilibria, unobserved heterogeneity in market characteristics could be another source of mis-specification for the identification and estimation of games. Both multiplicity and unobserved heterogeneity generate the same mixture structures, so they are observationally equivalent. Distinguishing the two has important economic and policy implications. Moreover, it is important to consider identification that incorporates both factors, such as in Aguirregabiria and Mira (2015) (hereafter referred as AM). The identification problem considered in this paper is a subset of AM. To construct a more general framework, AM imposes stronger assumptions, so that the order of the latent variable aggregating both factors can be matched across observed market characteristics. The AM assumption, however, does not necessarily hold in the setting considered here. I also propose a test with a null hypothesis of no unobserved heterogeneity using the fact that payoff primitives are over-identified with the presence of multiple equilibria. This paper contributes to several strands of literature, both theoretically and empirically. First, the proposed method complements the theoretical literature by acknowledging and addressing the possible multiple equilibria in games.3 Several approaches are implemented to tackle the multiplicity of equilibria in games. In games of incomplete information, for example, the existing literature intend to ignore the multiplicity of equilibria by assuming that the data are generated by a single equilibrium (see Seim (2006)). Some studies focus on testing the existence of multiple equilibria (see Hahn et al. (2015) and Otsu et al. (2016)). Some studies use the existence of multiple equilibria to identify the sign of the interaction term in the payoff functions(De Paula and Tang (2012)). In contrast, the present paper provides sufficient conditions to non-parametrically identify the underlying game structures allowing for multiple equilibria. Different information structures in games with multiple equilibria require different approaches. In games of complete information, for example, some studies impose restrictions on payoff functions so that certain quantities do not change when different equilibria are employed.4 Payoff primitives then can be consistently estimated by focusing on these quantities. Some studies turn to bound identification and estimation instead of point estimation (see Ciliberto and Tamer (2009)).5 Other studies show that the existence of an exclusion restriction with a large support can identify the 2

This result is consistent with the findings in Sweeting (2009). See De Paula (2012) for a survey of the recent literature on the econometric analysis of games with multiplicity. 4 See Berry (1992), Bresnahan and Reiss (1990), and Bresnahan and Reiss (1991). 5 Bounds estimation has also been used by Pakes et al. (2006), and Andrews et al. (2004). Berry and Tamer (2006) and 3

Berry and Reiss (2007) survey the econometric analysis of discrete games.

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payoff primitives uniquely(see, Tamer (2003) and Bajari et al. (2010a)). Using field data, this study also provides empirical evidence regarding multiple equilibria. First, the empirical findings support the existence of multiple equilibria. This finding suggests that researchers should be more cautious before making the assumption that the data are generated by a single equilibrium. This assumption could potentially lead to inconsistent estimations of payoff primitives. Secondly, the empirical findings show that the probabilities with which each equilibrium being selected differ across markets. These results shed lights on our understanding of the equilibrium selection mechanism, which is important for counterfactual analysis. The remainder of the paper is organized as follows. Section 2 outlines the static game framework and provides the non-parametric identification results. Section 3 describes the estimation procedure. Section 4 presents some Monte Carlo evidence for the methodology, and Section 5 presents an empirical application that focuses on radio stations’ decisions regarding the airing of commercials. Section 6 concludes. The Appendix contains the proofs, the figures, and the tables.

2

Non-parametric Identification of Static Discrete Games

2.1

Basic setup of static games

Consider a static simultaneous move game that involves N players. Player i, i ∈ N ≡ {1, ..., N } chooses an action ai out of a finite set A = {0, 1, ..., K}. Before making decisions, the individual player i obtains a (K + 1) × 1 vector of action-specific payoff shocks, i.e., i = {i (ai = 0), ..., i (ai = K)}, whose density is denoted as f (i ). These profit shocks are private information and are observable only to the player herself, while the density distributions are assumed to be common knowledge among all the players. Let a−i denote the actions of player i’s rivals, i.e., a−i = {a1 , ..., ai−1 , ai+1 , ..., aN }. s ∈ S captures all publically observable characteristics that affect players’ payoffs.6

The payoff for player i from choosing action ai is assumed to be additively separable as follows: Ui (ai , a−i , s, i ) = πi (ai , a−i , s) + i (ai ) Unlike in a standard discrete choice model, player i’s payoff depends not only on her own actions but also her rivals’. In particular, rivals’ actions enter player i’s payoff function directly. 6

Similar setups are studied in Aradillas-Lopez (2010).

5

Assumption 1. (Conditional Independence) The payoff shocks are identically and independently distributed (i.i.d) across actions and players, and the density distribution f (i ) has full support in the Euclidian space and is positive everywhere. The existing literature on estimation and inference in static games usually imposes the assumption of conditional independence of private information, see, e.g., Seim (2006), Aradillas-Lopez (2010), Sweeting (2009), De Paula and Tang (2012), Bajari et al. (2010b), and Ellickson and Misra (2008). Literature on dynamic games with incomplete information also imposes the same assumption. Given this independence assumption, player i’s payoff shocks reveal no information on those of her rivals. Let δi (i , s) : RK+1 × S → A be player i’s decision rule which prescribes an action for player i given her own private information, and δ = {δi (s, i )}i be a strategy profile. Using a strategy profile δ, I define a set of conditional choice probabilities (CCPs) p(s) = {pi (ai |s)}i such that Z I(δi (i , s) = ai )f (i )di , pi (ai |s) ≡ Pr(δi (i , s) = ai ) = i

where I(· · · ) is the indicator function. In what follows I define the payoff components for player i while assuming that other firms behave according to their respective strategies in δ. Let Πi (ai , s; δ) be player i’s expected payoff from choosing action ai and can be represented as Πi (ai , s; δ) =

X a−i

πi (ai , a−i , s)p−i (a−i |s).

Then, the Bayesian Nash Equilibrium can be defined as the follows. Definition 1. (BNE) For any s, the Bayesian Nash Equilibrium (BNE) is a set of strategies δ ∗ such that for any firm i and for any i ∈ RK+1 , δi∗ (i , s) = argmax{Πi (ai , s; δ ∗ ) + i (ai )}. ai ∈A

Note that for any set of strategies δ, in equilibrium or not, the payoff components depend on players’ strategies only through the CCPs p, associated with a strategy profile δ. Indeed, following Milgrom and Weber (1985), the equilibrium defined in the strategy space is equivalent to that defined in the probability space. Consequently, for any market characteristics s, the equilibrium conditions can be characterized in terms of CCPs as follows, pi (ai = k, s) = Pr {Πi (ai = k, s; p) + i (ai = k) > Πi (ai = j, s; p) + i (ai = j), ∀j 6= k} .

6

Following Hotz and Miller (1993), there exists a one-to-one mapping between the CCPs and the difference of individual players’ expected payoff. For any s, let p(a|s, π) collect CCPs of all players and π represent the payoff functions. The equilibrium conditions then lead to the following mapping: p(a|s, π) = Γ(p(a|s, π); s, π),

(1)

where Γ depends on the distribution of the payoff shocks. Assumption 1 guarantees that an equilibrium exists by Brouwer’s fixed point theorem. The equilibrium, however, may not be unique, because the equilibrium conditions consist of a system of nonlinear equations. The set of BNE associated with the payoff functions π = {πi (ai , a−i , s)}i , for any market characteristics s, can be characterized as Ω(s) = {p(a|s, π) : p(a|s, π) = Γ(p(a|s, π); s, π))}. I suppress the payoff primitives in Ω(s) when there is no ambiguity for easy exposition. Assume that the set of equilibria Ω(s) is discrete and finite, I then label the equilibrium as e∗ = e1 , e2 , e3 , etc, and represent the equilibrium set as ω(s), i.e., ω(s) = {e1 , e2 , e3 , ..., }, which provides the same information as the equilibrium set Ω(s). When there are multiple equilibria, modeling how the equilibrium is chosen is challenging even in theory, and lies outside the scope of this paper. Instead, this paper abstracts away from how equilibria are selected and assumes that there exists some exogeneous “mechanism” to determine the equilibrium. Assumption 2. (The Equilibrium Selection) Nature selects the equilibrium in each game. That is, the probability of a given equilibrium being selected is determined by a random mechanism. Furthermore, the model assumes there are no off-equilibrium outcomes. This random mechanism is represented by the equilibrium selection probability distribution pe (s) = {p(e∗ |s), e∗ ∈ ω(s)}. This assumption does not impose any restrictions on the rule of equilibrium selection. It does not rule out any types of “selection”, such as the equilibrium selection rule in Berry (1992) stating that the most profitable airlines have the opportunity to make entry decisions in advance of less profitable airlines.7 Note that this assumption allows nature to select certain equilibrium with a zero probability. That is, p(e∗ |s) = 0 for some e∗ ∈ ω(s). I refer to an equilibrium which nature selects with a positive probability as an active equilibrium. The set of active equilibria and the corresponding cardinality are 7

Note that Berry (1992) studies games with complete information.

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represented as ω a (s) = {e∗ , st : p(e∗ |s) > 0 and e∗ ∈ ω(s)} and Qs = ||ω a (s)||, respectively. As a result, I allow ω a (s) to be different from the overall equilibrium set ω(s), as is the cardinality of the two sets. I summarize the data generating process as follows {πi (ai , a−i , s), s ∈ S, Ω(s), ω(s), pe }.

2.2

Non-parametric Identification

Suppose one observes the actions of all players in cross-sectional markets m where m = 1, ...., M with characteristics sm ∈ S; that is, {am1 , ..., amN , sm }M m=1 . For ease of notation, I suppress the market characteristics s for now, and will reintroduce it when move to identify payoff primitives. In what follows I provide sufficient conditions to identify the number of equilibria first and next the equilibrium CCPs. The joint distribution of all players’ actions can be represented as follows, regardless of the number of equilibria in the data Pr(a1 , ..., aN ) =

X

e∗ ∈ω

=

Pr(a1 , ..., aN |e∗ ) Pr(e∗ )

X

e∗ ∈ω a

=

Pr(a1 , ..., aN |e∗ ) Pr(e∗ )

X Y

e∗ ∈ω a

i

Pr(ai |e∗ ) Pr(e∗ )

(2)

The first equality holds due to the law of total probability, the second equality holds because only active equilibria are relevant, and the last equality holds due to the conditional independence (assumption 1) among private information. The conditional independence assumption enables a simple test with the null hypothesis that the data are generated by a single equilibrium. If this null hypothesis is true, the marginal and joint distributions of players’ actions are the respective equilibrium counterparts. Then the above test is equivalent to the test with the null that players’ actions are independent. In particular, the data are generated by a single equilibrium is equivalent to the following condition, Pr(a1 , ..., aN ) = Pr(a1 )... Pr(aN ). (see De Paula and Tang (2012) for a formal test). This is because, if the conditional independence assumption is satisfied, the presence of multiple equilibria is the only reason that players’ actions are correlated. The correlation of players’ action provides more than just a test on the presence of multiple equilibria. Furthermore, this paper uses the correlation of players’ actions to identify the number of active equilibria, the equilibrium selection mechanism, and the equilibrium CCPs. 8

Note that identification requires at least three players. It would be interesting and worthwhile to consider identification for games with only two players. However, identification of a non-classical measurement error requires more than two measurements for the latent variable, which refers to the underlying equilibrium in the present setting. Thus, point identification is only feasible for games with at least three players. For games with only two players, point identification is infeasible but set identification instead can be explored. Moreover, lemma 1 below provides an inference concerning the lower bound of the number of active equilibria. In games with only two players, researchers can still conduct this lower bound inference, results of which can be used to test with a null hypothesis that the data are generated by a single equilibrium. If we fail to reject this null hypothesis, the conventional two-step estimator generates a consistent estimate of the payoff primitives. For identification, I first divide the N players into three groups. The first two groups are of equal size of n players, and the last and third group can have one or two players depending on whether N is odd or even. That is N = 2n + 1 or N = 2n + 2. Furthermore, I treat each group as a representative player and construct a “group action” variable for each group, denoted as ci , i = 1, 2, 3. For example, if N = 5, I can construct three groups of players with the first two groups consisting of 2 players and the last group consisting 1 player. Then I construct the group action c1 as follows: c1 = 1 if a1 = 0 and a2 = 0; c1 = 2 if a1 = 0 and a2 = 1; c1 = 3 if a1 = 1 and a2 = 0; c1 = 4 if a1 = 1 and a2 = 1. The group action variable for a general number of player n can be accomplished by the following function: c1 = (a1 + 1) + a2 (K + 1) + ... + an (K + 1)n−1 ∈ C ≡ {1, ..., (K + 1)n },

c2 = (an+1 + 1) + an+2 (K + 1) + ... + a2n (K + 1)n−1 ∈ C, and when N is odd, c3 = {a2n+1 }, or c3 = (a2n+1 + 1) + a2n+2 (K + 1) ∈ {1, ..., (K + 1)2 } otherwise.

I first provide conditions for identifying the number of equilibria using the joint distribution of the first two groups of players, which can be represented as: Pr(c1 , c2 ) =

X

e∗ ∈ω a

Pr(c1 |e∗ ) Pr(c2 |e∗ ) Pr(e∗ ).

(3)

Provided that the players choose their actions out of a finite set, the above connection between the observed joint distributions and the equilibrium CCPs involves (K + 1)2n equations and 2 × (K + 1)n × Q + Q unknowns. To make full use of all of available information, I introduce the following matrix notation and representation: Fc1 ,c2 = Ac1 |e∗ De∗ ATc2 |e∗ ,

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(4)

where Fc1 ,c2

≡ [Pr (c1 = k, c2 = j)]k,j ,

Aci |e∗

≡ [Pr (ci = k|e∗ = eq )]k,q , h i ≡ diag Pr(e∗ = e1 ) ... Pr(e∗ = eQ ) .

De∗

These matrices collect all possible values of the corresponding probabilities. In particular, matrix Fc1 ,c2 consists of joint distributions of actions of group 1 and 2, De∗ is a diagonal matrix with the probability of each active equilibrium being selected as the diagonal element, and matrix Aci |e∗ collects equilibrium CCPs for group i for all active equilibria, where i = 1, 2. The dimensions of these three matrices defined above are of (K + 1)n × (K + 1)n , (K + 1)n × Q, and Q × Q, respectively. An example of these matrices for games with three players (N = 3, c1 = {a1 }, c2 = {a2 } and c3 = {a3 }), is  binary actions, and only two active equilibria  Pr(c1 = 1, c2 = 1) Pr(c1 = 1, c2 = 2) , Fc1 ,c2 =  Pr(c1 = 2, c2 = 1) Pr(c1 = 2, c2 = 2)   Pr(ci = 1|e∗ = e1 ) Pr(ci = 1|e∗ = e2 ) . Aci |e∗ =  Pr(ci = 2|e∗ = e1 ) Pr(ci = 2|e∗ = e2 )



De∗ = 

Pr(e∗ = e1 )

0

0

Pr(e∗ = e2 )



,

Note that the number of active equilibria Q is unknown. As I will show, this number is identifiable

from the data under further assumptions. This identification result contrasts with the existing literature, where a single equilibrium is often assumed. I summarize the identification of the number of active equilibria in the following lemma. Lemma 1. The rank of the observed matrix Fc1 ,c2 serves as the lower bound for the number of active equilibria, i.e., Q ≥ Rank(Fc1 ,c2 ). Furthermore, the number of active equilibria is identified as Q = Rank(Fc1 ,c2 ), when the following conditions are satisfied: (1) (K + 1)n > Q; (2) both matrices Ac1 |e∗ and Ac2 |e∗ are full rank. 

Proof See Appendix B.

Note that the group action variable ci serves as measurements for the latent equilibrium index variable. Thus, inferring the number of equilibria requires enough variation of the measurements, indicating that either the number of equilibria n or the number of actions K is sufficiently large, as provided in condition (1). This condition seems to be restrictive, as it is likely to fail in a game with

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only 2 or 3 players and binary actions. In a 2 × 2 game, lemma 1 can only guarantee an inference as to whether the equilibrium is unique or multiple. Condition (2) is a full rank condition requiring that no active equilibrium CCPs are linearly dependent. This condition essentially implies that no active equilibrium is redundant. Provided that the number of active equilibria is identified, I show how to identify equilibrium CCPs of all players for all equilibria being active. In the measurement error literature, the latent variable shares the same support with its measurements, so the full rank condition indicates that the relevant matrices are invertible. In order to use the results in measurement error literature, I collapse the matrix Fc1 ,c2 by summing up columns and rows to create a new Q × Q full rank matrix Fc˜1 ,˜c2 . This cardinality collapsing is equivalent to generating a new action variable denoted as c˜i , which has exactly Q possible options. In particular, the collapsing can be obtained through a surjective function gi , i.e., c˜i = gi (ci ) : C → {1, ..., Q}. The following lemma provides one way to achieve a valid collapsing through summing up rows and columns of the original matrix. Lemma 2. For a (K + 1)n × (K + 1)n singular matrix Fc1 ,c2 with rank(Fc1 ,c2 ) = Q < (K + 1)n , there exists at least one way to construct a new Q × Q matrix, denoted as Fc˜1 ,˜c2 , by summing up rows and columns of Fc1 ,c2 so that it is of full rank, i.e., rank(Fc1 ,c2 ) = rank(Fc˜1 ,˜c2 ) = Q. 

Proof See Appendix B.

Lemma 2 provides a possible way to collapse the matrix, which is important for the purpose of identification. Note that there are more than one ways to collapse the matrix. The number of possible collapsing increases exponentially with the size of the support of the player’s actions, among which the number of valid collapsing depends on the properties of the original matrix. As long as there exists one valid collapsing, the identification argument can be carried through. Moreover, more than one collapsing leads to over-identification. Following the identification intuition in the literature of mis-classification errors, I state the identification in the following lemma. Lemma 3. (First Step Identification) With Assumptions 1- 2 and the conditions in lemma 1 satisfied, the equilibrium selection probability Pr(e∗ |s) and all players’ equilibrium CCPs {Pr(ai |e∗ , s), e∗ ∈

ω a , ∀i} are non-parametrically identified for every s. The identification is up to a permutation of the equilibrium labeling. 11



Proof See Appendix B.

Note that I do not impose the assumptions of distinctive eigenvalues or monotonicity, without which the equilibrium CCPs are not uniquely identified. Those assumptions are restrictive since the equilibrium is endogenous and unnecessary since its labeling does not convey economic meanings in the game setting. Furthermore, I will show later that this lack of unique identification does not prevent me from uniquely identifying payoff primitives. This is because equilibrium CCPs associated with different equilibria satisfy the same equilibrium conditions. So far I have shown how to identify the equilibrium CCPs conditional on the market characteristic s, denoted as Pr(ai |s, e∗ = e1 )....... Pr(ai |s, e∗ = eQs ), ∀i, e∗ ∈ ω a . Each set of equilibrium CCPs

{Pr(ai |s, e∗ )}i should satisfy the equilibrium conditions in equation (1). Now I state some regularity conditions in the existing literature for non-parametrically identifying the payoff primitives. As with the analysis in discrete choice models, it is impossible to uniquely identify payoffs associated

with all possible options, but only the difference of payoffs from the status quo. Normalization is necessary and is provided in the following assumption: Assumption 3. (Normalization) For all i and all a−i and s, πi (ai = 0, a−i , s) = 0. This assumption sets the mean utility from a particular choice as equal to zero, which is similar to an outside good in the discrete choice model. Moreover, it is impossible to non-parametrically identify payoff primitives and error distributions without extra assumptions (see, for example, Matzkin (1992)). Following the literature, I assume that the distribution of the payoff shocks is known. More specifically, to illustrate the identification I assume that the error terms follow a type I extreme-value distribution, which is unnecessary for nonparametric identification of the payoff primitives. With the normalization condition and the extreme value distribution of shocks, the equilibrium conditions become log pi (ai = k|s, e∗ ) − log pi (ai = 0|s, e∗ ) =

X a−i

πi (ai = k, a−i , s)p−i (a−i |s, e∗ )

∀k, s, e∗ , i.

(5)

The identification of payoff functions πi (ai = k, a−i , s) requires exclusion restrictions (see Bajari et al. (2010c) and Bajari et al. (2011)). If there are covariates that shift the payoff of one player, but not that of other players, all payoffs can be identified non-parametrically. As a result, I state the exclusion restriction assumption below. 12

Assumption 4. (Exclusion Restrictions) For each player i, the state variable can be partitioned into two parts denoted as si , s−i , so that only si enters player i0 s payoff; that is, πi (ai = k, a−i , s) ≡ πi (ai = k, a−i , si ). An example of exclusion restrictions is a covariate that shifts the profitability of one firm but that can be excluded from the profits of all other firms. Empirical studies commonly use firm-specific cost shifters as exclusion restrictions. For example, Jia (2008) and Holmes (2011) demonstrate that distance from firm headquarters or distribution centers is a cost shifter for big-box retailers such as Walmart. The existence of an exclusion restriction leads to the following equilibrium conditions log pi (ai = k|s, e∗ ) − log pi (ai = 0|s, e∗ ) =

X a−i

πi (ai = k, a−i , s)p−i (a−i |s, e∗ ) ∀i, s, e∗ , &k = 1, ..., K. (6)

The variation of s−i expands the total number of equations without adding the number of unknowns, and thus helps identify payoff functions πi (ai = k, a−i , si ) non-parametrically. Multiple equilibria actually can serve as an exclusion restriction with the equilibrium CCPs being identified. Specifically, it increases the number of equations on the order of the number of equilibria but not the payoff functions since the equilibrium CCPs differ across equilibria but different equilibria map with the same set of payoff functions. The presence of multiple equilibria alone, however, may not provide enough variation to identify the payoff non-parametrically. Specifically, in fixing s, there are K × Qs equations, which is magnified by the number of equilibria Qs , while there are K × (K + 1)N −1 unknowns for player i. Non-parametrically identifying the profit functions requires that the number of equations be greater than the number of unknowns (Qs ≥ (K + 1)N −1 ). On the other hand, the number

of equilibria needs to be relatively smaller (Qs < (K +1)n ) for its identification, as stated in lemma 1. It is unlikely that both conditions satisfied at the same time. Consequently, the multiplicity of equilibrium as an exclusion restriction alone cannot non-parametrically identify the payoff functions. The presence of multiple equilibria, however, does lessen the need for variation in extra exclusion restrictions.8 Theorem 1. (Identification of Payoff Primitives) Suppose Assumptions 1- 4 and the conditions given in lemma 1 are hold, the payoff functions are non-parametrically identified for static games with incomplete information. After identifying the payoff functions, I can investigate the equilibrium CCPs associated with the 8

The importance of multiple equilibria is shown when variation from available exclusion restrictions is not enough, or

there is no exclusion restriction at all (see Sweeting (2009)).

13

equilibrium being selected with a probability of zero can be obtained from solving all equilibria from the equilibrium conditions. The total number of equilibria then can be identified.

2.3

Multiple Equilibria versus Unobserved Heterogeneity

To identify the underlying game structure, I assume away any unobserved market-level characteristics, and only allow for multiple equilibria and action specific payoff shocks. With payoff shocks being conditionally independent, the correlation between players’ actions across games is driven by the presence of multiple equilibria. A discrete and finite unobserved market factor, however, also results in correlation of players’ actions. Both unobserved heterogeneity and multiple equilibria lead to similar finite mixture structures. Moreover, multiple equilibria can co-exist with unobserved heterogeneity. It is important but challenging to consider identification allowing both multiple equilibria and unobserved heterogeneity, even though the presence of both also resembles a similar mixture model. The difficulty lies in the label swapping issue. Unlike identification with only multiple equilibria, now the labeling of latent types does have economic meanings. The combination of endogenous equilibria and exogenous unobserved heterogeneity makes it more difficult to find monotonicity conditions or use the identification structure. More restrictive assumptions are required as shown in Aguirregabiria and Mira (2015). To construct a more general framework, AM imposes stronger assumptions, so that the order of the latent variable, which aggregates both factors, can be matched across observed market characteristics. The AM assumption, however, does not necessarily hold in the setting considered here. Even though this paper abstracts about unobserved heterogeneity, the identification result leads to a test on the existence of unobserved heterogeneity. In particular, the null hypothesis is with no unobserved heterogeneity, and the alternative hypothesis is with unobserved heterogeneity. When there is no unobserved heterogeneity, the presence of multiple equilibria results in over-identification of the payoff primitives, based on which the test can be conducted. Note that every set of equilibrium CCPs enables one estimate of the payoff primitives. The null hypothesis is equivalent to the fact that the payoff primitive associated with different equilibria are the same. If the null hypothesis is rejected, unobserved heterogeneity does exist.

14

3

Semi-parametric Estimation of Static Game

I propose a semi-parametric estimator following the identification procedure. In particular, the number of equilibria and the equilibrium CCPs are estimated non-parametrically, while the payoff functions are estimated parametrically. The asymptotical properties of these estimators are provided in the Appendix.

3.1

Non-parametric Estimation of Equilibrium Relevant Components

Note that the identification is constructive, so I propose an estimator following the identification procedure step by step. Given a sample {am1 , ..., amN , sm }M m = 1, I divide all players into three groups. In particular, N = 2n + 1 if N is odd, or N = 2n + 2 otherwise. The first two groups has n players, and the third group has 1 or 2. If s is discrete, I estimate the joint distributions of c1 and c2 using a simple frequency estimator, Pˆ r(c1 , c2 |s) =

1 M

P

m m m I(c1 = c1 , c2 P 1 m m I(s M

= c2 , sm = s) = s)

where I(·) is the indicator function. I estimate the joint distribution matrix Fc1 ,c2 by stacking the estimates of joint distributions of group 1 and 2’s actions as follows i h Fˆc1 ,c2 |s = Pˆ r (c1 = k, c2 = j|s)

k,j

.

I then estimate the number of active equilibria by estimating the rank of the joint distribution matrix Fc1 ,c2 , following the method developed in Robin and Smith (2000) using a sequence of tests.9 Specifically, the hypotheses are constructed as: H0r : Rank(Fc1 ,c2 ) = r against the alternatives H1r : Rank(Fc1 ,c2 ) > r with r = 1, 2, ..., (K + 1)n − 1. This test is based on a characteristic root (CR) statistics of the matrix quadratic form, denoted as CRT r . The sequence of tests starts with a null hypothesis of the rank of matrix Fc1 ,c2 |s being 1. If the null hypothesis is rejected, r is augmented by one and the test is repeated. When I fail to reject the null that the rank equals r for the first time, the rank of Fc1 ,c2 is estimated as r. To guarantee weakly consistency of the rank estimator, I adjust the asymptotic size αr of the CR test at each stage r to depend on the sample size M with a certain rate. The revised critical region at stage r is given by {CRT r ≥ cr1−αrM } with the critical value cr1−αrM along with an asymptotic size 9

See also Kleibergen and Paap (2006) and Camba-Mendez and Kapetanios (2009) for a review.

15

αrM under the null H0r : Rank(Fc1 ,c2 ) = r, r = 1, 2, ..., (K + 1)n − 1. The estimator for the number of equilibria Q then is defined as rˆ ≡ minr∈{1,...,(K+1)n −1} {r : CRT r ≥ ci1−αiM , i = 1, ..., r − 1, CRT r < cr1−αrM }. Note that the test is feasible only when the true rank is strictly less than (K + 1)n . If we fail to reject the null till (K + 1)n − 1, the matrix is full rank, i.e., Rank(Fc1 ,c2 ) = (K + 1)n . In this case, it is likely that the number of equilibria is greater than (K + 1)n . Thus, we cannot identify the number of equilibria exactly but its lower bound. With the number of equilibria being estimated, next I proceed to collapse joint distribution matrices. The only criteria for collapsing is that the resulting new matrix is of full rank. Since the dimensions are finite, there are a finite number of ways to collapse the matrix. A natural approach is to check all matrices resulting from collapsing by testing for full rank and screen out all the valid ones. One valid collapsing will lead to one estimate of the equilibrium CCPs. Using estimates from all valid collapsing could potentially improve estimation efficiency. However, screening out all valid collapsing is not trivial given that this number increases exponentially with the support of the action. For simplicity, this paper instead suggests selecting the collapsing matrix with the smallest condition number. The higher the condition number is, the more likely that the corresponding matrix is nonsingular. The condition number of a matrix can be computed easily and directly. Thus, even with potentially many matrices, the selection procedure is not costly. Given one valid collapsing, the estimation of the equilibrium CCPs follows exactly the identification procedure. First, eigen-decomposisition provides the equilibrium CCPs for the reduced-dimension actions of group 1 Ac˜1 |e∗ , based on the following equation: −1 Fc˜1 ,˜c2 ,c3 =k Fc˜−1 c2 = Ac˜1 |e∗ Dc3 =k|e∗ Ac˜1 |e∗ . 1 ,˜

(7)

The equilibrium CCPs for group 1 are estimated as Aˆc˜1 |e∗

= ψ(Fˆc˜1 ,˜c2 ,c3 =k Fˆc˜−1 c2 ), 1 ,˜

(8)

where ψ(·) is the eigenvector function. Normalization is obtained by the column sum equals to 1. The equilibrium selection probability can be estimated as ˆ e∗ = Aˆ−1 ∗ Fˆc˜ . D 1 c˜1 |e 16

(9)

The equilibrium CCPs for group 2 can be estimated as iT h ˆ e∗ )−1 Fˆc˜ ,˜c . Aˆc˜2 |e∗ = (Aˆc˜1 |e∗ D 1 2

(10)

Equilibrium CCPs for players in group 2 and 3, i.e., k > Q, can be estimated as h i−1 ˆ e∗ AˆTak |e∗ = Aˆc˜1 |e∗ D Fˆc˜1 ,ak .

(11)

Similarly, equilibrium CCPs for players in group 1, i.e., k <= Q, can be estimated as h i−1 ˆ e∗ AˆT ∗ . Aˆak |e∗ = Fˆak ,˜c2 D c˜2 |e

(12)

Note that the estimation involves only eigen-decomposition once. Since the index of the equilibrium can only be identified up to relabeling from each decomposition. Avoiding multiple decompositions circumvents the risk of resulting in different orderings of the equilibrium from each decomposition. As a result, the estimates of matrix Aˆa |e∗ have the same labeling of the equilibrium along the column. k

This is important since only the equilibrium CCPs associated with the same equilibrium for all players satisfy the equilibrium conditions. Specifically, we should plug in Pr(ak |e∗ = el ) , k = 1, ..., N into the equilibrium conditions. On the other hand, equilibrium CCPs associated with different equilibrium such as Pr(ak |e∗ = el ) and Pr(ak |e∗ = ek ), l 6= k, do not satisfy the equilibrium conditions at the same time.

3.2

Parametric Estimation of the Payoff Function

With the equilibrium CCPs being estimated, denoted as p˜(a|s, θ, e∗ ) = {˜ p(a1 |s, θ, e∗ ), ..., p˜(an |s, θ, e∗ )}, payoff functions can be estimated non-parametrically with exclusion restrictions. This paper instead parameterizes the payoff function and estimates the structural parameters using a least square estimator. I denote the parameterized payoff functions as πi (ai , a−i , s) = πi (ai , a−i , s; θ), and assume the market characteristics to be discrete with a cardinality of d, i.e., s ∈ {S1 , ...., Sd }. Following the one-to-one mapping between the CCPs and the expected payoff of individual players, the equilibrium conditions yield a mapping p(a|s, θ, e∗ ) = Γ(p(a|s, e∗ ); s, θ)

∀s & ∀e∗ .

(13)

The least square estimators are meant to solve for the structural parameters which minimize the distance between the components in the left- and right-hand sides of the above equations. Note that this equation 17

has to be satisfied for every s and every equilibrium. With the number of equations being greater than the number of parameters, a weight is assigned to each individual equation for minimization. Denote p˜M and Γ(ˆ p; θ) the vector collecting all pˆ(a|s, e∗ ) and all Γ(ˆ p(a|s, e∗ ); θ), respectively. Let WM be a P P symmetric positive definite matrix with a dimension of ((K + 1)n · s Qs ) × ((K + 1)n · s Qs ), which

may depend on the observations. A least square estimator associated with a weight matrix WM is a ˆ M ) to the problem solution θ(W ˆ M ) = argminθ θ(W

[ˆ p − Γ(ˆ p; θ)]0 WM [ˆ p − Γ(ˆ p; θ)] .

(14)

ˆ M ) brings the constraint closest to zero in the metric Thus, the asymptotic least squares estimator θ(W associated with the scalar product defined by WM . A simple example of the weight matrix WM is the identity matrix, which treats all constraints equally. Another example of the weighting matrix involves weighting each market type differently, according to the number of observations of each type.

4

Monte Carlo Simulation

This section is to examine the finite sample performance of the proposed estimator by simulating a simple game of five players with binary actions. In markets with characteristics s, homogeneous players decide whether to enter the market simultaneously. Suppose the payoffs from entry (1) or not (0) as follows π(ai = 1, a−i ; s) = αs + δ P

π(ai = 0, a−i ; s) = δ

P

j6=i I(aj

n−1 I(a j = 0) j6=i n−1

= 1)

+ i1 ,

+ i0 ,

where i1 and i0 are private shocks, and i.i.d. with the type one extreme value distribution. Given this specific payoff function, the number of players does not affect the equilibrium strategy. That is, only the fraction of players entering the market matters. I present all the symmetric equilibria for games with α = 0.04, δ = 2.5, and s = 1, 2, 3, 4 in figure 1. Markets s = 1, 2, 3, admit three equilibria, among which the middle one is unstable. Market s = 4 admits a unique equilibrium. I simulate players’ actions as follows. For markets with multiple equilibria, i.e., s = 1, 2, 3, I first simulate the equilibrium according to the equilibrium selection mechanism of each of the two stable equilibria being selected with a probability of 0.5. Given the realization of the equilibrium, I simulate each players’ actions based on the corresponding equilibrium CCPs. The Monte Carlo experiment 18

consists of a repetition of 500 with sample sizes of 300, 500, 800, 1000, and 1200. I then estimate the number of equilibria, the equilibrium CCPs, and the equilibrium selection probabilities for each market type separately. I present the estimation results with a sample size of 1200 in table 1. I then present the estimates of the model primitives using a least square estimator in table 2. Based on the results, the proposed estimators perform well in a sample of a moderate size. To compare the proposed method and the existing literature, I estimate the payoff primitives allowing for multiple equilibria and assuming that the data are generated by a single equilibrium, and present the results in table 3. The estimates by assuming a single equilibrium are problematic with the presence of multiple equilibria. First of all, these estimates are biased. Moreover, ignoring multiple equilibria results in an estimate with a sign opposite to that of the true parameter, which directs the inference to the wrong path. For example, firms are better off entering markets with a larger s in this simple framework. The unique equilibrium assumption, however, yields a counterintuitive estimate with a negative market effect. The unique equilibrium assumption is acceptable if the interaction effect is the sole focus. Still, the estimates are further from the true parameter compared to those estimates that consider multiplicity, but the sign is correct, and the bias is within a reasonable range. Again, this is just one simple example, but it raises the concern that making an ad hoc equilibrium assumption may introduce estimation errors. The size of the error depends on the whole structure, especially the equilibrium selection mechanism. For example, if players employ one typical equilibrium most of the time, the assumption that the data are generated by a single equilibrium could be a good approximation to the reality. However, without tackling the multiplicity issue, there is no prior information upon which to judge whether this is the case. Consequently, one should be cautious about making such assumptions.

5

Empirical Application

This section applies the proposed estimator to timing decisions for broadcasting commercial by radio stations with contemporary music formats (Contemporary Hit Radio (CHR)/Top 40, Country, Rock etc.). This section provides evidence on the existence of multiple equilibria.

19

5.1

Institution Background and Data

It is a common presumption that many listeners dislike commercials. They seek to avoid them by switching to other stations or opting out to other options, such as tapes or CDs. Advertisers prefer that different stations air their commercials at the same time to reduce commercial avoidance. However, stations may have different incentives, because the values of commercials are not based on the listenership of a particular commercial. Average commercial audiences are not measured. Instead, Arbitron, the radio rating company, estimates a station’s average audience by averaging over both commercial and noncommercial programming. As a result, the average audience might increase if stations play commercials at different times to keep listeners tuned in to the radios stations instead of seeking outside options. In reality, stations do tend to play commercials at the same time. Specifically, figure 2 presents the average proportion of stations playing commercials in each minute during two different hours of the day, and these commercial timing distributions are far from uniform. One possible explanation is that coordination increases stations’ commercial values. Another possible reason, however, is that common factors make different time slots for each hour particularly desirable for commercials. Knowledge of the radio station industry indicates that common factors do affect timing decisions. For example, the way that Arbitron computes listenership strongly affects stations’ commercial break decisions. Common factors, however, are not a perfect explanation for the clustering phenomenon. Suppose that common factors are indeed the underlying force behind the clustering phenomenon. Note that Arbitron uses the same methodology to compute listenerships. Consequently, one can expect that commercials are clustered in every market, and also at the same times across markets. This, however, is not the case. Figure 3 displays information for stations in two markets playing commercials during one particular hour. The distributions of commercial breaks in both markets have three peaks, which occur at noticeably different times. A similar situation exists in the aggregate distribution. Thus, the clustering of commercials at different times in different markets is not solely driven by any unobserved common factors. Another possible explanation for this phenomenon is the presence of multiple equilibria. Suppose in static games, stations strategically choose times to air their commercials. Stations coordinate to air their commercials at the same time to avoid listener switching. Multiple equilibria are present, and different equilibria are employed across markets. This rationalizes both the clustering in one market and

20

the different times of the clustering across markets. Allowing for multiple equilibria in a static game with commercial timing decision, this paper applies the methodology presented above to investigate whether there are multiple equilibria and how many of them exist in the data. The data used in this paper are also used in Sweeting (2009), which constructs the data on the timing of commercials by music radio stations in U.S. metro markets using hourly airplay logs collected by Medabase 24/7. The data are extracted from airplay logs that stations play on a minute-by-minute basis. I present the summary statistics in table 4. In summary, there are 144 markets in total. The number of stations in each market varies from 3 to 15 with a mean of 5.7. Each station has 236 observations, including 59 days, and each day there are two different hours, with one being ‘drivetime’ (hour = 16:00PM) and one ‘non-drivetime’ (hour=12:00PM). Please refer to Sweeting (2009) for a detailed description of the data.

5.2

Model Setup and Estimation

While actual commercial timing is continuous, a discrete feature exists in the schedule of commercials on music stations, because timing decisions involve planning the order of songs and commercial breaks. For example, the programmer considers the commercial breaks in the gaps between the songs. As a result, stations are modeled so as to play their commercials in finite time blocks simultaneously. Stations can play several sets of commercials at many different times during an hour. An estimation of games with this feature is beyond the scope of this paper. Instead, the choice of commercial breaks by stations is modeled as a simple binary choice game. Specifically, I use information about whether commercials are being played at two particular time interval in each hour, :48-:52 and :53-:57, denoted as option 0 and option 1, respectively. Stations not choosing either action are excluded. Following Sweeting (2009), I assume that stations are symmetric, and station i’s payoff for placing a commercial in time block t ∈ {0, 1} is defined as follows: P

j6=i I(aj = 1) + i1 , π(ai = 1, a−i ; s) = α(s) + δ(s) n−1 P j6=i I(aj = 0) π(ai = 0, a−i ; s) = δ(s) + i0 , n−1

where α(s) allows different average profit for stations in market s when they play their commercials in timing 1, δ(s) captures the coordination incentives in market characteristics s, and ’s represent the idiosyncratic private profit shocks, which the Stations receive before they make their timing decisions. 21

The ’s represent the fact that a station tends to play commercials at different times every day. This introduces variation due to the length of other programming, such as songs or travel news, can vary and be unpredictable. A station would not want to annoy its listeners by cutting short other programs to play commercials at precise times. I also assume it to be independent across actions, players and markets. Furthermore, it follows with a type one extreme value distribution. I consider two ways to control market heterogeneity. First, I divide markets into two types, ‘large’ and ‘small’, according to a rank by population. Specifically, the markets ranked in the top 30 according to population are treated as one type (‘large’), while the rest of the markets are another type (‘small’). This is because the larger the market, the more stations it has. Second, I divide the market into ‘drivetime’ and ‘non-drivetime’ based on the hour that the timing decision to be made. Commercials are planned for every hour, and different hours are treated differently. There are large differences between drivetime and non-drivetime. In-car listeners are more likely to switch stations to avoid commercials than those at home or at work, and there are more in-car listeners during drivetime. The strength of the incentive to coordinate depends on how much listeners dislike commercials and how easy it is for them to switch stations. Listeners might respond to commercials differently during driving time and non-driving time, because during driving time, they may easily switch stations with just one click. For example, in-car listeners switch stations 29 times per hour on average to avoid commercials (McDowell and Dick (2003)). Note that without multiple equilibria, the model is under-identified non-parametrically because there is one equation with two unknowns from the equilibrium condition when fixing s. Payoff-relevant exclusion restrictions do not apply here because stations are homogeneous. Consequently, we can only identify and estimate a further parameterizing payoff functions use variation across s. For example, let α(s) ≡ αs and δ(s) ≡ δs. In contrast, for market s with multiple equilibria, the coordination effect δ(s) is identified using multiple equilibria as an exclusion restriction, and so α(s) is identified. To estimate α(s) and δ(s), I need to first estimate the number of equilibria and the equilibrium CCPs for each market s. To estimate the number of equilibria, I divide all the players in three groups and construct the joint distribution matrix for the first two group representative players. However, there is a lot variation in the number of players across markets in the data. Note that I can pool markets together for estimation if they share the same market characteristics s. This is because only the proportion of radio stations’ timing enters the payoff function by the payoff function specification. I pool markets with more than four players together to do estimation instead of only using markets with the most number of

22

radio stations. This provides a sufficient number of observations. I then generate representative group players with two players as one group. That is, I generate c1 = {a1 , a2 } and c2 = {a3 , a4 }. These new representative action variables ci ∈ {1, 2, 3, 4}, i = 1, 2, which is with a cardinality of 4. I estimate the number of equilibria by conducting a sequence of tests with a null hypothesis that the rank of the joint distribution matrix Fc1 ,c2 has a rank of r, with r starting being 1. I find that there are either a unique equilibrium or two equilibria for market s. With the number of equilibria being estimated for each market s, I estimate the equilibrium CCPs for market s with two equilibria through a matrix decomposition. Note that players are homogenous here, so I only need to estimate the equilibrium CCPs for one player. Since the number of equilibria is two, and the players are homogeneous, the full rank condition is satisfied for any two players. As a result, I use data from three players to construct matrices in equation (7). Matrix decomposition generates an eigenvector matrix with the i column representing the equilibrium CCP associated with equilibrium i, i = 1, 2. I then estimate the payoff primitives using the moment conditions, with which there are two unknowns and two equations. I estimate δ(s) and α(s) by solving them from the equilibrium conditions. Note that the way I estimate the payoff primitives is very simple but consistent. Efficiency might be improved if I use the equilibrium CCPs of all players and use the least square estimator with a weighting matrix. For markets admitting a single equilibrium, I estimate the equilibrium CCPs directly from the data using the frequency estimator, but I donot estimate the payoff primitives δ(s) and α(s) since they cannot be identified.

5.3

Empirical Results

This subsection presents the empirical results in table 5, which provides evidence on the existence of multiple equilibria. The estimation results are consistent with those of Sweeting (2009). First of all, the number of equilibria is estimated to be two for the full market. Among the two equilibria being active, the equilibrium selection probabilities are a probability of 0.027 versus 0.973, indicating that the market favors one equilibrium a lot more than the other. With the presence of two equilibria, I can identify and estimate α and δ. Furthermore, I fail to reject the null that the difference in payoffs of the two choices α is zero, which indicates that both timing intervals are equally desirable for the stations. This insignificance is consistent with the fact that both time intervals are equally distant from the quarter-hours, which are known to be unattractive times for commercials. I control heterogeneity by dividing markets into ‘large’ and ‘small’ based on the city population and 23

present the results in columns 3-4 in table 5. From the estimation results, the data of ‘large’ market are generated by a single equilibrium, while the data of ‘small’ markets are generated by two equilibria. Similar to the estimates in full sample, players are inclined to employ one equilibrium more often than the other, with a probability of 0.819 versus 0.181. I also fail to reject the null hypothesis that α is zero, which is consistent with the finding in full markets. I control heterogeneity by dividing data of different hours (hour = 16:00PM vs 12:00PM) as separate markets, and provide the estimation results in columns 5-6 in table 5. A strong incentive to coordinate is the key that multiple equilibria exist, and the incentive should be greater during drivetime when there are more in-car listeners, which is supported by the estimation results. Markets of non-drivetime admit a single equilibrium, while markets of drivetime admit two equilibria. The two equilibria are similar to each other since the probability of airing the commercial at option 0 is with a probability of 0.522 and 0.478. The equilibrium selection probabilities are quite closed to each other, i.e., 0.436 vs 0.564. Moreover, I again fail to reject the null that both time intervals are equally desirable for airing commercials (α = 0).

6

Conclusion

In the setting of finite action games with incomplete information, this paper develops a methodology to non-parametrically identify the underlying game structure allowing for multiple equilibria. As an application of the proposed methods, I study the behavior of radio stations that strategically choose a time to air commercials and provide evidence of the existence of multiple equilibria. Dynamic games are not free of multiple equilibria either. A future direction may be to investigate identification in dynamic games as in Xiao (2015). With Markov perfect equilibria, each market employs the same equilibrium over time, and this provides us with a Markov process in a single market. If a time series continues long enough, consistent estimation can be obtained by carrying out the estimation in one individual market. However, pooling information across markets is sometimes necessary. Thus, a methodology dealing with multiple equilibria and unobserved market factors is important and necessary for empirical analysis.

24

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Appendix The appendix includes five sections, with tables and figures. Section A provides detailed comparison of the present paper with AM (2015). Section B presents the proof of the lemmas. Section C presents the asymptotical properties of the estimators. Section D discusses the identification when equilibrium coordination fails, and Section E provides the graphs and tables.

A

Comparison with Aguirregabiria and Mira (2015)

The paper is closely related to Aguirregabiria and Mira (2015) (hereafter AM), which focuses on distinguishing multiple equilibria from payoff-relevant heterogeneity in static games. The AM paper considers a more general model that nests the one considered in this paper. Specifically, AM provides identification results for static games with incomplete information allowing for both multiple equilibria and unobserved heterogeneity, while this paper only allows for multiple equilibria in the same setting. Also, both papers rely on the conditional independence assumption for identification. However, AM does not include this paper as a special case. First of all, AM imposes an extra assumption of independence of the unobservables and the instruments for identifying payoff primitives. This independence assumption enables further identification via circumventing the label swapping issue in the identification of the first step. Label swapping problem is more server when both multiple equilibria and unobserved heterogeneity are involved. In contrast, label swapping does not cause any problems for identification of payoff primitives in the present paper because equilibrium labeling has no any economic meaning, and thus does not affect the payoff of individual players. Consequently, this paper does not impose extra assumptions to identify payoff primitives, even though the identification of equilibrium CCPs is up to a permutation. Secondly, the extra assumption imposed by AM for permutation may not hold in some cases with multiple equilibria and unobserved heterogeneity. More importantly, those conditions cannot be tested, and so one may run into a problem of mis-specification. In contrast, even though the current paper cannot provide identification for a framework to consider both latent factors, it provides a test of whether or not unobserved heterogeneity is presented. Lastly, even though the identification proposed in the current paper and AM take advantage of the existing results from the measurement error/ finite mixture models, this application is not trivial,

29

because the cardinality of the measurement is greater than the latent variable in the game setting but equal in the mixture framework. The current paper discusses how to partition the action space and provides a step-by-step identification and estimation procedure. It also discusses the impact of equilibrium coordination failure and provides empirical evidence of the existence of multiple equilibria.

B

Proofs

Proof of Lemma 1 Based on the conditional independence assumption, the joint distribution of actions from two groups can be expressed as: Pr(c1 , c2 ) =

X e∗

Pr(c1 |e∗ ) Pr(c2 |e∗ ) Pr(e∗ )

Rewritten into matrix form: Fc1 ,c2 = Ac1 |e∗ DATc2 |e∗ By rank inequality Rank(Ac1 |e∗ DATc2 |e∗ ) ≤ min{Rank(Aa1 |e∗ ), Rank(DATa2 |e∗ )} = min{Rank(Aa1 |e∗ ), Rank(Aa2 |e∗ )}

(D is full rank)

≤ Q Thus, without any extra assumptions, I identify the lower bound of the number of equilibria, i.e., Rank(Fc1 ,c2 ) ≤ Q Furthermore, Rank(Ac1 |e∗ DATc2 |e∗ ) ≥ Rank(Aa1 |e∗ ) + Rank(DATa2 |e∗ ) − Q (by Sylvester’s rank inequality) = Q + Q − Q = Q (Aa1 |e∗ and Aa1 |e∗ are full rank) 

I conclude that Q = Rank(Fa1 ,a2 ).

Proof of Lemma 2 Suppose matrix Fc1 ,c2 has dimensions of (K + 1)n × (K + 1)n , and rank of Q.   Denote Fc1 ,c2 = l1 , ..., lQ , lQ+1 , ..., l(K+1)n where lk is the kth column vector. Since rank(Fc1 ,c2 ) = Q, among all (K + 1)n column vectors, there are at most Q of them that are linearly independent. Assume

30

l1 , ...., lQ , w.l.o.g, are linearly independent. Thus, for the rest columns lk , k > Q, there exists a series of λk1 , ....λkQ such that lk = λk1 l1 + λk2 l2 + .... + λkQ lQ . Consequently, I sum up all columns k = Q + 1, ...(K + 1)n and denote the new column as l−Q . That is, (K+1)n

l−Q ≡

X

(K+1)n

lk =

k=Q+1

X

(K+1)n

λk1 l1

+

k=Q+1

X

(K+1)n

λk2 l2

+ .... +

k=Q+1

≡ λ1 l1 + ... + λQ lQ ,

X

λkQ lQ

k=Q+1

where there must exist a λi > 0 because l1 , ...., lQ are positive by the nature of probability. Without loss of generality, assume λQ > 0. Next, I prove that collapsing all columns k > Q by adding all of them to column Q generates a (K + 1)n × Q matrix Fc1 ,˜c2 , which is of full column rank. That is Fc1 ,˜c2 = [l1 , ..., lQ + l−Q ] with the new Q column vectors l1 , l2 , ..., lQ + l−Q linearly independent. To prove the linear independence, I need to prove that for any η1 , ..., ηQ satisfying η1 l1 + η2 l2 + ..... + ηQ (lQ + l−Q ) = 0, I have η1 = η2 = ... = ηQ = 0. Plug l−Q = λ1 l1 + λ2 l2 + .... + λQ lQ back into above equation, leading to: (η1 + ηQ λ1 )a1 + (η2 + ηQ λ2 )a2 + ... + ηQ (1 + λQ )aQ = 0. Given that a1 , ...., aQ are linearly independent, all the linear coefficients of the above linear combination should equal zero. Thus, ηk +ηQ λk = 0,∀k = 1, ..., Q. Given that λQ > 0 by construction, ηQ (1+λQ ) = 0 implies that ηQ = 0. Then ηk = 0 for k = 1, ...Q. Thus, l1 , l2 , ..., lQ + l−Q are linearly independent, leading to a full column rank matrix Fc1 ,˜c2 . This is equivalent to generating a new action variable c˜2 by function g2 such that

  c if c < Q; 2 2 c˜2 = g2 (c2 ) =  Q if c2 ≥ Q.

Similarly, I can apply the same treatment to the row vectors by first determining the vector base, and then adding the rest rows to one row of the base, resulting in an invertible matrix with dimensions 

Fc˜1 ,˜c2 of Q × Q.

31

Proof of Lemma 3 First I show that group 1’s CCPs {Pr(˜ c1 |e∗ ), e∗ ∈ ω a } can be identified through eigenvalue-eigenvector decomposition. The joint distribution of three players’ actions satisfies the following equation: X

Pr(c1 , c2 , c3 ) =

e∗ ∈ω a

Pr(c1 |e∗ ) Pr(c2 |e∗ ) Pr(c3 |e∗ )P r(e∗ )

Fix c3 = k, matrix representation leads to Fc˜1 ,˜c2

= Ac˜1 |e∗ De∗ ATc˜2 |e∗

Fc˜1 ,˜c2 ,c3 =k = Ac˜1 |e∗ Dc3 =k|e∗ De∗ ATc˜2 |e∗

(B.1) (B.2)

Note that as matrices Ac˜1 |e∗ and Ac˜2 |e∗ both have full rank, post-multiplying Fc˜−1 c2 on both sides of 1 ,˜ equation (B.1) leads to the following main equation −1 Fc˜1 ,˜c2 ,c3 =k Fc˜−1 c2 = Ac˜1 |e∗ Dc3 =k|e∗ Ac˜1 |e∗ 1 ,˜

(B.3)

Equation (B.3) indicates that the observed matrix Fc˜1 ,˜c2 ,c3 =k , Fc˜−1 c2 and the unobserved matrix Ac˜1 |e∗ , 1 ,˜

Dc3 =k|e∗ , and A−1 c˜1 |e∗ are similar. The right-hand side of the equation above represents an eigenvalue-

eigenvector decomposition of the matrix on the left-hand side, with Dc3 =k|e∗ being the eigenvalue matrix

and Ac˜1 |e∗ being the eigenvector matrix (see Hu (2008)). The eigenvectors Pr(˜ c1 |e∗ ) are identified up to scale and permutation. The scale can be pinned down by the fact that column sum equals to 1 since each column of the eigenvector matrix Ac˜1 |e∗ is a probability distribution. The permutation issue cannot be solved without some monotonicity conditions, as in the measurement error literature. The identification of payoff primitive is invariant to the permutation of equilibrium orderings. Even though equilibrium labeling does not convey any economic meaning, the equilibrium has to be matched for different players’ CCPs for identifying the payoff primitives. To avoid mis-match, I rely on group 1’s equilibrium CCPs, which are identified from the eigen-decomposition, to recover the equilibrium selection probability and equilibrium CCPs of other players. Specifically, I fix the equilibrium labeling for group 1, and keep other players’ equilibrium ordering the same. Next I show that the equilibrium selection Pr(e∗ ) and the equilibrium CCPs for group 2 {Pr(˜ c2 |e∗ ), e∗ ∈

ω a } can be identified with the same equilibrium order as {Pr(˜ c1 |e∗ ), e∗ ∈ ω a }. The marginal distribution P of group 1’s action can be represented as P r(˜ c1 ) = e∗ ∈ωa P r(˜ c1 |e∗ )P r(e∗ ) with a matrix representation Fc˜1 = Ac˜1 |e∗ De∗ . Thus, the equilibrium selection mechanism can be identified through De∗ = A−1 c˜1 |e∗ Fc˜1 . 32

(B.4)

Furthermore, from equation B.1, I can identify the equilibrium CCPs for group 2 as T  Ac˜2 |e∗ = (Ac˜1 |e∗ De∗ )−1 Fc˜1 ,˜c2

(B.5)

Now I provide identification of the equilibrium CCPs for each individual player, which again has the same equilibrium ordering as group 1. For player k in group 2 and 3, i.e., k > Q, the joint distribution of c˜1 and ak is as follows Fc˜1 ,ak

= Ac˜1 |e∗ De∗ ATak |e∗ .

Since De∗ and Ac˜1 |e∗ are identified and invertible, Aak |e∗ can be identified as  −1 Fc˜1 ,ak . ATak |e∗ = Ac˜1 |e∗ De∗

(B.6)

Similarly, equilibrium CCPs for players in group 1, i.e., k <= Q, can be identified as h i−1 ATak |e∗ = Fak ,˜c2 De∗ ATc˜2 |e∗ .

(B.7)

An Example of Identification with Duplicate Eigenvalues Note that the identification is achieved via one eigenvalue-eigenvector decomposition and algebra manipulations. The existing literature imposes the assumption that eigenvalues are distinct because the identification is achieved via multiple decomposition and the distinctiveness of eigenvalues are used to distinguish the level of the latent variable. In our context, the labeling of equilibrium does not matter because the labeling of equilibrium does not convey any economic meanings. To illustrate the intuition, assume there are three players and ci is a binary variable, and the game −1 admits two equilibria. The identification equation is expressed as Fc˜1 ,˜c2 ,c3 =k Fc˜−1 c2 = Ac˜1 |e∗ Dc3 =k|e∗ Ac˜1 |e∗ 1 ,˜

where k = 0, 1. Decomposing the matrix on the left-hand side of the equation, which can be estimated from data, leads to the following representation −1 Fc˜1 ,˜c2 ,c3 =k Fc˜−1 c2 = [b1 b2 ]diag(h1 h2 )[b1 b2 ] 1 ,˜

where h1 , h2 are the two eigenvalues, and b1 , b2 are the corresponding eigenvectors. Duplicate eigenvalues indicate that h1 = h2 . Note that Fc˜1 ,˜c2 ,c3 =k Fc˜−1 c2 can be diagonalized by construction, which is not 1 ,˜ affected by the fact that its eigenvalues are duplicates. Since the labeling of equilibria does not imply any economic meaning, one can label arbitrarily one eigenvector as associated with equilibrium e1 (e∗ = e1 ), i.e., Pr(˜ c1 |e∗ = e1 ) = b1 and the other 33

eigenvector as equilibrium e2 , i.e., Pr(˜ c1 |e∗ = e2 ) = b2 . Also, Pr(c3 = k|e∗ = e1 ) = h1 = h2 = Pr(c3 =

k|e∗ = e2 ). As a result, the strategy profile associated with equilibrium e1 and e2 are {b1 , h1 } and {b2 , h2 }, respectively. I am going to show that whether I name {b1 , h1 } as strategy profile associated with equilibrium e1 or e2 is not going to affect the identification results of payoff primitives. Note that the two sets of equilibrium profiles ({b1 , h1 } and {b2 , h2 }) identified satisfy the same equilibrium conditions. Plugging both equilibrium profiles into the equilibrium conditions leads to: log pi (ai = 1|s, e∗ = e1 ) − log pi (ai = 0|s, e∗ = e1 ) = log pi (ai = 1|s, e∗ = e2 ) − log pi (ai = 0|s, e∗ = e2 ) =

X a−i

X a−i

πi (ai = 1, a−i , s)p−i (a−i |s, e∗ = e1 ) πi (ai = 1, a−i , s)p−i (a−i |s, e∗ = e2 )

The equilibrium index does not enter the payoff functions since multiple equilibria indicate multiple solutions from the equilibrium conditions. That is, the equilibrium index e∗ is not an argument in the payoff function πi (ai = 1, a−i , s). Payoffs can be identified whether I name strategy profile {b1 , h1 } as equilibrium profile associated with equilibrium e1 or e2 . To sum up, the equilibrium CCPs are not uniquely identified without the condition of distinctive eigenvalues. Payoff primitives, however, can be 

uniquely identified.

C

Asymptotic Property

This subsection summarizes the properties of the proposed estimators. I show the consistency of estimators of the number of equilibria Qs , the equilibrium CCPs, Pr(ai |e∗ , s) and the payoff primitives θ. I also prove that the estimators of equilibrium CCPs and payoff primitives are asymptotically normal. Note that the number of equilibria is estimated via a sequence of test. The test is constructed with the null hypothesis as H0r : Rank(Fc1 ,c2 ) = r against the alternatives H1r : Rank(Fc1 ,c2 ) > r with r = 1, 2, ..., (K + 1)n − 1. The consistency of the estimator of the number of equilibria relies on how the significance levels for the sequential tests are selected. To be specific, if the significance level αM is set to be a constant, the number of active equilibria estimated through this testing procedure will not be consistent, because the testing procedure rejects the true with a probability αM even as the sample size goes to infinity. To obtain consistent estimates for the number of active equilibria, the significance level αMs needs set to go to zero as the sample size Ms goes to infinity but that is not faster than a ˆ s as follows: given rate. I summarize the condition for consistency of the rank estimates for Q 34

ˆ s is weakely consistent, that is, Q ˆ s →p Lemma C.1. The estimator of the number of active equilibria Q Qs , if the asymptotic size for the sequential tests satisfies the following conditions. (i), αMs → 0; (ii), limMs →∞

lnαMs Ms

= 0. 

Proof See Robin and Smith (2000).

Note that it is challenging to derive an asymptotic property for the estimator of the rank. However, since the estimation of the number of equilibria serves as a model selection procedure, the consistency and asymptotic property of the equilibrium CCPs will not be affected given that the model selection is consistent (P¨ otscher (1991)). Thus, in what follows I discuss the consistency and asymptotic property of the estimators of equilibrium CCPs and payoff primitives taking the number of active equilibria as known. Lemma C.2. For any realization of an active equilibrium e∗ , the CCP estimators pˆ(ak |e∗ , s), ∀k, ∀s, are consistent and asymptotically normal. Proof Note that the joint distribution of group 1-3 is estimated via a a simple frequency estimator, i.e., Pˆ r(c1 , c2 |s) = Pˆ r(c1 , c2 , c3 |s) =

1 M

P

m m m m I(c1 = c1 , c2 = c2 , s = s) P 1 m m I(s = s) M P 1 m m m m m I(c1 = c1 , c2 = c2 , c3 = c3 , s M P 1 m m I(s = s) M

= s)

I define ρ0s = [vec(Fc˜1 ,˜c2 ,c3 =k ), vec(Fc˜1 ,˜c2 )] and ρˆs = [vec(Fˆc˜1 ,˜c2 ,c3 =k ), vec(Fˆc˜1 ,˜c2 )], where vec(F ) denotes the vector formed by collecting the entries of the matrix F in a single vector. Note that the vector ρ contains the same information as the matrices Fc˜1 ,˜c2 ,c3 =k and Fc˜1 ,˜c2 . Since every element in ρˆs is a −1/2

), and ρˆs is asymptotically normal: simple frequency estimator, I have ρˆs = ρ0s + Op (Ms √ M s (ˆ ρs − ρ0s ) →d N (0, Vs ) where Ms is the number of observation for market s. To prove consistency and asymptotic normality of the equilibrium CCPs estimators, I define B ≡ Fc˜1 ,˜c2 ,c3 =k Fˆc˜−1 c2 ≡ B(ρs ), so B is a matrix-valued function with ρs as input arguments. Note that a 1 ,˜ column in Ac˜1 |e∗ ,s is an eigenvector of B. That is, the equilibrium CCP for group 1 p(˜ c1 |e∗ , s) for a given s and a realization of equilibrium e∗ can be represented as:

p(˜ c1 |e∗ , s) := ψ(B) = ψ(ρs ), 35

and its corresponding estimates pˆ(˜ c1 |e∗ , s) := ψ(B) = ψ(ˆ ρs ), where ψ(·) is the eigenvector function, which is an analytical function as shown in Andrew et al. (1993) theorem 2.1. This eigenvector function is well-behaved around the true parameter. Moreover, it is known and nonstochastic. The continuity of ψ(·) implies that consistency of the vector ρˆs is sufficient for the consistency of the equilibrium CCPs pˆ(˜ c1 |e∗ , s). To show the asymptotic normality of the equilibrium CCPs p(˜ a1 |e∗ , s), Taylor expansion of ψ(ˆ ρs ) results in pˆ(˜ c1 |e∗ , s) − p(˜ c1 |e∗ , s) = ψ(ˆ ρs ) − ψ(ρ0s ) dψ = ( 0 )|ρ=ρ0s (ˆ ρs − ρ0s ) + Op (Ms−1/2 ). dρ

(C.1)

As a result, the asymptotic normality of the equilibrium CCP estimator for group 1 become √

M s (ˆ p(˜ c1 |e∗ , s) − p(˜ c1 |e∗ , s)) →d N (0, Σs ), 0

dψ where the variance matrix Σs = ( dρ V ( dψ )|ρ=ρ0s . 0 )|ρ=ρ0 s s dρ

The estimators for individual player’s equilibrium CCPs pˆ(ak |e∗ , s) are consistent and asymptotically

normal, because they are smooth functions of the estimator pˆ(˜ c1 |e∗ , s).



Note that for implementing the decomposition, one has to compute the inverse of the matrix Fa˜1 ,˜a2 , which may cause the estimation error to be large if the matrix is close to singular. The equilibrium CCPs estimated through eigenvalue-eigenvector decomposition and inversion automatically suffer from such estimation error. Let p collect all equilibrium CCPs, i.e., p = {(p(ai |e∗ , s))i , e∗ ∈ ω a (s) & s ∈ S}, and pˆ denote an estimator of p. By lemma C.2, pˆ is a consistent and asymptotically normal estimator of p. I represent the asymptotic distribution as follows √

M (ˆ p − p) →d N (0, Σ)

In order to characterize the asymptotic properties of the least squares estimators, I first state a number of assumptions in the following. Assumption C.1. Θ is a compact set. 36

Assumption C.2. the true value θ0 is in the interior of Θ. Assumption C.3. M → ∞, WM → W0 a.s. where W0 is a non-stochastic positive definite matrix. Assumption C.4. the functions π are twice continuously differentiable in θ. Assumption C.5. the matrix [Oθ Γ(p(θ0 ), θ0 )0 ]W0 Oθ0 Γ(p(θ0 ), θ0 ) is non-singular, where Oθ Γ(p(θ0 ), θ0 ) is the gradient of the mapping Γ(p(θ0 ), θ0 )] with respect to the second argument. Assumption C.6. θ is identified. (all of the assumptions listed in the text satisfied). Assumptions C.1.-C.5. are standard technical conditions to ensure the problem is well behaved. Those assumptions are easily satisfied. These assumptions are similar to those listed in Pesendorfer and Schmidt-Dengler (2008) but simpler as static games are a special case of dynamic ones with a discounted factor equals to 0. Thus, I state in the following that the asymptotic least square estimator of the payoff primitives isconsistent and asymptotically normal. Lemma C.3. Under Assumptions C.1.-C.6. and the equilibrium CCP estimators are consistent and asymptotically normal, the estimator θˆM of payoff primitives converges to θ0 and is asymptotically normal distributed with √

M (θˆM (WM ) − θ0 ) →d N (0, Ω(θ0 )),

as

M → ∞,

where Ω(θ0 ) = (Oθ Γ0 W0 Oθ0 Γ)−1 Oθ Γ0 W0 [I − Op0 Γ]Σ[I − Op Γ0 ]W0 Oθ0 Γ(Oθ Γ0 W0 Oθ0 Γ)−1 , I is an identity matrix, Op Γ is the gradient of the mapping Γ(p(θ0 ), θ0 )] with respect to the first argument, and the various matrices are evaluated at θ0 , p(θ0 ). Proof See Theorem 9.2 in Gourieroux and Monfort (1995).

D



Equilibrium Coordination Failure

This paper assumes that players in the same market have no problem coordinating on the same equilibrium. It is important to investigate the effect of coordination failure on the identification results. The identification of the number of equilibria may or may not be affected by coordination failure. When coordination fails, the number of equilibria can be identified using the proposed method if information consists of wo players is sufficient for the identification. Without loss of generality, suppose 37

there are two equilibria in a 2 × 2 game, denoted as eq1 = {Pr(a1 |e∗ = e1 ), Pr(a2 |e∗ = e1 )}, and eq2 = {Pr(a1 |e∗ = e2 ), Pr(a2 |e∗ = e2 )}. Coordination failure implies that in some markets player 1 employs eq1 and player 2 employs eq2, while in certain other markets player 1 employs eq2 and player 2 employs eq1. Consequently, the pooling data across markets consists of four strategy combinations: such as S1 ≡ {Pr(a1 |e∗ = e1 ), Pr(a2 |e∗ = e1 )}, S2 ≡ {Pr(a1 |e∗ = e2 ), Pr(a2 |e∗ = e2 )}, S3 ≡ {Pr(a1 |e∗ =

e1 ), Pr(a2 |e∗ = e2 )}, and S4 ≡ {Pr(a1 |e∗ = e2 ), Pr(a2 |e∗ = e1 )}.

I show below that the number of equilibria identified still equals to two. The observed joint distribution satisfies the following equation: Fa1 ,a2 = Aa1 |e∗ De∗ ATa2 |e∗ where Aa1 |e∗ De∗ Aa2 |e∗

=

Pr(a1 |e∗ = e1 ) Pr(a1 |e∗ = e2 ) Pr(a1 |e∗ = e1 ) Pr(a1 |e∗ = e2 ) | {z } | {z } | {z } | {z }

= diag =



S1

S2

S3

Pr(S1 ) Pr(S2 ) Pr(S3 ) Pr(S4 )



S4

Pr(a2 |e∗ = e1 ) Pr(a2 |e∗ = e2 ) Pr(a2 |e∗ = e2 ) Pr(a2 |e∗ = e1 ) {z } | {z } | {z } | {z } | S1

S2

S3

S4

! !

It is straightforward that   rank Pr(ai |e∗ = e1 ) Pr(ai |e∗ = e2 ) Pr(ai |e∗ = e1 ) Pr(ai |e∗ = e2 )   = rank Pr(ai |e∗ = e1 ) Pr(ai |e∗ = e2 )

As a result, coordination failure does not affect the identification of the number of equilibria if the information from two players is sufficient for the identification, because the number of equilibria reflects the variation of CCPs across different equilibria. Coordination failure alters the correlation between the two players, but it does not change the variation of CCPs for individual players. Even though the number of equilibria can be correctly identified, however, coordination failure does affect the identification of the equilibrium strategies and the equilibrium selection. If identification of the number of equilibria requires at least four players, the number of equilibria identified increases from Q to at most Ql with l as the number of players grouped. An example would be a binary game with four players and two equilibria denoted as {Pr(ai |e∗ = e1 ), i = 1, 2, 3, 4} and

{Pr(ai |e∗ = e2 ), i = 1, 2, 3, 4}. Group players 1 and 2 act as one representative player, and players 38

3 and 4 as the other representative player. For ease of notation, assume that players 2,3,4 always manage to coordinate, but player 1 might be off track sometimes. With coordination failure, the matrix constructed by equilibrium CCPs becomes 

Pr(a1 |e∗ = e1 )T ⊗ Pr(a2 |e∗ = e1 )T

  Pr(a |e∗ = e )T ⊗ Pr(a |e∗ = e ))T 1 1 2 2    Pr(a1 |e∗ = e2 )T ⊗ Pr(a2 |e∗ = e1 ))T  Pr(a1 |e∗ = e2 )T ⊗ Pr(a2 |e∗ = e2 )T

T      

In contrast, the matrix constructed by the equilibrium CCPs without coordination failure is represented as



Pr(a1

|e∗

= e1 ) ⊗ Pr(a2

|e∗

= e1 ) Pr(a1

|e∗

= e2 ) ⊗ Pr(a2

|e∗

= e2 )



The ranks of the above two matrices are not equivalent to each other. Thus, the number of equilibria cannot be identified with equilibrium coordination failure. Consequently, the equilibrium CCPs and payoff functions are not identifiable.

E

Graphs and Tables Figure 1: The Best Response Function for Different Values of s s=1, p*=0.1593,0.4601,0.8671 1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

45 degree line

0.1

best response function

0 0.1

0.2

0.3

1

0.4

0.5

0.6

0.7

0.8

0.2 45 degree line best response function

0.1

0.9

1

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

p

p

s=3, p*=0.2053,0.3601,0.8859

s=4, p*=0.8936

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.8

0.9

1

0.3

0.2

0.2 45 degree line best response function

0.1 0 0.1

s=2, p*=0.1780,0.4208,0.8772

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

45 degree line best response function

0.1 1

p

0 0.1

0.2

0.3

0.4

0.5

0.6

p

39

0.7

0.8

0.9

1

Figure 2: Timing Patterns for Commercials across Markets (Sweeting (2009))

&'(('$!("()*$+$!(",(#'*-+.!"/ 0(%'0!+-*!"!"#

%$&'(('$!("()*$+$!(",(#'*-+.!"/ 0(%'0!+-*!"!"#

%$01213456007839 8 2278 16 96 899268709 





 







!"#

%$







12















!"#

%$Figure 3: Timing Patterns for Commercials in Different Markets (Sweeting (2009))

0121345675228971  1359 3 56 3 9578 0893535705892 63785%9:4"42%#:;!"<2#= >%775>2"!:2#72#345

 !"#$%&'()( / . , + * 0**

0*/

0+*

0+/

0,*

0,/

0-* 12#345

0-/

0.*

0./

0/*

0//

0-/

0.*

0./

0/*

0//

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?@%>A5:45&6(B( / . , + * 0**

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40

Table 1: Monte Carlo Evidence

The Number of Eq p(a = 0|s, e∗ = e1 ) p(a = 0|s, e∗ = e2 ) The Eq Selection 1 2

s=1 Estimates 0.1594 (0.012) 0.8671 0.8673 (0.013) 0.5 0.5004 (0.017) DGP 2 0.1593

s=2 Estimates 0.1784 (0.014) 0.8772 0.8773 (0.0112) 0.5 0.4994 (0.018) DGP 2 0.1780

s=3 Estimates 0.2053 (0.012) 0.8859 0.8857 (0.0213) 0.5 0.4995 (0.018) DGP 2 0.2053

DGP 1 n/a

s=4 Estimates -

0.8936 0

0.8934 (0.004) 0 (0)

Sample size of each market type is 1200. The number in brackets is the standard deviation computed via bootstrap with 500 repetition.

Table 2: Monte Carlo Evidence: Model Primitives Strategic Interaction δ

true 2.5

Market Effect β

0.04

1

N=300 2.593 (0.135) 0.0409 (0.034)

N=500 2.556 (0.108) 0.0401 (0.029)

Sample Sizes N=800 N=1000 2.531 2.524 (0.066) (0.071) 0.0400 0.0401 (0.018) (0.021)

N=1200 2.518 (0.039 ) 0.0400 (0.011)

The number in brackets is the standard deviation computed via bootstrap with 500 repetition. Table 3: A Unique equilibrium vs multiple equilibria Strategic Interaction δ

DGP 2.5

Market Effect β

0.04

1 2

Unique Eq 2.8245 (0.0405) -0.0236 (0.0056)

Multiple Eq 2.5054 (0.0190) 0.0398 (0.0048)

Sample size of each market type is 1200. The number in brackets is the standard deviation computed via bootstrap with 500 repetition. Table 4: Summary Statistics Variable No. Players Timing Day Drivetime Market

Obs 92766 92766 92766 92766 92766

Mean 5.641 .499 31.745 1.518 1.572

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std. Dev 2.054 .489 17.723 1.518 1.378

Min 3 0 1 1 1

Max 15 1 59 2 2

Table 5: Estimates of Commercial Airing Strategies

The Number of Eq The Eq Selection: Pr(e∗ = e1 ) CCPs: Pr(a = 0|e∗ = e1 ) Pr(a = 0|e∗ = e2 ) α(s) δ(s) 1 2

All market 2 0.027 (0.176) 0.789 (0.188) 0.499 (0.043) -0.001 (1.049) 2.281 (0.797)

Market Size Large Small 1 2 1 0.181 (0.251) 0.506 0.638 (0.151) (0.124) 0.465 (0.039) -0.003 (0.952) 2.043 (0.441)

Time Drivetime Non-drivetime 2 1 0.436 1 (0.119) 0.522 0.520 (0.125) (0.004) 0.478 (0.184) 0.007 (3.758) 2.161 (0.860) -

The number in brackets is the standard deviation computed via bootstrap with 500 repetition. For markets s with a unique equilibrium, δ(s) and α(s) are not identified.

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