Bayesian Estimation of Discrete Games of Complete Information

Transcription

1 Bayesian Estimation of Discrete Games of Complete Information Sridhar Narayanan May 30, 2011 Discrete games of complete information have been used to analyze a variety of contexts such as market entry, technology adoption and peer effects. They are extensions of discrete choice models, where payoffs of each player are dependent on actions of other players, and each outcome is modeled as Nash equilibria of a game, where the players share common knowledge about the payoffs of all the players in the game. An issue with such games is that they typically have multiple equilibria, leading to the absence of a one-to-one mapping between parameters and outcomes. Theory typically has little to say about equilibrium selection in these games. Researchers have therefore had to make simplifying assumptions, either analyzing outcomes that do not have multiplicity, or making ad-hoc assumptions about equilibrium selection. Another approach has been to use a bounds approach to set identify rather than point identify the parameters. A third approach has been to empirically estimate the equilibrium selection rule. In this paper, we take a Bayesian MCMC approach to estimate the parameters of the payoff functions in such games. Instead of making ad-hoc assumptions on equilibrium selection, we specify a prior over the possible equilibria, reflecting the analyst s uncertainty about equilibrium selection and find posterior estimates for the parameters that accounts for this uncertainty. We develop a sampler using the reversible jump algorithm to navigate the parameter space corresponding to multiple equilibria and obtain posterior draws whose marginal distributions are potentially multi-modal. When the equilibria are not identified, it goes beyond the bounds approach by providing posterior distributions of parameters, which may be important given that there are likely regions of low density for the parameters within the bounds. When data allow us to identify the equilibrium, our approach generates posterior estimates of the probability of specific equilibria, jointly with the estimates for the parameters. Our approach can also be cast in a hierarchical framework, allowing not just for heterogeneity in parameters, but also in equilibrium selection. Thus, it complements and extends the existing literature on dealing with multiplicity in discrete games. We first demonstrate the methodology using simulated data, exploring the methodology in depth. We then present two empirical applications, one in the context of joint consumption, using a dataset of casino visit decisions by married couples, and the second in the context of market entry by competing chains in the retail stationery market. We show the importance of accounting for multiple equilibria in these application, and demonstrate how inferences can be distorted by making the typically used equilibrium selection assumptions. Our applications show that it is important for empirical researchers to take the issue of multiplicity of equilibria seriously, and that taking an empirical approach to the issue, such as the one we have demonstrated, can be very useful. Keywords: Discrete games, multiple equilibria, Bayesian estimation, Markov Chain Monte Carlo methods, reversible jump algorithm Graduate School of Business, Stanford University. sridhar.narayanan@stanford.edu. I thank Wes Hartmann for useful discussions, and the participants of the seminar at Goethe University and the Marketing Science Conference 2010 for useful comments. All errors are my own. 1

2 1 Introduction In a number of situations, agents payoffs from their actions are not independent of actions of other agents. For instance, the profits that a firm would earn from entering a market would depend on the entry decisions of its competitors, or the benefits to adopting a technology or a new product with network effects depend on adoption decisions of others. Discrete games have been applied to several such contexts where decisions are discrete or can be discretized. Discrete games of complete information were first applied by Bresnahan and Reiss (1990) and Berry (1992) in market entry contexts. The actions of agents are modeled as the Nash equilibrium outcomes of a static game where all agents are fully informed about the payoffs of all other agents. Subsequently, such games have been applied to product quality choice (Mazzeo, 2002), pricing strategy choice (Zhu, Singh, and Manuszak, 2009) and joint consumption (Hartmann, 2010) amongst others. A specific issue with such games is the absence of a one-to-one mapping between parameters and outcomes - a problem of multiplicity of equilibria. The extant literature has proposed ways to deal with multiplicity, ranging from the modeling of outcomes that are unique even in the presence of multiplicity (Bresnahan and Reiss, 1990; Berry, 1992), specification of a sequential rather than simultaneous move game (Berry, 1992), assuming an ad-hoc equilibrium selection rule (Hartmann, 2010), randomizing the equilibria (Soetevent and Kooreman, 2007), obtaining the identified set estimates rather than unidentified point estimates of parameters (Ciliberto and Tamer, 2009) and empirically estimating the equilibrium selection rule (Bajari, Hong, and Ryan, 2010). In this paper, we propose an alternative hierarchical Bayesian approach to deal with this issue of multiplicity, in which the uncertainty of the analyst over the multiple equilibria is accounted for and posterior parameter estimates obtained that reflect this uncertainty. The problem of multiple equilibria was explored in detail by Bresnahan and Reiss (1990, 1991) and Berry (1992) in their seminal studies of market entry. However, the idea itself has been pointed out earlier (for instance Heckman 1978). The main issue with estimation in the presence of multiple equilibria is that for a given set of parameters, observable covariates and unobservable variables, there is the potential for more than one equilibrium outcome. This causes the likelihood to be ill-defined because the probabilities of the potential outcomes for a particular observation add up to a value greater than 1. Such an econometric model is termed incoherent (Tamer, 2003) and cannot be directly estimated using a maximum likelihood procedure without simplifying assumptions. The literature on discrete games has made various simplifying assumptions, but these assumptions either reduce the scope of the problems that can be studied (e.g. the study of unique number of entrants rather than their identities in Bresnahan and Reiss (1990)) or can lead to inconsistency of estimates if the assumption is invalid (e.g. the assumed sequence of entry in Berry 1992, or the assumption of equilibrium selection in Hartmann 2010). An alternative approach has been to focus on estimating the bounds for parameters rather than point estimates since the bounds are uniquely determined for a given set of outcomes in the 2

3 presence of multiplicity (Tamer 2003; Ciliberto and Tamer 2009). While this approach has its appeal, one issue is that little is known about where the parameters lie within these bounds, reducing the utility of the estimates for any counterfactual analysis. A third approach has been to take an empirical approach to the problem of multiplicity, estimating the equilibrium selection rule (Bajari, Hong, and Ryan, 2010). However, this approach either requires the researcher to observe a set of variables that affect one player s payoffs, but can be excluded from the payoff functions of other players, or relies on an identification at infinity argument, where there are observations with sufficiently large values of covariates. In practice, it may be difficult to find such excluded variables in some contexts, or observations that meet the requirements for identification at infinity. For a detailed review of the empirical literature on discrete games, the reader is referred to Ellickson and Misra (2010). We develop a hierarchical Bayesian approach to estimate parameters of the payoff functions of the players in the presence of multiple equilibria. Essentially, this approach takes a Bayesian model selection route to addressing the problem of multiplicity, by conditioning the model on equilibrium selection. Conditional on equilibrium selection, the likelihood is well-defined, and hence the parameters of this conditional model can be estimated. Thus, the problem can be considered to be one where there are multiple (conditional on equilibrium selection) models, over which the analyst has uncertainty. We develop a sampling algorithm for this problem, using a reversible jump MCMC algorithm to sample the parameters over the multiple models corresponding to the multiple equilibria jointly with indicators for the models themselves. This procedure generates draws from the posterior distribution of parameters across the multiple equilibria, reflecting our uncertainty about equilibrium selection. The procedure has several appealing features. Compared to an approach that estimates bounds on parameters (Ciliberto and Tamer, 2009), this approach gives us posterior distributions of parameters, and therefore tells us the regions within the bounds with high posterior probability of the parameters. It also nests within it an equilibrium selection rule of the kind proposed by Bajari, Hong, and Ryan (2010), generating posterior estimates of the selected equilibrium in the data, provided the equilibrium is identified. In cases where the equilibrium is not identified, the parameter estimates reflect the analyst s prior uncertainty about equilibrium selection. It also nests within it equilibrium selection rules such as Pareto optimality (Hartmann, 2010). Thus, the proposed method provides a practical tool to deal with multiple equilibria in the estimation of discrete games, nesting within it many of the approaches used in the extant literature. We demonstrate our proposed algorithm first using simulated data. We show that it is able to recover parameters of a discrete game with more than 2 players with a high degree of accuracy. We further demonstrate through these simulations the downsides of making ad-hoc equilibrium selection mechanisms. We then apply the methodology to a social interaction game involving casino visits by married couples. We estimate the parameters of the model, and demonstrate how the parameter estimates as well as estimates of counterfactual simulations are biased when the presence of multiple equilibria is ignored or dealt with using an ad-hoc equilibrium selection rule. 3

4 We finally present an application to a competitive market entry game in the retail stationery market. Our estimates show that the commonly assumed equilibrium selection rule has low posterior probability in this market, raising questions about the assumption for other markets as well. The rest of the paper is organized as follows. We first set up an illustrative model to demonstrate the issue of multiplicity for two different contexts - one where players payoffs are positively affected by the presence of other players, and another where they are negatively affected. We briefly discuss existing approaches to estimating discrete games of complete information, including approaches that have been used to deal with multiplicity of equilibria. Next, we explain in detail our proposed Bayesian method to estimate the parameters of the conditional model and model indicators using a reversible jump MCMC sampler. We discuss conceptual issues related to the methodology, as well as issues related to implementation. We next demonstrate our methodology using simulated data, and then present our empirical application. We discuss the results of the empirical analysis, pointing in particular to the consequences of assumptions on equilibrium selection. We finally conclude. 2 Multiple equilibria in discrete games We first set up an illustrative model for a market entry game, modifying the discussion for a social interactions game subsequently. Assume that there are a total of i =1...I players in the game, and t =1...T outcomes observed for these players. In a market entry context, the players are firms considering entry in the T markets, and the outcomes represent the market structure in each market, i.e. which of the firms choose to enter and which do not. Let the payoff functions for firm i in market t be the following if it chooses to enter the market. π it = α i +X it β γ 1 y jt > 0 γδ 1 y jt > 1 γδ 2 1 y jt > 2...+ε it (1) j=i j=i j=i Here, α i is a player specific intercept and represents the intrinsic profitability of the firm. X it is a vector of exogenous covariates, representing factors like the population size and other demographic factors of market t, andβ is a vector of coefficients for these covariates. After this are a set of terms that represent the effect of the presence of competitors in market t. y it represents the decision of firm i to enter market t and takes the value 1 if the firm chooses to enter and 0 otherwise. The first of the competitive effect terms thus captures the impact on firm i s profits if it has one competitor in the market, and subsequent terms capture the effect of additional competitors. The parameter γ and δ are assumed to be positive, and thus the competitive effect is assumed to be negative. Finally, there is an additively separable error which is unobservable to the econometrician, but is observed by all the firms. This may represent unobserved fixed costs of entering the market, which is firm specific. The common knowledge of the errors amongst all players of the game makes this a complete information game. The 4

5 specification laid out above is similar to the one in Hartmann (2010) and captures the competitive effect in a parsimonious way. 1 Each firm considers its own entry decision, taking into account the optimal decisions of its competitors. It chooses to enter the market if it expects positive profits, and chooses to stay out of the market otherwise. The outcome is thus a Nash equilibrium of this game, with an equilibrium in pure strategies guaranteed with the restrictions on parameters that we have imposed (non negative γ and δ). As has been pointed out in the literature, such a game has multiple equilibria. To illustrate this point, we consider the two player case, and without covariates. The profit functions of the two firms are then given by π it = α i γ y i,t + ε it, i {1, 2} (2) where y i,t is the entry decision of the competing firm. Each firm enters the market if it expects positive profits. Let us first consider the case where neither firm enters market t. Thus, both y 1t and y 2t are 0 in this case. The profits for the firms in this situation are given by Thus, for neither firm to enter π 1t = α 1 + ε 1t < 0 π 2t = α 2 + ε 1t < 0 (3) ε 1t < α 1 ε 2t < α 2 (4) Firm 1 would enter and firm 2 would not enter the market if the former expected positive profits and the latter did not. In this condition y 1t =1and y 2t =0. Thus, the equilibrium conditions are given by ε 1t > α 1 and firm 2 would enter the market and firm 1 would not if ε 2t < α 2 + γ (5) 1 It is necessary to give structure to the competitive effect, particularly for small number of players since there are limited degrees of freedom available. For instance, in a two player game, one can estimate at most three parameters in addition to the coefficients for exogenous covariates - for instance two firm-specific intercepts and a competitive effect parameter. In a 3-player game, one can estimate at most seven parameters, and hence we would not be able to estimate the most general competitive effect even in this case, where there were for instance a different effect of each firm on each of its competitors, in addition to firm-specific intercepts. As the number of players increase, the degrees of freedom increase exponentially, giving more flexibility in specifying the competitive effects. 5

6 ε 1t < α 1 + γ ε 2t < α 2 (6) Both firms would enter the market if both expected positive profits and this corresponds to ε 1t > α 1 + γ ε 2t > α 2 + γ (7) The multiple equilibria in this case result from the fact the conditions in equations 5 and 6 are not mutually exclusive. The equilibrium conditions are depicted graphically in figure 1. Consider panel A of this figure. There are nine regions depicted in this panel, which we shall refer to as cells. These cells are labeled I through IX. These cells are defined by the various equilibrium conditions in equations 4 through 6. Now consider panel B of the figure. Cell I, which corresponds to neither firm entering (i.e. satisfying the equilibrium condition in 4) is shaded. Similarly panels C, D and E are graphical representations of the equilibrium conditions in 5, 6 and 7 respectively. Note from panels C and D that they share a cell - cell V, i.e. both equilibria are satisfied in this cell. This is the region of multiple equilibria, highlighted in panel F of the figure. This cell is consistent with two equilibria - either firm 1 alone enters, or firm 2 alone enters. Discrete games can also be set up to model social interactions, where players benefit from joint consumption of a product. A slightly modified version of the model used for analyzing market entry could be used for this context, although the equilibrium conditions, and the nature of multiplicity of equilibria in this case are distinctly different. Let the payoffs of the two players be given by π it = α i +X it β +γ 1 y jt > 0 +γδ 1 y jt > 1 +γδ 2 1 y jt > ε it (8) j=i j=i j=i Note that the main difference between the payoffs in this social interaction game and those in the market entry game in equation 1 is the sign of the interaction terms. In this case, consumption by another player increases the payoffs from consumption. Thus, for γ>0and δ>0, we are guaranteed the existence of an equilibrium in pure strategies for any realization of the errors ε it and any values of the covariates and the parameters. The equilibrium conditions for a two-player game with social interactions can be derived in a similar fashion as for the market entry game. Assuming there are no other covariates in the model, the payoffs are given by π it = α i + γ y i,t + ε it (9) 6

7 where y i,t is the consumption decision of the other player. Neither player consumes if both expect negative payoffs from consumption. The conditions for this equilibrium are ε 1t < α 1 ε 2t < α 2 (10) If player 1 consumes and player 2 does not, the expected payoff of the former is positive and that of the latter is negative. Hence, the conditions for this equilibrium are those for player 2 consuming alone are ε 1t > α 1 ε 2t < α 2 γ (11) ε 1t < α 1 γ and those for both players consuming jointly are ε 2t < α 2 (12) ε 1t > α 1 γ ε 2t > α 2 γ (13) The change in sign of the interaction effect in the payoff function has a non-trivial impact on the nature of the multiplicity of equilibria. In particular, the overlap now is between the equilibrium where neither player consumes and the one where both players consume. This is again best seen graphically. Figure 2 depicts the equilibrium conditions for the various outcomes. Panel A shows the nine regions defined by the equilibrium conditions in 10 through 13. Panel B of this figure shows us the conditions for the equilibrium where neither player consumes - this corresponds to cells I, II, IV and V. Panels C and D respectively show the conditions for player 1 consuming alone (Cell III) and player 2 consuming alone (Cell VII) respectively. Panel E shows the conditions for joint consumption by both players (Cells V, VI, VIII and IX). As can be seen in Panel F, multiplicity of equilibria arises from the fact that Cell V is consistent with two equilibria - neither player consuming and both of them consuming. With greater than two players in the game, the issue of multiplicity remains and the number of cells with multiple equilibria increases. While it is somewhat tedious to work out the equilibrium conditions for a large number of players by hand, it is straightforward to enumerate the equilibria using a computer. This is because irrespective of the values of the parameters, the cells that 7

8 correspond to more than one equilibrium are the same for a given number of players and a given type of game (i.e. market entry or social interactions). The addition of covariates to the model also does not fundamentally change the equilibrium conditions and the cells corresponding to the different equilibria. For instance, the equilibrium conditions for a market entry game are modified to the following. For nether firm to enter the market, the conditions are ε 1t < α 1 X 1t β ε 2t < α 2 X 2t β (14) Firm 1 would enter and firm 2 would not enter the market if the former expected positive profits and the latter did not. In this condition y 1t =1and y 2t =0. Thus, the equilibrium conditions are given by ε 1t > α 1 X 1t β and firm 2 would enter the market and firm 1 would not if ε 2t < α 2 X 2t β + γ (15) ε 1t < α 1 X 1t β + γ ε 2t < α 2 X 2t β (16) Both firms would enter the market if both expected positive profits and this corresponds to ε 1t > α 1 X 1t β + γ ε 2t > α 2 X 2t β + γ (17) It can be seen from these equilibrium conditions that covariates merely shift the boundaries of the cells corresponding to the various outcomes. 3 Estimating discrete games of complete information 3.1 Existing approaches There are several approaches that the literature has used to estimate complete information discrete games of the type laid out above. These approaches vary in how they deal with mul- 8

9 Figure 1: Equilibrium Conditions - Discrete Game of Market Entry 9

10 Figure 2: Equilibrium Conditions - Discrete Game of Social Interactions 10

11 tiplicity of equilibria. The early literature on market entry (Bresnahan and Reiss, 1990, 1991; Berry, 1992) recognized that while there are multiple equilibria in market entry games, the number of players is unique for any given realization of the errors and any set of values of the covariates and parameters. For the two player game for instance, it is clear from figure 1 that the multiplicity is between the equilibrium where firm 1 enters alone, or the one where firm 2 enters alone (cell V). Thus, if one were interested in modeling the number of firms that entered a market rather that their identities, there would be no problem of multiplicity. Estimation proceeds by specifying the distributions of the errors, which makes it possible to develop a maximum likelihood procedure to find parameter estimates. Another approach has been to make an ad hoc assumption about equilibrium selection. For instance, Hartmann (2010) estimates a game of social interactions to model the decisions of friends playing golf together. In such a game, the multiplicity is between no consumption by either player and joint consumption by both players in a two-player game. This approach relies on theoretical arguments to posit that agents choose the Pareto optimal outcome of both players choosing to consume over the Pareto dominated outcome of neither consuming. Thus, cell V in figure 2 is assumed to correspond entirely with the outcome of both players choosing to consume. With distributional assumptions on the errors, the likelihood of this model is now well-defined. A maintained assumption in this case is that the players are strictly better off by consuming the focal good than not consuming it. This could be violated, for instance, if the outside option involves consuming other goods, whose joint consumption gives greater payoffs than joint consumption of the focal good. For instance, if instead of playing golf together, the players could go to the movies together, it is not clear if the assumption on selection of the Pareto dominant equilibrium could be sustained. A third approach is to specify the sequence in which players move in a sequential rather than simultaneous move game. For instance, Berry (1992) estimates a sequential move discrete game in the airline market by making the assumption that the more profitable player moves first. Such an assumption gets rid of the issue of multiplicity, though it depends on observing the order in which firms entered. In many empirical applications of discrete games, this order is unobserved. The recent literature on discrete games has focused on taking an empirical approach to the issue of multiplicity. The methodology proposed by Ciliberto and Tamer (2009), for instance relies on the fact that equilibrium conditions of the kind in equations 4 through 7 define the necessary conditions that need to be met for each of the outcomes, though multiplicity implies that some of those conditions are not sufficient to define an outcome. These necessary conditions can be used to set up moment inequalities, that allow for set identification of the parameters even when the parameters are not point identified. This procedure identifies upper and lower bounds on the parameters which are obtained through two sets of moment inequalities for each parameter. These two sets of inequalities correspond respectively to regions of the error space leading to unique equilibria, and those including the regions of multiple equilibria. The predicted choice probabilities for various outcomes corresponding to these regions of the error 11

12 space are matched with the empirical choice probabilities to obtain estimates of the bounds on the parameters. Another empirical approach to dealing with the issue of multiplicity is to estimate the equilibrium selection probabilities simultaneously with other model parameters. In general, the equilibrium selection probabilities are not identified if we merely observe the actions of the agents. However, in the presence of exclusion restrictions, they are identified. This is the approach taken by Bajari, Hong, and Ryan (2010), where a set of excluded covariates, covariates that affect the payoffs of one player but not the others - is used to identify the probabilities for the various equilibria as well as the payoff parameters simultaneously. An example of excluded variables used in the literature is distance from the headquarters and warehouses of retailers in a market entry game - the rationale being that the distance from its own headquarters or warehouse to a focal market would impact the profits of a retailer in the market (through transportation and management costs that depend on distance) but would not directly affect the profits of its competitors (except through the entry decision of the retailer). Another example in the situation of joint consumption of a product by friends, for instance, is targeted promotions that are offered and can be availed by one player but not by the other Identification of equilibrium using exclusion restrictions In this sub-section, we develop the intuition for how exclusion restrictions can be used to identify the equilibrium selection probabilities. Assume a two player competitive game of the type depicted in Figure 1. Consider without loss of generality the case where variable X 1 affects the payoffs of firm 1, but not those of firm 2, and variable X 2 affects the payoffs of firm 2 but not of firm 1. The coefficients of X 1 and X 2 are respectively β 1 and β 2, which are assumed positive without loss of generality. Figures 3 and 4 graphically depict the effect of shift of covariates on the outcome probabilities for the two equilibria for such a game - the first (equilibrium 1) where Cell V is entirely allocated to firm 1 s entry, and the second (equilibrium 2) where Cell V is entirely allocated to firm 2 s entry. Panel A in these two figures depict the baseline situation, i.e. at given levels of the covariates and at some fixed set of values of the parameters. Now consider Panel B in the two figures, where the covariate for firm 1, X 1 is shifted to a lower value X1. The shift in the covariate shifts the cell boundaries to the right in the case of both equilibria. This shifts the probabilities of the various outcomes - in particular, it increases the probability that neither firm enters (Cell I expands) and decreases the probability that both firms enter the market (Cell IX contracts). Note that the effect of the shift in covariate on these two outcomes is identical for the case of both equilibria. The shift in covariate also shifts the probabilities for the remaining two outcomes - firm 1 entering alone and firm 2 entering 2 Bajari, Hong, and Ryan (2010) also discuss an identification at infinity approach to identify the equilibrium selection probabilities, which relies on observing local perturbations of covariates around values for which there is a unique equilibrium with probability approaching 1. Using an invariance assumption on the equilibrium selection rule, the probabilities of selecting the various equilibria are estimated using observations corresponding to other values of the covariates. In our Bayesian approach, we do not use such an approach and hence we do not discuss this in further detail. 12

13 alone. In the case of both equilibrium 1 and equilibrium 2, the probability of firm 1 entering the market alone decreases and that for firm 2 entering alone increases. However, the change in probabilities is different for the two equilibria as long as the errors are not uniformly distributed. This is because the specific areas that shift from being consistent with firm 1 entering alone to firm 2 entering alone are different, and these two areas would have different densities of the distribution of errors for any non-uniform distribution. Thus, the shift in covariates causes different substitution patterns for the two equilibria. However, this alone is insufficient to identify the equilibrium from the data, since a given substitution pattern arising out of the shift of X 1 could be rationalized by not just the two equilibria but also different sets of parameters. Thus, for instance, a shift in shares consistent with the parameters (α, β 1, γ) in Panel B of Figure 4 could be potentially rationalized by a a different set of parameters (α, β1, γ ) in Panel B of Figure 3, highlighting the fact that variation in this covariate alone cannot identify the equilibrium. We have a similar situation when we shift firm 2 s covariate X 2, as shown in Panel C of Figures 3 and 4. Another way to state this non-identification is that there are insuffficient moments to identify both the equilibrium and the parameters. Now consider the situation where we shift both covariates X 1 and X 2 simultaneously. This is depicted in Panel D of the two figures. We see once again that the shift in the probabilities of neither firm entering and both firms entering are identical for the two equilibria, and the shift in shares of the remaining two outcomes differ between the two equilibria. It can be seen from the figure that the changes in shares that result from the simultaneous shifting of the two covariates is not just a mere addition of the changes that take place when the two covariates are shifted separately. For instance, note that a small rectangular area at the bottom-left corner of Cell V becomes a part of the area for the new Cell 1 when the two covariates are shifted, and thus correspond to the outcome of neither firm entering the market. This area is not a part of the expanded Cell I for either the case where X 1 alone shifts (Panel B) or where X 2 alone shifts (Panel C). Similarly, the changes in the probabilities of the other outcomes in the case of both covariates shifting are not mere summations of the changes for each covariate shifting alone. This can be thought of as an interaction effect of the two covariates on the outcomes. In the case of the two outcomes that involve only one firm entering alone, this interaction effect is different for the two equilibria, since the regions involving this interaction effect are different for the two equilibria, with different densities of the error distribution (unless the distribution is uniform). Note, however, that due to the exclusion restriction, there is no direct interaction effect of the two covariates on the payoffs of the two firms. We thus have an additional set of moments that cannot be rationalized merely by a different set of coefficients. Thus, the equilibrium is identified when we observe the three different scenarios - where X 1 alone shifts, X 2 shifts and both variables shift. While we have presented a situation, for the purpose of illustration, where one covariate shifts, keeping the other covariate unchanged, the discussion can be extended to other patterns of variation in the data. In the absence of exclusion restrictions, the interaction effect of the shift of both covariates could be rationalized by an additional set of parameters (e.g. the effect of X 1 13

14 on firm 2 s payoffs and so on), and the equilibrium would be unidentified. Note also that the parameters and equilibria of discrete games of this nature are not identified non-parametrically - i.e. we have to impose assumptions on the distributions of the unobservables. This discussion illustrates how exclusion restrictions identify the parameters as well as the equilibrium. It can be trivially extended to the case of other games, for instance games of joint consumption and for games with a different number of participants. An important point to note is that the number of equilibria that can be identified depends on the number of moments that the exclusion restrictions make available for such identification - we cannot identify more equilibria than there are moments. This places restrictions on the nature of the empirical analysis that can be done. For instance, in a competitive game with just three players, there are hundreds of equilibria in pure strategies. Thus, one would need at least that many moments through the exclusion restrictions if one wished to identify which of these equilibria are played in a specific empirical context. Since in practice such a large number of moments is unlikely to be available, the empirical researcher would need to reduce the problem to a more feasible one - for instance, one could identify between a Pareto dominant and the set of all Pareto dominated equilibria in this case. Such restrictions are in practice necessary for any game that involves more than two players, since the number of equilibria increases exponentially in the number of players. 3.3 Bayesian approach Our proposed Bayesian approach to estimating discrete games of complete information starts by making distributional assumptions on the errors and specifying prior distributions for the parameters. Due to multiple equilibria, the likelihood is not well defined and hence a standard Markov Chain procedure such as a Metropolis Hastings algorithm is not feasible. However, conditional on a selected equilibrium, the likelihood is well-defined. The main idea behind our proposed methodology is to treat each selected equilibrium as a model, and to set up an estimation procedure that augments the parameter state space with a model state. Thus, the estimation routine navigates both parameter and model spaces, and generates joint posterior draws of parameters and model indicators. When the equilibrium is identified, for instance through the presence of excluded variables, the estimation routine generates posterior estimates of model parameters as well as model indicators. When the equilibrium is not identified, the posterior estimates of the model indicators would be the same as the prior indicators (up to a simulation error), but the procedure is able to uncover the posterior distribution of parameters that spans the various models and thus reflects the analyst s (prior) uncertainty about the equilibria. This procedure is related to the bounds procedure of Ciliberto and Tamer (2009) in that it provides estimates reflecting the full set of equilibria rather than selecting one of 14

15 Figure 3: Relationship of outcome probabilities with covariates - Equilibrium 1 15

16 Figure 4: Relationship of outcome probabilities with covariates - Equilibrium 2 16

17 them a priori. However, in place of the extremities of the set of parameter values, the proposed method obtains posterior distributions of the parameters and thus gives us information about where the posterior distributions of parameters have greater mass and where they do not. Like Bajari, Hong, and Ryan (2010), our procedure allows the probabilities corresponding to each equilibrium to be estimated when they are identified, but also allows for estimation of parameters when they are not identified. Further, we obtain parameter estimates corresponding to each of the equilibria, whereas the procedure in Bajari, Hong, and Ryan (2010) estimates one set of parameters. We first discuss a Bayesian approach to estimating a discrete game, with the use of an ad-hoc equilibrium selection rule, as has been often used in the literature. We then build on this to develop a methodology for dealing with multiplicity of equilibria Bayesian estimation with an ad-hoc equilibrium selection To illustrate the proposed methodology, we first start with the case of an entry game with two players, and subsequently extend it to cases with a greater number of players and to games of social interaction. Let the payoff functions of the two players be given by π it = α i + X it β γ y i,t + ε it (18) where the notation is the same as before. π it represents firm i s profits in market t. α i represents a firm specific intercept. The vector X it represents a set of exogenous covariates, which potentially includes a set of excluded variables. If it is an excluded variable, it takes the value 0 for all observations for the player whose payoff it is excluded from. The coefficient for the covariates is β, which includes the coefficients for the excluded variables. The interaction between firms is captured by the coefficient γ. Finally, ε it is assumed to be an error that is uncorrelated across markets. The conditions for entry are therefore 1 if π it > 0 y it = (19) 0 if π it < 0 Conditional on an equilibrium being selected, the model is a bivariate probit model. The equilibrium conditions are similar to those described earlier, except that covariates are included and the equilibrium selection rule is imposed. Note from figure 1 that there are two possible equilibria for a region of the error space (shown as Cell V in the figure) - one where firm 1 enters the market, and another where firm 2 enters the market. Take the case of the first equilibrium being selected, i.e. firm 1 enters but firm 2 does not when errors are in Cell V. This could result, for instance, from the true data generating process being one with sequential entry, with firm 1 making entry decisions before firm 2. In such a case, the regions of the error space corresponding to the four possible outcomes are below 1. (y 1t,y 2t )=(0, 0) (ε 1t,ε 2t ) Cell I 17

18 2. (y 1t,y 2t )=(0, 1) (ε 1t,ε 2t ) Cells IV/VII/VIII 3. (y 1t,y 2t )=(1, 0) (ε 1t,ε 2t ) Cells II/III/V/VI 4. (y 1t,y 2t )=(1, 1) (ε 1t,ε 2t ) Cell IX Note here that the boundaries of the cells would be modified from that in figure 1 due to the presence of covariates in the model. Thus α i would be substituted by α i + X it β in all the cell boundaries on both axes. For a given guess of parameters, the likelihood for a given observation is the sum of the probabilities of the errors being in the cells corresponding to that outcome. With a suitable distributional assumption on the errors, the likelihood can be evaluated. We assume that the unobservables are normally distributed with mean 0. i.e. ε it N (0, 1) (20) The variance of the error is fixed to 1 for identification purposes. In order to specify the likelihood for a particular observation, we need to find the probability of the unobservables being in one of the cells corresponding to the outcome for that observation. This probability is the sum of the probabilities for each of the cells for that outcome. The probability for the unobservables in any given cell j for the t th observation are given by ˆ P jt = f (ε t ) dε t (21) A jt where ε t is the vector of unobservables, f (ε t ) is the distribution of ε t and A jt denotes the region of the error space for the j th cell (i.e. the boundaries of the cell, which vary with observation because of variation in the covariates X it ). Since we have assumed ε t distributed to be distributed normally, this reduces to the following expression in our two-player example P jt = ˆ uj1t ˆ uj2t l j1t l j2t φ (ε 1t ) φ (ε 2t ) dε 1t dε 2t =[Φ(u j1t ) Φ(l j1t )] [Φ (u j2t ) Φ(l j2t )] (22) where l jit and u jit represent the lower and upper boundaries of the j th cell for ε it, φ ( ) represents the normal density function and Φ( ) represents the normal distribution function. The boundaries of the cells have been defined earlier. For instance, for cell I, l I1t = l I2t =, u I1t = α 1 X 1t β and u I2t = α 2 X 2t β. The boundaries of the other cells are similarly obtained. The likelihood of the t th observation is then L t (θ; X t,y t ) = 1(y 1t =0,y 2t = 0) (P I,t )+1(y 1t =0,y 2t = 1) (P IV,t + P VII,t + P VIII,t ) (23) +1 (y 1t =1,y 2t = 0) (P II,t + P III,t + P V,t + P VI,t )+1(y 1t =1,y 2t = 1) (P IX,t (24)) 18

19 where 1( ) is an indicator function, taking the value 1 if the conditions in it are true and 0 otherwise, y t is the vector of decisions, X t is the matrix formed by stacking all covariate vectors (X it ), and θ is the vector of all parameters The likelihood of the data is then L (θ; X, y) = t L t (θ; X t,y t ) (25) Here, y is the stacked vector of outcomes, and X is the stacked matrix of covariates across all observations. To complete the model, we need to specify prior distributions for the parameters - α 1, α 2, β and γ. Since γ is constrained to be positive, we reparametrize it as exp ( γ). Thus, θ = α 1 α 2 β γ (26) Let the prior distribution for θ be The posterior distribution of parameters is then given by θ N (µ, Σ) (27) f (θ X, y, µ, Σ) L(θ; X, y) f (θ µ, Σ) (28) The parameters can be estimated in an MCMC procedure that utilizes the Metropolis-Hastings algorithm (Chib and Greenberg, 1995). This involves generating a sequence of draws from a candidate density and then modifying it through a rejection step to achieve the detailed balance condition. This sequence of draws converges to the posterior distribution of parameters, allowing us to do inference or conduct counterfactual experiments. We have discussed the estimation of a market entry discrete game with two players, with the ad-hoc equilibrium selection rule of firm 1 entering in the case of multiple equilibria (where the two equilibria are of either firm 1 entering the market alone, or firm 2 entering it alone). But the methodology can be easily extended to more players. For instance, in a three player entry game, there are a total of 6 cells that are consistent with two equilibria each, and 2 cells that are consistent with three equilibria each. Conditional on X and ε (i.e. within a cell), there are up to 3 equilibria. However, there are a total 576 (= ) possible combinations of equilibria if we did not condition on X and ε. We henceforth refer to each such combination of cell-by-cell equilibria as an equilibrium profile. Conditional on choosing an equilibrium profile, there is a unique assignment of each cell to an outcome, and hence the likelihood for such a game is well-specified. Given the likelihood, a Metropolis Hastings algorithm for simulation from the posterior distributions of parameters is relatively straightforward to implement. The methodology can also be extended to other equilibrium selection mechanisms. For instance, the assumption in Berry (1992) is that in cases of multiple equilibria, the firms enter 19

20 in the order of profitability. This can be justified on the basis of an underlying assumption of sequential entry, where the firm with the greatest expected profits in a market enters first, followed by the next most profitable firm and so on. Thus, the equilibrium selection mechanism selects the equilibrium that maximizes industry profits market by market. Implementing the equilibrium selection rule described above requires a modification of the procedure we have described for a rule that allocates the entire cell with multiple equilibria to one of the outcomes. This is because within a cell, different regions would be allocated to different outcomes if we assumed that the more profitable firm entered. To see this, let us first discuss this in the context of the two-player market entry game described earlier and depicted in figure 1. The region for multiple equilibria is in Cell V. The two outcomes that this cell is consistent with involve either of the firms entering the market alone. Consider first the outcome of firm 1 entering, and consider for notational simplicity the case of no covariates. The equilibrium selection rule is that the more profitable firm enters the market in the case of multiple equilibria. Hence, it must be the case that the profits of firm 1 entering alone must be greater than the profits of firm 2 entering, given a realization of the errors. Thus, This implies π 1 (y 1 =1,y 2 = 0) >π 2 (y 1 =0,y 2 = 1) (29) α 1 + ε 1 >α 2 + ε 2 (30) Thus, the condition for firm 1 being the more profitable firm and hence the outcome of (y 1 =1,y 2 = 0) being picked over (y 1 =0,y 2 = 1) is ε 2 < α 2 + α 1 + ε 1 (31) This condition describes a straight line with intercept ( α 2 + α 1 ) and a slope equal to 1, i.e it is a 45 degree line intersecting the ε 2 -axis at ( α 2 + α 1 ). It is easy to see that it this line bisects the cell in two, running from the lower left corner to the upper right corner. In region under the line in Cell V, firm 1 enters the market and in the region above the line, firm 2 enters. This is shown graphically in figure 5. To estimate the model with this equilibrium selection rule (sequential entry in decreasing order of profitability), we need to ensure that the likelihoods corresponding to each outcome are correctly evaluated. The likelihood with this equilibrium selection rule is thus, L t (θ; X t,y t ) = 1(y 1t =0,y 2t = 0) (P I,t )+1(y 1t =0,y 2t = 1) P IV,t + P VII,t + P VIII,t + P V,t,(0,1) +1 (y 1t =1,y 2t = 0) P II,t + P III,t + P V,t,(1,0) + P VI,t +1 (y 1t =1,y 2t = 1) (P IX,t ) (32) 20

21 where P V,t,(0,1) is the probability of the errors being in the part of Cell V that is assigned to the equilibrium of firm 2 entering alone, and P V,t,(1,0) is the probability that the errors are in the remainder of Cell V. ˆ α1 +γ ˆ α2 +γ P V,t,(0,1) = φ (ε 2t ) dε 2t α 1 α 2 +α 1 +ε 1t = Φ( α 2 + γ)[φ( α 1 + γ) Φ( α 1 )] ˆ α1 +γ φ (ε 1t ) dε tt α 1 Φ( α 2 + α 1 + ε 1t ) φ (ε 1t ) dε tt (33) The integral in the above expression cannot be evaluated analytically, and hence is evaluated numerically. The probabilities of the errors being in each of the other cells remains the same as before. Thus, a similar estimation procedure as described earlier can be used, with the only difference being in the probabilities for the two outcomes of either firm entering the market alone. This procedure can easily be extended for more than 2 players. It is easy to verify for a three-player game of market entry, for instance, that with an equibrium selection rule that has the firms entering a market in decreasing order of profitability, each cell with multiple equilibria would similarly be equally divided between the different outcomes that are consistent with the cell. As in the two-player example illustrated above, the likelihood for each observation could be constructed through assignment of the cells partially to multiple outcomes, instead of being assigned exclusively to one outcome. To summarize, for any game with multiple equilibria,the likelihood is well-specified and a Metropolis Hastings procedure can be set up to sample from the posterior distribution of parameters as long as a particular equilibrium profile is selected, i.e. all the cells (or parts thereof) are uniquely assigned to specific equilibria Bayesian estimation with multiple equilibria In this sub-section, we discuss the Bayesian estimation of discrete games of complete information without any assumption on equilibrium selection. We have already seen that without an equilibrium selection rule, there is no unique mapping between some of the cells and outcomes, and thus no well-specified likelihood. Conditional on equilibrium selection, however, the likelihood is well-specified. Consider a particular equilibrium profile - i.e. a unique mapping of cells or parts of the cells to specific equilibria - to be a model. Thus, there is a series of models - one corresponding to each equilibrium profile, each of which can be estimated using a Metropolis Hastings algorithm. We introduce the reversible jump algorithm, which allows us to simulate posterior distributions of parameters across multiple models and discuss how we apply it to the context of discrete games with multiple equilibrium profiles corresponding to multiple models. 21

22 Figure 5: Equilibrium Conditions - Sequential Entry in Order of Profitability Reversible jump algorithm The reversible jump algorithm (Green, 1995; Green and Hastie, 2009) is a Markov Chain Monte Carlo algorithm that generates draws from a stationary distribution for an across-model state space which is potentially trans-dimensional. For instance, in modeling count data, there could be two potential models - a Poisson model, which is a single-parameter model and a Negative Binomial model, which has two parameters. The reversible jump algorithm is an extension of the Metropolis Hastings algorithm, which simultaneously draws a model indicator as well as parameters. Like the Metropolis Hastings algorithm, it does this by generating draws from a candidate density, which is modified through a rejection step to satisfy the detailed balance condition. Formally, let there be a countable set K of models, with the k being a model indicator and θ k being the parameter vector of dimension n k corresponding to the k th model. Let the data be represented by D. Let L (D k, θ k ) be the likelihood of the data corresponding to the k th model and p (θ k k) be the density of the prior for the parameters of this model. Also, let there be a prior across models, with its density 3 specified as p (k). Then the joint posterior of the model and parameters is given by π (k, θ k D) L(D k, θ k ) p (θ k k) p (k) (34) Let t denote a move type, including a forward move from (k, θ k ) to k θ k as well as the reverse move and let there be a countable set T of such moves. Let θ k have dimension n k. Let m t (k, θ k ) indicate the probability of this move, which could be dependent on both the current model and parameter values of the current state. The reverse probability would thus be m t k,θ k. For the forwards move, let us draw a vector of random numbers u with dimension 3 It is not necessary to have separate priors for θ k k and k - it is sufficient to specify a prior p (k, θ k ).However, for the applications we consider, it is natural to specify the two priors separately. 22