Bayesian estimation of discrete games of complete information

Abstract

Estimation of discrete games of complete information, which have been applied to a variety of contexts such as market entry, technology adoption and peer effects, is challenging due to the presence of multiple equilibria. In this paper, we take a Bayesian MCMC approach to this problem, specifying a prior over multiple equilibrium selection mechanisms reflecting the analysts uncertainty over them. We develop a sampler, using the reversible jump algorithm to generate draws from the posterior distribution of parameters across these equilibrium selection rules. The algorithm is flexible in that it can be used both in situations where the equilibrium selection rule is identified and when it is not, and accommodates heterogeneity in equilibrium selection. We explore the methodology using both simulated data and 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 demonstrate the importance of accounting for multiple equilibrium selection rules in these applications and show that taking an empirical approach to the issue, such as the one we have demonstrated, can be useful.

Appendix: Details of the algorithm

The algorithm for our sampler has the following steps:

• 1. The current draw is (k, θ _k ).

• 2. Generate a candidate draw of model k′ from the multinomial distribution Multinomial (p) where p is also the vector of prior probabilities for the various models.

• 3. Using a set of moment matching conditions, generate g(θ _k ). This is done by minimizing the distance between the moments for the state (k, θ _k ) k and $(k^{\prime},g(\theta_{k}))$.

• 4. Numerically evaluate $\frac{\partial g\left(\theta_{k}\right)}{\partial\theta_{k}}$.

• 5. Generate a draw for u from the normal distribution N(0, Σ _u )

• 6. Set the candidate parameter vector $\theta_{k^{\prime}}^{\prime}=g(\theta_{k})+u$

• 7. Calculate the acceptance probability $\alpha_{t}\left(\left(k,\theta_{k}\right),\left(k^{\prime},\theta_{k^{\prime}}^{\prime}\right)\right)$.

• 8. Generate μ ∼ Uniform(0, 1)

• 9. If $\mu<\alpha_{t}\left(\left(k,\theta_{k}\right),\left(k^{\prime},\theta_{k^{\prime}}^{\prime}\right)\right)$ accept the candidate draw $(k,\theta_{k^{\prime}}^{\prime})$, else set the new draw to be equal to the previous one (k, θ _k ).

1.1 Tuning the reversible jump algorithm

One of the important choices to be made in practically implementing a Metropolis Hastings algorithm is the candidate density. In the case of a random walk algorithm with a normal candidate, this choice boils down to the choice of the variance covariance matrix for the candidate move. In principle, this choice does not matter as long as the density fulfills certain regularity conditions, but in practice this is a crucial choice. Intuitively, if the candidate moves to a very far away location from the current draw, it might be in a region where the target density is very low, and hence the move would get rejected. If the step size is too small, such that the candidate draw has a very similar target density as the current draw, the probability of acceptance becomes very high. In both cases, information is lost due to high autocorrelation of the draws. This means that it would take a very large number of draws to dissipate initial conditions, and to traverse the regions where the target density is non-negligible. Thus, a significant part of practical implementation of a Metropolis Hastings algorithm involves tuning the algorithm.

The issue of tuning assumes even greater importance in the case of the reversible jump algorithm. This is because the algorithm traverses parameter spaces across models, which are not entirely comparable. At an extreme, consider the case where g(θ _k ) = θ _k for all moves, including across model moves. In principle, such a sampler would work, but the rejection rate of candidate draws might be unacceptable high and may render the algorithm practically useless. This is because the candidate draws generating by taking a deviation from the current draw may have very low target density for the candidate model, making it almost certain that the candidate draw would be rejected. The algorithm would not make across-model moves in such a case. The moment-matching step in our algorithm reduces the chance of a prohibitively high rejection rate, by ensuring that the target densities for the current draw and the transformed draw (i.e. g(θ _k ) are at least at similar levels. But tuning is still a non-trivial part of making the reversible jump algorithm work. One challenge is that a small step size for one model might be a large one for another. Hence, it would be attractive to come up with an adaptive version of the algorithm that has some degree of ’automatic’ tuning as has been attempted for the standard Metropolis Hastings algorithm (Roberts and Rosenthal 2001). These adaptive algorithms adjust the variance of the candidate draw to keep the acceptance rate close to a pre-specified target level. The problem with this approach is that there is typically no clear criterion (i.e. no generalizable choice of target acceptance rate) that can be used for automating the tuning, particularly in the case of the reversible jump algorithm. Thus, one would have to use relatively arbitrary criterion (such as an arbitrarily chosen ’optimal’ acceptance rate) to adaptively adjust the tuning. An alternative approach in the case of the Metropolis Hastings algorithm is to use a proposal that utilizes information about the gradient of the target density to generate efficient candidate draws in situations where rejection rates of the standard random walk algorithm are prohibitively high. This underlies the so-called Langevin algorithm (Roberts and Tweedie 1996; Roberts and Rosenthal 2006), which relies on generating draws that satisfies two conditions. First, it ensures that the acceptance rates are equal to 1 when the candidate draw is equal to the current draw. Second, it ensures that the gradient of the acceptance rate with respect to the candidate is equal to zero when the candidate is the current draw. A similar Langevin-like approach was introduced by Brooks et al. (2003) for the reversible jump algorithm, by proposing a draw that ensures that the acceptance rate is equal to 1 when candidate is a transformed version of the current draw, and by equating the gradient of the acceptance rate to zero with respect to a transformed version of the current draw to 0, at the current draw. One of the costs of this approach is that it can be computationally expensive, particularly in the context where the transformation function g(θ _k ) cannot be computed analytically.

We have tried both approaches in our Monte Carlo simulations and found in our context that an automatic tuning approach worked well, although it comes at the cost of using an arbitrary criterion such as the one in Roberts and Rosenthal (2001). We find that the sampler mixes quite well in this case, with an acceptance rate ranging from about 10 to 25 %; We use this procedure in our empirical applications.

1.2 Assessing convergence

Assessing convergence of any MCMC sampler can be tricky and often depends on intuitive methods such as plotting the chains of the parameter draws and looking at measures such as autocorrelations. The issue of assessing convergence becomes much more complicated in the case of the reversible jump algorithm due to the across-model parameter space it samples from, and due to the difficulties in assessing convergence of the model indicator itself. One way to assess convergence of parameters is to see if there is good mixing of the parameters within each model. Depending on the dimensions of the parameter vector and the number of models involved, this can be an onerous task to do manually. An alternative is to compute a summary statistic or moment for each set of parameters that is invariant across models and assess convergence of that measure both within and across models (Brooks and Giudici 2001). Assessing convergence of the model indicators can also be tricky since it is a discrete variable. One indicator of convergence would be that successive blocks of draws have very similar proportions of indicators for the various models or to use more formal tests of convergence such as the hypothesis tests suggested by Brooks et al. (2003).