Abstract
This paper outlines a Bayesian approach to estimating discrete games of incomplete information. The MCMC routine proposed features two changes to the traditional Metropolis–Hastings algorithm to facilitate the estimation of games. First, we propose a new approach to sample equilibrium probabilities using a probabilistic equilibrium selection rule that allows for the evaluation of the parameter posterior. Second, we propose a differential evolution based MCMC sampler which is capable of handling the unwieldy posterior that only has support on the equilibrium manifold. We also present two applications to demonstrate the feasibility of our proposed methodology.
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Notes
The probability space ℙ is the product of N K −unit cubes and is defined as \(\mathbb{P=}\prod\limits_{i=1}^{N}\mathbb{P}^{K}\), where
$$ \mathbb{P}^{K}=\big\{ \left( p_{1},...,p_{K}\right) \in \mathbb{R} ^{K}|\Sigma _{k=1}^{K}p_{k}=1\text{ and }p_{k}\geq 0\text{ for all }k\big\} $$\(\mathbb{S}\) and Θ are the state and parameter spaces respectively.
There are ongoing attempts to find all equilibria using Homotopy methods (Bajari et al. 2011) that appear promising albeit computationally expensive.
This is not required. We could just as easily alter the definition of \( \mathcal{D}\left( p_{m}^{e}\right) \) to be the Newtonian basin of attraction and use Newton’s root solving method to generate candidate equilibria. As long as every equilibria can be generated with positive probability our MCMC routine will converge.
One may opt to use other more efficient iterative methods to speed up convergence here. In our applications we use Mann iterations which seem to work well. Regular Picard iterations also gave essentially identical results but in some cases took a long time to converge. The reader is referred to the excellent book by Berinde (2007) for a review of Mann, Picard and other fixed point iteration methods and their properties.
Suppose we wanted to draw from a multinomial with probability \(\pi =\left( \frac{1}{2},\frac{1}{4},\frac{1}{4}\right)\). All we would need are weights \( \left( \text{say }w=\left( 50,25,25\right) \right) \) that are proportional to π and the knowledge that the maximum weight is bounded above by some value (say \(\bar{V}=100\)). Then by drawing an index uniformly \(\left( j{\kern1pt}\right) \) and accepting the draw only if \(w_{j}>U\left[ 0,\bar{V}\right] \) we will get draws exactly proportional to the requisite probabilities. In our case the indices are the equilibria that are generated non-uniformly (based on their domain of attraction). An R version of this algorithm for multinomial draws is available from the author upon request.
For more details on the data, the definition of markets etc. the reader is refered to Ellickson and Misra (2008).
For example, a simple two step estimator might offer reasonable starting values.
As a frame of reference, a traditional MH routine resulted in a <2 % acceptance rate after 200,000 iterations.
Standard convergence diagnostices such as R-statistic of Gelman and Rubin (1992) also confirm this. Trace plots and convergence diagnostics are available from the authors upon request.
During the time period of this data Target was not a major player.
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Misra, S. Markov chain Monte Carlo for incomplete information discrete games. Quant Mark Econ 11, 117–153 (2013). https://doi.org/10.1007/s11129-012-9128-5
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DOI: https://doi.org/10.1007/s11129-012-9128-5