Plan

In the lecture we review a new method for solution and structural estimation of dynamic stochastic games (DSGs) that deal with the fundamental problem of multiple equilibria. DSGs can, e.g., be used to model non-market interactions such as joint retirement decisions of spouses, and joint choice of residence and work location, as well as firms' choices of research and development, patent races, mergers, market entry and exit. They are increasingly used in antitrust cases and to develop policies to promote fair competition. In all these cases, the prediction of a DSG is an equilibrium, where each agent behaves optimally taking the behavior of other agents into account. Yet, the computational complexity and the existence of multiple equilibria put serious limits to inference using DSGs

Until recently, economists seemed content with finding a single equilibrium or proving that at least one equilibrium exists. Iskhakov, Rust and Schjerning (ReStud 2016) developed a new algorithm: recursive lexicographical search (RLS), which is guaranteed to find all ALL Markov perfect equilibria (MPE) for a specific class of games where states move in a directional manner. Using this algorithm we demonstrate deep problems of indeterminacy of economic models by finding millions of equilibria even in the simplest DSGs. The RLS algorithm is a computational breakthrough, but finding extremely many equilibria raises very serious concerns: How can we identify which equilibrium is observed in data, how can we determine which equilibrium plays out in actual situations? This is a deep problem of indeterminancy, which must be solved to make progress on DSG. I look forward to discuss this with you in class!

Having developed the RLS solution method, we then move on to estimating the dynamic game. Here I will present very recent work, where John Rust, Fedor Iskhakov and myself, develop a robust algorithm for computing the full solution maximum-likelihood estimator of a particular class of dynamic stochastic games with multiple equilibria, namely directional dynamic games. Our method allows for multiple equilibria having been played in data without making any assumptions on the equilibrium selection rule. It is based on Rust’s Nested Fixed Point estimator, but uses the Recursive Lexicographic Search (RLS) algorithm within the inner loop. Until recently, the NFXP approach have generally not been possible to implement for dynamic games with multiple equilibria since no algorithm was guaranteed to find all MPE. RLS provides this capability for directional dynamic games. We find that the full solution nested RLS estimator (NRLS) to be remarkably robust, computational fast and able to both obtain efficient MLE of the structural parameters and at the same time identify the equilibrium selection played in the data.

Application: Dynamics of Bertrand Price Competition with Cost-reducing Investments

Methods: Branch and Bound, RLS, NRLS

Preparation (in order of priority):

RLS algorithm: Iskhakov, Rust and Schjerning (ReStud, 2016)
NRLS: Iskhakov, Rust and Schjerning (Preliminary draft, 2020)
Leap-frogging paper: Iskhakov, Rust and Schjerning (IER, 2018)

Material:

Slides for RLS and Leapfrogging model (Video 1)
Slides for NRLS (NFXP combined with RLS) (Video 2)

Exercise (in MATLAB):

Solve leapfrogging model using RLS (Will not be covered in exercise class by Malene and will not be translated to Python, but for people who are intersted in RLS and games I thought this my be useful. I will answer questions regarding this at Piazza)

Problem set
MATLAB code to solve problem set

Video lectures:

Video 1: Recursive Lexicographic Search: Solving directional dynamic games for all Markov perfect equilibria

Video 2: Nested Recursive Lexicographical Search: Structural Estimation of Dynamic Directional Games with Multiple Equilibria