Recursive Lexicographical Search: Finding All Markov Perfect Equilibria of Finite State Directional Dynamic Games

Iskhakov, Fedor; Rust, John; Schjerning, Bertel

doi:10.1093/restud/rdv046

Cited by 22 publications

(26 citation statements)

References 37 publications

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“…In fact, we show, via numerical calculation of all MPE of example games using the recursive lexicographical search (RLS) algorithm of Iskhakov et al. (), that various types of leapfrogging equilibria are the typical outcome of the Bertrand investment game.…”

Section: Introductionmentioning

confidence: 94%

“…This implies that the Bertrand investment game is in the class of directional dynamic games (DDGs) defined in Iskhakov et al. (). This article introduced the RLS algorithm that is guaranteed to find all MPE of finite state DDGs provided certain conditions hold.…”

Section: The Modelmentioning

confidence: 99%

“…Iskhakov et al. () show that both of these conditions hold in the finite state version of the leapfrogging game, so the RLS algorithm can be used to find all MPE of the Bertrand investment game. In particular, Iskhakov et al.…”

Section: The Modelmentioning

confidence: 99%

“…In particular, Iskhakov et al. () develop a noniterative combinatorial algorithm based on the roots of second degree polynomial that finds all of the 1, 3, or 5 possible MPE of every stage game of the simultaneous move version of the investment game, and the 1 or 3 possible MPE of every stage game of the alternating move version.…”

Section: The Modelmentioning

confidence: 99%

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The Dynamics of Bertrand Price Competition With Cost‐reducing Investments

Iskhakov

Rust

Schjerning

2018

Int Economic Review

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Abstract:We extend the classic Bertrand duopoly model of price competition to a dynamic setting where competing duopolists invest in a stochastically improving production technology to "leapfrog" their rival and attain temporary low cost leadership. We find a huge multiplicity of Markov perfect equilibria (MPE) and show that when firms move simultaneously the set of all MPE payoffs is a triangle that includes monopoly payoffs and a symmetric zero mixed strategy payoff. When firms move asynchronously, the set of MPE payoffs is strictly within this triangle, but there still is a vast multiplicity of MPE, most of which involve leapfrogging.

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Section: Introductionmentioning

confidence: 94%

Section: The Modelmentioning

confidence: 99%

Section: The Modelmentioning

confidence: 99%

Section: The Modelmentioning

confidence: 99%

See 2 more Smart Citations

The Dynamics of Bertrand Price Competition With Cost‐reducing Investments

Iskhakov

Rust

Schjerning

2018

Int Economic Review

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“…Cabral and Riordan, 1994;Judd, Schmedders, and Yeltekin, 2012). Iskhakov, Rust, and Schjerning (2016) systemized this familiar procedure into an algorithm for computing all these games' equilibria. In the games considered, the state space can be partially ordered using primitive restrictions on state transitions: State B comes after state A if B can be reached from A but not the other way around.…”

Section: Introductionmentioning

confidence: 99%

Very Simple Markov-Perfect Industry Dynamics: Theory

et al. 2017

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This paper develops a simple model of firm entry, competition, and exit in oligopolistic markets. It features toughness of competition, sunk entry costs, and market-level demand and cost shocks, but assumes that firms' expected payoffs are identical when entry and survival decisions are made.We prove that this model has an essentially unique symmetric Markov-perfect equilibrium, and we provide an algorithm for its computation. Because this algorithm only requires finding the fixed points of a finite sequence of contraction mappings, it is guaranteed to converge quickly.

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Dynamic Programming, Numerical

Rust

2017

Wiley StatsRef: Statistics Reference Online

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This article reviews a large literature on numerical methods for finding approximate optimal or equilibrium solutions to sequential decision processes and dynamic games using the technique of dynamic programming, the name Bellman gave to a recursive procedure for solving complex decision problems through the process of backward induction. The main challenge is a problem Bellman called the curse of dimensionality , which is the exponential increase in computer power needed to solve increasingly complex, realistic dynamic programs. The article reviews the literature on information‐based complexity that has enabled computer scientists to rigorously formalize the curse of dimensionality in terms of worst case complexity bounds that increase exponentially in the number of continuous state and control variables entering the dynamic programming problem. For the most general classes of dynamic programming problems, this literature demonstrates that the curse of dimensionality is a fundamental problem that cannot be circumvented by any algorithm, no matter how clever. However, there are subclasses of dynamic programming problems that have special structure that can be exploited to enable us to break the curse of dimensionality in some cases, sometimes via randomized algorithms similar to Monte Carlo integration. The disadvantage of randomized algorithms is the stochastic noise in the approximate solutions, although this “noise” can be made as small as desired at the cost of higher computational effort. The article reviews a large literature on deterministic algorithms for solving finite and infinite horizon dynamic programming problems that are used in practice to provide accurate solutions to low‐to‐moderate dimensional problems. The article reviews a more recent literature called approximate dynamic programming (or neurodynamic programming ) that have been proposed to approximately solve more challenging high‐dimensional problems. These are typically iterative, stochastic methods that are inspired from the literature on artificial intelligence and reinforcement learning that include a method called Q‐learning , which can also be described as “model free” as it relies on methods of “training” from the reinforcement learning literature that only require simulated realizations of the controlled stochastic process rather than an explicit representation of this process (as a transition probability, for example). These methods were originally developed for problems with finite state and action spaces but have been extended to problems with continuous state and action spaces (or very many finite states and actions) using function approximation and nonlinear regression techniques including neural networks, hence the name neurodynamic programming. Although the analysis of the convergence of these types of methods to the true solution is incomplete and computable bounds between approximate and true solutions are often unavailable, there have been some impressive successful applications of these techniques including most recently Google's development of the AlphaGo program, which has defeated the world's best human player in the game of Go, a complex game whose number of possible board positions exceeds the number of atoms in the universe.

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Recursive Lexicographical Search: Finding All Markov Perfect Equilibria of Finite State Directional Dynamic Games

Cited by 22 publications

References 37 publications

The Dynamics of Bertrand Price Competition With Cost‐reducing Investments

The Dynamics of Bertrand Price Competition With Cost‐reducing Investments

Very Simple Markov-Perfect Industry Dynamics: Theory

Dynamic Programming, Numerical

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