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

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  • Fedor Iskhakov
  • John Rust
  • Bertel Schjerning

Abstract

We define a class of dynamic Markovian games, directional dynamic games (DDG), where directionality is represented by a strategy-independent partial order on the state space. We show that many games are DDGs, yet none of the existing algorithms are guaranteed to find any Markov perfect equilibrium (MPE) of these games, much less all of them. We propose a fast and robust generalization of backward induction we call state recursion that operates on a decomposition of the overall DDG into a finite number of more tractable stage games, which can be solved recursively. We provide conditions under which state recursion finds at least one MPE of the overall DDG and introduce a recursive lexicographic search (RLS) algorithm that systematically and efficiently uses state recursion to find all MPE of the overall game in a finite number of steps. We apply RLS to find all MPE of a dynamic model of Bertrand price competition with cost-reducing investments which we show is a DDG. We provide an exact non-iterative algorithm that finds all MPE of every stage game, and prove there can be only 1, 3, or 5 of them. Using the stage games as building blocks, RLS rapidly finds and enumerates all MPE of the overall game. RLS finds a unique MPE for an alternating move version of the leapfrogging game when technology improves with probability 1, but in other cases, and in any simultaneous move version of the game, it finds a huge multiplicity of MPE that explode exponentially as the number of possible cost states increases.

Suggested Citation

  • Fedor Iskhakov & John Rust & Bertel Schjerning, 2016. "Recursive Lexicographical Search: Finding All Markov Perfect Equilibria of Finite State Directional Dynamic Games," Review of Economic Studies, Oxford University Press, vol. 83(2), pages 658-703.
  • Handle: RePEc:oup:restud:v:83:y:2016:i:2:p:658-703.
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    File URL: http://hdl.handle.net/10.1093/restud/rdv046
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    References listed on IDEAS

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    1. Kenneth L. Judd & Philipp Renner & Karl Schmedders, 2012. "Finding all pure‐strategy equilibria in games with continuous strategies," Quantitative Economics, Econometric Society, vol. 3(2), pages 289-331, July.
    2. Doraszelski, Ulrich & Escobar, Juan, 2010. "A theory of regular Markov perfect equilibria in dynamic stochastic games: genericity, stability, and purification," Theoretical Economics, Econometric Society, vol. 5(3), September.
    3. Richard Bellman, 1957. "On a Dynamic Programming Approach to the Caterer Problem--I," Management Science, INFORMS, vol. 3(3), pages 270-278, April.
    4. Ritzberger, Klaus, 2002. "Foundations of Non-Cooperative Game Theory," OUP Catalogue, Oxford University Press, number 9780199247868.
    5. Fedor Iskhakov & John Rust & Bertel Schjerning, 2018. "The Dynamics Of Bertrand Price Competition With Cost‐Reducing Investments," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 59(4), pages 1681-1731, November.
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    Cited by:

    1. Simon Quinn & Tom Gole, 2014. "Committees and Status Quo Bias: Structural Evidence from a Randomized Field Experiment," Economics Series Working Papers 733, University of Oxford, Department of Economics.
    2. Jaap H. Abbring & Jeffrey R. Campbell & Jan Tilly & Nan Yang, 2018. "Very Simple Markov‐Perfect Industry Dynamics: Theory," Econometrica, Econometric Society, vol. 86(2), pages 721-735, March.
    3. Prüfer, Jens & Schottmuller, C., 2017. "Competing with Big Data," Discussion Paper 2017-006, Tilburg University, Tilburg Law and Economic Center.
    4. John Rust, 2014. "The Limits of Inference with Theory: A Review of Wolpin (2013)," Journal of Economic Literature, American Economic Association, vol. 52(3), pages 820-850, September.
    5. Doraszelski, Ulrich & Escobar, Juan F., 2019. "Protocol invariance and the timing of decisions in dynamic games," Theoretical Economics, Econometric Society, vol. 14(2), May.
    6. Axel Anderson & Jeremy Rosen & John Rust & Kin-Ping Wong, 2021. "Disequilibrium Play in Tennis," Working Papers gueconwpa~21-21-07, Georgetown University, Department of Economics.
    7. José-Antonio Espín-Sánchez & Álvaro Parra & Yuzhou Wang, 2018. "Equilibrium Uniqueness in Entry Games with Private Information," Cowles Foundation Discussion Papers 2126R, Cowles Foundation for Research in Economics, Yale University, revised May 2021.
    8. Kenneth Gillingham & Fedor Iskhakov & Anders Munk-Nielsen & John Rust & Bertel Schjerning, 2019. "Equilibrium trade in automobile markets," CESifo Working Paper Series 7650, CESifo.
    9. Kenneth Gillingham & Fedor Iskhakov & Anders Munk-Nielsen & John Rust & Bertel Schjerning, 2019. "Equilibrium trade in automobile markets," CESifo Working Paper Series 7650, CESifo.

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    JEL classification:

    • D92 - Microeconomics - - Micro-Based Behavioral Economics - - - Intertemporal Firm Choice, Investment, Capacity, and Financing
    • L11 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Production, Pricing, and Market Structure; Size Distribution of Firms
    • L13 - Industrial Organization - - Market Structure, Firm Strategy, and Market Performance - - - Oligopoly and Other Imperfect Markets

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