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A Newton collocation method for solving dynamic bargaining games

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  • John Duggan
  • Tasos Kalandrakis

Abstract

We develop and implement a collocation method to solve for an equilibrium in the dynamic legislative bargaining game of Duggan and Kalandrakis (2008). We formulate the collocation equations in a quasi-discrete version of the model, and we show that the collocation equations are locally Lipchitz continuous and directionally differentiable. In numerical experiments, we successfully implement a globally convergent variant of Broyden's method on a preconditioned version of the collocation equations, and the method economizes on computation cost by more than 50% compared to the value iteration method. We rely on a continuity property of the equilibrium set to obtain increasingly precise approximations of solutions to the continuum model. We showcase these techniques with an illustration of the dynamic core convergence theorem of Duggan and Kalandrakis (2008) in a nine-player, two-dimensional model with negative quadratic preferences.
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  • John Duggan & Tasos Kalandrakis, 2011. "A Newton collocation method for solving dynamic bargaining games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 36(3), pages 611-650, April.
    • Handle: RePEc:spr:sochwe:v:36:y:2011:i:3:p:611-650
    DOI: 10.1007/s00355-010-0513-2
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    1. Rubinstein, Ariel, 1982. "Perfect Equilibrium in a Bargaining Model," Econometrica, Econometric Society, vol. 50(1), pages 97-109, January.
    2. Marco Battaglini & Stephen Coate, 2007. "Inefficiency in Legislative Policymaking: A Dynamic Analysis," American Economic Review, American Economic Association, vol. 97(1), pages 118-149, March.
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    4. Tasos Kalandrakis, 2007. "Majority Rule Dynamics with Endogenous Status Quo," Wallis Working Papers WP46, University of Rochester - Wallis Institute of Political Economy.
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    8. Pohan Fong, 2008. "Endogenous Limits on Proposal Power," Discussion Papers 1465, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    9. Mas-Colell,Andreu, 1990. "The Theory of General Economic Equilibrium," Cambridge Books, Cambridge University Press, number 9780521388702, October.
    10. Pohan Fong & Daniel Diermeier, 2008. "Endogenous Limits on Proposal Power," 2008 Meeting Papers 521, Society for Economic Dynamics.
    11. John Duggan & Tasos Kalandrakis & Vikram Manjunath, 2008. "Dynamics of the Presidential Veto: A Computational," Wallis Working Papers WP56, University of Rochester - Wallis Institute of Political Economy.
    12. Kalandrakis, Anastassios, 2004. "A three-player dynamic majoritarian bargaining game," Journal of Economic Theory, Elsevier, vol. 116(2), pages 294-322, June.
    13. Banks, Jeffrey S. & Duggan, John, 2006. "A General Bargaining Model of Legislative Policy-making," Quarterly Journal of Political Science, now publishers, vol. 1(1), pages 49-85, January.
    14. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, April.
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    Cited by:

    1. Anesi, Vincent & Duggan, John, 2018. "Existence and indeterminacy of markovian equilibria in dynamic bargaining games," Theoretical Economics, Econometric Society, vol. 13(2), May.
    2. Alessandro Riboni & Facundo Piguillem, 2011. "Dynamic Bargaining over Redistribution in Legislatures," 2011 Meeting Papers 1320, Society for Economic Dynamics.
    3. Facundo Piguillem & Alessandro Riboni, 2015. "Spending-Biased Legislators: Discipline Through Disagreement," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 130(2), pages 901-949.
    4. , & , J., 2014. "Bargaining over an endogenous agenda," Theoretical Economics, Econometric Society, vol. 9(2), May.
    5. Zapal, Jan, 2020. "Simple Markovian equilibria in dynamic spatial legislative bargaining," European Journal of Political Economy, Elsevier, vol. 63(C).
    6. Anesi, Vincent, 2010. "Noncooperative foundations of stable sets in voting games," Games and Economic Behavior, Elsevier, vol. 70(2), pages 488-493, November.
    7. Duggan, John & Kalandrakis, Tasos, 2012. "Dynamic legislative policy making," Journal of Economic Theory, Elsevier, vol. 147(5), pages 1653-1688.
    8. Tasos Kalandrakis, 2016. "Pareto efficiency in the dynamic one-dimensional bargaining model," Journal of Theoretical Politics, , vol. 28(4), pages 525-536, October.
    9. Hülya Eraslan & Kirill S. Evdokimov & Jan Zápal, 2022. "Dynamic Legislative Bargaining," Springer Books, in: Emin Karagözoğlu & Kyle B. Hyndman (ed.), Bargaining, chapter 0, pages 151-175, Springer.
    10. Graham, Brett & Bernhardt, Dan, 2015. "Flexibility vs. protection from an unrepresentative legislative majority," Games and Economic Behavior, Elsevier, vol. 93(C), pages 59-88.
    11. Anesi, Vincent & Seidmann, Daniel J., 2014. "Bargaining over an endogenous agenda," Theoretical Economics, Econometric Society, vol. 9(2), May.
    12. Vincent Anesi & Daniel J Seidmann, 2012. "Bargaining in Standing Committees," Discussion Papers 2012-09, The Centre for Decision Research and Experimental Economics, School of Economics, University of Nottingham.

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