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A Newton collocation method for solving dynamic bargaining games


  • John Duggan


  • Tasos Kalandrakis


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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Suggested Citation

  • 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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    References listed on IDEAS

    1. Rubinstein, Ariel, 1982. "Perfect Equilibrium in a Bargaining Model," Econometrica, Econometric Society, vol. 50(1), pages 97-109, January.
    2. Mas-Colell,Andreu, 1990. "The Theory of General Economic Equilibrium," Cambridge Books, Cambridge University Press, number 9780521388702, March.
    3. 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.
    4. repec:cup:apsrev:v:83:y:1989:i:04:p:1181-1206_08 is not listed on IDEAS
    5. 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.
    6. Tasos Kalandrakis, 2007. "Majority Rule Dynamics with Endogenous Status Quo," Wallis Working Papers WP46, University of Rochester - Wallis Institute of Political Economy.
    7. Duggan, John & Kalandrakis, Tasos, 2012. "Dynamic legislative policy making," Journal of Economic Theory, Elsevier, vol. 147(5), pages 1653-1688.
    8. Daron Acemoglu & Georgy Egorov & Konstantin Sonin, 2012. "Dynamics and Stability of Constitutions, Coalitions, and Clubs," American Economic Review, American Economic Association, vol. 102(4), pages 1446-1476, June.
    9. Liqun Qi, 1993. "Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 227-244, February.
    10. Pohan Fong, 2008. "Endogenous Limits on Proposal Power," Discussion Papers 1465, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    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, 2000. "A Bargaining Model of Collective Choice," American Political Science Review, Cambridge University Press, vol. 94(01), pages 73-88, March.
    14. repec:cup:apsrev:v:90:y:1996:i:02:p:316-330_20 is not listed on IDEAS
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    Cited by:

    1. repec:the:publsh:2215 is not listed on IDEAS
    2. Alessandro Riboni & Facundo Piguillem, 2011. "Dynamic Bargaining over Redistribution in Legislatures," 2011 Meeting Papers 1320, Society for Economic Dynamics.
    3. Anesi, Vincent & Duggan, John, 0. "Existence and indeterminacy of markovian equilibria in dynamic bargaining games," Theoretical Economics, Econometric Society.
    4. Anesi, Vincent, 2010. "Noncooperative foundations of stable sets in voting games," Games and Economic Behavior, Elsevier, vol. 70(2), pages 488-493, November.
    5. Facundo Piguillem & Alessandro Riboni, 2015. "Spending-Biased Legislators: Discipline Through Disagreement," The Quarterly Journal of Economics, Oxford University Press, vol. 130(2), pages 901-949.
    6. Anesi, Vincent & Seidmann, Daniel J., 2014. "Bargaining over an endogenous agenda," Theoretical Economics, Econometric Society, vol. 9(2), May.
    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. Jan Zapal, 2014. "Simple Markovian Equilibria in Dynamic Spatial Legislative Bargaining," CERGE-EI Working Papers wp515, The Center for Economic Research and Graduate Education - Economics Institute, Prague.
    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.

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