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A Newton collocation method for solving dynamic bargaining games

Abstract

We develop and implement a collocation method to solve for an equilibrium in the dynamic legislative bargaining game of Duggan and Kalandrakis 2008, unpublished manuscript. We formulate the collocation equations in a quasi-discrete version of the model, and we show that they 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showcase these techniques with an illustration of the dynamic core convergence theorem of Duggan and Kalandrakis 2008, unpublished manuscript in a nine-player, two-dimensional model with negative quadratic preferences.

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Correspondence to John Duggan.

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Duggan, J., Kalandrakis, T. A Newton collocation method for solving dynamic bargaining games. Soc Choice Welf 36, 611–650 (2011). https://doi.org/10.1007/s00355-010-0513-2

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Keywords

  • Collocation Method
  • Ideal Point
  • Versus Versus Versus Versus Versus
  • Invariant Distribution
  • Policy Space