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Proximal Approach in Selection of Subgame Perfect Nash Equilibria

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In one-leader one-follower two-stage games, multiplicity of Subgame Perfect Nash Equilibria (henceforth SPNE) arises when the optimal reaction of the follower to any choice of the leader is not always unique, i.e. when the best reply correspondence of the follower is not a single-valued map. This paper concerns a new selection method for SPNE which makes use of a sequence of games designed using a proximal point algorithm, well-known optimization technique related to the so-called Moreau-Yosida regularization (Moreau 1965, Martinet 1972, Rockafellar 1976, Parikh and Boyd 2014 and references therein). Any game of the obtained sequence is a classical Stackelberg game (Von Stack- elberg 1952), i.e. a one-leader one-follower two-stage game where the best reply correspondence of the follower is single-valued. This mechanism selection is in line with a previous one based on Tikhonov regularization, in Morgan and Patrone (2006), but using the class of proximal point algorithms has a twofold advantage: on the one hand, it can provide improvements in numerical implementations and, on the other hand, it has a clear interpretation: the follower payoff function is modified subtracting a term that can represent a physical and behavioural cost to move (Attouch and Soubeyran 2009). The constructive method and its effectiveness are illustrated and existence results for the selection are provided under mild assumptions on data, together with connections with other possible selection methods.

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Paper provided by Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy in its series CSEF Working Papers with number 476.

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Date of creation: 21 Apr 2017
Handle: RePEc:sef:csefwp:476
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  1. M. Beatrice Lignola & Jacqueline Morgan, 2014. "Viscosity Solutions for Bilevel Problems with Nash Equilibrium Constraints," CSEF Working Papers 367, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy, revised 02 Oct 2014.
  2. Benoît Colson & Patrice Marcotte & Gilles Savard, 2007. "An overview of bilevel optimization," Annals of Operations Research, Springer, vol. 153(1), pages 235-256, September.
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