Proximal approach in selection of subgame perfect Nash equilibria

Francesco Caruso, Maria Carmela Ceparano and Jacqueline Morgan

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 presentation 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 Stackelberg 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.