Dynamics and Coalitions in Sequential Games

TL;DR

This paper studies the dynamics of strategy updates in sequential multi-player games, analyzing conditions under which these dynamics stabilize to Nash or subgame perfect equilibria, including the effects of coalitions and preference structures.

Contribution

It introduces a new turn-based dynamics that terminates at subgame perfect equilibria and extends lazy improvement to coalition formation, characterizing termination conditions.

Findings

Turn-based dynamics always terminate at subgame perfect equilibria.

Coalition-based lazy improvement may not terminate in general.

Termination depends on players' preferences and game structure.

Abstract

We consider N-player non-zero sum games played on finite trees (i.e., sequential games), in which the players have the right to repeatedly update their respective strategies (for instance, to improve the outcome wrt to the current strategy profile). This generates a dynamics in the game which may eventually stabilise to a Nash Equilibrium (as with Kukushkin's lazy improvement), and we argue that it is interesting to study the conditions that guarantee such a dynamics to terminate. We build on the works of Le Roux and Pauly who have studied extensively one such dynamics, namely the Lazy Improvement Dynamics. We extend these works by first defining a turn-based dynamics, proving that it terminates on subgame perfect equilibria, and showing that several variants do not terminate. Second, we define a variant of Kukushkin's lazy improvement where the players may now form coalitions to change strategies. We show how properties of the players' preferences on the outcomes affect the termination of this dynamics, and we thereby characterise classes of games where it always terminates (in particular two-player games).