  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).
