  Learning problems commonly exhibit an interesting feedback mechanism wherein
the population data reacts to competing decision makers' actions. This paper
formulates a new game theoretic framework for this phenomenon, called
"multi-player performative prediction". We focus on two distinct solution
concepts, namely (i) performatively stable equilibria and (ii) Nash equilibria
of the game. The latter equilibria are arguably more informative, but can be
found efficiently only when the game is monotone. We show that under mild
assumptions, the performatively stable equilibria can be found efficiently by a
variety of algorithms, including repeated retraining and the repeated
(stochastic) gradient method. We then establish transparent sufficient
conditions for strong monotonicity of the game and use them to develop
algorithms for finding Nash equilibria. We investigate derivative free methods
and adaptive gradient algorithms wherein each player alternates between
learning a parametric description of their distribution and gradient steps on
the empirical risk. Synthetic and semi-synthetic numerical experiments
illustrate the results.
