Evolutionary Game Theory Squared: Evolving Agents in Endogenously Evolving Zero-Sum Games

The predominant paradigm in evolutionary game theory and more generally online learning games is based on a clear distinction between population of dynamic agents that interact given fixed, static game. In this paper, we move away from the artificial divide games, to introduce analyze large class competitive settings where both they play evolve strategically over time. We focus arguably most archetypal game-theoretic setting---zero-sum (as well as network generalizations)---and studied dynamic---replicator, continuous-time analogue multiplicative weights. Populations compete against each other zero-sum competition itself evolves adversarially current mixture. Remarkably, despite chaotic coevolution prove system exhibits number regularities. First, has conservation laws an information-theoretic flavor couple behavior all games. Secondly, Poincare recurrent, with effectively possible initializations lying recurrent orbits come arbitrarily close their initial conditions infinitely often. Thirdly, time-average agent utility converge Nash equilibrium values Finally, provide polynomial time algorithm efficiently predict for any such coevolving