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We establish that algorithmic experiments in zero-sum games "fail miserably"to confirm the unique, sharp prediction of maxmin equilibration. Contradictingnearly a century of economic thought that treats zero-sum games nearlyaxiomatically as the exemplar symbol of economic stability, we prove that nomeaningful prediction can be made about the day-to-day behavior of onlinelearning dynamics in zero-sum games. Concretely, Multiplicative Weights Updates(MWU) with constant step-size is Lyapunov chaotic in the dual (payoff) space.Simply put, let's assume that an observer asks the agents playingMatching-Pennies whether they prefer Heads or Tails (and by how much in termsof aggregate payoff so far). The range of possible answers consistent with anyarbitrary small set of initial conditions blows up exponentially with timeeverywhere in the payoff space. This result is robust both algorithmically aswell as game theoretically:  1) Algorithmic robustness: Chaos is robust to agents using any of a generalsub-family of Follow-the-Regularized-Leader (FTRL) algorithms, the well knownregret-minimizing dynamics, even when agents mix-and-match dynamics, usedifferent or slowly decreasing step-sizes.  2) Game theoretic robustness: Chaos is robust to all affine variants ofzero-sum games (strictly competitive games), network variants with arbitrarylarge number of agents and even to competitive settings beyond these.  Our result is in stark contrast with the time-average convergence of onlinelearning to (approximate) Nash equilibrium, a result widely reported as "(weak)convergence to equilibrium".

Vortices Instead of Equilibria in MinMax Optimization: Chaos and Butterfly Effects of Online Learning in Zero-Sum Games