  We establish that algorithmic experiments in zero-sum games "fail miserably"
to confirm the unique, sharp prediction of maxmin equilibration. Contradicting
nearly a century of economic thought that treats zero-sum games nearly
axiomatically as the exemplar symbol of economic stability, we prove that no
meaningful prediction can be made about the day-to-day behavior of online
learning 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 playing
Matching-Pennies whether they prefer Heads or Tails (and by how much in terms
of aggregate payoff so far). The range of possible answers consistent with any
arbitrary small set of initial conditions blows up exponentially with time
everywhere in the payoff space. This result is robust both algorithmically as
well as game theoretically:
  1) Algorithmic robustness: Chaos is robust to agents using any of a general
sub-family of Follow-the-Regularized-Leader (FTRL) algorithms, the well known
regret-minimizing dynamics, even when agents mix-and-match dynamics, use
different or slowly decreasing step-sizes.
  2) Game theoretic robustness: Chaos is robust to all affine variants of
zero-sum games (strictly competitive games), network variants with arbitrary
large number of agents and even to competitive settings beyond these.
  Our result is in stark contrast with the time-average convergence of online
learning to (approximate) Nash equilibrium, a result widely reported as "(weak)
convergence to equilibrium".
