  We present the first study of a dynamical quantum game. Each agent has a
`memory' of her performance over the previous m timesteps, and her strategy can
evolve in time. The game exhibits distinct regimes of optimality. For small m
the classical game performs better, while for intermediate m the relative
performance depends on whether the source of qubits is `corrupt'. For large m,
the quantum players dramatically outperform the classical players by `freezing'
the game into high-performing attractors in which evolution ceases.
