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. The new field of quantum games is attracting significant interest [1][2][3][4]. We recently conjectured [5] that novel features should arise for quantum games with N ≥ 3 players. Benjamin and Hayden [6] subsequently created a Prisoner's Dilemma-like game for N = 3 with a highpayoff 'coherent quantum equilibrium' (CQE). Johnson [7] showed that this quantum advantage can become a disadvantage when the game's external qubit source is corrupted by a noisy 'demon'. So far, all studies have been restricted to static games. PACSHere we introduce an iterated version of the game, where players (agents) may modify their strategies based on information from the past -i.e. they 'learn' from their mistakes [8]. This evolutionary game produces highly non-trivial dynamics in both quantum and classical regimes, and represents the first step toward understanding iterated games employing temporal quantum coherence. Agents are provided with the minimum resources necessary for adaptability -specifically, each agent posesses a measure of her past success through a parameter $ m whose value reflects the payouts from recent rounds of the game. The fixed 'memory' parameter m effectively governs the number of rounds upon which $ depends (see later Eq. (1)), and turns out to be fundamental for deciding the relative superiority of the quantum and classical games. For small memory m, the classical game is superior (in the sense that the average payout per player is higher). For intermediate m, relative superiority is determined by the reliability of the external qubit source, while for large m the quantum game is radically superior due to evolutionary 'freezing' into a high-paying attractor. ( 1,-9,-9)( 1, 9, 9) ( 9, 9, 1) ( 9, 1, 9)

The Parrondo effect describes the seemingly paradoxical situation in which two losing games can, when combined, become a winning game [Parrondo, Harmer, and Abbott, Phys. Rev. Lett. 85, 24 (2000)]. Here, we generalize this analysis to the case where both games are history dependent, i.e., there is an intrinsic memory in the dynamics of each game. Results are presented for the cases of both random and periodic switching between the two games.

We present a detailed discussion of the role played by memory, and the nature of self-induced shocks, in an evolutionary population competing for limited resources. Our study builds on a previously introduced multi-agent system [Phys. Rev. Lett 82, 3360 (1999)] which has attracted significant attention in the literature. This system exhibits self-segregation of the population based on the 'gene' value p (where 0 ≤ p ≤ 1), transitions to 'frozen' populations as a function of the global resource level, and self-induced large changes which spontaneously arise as the dynamical system evolves. We find that the large, macroscopic self-induced shocks which arise, are controlled by microscopic changes within extreme subgroups of the population (i.e. subgroups with 'gene' values p ∼ 0 and p ∼ 1).

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