Evolutionary quantum game

Abstract

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)

Quantum Parrondo's games

“…If the entangling gate depends upon some parameter, then the classical game can be reproduced when this parameter is set to zero, representing no entanglement. In the present case this is problematic since the entangling gateĴ used by Eisert [4] and others [6,7,15,16,18] does not commute with the classical limit (all phases → 0) ofB, which was Eisert's motivation for the choice ofĴ. Thus this protocol would not reproduce the classical game when the phases are set to zero.…”

Quantum strategies win in a defector-dominated population

“…Marinatto et al [24] found a unique equilibrium for the Battle of the Sexes game, when entangled strategies were allowed. Later, evolutionarily stable strategies in quantum games and an evolutionary quantum game were also studied by Iqbal et al [25] and Kay et al [26] respectively. Moreover, quantum games have also been implemented using quantum computers [27,28,29].…”

Quantum entanglement, indistinguishability, and the absent-minded driver's problem

“…Hence, classically correlated physical systems (and specially anti-correlated systems) that could provide optimal expected pay-offs, are not allowed for decision-making. However, quantum states such as (4), (7), (8), (9), or those needed to implement all non-trivial vertices guarantee indistinguishability by fundamental laws of physics, while providing at the same time the desired correlations. According to quantum mechanics the quantum state of the N-qubit system is the most complete description of the system, in the sense that it provides the maximum possible information about any future experiments on these qubits.…”