Eur. Phys. J. B (2012) 85: 376 DOI: 10.1140/epjb/e2012-30398-1 Regular Article T HE EUROPEAN P HYSICAL JOURNAL B Evolution of quantum strategies on a small-world network Q. Li 1,2, a , A. Iqbal 2 , M. Chen 1 , and D. Abbott 2 1 College of Electrical Engineering, Chongqing University, Chongqing 400030, P.R. China 2 School of Electrical and Electronic Engineering, University of Adelaide, SA 5005, Australia Received 19 May 2012 / Received in final form 23 September 2012 Published online 19 November 2012 – c  EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2012 Abstract. In this paper, quantum strategies are introduced within evolutionary games in order to inves- tigate the evolution of quantum strategies on a small-world network. Initially, certain quantum strategies are taken from the full quantum space at random and assigned to the agents who occupy the nodes of the network. Then, they play n-person quantum games with their neighbors according to the physical model of a quantum game. After the games are repeated a large number of times, a quantum strategy becomes the dominant strategy in the population, which is played by the majority of agents. However, if the number of strategies is increased, while the total number of agents remains constant, the dominant strategy almost disappears in the population because of an adverse environment, such as low fractions of agents with dif- ferent strategies. On the contrary, if the total number of agents rises with the increase of the number of strategies, the dominant strategy re-emerges in the population. In addition, from results of the evolution, it can be found that the fractions of agents with the dominant strategy in the population decrease with the increase of the number of agents n in a n-person game independent of which game is employed. If both classical and quantum strategies evolve on the network, a quantum strategy can outperform the classical ones and prevail in the population. 1 Introduction Recently, the evolution of agents’ behavior in a popula- tion, in the framework of the evolutionary games on net- works, has attracted much interdisciplinary attention. As complex network theory developed, a shift from evolu- tionary games on regular lattices to evolutionary games on complex networks was observed [1], in particular on small world networks [2–7]. Moreover, the two-person Prisoner’s Dilemma (PD) and Snowdrift (SD) games have been widely studied. However, some situations such as collective actions of groups of individuals cannot be ab- stracted appropriately by two-person games. Therefore, n-person games as the generalization of two-person games offer new models for study of the collective behavior of interacting agents. For a n × m game, each of n agents chooses a strategy from m strategies simultaneously, and then each receives a payoff according to a payoff ma- trix of the game. Interesting results have been obtained for n-person games [8–11]. For example, Eriksson and Lindgren [8] investigated the cooperation driven by mu- tations when n-person PD games were employed; Chan et al. [9] reported results regarding the evolution of coop- eration in a well-mixed population performing n-person SD games. Surprisingly, the concept of evolutionary games has been extended to the microworld to describe interactions a e-mail: anjuh@cqu.edu.cn of biological molecules [12–15], a domain in which quan- tum mechanics defines the laws. In recent years, the new field of quantum game theory has emerged as the gener- alization of classical game theory. Due to quantum effects involved, quantum games exhibit new features that have no classical counterparts and it opens up new lines of re- search. For example, Meyer [16] first quantized the PQ penny flip game. His results showed that when an agent implements quantum strategy against the opponent’s clas- sical strategy, she/he can always defeat her/him and can thus increase her/his expected payoff. Eisert et al. [17] in- troduced an elegant scheme to quantize the PD game and demonstrated that the dilemma in the classical PD could be escaped when both agents resort to quantum strategies. Marinatto and Weber [18] gave a quantum model of the Battle of the Sexes game and found a unique Nash equi- librium (NE) for this game, when the entangled strategies were allowed, whereas the classical game has two NEs. Later, Iqbal and Toor [19] studied evolutionarily stable strategies in quantum games and Kay et al. [20] pre- sented an evolutionary quantum game. Moreover, quan- tum games were implemented using quantum comput- ers [21–23] and some related researches have also been performed [24–28]. For further background on quantum games, see references [29,30]. Additionally, in order to study the evolving behavior of agents in a population, a framework is often used [2–6], in which agents in the population are regarded as nodes