Cooperative quantum Parrondo’s games

Royal Society open science, 2018

Parrondo's paradox is ubiquitous in games, ratchets and random walks. The apparent paradox, devised by J. M. R. Parrondo, that two losing gamesandcan produce a winning outcome has been adapted in many physical and biological systems to explain their working. However, proposals on demonstrating Parrondo's paradox using quantum walks failed for a large number of steps. In this work, we show that instead of a single coin if we consider a two-coin initial state which may or may not be entangled, we can observe a genuine Parrondo's paradox with quantum walks. Furthermore, we focus on reasons for this and pin down the asymmetry in initial two-coin state or asymmetry in shift operator, either of which is necessary for observing a genuine Parrondo's paradox. We extend our work to a three-coin initial state too with similar results. The implications of our work for observing quantum ratchet-like behaviour using quantum walks are also discussed.

arXiv: Quantum Physics, 2020

The well-known counterintuitive phenomenon, where the combination of unfavorable situations can establish favorable ones, is called Parrondo's paradox. Here, we study one-dimensional discrete-time quantum walks, manipulating two different coins (two-state) operators representing two losing games A and B, respectively, to create the Parrondo effect in the quantum domain. We exhibit that games A and B are losing games when played individually but could produce a winning expectation when played alternatively for a particular sequence of the different periods. Moreover, we also analyze the relationships between Parrondo's games and quantum entanglement in our scheme. Along with the applications of different kinds of quantum walks, our outcomes potentially encourage the development of new quantum algorithms.

2002

Parrondo’s Paradox arises when two losing games are combined to produce a winning one. A history dependent quantum Parrondo game is studied where the rotation operators that represent the toss of a classical biased coin are replaced by general SU(2) operators to transform the game into the quantum domain. In the initial state, a superposition of qubits can be used to couple the games and produce interference leading to quite different payoffs to those in the classical case. pacs: 03.67.-a, 02.50.le keywords: quantum games, Parrondo’s paradox

A scheme for implementing bi-partite iterated quantum games in terms of the discrete-time quantum walk on the line is introduced. Several popular strategies of classical Game Theory give rise to families of quantum strategies which can be implemented in terms of one and two-qubit quantum gates. Some of the possibilities introduced by this connection between the quantum walk and iterated quantum games are discussed.

2002

Parrondo’s Paradox arises when two losing games are combined to produce a winning one. A history dependent quantum Parrondo game is studied where the rotation operators that represent the toss of a classical biased coin are replaced by general SU(2) operators to transform the game into the quantum domain. In the initial state a superposition of qubits can be used to couple the games and produce interference leading to quite different payoffs to those in the classical case. pacs: 03.67.-a, 02.50.le keywords: quantum games, Parrondo’s paradox

We study the impact of adaptive degrees of trust on the evolution of cooperation in the quantum prisoner's dilemma game. In addition to the strategies, links between players are also subject to evolution. Starting with a scale-free interaction network, players adjust trust towards their neighbors based on received payoffs. The latter governs the strategy adoption process, while trust governs the rewiring of links. As soon as the degree of trust towards a neighbor drops to zero, the link is rewired to another randomly chosen player within the network. We find that for small temptations to defect cooperators always dominate, while for intermediate and strong temptations a single quantum strategy is able to outperform all other strategies. In general, reciprocal trust remains within close relationships and favors the dominance of a single strategy. Due to coevolution, the power-law degree distributions transform to Poisson distributions. C ooperation is surprisingly a commonly observed behavior in human societies as well as in many other biological systems. It is surprising because cooperation means sacrificing individual benefits for the good of others, which inevitably compromises fitness in natural selection. According to Darwin, cooperation should thus die out in a competitive setting. The survival of cooperation has therefore attracted ample attention, and it has indeed become a significant topic studied within the theoretical framework of evolutionary games. From Nowak and May's seminal work 1 onwards, interesting results 2–9 have been reported using different types of games. In particular, it has been discovered that network structure is a key factor in the evolution of cooperation, which leads to a shift from studying evolutionary games on lattices to studying evolutionary games on complex networks 10–17 (for a review see 18). Moreover, coevolutionary rules were introduced to evolutionary games, which meant that focus was shifting from the evolution on networks to the evolution of networks. For example, the impact of different link-updating rules on the coevolution of strategy and structure 19–26 , population growth 27,28 , teaching activity 29–31 , as well as the mobility of players 32–36 were studied. The concept of evolutionary games has also been applied on very small scales, for example to describe interactions of biological molecules 37–40 , which motivates the consideration of the effects of quantum mechanics 41. Consequently, a new field of quantum game theory has emerged as the generalization of classical game theory. Notably, several phenomena without classical counterparts have been observed. For example, in the PQ penny flip game 42 , a player who implements a quantum strategy can increase his/her expected payoff. Moreover, when one player is restricted to classical strategies, while the other player is free to adopt quantum strategies, the quantum player can direct the outcome regardless of the classical player's strategy 43. When the Prisoner's Dilemma game is quantized 44 , the dilemma existing in the classical game is removed, if both players resort to quantum strategies in a restricted space. Moreover, there is a unique Nash equilibrium for the Battle of the Sexes game, if players are allowed to use entangled strategies 45. It has also been shown that quantum games are more efficient than classical games and a saturated upper bound for this efficiency is found 46. Further, when the previous performance of agents are considered, evolutionary quantum games 47 are found to be an appropriate description. If there are a small group of mutants using quantum strategies in the population, the results of how they invade a classical evolutionarily stable strategy (ESS) 48 and the role of quantum mechanical effects in ESS 49 have been investigated. Later, quantum repeated games 50 , quantum cooperative games 51 and quantum correlation

Physica A-statistical Mechanics and Its Applications, 2008

Iterated bipartite quantum games are implemented in terms of the discrete-time quantum walk on the line. Our proposal allows for conditional strategies, as two rational agents make a choice from a restricted set of two qubit unitary operations. Several frequently used classical strategies give rise to families of corresponding quantum strategies. A quantum version of the Prisoner's Dilemma in which both players use mixed strategies is presented as a specific example. Since there are now quantum walk physical implementations at a proof-of principle stage, this connection may represent a step towards the experimental realization of quantum games.

We study the coevolution of quantum and classical strategies on weighted and directed random networks in the realm of the prisoner's dilemma game. During the evolution, agents can break and rewire their links with the aim of maximizing payoffs, and they can also adjust the weights to indicate preferences, either positive or negative, towards their neighbors. The network structure itself is thus also subject to evolution. Importantly, the directionality of links does not affect the accumulation of payoffs nor the strategy transfers, but serves only to designate the owner of each particular link and with it the right to adjust the link as needed. We show that quantum strategies outperform classical strategies, and that the critical temptation to defect at which cooperative behavior can be maintained rises, if the network structure is updated frequently. Punishing neighbors by reducing the weights of their links also plays an important role in maintaining cooperation under adverse conditions. We find that the self-organization of the initially random network structure, driven by the evolutionary competition between quantum and classical strategies, leads to the spontaneous emergence of small average path length and a large clustering coefficient.

Physics Letters A, 2002

We study two forms of a symmetric cooperative game played by three players, one classical and other quantum. In its classical form making a coalition gives advantage to players and they are motivated to do so. However in its quantum form the advantage is lost and players are left with no motivation to make a coalition.

In this paper, quantum strategies are introduced within evolutionary games in order to investigate 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 different 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.