A. H. Toor

Quantum mechanics gives stability to a Nash equilibrium

We consider a slightly modified version of the Rock-Scissors-Paper (RSP) game from the point of view of evolutionary stability. In its classical version the game has a mixed Nash equilibrium (NE) not stable against mutants. We find a quantized version of the RSP game for which the classical mixed NE becomes stable.

Generalized quantization scheme for two-person non-zero sum games

We proposed a generalized quantization scheme for non-zero sum games which can be reduced to the two existing quantization schemes under an appropriate set of parameters. Some other important situations are identified which are not apparent in the two existing quantization schemes.

Quantum cooperative games

Abstract 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.

Backwards-induction outcome in a quantum game.

In economics duopoly is a market dominated by two firms large enough to influence the market price. Stackelberg presented a dynamic form of duopoly that is also called ‘leader-follower’ model. We give a quantum perspective on Stackelberg duopoly that gives a backwards-induction outcome same as the Nash equilibrium in static form of duopoly also known as Cournot’s duopoly. We find two qubit quantum pure states required for this purpose.

Dilemma and quantum battle of sexes

We analysed quantum version of the game battle of sexes using a general initial quantum state. For a particular choice of initial entangled quantum state it is shown that the classical dilemma of the battle of sexes can be resolved and a unique solution of the game can be obtained.

Darwinism in quantum systems

We investigate the role of quantum mechanical effects in the central stability concept of evolutionary game theory i.e. an Evolutionarily Stable Strategy (ESS). Using two and three-player symmetric quantum games we show how the presence of quantum phenomenon of entanglement can be crucial to decide the course of evolutionary dynamics in a population of interacting individuals.Abstract We investigate the role of quantum mechanical effects in the central stability concept of evolutionary game theory, i.e., an evolutionarily stable strategy (ESS). Using two and three-player symmetric quantum games we show how the presence of quantum phenomenon of entanglement can be crucial to decide the course of evolutionary dynamics in a population of interacting individuals.

Quantum games with correlated noise

We analyse quantum games with correlated noise through a generalized quantization scheme. Four different combinations on the basis of entanglement of initial quantum state and the measurement basis are analysed. It is shown that the quantum player only enjoys an advantage over the classical player when both the initial quantum state and the measurement basis are in entangled form. Furthermore, it is shown that for maximum correlation the effects of decoherence diminish and it behaves as a noiseless game.

Entanglement and dynamic stability of Nash equilibria in a symmetric quantum game

Abstract We study the evolutionary stability of Nash equilibria (NE) in a symmetric quantum game played by the recently proposed scheme of applying ‘identity’ and ‘Pauli spin-flip’ operators on an initial state with classical probabilities. We show that in this symmetric game dynamic stability of a NE can be changed when the game changes its form, for example, from classical to quantum. It happens even when the NE remains intact in both forms.

The effect of quantum memory on quantum games

We study quantum games with correlated noise through a generalized quantization scheme. We investigate the effects of memory on quantum games, such as Prisoners Dilemma, Battle of the Sexes and Chicken, through three prototype quantum-correlated channels. It is shown that the quantum player enjoys an advantage over the classical player for all nine cases considered in this paper for the maximally entangled case. However, the quantum player can also outperform the classical player for subsequent cases that can be noted in the case of the Battle of the Sexes game. It can be seen that the Nash equilibria do not change for all the three games under the effect of memory.

Evolutionarily Stable Strategies in Quantum Hawk-Dove Game

We quantize the Hawk–Dove game by using the most general form of a pure initial state to investigate the existence of pure and mixed evolutionarily stable strategies (ESS). An example is considered to draw a comparison between the classical and quantum version of the game. Our choice of the most general initial quantum state enables us to make the game symmetric or asymmetric. We show that for a particular set of game parameters where there exists only mixed ESS in the classical version of the game, quantization allows even a pure strategy to be an ESS for the symmetric game in addition to mixed ESS. On the other hand only pure strategy ESS can exist for the asymmetric quantum version of the Hawk–Dove game.