Azhar Iqbal
An example of a quantum game is presented that explicitly shows the impact of entanglement on the game-theoretical concept of evolutionary stability.
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In evolutionary game theory an Evolutionarily Stable Strategy (ESS) is a refinement of the Nash equilibrium concept that is sometimes also recognized as evolutionary stability. It is a game-theoretic model, well known to mathematical biologists, that was found quite useful in the understanding of evolutionary dynamics of a population. This chapter presents an analysis of evolutionary stability in the emerging field of quantum games.
Motivated by Buscemi's semi-quantum nonlocal game [PRL 108, 200401(2012)], We propose an entanglement witness game, a quantum game based on entanglement witness. Similar as the semi-quantum nonlocal game, the existence of entanglement shared by the players is necessary and sufficient for obtaining a positive average payoff in our entanglement witness game. Two explicit examples are constructed to demonstrate how to play our entanglement witness game and its relations with the CHSH nonlocal game.
We study a generalization of nonlocal games---which we call extended nonlocal games---in which the players, Alice and Bob, initially share a tripartite quantum state with the referee. In such games, the winning conditions for Alice and Bob may depend on outcomes of measurements made by the referee, on its part of the shared quantum state, in addition to Alice and Bob's answers to randomly selected questions. Our study of this class of games was inspired by the monogamy-of-entanglement games introduced by Tomamichel, Fehr, Kaniewski, and Wehner, which they also generalize. We prove that a natur…
We quantise the generalised Hawk-Dove Game. By restricting the strategy space available to the players, we show that every game of this type can be extended into the quantum realm to produce a Pareto optimal evolutionarily stable strategy. This equilibrium replaces the inefficient classical one when the entanglement prepared in the game exceeds a critical threshold value, which we derive analytically.
We investigate the 3-player quantum Prisoner's Dilemma with a certain strategic space, a particular Nash equilibrium that can remove the original dilemma is found. Based on this equilibrium, we show that the game is enhanced by the entanglement of its initial state.
The two-players N strategies games quantized according to the Eisert-Lewenstein-Wilkens scheme [1] are considered. It is shown that in the case of maximal entanglement no nontrivial pure Nash equilibrium exists. The proof relies on simple geometric properties of "chiral" group $SU (N)\times SU(N)$ and is based on considering the stability subgroup of the initial state of the game. The explicit forms of neither the gate operator nor the payoff matrix are necessary.
We give an explicit family of XOR games with O(n)-bit questions requiring 2^n ebits to play near-optimally. More generally we introduce a new technique for proving lower bounds on the amount of entanglement required by an XOR game: we show that near-optimal strategies for an XOR game G correspond to approximate representations of a certain C^*-algebra associated to G. Our results extend an earlier theorem of Tsirelson characterising the set of quantum strategies which implement extremal quantum correlations.
This paper unifies the concepts of evolutionary games and quantum strategies. First, we state the formulation and properties of classical evolutionary strategies, with focus on the destinations of evolution in 2-player 2-strategy games. We then introduce a new formalism of quantum evolutionary dynamics, and give an example where an evolving quantum strategy gives reward if played against its classical counterpart.
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.
We introduce a method of analyzing entanglement enhanced quantum games on regular lattices of agents. Our method is valid for setups with periodic and non-periodic boundary conditions. To demonstrate our approach we study two different types games, namely the prisoner's dilemma game and a cooperative Parrondo's game. In both cases we obtain results showing, that entanglement is a crucial resource necessary for the agents to achieve positive capital gain.
The last financial and economic crisis demonstrated the dysfunctional long-term effects of aggressive behaviour in financial markets. Yet, evolutionary game theory predicts that under the condition of strategic dependence a certain degree of aggressive behaviour remains within a given population of agents. However, as the consequences of the financial crisis exhibit, it would be desirable to change the 'rules of the game' in a way that prevents the occurrence of any aggressive behaviour and thereby also the danger of market crashes. The paper picks up this aspect. Through the extension of the …
A game-theoretic model of social preference and enlightened self-interest is formulated. Existence of symmetry and duality in the game matrices with altruistic social preference is revealed. The model is able to quantitatively describe the dynamical evolution of altruism in prisoner's dilemma and the regime change in prey-predator dynamics.
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