An introduction to quantum game theory

The application of the methods of quantum mechanics to game theory provides us with the ability to achieve results not otherwise possible. Both linear superpositions of actions and entanglement between the players' moves can be exploited. We provide an introduction to quantum game theory and review the current status of the subject.Comment: 8 pages, RevTeX; v2 minor changes to the text in light of referees comments, references added/update

Advantage of a quantum player over a classical one in 2x2 quantum games

We study a general $2 \times 2$ symmetric, entangled, quantum game. When one player has access only to classical strategies while the other can use the full range of quantum strategies, there are ``miracle'' moves available to the quantum player that can direct the result of the game towards the quantum player's preferred result regardless of the classical player's strategy. The advantage pertaining to the quantum player is dependent on the degree of entanglement. Below a critical level, dependent on the payoffs in the game, the miracle move is of no advantage.Comment: Revtex, 10 pages, 2 tables, 4 figures; v2 typo corrected in table 2, cosmetic changes to tables and figures, comment added to section VI E; v3 title changed to published title; minor mathematical errors in published version correcte

Signal acquisition via polarization modulation in single photon sources

A simple model system is introduced for demonstrating how a single photon source might be used to transduce classical analog information. The theoretical scheme results in measurements of analog source samples that are (i) quantized in the sense of analog-to-digital conversion and (ii) corrupted by random noise that is solely due to the quantum uncertainty in detecting the polarization state of each photon. This noise is unavoidable if more than one bit per sample is to be transmitted, and we show how it may be exploited in a manner inspired by suprathreshold stochastic resonance. The system is analyzed information theoretically, as it can be modeled as a noisy optical communication channel, although unlike classical Poisson channels, the detector's photon statistics are binomial. Previous results on binomial channels are adapted to demonstrate numerically that the classical information capacity, and thus the accuracy of the transduction, increases logarithmically with the square root of the number of photons, N. Although the capacity is shown to be reduced when an additional detector nonideality is present, the logarithmic increase with N remains.Comment: 7 pages, 2 figures, accepted by Physical Review E. This version adds a referenc

Nash equilibria in quantum games with generalized two-parameter strategies

In the Eisert protocol for 2 X 2 quantum games [Phys. Rev. Lett. 83, 3077], a number of authors have investigated the features arising from making the strategic space a two-parameter subset of single qubit unitary operators. We argue that the new Nash equilibria and the classical-quantum transitions that occur are simply an artifact of the particular strategy space chosen. By choosing a different, but equally plausible, two-parameter strategic space we show that different Nash equilibria with different classical-quantum transitions can arise. We generalize the two-parameter strategies and also consider these strategies in a multiplayer setting.Comment: 19 pages, 2 eps figure

Coalitions in the quantum Minority game: classical cheats and quantum bullies

In a one-off Minority game, when a group of players agree to collaborate they gain an advantage over the remaining players. We consider the advantage obtained in a quantum Minority game by a coalition sharing an initially entangled state versus that obtained by a coalition that uses classical communication to arrive at an optimal group strategy. In a model of the quantum Minority game where the final measurement basis is randomized, quantum coalitions outperform classical ones when carried out by up to four players, but an unrestricted amount of classical communication is better for larger coalition sizes.Comment: 12 pages, 1 figur

Experimental implementation of a four-player quantum game

Game theory is central to the understanding of competitive interactions arising in many fields, from the social and physical sciences to economics. Recently, as the definition of information is generalized to include entangled quantum systems, quantum game theory has emerged as a framework for understanding the competitive flow of quantum information. Up till now only two-player quantum games have been demonstrated. Here we report the first experiment that implements a four-player quantum Minority game over tunable four-partite entangled states encoded in the polarization of single photons. Experimental application of appropriate quantum player strategies give equilibrium payoff values well above those achievable in the classical game. These results are in excellent quantitative agreement with our theoretical analysis of the symmetric Pareto optimal strategies. Our result demonstrate for the first time how non-trivial equilibria can arise in a competitive situation involving quantum agents and pave the way for a range of quantum transaction applications.Comment: 9 pages, 5 figure

Equivalence between Bell inequalities and quantum Minority game

We show that, for a continuous set of entangled four-partite states, the task of maximizing the payoff in the symmetric-strategy four-player quantum Minority game is equivalent to maximizing the violation of a four-particle Bell inequality with each observer choosing the same set of two dichotomic observables. We conclude the existence of direct correspondences between (i) the payoff rule and Bell inequalities, and (ii) the strategy and the choice of measured observables in evaluating these Bell inequalities. We also show that such a correspondence between Bell polynomials (in a single plane) and four-player, symmetric, binary-choice quantum games is unique to the four-player quantum Minority game and its "anti-Minority" version. This indicates that the four-player Minority game not only plays a special role among quantum games but also in studies of Bell-type quantum nonlocality.Comment: v1 4 pages ReTeX, 2 figures (1 EPS); v2 11 pages LateX, 2 figures, changes to format, minor changes to wording (including title) and one new finding added on uniqueness of resul

Aspects of quantum game theory

Quantum game theory is an exciting new topic that combines the physical behaviour of information in quantum mechanical systems with game theory, the mathematical description of conflict and competition situations, to shed new light on the fields of quantum control and quantum information. This thesis presents quantizations of some classic game-theoretic problems, new results in existing quantization schemes for two player, two strategy non-zero sum games, and in quantum versions of Parrondo's games, where the combination of two losing games can result in a winning game. In addition, quantum cellular automata and quantum walks are discussed, with a history-dependent quantum walk being presented.Thesis (Ph.D.)--School of Electrical and Electronic Engineering , 2005