A non-cooperative Pareto-efficient solution to a one-shot Prisoner's Dilemma

The Prisoner's Dilemma is a simple model that captures the essential contradiction between individual rationality and global rationality. Although the one-shot Prisoner's Dilemma is usually viewed simple, in this paper we will categorize it into five different types. For the type-4 Prisoner's Dilemma game, we will propose a self-enforcing algorithmic model to help non-cooperative agents obtain Pareto-efficient payoffs. The algorithmic model is based on an algorithm using complex numbers and can work in macro applications.Comment: 14 pages, 3 figure

Quantum Bayesian implementation and revelation principle

Bayesian implementation concerns decision making problems when agents have incomplete information. This paper proposes that the traditional sufficient conditions for Bayesian implementation shall be amended by virtue of a quantum Bayesian mechanism. In addition, by using an algorithmic Bayesian mechanism, this amendment holds in the macro world. More importantly, we find that the revelation principle is not always right by using the quantum and algorithmic Bayesian mechanisms.

Two-agent Nash implementation: A new result

[J. Moore and R. Repullo, \emph{Econometrica} \textbf{58} (1990) 1083-1099] and [B. Dutta and A. Sen, \emph{Rev. Econom. Stud.} \textbf{58} (1991) 121-128] are two important papers on two-agent Nash implementation. Recently, [H. Wu, Quantum mechanism helps agents combat "bad" social choice rules. \emph{International Journal of Quantum Information}, 2010 (accepted). abs/1002.4294 ] broke through traditional results on Nash implementation with three or more agents. In this paper, we will investigate two-agent Nash implementation by virtue of Wu's quantum mechanism. The main result is: A two-agent social choice rule that satisfies Condition $\mu2$ will no longer be Nash implementable if an additional Condition $\lambda'$ is satisfied. Moreover, according to a classical two-agent algorithm, this result holds not only in the quantum world, but also in the macro world.Comment: 15 pages, 4 figure

Subgame perfect implementation: A new result

This paper concerns what will happen if quantum mechanics is concerned in subgame perfect implementation. The main result is: When additional conditions are satisfied, the traditional characterization on subgame perfect implementation shall be amended by virtue of a quantum stage mechanism. Furthermore, by using an algorithmic stage mechanism, this amendment holds in the macro world too.Comment: 16 pages, 3 figure

On the justification of applying quantum strategies to the Prisoners' Dilemma and mechanism design

The Prisoners' Dilemma is perhaps the most famous model in the field of game theory. Consequently, it is natural to investigate its quantum version when one considers to apply quantum strategies to game theory. There are two main results in this paper: 1) The well-known Prisoners' Dilemma can be categorized into three types and only the third type is adaptable for quantum strategies. 2) As a reverse problem of game theory, mechanism design provides a better circumstance for quantum strategies than game theory does.Comment: 6 pages, 2 figure

Quantum mechanism helps agents combat Pareto-inefficient social choice rules

Quantum strategies have been successfully applied in game theory for years. However, as a reverse problem of game theory, the theory of mechanism design is ignored by physicists. In this paper, we generalize the classical theory of mechanism design to a quantum domain and obtain two results: 1) We find that the mechanism in the proof of Maskin's sufficiency theorem is built on the Prisoners' Dilemma. 2) By virtue of a quantum mechanism, agents who satisfy a certain condition can combat Pareto-inefficient social choice rules instead of being restricted by the traditional mechanism design theory.Quantum games; Mechanism design; Implementation theory; Nash implementation; Maskin monotonicity

A classical algorithm to break through Maskin's theorem for small-scale cases

Quantum mechanics has been applied to game theory for years. A recent work [H. Wu, Quantum mechanism helps agents combat ``bad'' social choice rules. \emph{International Journal of Quantum Information}, 2010 (accepted). Also see http://arxiv.org/pdf/1002.4294v3] has generalized quantum mechanics to the theory of mechanism design (a reverse problem of game theory). Although the quantum mechanism is theoretically feasible, agents cannot benefit from it immediately due to the restriction of current experimental technologies. In this paper, a classical algorithm is proposed to help agents combat ``bad'' social choice rules immediately. The algorithm works well when the number of agents is not very large (e.g., less than 20). Since this condition is acceptable for small-scale cases, it can be concluded that the Maskin's sufficiency theorem has been broken through for small-scale cases just right now. In the future, when the experimental technologies for quantum information are commercially available, the Wu's quantum mechanism will break through the Maskin's sufficiency theorem completely.Quantum games; Prisoners' Dilemma; Mechanism design.

Two-agent Nash implementation: A new result

[Moore and Repullo, \emph{Econometrica} \textbf{58} (1990) 1083-1099] and [Dutta and Sen, \emph{Rev. Econom. Stud.} \textbf{58} (1991) 121-128] are two fundamental papers on two-agent Nash implementation. Both of them are based on Maskin's classic paper [Maskin, \emph{Rev. Econom. Stud.} \textbf{66} (1999) 23-38]. A recent work [Wu, http://arxiv.org/abs/1002.4294, \emph{Inter. J. Quantum Information}, 2010 (accepted)] shows that when an additional condition is satisfied, the Maskin's theorem will no longer hold by using a quantum mechanism. Furthermore, this result holds in the macro world by using an algorithmic mechanism. In this paper, we will investigate two-agent Nash implementation by virtue of the algorithmic mechanism. The main result is: The sufficient and necessary conditions for Nash implementation with two agents shall be amended, not only in the quantum world, but also in the macro world.Quantum game theory; Mechanism design; Nash implementation.