Quantum Strategies

Abstract

We consider game theory from the perspective of quantum algorithms. Strategies in classical game theory are either pure (deterministic) or mixed (probabilistic). We introduce these basic ideas in the context of a simple example, closely related to the traditional MATCHING PENNIES game. While not every two-person zero-sum finite game has an equilibrium in the set of pure strategies, von Neumann showed that there is always an equilibrium at which each player follows a mixed strategy. A mixed strategy deviating from the equilibrium strategy cannot increase a player's expected payoff. We show, however, that in our example a player who implements a quantum strategy can increase his expected payoff, and explain the relation to efficient quantum algorithms. We prove that in general a quantum strategy is always at least as good as a classical one, and furthermore that when both players use quantum strategies there need not be any equilibrium, but if both are allowed mixed quantum strategies there must be.

Quantum dating market

“…In the case of only one woman and one man, for example, classic and quantum will not have any advantage on searching and the dating success difference for the first game will depend only on that woman preferences, that is, if P c > P q then D/T < 0 and the quantum player will do better when P q > P c . Similar chances for both players is not usual in most quantum games, such as, for example the coin flip game introduced by Meyer [1] where the quantum player always beats the classic player in a "mano a mano" game. For a two women set Q uses only one step, but C needs two steps to find the right partner.…”

“…An example is provided by so-called quantum games, introduced by Meyer [1] and Eisert et al [2], that allow effects which are impossible in a classical setting.…”

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

“…Quantum game theory was initially applied to classical game situations [1,2,3,4,5,6,7,8] and to markets [9,10], the traditional motivator of game theory [11]. In addition, it has been used to shed light on the role of entanglement [12,13,14,15,16], decoherence and noise [17,18,19,20,21], and quantum correlations [22,23,24,25], among other phenomena [26,27,28].…”

Quantum cat's dilemma: an example of intransitivity in a quantum game

“…It seems that the methods of quantum cryptography that are being presently worked out will remain safe even in the times of the quantum computers [8]. The combination of the research methods of both information and game theories results in emerging of the new mysterious field -quantum game theory, in which the subtle quantum rules characterizing the material world determine ways of controlling and transformation of information [9,10,11,12,13]. In the quantum game formalism, pure strategies correspond to the vectors of Hilbert space (to be more precise: the projective operators on subspaces determined by these vectors).…”