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arXiv:quant-ph/0209121 · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences

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

Adrian P. Flitney, Derek Abbott

Published Sep 16, 2003• 93 citations
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Abstract

We study a general 2 × 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 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 pay–offs in the game, the miracle move is of no advantage.

DOI
10.1098/rspa.2003.1136

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We study a general 2 × 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 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 pay–offs in the game, the miracle move is of no advantage.
We study a general 2times22 \\times 22times2 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.
This paper studies sequential quantum games under the assumption that the moves of the players are drawn from groups and not just plain sets. The extra group structure makes possible to easily derive some very general results, which, to the best of our knowledge, are stated in this generality for the first time in the literature. The main conclusion of this paper is that the specific rules of a game are absolutely critical. The slightest variation may have important impact on the outcome of the game. It is the combination of two factors that determine who wins: (i) the sets of admissible moves for each player, and (ii) the order of moves, i.e., whether the same player makes the first and the last move. Quantum strategies do not a priori prevail over classical strategies. By carefully designing the rules of the game the advantage of either player can be established. Alternatively, the fairness of the game can also be guaranteed.
We examine the advantages that quantum strategies afford in communication-limited games. Inspired by the card game blackjack, we focus on cooperative, two-party sequential games in which a single classical bit of communication is allowed from the player who moves first to the player who moves second. Within this setting, we explore the usage of quantum entanglement between the players and find analytic and numerical conditions for quantum advantage over classical strategies. Using these conditions, we study a family of blackjack-type games with varying numbers of card types, and find a range of parameters where quantum advantage is achieved.
A version of the Monty Hall problem is presented where the players are permitted to select quantum strategies. If the initial state involves no entanglement the Nash equilibrium in the quantum game offers the players nothing more than that obtained with a classical mixed strategy. However, if the initial state involves entanglement of the qutrits of the two players, it is advantageous for one player to have access to a quantum strategy while the other does not. Where both players have access to quantum strategies there is no Nash equilibrium in pure strategies, however, there is a Nash equilibrium in quantum mixed strategies that gives the same average payoff as the classical game.
Game theory is the mathematical framework for analyzing strategic interactions in conflict and competition situations. In recent years quantum game theory has earned the attention of physicists, and has emerged as a branch of quantum information theory [1]. With the aid of entanglement and linear superposition of strategies, quantum games are shown to yield signifcant advantage over their classical counterparts. In this paper we explore optimal and equilibrium solutions to quantum minority games. Initial states with different level of entanglement are investigated. Focus will be on 4 and 6 player games with some N-player generalizations.
Game theory is the mathematical framework for analyzing strategic interactions in conflict and competition situations.In recent years quantum game theory has earned the attention of physicists, and has emerged as a branch of quantum information theory [1].With the aid of entanglement and linear superposition of strategies, quantum games are shown to yield significant advantage over their classical counterparts.In this paper we explore optimal and equilibrium solutions to quantum minority games.Initial states with different level of entanglement are investigated.Focus will be on 4 and 6 player games with some N -player generalizations.