A combinatorial game is a deterministic game with no hidden information played between two opponents such as tic-tac-toe, checkers or chess. In this paper we extend combinatorial games to the quantum setting, by first revisiting and reformulating existing theory of classical combinatorial games. We investigate in which case a quantum opponent has an advantage over a classical one. Surprisingly, our instantiation of Zermelo's classical theorem in the quantum setting shows that the effects of quantum mechanics do not convey an advantage against a classical player that plays a perfect classical strategy. In a more realistic scenario, when the classical player makes mistakes, we show how the quantum opponent can amplify the mistake to increase their chance of winning. Our theory has application beyond the mere playing of board games and can be used as a tool in finite deterministic adversarial models with perfect information.
A combinatorial game is a deterministic game with no hidden information played between two opponents such as tic-tac-toe, checkers or chess. In this paper we extend combinatorial games to the quantum setting, by first revisiting and reformulating existing theory of classical combinatorial games. We investigate in which case a quantum opponent has an advantage over a classical one. Surprisingly, our instantiation of Zermelo's classical theorem in the quantum setting shows that the effects of quantum mechanics do not convey an advantage against a classical player that plays a perfect classical strategy. In a more realistic scenario, when the classical player makes mistakes, we show how the quantum opponent can amplify the mistake to increase their chance of winning. Our theory has application beyond the mere playing of board games and can be used as a tool in finite deterministic adversarial models with perfect information.
In this paper, we propose a Quantum variation of combinatorial games, generalizing the Quantum Tic-Tac-Toe proposed by Allan Goff. A combinatorial game is a two-player game with no chance and no hidden information, such as Go or Chess. In this paper, we consider the possibility of playing superpositions of moves in such games. We propose different rulesets depending on when superposed moves should be played, and prove that all these rulesets may lead similar games to different outcomes. We then consider Quantum variations of the game of Nim. We conclude with some discussion on the relative interest of the different rulesets.
In this paper, we perform a minimalistic quantization of the classical game of tic-tac-toe, by allowing superpositions of classical moves. In order for the quantum game to reduce properly to the classical game, we require legal quantum moves to be orthogonal to all previous moves. We also admit interference effects, by squaring the sum of amplitudes over all moves by a player to compute his or her occupation level of a given site. A player wins when the sum of occupations along any of the eight straight lines we can draw in the 3 × 3 grid is greater than or equal to 3. We play the quantum tic-tac-toe first randomly, and then deterministically, to explore the impact different opening moves, end games and different combinations of offensive and defensive strategies have on the outcome of the game. In contrast to the classical tic-tac-toe, the deterministic quantum game does not always end in a draw. In contrast also to most classical two-player games of no chance, it is possible for player 2 to win. More interestingly, we find that player 1 enjoys an overwhelming quantum advantage when he opens with a quantum move, but loses this advantage when he opens with a classical move. We also find the quantum blocking move, which consists of a weighted superposition of moves that the opponent could use to win the game, to be very effective in denying the opponent his or her victory. We then speculate what implications these results might have on quantum information transfer and portfolio optimization.
We propose a quantum version of Tic-Tac-Toe which accurately reflects the inherent probabilistic nature of the measurement principle in quantum mechanics. We then formulate a quantum strategy which allows a quantum player to consistently win over a classical player, with a certain probability. This result can be seen as another proof of the superior computational power of a quantum system with respect to a classical one. Our investigation also reveals that the non-determinism and complexity introduced by the principles of quantum mechanics into even the most simple games make brute-force strategies considerably more difficult to implement. Consequently, games in which machines have gained the upper hand over humans may be made fair again by upgrading them to a quantum level.
Quantum games offer situations where quantum information theory may help in solving or improving games with lack of information. For example, one may rely upon properties of entanglement for showing the superiority of a quantum player who is allowed to use quantum operations against a classical player (penny flipover [5]). Quantum communication properties enter into play in cooperative games (prisoners game, [3], guessing number [10]). An introduction to this field and an overview of important results are proposed in [7, 8]. But games may also help to make notions easier to understand, and new ways of reasoning to apprehend. The objective of this paper is to formalize and study a simple game with qubits using quantum notions of measurement and superposition, while keeping a simple formalism so that knowing quantum mechanics is not necessary to play the game. We generalize Conway's classical octal games to quantum octal games, and we solve the quantum combinatorial game QO.07 by giving a winning strategy for it. A playable version of the studied qubit game is available at the address .
Game theory is the study of strategic competition used in many branches of science. In general, a game is a match between two or more players and each wants to maximize his or her score. In a game, the player’s score depends on their strategies. The goal is to find the optimized decision, with regard to the payoff, in conformity with the rules of the game. This thesis considers the games with two players who know both their own and their rival’s payoff. However, they do not know the strategy chosen by the opponent.\nIn a classical game, the options of players are referred to as cooperation and defect. Quantum game theory is a generalization of classical game theory. In quantum games a linear combination of two options specify the strategy based on qubit. The quantum domain is characterized by entanglement. The quantum entanglement coefficient (y) varies from 0 to . The game is in the classical regime when y is smaller than a threshold. Quantum entanglement is intrinsically non-local.\nIn an iterated game tournament, 63 different strategies contest with four known strategies. Depending on the result, they are divided to a number of groups. The competition specifies which group of strategies can obtain a higher score.\nOther types of games are discussed in the thesis. They are known as incomplete information games. In these games one player has two payoff matrixes, and the rival does not know which of them is used. We discuss the advantage of using quantum strategy over the classical game, particular in avoiding the dilemma in classical games.
The last two decades have witnessed a rapid development of quantum information processing, a new paradigm which studies the power and limit of "quantum advantages" in various information processing tasks. Problems such as when quantum advantage exists, and if existing, how much it could be, are at a central position of these studies. In a broad class of scenarios, there are, implicitly or explicitly, at least two parties involved, who share a state, and the correlation in this shared state is the key factor to the efficiency under concern. In these scenarios, the shared \emph{entanglement} or \emph{discord} is usually what accounts for quantum advantage. In this paper, we examine a fundamental problem of this nature from the perspective of game theory, a branch of applied mathematics studying selfish behaviors of two or more players. We exhibit a natural zero-sum game, in which the chance for any player to win the game depends only on the ending correlation. We show that in a certain classical equilibrium, a situation in which no player can further increase her payoff by any local classical operation, whoever first uses a quantum computer has a big advantage over its classical opponent. The equilibrium is fair to both players and, as a shared correlation, it does not contain any discord, yet a quantum advantage still exists. This indicates that at least in game theory, the previous notion of discord as a measure of non-classical correlation needs to be reexamined, when there are two players with different objectives.