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British Journal of Mathematics & Computer Science

Equilibria in Quantum Three-Player Dilemma Game

Aden Ahmed

Published Jan 10, 2013• 2 citations
Publisher

Abstract

After a brief exploration of certain topological features of the unit octonions, we identify the Nash equilibria in a quantized version of three-player dilemma game using an octonionic representation of the payoff function. This representation is an important tool for the relative ease in working with octonionic arithmetic as opposed to multi-variant tensors and provides a fresh computational framework that can be utilized for the classification of Nash equilibria in generic three-player two-strategy games. While the full classification of equilibria remains a goal of future research, our representation has already established certain family of equilibria in a quantized version of three-player dilemma game.

DOI
10.9734/bjmcs/2013/2805

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After a brief exploration of certain topological features of the unit octonions, we identify the Nash equilibria in a quantized version of three-player dilemma game using an octonionic representation of the payoff function. This representation is an important tool for the relative ease in working with octonionic arithmetic as opposed to multi-variant tensors and provides a fresh computational framework that can be utilized for the classification of Nash equilibria in generic three-player two-strategy games. While the full classification of equilibria remains a goal of future research, our representation has already established certain family of equilibria in a quantized version of three-player dilemma game.
We present an effect on classical games that is obtained by replacing the notion of probability distribution with the notions of quantum superposition and measurement. Our particular focus will be on two and three player games where each player has precisely two pure strategic choices. Games in normal form are represented as "payoff" functions. Game quantization requires the extension of these functions to much larger domains. The main result of this work is the co-ordinatization of these extended functions by either the quaternions or octonions in order to obtain computationally friendly versions of these functions. This computational capability is then exploited to analyze and potentially classify the Nash equilibria in the new extended games with occasionally counter intuitive results.
For any two-by-two game $\G$, we define a new two-player game $\G^Q$. The definition is motivated by a vision of players in game $\G$ communicating via quantum technology according to a certain standard protocol originally introduced by Eisert and Wilkins [EW]. In the game $\G^Q$, each players' strategy set consists of the set of all probability distributions on the 3-sphere S3S^3S3. Nash equilibria in this game can be difficult to compute. Our main theorems classify all possible equilibria in $\G^Q$ for a Zariski-dense set of games $\G$ that we call {\it generic games}. First, we show that up to a suitable definition of equivalence, any strategy that arises in equilibrium is supported on at most four points; then we show that those four points must lie in one of a small number of geometric configurations. One easy consequence is that for zero-sum games, the payoff to either player in a mixed strategy quantum equilibrium must equal the average of that player's four possible payoffs.
Quantum games with incomplete information can be studied within a Bayesian framework. We analyze games quantized within the EWL framework [Eisert, Wilkens, and Lewenstein, Phys Rev. Lett. 83, 3077 (1999)]. We solve for the Nash equilibria of a variety of two-player quantum games and compare the results to the solutions of the corresponding classical games. We then analyze Bayesian games where there is uncertainty about the player types in two-player conflicting interest games. The solutions to the Bayesian games are found to have a phase diagram-like structure where different equilibria exist in different parameter regions, depending both on the amount of uncertainty and the degree of entanglement. We find that in games where a Pareto-optimal solution is not a Nash equilibrium, it is possible for the quantized game to have an advantage over the classical version. In addition, we analyze the behavior of the solutions as the strategy choices approach an unrestricted operation. We find that some games have a continuum of solutions, bounded by the solutions of a simpler restricted game. A deeper understanding of Bayesian quantum game theory could lead to novel quantum applications in a multi-agent setting.
This paper investigates Nash equilibria in pure strategies for quantum approach to the Prisoner's Dilemma. The quantization process involves extending the classical game by introducing two additional unitary strategies. We consider five classes of such quantum games, which remain invariant under isomorphic transformations of the classical game. For each class, we identify and analyse all possible Nash equilibria. Our results reveal the complexity and diversity of strategic behaviour in the quantum setting, providing new insights into the dynamics of classical decision-making dilemmas. In the case of the standard Prisoner's Dilemma, the resulting Nash equilibria of quantum extensions are found to be closer to Pareto optimal solutions than those of the classical equilibrium.