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arXiv:0808.1391 · International Journal of Quantum Information

OCTONIONIZATION OF THREE PLAYER, TWO STRATEGY MAXIMALLY ENTANGLED QUANTUM GAMES

ADEN OMAR AHMED, STEVEN A. BLEILER, FAISAL SHAH KHAN

Published Apr 1, 2010• 11 citations
arXivHTMLPDF

Abstract

We develop an octonionic representation of the payoff function for three player, two strategy, maximally entangled quantum games in order to obtain computationally friendly version of this function. This computational capability is then exploited to analyze and potentially classify the Nash equilibria in the quantum games.
DOI
10.1142/s0219749910006344

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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.
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.
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.