Finite Automata Capturing Winning Sequences for All Possible Variants of the PQ Penny Flip Game

Abstract

Abstract:The meticulous study of finite automata has produced many important and useful results. Automata are simple yet efficient finite state machines that can be utilized in a plethora of situations. It comes, therefore, as no surprise that they have been used in classic game theory in order to model players and their actions. Game theory has recently been influenced by ideas from the field of quantum computation. As a result, quantum versions of classic games have already been introduced and studied. The PQ penny flip game is a famous quantum game introduced by Meyer in 1999. In this paper, we investigate all possible finite games that can be played between the two players Q and Picard of the original PQ game. For this purpose, we establish a rigorous connection between finite automata and the PQ game along with all its possible variations. Starting from the automaton that corresponds to the original game, we construct more elaborate automata for certain extensions of the game, before finally presenting a semiautomaton that captures the intrinsic behavior of all possible variants of the PQ game. What this means is that, from the semiautomaton in question, by setting appropriate initial and accepting states, one can construct deterministic automata able to capture every possible finite game that can be played between the two players Q and Picard. Moreover, we introduce the new concepts of a winning automaton and complete automaton for either player.

The Connection between the PQ Penny Flip Game and the Dihedral Groups

“…Another related problem, namely that of quantum gambling based on the Nash-equilibrium was examined in [10]. The association of every finite variant of the PQ penny flip game with finite automata, so that strategies are words accepted by the corresponding automaton, was established in [11]. In that work, the underlying assumption was that Q will always use the Hadamard operator.…”

“…This was accomplished in [9] where the authors exploited entanglement in a clever way, so that whether the system ends up in a maximally entangled or separable state determines the outcome. In [11], it was shown that all possible finite extensions of the PQG can be expressed in terms of simple finite automata, provided that the allowable moves of Picard are either I or F and Q always uses the Hadamard transform H. The focus of the present investigation is enlarging the operational space of the game. Therefore, when considering extensions of the original PQG, not only the assumptions ( 24) and (41), i.e., M P = {I, F} and M Q = U(2) are valid, but, additionally, it is supposed that:…”

“…Figure11. This figure depicts two winning strategies for Q; two n-round games where Q makes the first and the last move.…”

Quantum Conditional Strategies and Automata for Prisoners’ Dilemmata under the EWL Scheme

“…Automata in general have already been shown to be related to games and strategies in various works [31,32,38,39]. Andronikos et al established a sophisticated connection between finite automata and the PQ Penny game, constructing automata for various interesting variations of the game [39]. The proposed semiautomaton is capable of capturing the game's finite variations, introducing the notion of the winning and the complete automaton for either player.…”

Quantum Conditional Strategies for Prisoners' Dilemmata Under the Ewl Scheme

“…Automata in general have already been shown to be related to games and strategies in various works [26,27,33,34]. Andronikos et al established a sophisticated connection between finite automata and the PQ Penny game, constructing automata for various interesting variations of the game [34]. The proposed semiautomaton is capable of capturing the game's finite variations, introducing the notion of the winning and the complete automaton for either player.…”