Application of the Eisert–Wilkens–Lewenstein quantum game scheme to decision problems with imperfect recall

Abstract

We investigate implementations of the Eisert-Wilkens-Lewenstein scheme of playing quantum games beyond strategic games. The scope of our research are decision problems, i.e., one-player extensive games. The research is based on the examination of their features when the decision problems are carried out via the EWL protocol. We prove that unitary operators can be adapted to play the role of strategies in decision problems with imperfect recall. Furthermore, we prove that unitary operators provide the decision maker possibilities that are inaccessible for classical strategies.

Quantum Games with Strategies Induced by Basis Change Rules

“…It has found application in many branches of game theory: from simple 2 × 2 games [2] to evolutionary games [3] and games with imperfect recall [4]. The players' strategies in this scheme are identified with unitary operators whose range is crucial for the result of a game.…”

Quantum Penny Flip game with unawareness

“…Bob has two strategies equivalent to ones in the classical game. Moreover, looking at (11) we see that HH is Alice's winning strategy.…”

“…In fact, (11) turns out to provide Bob with much richer description of the game than he actually has. Making his strategic decision based on (11), Bob finds that Alice has the additional action H, and consequently the winning strategy. Perhaps, Bob does not know that H is the Hadamard matrix or he does not even realize that he is to play the quantum game.…”

“…Quantum counterparts of games with unawareness have not been studied yet. Papers on quantum games with incomplete information concerned only Bayesian games [6], [7], [8], [9] and games with imperfect recall [10], [11]. Our project is the first attempt to use the notion of game with unawareness in theory of quantum games.…”

Quantum Approach to Cournot-type Competition

“…The first attempt to describe a game in the quantum domain applied to a simple coin tossing game [1] and 2 × 2 bimatrix games [2,3]. Shortly after that quantum game theory has found applications in various fields including decision sciences [4][5][6], financial theory [7][8][9] or mathematical psychology [5]. One of the economic applications concerns duopoly problems.…”