Quantumlike representation of extensive form games: Probabilistic aspects
Abstract
We consider an application of the mathematical formalism of quantum mechanics outside physics, namely, to game theory. We present a simple game between macroscopic players, say, Alice and Bob (or in a more complex form—Alice, Bob, and Cecilia), which can be represented in the quantumlike (QL) way—by using a complex probability amplitude (game's "wave function") and noncommutative operators. The crucial point is that games under consideration are so-called extensive form games. Here the order of actions of players is important; such a game can be represented by the tree of actions. The QL probabilistic behavior of players is a consequence of incomplete information (which is available to, e.g., Bob) about the previous action of Alice. In general one could not construct a classical probability space underlying a QL game. This can happen even in a QL game with two players. In a QL game with three players Bell's inequality can be violated. The most natural probabilistic description is given by the so-called contextual probability theory completed by the frequency definition of probability.
- Publication:
-
Journal of Mathematical Physics
- Pub Date:
- July 2007
- DOI:
- 10.1063/1.2752012
- arXiv:
- arXiv:0705.1260
- Bibcode:
- 2007JMP....48g2107K
- Keywords:
-
- 03.65.Ca;
- 03.65.Ud;
- 02.50.Le;
- 02.50.Cw;
- Formalism;
- Entanglement and quantum nonlocality;
- Decision theory and game theory;
- Probability theory;
- Quantum Physics
- E-Print:
- J. Math. Phys., 48, 072107 (2007)