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Palgrave Macmillan UK eBooks

The Intrinsic Quantum Nature of Classical Game Theory

Yohan Pelosse

Published Jan 1, 2017• 1 citation
Publisher

Abstract

The notion of mixed strategy, as originally introduced by Von Neumann and Morgenstern ( 1944 ) is a basic ingredient of classical game theory. Yet, as pointed out by von Neumann and Morgenstern themselves, the idea that a rational player may have to use a randomizing device, such as a coin flip, to decide on their actions poses some insuperable conceptual difficulties. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI
10.1057/978-1-137-49276-0_4

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•Jan 1, 2000•0 citations
The notion of mixed strategy, as originally introduced by Von Neumann and Morgenstern ( 1944 ) is a basic ingredient of classical game theory. Yet, as pointed out by von Neumann and Morgenstern themselves, the idea that a rational player may have to use a randomizing device, such as a coin flip, to decide on their actions poses some insuperable conceptual difficulties. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
The mathematical theory of games in the form known to economists, social scientists and biologists was developed by John von Neuman and Oskar Morgenstern. Their framework has been deepened and generalized over the past seven decades. Psychologists use classical probability theory in which rational choice is built in, and they interpret any departure from classical probabilities as irrationality. It has been suggested that an alternative interpretation would be to retain rationality by giving up classical probability theory in favour of quantum probability theory. Some game theorists have demonstrated that a tit-for-tat strategy evolves in the repeated Prisoners’ Dilemma setting where one player responds back in the same coin to the other after seeing the letter’s move. Co-operation is generated at times since ‘credible’ threat can be created in repeated games. The main difference between quantum and classical probability theories is the use of Hilbert space representations of states rather than set theory.
We consider game theory from the perspective of quantum algorithms. Strategies in classical game theory are either pure (deterministic) or mixed (probabilistic). While not every two-person zero-sum finite game has an equilibrium in the set of pure strategies, von Neumann showed that there is always an equilibrium at which each player follows a mixed strategy. A mixed strategy deviating from the equilibrium strategy cannot increase a player's expected payoff. We show by example, however, that a player who implements a quantum strategy can increase his expected payoff, and explain the relation to efficient quantum algorithms.
This chapter gives an introduction to a chain of results attempting to exclude deeper layers underneath quantum mechanics that restore some form of classical physics: ‘[Such results] more or less illustrate the ways along which some opponents might hope to escape Bohr’s reasonings and von Neumann’s proof and the places where they are dangerously near breaking their necks.’ (Groenewold, 1946, p. 454). In so far as they are mathematically precise, such no-go results have their roots in von Neumann’s 1932 book, which gave rise to two traditions that were often in polemical opposition to each other. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
An outline is given of the interpretation of quantum mechanical expressions if one assumes that what is called Quantum Mechanics today, refers to an underlying reality as in Chap. 2 , even if many authors claim to have evidence to the contrary, as is usually brought forward. If there exists a sub-microscopic classical world of real things, is that the same classical world as the world of large things such as planets, rocks and people? How does the notion of probability enter into quantum mechanics? These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
In these lecture notes we investigate the implications of the identification of strategies with quantum operations in game theory beyond the results presented in [J. Eisert, M. Wilkens, and M. Lewenstein, Phys. Rev. Lett. 83, 3077 (1999)]. After introducing a general framework, we study quantum games with a classical analogue in order to flesh out the peculiarities of game theoretical settings in the quantum domain. Special emphasis is given to a detailed investigation of different sets of quantum strategies.