Density Matrix Formalism of Game Theory
A new representation of Game Theory is developed in this work.
State of players is represented by a density matrix, and payoff
function is a set of hermitian operators, which when applied onto
the density matrix gives the payoff of players. By this formalism,
a new way to find the equilibria of games is given by generalizing
the thermodynamical evolutionary process leading to equilibrium in
Statistical Mechanics. And in this formalism, when quantum objects
instead of classical objects are used as the objects in the game,
it naturally leads to the Traditional Quantum Game, but with a
slight difference in the definition of strategy state of players:
the probability distribution is replaced with a density matrix.
Further more, both games of correlated and independent players can
be represented in this single framework, while traditionally, they
are treated separately by Non-cooperative Game Theory and
Coalitional Game Theory. Because the density matrix is used as
state of players, besides classical correlated strategy, quantum
entangled states can also be used as strategies, which is an
entanglement of strategies between players, and it is different
with the entanglement of objects' states as in the Traditional
Quantum Game. At last, in the form of density matrix, a class of
quantum games, where the payoff matrixes are commutative, can be
reduced into classical games. In this sense, it will put the
classical game as a special case of our quantum game.
First, in order to apply Statistical Mechanics onto Game Theory
Second, from the requirement of games on quantum objects
The object of Game Theory is a game, a multi-player decision
making situation, usually with conflicts between players. For
example, in a Penny Flipping Game (PFG), two players play with a
coin, say initially with head state. The strategies can be used
by players are Non-flip and Flip, which, in the language of
Physics, are operators acting on the coin. The payoff is defined
such as player 1 wins one dollar for head state after both
players applied their strategies, and lose one dollar for tail
state. For such static strategy games, Nash Theorem has given a
closed conclusion that at least one mixture-strategy Nash
Equilibria (NE) exists for any games. Here the NE is defined that
under such state no more players will like to change its own
strategy state, and mixture strategy is defined as a probability
distribution function (PDF) over the strategy space of every
Our question is how about we replace the two-side coin here with a
½-quantum spin? What's the effect of this on Game
Theory? It's still a game-theory question. Players can still
choose strategies to act on the spin, although they have much more
choices now. Compared with Non-flip and Flip , in Quantum
Mechanics, any unitary 2×2 matrices can be used as
operators, and I, X, Y, Z are the four typical
matrices of them. Now the Game Theory must answer how to define
the strategy state for this game, how to define NE, and the
existence of NE. At last, we have to ask whether such game can be
studied within the framework of Traditional Game Theory (TGT), or
should we develop a new framework but still with the same spirit
of Game Theory? In this work, we will construct a new framework,
which can be used both TGT and the game on quantum objects, named
Quantum Game Theory (QGT).
In this work, we introduce a framework of new mathematical
representation of Game Theory, including static classical game and
static quantum game. The idea is to find a set of base vectors in
strategy space and to define their inner product so as to form
them as a Hilbert space, and then form a Hilbert space of system
state. Basic ideas and formulas in Game Theory have been
reexpressed in such a space of system state. This space provides
more possible strategies than traditional classical game and
quantum game. All the Game Theories have been unified in a new
representation and their relation has been discussed. It seems
that if the quantized classical game has some independent meaning
other than traditional classical, a payoff matrix with non-zero
off-diagonal elements is required. On the other hand, when such
new representation is applied onto quantum game, the payoff matrix
gives non-zero off-diagonal elements. Also in the new
representation of quantum games, a set of base vectors are
naturally given from the quantum strategy (operator) space. This
gives a kind of support for our approach in classical game. Ideas
and technics from Statistical Physics can be easily incorporated
into Game Theory through such a representation. Such incorporation
gives some endogenous method for refinement of Equilibrium State
and some hits to simplify the calculation of Equilibrium State.
Kinetics Equation and thermal equilibrium has been introduced as
an efficient way to calculate the Equilibrium State. Although we
have gotten some successful experience on some trivially cases,
the progress of such a dynamical equation for the general case is
still waiting for more exploration.
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