Study Of Some Issues In The Evolutionary Game Theory And Quantum Game Theory

Evolutionary game theory and quantum game theory are new kinds of analytical theory of games which are developed on the basis of classical game theory. Evolutionary game theory takes the assumption of bounded rationality for the players, characterizes the path to the equilibrium by evolutionary dynamics and it also provides a good analytical framework for the study of cooperative behavior in competitive systems. Quantum game theory is developed by the integration of quantum information theory and game theory. It extends the classical game theory to the domain of quantum probability, introduces the quantum concepts such as superposition, entanglement and interference to the classical game theory and expands the classical strategy space, so the methods and scope of game theory are extended by quantum game theory.Cooperative behavior in competitive systems can be studied with methods in evolutionary game theory. In the problem of network reciprocity in evolutionary game theory, some precise results about evolutionary game on one-dimensional cycle can be obtained mathematically since the network structure is simple. When mutation is present, the dynamics can be discussed with the method of stationary IBD probabilities.In fact, the evolutionary dynamics can also be studied from the view of stochastic stability:if the mutation is considered as the impact from outside, whether some of the dynamics properties are robust with the impact? Whether the results when mutation rate goes to zero are in consistence with the results when mutation is absent? For stochastic system, the stability is not always guaranteed, so it is necessary to check the above questions in detail mathematically although they are straightforward.As for the general theoretical framework of quantum game theory, the Hamiltonian formalism of game theory is a new kind of theory which provides particular ideas and methods. The theory still needs to be discussed and optimized further, the difference and link between the new theory and the classical game theory or the traditional quantum game theory also need to be analyzed and described further. If M-W quantum game model is discussed in the Hamiltonian formalism of game theory, what new result can be obtained is unknown.One of the most important problems in quantum game theory is the impact of quantization on the evolutionary stability of nash equilibrium. There are some deficiencies in the existing research. For the ESS of quantized 2×2 symmetric game, the discussion based on general representation of the initial state is lack. On the issue of existence of ESS in the style of mixed strategy in quantized RSP game in different initial quantum states, the existing research is piecemeal and incomplete. When the initial M-W quantum game is extended to the 3×3 situation, there are some deficiencies about the widely used method in existing research because some good properties in the initial M-W quantum game are destroyed after extension. A new method proposed by Piotr extended the initial M-W quantum game to N×N situation and met the above shortfall. We think the widely used method in existing research should be replaced by the new method in related discussion.For the concept of mixed strategy in classical game theory, the probability of the strategy which is chosen by players may be impacted by some uncertain factors in real situation. In order to describe the uncertainty, a new kind of strategy so called belief strategy is proposed recently on the basis of D-S evidence theory. With the concept of belief strategy the concept of traditional ESS should be adjusted accordingly, so it is necessary to discuss the ESS under the concept of belief strategy for quantum game.On the above issues, this dissertation made the following research and discussion:First, the stability of dynamics properties of evolutionary game on one-dimensional cycle with tiny mutation rate is discussed. With some additional assumption we get the ergodic markov chain's stationary distribution which can describe the evolutionary dynamics of the system when mutation goes to zero. The stationary distribution shows that, as the markov chain's absorbing state when mutation is absent, the two states of the system in which all agents in the population choose the same strategy are robust when they meet the impact of tiny mutation. Furthermore, in small mutation rate limit and large population size limit, the result about the domination of strategy is obtained. Applying these results to the prisoner's dilemma game on one-dimensional cycle one can see that the shifting rule can make cooperation to be domination strategy so the cooperative behavior can emerge in the population.Second, the M-W quantum game model in the Hamiltonian formalism of game theory is discussed. The model framework and analytical results about equilibrium for the initial M-W quantum game are introduced, then the Hamiltonian formalism of game theory is discussed and the M-W quantum game is discussed in the new formalism. We introduce the basic idea and concept of the Hamiltonian formalism of game theory, express the mathematical representation in detail, and redescribe the classical games in mixed strategies and the traditional quantum games in the new representation. On the basis of the above work, we discuss the M-W quantum game in the Hamiltonian formalism of game theory. From the view of the density matrix of the quantum strategy and the payoff operator of player, the traditional M-W quantum game is concerned with the situation in which all the off-diagonal elements of the density matrix and the payoff operator are zero, but the model in our discussion is concerned with the situation in which the off-diagonal elements of the density matrix and the payoff operator are nonzero, so the situation in our discussion is different with the traditional M-W quantum game. The global equilibrium state of the game is also discussed when the system state is described by entanglement.Third, the impact of quantization on the evolutionary stability of Nash equilibrium is discussed by the concept of evolutionarily stable strategy (ESS). We obtained some results about the existence and the detailed mathematical representation of the ESS in different situations for the quantized hawk-dove game, RSP game and repeated prisoner's dilemma game. For the quantization of hawk-dove game, our discussion is on the basis of general representation of the initial quantum state. For the quantization of 3 x 3 symmetric game, we use a new method proposed by Piotr which extends the M-W quantum game to Nx N situation. On the issue of existence of ESS in the style of mixed strategy in quantized RSP game in different initial quantum states, our discussion is clear and complete. At last, the ESS under the concept of belief strategy which is based on D-S evidence theory is discussed for the quantized hawk-dove game and some results about the existence and the detailed mathematical representation are obtained.