In this paper, we show the analysis of price changes in artificial double auction markets consisting of multi-agents who learn from past experiences based on the Genetic Programming (GP) and its applications. For simplicity, we focus on the double auction in an electricity market. Agents in the market are allowed to buy or sell items (electricity) depending on the prediction of situations. Each agent has a pool of individuals (decision functions) represented in tree structures to decide bid price by using the past result of auctions. A fitness of each individual is defined by using successful bids and a capacity utilization rate of production units for a production of items, and agents improve their individuals based on the GP to get higher return in coming auctions. In simulation studies, changes of bid prices and returns of bidders are discussed depending on demand curves of customers and the weight between an average profit obtained by successful bids and the capacity utilization rate of production units. The validation of simulation studies is examined by comparing results with classical models and price changes in real double auction markets. Since bid prices bear relatively large changes, we apply an approximate method for a control by forcing agents stabilize the changes in bid prices. As a result, we see the stabilization scheme of bid prices in double auction markets is not realistic, then it is concluded that the market contains substantial instability.

The paper deals with the estimation method of system equations of dynamic behavior of an input-pricing mechanism by using the Genetic Programming (GP) and its applications. The scheme is similar to recent noise reduction method in noisy speech which is based on the adaptive digital signal processing for system identification and subtraction estimated noise. We consider the dynamic behavior of an input-pricing mechanism for a service facility in which heterogeneous self-optimizing customers base their future join/balk decisions on their previous experiences of congestion. In the GP, the system equations are represented by parse trees and the performance (fitness) of each individual is defined as the inversion of the root mean square error between the observed data and the output of the system equation. By selecting a pair of individuals having higher fitness, the crossover operation is applied to generate new individuals. The string used for the GP is extended to treat the rational form of system functions. The condition for the Li-Yorke chaos is exploited to ensure the chaoticity of the approximated functions. In our control, since the system equations are estimated, we only need to change the input incrementally so that the system moves to the stable region. By assuming the targeted dynamic system f(x(t)) with input u(t)=0 is estimated by using the GP (denoted (x(t))), then we impose the input u(t) so that xf= (t+1)=(x(t))+u(t) where xf is the fixed point. Then, the next state x(t+1) of targeted dynamic system f(x(t)) is replaced by x(t+1)+u(t). We extend ordinary control method based on the GP by imposing the input u(t) so that the deviation from the targeted level xL becomes small enough after the control. The approximation and control method are applied to the chaotic dynamics generating various time series based on several queuing models and real world data. Using the GP, the control of chaos is straightforward, and we show some example of stabilizing the price expectation in the service queue.

This paper deals with the control of chaotic dynamics by using the approximated system equations which are obtained by using the Genetic Programming (GP). Well known OGY method utilizes already existing unstable orbits embedded in the chaotic attractor, and use linearlization of system equations and small perturbation for control. However, in the OGY method we need transition time to attain the control, and the noise included in the linealization of equations moves the orbit into unstable region again. In this paper we propose a control method which utilize the estimated system equations obtained by the GP so that the direct nonlinear control is applicable to the unstable orbit at any time. In the GP, the system equations are represented by parse trees and the performance (fitness) of each individual is defined as the inversion of the root mean square error between the observed data and the output of the system equation. By selecting a pair of individuals having higher fitness, the crossover operation is applied to generate new individuals. In the simulation study, the method is applied at first to the artificially generated chaotic dynamics such as the Logistic map and the Henon map. The error of approximation is evaluated based upon the prediction error. The effect of noise included in the time series on the approximation is also discussed. In our control, since the system equations are estimated, we only need to change the input incrementally so that the system moves to the stable region. By assuming the targeted dynamic system f(x(t)) with input u(t)=0 is estimated by using the GP (denoted (x(t))), then we impose the input u(t) so that xf=(t+1)=(x(t))+u(t) where xf is the fixed point. Then, the next state x(t+1) of targeted dynamic system f(x(t)) is replaced by x(t+1)+u(t). The control method is applied to the approximation and control of chaotic dynamics generating various time series and even noisy time series by using one dimensional and higher dimensional system. As a result, if the noise level is relatively large, the method of the paper provides better control compared to conventional OGY method.

Hogg and Huberman have proposed a strategy for stabilizing chaotic multi-agent systems. This paper applies their strategy to a resource allocation problem in a manufacturing system consisting of two machines and two types of parts. These part-types conflict each other over resource allocation. We introduce a discrete-time model of the system by using game theory, and examine stability and bifurcation phenomena of its fixed point. We show by computer simulation that chaotic behaviors are observed after successive occurrence of period-doubling bifurcations. It is also shown that the optimal state of the system is stabilized by a reward mechanism.