Recently, the investigation of minority games has received considerable attention as an interdisciplinary topic between physicists and economists. Among many outstanding topics, the evolutionary minority game, adaptive minority game, multi-choice minority game, $-game model, and grand canonical minority game have been the focus of attention. Challet et al. reported that the dynamical behavior of stock prices are characterized by anomalous fluctuations and exhibited a fat-tailed distribution and a long-range correlation. The dynamical behavior of statistical quantities between different types of trader has been extensively studied in the minority and majority games and $-game. Furthermore, the grand canonical minority game has been shown to reveal important stylized facts regarding financial market phenomenology such as return changes and volatility clusterings. Many researchers have mainly treated Arthur’s bar model, the seller and buyer model in financial markets, and the passenger problem in subways and buses. In various models, several methods of rewarding an agent’s strategies have been discussed and compared the resulting behaviors of the configurations in the minority game theory. De Almeida and Menche studied two options rewarded in standard minority games using adaptive genetic algorithms, and their result was found to be similar to that of standard minority games. Kim et al. analyzed the minority game for patients that is inspired by de Almeida and Menche’s problem. In particular, they applied the minority game theory to the numbers of both general patients and reserved patients in order to discuss numerically standard deviation and global efficiency. Until now, the minority game theory has been extensively applied to novel investigations for universal properties based on statistical concepts and methods in financial models. To our knowledge, it is of fundamental interest to estimate numerically and analytically the statistical quantities using the minority game theory. In this paper, we present the minority game theory for the transaction numbers of two types of KTB futures (KTB409 and KTB503) in the Korean futures exchange market. We discuss market payoffs and statistical quantities in the game theory. Using the minority game payoff, we numerically estimate standard deviation, global efficiency, and autocorrelation for the particular strategy. First of all, let us introduce the dynamical mechanism of both the minority and majority games. We assume that the agents N can decide independently whether to buy or sell the stock in round m. When the information and the strategy respectively take the value ðtÞ and S at time t, the action of the i-th agent is presented in terms of ai;s ðtÞðtÞ. Since an agent can submit an order ai;s ðtÞðtÞ 1⁄4 1 (buy) or ai;s ðtÞðtÞ 1⁄4 1 (sell), and the aggregate value, i.e., the sum of all action of agents, is given by AðtÞ 1⁄4 PN i1⁄40 ai;s ðtÞðtÞ. The statistical quantities such as volatility, global efficiency, and autocorrelation can be also found from the aggregate value. The payoffs Ui;s ðtÞ and Ui;sðtÞ for minority game and majority game models are, respectively, represented in terms of
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