Income and Wealth Distributions from Stochastic Strategy Minority Game

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Introduction
The price°uctuations of a commodity in a market are the result of the imbalance in its demand and supply. Similarly, price°uctuation of a stock in a stock-market is the result of imbalance of traders' interests in buying and selling a stock. If more people This is an Open Access article published by World Scienti¯c Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited. are interested in selling than buying, then the price of that stock goes down and vice versa. It is therefore often compared with the Minority Game problem [1][2][3] where the players are asked to make a choice between two options at each time step. The players ending up in the minority receive a positive pay-o®, much like the smaller number of buyers in a market with more sellers.
The Minority Game is played between N independent players making parallel decisions. The number N is taken to be odd, to make an unambiguous minority at each step. The objective of the game is to divide the population almost equally between the two choices and hence to maximize the resource utilization and also to do that within a short time. This is a variant of the El Farol bar problem 4 and has a vast literature in dealing with this objective that concerns with various strategies and their merits and demerits. In particular, stochastic strategies have been useful in maximizing the resource utilization 5,6 (see Ref. 7 for a review).
Our¯rst objective in this study is to de¯ne the minority in a more realistic way in terms of imbalance in demand and supply rather than just the number of people opting for a choice. In the traditional variant, it is e®ectively assumed that the selling and purchasing power of every agent is the same, which is far from realistic. It is the amount of stock to be sold and the amount of money involved in buying those stocks that determine the imbalance in demand and supply and hence the`minority'.
Our second objective is to check the distributions of income and wealth (amount of stock and/or money) among the agents after the game is played repeatedly for a long time. The income and wealth distribution among people in a society is another long studied problem. Empirical laws, such as the Pareto's law, 8 are known to be valid across many nations and at di®erent times. 9 It is therefore very natural to demand that a well known form of wealth exchange, such as the stock market dynamics, should respect such empirical laws. There have been very few attempts in the past along this line, 10,11 which at the end did not recover the wealth and income distributions very well.
In the following sections, we¯rst de¯ne the new version of the Minority Game model and then discuss the nature of the income and wealth distributions in those models. Finally, we discuss some open questions and conclude.

Model
We consider the dynamics of a single stock and N agents (see Fig. 1 for a schematic diagram) making repeated choice of either buying or selling that stock. The state of each agent (i) at every time step (t) is given by three variables, the amount of stock s i ðtÞ and the amount of money m i ðtÞ she has at that time and a third variable that shapes the dynamics is the`saving propensity' i . This is similar to the case in the kinetic exchange models of wealth exchanges, [12][13][14][15] where a saving propensity was introduced from a realistic perspective of that occurring in real life. At every time step, the agents must choose between buying (choice B) or selling (choice S). In other words, they invest a fraction of their money [Ám i ¼ ð1 À i Þm i ðtÞ] or stock [Ás i ¼ ð1 À i Þs i ðtÞ] at each step. Then, the total money invested MðtÞ ¼ P i2B Ám i ðtÞ is redistributed between the stock investors in the proportion of their investments, i.e., the agent with an investment of stock Ás i ðtÞ will get back money m i ðÁs i ðtÞÞ ¼ MðtÞ Ás i ðtÞ SðtÞ , where SðtÞ ¼ P i2S Ás i ðtÞ. Similarly, for an agent investing money Ám i ðtÞ there will be a return in terms of stock s i ðÁm i ðtÞÞ ¼ SðtÞ Ám i ðtÞ MðtÞ . Note that throughout this work we have considered the choice between buying and selling at each time to be completely random. Of course, this is not true in reality. However, optimization of this step is very di®erent from the traditional Minority Game problem, since it is not the number of agents but their amount of investments that is determining the minority here.
The only parameter therefore, is the form of the saving propensity. In the following, we measure the distribution of stock and money among the agents at steady state for various form of saving propensity and also the distribution of`income' at each time step of the agents (money earned by the agents at each step) who gets money. We consider the cases of constant saving fraction, where each agent saves a fraction of her wealth and also the distributed savings, where the variables i are random for each agent but¯xed in time. In all cases, however, we consider the saved value of saving propensity for a given agent for both stock and money.
Considering the fact that only a single stock is traded and also the choices are made randomly, this model is of course far from the reality. Nevertheless, the model in this simplest form gives many realistic features which we will discuss here. Further extensions are to be taken up elsewhere.

Wealth distribution between the agents
The agents only invest a fraction ð1 À i Þ of their stock or money each time and that is chosen randomly. Due to the presence of only one stock in the market, stock and money are symmetric in this case. Therefore, the wealth distribution can be measured by measuring the distribution of either one of those. After a su±cient relaxation time, the distribution function of stock/money comes to a steady state. Figure 2 depicts the results of such distributions for di®erent constant values of as well as when they are uniformly distributed in [0:1]. The case of zero saving leads to an absorbing state at the end. This is because, some agents can eventually get completely broken, and that fraction will subsequently increase,¯nally concentrating the entire wealth to one agent. For non-zero but¯xed values of , the distribution is either exponential or gamma-like depending on whether is less than or greater than 0:5. This is somewhat surprising in the sense that even for a¯nite savings, most people have very low wealth, if < 0:5 (see Fig. 3). This gives rise to a critical saving fraction c for the population to have zero density at zero wealth.
The most interesting case is the one with distributed saving propensity, i.e., the saving propensities i are di®erent for each agent and are taken from a distribution uniform in [0:1]. It has two power-law regions, separated by the average wealth 1. This means, agents who are much richer than the average, follow a power law distribution with exponent À2 (this part can depend on the distribution of the saving propensities). This is in very good agreement with the empirical Pareto-law. Similarly as before, the average wealth and income increase with increasing saving propensities in the society (see Figs. 5 and 6).

Distribution of income
The agents who invest (sell) stock, get back money in proportion of their investments. This constitute their`income' at each time step. We measure the size distribution of these incomes in the steady state. Figure 4 represents such a distribution for¯xed . It looks similar to the wealth distribution. In the same¯gure, we plot the Gini-index for di®erent values of , which is a measure of the inequality in the income Ref. 16 (see also Ref. 17 for other measures and comparison with World Bank data). For distributed saving, however, the income distribution (shown in Fig. 7) is di®erent from the wealth distribution. Nevertheless, it has a power-law tendency before dropping o® with a cut-o®. The Gini-index for this case is about 0.54 and is in the range typically observed in di®erent biased countries 18.

Price°uctuation
The price of the stock will°uctuate as the investments vary at each time. The price SðtÞ is of course de¯ned as the ratio of the total invested stock and total invested money at each time step. We compute the log-return r ¼ logðSðtÞ=Sðt À 1ÞÞ and the distribution of the log returns for both¯xed (Fig. 8) and distributed saving (Fig. 9).  It matches with the normal distribution since the process here is completely random, i.e., there is no temporal correlation between the events. This is certainly not the case for the real stock markets, where the distribution di®ers from normal.

Conclusions
We have proposed a Minority-Game-like model for the stock market dynamics. The players have two choices of either investing money or stocks at each time step. Then, the invested money is returned to the stock investors and the invested stock is returned to the money investors in proportion to their respective investments. If the total invested stock is less than the total invested money, then the stock investors are in minority and receive a higher pay-o® and vice versa. Therefore, the minority is determined by the amounts of investments and not by the number of investors. As a gross simpli¯cation, we have assumed all players to be random traders. This simple model reproduces several features of income and wealth distributions. Particularly, when the agents invest a random fraction of their stock or money at each step (saving fractions drawn from uniform distribution), the steady state distribution of the accumulated wealth has a power-law. Furthermore, the distribution of income, i.e., the money earned by the stock investors at each step, is also a powerlaw. These features are in accordance with the Pareto's law, which is a widely held observation for various economies. The measure for income inequality, viz the Gini coe±cient has a value > 0:4 indicating a biased society. The bias is entering the model in the sense that the riches have most of the investment and therefore have most of the return. On the other hand, a poor agent does not win much even after ending up in the winning side. This e®ect accumulates in giving a biased wealth distribution (see Fig. 5). An obvious future direction is to investigate the e®ects of having rational agents and multiple types of stocks.