Self organization in a minority game: the role of memory and a probabilistic approach

A minority game whose strategies are given by probabilities p, is replaced by a 'simplified' version that makes no use of memories at all. Numerical results show that the corresponding distribution functions are indistinguishable. A related approach, using a random walk formulation, allows us to identify the origin of correlations and self organization in the model, and to understand their disappearence for a different strategy's update rule, as pointed out in a previous workComment: 9 pages and 4 figure

Quenching and Annealing in the Minority Game

We report the occurrence of quenching and annealing in a version of the Minority Game (MG) in which the winning option is to join a given fraction of the population that is a free, external parameter. We compare this to the different dynamics of the Bar Attendance Model (BAM) where the updating of the attendance strategy makes use of all available information about the system and quenching does not occur. We provide an annealing schedule by which the quenched configuration of the MG reaches equilibrium and coincides with the one obtained with the BAMComment: 8 pages, 4 figure

Thermal treatment of the minority game

We study a cost function for the aggregate behavior of all the agents involved in the Minority Game (MG) or the Bar Attendance Model (BAM). The cost function allows to define a deterministic, synchronous dynamics that yields results that have the main relevant features than those of the probabilistic, sequential dynamics used for the MG or the BAM. We define a temperature through a Langevin approach in terms of the fluctuations of the average attendance. We prove that the cost function is an extensive quantity that can play the role of an internal energy of the many agent system while the temperature so defined is an intensive parameter. We compare the results of the thermal perturbation to the deterministic dynamics and prove that they agree with those obtained with the MG or BAM in the limit of very low temperature.Comment: 9 pages in PRE format, 6 figure

Order and disorder in the Local Evolutionary Minority Game

We study a modification of the Evolutionary Minority Game (EMG) in which agents are placed in the nodes of a regular or a random graph. A neighborhood for each agent can thus be defined and a modification of the usual relaxation dynamics can be made in which each agent updates her decision scheme depending upon the options made in her immediate neighborhood. We name this model the Local Evolutionary Minority Game (LEMG). We report numerical results for the topologies of a ring, a torus and a random graph changing the size of the neighborhood. We focus our discussion in a one dimensional system and perform a detailed comparison of the results obtained from the random relaxation dynamics of the LEMG and from a linear chain of interacting spin-like variables at a finite temperature. We provide a physical interpretation of the surprising result that in the LEMG a better coordination (a lower frustration) is achieved if agents base their actions on local information. We show how the LEMG can be regarded as a model that gradually interpolates between a fully ordered, antiferromagnetic system and a fully disordered system that can be assimilated to a spin glass.Comment: 12 pages, 8 figures, RevTex; omission of a relevant reference correcte

Temporal oscillations and phase transitions in the evolutionary minority game

The study of societies of adaptive agents seeking minority status is an active area of research. Recently, it has been demonstrated that such systems display an intriguing phase-transition: agents tend to {\it self-segregate} or to {\it cluster} according to the value of the prize-to-fine ratio, $R$. We show that such systems do {\it not} establish a true stationary distribution. The winning-probabilities of the agents display temporal oscillations. The amplitude and frequency of the oscillations depend on the value of $R$. The temporal oscillations which characterize the system explain the transition in the global behavior from self-segregation to clustering in the $R<1$ case.Comment: 5 pages, 5 figure

Criticality and finite size effects in a simple realistic model of stock market

We discuss a simple model based on the Minority Game which reproduces the main stylized facts of anomalous fluctuations in finance. We present the analytic solution of the model in the thermodynamic limit and show that stylized facts arise only close to a line of critical points with non-trivial properties. By a simple argument, we show that, in Minority Games, the emergence of critical fluctuations close to the phase transition is governed by the interplay between the signal to noise ratio and the system size. These results provide a clear and consistent picture of financial markets as critical systems.Comment: 4 pages, 4 figure

Strategy updating rules and strategy distributions in dynamical multiagent systems

In the evolutionary version of the minority game, agents update their strategies (gene-value $p$) in order to improve their performance. Motivated by recent intriguing results obtained for prize-to-fine ratios which are smaller than unity, we explore the system's dynamics with a strategy updating rule of the form $p \to p \pm \delta p$ ($0 \leq p \leq 1$). We find that the strategy distribution depends strongly on the values of the prize-to-fine ratio $R$, the length scale $\delta p$, and the type of boundary condition used. We show that these parameters determine the amplitude and frequency of the the temporal oscillations observed in the gene space. These regular oscillations are shown to be the main factor which determines the strategy distribution of the population. In addition, we find that agents characterized by $p={1 \over 2}$ (a coin-tossing strategy) have the best chances of survival at asymptotically long times, regardless of the value of $\delta p$ and the boundary conditions used.Comment: 4 pages, 7 figure

Self-Segregation vs. Clustering in the Evolutionary Minority Game

Complex adaptive systems have been the subject of much recent attention. It is by now well-established that members (`agents') tend to self-segregate into opposing groups characterized by extreme behavior. However, while different social and biological systems manifest different payoffs, the study of such adaptive systems has mostly been restricted to simple situations in which the prize-to-fine ratio, $R$, equals unity. In this Letter we explore the dynamics of evolving populations with various different values of the ratio $R$, and demonstrate that extreme behavior is in fact {\it not} a generic feature of adaptive systems. In particular, we show that ``confusion'' and ``indecisiveness'' take over in times of depression, in which case cautious agents perform better than extreme ones.Comment: 4 pages, 4 figure

Continuous transition of social efficiencies in the stochastic strategy Minority Game

We show that in a variant of the Minority Game problem, the agents can reach a state of maximum social efficiency, where the fluctuation between the two choices is minimum, by following a simple stochastic strategy. By imagining a social scenario where the agents can only guess about the number of excess people in the majority, we show that as long as the guess value is sufficiently close to the reality, the system can reach a state of full efficiency or minimum fluctuation. A continuous transition to less efficient condition is observed when the guess value becomes worse. Hence, people can optimize their guess value for excess population to optimize the period of being in the majority state. We also consider the situation where a finite fraction of agents always decide completely randomly (random trader) as opposed to the rest of the population that follow a certain strategy (chartist). For a single random trader the system becomes fully efficient with majority-minority crossover occurring every two-days interval on average. For just two random traders, all the agents have equal gain with arbitrarily small fluctuations.Comment: 8 pages, 6 fig

Theory of Phase Transition in the Evolutionary Minority Game

We discover the mechanism for the transition from self-segregation (into opposing groups) to clustering (towards cautious behaviors) in the evolutionary minority game (EMG). The mechanism is illustrated with a statistical mechanics analysis of a simplified EMG involving three groups of agents: two groups of opposing agents and one group of cautious agents. Two key factors affect the population distribution of the agents. One is the market impact (the self-interaction), which has been identified previously. The other is the market inefficiency due to the short-time imbalance in the number of agents using opposite strategies. Large market impact favors "extreme" players who choose fixed strategies, while large market inefficiency favors cautious players. The phase transition depends on the number of agents ($N$), the reward-to-fine ratio ($R$), as well as the wealth reduction threshold ($d$) for switching strategy. When the rate for switching strategy is large, there is strong clustering of cautious agents. On the other hand, when $N$ is small, the market impact becomes large, and the extreme behavior is favored.Comment: 5 pages and 3 figure