Collaborate,compete and share

We introduce and study a model of an interacting population of agents who collaborate in groups which compete for limited resources. Groups are formed by random matching agents and their worth is determined by the sum of the efforts deployed by agents in group formation. Agents, on their side, have to share their effort between contributing to their group’s chances to outcompete other groups and resource sharing among partners, when the group is successful. A simple implementation of this strategic interaction gives rise to static and evolutionary properties with a very rich phenomenology. A robust emerging feature is the separation of the population between agents who invest mainly in the success of their group and agents who concentrate in getting the largest share of their group’s profits.     

The $-game

We propose a payoff function extending Minority Games (MG) that captures the competition between agents to make money. In contrast with previous MG, the best strategies are not always targeting the minority but are shifting opportunistically between the minority and the majority. The emergent properties of the price dynamics and of the wealth of agents are strikingly different from those found in MG. As the memory of agents is increased, we find a phase transition between a self-sustained speculative phase in which a “stubborn majority” of agents effectively collaborate to arbitrage a market-maker for their mutual benefit and a phase where the market-maker always arbitrages the agents. A subset of agents exhibit a sustained non-equilibrium risk-return profile. Received 5 June 2002 / Received in final form 21 November 2002 Published online 27 January 2003 RID="a" ID="a"e-mail: sornette@unice.fr RID="b" ID="b"CNRS UMR7536 RID="c" ID="c"CNRS UMR6622     

Minimal agent based model for financial markets I

We introduce a minimal agent based model for financial markets to understand the nature and self-organization of the stylized facts. The model is minimal in the sense that we try to identify the essential ingredients to reproduce the most important deviations of price time series from a random walk behavior. We focus on four essential ingredients: fundamentalist agents which tend to stabilize the market; chartist agents which induce destabilization; analysis of price behavior for the two strategies; herding behavior which governs the possibility of changing strategy. Bubbles and crashes correspond to situations dominated by chartists, while fundamentalists provide a long time stability (on average). The stylized facts are shown to correspond to an intermittent behavior which occurs only for a finite value of the number of agents N. Therefore they correspond to finite size effects which, however, can occur at different time scales. We propose a new mechanism for the self-organization of this state which is linked to the existence of a threshold for the agents to be active or not active. The feedback between price fluctuations and number of active agents represents a crucial element for this state of self-organized intermittency. The model can be easily generalized to consider more realistic variants.     

”Illusion of control” in Time-Horizon Minority and Parrondo Games

Human beings like to believe they are in control of their destiny. This ubiquitous trait seems to increase motivation and persistence, and is probably evolutionarily adaptive [J.D. Taylor, S.E. Brown, Psych. Bull. 103, 193 (1988); A. Bandura, Self-efficacy: the exercise of control (WH Freeman, New York, 1997)]. But how good really is our ability to control? How successful is our track record in these areas? There is little understanding of when and under what circumstances we may over-estimate [E. Langer, J. Pers. Soc. Psych. 7, 185 (1975)] or even lose our ability to control and optimize outcomes, especially when they are the result of aggregations of individual optimization processes. Here, we demonstrate analytically using the theory of Markov Chains and by numerical simulations in two classes of games, the Time-Horizon Minority Game [M.L. Hart, P. Jefferies, N.F. Johnson, Phys. A 311, 275 (2002)] and the Parrondo Game [J.M.R. Parrondo, G.P. Harmer, D. Abbott, Phys. Rev. Lett. 85, 5226 (2000); J.M.R. Parrondo, How to cheat a bad mathematician (ISI, Italy, 1996)], that agents who optimize their strategy based on past information may actually perform worse than non-optimizing agents. In other words, low-entropy (more informative) strategies under-perform high-entropy (or random) strategies. This provides a precise definition of the “illusion of control” in certain set-ups a priori defined to emphasize the importance of optimization. An erratum to this article is available at .     

Evolutionary ultimatum game on complex networks under incomplete information

This paper studies the evolutionary ultimatum game on networks when agents have incomplete information about the strategies of their neighborhood agents. Our model assumes that agents may initially display low fairness behavior, and therefore, may have to learn and develop their own strategies in this unknown environment. The Genetic Algorithm Learning Classifier System (GALCS) is used in the model as the agent strategy learning rule. Aside from the Watts-Strogatz (WS) small-world network and its variations, the present paper also extends the spatial ultimatum game to the Barabási-Albert (BA) scale-free network. Simulation results show that the fairness level achieved is lower than in situations where agents have complete information about other agents’ strategies. The research results display that fairness behavior will always emerge regardless of the distribution of the initial strategies. If the strategies are randomly distributed on the network, then the long-term agent fairness levels achieved are very close given unchanged learning parameters. Neighborhood size also has little effect on the fairness level attained. The simulation results also imply that WS small-world and BA scale-free networks have different effects on the spatial ultimatum game. In ultimatum game on networks with incomplete information, the WS small-world network and its variations favor the emergence of fairness behavior slightly more than the BA network where agents are heterogeneously structured.     

Mechanistic origin of dragon-kings in a population of competing agents

We analyze the mechanistic origins of the extreme behaviors that arise in an idealized model of a population of competing agents, such as traders in a market. These extreme behaviors exhibit the defining characteristics of ‘dragon-kings’. Our model comprises heterogeneous agents who repeatedly compete for some limited resource, making binary choices based on the strategies that they have in their possession. It generalizes the well-known Minority Game by allowing agents whose strategies have not made accurate recent predictions, to step out of the competition until their strategies improve. This generates a complex dynamical interplay between the number V of active agents (mimicking market volume) and the imbalance D between the decisions made (mimicking excess demand). The wide spectrum of extreme behaviors which emerge, helps to explain why no unique relationship has been identified between the price and volume during real market crashes and rallies.     

Playing the hypothesis testing minority game in the maximal reduced strategy space

Hypothesis Testing Minority Game (HMG) is a variant of the standard Minority Game (MG) that models the inertial behavior of agents in the market. In the earlier study of our group, we find that agents cooperate better in HMG than in the standard MG when strategies are picked from the full strategy space. Here we continue to study the behavior of HMG when strategies are chosen from the maximal reduced strategy space. Surprisingly, we find that, unlike the standard MG, the level of cooperation in HMG depends strongly on the strategy space used. In addition, a novel intermittency dynamics is also observed in the minority choice time series in a certain parameter range in which the orderly phases are characterized by a variety of periodic dynamics. Remarkably, all these findings can be explained by the crowd-anticrowd theory.     

Mechanisms for similarity based cooperation

Cooperation based on similarity has been discussed since Richard Dawkins introduced the term “green beard” effect. In these models, individuals cooperate based on an aribtrary signal (or tag) such as the famous green beard. Here, two different models for such tag based cooperation are analysed. As neutral drift is important in both models, a finite population framework is applied. The first model, which we term “cooperative tags” considers a situation in which groups of cooperators are formed by some joint signal. Defectors adopting the signal and exploiting the group can lead to a breakdown of cooperation. In this case, conditions are derived under which the average abundance of the more cooperative strategy exceeds 50%. The second model considers a situation in which individuals start defecting towards others that are not similar to them. This situation is termed “defective tags”. It is shown that in this case, individuals using tags to cooperate exclusively with their own kind dominate over unconditional cooperators.     

SIR model of epidemic spread with accumulated exposure

We study an extended and modified SIR model of epidemic spread in which susceptible agents during interactions with infectious neighbors are exposed to the disease and can consequently become infectious. The studied model is extended to include heterogeneity of interactions which is modelled assuming random character of the dose accumulated by susceptible agents in every interaction with infectious neighbors. When the accumulated exposure is larger than the individual’s resistance, an agent becomes infectious and consequently introduces a new source of an epidemic which is capable of passing the disease further. We study statistical properties characterizing the course of an epidemic. The examination of the modified SIR model reveals a possible “resonant activation”-like behavior of the system in the duration of the epidemic outbreak and a possible bistable behavior of the model with accumulated exposure. Furthermore, the linear scaling of the duration of the epidemic with the system size for a wide range of the model parameters is recorded.     

Crowd-anticrowd theory of multi-agent market games

We present a dynamical theory of a multi-agent market game, the so-called Minority Game (MG), based on crowds and anticrowds. The time-averaged version of the dynamical equations provides a quantitatively accurate, yet intuitively simple, explanation for the variation of the standard deviation (`volatility') in MG-like games. We demonstrate this for the basic MG, and the MG with stochastic strategies. The time-dependent equations themselves reproduce the essential dynamics of the MG. Received 28 August 2000 and Received in final form 23 September 2000     

Efficiency based strategy spreading in the prisoner’s dilemma game

In contrast to well-mixed populations, discrete interaction patterns have been shown to support cooperation in the prisoner’s dilemma game, and a scale-free network topology may even lead to a dominance of cooperation over defection. The majority of studies assumes a strategy adoption scheme based on accumulated payoffs. The use of accumulated payoffs, however, is incompatible with the integral property of the underlying replicator dynamics to be invariant under a positive affine transformation of the payoff function. We show that using instead the payoff per interaction to determine the strategy spread, which has been suggested recently and recovers the required invariance, results in fundamentally different dynamical behavior under a synchronized strategy adoption considered here. Most notably, in such an efficiency based scenario the advantage of a scale-free network topology vanishes almost completely. We present a detailed explanation of the fundamentally altered dynamical behavior.     

Study on composite photonic crystal patch antenna

The paper investigated a composite photonic crystal patch antenna by using the method of finite difference time domain (FDTD). The results show that there exists a wave resonance state at 2.635 GHz, where the real part of the permittivity and permeability are all negative; its refraction index is –1. The effect has largely enhanced the electromagnetic wave’s resonance intensity, and has improved the localized extent of electromagnetic energy obviously in such photonic crystal structure (PBG), resulting in a higher antenna gain, a lower return loss, and a better improvement of the antenna’s characteristics. Due to such the advantages, the use of patch antennas can be extended to such fields as mobile communication, satellite communication, aviation, etc.     

Motion of influential players can support cooperation in Prisoner’s Dilemma

We study a spatial Prisoner’s dilemma game with two types (A and B) of players located on a square lattice. Players following either cooperator or defector strategies play Prisoner’s Dilemma games with their 24 nearest neighbors. The players are allowed to adopt one of their neighbor’s strategy with a probability dependent on the payoff difference and type of the given neighbor. Players A and B have different efficiency in the transfer of their own strategies; therefore the strategy adoption probability is reduced by a multiplicative factor (w < 1) from the players of type B. We report that the motion of the influential payers (type A) can improve remarkably the maintenance of cooperation even for their low densities.     

Core percolation in random graphs: a critical phenomena analysis

We study both numerically and analytically what happens to a random graph of average connectivity α when its leaves and their neighbors are removed iteratively up to the point when no leaf remains. The remnant is made of isolated vertices plus an induced subgraph we call the core. In the thermodynamic limit of an infinite random graph, we compute analytically the dynamics of leaf removal, the number of isolated vertices and the number of vertices and edges in the core. We show that a second order phase transition occurs at α = e = 2.718 ... : below the transition, the core is small but above the transition, it occupies a finite fraction of the initial graph. The finite size scaling properties are then studied numerically in detail in the critical region, and we propose a consistent set of critical exponents, which does not coincide with the set of standard percolation exponents for this model. We clarify several aspects in combinatorial optimization and spectral properties of the adjacency matrix of random graphs. Received 31 January 2001 and Received in final form 26 June 2001     

Incomplete strategy effect in Minority Game

We perform simulations with the extended Incomplete Minority Game model, and find that limited incompleteness in strategies can improve the system efficiency greatly. There is a critical p_Ic such that agents in a system can get the best cooperation. The critical strategy density value α_c decreases monotonously with the increase of p_I. We also mix incomplete strategy agents with standard strategy agents to study the effect of incompleteness distribution on system properties and group average profit. Concentration of incompleteness weakens its effect. When p_I is close to zero, agents using incomplete strategies averagely have better profits than those using standard strategies.     

Modeling wealth distribution in growing markets

We introduce an auto-regressive model which captures the growing nature of realistic markets. In our model agents do not trade with other agents, they interact indirectly only through a market. Change of their wealth depends, linearly on how much they invest, and stochastically on how much they gain from the noisy market. The average wealth of the market could be fixed or growing. We show that in a market where investment capacity of agents differ, average wealth of agents generically follow the Pareto-law. In few cases, the individual distribution of wealth of every agentcould also be obtained exactly. We also show that the underlying dynamics of other well studied kinetic models of markets can be mapped to the dynamics of our auto-regressive model.     

Pattern formation,social forces,and diffusion instability in games with success-driven motion

A local agglomeration of cooperators can support the survival or spreading of cooperation, even when cooperation is predicted to die out according to the replicator equation, which is often used in evolutionary game theory to study the spreading and disappearance of strategies. In this paper, it is shown that success-driven motion can trigger such local agglomeration and may, therefore, be used to supplement other mechanisms supporting cooperation, like reputation or punishment. Success-driven motion is formulated here as a function of the game-theoretical payoffs. It can change the outcome and dynamics of spatial games dramatically, in particular as it causes attractive or repulsive interaction forces. These forces act when the spatial distributions of strategies are inhomogeneous. However, even when starting with homogeneous initial conditions, small perturbations can trigger large inhomogeneities by a pattern-formation instability, when certain conditions are fulfilled. Here, these instability conditions are studied for the prisoner’s dilemma and the snowdrift game. Furthermore, it is demonstrated that asymmetrical diffusion can drive social, economic, and biological systems into the unstable regime, if these would be stable without diffusion.     

Network evolution based on minority game with herding behavior

The minority game (MG) is used as a source of information to design complex networks where the nodes represent the playing agents. Differently from classical MG consisting of independent agents, the current model rules that connections between nodes are dynamically inserted or removed from the network according to the most recent game outputs. This way, preferential attachment based on the concept of social distance is controlled by the agents wealth. The time evolution of the network topology, quantitatively measured by usual parameters, is characterized by a transient phase followed by a steady state, where the network properties remain constant. Changes in the local landscapes around individual nodes depend on the parameters used to control network links. If agents are allowed to access the strategies of their network neighbors, a feedback effect on the network structure and game outputs is observed. Such effect, known as herding behavior, considerably changes the dependence of volatility σ on memory size: it is shown that the absolute value of σ as well as the corresponding value of memory size depend both on the network topology and on the way along which the agents make their playing decisions in each game round.     

Kelly criterion revisited: optimal bets

Kelly criterion, that maximizes the expectation value of the logarithm of wealth for bookmaker bets, gives an advantage over different class of strategies. We use projective symmetries for a explanation of this fact. Kelly's approach allows for an interesting financial interpretation of the Boltzmann/Shannon entropy. A “no-go” hypothesis for big investors is suggested.     

Collective Strategy Condensation: When Envy Splits Societies

Human societies are characterized by three constituent features, besides others. (A) Options, as for jobs and societal positions, differ with respect to their associated monetary and non-monetary payoffs. (B) Competition leads to reduced payoffs when individuals compete for the same option as others. (C) People care about how they are doing relatively to others. The latter trait—the propensity to compare one’s own success with that of others—expresses itself as envy. It is shown that the combination of (A)–(C) leads to spontaneous class stratification. Societies of agents split endogenously into two social classes, an upper and a lower class, when envy becomes relevant. A comprehensive analysis of the Nash equilibria characterizing a basic reference game is presented. Class separation is due to the condensation of the strategies of lower-class agents, which play an identical mixed strategy. Upper-class agents do not condense, following individualist pure strategies. The model and results are size-consistent, holding for arbitrary large numbers of agents and options. Analytic results are confirmed by extensive numerical simulations. An analogy to interacting confined classical particles is discussed.