Evolutionary minority games with small-world interactions

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Abstract

In the evolutionary minority game (EMG), agents compete for a limited resource and are rewarded if they correctly select the minority behaviour. At each time step, agents make their decision based on the aggregate history of past moves and an internal parameter—the probability that the individual follows the given strategy. In this study, the effects of strategic imitation among agents are examined. Here, I combine and extend previous work using local information transmission mechanisms to promote coordination in the population. Extensive numerical simulations using different network architectures, ranging from regular lattices to random networks, are used to investigate the population dynamics. The results suggest that agents sharing information in small-world networks can coordinate their behaviour more effectively than agents playing the standard EMG. However, both the network re-wiring probability and level of imitation significantly impact on performance.

Introduction

Agent-based models of complex adaptive systems provide valuable insights into the emergent properties and large-scale effects of locally interacting components [1]. One model that combines the general properties of such systems is the minority game (MG) [2] inspired by the El Farol bar-attendance problem [3]. In the MG, a population of agents with bounded rationality compete for a limited resource. The traditional game involves N players (agents) forced to make a binary decision: 0 (e.g. go to the bar, or take route A) or 1 (e.g. do not go to the bar, or take route B). The agents share a common “memory” or look-up table, containing the outcomes from the m most recent occurrences. The resulting 2m possible histories are then used to generate strategies for making an appropriate binary choice. At each time step, an agent receives one point if their decision places them in the minority and loses a point otherwise. The game evolves as the agents modify their behaviours (or strategies) based on the previous experiences.

Johnson and co-workers [4] introduced an extension to this basic game—the evolutionary minority game (EMG)—where every agent employs the same strategy, based on the m most recent occurrences. However, the differentiating factor between the agents is a gene parameter p characterising the probability that the agent takes an action based on the prediction of the strategy. That is, the probability p is the chance that an agent decides to follow the strategy's prediction, and 1-p is the chance that the agent decides to act opposite to the current trend. If an agent's utility (number of successes) falls below some threshold d, their p-value is mutated. In this sense, each agent tries to learn from past mistakes and modifies their strategy in order to survive.

Typically, in the EMG, the agents do not have direct interactions. As such, the models are really mean-field descriptions. In contrast, in many real-world scenarios agents can combine local information, accessed via dialogues with their peers or local consultant, with public information in order to make decisions. Subsequently, some researchers have studied variations of both the MG and EMG with local interactions [5], [6], [7], [8], [9], [10]. The results reported indicate that local communication within the agent population may improve the efficiency of the systems. It should be noted, however, that a detailed study of how coordination in the EMG is related to the underlying communication topology of agents playing the game is lacking.

In this study, local information transmission mechanisms are extended and combined with recent studies in complex network theory in order to investigate the population dynamics of the EMG. Here, the agents playing the EMG are mapped to the nodes of a small-world network [11]. The fundamental rules of the EMG have not changed. However, when an agent's utility falls below the threshold d, the agent basically “starts again”. That is, its utility is set to zero and the agent is forced to modify its p-value. The new gene is a mutated copy of the p-value of the local neighbour with the highest utility. The rationale behind this approach is based on the fact that network topology significantly influences the dynamical behaviour in ecological and social networks [11], [12], [13]. It is to be expected that the local information available to the agents differs across the network. This in turn will allow for the formation of alternative clusters of like individuals and consequently a reduced fluctuation in the number of agents correctly selecting the minority group.

Section snippets

EMG description

A simple example illustrating the basic functionality of the EMG was given by Johnson et al. [4]. Consider the following look-up table (xyz)w, containing the outcomes from the m=3 most recent occurrences. Here, the bit string (xyz) represents the corresponding sequence and the outcome w. An example memory would comprise (000)1, (001)0, (010)0, (011)1, (100)0, (101)1, (110)0, (111)1. In this scenario, a sequence of three 0s in the past was followed by a 1. Therefore, the look-up table available

Small-world connections in the EMG

The EMG model presented in this paper can be represented as a dynamic network of interconnected agents sending signals to other agents (in the local neighbourhood), with global feedback available to all agents based on aggregate measures. Here, the agents occupy nodes of alternative network architectures, ranging from regular lattices to random networks. Before describing the enhanced EMG model in detail, small-world networks are introduced.

Simulations and results

Extensive computational simulations were carried out to investigate the population dynamics of the games played. All experiments were performed on a network consisting of N=31×31 nodes. The small-world networks were generated by systematically varying the value of λ from 0 to 1, starting from a 2-D regular lattice base. For each network architecture, the value of R was varied across the range R=0 to R=2. The common “memory” or look-up table bit string length was set to m=3 and the utility

Discussion

In the EMG, agents with limited information and rationality compete for a finite resource and are rewarded when they select the minority group. Agents have their own internal mechanism/strategy used to make a decision. Typically, individual agents react to the decisions of other agents, which often results in volatile aggregate behaviour that is far from efficient. Consequently, it is possible to describe the system dynamics at different levels: the microscopic level, where the decisions of the

Conclusion

In this paper, I have discussed an extension to the EMG which preserves the basic parameters of the standard game. Here, a framework for modelling individual interactions and decision-making was introduced based on small-world networks. The simulation results suggest that system efficiency depends both on the level of interactions between agents as well the mode of learning adopted by the agents. The population dynamics displayed were driven not only by the imitation of a neighbour's successful

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