Physica A: Statistical Mechanics and its Applications
Evolutionary minority games with small-world interactions
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 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 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 , containing the outcomes from the most recent occurrences. Here, the bit string represents the corresponding sequence and the outcome . An example memory would comprise , , , , , , , . 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 nodes. The small-world networks were generated by systematically varying the value of 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 to . The common “memory” or look-up table bit string length was set to 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
References (17)
- et al.
Emergence of cooperation and organisation in an evolutionary game
Physica A
(1997) - et al.
Self-segregation and enhanced cooperation in an evolving population through local information transmission
Physica A
(2003) - et al.
The evolutionary minority game with local coordination
Physica A
(2004) - et al.
The minority game with interactions
Physica A
(2004) - et al.
Cooperation in the minority game with local information
Physica A
(2000) - et al.
Coordination of decisions in a spatial agent model
Physica A
(2002) Hidden Order: How Adaptation Build Complexity
(1995)Bounded rationality and inductive behavior (the El Farol problem)
Amer. Econ. Rev.
(1994)