Particle Swarm Optimization in Agent‐Based Economic Simulations of the Cournot Market Model

• Author: Michael K. Maschek
• Date: 01 April 2015
• DOI: http://doi.org/10.1002/isaf.1367
• Published date: 01 April 2015

PARTICLE SWARM OPTIMIZATION IN AGENT-BASED

ECONOMIC SIMULATIONS OF THE COURNOT MARKET MODEL

MICHAEL K. MASCHEK*

University of the Fraser Valley, Abbotsford, BC Canada

SUMMARY

The numerous variations of the particle swarm optimization (PSO) algorithm originally proposed by Kennedy and

Eberhart (1995. Particle swarm optimization. In Proceedings of the IEEE International Conference on Neural

Networks IV. IEEE: Piscataway, NJ; 1942–1948) have proven to be powerful optimization methods that rely on

exploiting simple analogues of social interaction. In this study, PSO is adopted in lieu of the social or individual

evolutionary learning algorithms as a model of individual adaptation in an agent-based computational model. In

this examination of the simple Cournot market framework, each agent’s individual strategy evolves according to

the PSO algorithm. The model is one in which agents’strategies must adapt interdependently. That is, a change

in one particle may not only affect its performance but also other particles within the same swarm simultaneously.

The dynamics and convergence properties associated with this model are compared with those where evolutionary

learning algorithms are employed. Similar to evolutionary learning, convergence to equilibrium is dependent on the

scope of learning, social or individual. While convergence is dependent on some of the algorithm parameters,

prices resulting from the individual PSO are nearest the Cournot equilibrium and those from social PSO are nearest

the Walrasian equilibrium in all cases. For particular parameterizations, certain advantages over evolutionary algo-

rithms exist: in the main, decreasing volatility in market prices does not require an election operator or the addition

of free parameters through two-level learning. Copyright © 2015 John Wiley & Sons, Ltd.

Keywords: agent-based computational economics; Cournot oligopoly; genetic algorithm; particle swarm

optimization

1. INTRODUCTION

Two models of learning and adaptation have been frequently featured in agent-based computational

economic (ACE) models. Genetic algorithm (GA) learning is the best known representation of a class

of direct random search methods referred to as evolutionary algorithms. These algorithms describe the

evolution of a population of rules or beliefs in response to experience. The success of a particular rule is

referred to as its fitness and is determined according to a particular fitness function. Rules whose appli-

cation has been more successful in terms of this fitness are more likely to be represented in the popu-

lation. Heterogeneity is introduced to the population through the evolutionary operators referred to as

mutation and crossover. Work featuring this form of adaptation includes Arifovic (1994, 1996);

Axelrod (1987); Bullard and Duffy (1999); Reichman (2001) and Vriend (2000). Alternatively, rein-

forcement learning (RL) has been utilized as an alternative to GA learning. Here, adaptation is

modelled as a finite-state Markov decision process where the likelihood of selecting any particular

* Correspondence to: Michael K. Maschek, Economics, University Of The Fraser Valley, 33844 King Road, Abbotsford, V2S

7M8, British Columbia, Canada. E-mail: michael.maschek@ufv.ca

Copyright © 2015 John Wiley & Sons, Ltd.

INTELLIGENT SYSTEMS IN ACCOUNTING, FINANCE AND MANAGEMENT

Intell. Sys. Acc. Fin. Mgmt. 22, 133–152 (2015)

Published online 16 April 2015 in Wiley Online Library (wileyonlinelibrary.com)DOI: 10.1002/isaf.1367

strategy increases as its performance against the strategy space of rivals improves (Erev & Roth, 1998;

Kutchinski, Uthmann, & Polani, 2003).

In this study, an alternative to these two frequentlyimplemented models of adaptation is investigated:

a stochastic optimization technique referred to as particle swarm optimization (PSO). Proposed by

Kennedy and Eberhart (1995), instead of constructing a representation of purely individual cognitive

abilities, it is aimed at producing computational intelligence by exploiting simple analogues of social

interaction. It is based on a number of simple entities, referred to as particles, that are evaluated in

terms of their ability to solve some problem (the objective function). The movement of each particle

results from comparing its own current and best solutions with those of one or more other members

of the local population (referred to as the swarm) with some randomness. Re-evaluation occurs itera-

tively following each particle’s movement. Previous investigations have shown that PSO manages to

locate a global optimum faster than with use of GAs (Hassan, Cohanim, de Weck, & Venter, 2005;

Mouser & Dunn, 2005; Panda & Padhy, 2007).

Implementation of this algorithm for ACE modelling and simulation has only occurred very recently.

In the work of Zhang and Brorsen (2009), the PSO algorithm was utilized in an ACE model of a Cournot

oligopsony market. They illustrate that artificial agents with the PSO learning algorithm do find the op-

timal strategies predicted by theory. They argue that the PSO algorithm is simpler and more robust to

algorithm parameter changes than the GA method utilized in many previous studies. In their model,

agents utilizing the PSO learning algorithm do so independently. That is, they adjust the PSO algorithm

for noncooperative games by constructing multiple parallel markets. This allows for each agent to have

its own clones in every market; each agent has a separate ‘flock of birds’that trade independently and

simultaneously in separate markets. As such, the performance of one clone is independent of all other

clones in an agent’sflock. It is argued that this independence is necessary in order to implement the

PSO algorithm in an agent-based model where all agents solve their own optimization problem under

dynamic conditions in which one agent’s profit depends on the actions of all other agents. However, pre-

vious implementations of the social GA algorithm in economic environments characterized by such stra-

tegic interaction have been successful. In the social GA, each agent is associated with a single gene

representing (encoding) their strategy. Arifovic (1994) first illustrated that the implementation of such

a learning model in the Cobweb market model resulted in convergence to the Walrasian equilibrium

(WE). The GA learning algorithm does not seem to be encumbered by the fact that the performance

of individual genes is interdependent. In this study, the implementation of the PSO learning algorithm

will occur in the Cournot oligopoly model in which, similar to the work of Arifovic, the performance

of individual clones is interdependent. Multiple parallel markets are not utilized in order to assess the

performance of the PSO in an environment characterized by strategic interaction within a‘flock’.

The economic model implemented in this study has been extensively investigated in previous work

utilizing the GA learning algorithm. Vriend (2000) first noted that when two distinct forms of the GA

algorithm were utilized, convergence to different economic equilibria can result. Specifically, the social

GA algorithm, where each individual is represented by a single encoded rule, resulted in convergence

to the WE, whereas the individual GA algorithm, where each individual is associated with a distinct set

of encoded rules, exhibited convergence to the Cournot equilibrium (CE). The difference between these

two outcomes results from what is referred to as the spite effect:afirm will not adopt a strategy that

increases its performance if it simultaneously improves the performance of its competitors by even

more. This spite effect is observed in all deviations from the WE and where learning possesses a social

dimension whose dynamics are based on relative performance. In a subsequent study utilizing the same

framework, contradictory results were found by Arifovic and Maschek (2006), where both social

and individual GA learning resulted in convergence to the WE. However, this contradiction was

134 M. K. MASCHEK

Copyright © 2015 John Wiley & Sons, Ltd. Intell. Sys. Acc. Fin. Mgmt., 22, 133–152 (2015)

DOI: 10.1002/isaf