Abstract
This work is concerned with the possible impact binary encoding of strategies may have on the performance of genetic algorithms popular in agent-based computational economic research. In their recent work, Waltman et al. (J Evol Econ 21(5): 737–756, 2011) consider binary encoding and its possible contribution to a phenomenon referred to as premature convergence; the observation that different individual runs of the genetic algorithm can lead to very different results. While Alkemade et al. (Comput Econ 28(4): 355–370, 2006), (Comput Intell 23(2): 162–175, 2007), (Comput Econ 33(1): 99–101, 2009) argue that premature convergence is caused by insufficient population size, Waltman et al. argue that this phenomenon depends crucially on strategies being encoded in binary form. This conclusion is based on their illustration that premature convergence can be avoided even in simulations with small populations so long as real, rather than binary, encoding of strategies is utilized. Utilizing their methodology, we return to the consideration of the cause of premature convergence. After robustness checks with respect to the length of the binary string used for encoding, the fitness function, and the form of mutation, it is concluded that an alternative specification of mutation may also alleviate the occurrence of premature convergence. It is argued that this alternative form of mutation may be more appropriate in a wider range of problems where real encoding of strategies may not prove sufficient.
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Notes
In addition to Waltman et al., the consideration of the impact binary encoding has on simulation results has also been studied extensively by Dawid (1996). However, as argued by Waltman et al., the methodology utilized herein differs significantly from that work. Dawid’s focus is on evolutionary algorithms with large population specifications and aggregate results averaged over many simulation runs. Waltman et al. utilize a small population level and focus on comparing results of individual runs. Additionally, the crossover operator is not important in the Waltman et al. consideration, while it is significant in the Dawid approach.
For simplicity, the crossover operator will not be employed. In their analysis of the EA in this environment, Waltman et al. (2011) found no significant difference between the algorithm with, and without the crossover operator. In an earlier work, Waltman and Van Eck (2009b) showed mathematically that, if the mutation rate is small and some technical assumptions are satisfied, the effect of the crossover operator on results produced by the EA tend to be negligible in the long run.
We utilize the parameterization of \(k=30\) because it has been used in previous works where strategies were encoded using binary strings. However, a smaller string length of \(k=13\) was also utilized in order to capture a more favorable proportion of profit increasing single bit mutations from the stable quantity of 64 (refer to Fig. 1). However, no significant difference appeared in the results to those for \(k=30\).
Keeping the likelihood that a rule will remain unaffected by mutation at a particular level may be in an effort to match observed real-world outcomes or behaviour.
An alternative to the two-parameter mutation operator utilized in the above analysis is one where the binary string is drawn according to a uniform distribution over the entire strategy space when mutation occurs (again, occurring with probability \(p_{m}\)).
For the specification of \(s\) utilized by Waltman et al., only 2.7 mutations in 1,000 would exceed the upper or lower bound given by \([q_{old}-3,q_{old}+3]\). On average, only 400 mutations occur in a simulation 10,000 periods long, with 4 strategies each undergoing mutation whose likelihood is determined according to \(p_{m}=0.01\).
The software used to obtain the results described in this section is available online at http://sites.google.com/site/michaelkmaschek/software
Though not presented, the characteristics of individual runs for the EA2, EA3, and EA4 simulations are indifferentiable from the baseline EA1 (presented in Fig. 2).
Interestingly, while increasing the binary string length used to encode rules (EA2) or implementation of the effective fitness function (EA3) cannot solve the problem of premature convergence at low levels of mutation, there are significant performance improvements associated with these changes to the algorithm at higher mutation rates. At mutation rates utilized in EA5 (\(p_m = 0.01\)), implementing both of these changes simultaneously (as in EA4) increases the proportion of simulations whose firms’ average production is associated with the stable quantity of 50 by approximately 20 percent. However, other than this performance improvement, these simulations look qualitatively similar to EA5 and these proportions are still significantly lower than those associated with EA6.
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Maschek, M.K. Economic Modeling Using Evolutionary Algorithms: The Influence of Mutation on the Premature Convergence Effect. Comput Econ 47, 297–319 (2016). https://doi.org/10.1007/s10614-015-9485-8
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DOI: https://doi.org/10.1007/s10614-015-9485-8