Selection without multiple replicators?

John W. Pepper

Santa Fe Institute,

1399 Hyde Park Road, Santa Fe NM 87501,

U.S.A.

jpepper@santafe.edu

http://www.santafe.edu/~jpepper

Thorbjørn Knudsen

Department of Marketing/ School of Business and Economics,

University of Southern Denmark/ Odense University,

Campusvej 55, DK-5230 Odense M,

Denmark

tok@sam.sdu.dk

http://www.sam.sdu.dk

Abstract

Hull et al.’s construction of operant learning as an instance of selection gives rise to problems that weaken this application of selection theory beyond acceptable limits. We point out that the most fundamental is a disregard for the need to include multiple concurrent replicators in any definition of selection, and indicate how this problem may be solved.

Of the three phenomena considered in the target article, the analysis of operant learning is the most challenging and problematic. Some of the obstacles are due to our lack of knowledge. For example, it is not clear what would play the role of replicators in operant learning, or how feedback between environmental interaction and differential replication would be mediated. But an even more fundamental issue concerns the role of multiple concurrent replicators that differ in their replication rates. We will focus on this point because we believe it can be resolved in principle, without any need for further empirical studies of behavior or neurobiology.

The issue is raised by the authors in their introduction: “In operant learning selection occurs only with respect to sequences of environmental interaction rather than with respect to numerous concurrent alternatives. Is this difference sufficient to disqualify it as a case of selection?”

The question would seem to be a matter of definition. In their general description of selection, the authors define it as “repeated cycles of replication, variation, and environmental interaction so structured that environmental interaction causes replication to be differential.” The standard interpretation of the word ‘differential’ is that in each cycle, multiple replicators differ in the extent to which they replicate. Hull et al. seem to concur with this interpretation, stating that, “Variants must be linked to proliferation so that at any one time, numerous alternatives are available for selection.”

Thus it would seem clear that selection requires multiple replicators by definition. When the authors discuss operant learning, however, they frame the theory in terms of a single replicator at any given time. They clearly recognize the problem this raises: “If the environment must have multiple and differing copies of a replicator concurrently available for selection to occur, operant behaviors seems definitionally excluded.” However, they go on to conclude rather surprisingly that this requirement does not apply. They reason that, “there appears to be no reason to assume that all replication processes involve concurrently existing events or objects.” That is certainly true, but replication is not the same as selection. With little further discussion of the problem, they decide in thier conclusion that, “replicators that do not proliferate in this way also count as instances of selection.”

This leaves unresolved contradictions between the authors’ general definition of selection and their specific conclusion about operant learning. This new interpretation of selection also includes a much wider range of phenomena than the original definition. For example, imagine a bird that molts its feathers each year, then produces a new “generation” of feathers that is similar but not identical to the previous year’s. This would fit the description of a process that retains features of an object across generations, with a mechanism of variation to introduce novelty. But surely we would not want to call this selection. Indeed this seems a clear example of the mere persistence of patterns, which the authors in their introduction explicitly exclude. Even if the sequence of plumages showed “improvement” over time by some criterion, it would not be by means of differential replication, and it would not be through a process of selection.

Is the definition of selection ultimately a matter of taste or semantics, with one answer being as good as another? It is not, in part because the theory of selection has developed beyond mere verbal argument. A tradition of rigorous mathematical description of the selection process in biology provides a foundation for developing a general theory. All formal representations of selection are quite explicit about the requirement for variation among multiple concurrent alternatives. This includes Fisher’s “fundamental theorem of natural selection” (Fisher 1958), the Price equation (Price 1970), and the replicator equation (Schuster & Sigmund 1983). Here the role of variation is not just qualitative, but appears as a quantitative expression of the rate of change as a function of the genetic or phenotypic variance present at a given point in time.

To pursue the biological analogy, imagine an organism that consistently produces a single offspring and then dies. If we allow for heritability and mutation, and analyze this situation using the mathematics of selection theory we will inevitably conclude, quite correctly, that natural selection does not occur because there are no variants to select among. The lineage either persists or ends, but it will not generate adaptation. The same reasoning applies to operant behavior, for the same reasons.

Does this mean that the project of explaining operant learning as a selective process is doomed to failure? We do not believe so, because there is no need to envision the process as involving only a single replicator at any given time. If replicators consist of specific neural configurations that produce tendencies or proclivities for certain behaviors, it is not hard to imagine a population of such replicators that compete for the opportunity to be expressed as behaviors (interactors), and to be thereby strengthened or weakened according to their relative “success” (e.g., in eliciting positive affect). It is also not hard to envision that stronger neural configurations would be more likely both to persist and to spawn variants.

The formal structure of selection theory can be applied to such a scenario without any major conceptual obstacles. We could even envision the possibility that selection occurs among a set of concurrent alternatives previously generated in sequential order. Given a straightforward conceptual solution, what remains is only that the empirical aspects be clarified and tested. Indeed, a start has already been made on developing the quantitative selection theory developed in biology into a broader account of selection in general (Price 1995), and of learning in particular (Frank 1997). We think this approach holds considerable promise, and we urge the authors as well as other workers to forge ahead on this exciting endeavor.

References

Fisher, R. A. (1958). The Genetical Theory of Natural Selection. 2nd ed. New York: Dover.

Frank, S. A. (1997). The design of adaptive systems: Optimal parameters for variation and selection in learning and development. J. theor. Biol., 184, 31-39.

Price, G. R. 1970. Selection and covariance. Nature, 227, 520-521.

Price, G. R. 1995. The nature of selection. J. theor. Biol., 175, 389-396

Schuster, P. & Sigmund, K. 1983. Replicator dynamics. J. theor. Biol., 100, 533-538.