June 14, 2012

Does arbitrage exist in biological systems?

In this post, I will be discussing a concept from economics called arbitrage in the context of biological and ecological systems. In economics, arbitrage is the act of exploiting so-called inefficiencies in the marketplace, usually by finding the maximum payoff in one market and the minimum cost in another [1]. This is similar to a so-called minimax strategy commonly used in behavioral ecology [2] and evolutionary optimization [3]. This also relates to the maximum power principle in the systems ecology literature [4]. However, biological arbitrage may allow for evolutionary tradeoffs that can be balanced, providing an optimum that is highly beneficial to the species in question.

Figure 1. A schematic that defines biological arbitrage in an ecological food web. The arbitrer in this example is the prey species.

An example of arbitrage from biology (ecological) is shown in Figure 1. In this case, the prey species conducts a mimimax-like search of foraging costs with regard to the current selection pressures introduced by predation. This sounds like what happens in every predator-prey interaction. However, the difference here is a single species (or organism) is a member of two markets simultaneously: a market of consumers, and a market of producers. These consumers and producers can be any set of biological agents, and usually results in an energetic hierarchy (e.g. set of trophic levels, as in Figure 1). In this case, a market is a specific set of interaction strategies. This market is defined by transactions and transaction costs [5]. Aside from transaction costs, every member of the market must pay a barrier to entry. If the market is nocturnal foraging, there are costs associated with this, and not every organism or species can pay.

Figure 2. A consumer-producer relationship with regard to energetic abundance provided by the producer.

In the case of Figure 2, the producer makes both a high nutrient content per berry and a large number of berries. Because of this, the consumer is able to do more with less. Specifically, the selection pressure on the consumer population is relaxed that allows even consumers with highly inefficient metabolism to flourish. This might provide a mechanism for sub-optimal biological traits to flourish and even become predominant [6]. Another possibility is that cultural behaviors may have emerged as a form of biological arbitrage. Cultural behaviors allow humans to adapt to and survive in a wide variety of ecosystems [7] by both developing new resources (e.g. innovative means of extraction or technology) and reducing the cost associated with resource extraction.

Figure 3. A schematic that defines biological arbitrage in an animal tissue. In this example, the arbitrer [8] is the tissue.

The self-assembly and maintenance of tissue microenvironments may also be better understood by using the concept of arbitrage. Figure 3 shows this trophic, hierarchical relationship and the proposed arbitrage that may exist between levels of organization. In this case, the “profit” is not made via predator-prey relationships, but rather through engaging in coordinated and other collective behaviors which minimize energy expenditure and maximize the information available to individual cells. The collective behavior of individual cells in a physiological system can be extended to individual organisms, or even social groups and populations.

Figure 4 shows a hypothetical distribution of costs and payoffs associated with a single individual or hierarchical level. These costs and payoffs are embedded in a hierarchy, and can involve both energy and information [9]. While in many theories it is the mean value that is of interest, in this case we care about the shared extreme value (e.g. overlapping tails of the distribution) [10]. In addition, these distributions do not have to be uniformly distributed (as shown in Figure 4). Such non-uniform distributions are expected for processes that involve patchy resources or asymmetrical hierarchies.

Figure 4. Hypothetical distributions of costs and payoffs for each hierarchical level. Blue distribution represents the cost function and the red distribution represents the payoff function. In the inset at right, the black lines demonstrate the optimal point which maximizes both. Overlap (region where red AND blue parts of the distribution exist) allows for the biological unit of interest to engage in arbitrage. Of course, this is a highly conceptual and idealized model (axes are in arbitrary units).

Why is arbitrage such a potentially powerful mechanism for enforcing biological organization? Perhaps by serving the same function as it does in economic markets. In economics, arbitrage allows for sellers to recoup costs associated with entering a new marketplace [11]. To see the biological analogy, imagine an organism that engages in an alternate foraging strategy that has the potential to unlock new resources but does not guarantee a high rate of success. The relatively high cost of this strategy can be offset by evolving adaptations that reduce the likelihood of being predated upon, such as a toxic defense mechanism. 

Hopefully, this post provides a synthesis of ideas from disparate fields of study. And while the analogies are sometimes incomplete, there are supporting concepts from the areas of economics, applied mathematics, and statistics which might be able to fill in the gaps. What I have covered here is just a quick, first attempt at understanding this idea. As always, comments and feedback are welcome and appreciated.

[1] Bjork, T. (2004). Arbitrage Theory in Continuous Time. Oxford University Press, Oxford, UK.

An introduction to arbitrage can be found here, and from a financial standpoint will be able to explain it better than I will attempt here.

[2] Stewart-Oaten, A. (1982). Minimax strategies for a predator-prey game. Theoretical Population Biology, 22, 410-424.

* for information on optimal foraging (a related approach), please see: Charnov, E.L. 1976. Optimal foraging: the marginal value theorem. Theoretical Population Biology, 9, 129-136. In this paper, the author examines what determines the optimal length of time for exploration of resource patches.

* for information on the limits of optimal foraging, please see: Guyader, S. and Burch, C.L. (1976). Optimal foraging predicts the ecology but not the evolution of host specialization in bacteriophages. PLoS One, 3(4), e1946.

[3] Parpas, P. and Rustem, B. (2001). Algorithms for minimax and expected value optimization. Handbook of Computational Econometrics, Chapter 4. D.A. Belsley and E.J. Kontoghiorghes eds. Wiley, New York.

For an explanation of minimax theory as originally developed by John von Newmann in the context of game theory, read this link.

For the relationship between the Nash equilibrium, minimax, and game theory, please see: Hofbauer, J. and Sigmund, K. (2003). Evolutionary Game Dynamics. Bulletin of the American Mathematical Society, 40(4), 479-519.

This may not be made clear by this post, but approximation of the minimax strategy or return by a biological agent may oftentimes be non-convex. For a high-level treatment of minimax theory with reference to non-convex problems, please see: Du, D-Z. and Pardalos, P.M. (1995). Minimax and Applications. Kluwer, New York.

[4] Odum, H.T. and Brown, M.T. (2007). Environment, Power and Society for the Twenty-First Century: The Hierarchy of Energy. Columbia University Press.

[5] I have likely not done the idea of biological markets justice. For more information, please see the following references:

* Noe, R. and Hammerstein, P. (1995). Biological markets. Trends in Evolution and Ecology, 10(8), 336-339.

* Norscia, I., Antonacci, D., and Palagi, E. (2009). Mating first, mating more: biological market fluctuation in a wild prosimian. PLoS One, 4(3), e4679.

[6] To develop a point of view on this, I took loose inspiration from research demonstrating the limits of arbitrage with respect to market efficiency. Please see the following citations from the finance literature:

* Shleifer, A. and Vishny, R.W. (1997). The Limits of Arbitrage. Journal of Finance, 52(1), 35-55.

* Stein, J.C. (2005). Why are most funds open-end? Competition and the limits of arbitrage. Quarterly Journal of Economics, 120(1), 247-272.

[7] Boyd, R. and Richerson, P.J. (2005). The Origin and Evolution of Cultures. Oxford University Press, Oxford, UK.

[8] I am defining an arbitrer in a biological system as a constituent of an intermediate hierarchical level. In an ecological food web, a top predator or primary producer could not engage in biological arbitrage, because an agent in either of these roles cannot maintain simultaneous relations with producers and consumers. In this example, we can see that arbitrage may require a trophic middleman (although this requirement may not be absolute).

[9] Alicea, B. (2008). Hierarchies of Biocomplexity: modeling life’s energetic complexity. arXiv:0810.4547.

[10] In mathematics and finance, this is also known as a copula. For more information on copulas, please see: Nelsen, R.B. (2006). An Introduction to Copulas. Springer, New York.

[11] This is a common theme in the economics and finance literature. Companies will often engage in arbitrage to recoup losses incurred from entering a new market. In this case, the recovery is purely monetary. In a biological context, the losses and recovery could be either energetic, informational, or both.

* To date, I have not been able to find any specific references to “arbitrage” in the biological literature. Academically speaking, the idea comes mainly from finance and economics, so the ideonational translation may be a bit rough.

BONUS: you may or may not be surprised, but the legal literature surrounding the term arbitrage can also involve predators (of the unethical, human kind).

1 comment:

  1. Um, I'm confused, I don't think the examples you describe constitute arbitrage. Arbitrage refers to simultaneously buying the same (or a very similar) good in one market and selling it in another, which results in a "risk-free" profit when the good trades at different prices in different markets (or other conditions hold). Arbitrage isn't really about finding the optimal solution to a trade-off, or the optimal balance of costs and benefits. Maybe I'm misunderstanding what you're getting at?