Tuesday, July 4, 2017

Peloton theory and contractual relations



A contract between parties is a cooperative venture.  The contracting parties seek to exchange something of value from the bargain; otherwise they would not enter it.  There are many theories of contractual relations (e.g. as noted in 1), which I do not propose to discuss in detail.  What I do want to explore, however, are some of the resource-related conditions under which parties are likely to enter contracts.  Similarly, I wish to explore certain conditions under which parties are unlikely to enter contracts, even if the parties are willing and able to enter such contracts and would choose to do so if the appropriate conditions for doing so existed.

More specifically, I wish to explore the hypothesis that contracts tend to be formed under conditions of relative personal or business prosperity, and that as parties' prosperity diminishes, the likelihood that they will enter a contract also diminishes. Another way to state this is that as parties' resources become increasingly strained, so diminishes the likelihood that parties will form contracts.

There is nothing particularly novel about this; indeed it may seem obvious that parties must have some minimum level of affluence in order to contemplate entering contracts of any kind. However, from what limited research I have done, there does not appear to be much literature that identifies the threshold level of resource scarcity at which parties will choose not to enter the kind of cooperative bargain that a contract entails. The actual presence of such a threshold does seem to be implied by the work of Ronald Coase, for example, some of whose work (2, 3) I will look at here in more detail, and no doubt many others.  Yet the question of how and where that threshold occurs, if it exists, seems to be rather an open one, and this is the subject of this present exploration.  I aim to elucidate this problem by what I suggest are simple analogous dynamics: the dynamics of bicycle pelotons.


Enter peloton theory

Pelotons are groups of cyclists coupled by the energy saving benefits of drafting. Commonly known among competitive cyclists, drafting occurs when a cyclist follows sufficiently near another in a region of reduced air pressure; this permits the following cyclist to ride with reduced power output at the pace set by the leading cyclist.  

A peloton may seem to be an odd source of inspiration for understanding aspects of contract theory. However, it is not as far-fetched as it may seem.  For instance, the strategies of cyclists in mass-start bicycle races (and who therefore form pelotons) are highly amenable to game-theory analysis, which involves implied bargains, cooperation and defection (Mignot, 2016).  Any sort of bargain is contractual in nature due to the parties’ exchange of some consideration of value.

Such strategies and game theory analysis would thus be an obvious source of any analogies between contract theory and pelotons. However, I propose to drill down farther to look beneath the deliberate and conscious strategies that cyclists might employ, toward basic physical principles that underlie unconscious self-organizing peloton behaviors.  These “unconscious” behaviors are global, or collective, in the sense that they emerge holistically among the entire peloton, independently of the behavior of individual cyclists.  In this respect, collective peloton behavior cannot be directly predicted by analyzing the behavior of an individual cyclist in isolation.

The basic physical principles that drive collective peloton behavior are differentials in cyclists’ maximal sustainable outputs, or their fitness or metabolic capacities, as mediated by the energy saving benefits of drafting.  In (5, 6) we describe phases of peloton behavior that emerge across domains of cyclists' metabolic outputs. Between these domains, or phases, are certain collective output thresholds that separate these phases, identifiable by their different descriptions of quantifiable global behaviors.  

In (5, 6) we describe two primary phases by their topologies and energetic principles: a stretched formation in which the cyclists ride at near maximal-sustainable outputs (MSO); a compact formation, in which cyclists collectively ride below their MSOs.  We also describe the compact formation as a convective phase, for reasons I don’t discuss here. We can quantify the threshold between these phases as corresponding to the quantity of energy saving generated by drafting (5, 6).  

In (7) I proposed an alternative description for these phases, and introduced the notion of "protocooperative behavior" (which, it should be noted, is different from "protocooperation").  This describes the range of self-organizing peloton dynamics that includes the above-noted primary phases, but described differently as 1), a cooperative phase; and 2) a predominantly free-riding phase. The cooperative phase is the same as the compact or convective phase, and occurs at sufficiently low collective power or metabolic outputs so as to permit cyclists to share time in the highest cost non-drafting positions; while the free-riding phase is the stretched phase that emerges at a comparatively high collective metabolic output -- in this phase cyclists are forced to maximize time in drafting positions when it is physiologically impossible for cyclists to share time in the highest-cost front position. Of course, if the collective speed drops, cyclists can then generally resume sharing time in the front (non-drafting) positions, in which case the cooperative phase is re-established.  As stated, these dynamics emerge naturally from the coupled metabolic outputs of the cyclists as facilitated by the power reductions obtained by draftinga,b

In (7) I proposed certain principles of protocooperative behavior, including that:
  
·    Protocooperative behavior emerges in pelotons as a function of differentials in cyclists’ energetic requirements, whose outputs are coupled by the energy saving mechanism of drafting.

·      At comparatively low speeds when cyclists’ capacity to pass others is abundant, there is a tendency for cyclists to share time spent in the highest cost non-drafting positions. This tendency diminishes as collective speeds increase and free-riding (i.e. extended exploitation of the lowest-cost positions) increases. 

·      There is a quantifiable threshold between the cooperative phase and the predominantly free-riding phase. This “protocooperative threshold” exists as a function of the quantity of energy saved by drafting, and the difference between the output of the leading rider and maximum capacities of the following riders. Coupled cyclists thus approach this threshold when following riders approach their maximal sustainable outputs at the speed set by the front rider, even when drafting, since the follower’s ability to pass the front rider by accelerating is reduced to a near-impossibility. And, as indicated, below this threshold, cooperative passing behavior and the sharing of costly front positions occurs more abundantly; and above this threshold, free-riding is generally a physiological necessity.

Perhaps one of the potentially more controversial, and counter-intuitive, implications of this theoretical framework is that cooperation does not emerge in times of high stress, as might be the intuitive conclusion. Rather, I am suggesting that cooperation tends to emerge when resources are plentiful. In other words, cooperation is a luxury and when individuals are strained to their maximum capacities and their available resources are severely limited, individuals are less likely to cooperate and tend to act predominantly in their self-interest. This is counter to the intuitive conclusion that stressful conditions necessitate or precede cooperation to alleviate the collective stress (at present I don’t have a good citation for this).

At this point I confess that I cannot cite exhaustive evidence to support the assertion that cooperation is a luxury; for this reason I embark only on an exploration for such evidence, some of which appears to be found among elements of contract theory. All of this a developing work in progress.


Connecting peloton theory and contract theory

So how does peloton protocooperative behavior shed any light on the hypothesis that parties to a contract will execute their agreements in conditions of relative prosperity, and that there is an identifiable threshold between cooperative contracts and free-riding contractual relations?

     First let us turn to the work of Ronald Coase, and his seminal work, The Nature of the Firm (1937).  Coase earned a Nobel prize in 1991 for work that originated in his 1937 paper (8). 

Coase describes the function of firms and how they emerge in a free market to reduce “transaction costs”, or costly inefficiencies in contractual relations. Coase describes it this way (p. 390):
    
The main reason why it is profitable to establish a firm would seem to be that there is a cost of using the price mechanism.  The most obvious cost of “organising” production through the price mechanism is that of discovering what the relevant prices are.  This cost may be reduced but it will not be eliminated by the emergence of specialists who will sell this information.  The costs of negotiating and concluding a separate contract for each exchange transaction which takes place on a market must also be taken into account. Again, in certain markets, e.g. produce exchanges, a technique is devised for minimising these contract costs; but they are not eliminated.  It is true that contracts are not eliminated when there is a firm but they are greatly reduced. A factor of production (or the owner thereof) does not have to make a series of contracts with the factors with whom he is co-operating within his firm, as would be necessary, of course, if this co-operation were as a direct result of the working of the price mechanism. For this series of contracts is substituted one.
    
   In view of Coase’s description of the emergence of the firm as a means of reducing inefficient transaction costs, I suggest, albeit perhaps over-simplistically, that the emergence of a firm is analogous to the cooperative phase of peloton dynamics at comparatively low collective output.  In this view, a firm emerges when there is some comparative abundance of resources and energy available to the managers and employees who comprise the firm. The firm thus precedes and/or prevents the incremental increase in transaction costs required for a series of individual contracts, which costs potentially increase until contracting parties reach the limits of their financial or operational resources.   

Thus, on one hand, it seems fair to say that that the firm emerges not as a response to high transaction costs, but prior to the point at which parties incur such costs. On the other hand, it is also perhaps fair to say that the emergence of the firm is indeed a rational and expected response to the high transaction costs of a series of costly individual contracts. If it is the latter, then the firm represents the collapse, if you will, of the high-output free-riding phase that occurs in times of high-transaction cost contracts. Under both these perspectives, the peloton analogy is sustainable, but perhaps it is not entirely consistent with a strict view that cooperation emerges only in times of luxury. Perhaps we are left with a kind of chicken-or-egg scenario that is impossible to resolve.


Oscillating contractual conditions

At this juncture, it is premature, based on this limited exploration, to conclude with any degree of certainty that efficient contractual conditions, such as the emergence of firms, arise in times of relative luxury. However, we appear to be left with a viable analogue of oscillating contractual conditions: high transaction cost relationships are like the stretched phase of peloton behavior in which resources are strained to their maximum availability; the emergence of firms is like the lower-cost cooperative (or compact, or convective) phase of peloton dynamics. In a given economy, we may imagine that these conditions will oscillate over time, very much as the sorts of oscillating phase dynamics we observe in pelotons.

At this point, I suggest that the peloton analogy goes farther than merely illustrating a point about oscillating contractual relationships in a broad economic context.  Rather, I suggest that the analogy describes a real theoretical framework for investigating the presence of quantifiable energetic-related thresholds of contractual relations.  Obviously, a lot more work needs to be done to develop this, however.

     With this limited and simplistic foray into some new explorations, I will leave off.  In a further installment, I’ll look at Coase’s discourse on “The Problem of Social Cost”, and how contracts for compensation and litigation damages tend to equalize differentials in parties’ economic positions. I argue that equalization of these differentials is rather like the effect of the energy saving mechanism of drafting in pelotons, by which cyclists equalize their output differentials. I’ll also consider standard form contracts, otherwise known as adhesion contracts, how they tend to reduce transaction costs and promote free-riding (Waddams et al., 2000) and identify analogous peloton behaviors.

___________________________

a As noted, here I am not concerned with dynamics that result from cyclists' deliberate strategies. Some might argue that it is impossible to separate cyclists' conscious strategies from the collective dynamics of the peloton. However, the fact that we can simulate basic collective behavior of pelotons and their phase dynamics, as we have done in (6) based on a few basic parameters and without introducing volitional strategies, strongly suggests that certain basic peloton dynamics self-organize independently of these volitional strategies.  

b Some might point to a third phase that emerges at very low metabolic outputs, when free riding is also abundant because riders do not tend to share costly front positions. To the extent such a phase exists, it is distinguishable from high-output free-riding that occurs by physiological necessity. In this high-output phase, cyclists are physically unable to share the front position and must free ride to sustain the pace set by the front rider. In the low-output phase, cyclists have a choice to share the front position and, should they do so, a general increase in speed is observed as cyclists attempt to shift positions within the peloton to achieve front positions or other non-drafting positions, such as while moving up along perimeters -- hence generating the compact, or cooperative phase.

References

1.  Waddams, S.W., Trebilcock, M.J., Waldron, M.A. 2000. Cases and Materials on Contracts. Ch 1. Perspectives on Contract Law.  Edmond Montgomery Publications Limited, Toronto, Canada.

2.  Coase, R.H., 1960. The problem of social cost. The Journal of Law and Economics, (3), 1-44.

3.   Coase, R.H., 1937. The nature of the firm. Economica, New series, 4(16), 386-485.

4. Mignot, J.F., 2016. Strategic Behavior in Road Cycling Competitions. In The Economics of Professional Road Cycling (pp. 207-231). Springer International Publishing.

5.  Trenchard, H., Richardson, A., Ratamero, E. and Perc, M., 2014. Collective behavior and the identification of phases in bicycle pelotons. Physica A: Statistical Mechanics and its Applications405, pp.92-103.

6.   Trenchard, H., Ratamero, E., Richardson, A. and Perc, M., 2015. A deceleration model for bicycle peloton dynamics and group sorting. Applied Mathematics and Computation251, 24-34.

7.  Trenchard, H. The peloton superorganism and protocooperative behavior. Applied Mathematics and Computation 270 (2015): 179-192.


8.       http://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/1991/coase-lecture.html











Monday, June 12, 2017

Trilobite pelotons and the variation range hypothesis

Our new paper on "Trilobite pelotons" published by the journal, Paleontology, is now out.

Although it remains a hypothesis, the findings of the paper support the "variation range hypothesis", the notion that there is a relationship between the energy saving quantity and the size range among a group of organisms where there is an energy saving mechanism. I first explored this (with Matjaz Perc) across a range of organisms in our paper "Energy saving mechanisms in biological systems".

The variation range hypothesis is simple and intuitive. Weaker individuals can sustain speeds set by stronger individuals by exploiting the available energy saving mechanism. So, these individuals can be weaker to a degree that is roughly proportionate to the energy saving quantity.  In the paper, we summarize it this way:

"The variation range hypothesis posits that the size range among individuals in groups corresponds proportionately to the energy saving quantity (as a per cent) because weaker, smaller, individuals sustain speeds of stronger, larger, individuals by exploiting the energy saving mechanism. We determine size range by

SR = [(BLmax – BLmin) / BLmax] * 100,                      (1)

where SR is size range, and BL is body-length. As discussed further, this allows comparisons between size ranges and energy saving as a per cent, shown as equation (8). Individuals too small to fit within this range become isolated from the group and may perish, or form sub-groups of narrower size ranges, as has been modelled by Trenchard et al. (2015) in the context of bicycle pelotons.

In the wider context of migration as a factor that drives speciation (Winker 2000), we consider the possibility that this form of group-sorting may contribute to migratory divergence and to reproductive isolation, as proposed by Delmore et al. (2012). Speyer & Brett (1985) reported that individuals within trilobite groups generally fall within a rather narrow size range, a finding which tends to support the variation range hypothesis. Moreover, the presence of any similarly sized trilobites of other species mixed with clusters of another species also tends to support the variation range hypothesis, since such individuals would likely have possessed similar power and speed capacities."

Sunday, July 24, 2016

Wednesday, May 11, 2016

Our new paper: Trenchard H., Perc, M. "Equivalences in biological and economical systems: Peloton dynamics and the rebound effect."



Our paper has just been published in PLOS ONE. This paper foreshadows what I anticipate will ultimately be a more important paper about energy savings mechanisms in biological systems, also co-authored with Matjaz Perc, currently under consideration for publication. When that paper comes out, I'll have more to say about some implications of what we discuss there in relation to what we discuss in this PLOS ONE paper. 

On a related side note, I am also hoping to hear definitive word soon on another paper prepared in collaboration with Andrew Renfree and Derek Peters from the University of Worcester, in which we apply our peloton model to groups of runners and test the effects of drafting on certain collective running dynamics. I have also now resumed work on a collaborative analysis of fish schooling dynamics that applies the peloton model.

So, in a sense this new PLOS ONE paper is a companion piece to the larger review paper in which we review literature on energy savings mechanisms in natural systems, and wherein we identify certain principles of the collective dynamics of pelotons that are common to other natural systems.

It is difficult to anticipate how our PLOS ONE paper may be viewed, if it gets much attention at all. I don't deny that we are somewhat perilously pushing the envelope of the peloton analogy into the realm of economics. Broadly speaking, some might argue that our attempts at identifying commonalities between certain economic parameters and biological ones are naive and audacious. There are risks to this interdisciplinary endeavor, but scientific breakthroughs are impossible without such risks and occasional failures. Regardless, clearly PLOS ONE and the reviewers have judged the analysis of sufficient merit to publish it for further appraisal in the wider cauldron of academic consideration.

A brief summary:
Our model of the rebound effect is premised on four main factors: 1. the price of the energy service, externally imposed upon the consumer; 2. the consumer’s maximum capacity (or budget) to pay for the energy service; 3. the reduction in cost to the consumer due to some energy service efficiency, as a percentage; 4. the rebound quantity, as a percentage. In our paper we identify the first three as the primary ones, but the fourth one is obviously of critical importance and discussed in detail in our paper.

These factors all have equivalences in peloton dynamics, respectively: 1. The speed set by a pacesetter in a non-drafting position (akin to the imposed cost of the service to the consumer); 2. The maximum sustainable output of a following cyclist in a drafting position (akin to the consumer’s available budget); 3. The energy savings quantity due to drafting, as a percentage (akin to the efficiency in the energy service that reduces the service cost to the consumer by some percentage); 4. Potentially some surplus energy facilitated by the energy savings mechanism of drafting that permits the cyclist to achieve higher speeds by drafting than she could without the drafting benefit (akin to the fraction of the consumer’s budget that has been freed for the consumer to purchase more of the energy service as a result of the reduction in cost of the energy service, than previously used).

By adapting a basic equation that describes the relationship of these factors in peloton dynamics, we have derived an analogous equation that describes the relationship between these four economic factors. Thus a ratio that models these factors allows us to identify two main thresholds that we have also identified in peloton dynamics.  The first one is clear: a decoupling threshold between the price of an energy service and a consumers ability to pay for it; the second is less clear but is based on upon the application of an analogous threshold in pelotons: the "protocooperative" threshold between one phase of collective behavior in which cyclists can share costly front positions, and a second phase of collective behavior in which cyclists can sustain the speed of a pacesetter only by exploiting the energy savings mechanism (drafting) but cannot pass the pacesetter in order to share the costly front position.

Our paper has just been published
Our paper has just been publishedI anticipate some criticisms of our approach and more particularly to our interpretation of the model. However, I feel it is prudent at this point first to hear what the criticisms may be before I try to anticipate and enumerate them here. My sense at this early stage is that the rebound ratio equation itself is defensible and, if there are problems in the paper, it isn't in the model itself, but in how best to interpret it. Our present interpretation of it and the effects of it may need to be modified in light of feedback and further analysis. 

If in the passage of time some or other aspects of our analysis turn out to be flawed, and if nothing else is achieved by our new paper in PLOS ONE, I hope that this singular point will stand the test of time: certain principles of peloton dynamics have application across a wide variety of systems, including not only a wide range of natural biological systems, but also among human-centric economic ones.  

Saturday, March 12, 2016

Come scientists and academics, publish while you still can; the times they are changing (with apologies to Bob Dylan).


It has been my intention to keep this blog focussed solely on peloton dynamics and its analogs, but I thought I would stray briefly across the boundaries and throw in my two cents worth about the historic third match yesterday between AlphaGo and master Go player Lee Sedol, summarized in this Wired article.

While I had not originally planned to watch the live game on the internet, serendipity led me to witness an unforgettable 3 1/2 hour game and move-by-move analysis by Michael Redmond, an American professional Go player, who has achieved the highest rank of this Asian dominated game. Certainly without Redmond's capable and animated analysis, the game would have been lost on me, although I could follow along roughly with my own crude evaluation of how the balance of advantage was unfolding. Redmond was accompanied by Go E-Journal Managing Editor, Chris Garlock, whose bias in favor of Sedol was palpable and contagious.

With Korean Lee Sedol down two games to nil in the best of 5 match, the third game was do or die for Sedol, and an historical milestone for the power of artificial intelligence, and for the AI and Go communities.

I am not a Go player, but growing up I learned the rules and played a couple of games with my brothers, who played among themselves enough to become competent players. I certainly remember the Ko rule, a situation in which players can alternate taking a single surrounded stone, but who must play an intermediate move elsewhere before mirroring the Ko move. Michael Redmond remarked that an AI Go player version from a few years ago ran into difficulty around Ko moves, which the underlying computer algorithm, based on Monte Carlo simulation methods, was not well equipped to handle. Redmond remarked, however, that even in October of 2015, AlphaGo demonstrated competence around Ko moves. Despite this, Redmond noted there were rumors that colleagues had advised Lee Sedol to induce AlphaGo into error by drawing AlphaGo into Ko moves that might be difficult for the computer algorithms.

Each player was allotted a total of 2 hours for their moves, with three 1-minute overtime periods per player, which re-started if a move was made before the expiration of the 1-minute period. With about 40 minutes remaining on AlphaGo's clock, Sedol was down to his three overtime periods, and Redmond was predicting an AlphaGo victory of about 60 to 30 (total territory claimed).

With one 1-minute period remaining (again, re-started if a move was made before expiry), Sedol displayed remarkable brilliance under enormous pressure by deftly guiding AlphaGo into a sequence of Ko moves. Many times Sedol placed his stone with one or two seconds remaining. Meanwhile, in the commentator's box, Redmond's hands were flying across the working board, demonstrating variations in play, computing the relative "liberties" (viable options surrounding a critical point of play), and continually declaring that AlphaGo had the advantage. The game was becoming increasingly complex, and the likelihood of Sedol making a crucial mistake was frighteningly high, while AlphaGo still had several minutes of regular play time in hand.

What Sedol did was utterly brilliant, but what I witnessed was almost a kind of AlphaGo mockery. Twice Sedol and AlphaGo exchanged Kos, until AlphaGo surprisingly placed a stone in an uncontested opposite corner of the board. Redmond and Garland, scrutinizing variations, missed seeing AlphaGo's move, and when they looked back at the actual game board to the locus of Ko exchanges, they were momentarily confused as to where AlphaGo had placed its stone. Earlier, Redmond pointed out that once AlphaGo computes the probabilities of an advantage in a particular region of play, if it "feels" it is ahead, it will place a move elsewhere on the board. At an earlier stage of play, AlphaGo made another such move, which Redmond said actually allowed Sedol to recover from a losing position.

On one hand we see how amazingly brilliant is the mind of Sedol, a man with a human brain, knowing that he can apply human pattern recognition and intuition, against a massively powerful computer, programmed with learning algorithms that can play itself continuously and update optimal solutions over the course of millions of iterations. Yet, when AlphaGo made its seemingly casual move in the uncontested corner in the closing moves of the game, preceding Sedol's resignation a few moves later, for me it was a crushing recognition that artificial intelligence has advanced into a realm in which there are no longer problems or fields of human inquiry that cannot be solved by artificial intelligence.

For instance, Google has access to vast libraries of scientific and academic journals, in addition to massive quantities of data that are constantly, and at increasing rates, being uploaded into the Cloud. With such data and access to knowledge, algorithms need only be implemented to ask questions about what information or solutions are missing from the scientific literature, and then in turn to synthesize the vast store of available information in conjunction with enormous quantities of data, and to answer itself the questions that it poses.

This makes me anxious. I don't know how scientists generally feel, but in the context of my own miniscule contribution to human knowledge, whatever that may be, I sense a sudden and sky-high spike in urgency now for humanity to maximize its creative resources in the sciences and all of academia. Study, learn, be bold and creative and push the boundaries of knowledge now; discover, cogitate and publish while you can before your quests and thirst for knowledge are quenched far more rapidly and adroitly by machines with names like DeepBlue and DeepMind. Take the chance now, or it won't come again.

While there will always be ways for humans to satisfy their intellectual hunger, to justify their lives, to seek their own unique place amid the universal struggle to balance suffering and happiness, to me the defeat of Lee Sedol by AlphaGo represents a cross-roads for science and the human quest for discovery, as the form of the human contribution to science is bound to look very different in the coming years.



Sunday, October 25, 2015

Spider pelotons


How are pholcid spider collectives like a bicycle peloton?
Hugh Trenchard

Rebecca Boyle, for the New Scientist,  reported quite nicely one of the essential aspects of the peloton model discussed in my recent paper on protocooperative behavior [1] (free pre-print), which is that the model strips away the deliberate racing strategies and tactics of cyclists, and shows how certain collective peloton formations are dominated by basic physical/physiological principles.

Boyle also notes very well how these basic physical principles drive the changing shape of a peloton -- which was earlier presented in two collaborative papers [2,3] -- but she is circumspect in relation to other key points of my paper and only hints at something potentially more important in it. At the same time the headline emphasizes only the obvious point that weaker cyclists can gain advantage by drafting.

Key elements, not discussed in the New Scientist article, include that my peloton model shows: 1. a mechanism for how groups divide into smaller groups; 2. the threshold between cooperative behavior and predominantly free-riding behavior, and the features of these phases of behavior; and why these factors are significant in an evolutionary sense.

I have a lot of work ahead to show how the principles outlined in my paper may be applied to other biological systems, which is really the Holy Grail of my research. Rebecca Boyle intimates this in referencing fish schools and the comments of  James Herbert-Read and Shaun Killen. And while there is voluminous work ahead to show how peloton principles apply to schools of fish and flocks of birds, it is also instructive to consider how the principles apply in systems that are less obviously physically similar to a peloton.

So, here I explore, just a little, how peloton principles apply to a less obvious analogue: social spiders. Unlike the uniformity of movement we see in schools and flocks, spiders tend to move about in different directions and may be separated by considerable distances, and there are few obvious similarities in group shapes and formations, nor is there an obvious energy savings mechanism among spiders, like drafting.

So the question is, how is this:


Fig 1. South American social spiders feeding on katydid 

...like this?

Fig. 2. Tour de White Rock men's pro-1,2 criterium (2013)

There are some superficial similarities: they are both groups of organisms; individuals are assembled fairly near each other, and they both have cool shadows!

I suggest there are also deeper similarities. To see how, let's look at some research by  Elizabeth Jakob about  group dynamics of pholcid spiders [4]. I'm no spider expert, but note that pholcids are not the same spiders as the social spiders in Fig. 1; as I understand it, pholcids are more like the daddy long-legs we commonly see around B.C. I used Figure 1 because it shows a nice collective of spiders*.

For pholcids, groups can be as large as 15, but the most common composition is two sharing a web -- one small and one large spider [4].

Paraphrasing from the Abstract [4] Jakob conducted experiments to examine the role of individual variation in the dynamics of pholcid group formation. Pholcids (specifically H. pluchei) either share webs or live alone, and individuals shift frequently between these conditions. A spider's decision to move is influenced by its size and recent feeding success. Small hungry spiders joining a web already occupied by a larger one were more likely to abandon the web and build their own than small well-fed spiders. The reverse is true for large hungry spiders, which were more likely to stay in a web already occupied than a large well-fed spider.

Jakob explains why, at p. 18:
Hungry small spiders, closer to the bottom of their energetic reserves, may be more sensitive to the negative effects of competition, and thus more willing to take the risk of abandoning a web for potential gain...The behavior of the larger spiders is less obvious: why do well-fed spiders leave groups? One hypthesis is that H. pluchei judge each others' fighting ability by weight. Spiders that are very similar in size fight most intensely (Jakob 1994). Well fed instar 5 spiders have been close enough to the resident's weight that high-intensity potentially dangerous fights were likely.
For more background, Jakob reported in [5] that for H. pluchei (quoting from the Abstract):
The main benefit of group living is likely to be the reduction of the cost of web building when spiderlings take advantage of webs built by larger conspecifics. 
So this is where we can try some peloton analysis.

In [1-3] we apply an equation that I call the "peloton convergence ratio" (PCR), which is really just a description of the coupled, interactive, relationship between drafting and non-drafting cyclists.
PCR = (P_front * d) / MSO_follow
  • P_front is the power output of the leading, non-drafting, cyclist, who sets the pace for the group. 
  • d is the drafting coefficient (i.e. the ratio of output of the drafting cyclist to the output of the non-drafting cyclist; the actual energy savings is 1-d). It could just as easily be called the "power-output coefficient for coupled cyclists", and is a ratio that describes the relative energetic costs between them, given that a drafting cyclist's output is less than that of the pace-setter. This conceptual subtlety is important when we identify analogues in other systems. 
  • MSO_follow is the maximum sustainable power output of the following, drafting cyclist. 

The MSO_follow of the following rider can vary; it need not be static. In other words, MSO_follow can change according to fatigue or hunger. While we did mention the variability of MSO, in none of [1-3] did we actually model it. However, such a variable MSO is equivalent to changing the speed of the front rider: as the front rider accelerates, the closer that puts the follower to his/her MSO (assuming the follower is seeking to hold the pace of the pacesetter). Similarly, if you reduce the rider's MSO due to sudden fatigue, or the "bonks" due to hunger, he/or she is driven to his/her MSO at a lower speed.

When PCR > 1, coupled cyclists diverge. This simply means that the front cyclist is riding faster than the follower can sustain, even by drafting. 

For a small hungry pholcid who decides to abandon the web, we may describe the condition under which it leaves as SCR < 1 (where SCR is "spider convergence ratio").  This is the inverse of the cyclist situation and I will show why, but SCR, like PCR, is still a description of the energetic relationship between the spiders sharing the web, and SCR < 1 describes the threshold at which the smaller pholcid is forced to diverge from that energetic relationship. We can think of this relationship in terms of two pholcids sustaining their individual energetic requirements given that food is distributed between them; and given the smaller spiders' food intake is determined by the larger spider; and in view of energetic costs that are saved by the smaller spider.

In this case, the energy savings is enjoyed by the smaller spider because it saves the cost of constructing its own web by joining a web already constructed by the larger spider.  The food energy permitted for the smaller spider's consumption is set by the larger spider, who dominates the web. To understand how these factors might work for spiders and how we might model them, let's first outline some other pholcid behaviors that Jakob identified.

In [5] Jakob observed that (quoting from the Abstract): 
...when prey were introduced into group webs, the largest spider in the web that detected the prey won the prey approximately 77% of the time. In spite of this cost, spiderlings were found in groups more often than expected by chance.
Also, in [4] Jakob observes that the energy cost of generating silk to make a web is equivalent to about 9 flies, or an average of 4-5 days of foraging. Spiders normally capture 1-2 flies/day.  Young pholcids can survive up to 3 weeks without without food (21 days), and adults up to two months [4] (60 days).

Now we can further explore the notion of the energy savings quantity and what I call the "StarvationCoefficient". If we say that the StarvationCoeff is the equivalent of the drafting coefficient for cyclists, intuitively it doesn't seem to match: the drafting coefficient implies a reduction in output for the drafting cyclist relative to the lead cyclist, and how does this equate to a spider that saves the cost of building its own web by joining one already constructed?

Consider that the drafting coefficient is really a ratio of the relative costs of a leading and drafting cyclist. The cost is higher for the leading cyclist in terms of power output, and lower for the drafting cyclist. The actual energy savings for the drafting cyclist is 1 - d. For example, if the power output for a drafting cyclist is 300W and the power output for the pace-setter is 400W, then the drafting coefficient is 0.75 and the energy savings 25%.

For pholcids (or any organism for that matter), costs are daily energetic demands that prevent the ultimate cost of death by starvation. If we know how long a spider can go without food before death by starvation, and if we know the costs incurred in building a web, we can show the cost of web construction as a ratio of the ultimate cost of starvation.

I calculate the proportionate energy saving to a young pholcid by not building its own web is between 19% (4/21 days) and 23% (5/21 days), and between 6% (4/60 days) and 8% (5/60 days) for an adult. These values represent the energy saved by not building your own web, while the StarvationCoefficients -- the ratio of the reduced time to starvation due to time lost taken to build your own web, to the total cost of living to starvation (1 - energy savings) -- are 0.81, 0.77, 0.94, and 0.92 respectively.

There may be better ways of determining the energy savings benefit of not building a web, but it seems the time saved from imminent starvation is a reasonable quantity by which to develop a model that accounts for the energy savings of not building a web. There may be other factors too that contribute to energy savings or costs, but let's use this StarvationCoeff and the 19% and 8% energy savings and corresponding StarvationCoefficients to develop a model in principle.

Now let's consider the PCR equation and how it is equivalent to the pholcid situation. First, PCR describes the coupled relationship between a lead cyclist and a following drafting cyclist. The lead cyclist sets the pace for the follower, while that pace is some proportion of the maximum ability of the follower.  For pholcids, it makes sense to consider the larger spider who has constructed its own web as the "lead" spider who dominates the web and dictates what proportion of the total food available goes to the smaller spider. In this way, the larger spider "sets the pace" for the smaller spider in terms of food energy available. As noted, Jakob found the dominant spider won food 77% of the time [5]. So, roughly speaking, if the web catches 2 flies/day, the big spider will get 1.5 spiders, while the small spider gets 0.5 flies/day. This may or may not meet the needs of the small spider. But if 0.5 flies/day is insufficient for its needs, eventually the small spider will reach a threshold point of hunger and will be forced to abandon the shared web in order to make its own.

So for pholcids, 
SCR =  EnergySet_lead / (EnergyRequired_follower * StarvationCoefficient_follower)
  • SCR is "spider convergence ratio". 
  • EnergySet_lead is the number of flies/day allowed for the smaller spider as determined by the larger spider.
  • StarvationCoefficient_follower is the energy cost in days without food as a proportion of the total cost (starvation) in days to starvation, minus the number of days saved by not building a web (or some other energy saving not considered here); i.e. StarvationCoefficient = (1 - energy saved by not building a web).
  • EnergyCost_follower is the minimum number of flies required per day for the smaller spider to sustain life.
So for pholcids, let's use energy savings 19% for young ones (StarvationCoeff = 0.81), and 8% for adults (StarvationCoeff = 0.92). So let's say the small pholcid on a web gets 0.5 flies/day, as determined by the large spider, but it actually requires 1 fly/day for sustenance. 
SCR = (0.5 flies/day) / (1 fly/day * 0.81) 
SCR = 0.62
In this case, since SCR < 1, the small pholcid is below its daily energy requirement even as offset by the energy saved by not building its own web, and because SCR < 1 indicates spider divergence, the smaller spider will abandon the shared web to make its own.

Note the PCR and SCR thresholds are inverted. For cyclists the PCR equation denominator reflects the following cyclists' maximum output;  the numerator is the required output set by the front cyclist, and so the drafting coefficient reflects the reduction of the required output. For spiders, the denominator is the minimum energy requirement for living. Also, for SCR, the energy savings component implies that the spider's minimum daily food requirement is less than without the energy savings component, so we multiply the energy savings by the minimum requirement in the denominator, and not by the food quantity actually permitted it by the larger spider (in the numerator).

So despite that the PCR and SCR divergence thresholds are inverted, the parameters as between cyclists and spiders are essentially the same in that they both include: 1. an energy quantity determined by a leading agent; 2. an energy savings mechanism and quantity; 3. an energetic limitation for the follower (either a maximum output, as for cyclists, or a minimum food requirement, as for spiders).

Let's see how this works for an adult spider whose StarvationCoeff is 0.92, but which needs say 1.5 flies/ day, but is only allowed 0.5 flies/day by the larger spider.
SCR = (0.5 flies/day) / (1.5 flies/day * 0.92) 
SCR = 0.36
Here SCR is even smaller than for the example above and, according to this result, this spider will likely have chosen to leave well before it got to this state of hunger.

What happens if either spider is fed before it wanders onto their neighbors' webs? Being fed means that current daily food requirement decreases. Let's say our adult spider is sufficiently well-fed so that for a few days its daily requirement is now only 0.1 flies/day. Say it's still permitted 0.5 flies/day by the dominant larger spider.
SCR = (0.5 flies/day) / (0.1 flies/day * 0.92) 
SCR = 5.43
Here SCR > 1, and the spider chooses to stay, since its needs are comfortably met. Eventually, as it burns off the food energy from its current state of satiety, the daily requirement will begin to increase again, and the SCR value will fall.

Now it becomes easy to find the rate of underfeeding the smaller spider will tolerate before it abandons the web. We simply apply the situation where SCR = 1, which is the point at which the equation states that the spiders will diverge, or the point at which the smaller spider's daily needs are not met, accounting for the energy saved by not building its own web. So with this we can show how hungry the smaller spider will become at its current rate (if 0.5 flies a day is insufficient to meet its daily caloric needs) before it leaves by determining the number of flies/day in the numerator when SCR=1. Let's say our adult smaller spider requires 1 flies/day for sustenance with the same StarvationCoeff of 0.92.  How many flies/day will it tolerate before it decides to abandon the web?
1 =  (x flies/day) / ( 1 * 0.92)
x= 0.92
Here, even though the spider requires 1 fly/day, it will tolerate 0.92 flies/day -- less than its daily requirement -- for some time because of the energy saved by not having to build its own web. Of course it can't sustain this indefinitely. As the spider becomes hungrier, its daily requirement also increases. At some point it will not tolerate only 0.92 flies/day.

In any event, this all seems somewhat promising, though there is more data in Jakob's paper [4] I can test against the framework here, and further work is required generally to develop a more complete model. Also, in terms of protocooperative behavior, a question arises as to how long a spider will go before it relocates, given its willingness to tradeoff the cost of building its own web for receiving less than daily required food needs.

To conclude, there is some promise in adopting an equation similar to the peloton PCR for pholcid coupling behavior. A spider coupling equation, SCR, involves a description of the energetic requirements of coupled spiders that is similar to that of the coupled energetic requirement of cyclists where there is an energy savings quantity. Namely, SCR incorporates: 1. an energy savings quantity; 2. an energetic output that is determined by a "lead" spider, and; 3. an energy quantity limitation for the following spider. These are the essential features of the peloton PCR coupling equation. The SCR equation may allow predictions of food quantities tolerated by pholcids before leaving webs and other related predictions. The apparent application of an SCR equation supports the universality of principles that determine the collective behavior of pelotons.

References

[1] Trenchard, H. The peloton superorganism and protocooperative behavior. Applied Mathematics and Computation Volume 270, 1 November 2015, pp. 179–192
[2] Trenchard, H., Ratamero, E., Richardson, A., Perc, M. A deceleration model for bicycle peloton dynamics and group sorting Applied Mathematics and Computation Volume 251, 15 January 2015, 24–34
[3] Trenchard, H.,Richardson, A.,Ratamero, E.,Perc, M., Collective behavior and the identification of phases in bicycle pelotons, Physica A: Statistical Mechanics and its Applications Volume 405, 1 July 2014, pp. 92–103
[4] Jakob, E. Individual decisions and group dynamics: why pholcid spiders join and leave groups, Animal Behaviour Volume 68, Issue 1, July 2004, pp.9–20
[5] Jakob, E. Costs and benefits of group living for pholcid spiderlings: losing food, saving silk, Animal Behaviour Volume 41, Issue 4, April 1991, pp. 711–722

* Figure 1 also comes with a creative commons licence.

Image credit (Fig. 1)
South American social spiders feeding on katydid
© Ken Preston-Mafham / Premaphotos Wildlife
Premaphotos Wildlife
Amberstone, 1 Kirland Road, Bodmin, Cornwall
PL30 5JQ United Kingdom
Tel: +44 (0) 1208 78 258
Fax: +44 (0) 1208 72 302
library@premaphotos.co.uk
http://www.premaphotos.com

Figure 2 is from video footage taken by either myself or Ash Richardson, I'm not sure which.

Saturday, August 29, 2015

The peloton superorganism and protocooperative behavior - a brief overview


My latest paper The peloton superorganism and protocooperative behavior is published at last. Obviously any published paper requires a lot of work, but for me this one has required the most so far. I began writing this in late 2014, and it was accepted for publication in early August 2015 after the usual process of review and revisions. I would like to think that while this one was more work than my previous papers, it will have the most impact.

I. Protocooperative behavior

Basically the paper argues for a new concept that I call "protocooperative behavior", or PB for short. While the evidence I present is in relation to bicycle pelotons, I suggest that it applies to any biological system in which there is some energy savings mechanism, such as may be found in bird flocks, or fish schools, among others.

Two phases of PB

PB is defined by two phases of behavior: a phase in which cyclists (or other organisms) proceed at a sufficiently low output or speed for cyclists to pass each other and to share the most costly front position(s); a second phase in which cyclists can maintain the speeds of stronger front riders, but cannot pass them. In the low speed phase, the peloton is high density, and in the higher speed phase, the peloton is stretched in single file. At a second threshold, cyclists decouple and diverge.

Phase 1: Low speed, passing and cooperative sharing of costly position

In the low-speed phase, cyclists can pass because to do so is well within their metabolic capacity. To simulate passing, simulated cyclists are programmed to accelerate randomly within a range of speeds up to their maximum capacities. In this low speed phase, cyclists naturally share the most costly high-drag front positions.


Phase 2. High speed, no-passing, free-riding phase

As speeds increase, cyclists' capacity to pass diminishes until a threshold is reached when they cannot pass at all, yet they can still keep pace with stronger riders ahead (the stretched phase). This is possible due to the power output reductions afforded by drafting. In the stretched phase, cyclists are free-riders by physiological necessity, not by choice.


Individual and team strategies need not be modeled

These behaviors -- passing behavior/ sharing the most costly front position, and maintaining speed of stronger rider while being unable to pass -- need not be modeled by strategic probabilities or game theory. The behaviors self-organize from principles of energy savings, cyclists' maximal capacities, their current output, and a deceleration parameter that is triggered when cyclists are effectively driven over their maximal outputs. This is one of the main features that distinguishes this model from more standard models of cooperative behavior; i.e. no strategies are introduced into the model to generate cooperative behavior.

In that vein, team dynamics are not modeled, so circumstances in which teams might dominate the front at low speeds are not considered. This is important because, in natural biological collectives, we are more likely to see random passing based on inherent natural capacities, and less likely to see "teams" motivated by some tactical reason to dominate given positions. We may see something like that in nature, but I suggest such team-like domination would be observed among a comparatively advanced evolutionary stage than the primitive dynamics I am modeling.

The threshold between the phases

There is a clear threshold between a high-density passing phase, and a stretched phase. This threshold is demarcated by the equivalent of the coefficient of drafting. My paper sets out the details of how this works, but simply put, it is a function of the relationship between the coupled outputs of the cyclists, their maximum capacities, and the energy saved by drafting. I illustrate the transition here:

Figure 1. The protocooperative behavior threshold (pbt), approximately equivalent to the coefficient of drafting (d). See paper for details relating to "PCR" and Figure 2; Appl. Math and Computation Vol. 270, 1 Nov.  2015, Pages 179–192.


Cyclists sorting into ranges of output capacities

Additionally, cyclists (or any organism) exhibit a heterogeneous range of maximal outputs (i.e. they are not all the same). If the range of outputs is sufficiently broad, when the strongest riders drive peloton speeds to near maximums, the peloton tends to sort into subgroups in which the range of outputs is equivalent to the energy saved by drafting. My paper details why this is so.  Below I illustrate the group sorting hypothesis.

Figure 2.  Illustrating group sorting where, after speeds are driven to maximal speeds by the strongest riders, the range of maximal capacities within groups corresponds to the energy savings quantity.

II. Significance 

1. The two phases of PB and the threshold between them suggests a primitive form of cooperative behavior that does not rely on any evolutionary strategy (hence "proto" cooperative behavior) -- this proto-behavior may precede other evolutionary mechanisms for cooperation, such as kin selection, reciprocity and others; i.e. it is simply a function of changing speeds and organisms' outputs, coupled by an energy savings mechanism.

2. In its most primitive form, cooperation can't occur unless group members are below a critical threshold of outputs; i.e. they must have some spare energetic resources before they can cooperate, a la the quote from Roosevelt in the paper. So, if resources are strained such that group members are taxed to their physical limits, they will not physically be able to cooperate. This suggests that cooperation evolves in circumstances of "luxury".

As discussed,  the critical threshold for "luxury" corresponds to the variation range of maximal outputs that corresponds to the energy savings quantity (1 - d, where d is the energy savings coefficient): when organisms operate below this output threshold, they can cooperate; when above it they can free-ride, but cannot cooperate (Figs 1 and 2), up to a second de-coupling threshold.

So, as group member outputs are increased (and by similar process, resources become scarce), fewer and fewer among the group are within their limits of "luxury" and capable of cooperation; i.e. the strongest cooperate, while the weaker engage in free-riding behavior. Conversely, as outputs fall, or resources become more abundant, cooperation tends to be more widespread, even if it is less necessary for the survival of the group.

I have not researched the literature to find support for this, but it may be seen to be somewhat at odds with some current thinking -- for example, the "ecological constraints hypothesis" suggests that cooperative parenting occurs when resources are scarce [1] rather than when resources are more abundant, as I have suggested. While I need to study the literature a lot more, there may not be an inconsistency at all: what we may be seeing in circumstances of ecological constraint are situations when cooperation is narrowed to the stronger members of the group, while there is an increase in the number of free-riders. So, as a simplistic illustration, the weaker, young members may be fed by the cooperating stronger "parents" for an increased period of time (greater free-riding) before the young are permitted to feed on their own.

3.  In some models, such as that of Aviles [2], a cooperation parameter, y, is introduced which, when adjusted, generates different kinds of collective behavior.  However, at least insofar as [2] is set out, there is little or no explanation of what criteria are required to adjust the y parameter.  The peloton model suggests criteria for tuning that parameter as a function of individual maximal capacities, current outputs, and the energy savings quantity.

4. By modeling a mechanism for group sorting and the resulting range of differences among members of each group, we have a testable mechanism for niche formation, speciation, and group heterogeneity.

Although I didn't say this directly in the paper, here I go so far as to suggest that many branches in the evolutionary tree can be traced to this sorting mechanism. The model suggests an evolutionary mechanism that permits for the shedding of weak group members who are therefore readily susceptible to predators, or who become isolated and lose the opportunity to reproduce.  Where whole groups divide and separate, members of each may reproduce, but possibly in very different environments; e.g. one group might "make it over the Himalayas" (thank-you Ross Hooker for the illustration) into the rain-forest, while another may may end up in the high mountains. Indeed it may be possible to show mathematically how the entire range of differences among all existing species conforms to energy savings quantities at critical group sorting points. This would not be an easy task by any means, but it would have to begin with identifying the various energy savings mechanisms enjoyed by different organisms, and the energy savings quantities.

Of course, it is not necessarily true that all biological collectives enjoy energy savings mechanisms. In fact a recent study that shows pigeon flocks involve increased energetic costs due to positional adjustments especially during turning motions [3]*.  Still, it is apparent that such mechanisms do appear in many species. Even collections of bacteria tend to move faster than individuals [4] which suggests an energy savings mechanism.

In terms of other applications, there are other types of energy savings mechanisms. For example, any sort of leader-follower situation involves some sort of energy saving for the follower. The person who tramps snow first saves energy for the follower; the person who cuts through the forest makes it easier for those who come after.  A teacher saves a student energy because the student does not have to reinvent the wheel, as it were. These are but a few examples. While PB and group sorting for such situations may be more complicated, it is open to consider how the principles I present may be broadly applied.

[1] B.J. Hatchwell and J. Komdeur  Ecological constraints, life history traits and the evolution of cooperative breeding. (2000) 59(6):1079-1086.
[2] Aviles, L. "Cooperation and non-linear dynamics: An ecological perspective on the evolution of sociality." Evolutionary Ecology Research (1999), 1: 459-477
[3] Usherwood, James R., et al. "Flying in a flock comes at a cost in pigeons."Nature 474.7352 (2011): 494-497. 

[4] Cisneros, Luis H., et al. "Dynamics of swimming bacteria: Transition to directional order at high concentration." Physical Review E 83.6 (2011): 061907.  http://www.physics.arizona.edu/~kessler/micro/PhysRevE.83.061907.pdf


* Usherwood et al. seem to focus on energetic costs incurred in banking and turning, as they demonstrate in this video: https://www.youtube.com/watch?v=ssUslfD47l0. I've observed pigeons to fly several km at a stretch in roughly mean straight trajectories, and there are pigeon races over hundreds of km, meaning they are capable of long flight: https://en.wikipedia.org/wiki/Pigeon_racing.  I would be interested to see if the same principles of increased costs in flock formations apply in these circumstances.