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.


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