Contextual Contracts: On a Context-sensitive Approach to Contract Theory

Abstract

I propose a method for introducing ‘context’ within the contractual environment based on a simple and intuitive application of control theory. The approach permits looking at several interesting practical features of contracts, be they incomplete, complete or smart, within a single framework. I define a contextual environment with the help of an interaction between three distinct spaces: a market-based contractual space framed within a formal institutional space and an even larger cultural space. Each space is characterized by a governing law built on a selection of control mechanisms that differ in their approach as well as their reliance on information generated from feedback mechanisms. I suggest how these governing laws tie the contextual spaces together and present some ideas on how they evolve through their interactions with other spaces.

JEL: D20, D86, P50, Z10

Keywords

Contract theory institutions control process theory culture context

Background and Motivation

The two key sources of incompleteness in contracts are the indescribability of future states and the concerns over the verifiability of outcomes by third parties.¹ The fundamental theoretical critique of this premise is that an appropriately devised implementation mechanism for preference revelation can ameliorate concerns arising from the abilities of parties participating in a contract to predict aspects pertaining to future states. Such a mechanism makes the indescribability of future states irrelevant to whether current contracts can be written comprehensively enough to reduce concerns about adverse incentives in investment, the prospects of renegotiation or the possibility of contractual breakdown (Maskin & Tirole, 1999). This bolsters the alternate perspective of contracts that are ‘complete’ in the sense that they do not require renegotiation since they perfectly foresee all variables that eventuate in all contingencies.

Yet, it is incontrovertible that incomplete contracts do exist in practice.² A possible explanation is that they are unavoidable in complex environments, which are, in general, environments where the sources for holdup are myriad. Therefore, making a contract verifiable by a court would require considering an expanding set of possible situations that might impact investment incentives (Segal, 1999). It has also been suggested that the incompleteness of a contract might be seen as a deliberate mechanism in and of itself; it can serve as a focal point within a contract to generate desirable behaviour. An incomplete contract can set expectations for involved parties (Hart & Moore, 2008).

A principal point of difference between the approach of mechanism design and complete contracts and that of incomplete contracts and theories of the firm are in the source of the rules that govern behaviour in strategic contexts. The former adopts an approach whereby some ‘social planner’ can design a mechanism to effect the maximization of a given social welfare function (and implements the associated rules), whereas the latter allows the parties to a contract to devise rules that reflect their extant bargaining powers.

However, this debate is largely ‘context free’; partly, this is necessary since the removal of context permits useful generalizations and baseline models, but partly it is also because context is simply too far beyond the purview of economics alone. Whatever the reason, it is indubitably a shocking lacuna, since it is plainly obvious that context can and does matter to contracts for a number of reasons.

First, ignoring context perforce necessitates relegating the impact of second-best institutions on contracts. These institutions come into being as a result of the inability of the ideal first-best institutions to provide a set of rules that accord with a specific environment representing positive frictional costs. In a real-world setting with second-best institutions, new rules are often discovered, old ones are often modified, and first-best rules are sometimes adopted, but they are also often rejected outright and sometimes contextualized.

Second, the contextual environment and the market interact through the contractual process. Adjustments in the contract occur via a process of tatonnement by both agents and social planners over the position of several contextual variables that span the polity, economy and society, and cannot be taken as fixed in either time or space.

In other words, what I propose in this article is that contracts are incomplete because ‘experience’ bridges the space between the extant conditions under which the contract was written and those that impinge on the contractual environment over the duration of the contract.³ Likewise, in an environment of second-best institutions, social planners may only implement mechanisms involving adjustments to rules that do not contravene those rules that enable the instantiation of the institutions themselves. In other words, endogeneity in the contractual environment between the sources of rules that enable contracting in the first place is what makes context relevant to the nature of the contract itself, as well as providing a role for experience in learning how to contend with its nuances.

Therefore, the contention outlined in this article is that the difference in perspective between implementation and incomplete contracts is useful to the study of context, and, indeed, the opposite is arguably true in equal measure; some differences in perspectives on contracts might be resolved by appealing to the institutional and social context of a market. I suggest how an overarching framework that binds both these approaches within the backdrop of a contextual environment might be modelled.

Information plays an important and multifaceted role in the approach presented here. I develop a contextual environment as a composite of arenas or ‘spaces’ with increasing orders of change in time, each of which is governed by a class of governing laws that form the basis for tuning the control effort exerted within a contract. A contract, in this article, is seen as either a formal or an implicit agreement for the exercise of control towards the achievement of some target; institutions help contextualize contracts by essentially only permitting a subset of control options among all the possible variants that might exist. Once a particular option is selected, it then behaves like a regulating valve for control effort, metering the degree of information that it permits within a contractual space usable for the adjustment of the terms of control. At this level, contracts are incomplete because the control option has to be managed and altered—that is to say expertly tuned—through experiential knowledge to perform at its best.

The approach I advocate for contextual contracts in this article borrows from control process engineering (CPE), which is a branch of engineering that deals with controlling a vast variety of dynamical engineered systems using programmed controllers, from autopilot systems in aircraft to the movements of robots. The benefits of using this approach are principally two. First, CPE methods are now sophisticated enough to comfortably deal with out-of-equilibrium dynamic systems using several methods that allow subjective and contextual experience, such as intelligent controllers and fuzzy learning. This prevents their application to contract theory from being seen as overly deterministic. Second, and on the other hand, the rapidly increasing adoption of blockchain technology has enabled the creation of fully decentralized and autonomous organizations, as well as partially autonomous applications within enterprise environments.⁴ Such ventures are built on the premise of ‘smart’ contracts, which should be seen as contracts that seek to automate the obligations between the parties to a traditional contract. The contextual contract approach in this article—built on the principles of control engineering—provides a useful conduit for examining such contracts just as flexibly as it does for traditional contracts.

The relevance of an institutional context to the contractual environment is not unrecognized in the literature. Two articles, that also provide useful surveys of the literature, are especially illustrative. Rodrik (2008) suggests, significantly, how a misplaced emphasis on creating a set of first-best institutions misses variances in contractual performance. Schwartzstein and Shleifer (2013), in examining investment decisions by firms in socially useful projects, suggest regulation and litigation as distinct institutional forms. Whether or not compliance with regulation should absolve a firm from litigation naturally impinges on the types of contracts a firm would willingly entertain. The approach in this article is closest in spirit to Tirole (2009), where contractual incompleteness rests on costs associated with acquiring information that would reduce the possibility for renegotiation. These cognitive efforts entail transactions costs, thus leaving the scope for contractual incompleteness to remain. I liken these transactions costs to a larger set of frictional costs and build more character into the context by suggesting an approach that, I hope, will encourage cross-disciplinary research in this broad area.

The Contextual Contract Environment

The need to reach across disciplinary lines in addressing the topic of a contractual environment becomes evident when we see ‘context’ as being defined by factors that are beyond the usual scope of economics.

To motivate our foray across a disciplinary premise for our set-up, it is worth considering a broad analytic framework, such as the one proposed in Williamson (2000). Taking stock of the state of knowledge in institutional economics, the article proposed four levels of social analysis in a hierarchical set-up, where the topmost level frames the constraints on the levels below.

At the top of the hierarchical structure lies a social embeddedness level that concerns itself with norms, culture and tradition. The word ‘embeddedness’ has usefulness in its ability to signal that the frequency of change at this level is very gradual, measured at the broadest timescale, perhaps centuries. Below this level is the formal institutional environment, providing the formal rules of the game; here, the frequency of change is measurable in decades. The third level from the top is the level of governance, comprising contracts that frame transactions, and the proposed frequency for change here is in the vicinity of a year. At the lowest level of the hierarchy is the level of resource allocation, where prices align incentives on a continuous basis.

This separation of institutional effects on the basis of orders of change is useful to describing context precisely because it does not preclude the possibility of feedback and interaction between different categories of social change in determining institutional outcomes. It becomes possible to accommodate a complex pattern of endogeneity across these determinants of a context and, over time, the contextual environment for contracts can also be expected to evolve in a variegated manner. This is a necessary premise, since research has clearly demonstrated an endogeneity on a number of levels: between culture—defined, broadly, as a set of beliefs and informal rules—and institutions (a useful overview of this strand of work can be found in Alesina and Giuliano (2015)); between certain political institutions and market outcomes (e.g., Acemoglu & Robinson, 2006); and between cultural and market outcomes more directly (e.g. Tabellini, 2010).

However, in developing the contextual environment for examining contracts, we only need focus attention at the top three levels of the Williamsonian hierarchy in this article.⁵ Figure 1 presents a visual illustration of the approach to a contextual environment, comprising three spaces; rather than traditional crisp sets, these spaces should instead be understood as sets with permeable boundaries permitting interaction and feedback in determining the relevant rules that frame a given contract. We shall see that it is the interplay among these spaces that helps determine the nature of the ‘game’ that frames the contract; moreover, this interplay can usefully be understood as a process for the modulation of information feedback. The contractual space, for example, interacts with the cultural space directly as well as indirectly through the institutional space. That such an interaction is worth examining has not the escaped notice of contract theorists who, for example, examine how social norms affect the effectiveness of traditional monetary rewards in an agency framework; Fischer and Huddart (2008) serve as an interesting example.

Figure 1.

Source: Author’s own.

The contractual space (CoS) is characterized by relationships that are based on some underlying market, generally involving an explicit or shadow price. Employment contracts, insurance contracts, contracts between suppliers and retailers, between landlords and tenants, between firms and investors, and so on, are all examples of contracts that would manifest themselves within the contractual space. All contracts within this space are enforced by either formal or informal institutional arrangements.

The institutional space (IS) comprises formal institutions. The type and specific features of the political and legal systems, the quality of bureaucracy, the effectiveness of regulators, the nature of organized civil society organizations, and so on, are relevant in defining the institutional space. Contracts are enshrined formally—a constitution or a law, for instance—and endure over longer periods of time.

Finally, the cultural space (ClS) is where ‘contracts’ are implicit, informal and of the longest duration. Factors, including culture, language, religion, unwritten conventions and traditions, and historical precedent, are relevant at the level of the cultural space.

Indeed, both the IS and the ClS are part of a broader complex of institutional arrangements that, collectively, provide both formal and informal rules and place explicit as well as implicit constraints on behaviour in a society (North, 1990, 2005). Their success at achieving the objective of enforcing rules permits all forms of contracts—at all levels of analysis in an economy—to come into existence and flourish. However, the key difference between them can be understood in terms of categorizing ‘institutional’ forms based on their relative embeddedness within the overall matrix of institutional arrangements; this difference can also be understood in terms of an institutional stickiness (Boettke, Coyne & Leeson, 2008), which captures the idea that there is an inherent inertia in attempting to alter institutional arrangements, and that this is a principal reason for why formal institutions are often unable to supplant informal norms.

The demarcations of contracts into the spaces are sometimes crisp, though often may be fuzzy; in other words, a contract’s reliance on a particular space, and type of institutional arrangement, can be seen as probabilistic rather than discrete. As a result of this, at least three further classes of transactions can be identified.

First, those transactions that are between the cultural space and the contractual space (which we can denote by ClS ) CoS). These might be market transactions that do not rely on formal markets or formal contracts, but on informal norms and traditions. Social capital and interpersonal trust are often relevant as real alternatives to the official enforcement of contracts, and this is a fundamental premise relevant to such contracts. Second, transactions between the institutional space and the contractual space (IS ) CoS) are the more traditional contractual market transactions that are expressly sanctioned and enforced by an institution belonging to the formal institutional space. Finally, transactions that are between formal and informal institutions may also be considered (ClS ) IS). An example of this might be studying the impact of religious adherence on the stability and type of political institutions within a country, or establishmentarianism more generally.

In keeping with the spirit of the Williamsonian hierarchy, the overall contextual environment consists of all three spaces such that the institutional space is constructed within a cultural space and the institutional space, in turn, constricts some though not necessarily all of the contractual space. The figure underscores the idea that this approach rests, principally, on second-best institutions that are a function of their cultural context and, significantly, that this also creates a second-best contractual space, where even traditional market contracts are influenced by both their contextual institutions and their cultural context.

Any approach to modelling the dynamics of this framework to contextual contracts is complicated principally by a need to allow for a rich characterization of the context within each space; moreover, the model should also then tractably enable adjustments within each space based on agents learning the characteristics of their contextual environment.

It is worth recapping two observations that are relevant in sorting out the merits of any proposed modelling approach for contextual contracts. First, the orders of magnitude for change in the spaces differ, becoming larger from the contractual space to the cultural space. This is obviously a crucial element to the contextualization of the spaces since their dynamics are delimited by change within the higher spaces. Second, the role of information, and, more specifically, that of feedback from a contractual arrangement that affects asymmetries in information, is critical to understanding how the spaces interconnect. Broadly, information feedback generated in each of the spaces governs learning, thereby permitting adjustments to the manner in which contracts operate within each space, and whether and how the spaces evolve over time.

In the third section, I first outline the nature of an overarching modelling approach, based on CPE, that helps us develop a framework suitable to being readily contextualized. Such contextualization can occur in two broad ways. First, by virtue of providing a set of rules—what I call a governing law—that each of the spaces provides for contractual arrangements that take place within each space or those that involve more than one space. We shall see in the following section that this admixture of spatial frameworks applicable to contracts shapes the contextual environment that inheres to a contract, and also enables us to deal simply with the observations on differences in institutional embeddedness and information feedback that we have made above.

Second, in the fourth section, we turn to an examination of how the model can be adjusted to accommodate learning; doing so makes it possible to incorporate a role for experience, expertise and subjective assessment of the specific context relevant to a given contract.

The Modelling Approach

The proposed model is an adaptation of a control process model (often known as CPE), commonly used in engineering dynamical systems.⁶ The fundamental application of CPE is where the error in a process variable is modulated by exerting a control effort. An illustrative application for CPE is for the cruise control function on a vehicle. The user inputs a desired target speed or ‘setpoint’ and an algorithm draws on the vehicle’s accelerative systems (viz. the throttle and drivetrain) and decelerative systems (viz. brakes and drivetrain) to eliminate the discrepancy between the setpoint and the actual velocity of the vehicle.

Here, we can adapt the most widely used class of control algorithms in engineered systems, known as PID control, and develop a parallel class of control algorithms, termed PRM, for the governing laws that operate in the spaces of the contextual environment.

There are numerous benefits to using a CPE approach that augur well for our purpose. First, and principally, the advantage is that it has been field-tested in numerous applications involving highly complex systems that are characterized by error and disequilibrium rather than stable equilibrium. While there exist other techniques to model complexity in dynamical systems, the benefit of CPE is that it focuses on the aspect of control, which is a key feature of contracts in our construction. Second, it mimics the problem at hand very closely. As we shall see below, the structure of the model recreates a contextual environment. The longer-term functioning of the process relies on the prior specification of an overarching rule drawn from a class of rules. Finally, the shorter-term functioning of the system relies on tuning the rule towards a specific target, and, most crucially, this tuning can be done based on user experience and expert knowledge.

Contractual Governing Laws as Control Processes

Any contractual arrangement relies on a set of enforceable laws. In much of economics, such laws are usually taken to be exogenously provided by a set of first-best institutions; the contractual space in our framework acknowledges this approach by permitting the direct provision of such laws from the institutional space or from the cultural space.

However, as discussed above, beyond contracts between economic actors alone, political and social contracts can also exist in a society. Therefore, we begin with the simplifying assumption that all contracts rest on what we can call a governing law, ψ. Further, we make the assumption that the governing law is provided by either one or more spaces depending on the entities involved in the contract.

Definition 1 A governing law is either intraspatial or interspatial. It is intraspatial if the law pertains to contractual transactions between agents operating within the same space and interspatial if at least two spaces are involved in the transaction.

Therefore, for a given employment contract, the relevant governing law might be ψ^CoS (e.g. for a worker contracted for a short period) or it may rely on ψ^IS"CoS (say, in a contract over the career of a worker and a labour union), or perhaps even ψ^ClS"CoS (e.g. if a multinational firm modifies employment contracts to reflect the cultural traditions of the countries where its subsidiaries are located).

Figure 2.

Source: Author’s own.

The governing laws can further be indexed by a country or region and then extended to reflect contractual relationships that exist between countries or regions so that $ψ_{i \to j}^{C l S \to C o S}$ may define a governing law relevant for the contractual space in j provided by the cultural space in i.

We shall see that these laws govern the types of adjustments that can be made to control adverse behaviour in a contractual relationship, and their specific form and approach vary according to context.

Control Mechanisms

Generally, all forms of governing laws can be classified depending on whether they possess one or more of the following mechanisms for exerting control in their contractual arrangement. Naturally, several features in an instantiated contract may replicate one or more among these three mechanisms for different processes—or perhaps even the same process—as stipulated by the contract.

Proportional (P), known as proportional control in CPE: This is the main mode of control since, by nature, a governing law in any institutional context permits the exertion of a control effort that is in proportion to the difference between some desired goal and the state of the process involving a defined set of agents. The mechanism of control in a contract employing proportional control would be such that party A only undertakes some control action against party B if adverse behaviour, or, indeed, the agent’s type, has been positively identified, and the magnitude of the corrective action is proportional to the size of the measured discrepancy.

Memory (M): The problem with a governing law based solely on proportional corrections is that the resulting contractual arrangement would perforce have to permit some degree of sustained discrepancy between the desired goal and the process. This would be the case if B has any incentive at all to not operate at the desired level when there is no corrective action undertaken by A, or if there are frequent external disturbances that prevent a proportional control response from being effective. Therefore, A would need to keep exerting corrective control even when B is at the desired level.

When this is a significant concern, A either must tolerate continually exerting some level of corrective effort at some positive cost or permit B to shade on performance whenever corrective efforts are 0. Clearly, if the costs of corrective efforts exceed the damage done by such shading, then doing nothing is optimal for A. Therefore, a governing law that relies entirely on proportional control would be appropriate only in those cases when such control effort can be exerted constantly and at low cost.

Alternatively, a contract can be devised to allow A to provide corrective effort based on memory of past performance, which is a feature that is based on the idea of an ‘integral control’ in CPE. The time over which a bias might exist can be adjusted in such a manner that, periodically, A exerts a larger control effort based on the discrepancy that arises in the process and is attributable to B’s adverse behaviour.

Rate (R): It is logical that, in conjunction, a governing law for a contract that is based on both proportional- and memory-based control would work efficiently in order to curb discrepancies arising from adverse behaviour. However, a contract might also include more sophisticated features that are based on the rates of increase in the discrepancy. When a rate control is considered for a governing law, not only does it increases the need for A to be informed continually of B’s behaviour in the process, increasing A’s monitoring costs, but it also raises the requirement for better quality of feedback on the error between the desired target and the process of the contract. For these reasons, rate control is seldom used in contractual arrangements and is usually associated with more sophisticated (i.e., experienced or even automated) controllers.

We can now turn to the two further definitions that are needed to characterize the idea of governing laws.

Definition 2: A governing law is any combination of the control mechanisms (such as PM, P only, M only or PMR)

The generic form for a PMR governing law in the specification of a controller resembles

ψ_{t} = P e_{t} + M (\int_{0}^{t} e_{t} d t) + R (\frac{d}{d t} e_{t}),

(1)

where ψ_t is the corrective effort generated from the governing law at time t and P, M and R are known as the tuning constants. e_t is the error between party A’s desired target and the actual position of the process that involves effort by party B. Therefore, e_t = ϕ^d – ϕ_t.

Definition 3: An interspatial governing law sets a control band comprising an inner band of control ψ_in,t and an outer band ψ_ou,t. The spatial context of the higher order of change forms the outer band and the spatial context of lower order of change forms the inner band.

The bands are defined such that L_in < L_ou, where L is the contextual frictional cost (see Section 2.2.3).

Contracts

It is now worth reemphasizing the perspective we suggest here, by defining contracts contextually.

Definition 4: A contract is an application of any specific instantiated governing law over one or more observable variables with specific values for the tuning constants between two or more parties belonging to one or more spatial contexts.

Therefore, first, contracts can exist between parties who may or may not belong to the same spatial context. If they do not belong to the same space, then more than one governing law become relevant to the contract. Specifically, according to Definition 3, the governing law of the spatial context of higher order of change forms the outer band and the governing law of spatial context of lower order of change forms the inner band for the control process. Second, and fundamentally, a contract is a compact between at least two parties over the use of control in a given dynamical system. Parties use contracts as devices to assist in tuning the control mechanism specifications. They then refine their expectations from the contract based on their experiences.

Contextual Frictional Cost: Leverage

Control process theory provides useful guidance on the behaviour of practical systems based on two features that are intrinsic to our characterization of frictional costs in the contextual environment. These relate to time horizons over which the system is designed to operate (alternatively, the change order) and the notion of inertial change.

First, the order of magnitude is clearly important to the time it takes for a single control loop to be completed and, consequently, the length of time required for the generation of feedback on the errors. With unforeseen contingencies and a highly volatile environment, a longer horizon for a governing law reduces its ability to successfully account for the variations. Uncertainty about future events may indeed become so high that no control model based on the extant governing law may seem appropriate. This relates back to the role of feedback driving the control model. When uncertainty about the nature of feedback rises to such a degree that it loses all its value, the model is untunable.

Second, the idea of inertial change is central to control theory. Concepts such as ‘dead time’ and ‘stiction’ in control process theory are necessary since they recognize that a degree of control effort is usually required prior to seeing any movement in the output variable. This creates challenges in tuning any model appropriately. Likewise, such inertial friction is pivotal to characterizing context as well; I call it leverage to emphasize the fact that it is a variable that describes the purchase one gets from a given amount of effort.

Intuitively, the control action exerted by party A may not smoothly translate into the movement of the variable of interest pertaining to party B’s behaviour simply due to the fact that B may require an unknown level of action to motivate change, and, further, this resistance changes by context. This is also one of the clearest demonstrations of this approach’s ability to deal with context due to its recognition that A, with experience and knowledge of context, may improve at understanding how to motivate such change quicker by tuning the features of their contract more effectively.

Note that a contract’s leverage is derived from the governing law. For instance, for a contract that relies solely on tuning the proportional control mechanism for all efforts exerted by A, the leverage, L, determines the proportional control effort A needs to exert for a given discrepancy created by B’s adverse behaviour. A higher leverage would enable A to generate a higher realized control effort for a given amount of discrepancy with the same effort than if leverage was lower. Similarly, a contract employing memory would benefit from a higher leverage coefficient in the contract in order to amplify the effort exerted by A.

Features of the Governing Law

This section is intended for those readers who wish to examine how the elements of the PRM process, as the bases of a governing law, might work within any contract, regardless of which space they inhere to; readers can opt to skip this technical section without losing any thrust in the argument of this paper.

Assume that, at time t, the control effort exerted by A is κ_t and this still enables B to engage in some adverse or opportunistic behaviour, ϑ_t > 0, which results in a deviation from the ex ante optimally expected action by some ϕ_t units. So ϑ_t = ξϕ_t, where ξ > 0. A continually exerts control effort so long as ϕ_t > ϕ^d where ϕ^d is some tolerable level of deviation in observed behaviour that B is permitted by ψ.⁷

Such a contractual context ψ would have a leverage that measures how much ϕ_t changes when A’s exerted control effort changes. It is measured by

L = ϕ_{t} / κ_{t}

(2)

Therefore, L can be seen as B’s perception of the baseline constraint on ϑ_t when A’s control effort is κ_t.

Proportional

Control in this contractual relationship control is proportional and works so that

ϕ_{t} = L_{p} . κ_{t}

(3)

However, A’s control mechanism itself has an intrinsic efficiency, J, that determines control effort at any given time

ψ_{t} = J (ϕ_{\max} - ϕ_{t}),

(4)

where ϕ_max is a constant that describes the maximum distance that B can deviate when the level of corrective output that A can exert is at its highest possible value. So the quantity (ϕ_max ϕ_t) that enters A’s observation mechanism as an input gains strength by some factor J before it is outputted to the process as the corrective control output ψ_t.

Therefore, Equation (4) accounts for differing abilities; A might have contexts to ameliorate situations of incentive incompatibility using the extant. This control equation can also be rewritten as

ψ_{t} = J (ϕ^{d} - ϕ_{t}) + ψ_{B}

(5)

where ϕ^d is achieved when ϕ_t equals the setpoint.

ψ_B = J (ϕ_max – ϕ^d) is a bias constant that exists to counter inertia in the control process. It provides a measure for the control effort level; A would need to exert in order to ensure ϕ_t remains at its desired level in the absence any exogenous disturbance. Equation (5) thus shows how A’s control effort is a result of the error between the process variable at the theoretical goal in the presence of a bias as well as an intrinsic efficiency of the effort working through a particular contextual contract.

This form of control behaviour is called a proportional control system due to the leverage coefficient. The greater the error, the greater the control effort, and as long as error remains, A will need to continue to output a corrective level of control.

Memory

In the presence of disturbances, a governing law that restricts A to solely a proportional output of control effort will leave a steady-state error in ϕ(t). The agent, perceiving a proportional-only institutional control structure, could in fact increase ϑ(t) by undertaking strategic behaviour that increases the level of disturbance as perceived by the controller.

Proportional-only control can lead to persistent problems. To see this, simply assume that L = J = ψ_B = 1.⁸ If the tolerable deviation goal requires ϕ^d = 3 and ϕ_t = 2, this implies a deviation of ϕ^d – ϕ_t = 1. A’s control mechanism will amplify the control level to 2. However, since the output level of 2 will, in turn, cause ϕ_t to remain at 2, A will make no further changes and the error in deviation will remain at 1.

For a controller, A, who is permitted to exert control effort using memory of past behaviour, this process can be improved on. An integral term can solve this problem by generating output that is proportional not to the present deviation, but to accumulated deviations. Thus,

ϕ_{t} = \int^{​} L_{M} \cdot ψ ​_{t} d t,

(6)

which shows that deviation depends not only on the level of perceived control output but also on the length of time the control is applied. This form of control structure will not generally admit a steady-state error.

Integrative correctives are susceptible to the problem (known as the closed-loop problem) that a large leverage will allow aggressive corrective control usage to transform B’s deviation into another large deviation in the opposite direction, which may be counterproductive. The contract is then likely to suffer from the problem of oscillating between maximum and minimum control needed to be exerted by A. Proportional correctives would be unlikely to cause this, but on their own they are prone to bias.

Rate

Working together, the proportional memory control mechanisms are likely to be effective in controlling B’s behaviour, but it might still take too long to compensate for disturbances that impact the environment. A corrective effort that adjusts control by the rate of deviation is even more effective since it would generate a corrective control based on the time derivative of B’s deviation in behaviour from some ideal.

ψ_{t} = L_{R} (\frac{d}{d t} e_{t})

(7)

Rate control represents a very sophisticated control procedure, sensitive even to measurement error, but naturally requires a high degree of information. In a perturbed system, such a method of control would generate one large corrective control action to begin eliminating deviation immediately. If, for instance, a surge in ϑ(t) occurs at time t₀, the time derivative function of the agent’s deviation would demonstrate an even larger surge at t₁ since the derivative of a step function is an even more pronounced impulse function. Conversely, closer to the target, this sort of control would slow the output of control effort down. In short, it is a corrective mechanism that only a more informed controller could employ. The problem with a rate-based control mechanism is that it also crudes on its own without memory and proportional correctives and would make for a poor governing law for a contract.

On Experience and Subjective Learning across Spaces

PID control processes for engineering dynamical systems are often deterministically driven by some physical process. There need not be any role for agency, let alone learning and practical experience. However, this approach has the added benefit of also opening some interesting doors to accommodate learning and experience, making them especially attractive for contextualizing contracts.

We now turn to an overview of two promising avenues for such modelling approaches.

FLC

A commonly used model for PID tuning a dynamical system with subjective information is known as fuzzy logic control (FLC).

Fuzzy sets are sets with smooth boundaries rather than crisp and clearly demarcated ones that we are used to when we represent them on Venn diagrams. This is achieved simply by defining the membership of an element to a fuzzy set A—represented by a membership function µ_A—in terms of ‘degrees’, bounded between 0 (implying ‘no membership’) and 1 (suggesting ‘complete membership’). This function usually assumes a triangular or trapezoidal shape to enable an uncomplicated representation of the smooth edges at the set boundaries. However, quite significantly, the most important rationale for characterizing imprecision with a non-distichous membership function in fuzzy logic is to enable the use of a ‘term set’ comprising subjective linguistic descriptors.

Let us examine how fuzzy control for a governing law might work with a brief example.

The objective for the controlling actor remains to reduce the error in the controlled agent’s behaviour through the application of control effort. The error is first normalized using a normalization factor (derived as the inverse of the maximum permissible error) so that the normalized error is e _t = α_ee_t and the normalized change in error over a time period is

d e_{- t} = α_{d e} d e_{t'}

(8)

Table 1.

Fuzzy Control Rules

e_t	H	N	N	H
de_t	N	N	H	H
dc_t	S	R	S	P

Source: Author’s own.

Figure 3.

Source: Author’s own.

To generate the change in control output dc_t, the membership functions for the errors are used to create subjective rules, which are usually stated in a table, such as shown in Table 1. Here, H = High Error; N = Negative Error; P = Punishment; S = Status Quo and R = Reward. Then, using fuzzy set logic to combine the linguistic values for error and change in error, the linguistic change in control output is generated. Examples are shown in Figure 3. The process is completed by denormalizing the fuzzy control output using a factor, such as the inverse of the maximum permissible control effort.

Network Learning

Another benefit of using this approach to contextualizing contracts by using a governing law is that the governing law is shared across all members within a space. It thus enables the option of examining the structure of the spatial context in the language of networks. If all entities within a space can be represented by nodes, then direct links between nodes might represent an implicit or formal basis for a contractual relationship, but, interestingly, indirect links can also influence the behaviour of a node as can the links across the spaces in the contextual environment.

Consider some general approaches as illustrative suggestions. One manner of proceeding would be to assume that there are n players in each space and that they are all connected with one another by virtue of being members in that space. Each player has one of two choices she could make: if she plays by the status quo governing law, $\tilde{ψ}$ , she continues receiving the current benefit; if she adopts a rival, and potentially innovative governing law option, $\hat{ψ}$ , she gets a higher benefit with probability p or loses everything with probability (1 – p).

It can be shown for agents with Bayesian updating in their beliefs that, if p ≠ ¹/₂ then, over a long enough period of time, everyone within the space would eventually converge to adopting the same ψ. However, the result does not guarantee that convergence will necessarily be towards adopting the better option. It is intuitive to see that this is possible with pessimistic enough or sufficiently risk-averse agents and with a size of p such that the entities all stick with $\tilde{ψ}$ even if p < ¹/₂, and is thus the inferior governing law, while may not all swap to $\hat{ψ}$ even when p > ¹/₂ (Bala & Goyal, 1998).

Alternatively, more emphasis can be placed on the context of the network structure by considering a different modelling approach, benefit of which is in assessing the comparative tendencies across spaces for convergence from one of the governing laws to another directly given only some starting beliefs and the diffusion of information based on the propensities of actors to incorporate the beliefs of neighbouring nodes as well as relying on their own beliefs. This approach places a stronger reliance on context if one believes that initial propensities to rely on information from neighbours is sticky over time and particular to the context in which the contract is operating. The approach calls for specifying a ‘trust’ matrix T that contains directed weights between the n actors in the space. The starting beliefs actor i possesses are b_i (0) and can then simply be evolved over time through an updating process: b_i (t) = Rj T_ij b_j (t – 1), where R_j T_ij = 1 and T _ii > 0. Convergence operates similarly and is captured by b(t) = T^tb (0). Interestingly, the speed of convergence depends on the propensities of actors to trust others apart from themselves as well as the number of actors who hold influence within the space (DeGroot, 1974).

Finally, it is possible to examine the role of strategic behaviour more directly by examining the idea of the ‘cohesiveness’ of the contextual spaces. The cohesiveness of a subset of nodes on a network is determined simply by the maximal fraction of neighbours of a node within the set that remains within the set as opposed to its complement. In contrasting across spaces or subsets of nodes within a space, this is a useful concept to appeal in examining the role of context, since, among much else, there is a close relationship between cohesiveness and homophily.

Concluding Remarks

We have outlined the idea of using control process theory in the ambition to define the idea of a contextual contract and to assist a transdisciplinary approach to contract theory. The framework outlined in this article has the benefit of serving as a useful platform for collaboration between engineers and economists at a theoretical level on contract design, and with managers and users, more generally, at a practical level. At the very least, it explicitly allows the notion of context to be methodically integrated into this area of research. The overall approach relies on characterizing governing laws based simply on the type of control mechanisms that they permit for the control of adverse behaviour in implicit or formal contractual relationships. Difference in these mechanisms captures the essential elements of context both ex ante, but especially ex post, and create variance in the desired outcome.

Admittedly, this is a diverse enterprise, since the range of such contracts varies markedly, from implicit cultural contracts between a subpopulation and a transnational institution to a smart contract for a mobile app. However, it is also undeniable that social scientists routinely engage with such questions. Theories of democratization are illustrative in this regard. A case in point is the question of whether democracy is incompatible with the Islamic religion (a ‘civilizational’ question addressed as a relationship between the cultural and the institutional spaces) as has been argued by Huntington (1997), or whether it is the fact that oil dependence creates a rentier state and thus limits open and participatory government, which is a question better addressed between the institutional and the contract space (Ross, 2001).

Innovation is another illustrative example; it is still poorly understood why innovation is so rampant and fertile in some parts of the world and woefully absent in others. Factors proffered as relevant clearly run the gamut of all the spaces within the contextual environment, yet they are rarely seen as being relevant to a single eclectic portrayal of explanations rooted in context. Within the contractual space, the emphasis is on immediate incentives. Therefore, the focus here might be on the types of incentives generated by employment contracts and inter-firm relational contracts as well as from the incentives created from mergers and acquisitions that influence the size of the firm and the scope of its activities. Innovation is often also studied from the perspective of the institutional space. Here, the focus tends to be on broader incentives that usually pertain to the quality of institutions, the quality of regulation, copyright and patent law, FDI and trade arrangements, political regime characteristics, and so on. At the level of the cultural space, innovation is studied as the long-term conduciveness of socio-cultural environments to innovation. The long-term process of von Neumann’s technological singularity, for instance, covers an extremely long horizon and is, in the end, a proposition on the direction of innovation.

By examining a variety of factors simultaneously and incorporating rich insight within each space from experts into the specification of governing laws for contextual contracts, much quicker and more fruitful advance is likely possible.

Footnotes

Notes

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