2. The rules account

The most popular and widely cited characterization of social institutions can be found in the opening paragraphs of Douglass North's monograph on Institutions, Institutional Change and Economic Performance:

Like many other scholars, North does not say explicitly whether he is giving a definition, an empirical description, or whether he is introducing an idealized theoretical concept for the study of institutions. Our first goal, therefore, will be to identify and lay bare the ‘conception’ or ‘account’ of institutions that is implicit in his work and in the work of other scientists.

The rule-based conception belongs to a venerable tradition that goes back to the founders of modern social science.Footnote ² It states what institutions are (they are rules) and what they do (they facilitate human interactions). Consider marriage for instance. Married couples have rights and obligations that indicate what they must and must not do when they engage in certain activities. In most Western countries both husband and wife are responsible for procuring the material resources to support their family, for example. They are both responsible for their kids’ welfare and education; they have a mutual right of sexual monopoly, and they are committed to support each other in case of need. The reason why such rules exist is fairly obvious: they help husband and wife attain goals that would be more difficult to accomplish if they acted independently, in an uncoordinated manner.

This idea can be generalized to many other cases: institutional economists like North have used the rules conception to study the way in which institutions facilitate economic growth, for example. Accountancy rules foster transparency and trust; bankruptcy rules reduce uncertainty when businesses fail; property rights encourage investments, and so forth. By inventing and following new rules people can overcome the natural obstacles that limit production, trade, and more generally hinder the welfare of a society. Another virtue of the rules account is that it is closely related to policy. Rules often emerge by trial and error and spread spontaneously by imitation, but they can also be designed and implemented by an authority by issuing laws and decrees.

Many rules, however, are never followed even though they are formally included in the legal system. In May 2010, for instance, ten French ministers proposed to repeal a law that forbids women to wear trousers. The law had been in place since 1799, although no one had tried to implement it for a long time. Rules like the French ban on trousers are ineffective, and raise a difficult problem for rule-based accounts of institutions. Why are some rules followed, while others are not?

In the case of the French ban on trousers, the law was simply forgotten. But some formal rules such as the speed limit are rarely observed even though they are universally known. In many North-American states many cars drive between 65 and 75 mph, for example, in spite of an official speed limit of 65 mph (Greif and Kingston, Reference Greif, Kingston, Schofield and Caballero2011). So clearly the formal rule is not effective – the real, informal speed limit is somewhere around 75 mph. But to say that 65 mph is not the ‘real’ rule leaves several important questions unanswered: What distinguishes ‘real’ from merely ‘nominal’ rules? What is the difference between the 65 mph rule and the 75 mph rule? Why do people comply with the latter but not with the former?

A sketchy explanation may go like this: the nominal rule is just a signal that indicates roughly what kinds of behaviours are expected, but no one believes that it will be followed strictly. It would be pointless for the police to fine all the drivers who exceed the official limit by a small margin (those who drive at 67 mph, say). It may be wise to sanction only major violations of the nominal rule and implement a stochastic strategy: fine every car speeding at 75 mph or more; fine some cars speeding around 75 mph; fine no car speeding at 65–70 mph. This would work reasonably well and would ensure that most people do not exceed 75 mph. Drivers have an incentive not to exceed 75 mph; the police has an incentive to tolerate those who do not exceed the 75 mph limit. If a naïve observer were to look at the traffic flowing down the highway, she would conclude that the effective speed limit is roughly 75 mph: everybody believes that one should not exceed that limit, and everybody's behaviour confirms that belief. The system is in equilibrium.

The preceding line of argument puts some pressure on the idea that institutions are rules. It suggests that they are perhaps better conceived of as actions that people have an incentive to make. As a consequence, one might think that an account based on the concept of equilibrium can incorporate incentives and make rules redundant. Rules cannot be institutions, the thought would be, because by themselves they lack the power to influence behaviour.Footnote ³

3. The equilibria account

The idea that institutions are equilibria of strategic games is central to another account of institutions that has been prominent in the literature for the last three decades.Footnote ⁴ Theories within the equilibria approach view institutions as behavioural patterns or regularities. For example, Andy Schotter – a prominent game theorist and experimental economist – defines institutions as ‘regularities in behaviour which are agreed to by all members of a society’ (Reference Schotter1981: 9). Such regularities ‘can be best described as non-cooperative equilibria’ of strategic games (Reference Schotter1981: 24), because out-of-equilibrium actions are unstable and are unlikely to be repeated in the course of many interactions.

An equilibrium in game theory is a profile of strategies (or actions), one for each player participating in a strategic interaction. Each action may be described by a simple sentence of the form ‘choose X’ or ‘do Y’. The defining characteristic of an equilibrium – what distinguishes it from other profiles – is that each strategy must be a best response to the actions of the other players or, in other words, that no player has an incentive to change her strategy unilaterally. If the others do their part in the equilibrium, no player can do better by deviating. Those who defend a pure equilibria account hold that institutions can be equated with equilibria that have certain properties. They maintain that recourse to rules is not needed. We will argue that this will not do, because rules play an essential role in achieving those equilibria that form institutions.

The first major breakthrough in the equilibria approach is due to David Lewis. Lewis (Reference Lewis1969) proposed to model conventions as solutions to coordination games with multiple equilibria. His analysis focused on games with symmetric equilibria in which the players do not strongly prefer to converge on one rather than another solution. A classic example is the ‘driving game’: drivers do not particularly care about keeping right or left, provided everybody does the same. The theory, however, can easily be generalised to other cases, where the payoffs are asymmetric and the players have different preferences about the outcomes. Here we choose an example that has been discussed in some depth in the literature, and that provides a simple model for the institution of private property.

The use of resources such as land raises a coordination problem if interests are served badly by two persons trying to use the same piece of land. The optimal solution in such cases is that one uses the land, perhaps to grow a crop, and the other abstains from using it to graze her cattle. The game of private property can be represented in strategic form using a matrix known as ‘hawk-dove’ in biology, and ‘chicken’ in economics.Footnote ⁵ For every piece of land, the players have to make a decision: in Figure 1 the strategy U stands for ‘use’, NU for ‘not use’. If they both decide to use the same land, the players will end up fighting, which is the worst outcome for all (0, 0).Footnote ⁶ If they both abstain, they will not clash but will miss the opportunity to use the land (1, 1). The best solution is to converge on one of the two equilibria in the top-right and bottom-left corners, where one player uses the land and the other lets him use it.

Figure 1. The private property game (hawk-dove).

The property game is a problem of coordination with asymmetric equilibria, depending on who is going to give way. But since the players are perfectly identical, why should one of them accept a lower payoff? Notice that the only symmetric solutions here are not only inefficient, but are not even equilibria of the game. As a consequence we should expect some player to deviate unilaterally sooner or later.Footnote ⁷

An obvious solution in such circumstances is to adopt a correlation device. A correlation device is a signalling mechanism that the players can use to coordinate their actions. Think of a traffic light, for example, indicating by means of different colours (red/green) who has the right to cross a busy road at each particular moment. In general, correlation devices need not be artificial tools built for a specific purpose. Any external mechanism may do, provided it sends reliable and correlated signals to the players. In the case of property, for example, the players may rely on a simple pre-emption device: whoever occupied the land first has the right to use it. The temporal order of occupation, or the sequence of the claims made by the players, is used as correlation device. Except in rare cases, this device provides unambiguous, correlated signals to the players. If they all follow this simple mechanism, fights should be avoided and coordination should run smoothly. No agent is served better by acting differently, on the assumption that the others follow the signal, which implies that the set of actions is a correlated equilibrium.Footnote ⁸

Technically speaking, this solution involves a set of conditional strategies. Each player conditions her move (U or NU) on her temporal and physical location relative to the piece of land. If she arrived first, the player uses the land, if she did not, she lets the other player use it. In a series of repeated encounters, the average payoff the players achieve will depend on the probability of occupying the land first. If the probability is roughly equal, for example, they will both achieve an average payoff of 1.5 units in the long run. But even if the outcome does not respect perfect equality, the correlated equilibrium of the property game tends to be more egalitarian than either of the two asymmetric equilibria of the hawk-dove game with uncorrelated (or unconditional) strategies.

This is similar to the solution of other problems analysed by Lewis (Reference Lewis1969), such as the driving game that we have mentioned earlier. In that case, the drivers condition their choices on the history of play. The only difference is that in the driving game the conditional strategy does not lead to a substantially different outcome than any of the two unconditional strategies (‘keep right’, ‘keep left’). In the property game in contrast it creates a new behavioural pattern, for none of the unconditional strategies can deliver symmetric payoffs. This capacity – the capacity to create new outcomes – is an important feature of many institutions, as we shall see shortly.

We will assume, for the sake of the argument, that this story gives an adequate (albeit simplified) account of the institution of property. Since real property rights involve more than the right to use, we shall use a star symbol to distinguish our simple proto-institution from its real-world counterpart. Property* is a correlated equilibrium of the hawk-dove game, in which one player uses and the other one refrains from using a piece of land, according to the pre-emption system.

Not all equilibria can be institutions, however. Two features of the private property game are important: first, it is a coordination problem; and second, the solution requires that the players correlate their strategies. The significance of coordination can be illustrated using a prisoner's dilemma game (Figure 2 on the left). Consider mutual defection. The pair of strategies (D, D) is an equilibrium, but intuitively it is not an institution. Why? The reason is that each agent can implement the rule independently. There is no need to coordinate strategies. In fact there is no reason to even think about the action of the other player: whatever she does, it is optimal to defect. That's why mutual defection in this game is often taken to represent the proto-typical failure of sociality.

Figure 2. A prisoner's dilemma and a stag hunt game.

To appreciate the significance of correlation, consider the stag hunt game on the right of Figure 2: both (S, S) and (H, H) are equilibria of this game. But (H, H) does not require that the players correlate their strategies. The minimum payoff is guaranteed, so one does not have to pay attention to what the other player does. Since in (S, S) correlation is crucial, but in (H, H) it is not, only the former equilibrium is an institution.

4. The rules-in-equilibrium account

So institutions must be correlated equilibria of coordination games with multiple equilibria. In a correlated equilibrium, as we have seen, the strategy of each player is conditioned on an event or signal sent by a coordination device. In order to achieve a satisfactory definition of institutions, however, we must introduce a third condition: representation, to capture the idea that the players must be able to represent the equilibrium in symbolic form. As we discuss in Section 5, this can be facilitated by so-called constitutive rules that have special symbolic significance. The reason why this third condition of representation is needed is that the notion of correlated equilibrium is far too permissive and would let in too many behavioural patterns that we would not intuitively consider institutions.

So-called animal conventions are a paradigmatic example. Consider Pararge aegeria, a butterfly living in the woodlands of Asia and Europe. Male butterflies patrol the patches of sunlight that appear on the woodland's floor, where they mate with females after a brief courtship. When a male enters a sunspot that is already occupied by another male, the incumbent attacks it. After a brief skirmish, the defeated butterfly leaves the spot. Remarkably, the intruder is nearly always defeated, and the incumbent nearly always retains its territory.Footnote ⁹

Pararge aegeria play correlated equilibria, and similar behavioural patterns have been observed in swallowtails, baboons, and lions. The standard interpretation is in terms of a repeated hawk-dove game. As a solution, these species have evolved pairs of strategies (‘conventions’) that minimize damage by granting the territory and the mating opportunity to the incumbent after a ritual contest. The biologist John Maynard-Smith (Reference Smith1982), who first used game theory to explain such behaviour, has called it ‘bourgeois equilibrium’.

Non-human animals solve coordination games using correlation devices, but animals do not have institutions. Since there are in nature a few examples of anti-bourgeois equilibria, the expression ‘animal convention’ seems to be more appropriate. Whichever equilibrium has been selected, however, the important point is that the strategies are biologically implemented or – in a broad sense of the term – genetically encoded in each species. A group of butterflies cannot coordinate on anything but who occupied the sunspot first. They can play this particular strategy only. They cannot invent a new equilibrium. Humans, in contrast, can: they hook onto different correlations, invent constantly new strategies, and dramatically enlarge the number of possible equilibria.

What distinguishes human institutions from the correlated equilibria of Pararge aegeria? The answer seems to be that butterflies react only to a narrow set of signals, such as who enters the spotlight first. A simple mechanism that links one type of stimulus with one type of behaviour guarantees coordination. More complex creatures in contrast are able to decouple stimulus and behaviour. They do so by adding an intermediate state – a representation of the environment – that they use to condition their behaviour (Sterelny, Reference Sterelny2003). Moreover, such complex creatures can condition their strategies on many different representations – many signals and many correlation devices. In the case of humans, we say that they can follow different rules. These rules are representations in symbolic form of the strategies that ought to be followed in a given game. Just like the rules account without equilibria is incomplete, so is an equilibrium-based account without rules. A satisfactory theory must combine the best features of both.

Notice that the concept of a rule is ambiguous. Sometimes we use rules to describe, and sometimes to prescribe behaviour; occasionally we use them to do both things at once. But these functions are conceptually distinct. Let us distinguish between agent-rules (or a-rules for short) and observer-rules (o-rules), respectively. An observer formulates an o-rule mainly to summarize others’ behaviour; an agent formulates an a-rule to summarize and to guide her own behaviour.Footnote ¹⁰ Equilibrium theories are observer theories, and so the actions of the players are described from an observer's point of view only. Formulating rules, however, may also facilitate convergence on an equilibrium, so an adequate theory must capture the fact that rules are used both to represent and to influence behaviour.

These insights can be combined into a coherent whole by stipulating that institutions are rules in equilibrium, where the rules are summarized by the agents using some kind of symbolic representation. According to Avner Greif and Christopher Kingston, for example

In a similar vein, Masahiko Aoki (Reference Aoki2007, Reference Aoki2011) emphasizes the importance of public representations or social cognitive artefacts. He proposes the following definition:

In a nutshell, the rules represent equilibria (or parts of equilibria) and help the players to exploit a particular correlation device. Let us see how this account works in the simple case of property*. Recall that the players (P1 and P2) use pre-emption as a correlation device. The correlated equilibrium in the game of property* is the pair of strategies:

From the point of view of an external observer, the convention that regulates property* takes the form of a regularity that corresponds to a correlated equilibrium in the hawk-dove game. But each strategy in this profile also takes the form of a rule that dictates each player what to do in the given circumstances. Each player, therefore, will perceive the institution as a prescription to use the land if the circumstances are ‘right’. Since the two strategies are formulated as rules, clearly the equilibrium is a set of rules – one for each player – that, as North puts it, ‘establish a stable structure to human interaction’ (Reference North1990: 6).

Unlike in ‘pure’ rules-based theories, the concept of equilibrium is central in this account. But unlike ‘pure’ equilibrium-based theories, this account brings at centre stage the representation of the equilibrium strategies by means of symbolic markers (rules). Aoki (Reference Aoki2001, Reference Aoki2007) in particular has emphasized that symbolic markers summarize the properties of equilibria. Institutions help individual players not only to reach coordination, but also to economize cognitive effort. As we shall see shortly, one way of doing this is simply by means of theoretical terms that are used to encompass an entire class of rules under the umbrella of a single institution. This process – the naming or baptizing of institutions – has been analysed in depth by philosophers and will be discussed in detail in the remaining sections of this paper.

Before doing so, let us pause briefly and comment on what has been achieved thus far. We have argued that both the equilibrium approach and the rules approach can capture certain aspects of institutions, but that a proper account requires a combination of both. Furthermore, we have proposed a unified theory that does indeed combine both. The fact that this can be done reveals that they are not inconsistent, but complementary. We have shown that existing theories fail to do justice to the role of either coordination, or correlation, or representation. The unified theory, in contrast, provides an adequate explanation of institutions, because it incorporates all three dimensions.

As it combines insights from both approaches, the explanatory power of the unified theory is larger than that of theories that belong to either one of them. One advantage is explanatory efficiency: the theory explains more aspects of institutions than its rivals. However, explanatory power is not only a matter of convenience. It also serves to reveal that apparent diversity can be traded for an appreciation of the actual unity of the social world (Mäki Reference Mäki2001: 502–03). This can be done by increasing explanatory depth or explanatory integration (Ylikoski and Kuorikoski Reference Ylikoski and Kuorikoski2012). Providing a mechanism, as we have done when discussing correlation, increases explanatory depth. The unified theory can answer more questions than the two original approaches could do even in combination. As a consequence, it provides a higher degree of understanding of the nature of institutions.

We have seen that some theorists have already tried to integrate aspects from the rules and equilibria approaches, and we have followed their lead until now. But in the next few sections we will take a crucial step forward: we will incorporate a third approach that focuses on the representational or symbolic dimension of institutions, and which has become increasingly influential during the last two decades, especially in philosophical circles. Although the approach is a variant of the rule-based account of institutions, it attempts to explicate institutions using a very different kind of rule that, instead of merely regulating behaviour, creates the very possibility of new types of behaviour. Our goal in the remaining part of this paper is to demonstrate that this approach – the constitutive rules approach – can be encompassed within the theory of institutions as rules-in-equilibria.Footnote ¹¹

5. The constitutive rules account

The best-known proponent of the constitutive rules approach is John Searle, the author of a widely discussed book on The Construction of Social Reality (Reference Searle1995; see also Reference Searle1969; Reference Searle2005, Reference Searle2010). In an article entitled ‘What Is an Institution?’ Searle claims: ‘an institution is any system of constitutive rules of the form X counts as Y in C’ (Reference Searle2005: 10; see also Reference Searle1969: 51). Searle contrasts constitutive rules to regulative rules that have as their syntax ‘do X’, or ‘if Y, do X’. Since the actions or strategies that appear in game-theoretic accounts of institutions have precisely this form, regulative rules play a key role in the rules-in-equilibrium approach. So Searle's distinction is meant to suggest that there is a deep hiatus between his approach and the accounts of institutions found in the social science literature.

A central claim of the constitutive rules approach is that institutions exist only because we believe they exist. Our beliefs are thought to play a constitutive role with respect to institutional actions. The constitutive view, however, is not restricted to actions. In addition to actions, institutions often involve objects (like money, university buildings), persons (police officers, presidents), and events (declarations of war, graduations). The constitutive view applies to items from all of these ontological categories. In the case of money, for example, a constitutive rule is: ‘Bills issued by the Bureau of Engraving and Printing (X) count as money (Y) in the United States (C)’ (1995: 28).Footnote ¹² The schematic letter X can be replaced by predicates that apply either to actions or to items from several other ontological categories. But what does the letter Y refer to, exactly?

Money, according to Searle, is an example of a status function. By accepting certain entities as money we assign the status function of being a means of exchange to these entities. For the purposes of our analysis, it will be useful to break Searle's formula in two parts, introducing the twin notions of ‘base rule’ and ‘status rule’. A status rule explicates what it means to have a certain status. The status rule of money, for instance, is ‘money is a medium of exchange’. A base rule specifies what it takes to have a certain status. In certain contexts, an item has to be a shell in order to be money; in others it has to be a disc of metal, and so forth. In more theoretical terms, base rules concern the ontological basis or constitution base of statuses. Status rules, in contrast, are meaning rules and concern the definition of status terms. They concern the behaviour that the status regulates, or the rights and obligations that the status entails.

The ‘counts as’ phrase that appears in Searle's formulation of a constitutive rule can now be interpreted more precisely by relating it to base rules. In medieval Finland, for example, squirrel pelts were money (Tuomela, Reference Tuomela2002). Searle would say that the constitutive rule of money in medieval Finland was ‘Squirrel pelts count as money in Finland’. We suggest interpreting the ‘counts as’ phrase as follows:

\begin{equation*} {\rm X}\;{\rm counts}\;{\rm as}\;{\rm Y}\; \leftrightarrow \;{\rm X}\;{\rm is}\;{\rm collectively}\;{\rm accepted}\;{\rm as}\;{\rm Y} \end{equation*}

and

\begin{equation*} {\rm X}\;{\rm is}\;{\rm collectively}\;{\rm accepted}\;{\rm as}\;{\rm Y}\; \leftrightarrow \;{\rm X}\;{\rm is}\;{\rm Y}{\rm .} \end{equation*}

In our terminology, the base rule of money relevant to medieval Finland takes the following form: ‘In Finland, squirrel pelts are money’. This base rule applied in the Middle Ages because it was collectively accepted to apply.

The notion of collective acceptance has been proposed in relation to that of collective intentionality – roughly, the intentionality exhibited by social groups.Footnote ¹³ Standard game-theoretic approaches do not deploy such a notion. However, collective acceptance can be dissociated from the notion of collective intentionality and defined in general terms as whatever set of intentional states is needed for institutions. Thus, the standard game-theoretic notions of preferences, expectations and common knowledge may qualify as a kind of collective acceptance.Footnote ¹⁴

With this proviso in mind, let us address the distinction between regulative and constitutive rules. For the sake of concreteness, it is convenient to focus on a specific example, so we shall return to our proto-institution of property*. In Section 4 we analysed this institution by means of the correlated equilibrium (or pair of strategies):

where the players use the pre-emption system to coordinate their actions. We have remarked earlier that these two strategies appear to the relevant players as rules that guide and constrain their actions in the game of private property (hawk-dove). These rules are regulative rules, in Searle's language, and for simplicity they can be summarized by means of a single principle:

Notice that this rule does not include a label for or name of the institution. Suppose we now introduce the term ‘property*’ as follows: we say that what it takes for a piece of land to become someone's property* is that she is the first to occupy it. Furthermore, we say that what it is or means for a piece of land to be someone's property* is that she has the right to its exclusive use. By so doing we have split the regulative rule in two parts and used the term ‘property*’ to turn these parts into complete sentences: the first one says that a piece of land is the property* of the person who is the first to occupy it; the second one that if a piece of land is someone's property*, she has the right to its exclusive use.

Another way to put it is that we have transformed the regulative rule [R] in two rules, [B] and [S], respectively:

The [C] rule has the typical structure of Searle's ‘X counts as Y in C’ formula, provided that (a) the expression ‘counts as’ is interpreted in terms of conditions of acceptance, as proposed earlier; (b) the Y term is unpacked so as to make the content of the status function explicit by means of the status rule. A constitutive rule, once these two points have been made explicit, has the following structure: ‘If C then X is Y, and if Y then Z’, where ‘if Y then Z’ is a status rule that specifies the actions that are made available to the relevant individuals. The view that regulative rules can be transformed into constitutive rules using this XYZ schema and via the introduction of terms such as ‘property*’ is what we call (following Hindriks Reference Hindriks2005, Reference Hindriks2009) the transformation view of constitutive rules.

The same procedure can be used to introduce other terms, referring, for example, to institutional roles. We may create a rule stating that ‘The person who is the first to occupy a piece of land owns* it’ and another one stating that ‘An owner* has the right to exclusive use of her land’, for example. Transforming a regulative rule by introducing institutional terms such as ‘owner*’ or ‘property*’ is very convenient. On the supposition that the stipulated usage of the term is generally accepted, the simple claim that a particular piece of land is my property* conveys a lot of information. It presupposes that I was the first to occupy it and it means that I have the right to its exclusive use. Thus, in line with the rules-in-equilibrium approach, the representation of the equilibrium in symbolic form has the advantage of cognitive economy, especially in those cases where several rules are used to govern interrelated strategic interactions. But apart from this, no big changes are implied as far as the original rule is concerned. In particular, behaviour in accordance to the original rule [R] is extensionally equivalent to behaviour in accordance to the content of the rules [B] and [S] that employ the term ‘property*’.

6. Transformation, elimination, and the reference of theoretical terms

The argument presented in the previous section, if correct, entails that the rules-in-equilibrium and the constitutive rules approaches are perfectly consistent. Constitutive rules are linguistic transformations of regulative rules. Such transformations rely on the introduction of a new term that is used to name an institution. In the end, constitutive rules are nothing but (systems of) regulative rules augmented by the introduction of theoretical terms.

In this section we address a worry one might have about the transformation view, and we review a number of virtues of the rules-in-equilibrium approach. To begin with the former, it may be argued that some transformations introduce qualitative changes that preclude consistency between the rules before and after the transformation. To understand this worry, let us draw an analogy with theoretical revolutions in science: a paradigm shift is generated sometimes by introducing new theoretical terms and abandoning some terms that played a key role in an old scientific theory (Kuhn Reference Kuhn1970). The introduction of new terms may change the meaning of the original terms that survive the transformation, and as a consequence the post-transformation theory may be inconsistent with the pre-transformation theory.Footnote ¹⁵ So, we need to find transformation criteria that guarantee consistency.

Belnap's (Reference Belnap1993) criteria for rigorous definitions can play this role. Belnap argues that, in order for a definition to be rigorous, it should satisfy the criteria of eliminability and conservativeness. The criterion of eliminability requires ‘that the defined term be eliminable in favour of previously understood terms’, and the criterion of conservativeness demands ‘that the definition not only not lead to inconsistency, but not lead to anything – not involving the defined term – that was not obtainable before’ (Belnap 1993: 117). In other words, a definition of a term is rigorous if we can do without it, and if it does not entail anything new – anything that is qualitatively different from what can be expressed by only using terms previously understood. We will say that the addition of a term is a conservative transformation of the theory in which it is used, if the definition of that term is rigorous in this sense.Footnote ¹⁶

The core of the transformation in the case of institutions is the introduction of a Y-term. This introduction as we have seen leaves all the features of the rules-in-equilibrium account intact. In other words, the transformation does not introduce any alterations that are in conflict with the theory that explicates institutions using only rules in equilibrium (strategies). Another way to put it is that constitutive rules are regulative rules with special features. In particular, they are regulative rules that have been split in two parts using Y-terms to turn the parts into complete sentences. And the definitions of Y-terms are rigorous in Belnap's sense: they do not lead to anything that could not be obtained before. Before the introduction of the relevant Y-term the link between the two parts was internal to the regulative rule. After its introduction, the link is forged by the fact that the Y-term figures both in the base rule and in the status rule. This implies that definitions of Y-terms are conservative in Belnap's sense.

Y-terms are also eliminable. A constitutive rule can be transformed into a regulative rule by reversing the transformation process outlined above. The first step is to eliminate the Y-term, and the second to join the resulting parts to form a complete sentence. In other words, the thing to do is to move back from [C] to [B] and [S], and from these to [R]. Thus, the definition of a Y-term also satisfies the criterion of eliminability. This implies that at the level of reference there are no substantial changes: the behaviour implied by [R] is extensionally equivalent to that involved in following [B] and [S]. Given that nothing that conflicts with the rules-in-equilibrium account has been introduced, we can conclude that the constitutive rules approach and the rules-in-equilibrium approach are perfectly consistent.

As conservative transformations do not involve qualitative changes of the theory at issue, why should we bother to introduce the theoretical terms in the first place? In other words, one might worry that this argument shows too much: if the rules-in-equilibrium and the constitutive rules accounts are in a sense equivalent to one another, one may conclude that the constitutive rules account has nothing to offer that the rules-in-equilibrium account does not have. This conclusion, however, would be hasty. Constitutive rules are useful theoretical constructs that help us understand several important features of institutions. In the second half of this section, we highlight four virtues of the rules-in-equilibrium approach that are closely associated to the use of theoretical terms. We focus in particular on four virtues, two of which are linguistic and two ontological. These virtues exhibit the explanatory power of our unified theory of institutions: they reveal some more of the advantages of the unification that we achieve in this paper.