2. Conventions and reason to believe

To appreciate the depth of Lewis’s thought, two points need to be made. First, his analysis was only originally meant to apply to coordination problems. However, in almost no case does Lewis rely on properties of an action other than its being optimal given an agent’s beliefs. The same framework can therefore be applied to a much broader set of interactions.

Second, although Lewis is commonly credited with being the first to introduce the concept of common knowledge, ^{Footnote 1} his work actually concerns beliefs rather than knowledge (Cubitt and Sugden Reference Cubitt and Sugden2003; Sillari Reference Sillari2005, Reference Sillari2008). More precisely, Lewis’s interest lay in assessing the modes of reasoning that underpin beliefs and behaviour. The framework is that of epistemic rationality: by saying that agent i has reason to believe proposition x, we mean that i is justified in believing that x is true by virtue of the reasoning he endorses and the evidence he possesses. Lewis eventually admitted that the term ‘common knowledge’ used in his definition of convention ‘was unfortunate, since there is no assurance that it will be knowledge, or even that it will be true’ (Lewis Reference Lewis1978: 44, cit. in Sillari Reference Sillari2008). To avoid confusion, hereafter I use the term ‘common reason to believe’ (CRtB).

Lewis’s (Reference Lewis1969) definition of CRtB goes as follows. Members of a population P have common reason to believe a proposition x if and only if some state of affairs A holds such that:

1. (i) everyone in P has reason to believe that A holds;

2. (ii) A indicates to everyone in P that everyone in P has reason to believe that A holds;

3. (iii) A indicates to everyone in P that x.

As suggested by Cubitt and Sugden (Reference Cubitt and Sugden2003), states of affairs can be thought of as Savage’s states of the world – ‘description[s] of the world, leaving no relevant aspect undefined’ (Savage Reference Savage1954: 9). When conditions (i)–(iii) are met, A is said to be a basis for common reason to believe. This, together with what Lewis refers to as ‘suitable ancillary premises’ about agents’ rationality and their beliefs about each other’s reasoning, allows the development of higher-order reason to believe.

The mutual consistency of individual expectations is of crucial importance in sustaining conventions. For expectations to be consistent, each individual must have reason to believe that everyone else will adhere to a regularity in behaviour. If agents have a conditional preference for conformity, then each one is motivated to behave in ways which in turn confirm other people’s expectations. The challenge is therefore to identify the conditions that allow the development of mutually consistent beliefs about future behaviour. An obvious way of producing first- and higher-order reason to believe, Lewis observed, consists in explicitly agreeing to stick to an action or course of action. Another important factor influencing beliefs and behaviour is salience, namely the perception that some elements of one’s view of a situation stand out from others. Finally, in the case of repeated interactions, precedence may emerge as a special kind of salience, resulting in a tendency to take those actions which have proven successful in the past. ^{Footnote 2}

2.1. On symmetric reasoning

What individuals have reason to believe depends on their logic of reasoning and the information on which they assess situations. In particular, Lewis’s common reason to believe can only obtain when all members of a group have reason to believe that they all share the same inductive standards and background information. It therefore cannot be understood from a purely individualist perspective but requires people to endorse lines of reasoning of the kind ‘Since I have reason to believe x, so may others’. ^{Footnote 3}

Lewisian models, however, often go beyond this and assume that agents actually share inductive standards. For example, Hédoin (Reference Hédoin2014) explicitly postulates that individuals commonly know that they share the same modes of reasoning with respect to a given set of situations. Lewis’s original framework, too, is generally interpreted as requiring common inductive standards. For Sillari (Reference Sillari2005: 391), ‘in some cases inductive standards are shared by groups of agents, as, for example, in those cases in which common reason to believe comes about’, and Bicchieri (Reference Bicchieri2006: 37) observes that ‘Lewis’s argument is crucially dependent on assuming shared inductive standards’. Similarly, Vanderschraaf (Reference Vanderschraaf1998: 362) claims that ‘a crucial assumption in Lewis’s analysis of common knowledge is that agents know they share the same rationality, inductive standards, and background information with respect to a state of affairs’.

Lewis’s definition of CRtB does not necessarily require reasoning to be identical across all agents. For instance, it allows individuals to infer the same conclusion from the same premise through somewhat different paths. Also, it does not mean to suggest that all inductive inferences need to be shared; the condition only applies to a particular set of propositions which are relevant to the situation being considered. However, Lewis’s definition does require individuals to reason symmetrically: if a proposition (‘A holds’) indicates another proposition (x) to a member of P, then it does the same to all other members of the population (Hédoin Reference Hédoin2014, Reference Hédoin2017; Vanderschraaf and Sillari Reference Vanderschraaf, Sillari and Zalta2014). This requirement is controversial; Sillari (Reference Sillari2008: 31) acknowledges it as being ‘far from innocuous’, and for Aoki (Reference Aoki2011: 27) ‘it is admittedly strong and may often fail to apply’.

This symmetry in agents’ reasoning arises in Lewis’s framework as a consequence of the fact that no attempt is made to distinguish between the world as it is and the world as individuals see it. Lewis’s and Lewisian models implicitly assume that states of affairs are public and perfectly self-revealing, implying that all agents hold the same, correct view of A (Cubitt and Sugden Reference Cubitt and Sugden2003). If A obtains, then each agent has reason to believe that A holds; this, in turn, leads everyone to infer x. A graphical representation of the perception-inference process underlying Lewis’s CRtB is shown in the left-hand panel of Figure 1 (ignore the right-hand panel for now; we will return to it later).

Figure 1. Perception-inference processes.

When is an event public and self-revealing? As an example, consider the coordination problem discussed by Lewis (Reference Lewis1969: 52) to introduce conventions by agreement:

[s]uppose the following state of affairs – call it A – holds: you and I have met, we have been talking together, you must leave before our business is done; so you say you will return to the same place tomorrow. Imagine the case. Clearly, I will expect you to return. You will expect me to expect you to return. I will expect you to expect me to expect you to return.

This state of affairs can reasonably be seen as capable of generating higher-order expectations through roughly common lines of thought: (i) my view of A is the same as yours (you had to leave and you said you will return to the same place tomorrow), and both you and I have reason to believe that this is indeed the case; (ii) A indicates to both that both have reason to believe that A holds; and (iii) A indicates to both that you will return. Therefore, A is a basis for CRtB in Lewis’s sense.

In other cases, and especially when individuals are driven by different motives or interpret the same situation differently, the symmetry-in-reasoning condition may not hold. Yet agents may still form consistent higher-order beliefs. Cubitt and Sugden (Reference Cubitt and Sugden2003) offer useful insights in this regard by introducing the concept of ‘distributed reason to believe’. In discussing why American drivers drive on the right side of the road, they observe that no driver can have direct knowledge of other people’s experiences and driving habits, and that each agent’s decisions are made on the basis of experience of a particular sample of American drivers. This makes it natural to think of drivers’ expectations as being licensed by mental state inferences that differ slightly from one another. In a related vein, Cubitt and Sugden (Reference Cubitt and Sugden2014) present a model where each agent endorses a private ‘reasoning scheme’. Reasoning schemes are assumed to share a core of ‘common reason’, which consists of those modes of reasoning that are shared across individuals and that each attributes to everyone in the population. However, apart from that, agents’ reasonings may differ to a large extent. The model presented here builds on these ideas by introducing elements of framing theory into Lewis’s framework.

3. The model

3.1. Some formal apparatus

3.1.1. Reason to believe and indication

The notation follows that of Cubitt and Sugden (Reference Cubitt and Sugden2003) and adapts it to a possible worlds setting. Let P = {1,…,n} be a finite population of agents and let ${\mathcal A}$ be a non-empty set of possible worlds or states of affairs. A proposition is a subset $x \subseteq {\mathcal A}$ , and the set of all possible propositions is given by the power set ${\mathbb P} \left( {\mathcal A} \right)$ . If $A \in {\mathcal A}$ belongs to x, then the proposition x is true in world A. For each agent $i \in P$ , let ${{\cal R}_i}$ be a modal operator from ${\mathbb P} \left( {\mathcal A} \right)$ to itself. ${{\cal R}_i}\left( x \right)$ contains those worlds in which i has reason to believe x. Reason to believe can be nested to arbitrary depths; I write ${{\cal R}_i}\left[ {{{\cal R}_j}\left( x \right)} \right]$ to denote the proposition that i has reason to believe that j has reason to believe x, and I write ${{\cal R}^P}\left( x \right)$ to mean that ${{\cal R}_i}\left[ {{{\cal R}_j}\left[ {{{\cal R}_k}\left[ { \ldots \left[ {{{\cal R}_n}\left( x \right)} \right] \ldots } \right]} \right]} \right]$ is true for all finite sequences $i,j,k, \ldots ,n \in P$ .

Lewis’s indication relation is captured by a three-place modal operator, ind, where $x\;{\rm{in}}{{\rm{d}}_i}\;y$ is the proposition that if i has reason to believe x, i thereby has reason to believe y. To write $A \in \left( {x\;{\rm{in}}{{\rm{d}}_i}\;y} \right)$ is to say that in the context of state of affairs A, having reason to believe x provides i with reason to believe y. Put another way, in world A – and possibly in many or even all other worlds – the logic of reasoning that i endorses justifies an inference from x to y. However, there may exist $j \ne i \in P$ and $A' \ne A \in {\mathcal A}$ for which this does not hold, meaning that indication has no universal validity. The indication relation is taken to be reflexive: every proposition x always indicates to i that x, that is $x\;{\rm{in}}{{\rm{d}}_i}\;x$ holds tautologically.

The following axioms are assumed. Let x, y and z be propositions; for any distinct i and j in P:

(1) $$\left[ {{{\cal R}_i}\left( x \right) \cap \left( {x\;{\rm{in}}{{\rm{d}}_i}\;y} \right)} \right] \subseteq {{\cal R}_i}\left( y \right),$$

(2) $$\left[ {\left( {x\;{\rm{in}}{{\rm{d}}_i}\;{{\cal R}_j}\left( y \right)} \right) \cap {{\cal R}_i}\left( {y\;{\rm{in}}{{\rm{d}}_j}\;z} \right)} \right] \subseteq \left[ {x\;{\rm{in}}{{\rm{d}}_i}\;{{\cal R}_j}\left( z \right)} \right].$$

Rule (1) states that if x indicates y to agent i, then reason to believe x implies reason to believe y. Rule (2) says that if x indicates to i that j has reason to believe y, and if i has reason to believe that y indicates to j that z, then x indicates to i that j has reason to believe z. For example, let x, y and z be the propositions ‘the thunder is getting louder’, ‘it may rain soon’, and ‘it is a good idea to take an umbrella when leaving the house’, respectively. According to (2), if the thunder indicates to i that j has reason to believe that it may rain soon, and if i has reason to believe that the possibility of showers indicates to j that it is a good idea to take an umbrella, then having reason to believe that the thunder is getting louder indicates to i that j has reason to believe that they should leave the house with an umbrella.

3.1.2. Framing structures

A frame is composed of bundles of concepts that are used in an individual’s contextual representation of a situation. For Turner (Reference Turner2001: 13, 101), ‘basic mental operations operate over … frames’, and ‘it is natural to assume that decision making in any specific situation will depend on what frames are used by the decision maker as conceptual inputs’. Moreover, interacting agents typically base their decisions on their perceptions and assessments of other people’s views. Based on these observations, I define a framing structure as a tuple:

$${\mathcal F}:=\langle P,{\mathcal A},{\left\{ {{f_i},{f_{i;j}}} \right\}_{j \ne i \in P}}\rangle ,$$

where for every $i \in P$ , the function ${f_i} : \mathcal A \to {\mathbb P} \left( {\mathcal A} \right)$ maps world $A \in {\mathcal A}$ to a proposition ${f_i}\left( A \right)$ that represents i’s view of A. This introduces a distinction between the world as seen from the external viewpoint of the modeller, A, and the world as each agent sees or perceives it, ${f_i}\left( A \right)$ . Different frames may reflect differences in culture, cognitive ability or experience. In a related fashion, for every pair of individuals i and j, the function ${f_{i;j}}: {\mathbb P} \left( {\mathcal A} \right) \to {\mathbb P} \left( {\mathcal A} \right)$ maps ${f_i}\left( A \right)$ to another proposition, ${f_{i;j}}\left( \,{{f_i}\left( A \right)} \right)$ . The latter represents what i conjectures j’s frame to be, given that i’s own frame is ${f_i}\left( A \right)$ . ^{Footnote 4}

Several points are worth making here. First, if ${f_i}\left( A \right) = {f_i}\left( {A'} \right)$ holds for any distinct $A,A' \in {\mathcal A}$ , then the two worlds are frame-equivalent, meaning that i cannot discriminate A from $A'$ . Second, for all members of P to hold the same view of A, it must be that ${f_i}\left( A \right) = {f_j}\left( A \right)$ for every i and j in P. Third, in the particular case where ${f_i}\left( A \right) = {f_{i;j}}\left( \,{{f_i}\left( A \right)} \right) = A$ for all i and j, the frame-dependent model reduces to the symmetric Lewisian model. As we have seen, this may be a strong condition to impose; in general, i’s and j’s frames of A will be similar in some (perhaps many) respects and different in others. Likewise, i’s frame and i’s conjecture of j’s frame will typically be correlated with one another, but may also differ in some important respects. For instance, in our earlier example, some gang members view their group’s behaviour as morally wrong but conjecture that other members take pride in violence. Fourth, ‘framing is logically prior to believing’ (Bacharach Reference Bacharach, Dimitri, Basili and Gilboa2003: 66): if ${f_i}\left( A \right)$ and ${f_{i;j}}\left( \;{{f_i}\left( A \right)} \right)$ do not contain proposition y, then agent i cannot have beliefs about y when representing A to himself. Moreover, i remains unaware of this because y has not occurred to him at all.

Finally, the components of an individual’s frame may be seen as being characterized by different degrees of salience and choice relevance. The salience of a proposition is related to the frequency with which that proposition is employed in agents’ representations of a situation (Bacharach Reference Bacharach, Dimitri, Basili and Gilboa2003, Reference Bacharach, Gold and Sugden2006), whereas choice relevance refers to the ability to influence decisions (Gold Reference Gold, Okasha and Binmore2012). Both salience and choice relevance are typically time- and context-dependent.

3.2. Breaking the symmetry

A more general definition of basis for CRtB can now be given.

Definition 1 (Basis for common reason to believe). Let x be a proposition. A state of affairs A is a basis for common reason to believe in P that x if and only if:

(3) $$\hskip -54pt \forall \;i \in P:\quad A \in {{\cal R}_i}\left[ {{f_i}\left( A \right)} \right],$$

(4) $$\hskip -40pt\forall \;i \in P:\quad A \in \left[ {{f_i}\left( A \right)\;{\rm{in}}{{\rm{d}}_i}\;x} \right],$$

(5) $$\forall \;i,j \in P:\quad A \in \left[ {{f_i}\left( A \right)\;{\rm{in}}{{\rm{d}}_i}\;{{\cal R}_j}\left[ {{f_{i;j}}\left( \;{{f_i}\left( A \right)} \right)} \right]} \right].$$

Condition (3) states that each $i \in P$ frames A as ${f_i}\left( A \right)$ and has reason to believe that ${f_i}\left( A \right)$ holds. Thus, A is neither perfectly nor identically self-revealing. Note that it may be that $A \;\notin \;{f_i}\left( A \right)$ (so that agent i completely misperceives A) and yet $A \in {{\cal R}_i}\left[ {{f_i}\left( A \right)} \right]$ (meaning that i has reason to believe in his view of A). This is because, for any proposition y, the reason to believe operator need not satisfy the truth axiom ${{\cal R}_i}\left( y \right) \Rightarrow y$ . Condition (4) says that ${f_i}\left( A \right)$ indicates to each i that x, i.e. that agents’ modes of reasoning license an inference from their view of A to x. The perception-inference process corresponding to conditions (3) and (4) is represented in the right-hand panel of Figure 1. If condition (5) holds for some A, then for each pair of individuals $i,j \in P$ , having reason to believe ${f_i}\left( A \right)$ indicates to i that j has reason to believe ${f_{i;j}}\left( \;{{f_i}\left( A \right)} \right)$ – the latter being i’s conjecture of j’s view of A, which may differ significantly from j’s actual view.

Sometimes (as in the example of Section 4) agents might have reason to believe that other members of P have the same view of A as themselves, or at least that they do so in all relevant respects. If this is the case, then ${f_i}\left( A \right) = {f_{i;j}}\left( \;{{f_i}\left( A \right)} \right)$ for all i and j, and (5) reduces to:

(5’) $$\forall \;i,j \in P:\quad A \in \left[ {{f_i}\left( A \right)\;{\rm{in}}{{\rm{d}}_i}\;{{\cal R}_j}\left[ {{f_i}\left( A \right)} \right]} \right],$$

which reflects a mutual ascribing of common views. The condition in (5’) states that ${f_i}\left( A \right)$ indicates to each i that others have reason to believe ${f_i}\left( A \right)$ as well. Again, this by no means implies an actual commonality of views.

Our ancillary condition for the development of common reason to believe is given in the definition below.

Definition 2 (Ascription of mutually consistent inductive standards). Let y be a proposition. In state of affairs A, members of P ascribe mutually consistent inductive standards to each other if:

(6) $$\forall \;i,j \in P,\;\forall \;y:\quad A \in \left[ {{f_i}\left( A \right)\;{\rm{in}}{{\rm{d}}_i}\;y} \right] \Rightarrow A \in {{\cal R}_i}\left[ {{f_{i;j}}\left( \;{{f_i}\left( A \right)} \right)\;{\rm{in}}{{\rm{d}}_j}\;y} \right].$$

The intuition behind (6) is that although recognizing that other agents’ views differ from his own, an individual may have reason to believe that these views all justify an inference to proposition y. For instance, in Example 1 earlier: (i) Bob’s view indicates to him that meetings with Alice are not going to stop any time soon; (ii) Bob is aware that Alice sees him as a friend, i.e. that Alice’s view differs from his own; and yet (iii) Bob has reason to believe that Alice’s view (as he believes it to be) indicates to her that the meetings are going to continue. More precisely, to say that (6) holds for some A is to say that, for all i and j in P, if ${f_i}\left( A \right)$ indicates any proposition y to i, then i has reason to believe that j’s view of A (as i conjectures it to be!) indicates to j that y. As in Cubitt and Sugden (Reference Cubitt and Sugden2003: 189, 192), this condition is stronger than is strictly necessary but makes it possible to keep technicalities from dominating the exposition. In the following, I do not require the material implication to hold for all ys but only for a limited set of propositions required for the development of higher-order reason to believe.

Definition 2 encompasses cases in which agents have reason to believe that others share the same inductive standards, that is, in which:

(6’) $$\forall \;i,j \in P,\;\forall \;y:\quad A \in \left[ {{f_i}\left( A \right)\;{\rm{in}}{{\rm{d}}_i}\;y} \right] \Rightarrow A \in {{\cal R}_i}\left[ {{f_i}\left( A \right)\;{\rm{in}}{{\rm{d}}_j}\;y} \right].$$

The condition in (6’) states that if ${f_i}\left( A \right)$ indicates any proposition y to i, then i has reason to believe the same to hold true for every other person in P. This corresponds to Lewis’s ‘suitable premise’ about agents’ inductive standards, and dovetails with a condition given by Perea (Reference Perea2007) to discuss the epistemic foundations of Nash equilibrium. As Perea shows, in games with at least three players, Nash equilibrium requires each player to have ‘projective beliefs’. Simply put, if i has projective beliefs, then his belief about j’s behaviour is the same as his belief about k’s belief about j’s behaviour.

The conditions in the above definitions, together with axioms (1) and (2), yield the following proposition, which is a frame-dependent version of Lewis’s main result. If a state of affairs A is a basis for common reason to believe in P that x, and if members of P ascribe mutually consistent inductive standards to each other, then each member of the population can form higher-order beliefs about x.

Proposition. If the conditions in Definitions 1 and 2 hold for some A , then $A \in {{\cal R}^P}\left( x \right)$ .

Proof. The proposition follows immediately from applying the definitions to the chain of logical implications used by Cubitt and Sugden (Reference Cubitt and Sugden2003: app. 1) to formalize Lewis’s CRtB.

(P1.1) $$\hskip -120pt \forall \;i \in P:\quad A \in {{\cal R}_i}\left[ {{f_i}\left( A \right)} \right]\quad {\rm{from}}\,{\rm{(3)}}$$

(P1.2) $$\hskip -60pt \forall \;i,j \in P:\quad A \in \left[ {{f_i}\left( A \right)\;{\rm{in}}{{\rm{d}}_i}\;{{\cal R}_j}\left[ {{f_{i;j}}\left( \;{{f_i}\left( A \right)} \right)} \right]} \right]\quad {\rm{from}}\,{\rm{(5)}}$$

(P1.3) $$\hskip -100pt\forall \;i \in P:\quad A \in \left[ {{f_i}\left( A \right)\;{\rm{in}}{{\rm{d}}_i}\;x} \right]\quad {\rm{from}}\,{\rm{(4)}}$$

(P1.4) $$\hskip -70pt\forall \;i \in P:\quad A \in {{\cal R}_i}\left( x \right)\quad {\rm{from}}\,{\rm{(P1}}{\rm{.1),(P1}}{\rm{.3)}}\,{\rm{and}}\,{\rm{(1)}}$$

(P1.5) $$\hskip -55pt \forall \;i,j \in P:\quad A \in {{\cal R}_i}\left[ {{f_{i;j}}\left( \;{{f_i}\left( A \right)} \right)\;{\rm{in}}{{\rm{d}}_j}\;x} \right]\quad {\rm{from\;(P1}}{\rm{.3)}}\,{\rm{and}}\,{\rm{(6)}}$$

(P1.6) $$\hskip -10pt \forall \;i,j \in P:\quad A \in \left[ {{f_i}\left( A \right)\;{\rm{in}}{{\rm{d}}_i}\;{{\cal R}_j}\left( x \right)} \right]\quad {\rm{from\;(P1}}{\rm{.2),(P1}}{\rm{.5)}}\,{\rm{and}}\,{\rm{(2)}}$$

(P1.7) $$\hskip -65pt \forall \;i,j \in P:\quad A \in {{\cal R}_i}\left[ {{{\cal R}_j}\left( x \right)} \right]\quad {\rm{from\;(P1}}{\rm{.1),(P1}}{\rm{.6)}}\,{\rm{and}}\,{\rm{(1)}}$$

(P1.8) $$\hskip -45pt \forall \;i,j,k \in P:\quad A \in {{\cal R}_i}\left[ {{f_{i;j}}\left( \;{{f_i}\left( A \right)} \right)\;{\rm{in}}{{\rm{d}}_j}\;{{\cal R}_k}\left( x \right)} \right]\quad {\rm{from}}\,{\rm{(P1}}{\rm{.6)}}\,{\rm{and}}\,{\rm{(6)}}$$

(P1.9) $$\hskip -05pt \forall \;i,j,k \in P:\quad A \in \left[ {{f_i}\left( A \right)\;{\rm{in}}{{\rm{d}}_i}\;{{\cal R}_j}\left[ {{{\cal R}_k}\left( x \right)} \right]} \right]\quad {\rm{from}}\,{\rm{(P1}}{\rm{.2),}}\,{\rm{(P1}}{\rm{.8)}}\,{\rm{and}}\,{\rm{(2)}}$$

(P1.10) $$\hskip -60pt \forall \;i,j,k \in P:\quad A \in {{\cal R}_i}\left[ {{{\cal R}_j}\left[ {{{\cal R}_k}\left( x \right)} \right]} \right]\quad {\rm{from}}\,{\rm{(P1}}{\rm{.1),}}\,{\rm{(P1}}{\rm{.9)}}\,{\rm{and}}\,{\rm{(1)}}$$

$$ \ldots {\rm{and}}\,{\rm{so}}\,{\rm{on,up\; to}}\;A \in {{\cal R}^P}\left( x \right).$$

Line (P1.8) follows from the fact that (P1.6) holds for any pair of individuals in P. Thus, having reason to believe ${f_i}\left( A \right)$ indicates to i that k has reason to believe x. Lines (P1.4), (P1.7) and (P1.10), together with the subsequent steps, establish the result.□

The proposition relaxes Lewis’s theorem by providing weaker sufficient conditions for the development of higher-order reason to believe. The distinction between a state of affairs, an individual’s private view, and an individual’s beliefs about other agents’ views allows us to distinguish between two kinds of heterogeneity in reasoning. First, different agents may frame A in ways that differ considerably from one another, and yet all infer x. Second, individuals may have reason to believe that others’ frames and inferences differ from their own. Furthermore, the beliefs that agents have about others’ beliefs may be at odds with those agents’ actual beliefs. When this is so, then as long as the course of action is consistent with their reasoning, agents do not receive any evidence that their views are inaccurate and their beliefs false. The following example illustrates this point.

4. An extended example

The understanding of the American Revolution put forward by Jack Rakove, Barry Weingast and co-authors (Rakove Reference Rakove1996; Rakove et al. Reference Rakove, Rutten and Weingast2005; de Figueiredo et al. Reference de Figueiredo, Rakove and Weingast2006) has some interesting similarities with the framework developed here. As historians have long recognized, the revolutionary crisis was ‘the product of … dimly perceived and rapidly changing thoughts and situations’, and its escalation ‘becomes comprehensible only when the mental framework … into which the Americans fitted the events of the 1760s and 1770s is known’ (Wood Reference Wood1966: 14, 23). Moreover, the century of Anglo-American cooperation that preceded the 1775–83 war offers an example of regularity in behaviour that is conventional (i.e. customary, expected by each party, and characterized by a conditional preference for conformity) but cannot be adequately explained by symmetrical Lewisian models.

Underlying cooperation, there were profound disagreements about the nature of the rule of law and the division of authority between metropole and colonies, of which neither Americans nor British were fully aware (Reid Reference Reid1976, Reference Reid1995; Greene Reference Greene1986; Zuckert Reference Zuckert2005). Long-established practices had been based on contrasting interpretations of the American position within the British imperial order, which did not come to light until Britain’s parliamentary intervention in colonial affairs became a matter of paramount importance.

The colonists understood the constitution of the empire as a system of common law rules built on precedent. Between the late 17th and the mid-18th centuries, American control over domestic affairs had continued to grow towards what Greene (Reference Greene1991) has defined a ‘negotiated authority’: the motherland had retained power over matters related to security and international trade, while colonies had come to administer internal affairs that ranged from religion to taxation and the enforcement of contracts. Moreover, in the eyes of the colonists, well-established customs and self-government practices had to be given a pre-eminent legal status. The Americans ‘were espousing the … theory of a constitution of customary restraints on arbitrary power, a theory that seemed to them to be more compatible with – and more explanatory of – their own constitutional experience both within the colonies and with regard to metropolitan-colonial relationships’ (Greene Reference Greene1986: 71). The fact that the British had long ceased to directly intervene in colonial domestic matters was seen as the foundation for a constitutional claim; sovereignty within the empire, colonists believed, was divided, and the metropole lacked the authority to regulate colonial domestic affairs.

However, on the other side of the Atlantic, a different view of common law constitutionalism had been maturing. By the middle of the 18th century, legislative bodies had considerably extended their control over the executive authority of the Crown. The King-in-Parliament principle had been firmly established, and parliamentary sovereignty had emerged as the core of the British constitution. The Houses had been vested with the power to create, amend or abolish any law. Furthermore, the British saw the division of authority between motherland and colonies as motivated by mere expediency. For them,

The British view had therefore little to do with the American ideas of shared sovereignty and self-government practices as acquired rights which restricted parliamentary authority. British policymakers saw sovereignty as indivisible, and the Houses as having supreme legal power in both motherland and colonial territories.

For decades, Anglo-American relations remained characterized by a mutual ignorance of these different, and difficult to reconcile, views. This caused both parties to rely on incorrect but self-confirming beliefs. For the British, what underpinned cooperation was the threat posed by France, both in Europe and overseas. The metropole’s accommodating stance towards the American push for autonomy was largely motivated by the need to maintain a solid front against the French. The colonists feared the French and their native allies, too, but an equally compelling reason to cooperate was provided by their being fine with the political independence they had gained. As North (Reference North, Weingast and Wittman2005: 1006) put it, ‘to the extent that Americans thought about the metropole, it was that Britain was a benign, if remote, presence’. No evidence emerged to reveal that the two parties’ constitutional ideas were contradictory and their beliefs inaccurate.

Figure 2 represents Anglo-American cooperation as the equilibrium of a coordination game. Upper and lower case letters represent the choices of, and the payoffs to, the British and the Americans, respectively. B, S, T and W stand for best, second best, third best and worst; C and D denote cooperation and defection. For the British, to cooperate means not intervening in American domestic affairs. For the Americans, it means not revolting against the British. The two parties’ preference orderings over the possible outcomes are, respectively, $\left( {C,c} \right){ \succ _{UK}}\left( {D,c} \right){ \succ _{UK}}\left( {D,d} \right){ \succ _{UK}}\left( {C,d} \right)$ and $\left( {C,c} \right){ \succ _{US}}\left( {C,d} \right){ \succ _{US}}\left( {D,d} \right){ \succ _{US}}\left( {D,c} \right)$ . The game has two Nash equilibria in pure strategies, ${(C,c)}$ and ${(D,d)}$ .

Figure 2. The Anglo-American cooperation game.

This representation reveals very little about players’ views and (mis)conjectures about the opponent’s view. To apply our framework, let A denote Anglo-American relations (as described above and as seen by an omniscient external observer) at some point in time between the 1690s and the 1760s, and let ${f_{US}}\left( A \right)$ and ${f_{UK}}\left( A \right)$ denote the contextual features of A that the Americans and the British considered relevant. For ease of exposition, and coherently with Rakove et al.’s narrative, suppose that conditions (5’) and (6’) hold, reflecting that neither party had reason to expect the other to view A in a significantly different manner from himself. ^{Footnote 5} The American and British frames of A can be thought of as:

$${f_{US}}(A) = \left\{ {{\rm{common}}\;{\rm{law}}\;{\rm{constitutionalism}}\;{\rm{built}}\;{\rm{on}}\;{\rm{precedent}};} \right.{\mkern 1mu} {\rm{ crcolonial}}\;{\rm{self}} - {\rm{governance}}\;{\rm{as}}\;{\rm{an}}\;{\rm{acquired}}\;{\rm{right}};{\mkern 1mu} {\rm{ crnegotiated}}\;{\rm{authority}}\;{\rm{as}}\;{\rm{a}}\;{\rm{satisfying}}\;{\rm{status}}\;{\rm{quo}}\},$$

$${f_{UK}}(A) = \left\{ {{\rm{parliamentary}}\;{\rm{supremacy}};} \right.{\rm{ cr}}{\mkern 1mu} {\rm{colonial}}\;{\rm{self}} - {\rm{governance}}\;{\rm{as}}\;{\rm{a}}\;{\rm{matter}}\;{\rm{of}}\;{\rm{policy}}\;{\rm{choice}};{\rm{ cr}}{\mkern 1mu} {\rm{negotiated}}\;{\rm{authority}}\;{\rm{necessary}}\;{\rm{under}}\;{\rm{the}}\;{\rm{French}}\;{\rm{threat}}\}.$$

Finally, let x be the proposition ‘the existing status quo should be maintained’. Table 1 summarizes the modes of reasoning that underlay the two parties’ behaviour. The first three lines of the table show that despite the significant differences in the British and American views of A, the latter represented a basis for common reason to believe x. Line four follows from taking (6’) as the ancillary condition allowing higher-order beliefs to come about.

Table 1. The reasoning schemes underpinning Anglo-American relations

France’s defeat in the Seven Years’ War was a major environmental shock that produced a change in the two parties’ willingness to cooperate and eventually led to open conflict. For the first time in years, Britain sought to directly intervene in colonial domestic affairs in order to cover the costs of war. Most laws, such as the 1764 Sugar Act and the 1765 Stamp Act, introduced relatively small taxes. Yet they astounded the colonists because they were inconsistent with their long-held beliefs. The American view of common law constitutionalism made the colonists interpret the British interventions as unlawful. The British, in turn, interpreted the American discontent as fading loyalty towards the empire, and reacted by suspending the New York Colonial Assembly. As events unfolded and the revolutionary crisis escalated, the cooperative equilibrium rapidly fell apart.

5. Related literature

The case of pre-1760s Anglo-American relations can be recognized as an example of ‘spurious unanimity’, namely a situation characterized by unanimity of preferences without unanimous reasons (Mongin Reference Mongin2016). It is also a case of self-confirming behaviour, a concept introduced into the economics literature by the aforementioned work of Hayek (Reference Hayek1937: 51): ‘we may … very well have a position of equilibrium only because some people have no chance of learning about facts which, if they knew them, would induce them to alter their plans’. In this same line of thought we have Hahn (Reference Hahn1977), who describes an economy in which agents formulate conjectures about market conditions from the private signals they receive. Their hypotheses may be imprecise and yet lead to a ‘conjectural equilibrium’, i.e. a set of mutually consistent signals and individual actions that confirm and induce each other.

In game theory, the concept of self-confirming equilibrium was introduced by Fudenberg and Levine (Reference Fudenberg and Levine1993) as a coarsening of Nash equilibrium such that no agent ever observes an action that contradicts his beliefs – which need not be correct on off-equilibrium paths. Battigalli et al. (Reference Battigalli, Cerreia-Vioglio, Maccheroni and Marinacci2015) study self-confirming equilibria in situations where agents do not know the probability model underlying the variables affecting their choices. If the game is played repeatedly in a stationary environment, then agents can learn the distribution of payoffs associated with observed strategy profiles. However, ambiguity about unchosen strategies persists and makes them less appealing. Schipper (Reference Schipper2018) introduces a notion of self-confirming equilibrium that applies to games with unawareness, that is games where individuals are not aware of some of the actions that can be taken by others. He shows that rational play can enrich agents’ information sets, and characterizes the game as the endogenous result of strategic interaction and learning. An epistemic game-theoretic approach to reasoning in games with unawareness has been proposed by Perea (Reference Perea2017), who studies the concept of common belief in rationality in that setting.

The role of agents’ views is also central to the analysis of mutually acceptable behaviours by Greenberg et al. (Reference Greenberg, Gupta and Luo2009). The authors show that, even if agents view the same interaction problem as different extensive form games, they may follow a common course of action. Kaneko and Atsui (Reference Kaneko and Atsui1999) and Kaneko and Kline (Reference Kaneko and Kline2008) propose an inductive framework to explain how individuals develop and modify their views of the strategic environment. In their models, agents do not have a priori knowledge of the structure of the game they play, and gather information by means of occasional random moves.

Another means of studying conventions is through evolutionary game-theoretic models, which rely on random perturbations to introduce variations in the frequencies with which strategies are played by individuals in a population (e.g. Young Reference Young1998; Skyrms Reference Skyrms2014). Information processing is assumed to be costly, and individuals myopically adapt to the behaviour of other agents. Models of this kind allow one to investigate the roles of mutation and adaptation in determining long-run behavioural patterns, but also result in a downplaying of reasoning. This makes the evolutionary approach a valuable complement to Lewisian models of behaviour, but not a substitute for them.

6. Concluding remarks

Ideas from psychology and cognitive sciences have recently started to permeate theories of decision making, resulting in a variety of models that emphasize the role of mental states as drivers of behaviour. In a related field of inquiry, Lewisian models have been investigating how experience and reasoning can allow behavioural regularities to emerge.

Building on an ambiguity in previous research about whether or not conventions require symmetric reasoning, this paper sought to expand Lewis’s framework by incorporating the concept of ‘frames’ in it. I stressed that different agents can understand the same situation differently, and that individuals may believe others’ beliefs to differ from their own. This diversity pervades our lives, as shown by examples ranging from day-to-day interactions to deviant behaviours and international relations. The main contribution of the paper is to show that individuals can develop consistent higher-order expectations even when their modes of reasoning differ substantially from one another. It also shows that beliefs about other people’s views and inferences may not be true, and that self-confirming patterns of behaviour may emerge that are sustained by false beliefs.

Needless to say, many of these points would benefit from further study. For example, the topological properties of framing structures have largely been left unspecified, and a formal definition of when agents perceive a particular feature of a situation as salient or choice-relevant is still missing. Finally, embedding the model in a dynamic framework would help explain how the revision of frames and beliefs works. Research along these lines could provide important new insights into the reasoning processes that motivate behaviour.