2. COMMON KNOWLEDGE OF RATIONALITY IN A BAYESIAN MODEL

In this section, we present a Bayesian model of CKR, based on that of Aumann (Reference Aumann1987).

We consider the class G of finite, normal-form games of complete information, interpreted as one-shot games. For any such game, there is a finite set N = {1, . . ., n} of players, with typical element i and n ≥ 2; for each player i, there is a finite, non-empty set of (pure) strategies S_i, with typical element s_i; and, for each profile of strategies s = (s ₁, . . ., s_n), there is a profile u(s) = (u ₁[s], . . ., u_n[s]) of real-valued and finite utilities. The set S ₁ × . . . × S_n is denoted S; the set S ₁ × . . . × S_{i −1} × S_{i +1} × . . . × S_n is denoted S _−i and its typical element by s _−i. We impose that, for all distinct i, j, ∈ N, S_i ∩ S_j = ∅. This condition has no substantive significance, but imposes a labelling convention that the strategies available to different players are distinguished by player indices, if nothing else. This convention allows a conveniently compact notation in later sections.

Aumann's modelling strategy is Bayesian in the sense that agents’ beliefs are described by subjective probabilities, defined on some set of states and updated from initial priors in the light of known information. Thus, we define a Bayesian model, for any game in G, so that it specifies all of the following: a set of states of the world; players’ behaviour; players’ knowledge; players’ subjective beliefs; and a standard of decision-theoretic rationality.

Uncertainty is represented by means of a finite, non-empty, universal set Ω of states, whose typical element is denoted ω. A set of states is an event.

Players’ behaviour is represented by a behaviour function b(.), which assigns a profile of strategies b(ω) = (b ₁[ω], . . ., b_n[ω]) to each state ω, to be interpreted as the profile of strategies that are in fact chosen by the players at ω. Stochastic choice (such as mixed strategies) is represented as choice that is conditioned on random events. For each profile s of strategies and each strategy s_i, we define the events E(s) = {ω ∈ Ω | b(ω) = s} and E(s_i) = {ω ∈ Ω | b_i(ω) = s_i}. Let S* = {s ∈ S | E(s) ≠ ∅} and S_i* = {s_i ∈ S_i | E(s_i) ≠ ∅}. S* (respectively S_i*) is the set of strategy profiles (respectively strategies for i) included in the Bayesian model. Thus, a Bayesian model specifies a binary partition of each player's strategy set S_i, the elements of which are the set of included strategies S_i* and the set of excluded strategies S_i\S_i*. By construction, each S_i* is non-empty.

Players’ knowledge is represented by an information structure I = (I₁, . . ., I_n). For each player i, I_i is an information partition of Ω, representing what i knows at each state. K_i(E), where E is an event, is the event {ω ∈ Ω| ∃E′ ∈ I_i: (ω ∈ E′) and (E′ ⊆ E)}.Footnote ⁴ If ω ∈ K_i(E), we say ‘i knows E at ω’. An event E is Bayesian common knowledge at ω if ω is an element of all events of the finitely-nested form K_i(K_j(. . . K_k(E). . .)). (This is the formal definition of ‘common knowledge’ used in the model. We use the qualifier ‘Bayesian’ to distinguish this theoretical construct from the intuitive concept.) Since Ω is the universal set, it follows that, for every player i and every state ω, i knows Ω at ω. Thus, Ω is Bayesian common knowledge at every state.

For any player i, a prior is a function π_i: Ω → [0, 1] satisfying the conditions (i) Σ_ω∈Ω π_i(ω) = 1; and (ii) for every event E ∈ I_i, there exists some state ω ∈ E, such that π_i(ω) > 0. π_i(ω) is interpreted as a subjective probability. We extend this notation to events by defining, for each event E, π_i(E) = Σ_ω∈E π_i(ω). Posterior probabilities, conditional on events, are defined from priors by means of Bayes's rule. Condition (i) is the obvious condition that prior probabilities sum to unity; condition (ii) guarantees that posterior probabilities, conditional on what player i knows at any state, are well-defined. The latter condition is required if the standard of rationality developed below and drawn from Aumann (Reference Aumann1987) is itself to be well-defined.

To represent the normative standard of practical rationality that is to be built into the model, we define a choiceworthiness function for each player i as a function χ_i: Ω → (S_i), where (S_i) denotes the power set of S_i, satisfying two restrictions. First, χ_i(ω), the set of strategies that are choiceworthy for i at ω, is non-empty for all ω. Second, for all E ∈ I_i, for all ω, ω′ ∈ E: χ_i(ω) = χ_i(ω′). The interpretation is that χ_i(ω) is the set of strategies which, according to the standard of rationality, may be chosen by i at ω. The first restriction stipulates that, in every state, there is at least one choiceworthy strategy; the second that what is choiceworthy for a player cannot differ between states that he is unable to distinguish.

In Aumann's case, the standard of rationality is subjective expected utility maximization. Consider any player i. For any s ∈ S, for any s_i′ ∈ S_i, let σ_i(s, s_i′) denote the strategy profile created by substituting s_i′ for s_i in s (i.e. σ_i[s, s_i′] = [s ₁, . . ., s_{i −1}, s_i′, s_{i +1}, . . ., s_n]). For any prior π_i, for any state ω′, for any E ∈ I_i, let π_i(ω′|E) denote the posterior probability of ω′, given E. For each player i, for each state ω, a strategy s_i is SEU-rational for i at ω with respect to the information partition I_i and prior π_i if, for each strategy s_i′ ∈ S_i, ∑_ω′∈E π_i(ω′|E) (u_i[σ_i(b[ω′], s_i)] – u_i[σ_i(b[ω′], s_i′)]) ≥ 0, where E is the event such that ω ∈ E ∈ I_i. Thus, s_i is SEU-rational for i at ω if it maximizes expected utility for i, conditional on his prior beliefs updated by his information at ω.

We define a Bayesian model of a particular game as an ordered quintuple <Ω, b(.),I, π, χ>, where Ω is a finite, non-empty set of states and b(ω) = (b ₁[ω], . . ., b_n[ω]),I= (I₁, . . ., I_n), π = (π₁, . . ., π_n) and χ = (χ₁, . . ., χ_n) are, respectively a behaviour function, an information structure, a profile of priors and a profile of choiceworthiness functions defined with respect to Ω and the game, such that the following three conditions are satisfied:

Choice Rationality requires that, at each state, each player's actions are consistent with whatever standard of decision-theoretic rationality is being modelled. SEU-Maximization stipulates that the standard of rationality is the maximization of subjective expected utility. Knowledge of Own Choice imposes the obvious restriction that, at each state, every player knows the pure strategy that he chooses.Footnote ⁵ Our definition of a Bayesian model is equivalent to that of Aumann (Reference Aumann1987), except that we do not impose (or rule out) Aumann's assumption of common priors (that is, that for all i, j ∈ N, π_i = π_j). Aumann (12–15) notes that, unlike the other features of his model, it is not essential to a Bayesian representation of CKR; he imposes it only to generate sharp results.

The following result is implied by the analysis of Aumann (Reference Aumann1987):

Theorem 1. For every game in G, a Bayesian model exists.

This theorem shows that, for every game in G, the concept of a Bayesian model is an internally consistent representation of CKR. In particular, in any such model, Ω is a universal set of states at each of which some profile of strategies is played that contains only choiceworthy strategies; and S* is the set of profiles played at states in Ω. As Ω is Bayesian common knowledge at all states and Ω = ∪_{s ∈S*}E(s), there is Bayesian common knowledge at all states of the event that a profile in S* is played.

3. THREE PUZZLES

Given Theorem 1, it is natural to ask whether further conditions can be imposed on Bayesian models. Our concern is with two restrictions on player's beliefs. To formulate them, we use the following concepts: A prior π_i for player i is independent if, for each profile s = (s ₁, . . ., s_n) of included strategies, π_i(E[s]) = Π_{j ∈N} π_i(E[s_j]). We extend our notation for posterior probabilities to those on events by using π_i(E′|E), where E′ ⊆ E and E is an element or union of elements of I_i, to denote player i's posterior probability of E′ given E. We can now state:

Independence rules out the possibility that some player i believes that the choices of players from among their included strategies are correlated with one another.Footnote ⁶ Although Aumann's (Reference Aumann1987) Bayesian model of CKR allows correlation of strategies between players, game theory needs to be able to model situations in which the players have no mechanisms for achieving such correlation or grounds for believing in it. If the representation of CKR is to apply to such cases, it must be possible to impose Independence on the model.

Caution requires that, if some strategy s_i is choiceworthy for player i at state ω then, at the same state, other players must assign strictly positive posterior probability to s_i being played. The motivation for the condition rests on two background assumptions. The first is the implicit assumption, integral to Aumann's approach, that each player knows, in an informal sense, the information partition and prior of each other player (see Aumann Reference Aumann1987: 9–10). From this, player j should also be taken to know, in the same informal sense, the choiceworthy strategies for player i at every state ω. The second background assumption is that the tie-breaking mechanism that player i uses to discriminate between options which, according to the standard of rationality, are equally choiceworthy is private to him. Since tie-breaking occurs only when rationality fails to determine what should be chosen, the properties of a tie-breaking mechanism must be non-rational. If the representation of CKR is to apply to cases in which non-rational tie-breaking rules are private, it must be possible to impose on the model the background assumption that they are. To do this would be to require that, if s_i is choiceworthy for player i at state ω, then the element of (each other) player j's information partition that contains ω must contain some state(s) at which s_i is played. Given these background assumptions, Caution expresses the following principle of prudential belief for each player j ≠ i: If s_i is choiceworthy for player i at state ω then, given what player j knows at ω, j's beliefs should allow that s_i might be played.

Note that there is no corresponding prudential argument in relation to j's beliefs about a strategy s_i that is not choiceworthy for i, as (by Aumann's implicit assumption) j is aware of such non-choiceworthiness. If j knows, in the informal sense, that s_i is not choiceworthy for i at any ω, then an Aumann-like understanding of CKR (captured by Choice Rationality and SEU-Maximization) should not allow j to assign positive probability to s_i. By defining the requirement of Caution relative to choiceworthy strategies, we follow Börgers and Samuelson (Reference Börgers and Samuelson1992) and Pearce (Reference Pearce1984).Footnote ⁷ As with Independence, we are not suggesting that Caution is an implication of Bayesian rationality, but merely that it represents a case of potential interest for the modelling of CKR.

To be more precise, we interpret Bayesian models which satisfy both Independence and Caution as attempting to represent common knowledge of the following standard of practical rationality: each player's beliefs assign independent probabilities to other players’ strategies, zero probability to strategies regarded as not rationally playable, and strictly positive probability to all strategies regarded as rationally playable; and each player maximizes expected utility relative to these beliefs. We call this standard that of independent cautious expected utility maximization (or the ICEU standard, for short).Footnote ⁸ Thus, a Bayesian model which satisfies Independence and Caution is an ICEU Bayesian model.

It would be puzzling if an otherwise coherent representation of CKR could not accommodate the view of rationality embedded in the ICEU standard without giving rise to paradoxes or impossibility. However, that is how matters turn out. We use three games to illustrate this.

Our first exhibit, illustrating the Proving Too Much Paradox, is Game 1.Footnote ⁹

Proposition 1. ICEU Bayesian models of Game 1 exist; and, in every such model, S ₁* = {first} and S ₂* = {left, right}.

Proposition 1 is paradoxical because S ₁* = {first} implies that first is choiceworthy at every state and the only strategy that Player 1 may play. The strategy first can only be choiceworthy at every state if Player 1's posterior probability on left is always greater than or equal to 2/3 (since otherwise second would be strictly better than first). But, at every state, Player 1 knows that Player 2 is indifferent between left and right. Clearly, Player 1 might believe that there is a tendency for Player 2's tie-breaking mechanism to resolve in favour of left even when, as in the situations that ICEU Bayesian models are intended to represent, that mechanism is private. The puzzling feature of Proposition 1 is that it implies that, even when the tie-breaker is private, Player 1 must believe it to have that tendency. In this sense, we seem to have proved too much.

In more general terms, the Proving Too Much Paradox takes the following form. The property that some specific strategy is (or is not) choiceworthy holds for all ICEU Bayesian models, with the apparent implication that common knowledge of this property is implied merely by the assumption that the ICEU standard of rationality is common knowledge; but there seems to be no way that the players could arrive at that conclusion, using only the reasoning resources attributed to them by that assumption.Footnote ¹⁰

Our second exhibit illustrates another way in which a Bayesian modelling approach can seem to generate paradoxical implications. This game is named in memory of a method of marking lanes on single-carriageway roads which was once common in Britain. There were three lanes, one for slow traffic in each direction, and a central lane designated for overtaking in both directions. The players are drivers travelling in opposite directions who have simultaneous overtaking opportunities; each is indifferent between staying in his slow lane and overtaking successfully. Each player i chooses whether to pull out to overtake (out_i) or not to do so (in_i):

Proposition 2. ICEU Bayesian models of Game 2 exist; and, in every such model, either (i) S ₁* = {in ₁} and S ₂* = {in ₂, out ₂} or (ii) S ₁* = {in ₁, out ₁} and S ₂* = {in ₂}.

Proposition 2 implies that, in every ICEU Bayesian model of Game 2, one of the two players plays the ‘risky’ strategy (out) in some states, while the other plays it in none. The structure of the game is entirely symmetrical with respect to the two players. But, if one player plays out in some states and the other plays it in none, there must be some asymmetry that tells one and only one player that they may play out. The apparent implication of Proposition 2 is that the existence of such an asymmetry is implied merely by the assumption that the ICEU standard of rationality is common knowledge; but there seems to be no way in which the players could discover that asymmetry using only the knowledge attributed to them by that assumption, since both the assumption and the game are symmetric. Of course, there is no paradox in the idea that there could be common knowledge of an asymmetry in what rationality requires of the players, grounded on information external to the formal description of the game. The Three-lane Road Paradox is the apparent demonstration that a conception of CKR implies that there must be such knowledge.

Neither Game 1 nor Game 2 yields an outright inconsistency in the conditions that define an ICEU Bayesian model. In fact, these conditions are mutually consistent for every two-player game in G.Footnote ¹¹ However, an inconsistency can be shown using a game introduced by Cubitt and Sugden (Reference Cubitt and Sugden1994), which can be thought of as a three-player extension of Game 2.Footnote ¹² We call the inconsistency shown by this game the Tom, Dick and Harry Paradox to match the story described in that paper. Tom (player 1), Dick (player 2) and Harry (player 3) are guests in an isolated hotel. Tom is trying to avoid Dick, Dick to avoid Harry, and Harry to avoid Tom; yet, there is no alternative to taking their evening meal in the hotel. Guests who eat in the restaurant (out) will meet each other, whereas those who eat in their rooms (in) will not meet any others.

The paradox consists in the fact that there is no ICEU Bayesian model of Game 3, which constitutes a proof of the following result:

Theorem 2. There are games in G for which no ICEU Bayesian model exists.

Theorem 2 establishes that there is at least one standard of rationality, namely ICEU, that cannot be represented without contradiction in a Bayesian model. However, we also know from Theorem 1 that there is at least one standard, namely SEU-maximization, that can be so represented. So should one simply conclude that the Bayesian modelling strategy has been vindicated and that ICEU is unacceptable? We think not.

Considered in its own right (rather than in the context of Bayesian modelling), the view of rationality embedded in ICEU seems to be internally coherent. The normative issue of adjudicating between alternative standards of rationality seems orthogonal to the modelling issue of how to represent a world in which some standard of rationality is common knowledge. Thus, whether or not one believes that obeying the ICEU standard is a normative requirement of rationality, it is puzzling that common knowledge of that standard cannot always be represented in a Bayesian model.

Of course, it is open to a game theorist to treat the Bayesian modelling strategy as a fixed point, and to stipulate that the only relevant standards of rationality are those that can be represented in such models. But that would leave unanswered the theoretical question of why the Bayesian approach cannot represent common knowledge of an apparently coherent standard of rationality. And in the absence of a satisfactory answer to that question, privileging that approach seems an arbitrary manoeuvre, and all the more so if common knowledge of the ICEU standard can be represented by a different approach.

4. REASONING SCHEMES

We now develop our Lewisian rendition of CKR in which players’ reasoning is represented explicitly. Such representation requires different primitives. As a first step, we introduce our representation of a mode of reasoning.

We model a mode of reasoning as a structure defined in relation to some domain P of sentences, to be interpreted as well-formed formulae of a formal language. Implicitly, these sentences express propositions which may either be taken as given or inferred within a given mode of reasoning. We use p, q, r as sentence variables, that is as placeholders for unspecified sentences in P. In this Section, we impose only very general conditions on P. The logical connectives ¬, ∧, and ⇒, defined by the rules of classical logic, are used, respectively, for negation, conjunction and material implication. We impose throughout that P is closed under the formation of negation, conjunction and material implication, i.e. that if p, q ∈ P then ¬p, p ∧ q, p ⇒ q ∈ P.

An inference rule in domain P is a two-place instruction of the form «from . . ., infer . . . », where the first place is filled by a non-empty, finite subset of P (whose elements are the premises of the rule) and the second place by an element of P (the conclusion of the rule).

An inference structure is a triple R = <P, A(R), I(R)>, where P is the domain in which reasoning takes place, A(R) ⊆ P is the set of axioms of R, and I(R) is a set of inference rules in domain P. The set T(R) of theorems of R is defined inductively as follows. We define T ₀(R) = A(R). For k ≥ 1, T_k(R) is defined as T_{k −1}(R) ∪ {p ∈ P| p is the conclusion of an inference rule in I(R), all of whose premises are in T_{k −1}(R)}. Then T(R) = T ₀(R) ∪ T ₁(R) ∪ . . . .

Below, we will use a particular class of inference structures called ‘reasoning schemes’ to represent modes of reasoning that are accessible to the players of a game. The game-theoretic content of a reasoning scheme R will be represented by properties of P, A(R) and I(R). P will be defined in terms of a particular formal language, in which sentences refer to properties of strategies in a game; and principles of game-theoretic reasoning will be represented as axioms and inference rules of R. Thus, we will include inference rules which represent such principles, but are not licensed by classical logic – for example, rules which infer what it would be rational for one player to do from a prediction about what another player will do. Our main interest is in the implications for players’ reasoning of specific game-theoretic axioms and inference rules. But since we are modelling rationality, it is incumbent on us also to endow players with the capability to make inferences that are licensed by classical logic. As a means of achieving this, we proceed as follows.

Throughout our analysis, we will use the term ‘logically’ only to refer to operations licensed by classical logic. Thus, we will say, for any non-empty, finite subset Q = {q ₁, . . ., q_m} of P and any p ∈ P, that p is logically entailed by Q if q ₁ ∧ . . . ∧ q_m ∧ ¬p is a contradiction in classical logic (i.e. if every truth-value assignment renders it false); and that an inference rule is logically valid if its conclusion is logically entailed (in this sense) by the set of its premises. For any inference structure R, we will say that I(R) contains the rules of valid inference if, for every non-empty, finite Q ⊆ P and every p ∈ P, if p is logically entailed by Q then «from Q, infer p» ∈ I(R). We will say that a set of sentences is consistent if it has no finite subset the conjunction of all elements of which is a contradiction in classical logic.

An inference structure R, such that A(R) is non-empty and I(R) contains the rules of valid inference, is a reasoning scheme. Thus, for every reasoning scheme R, every tautology in P is an element of T(R). We will say that a reasoning scheme R is consistent if T(R) is consistent. Our aim is to model CKR in terms of reasoning schemes that are consistent. But, to demonstrate the feasibility of this goal, we need to use a modelling framework in which consistency can be proved. For this reason, we do not impose consistency as part of the definition of a reasoning scheme.

We will say that a person endorses a reasoning scheme R if he takes its axioms to be true and accepts the authority of its inference rules; a person who endorses R has reason to believe each of its theorems. Our analysis will be about what players have reason to believe about one another's strategy choices, given that they endorse particular kinds of reasoning schemes. Because we take this approach, we will never need to consider whether the theorems of some reasoning scheme are ‘really’ true or false.

5. COMMON REASONING IN A POPULATION

Our approach is to model CKR among a population of agents as the existence of a core of shared reasoning which is endorsed by each agent in the population and is commonly attributed to other such agents. It allows us to represent the idea that each individual maintains a distinction between (on the one hand) what everyone has reason to believe, given the axioms and inference rules that everyone endorses and (on the other hand) what he has reason to believe, given the axioms and inference rules that he endorses. Thus, given a finite, non-empty, population N = {1, . . ., n} of agents, we postulate the existence, for each agent i, of a reasoning scheme R_i of private reason which i endorses, and the existence of a reasoning scheme R* of common reason. Of each sentence p that is a theorem of R*, we will say that there is common reason to believe p.

As a step in defining the formal language of our models, we introduce the following piece of syntax. If p is a sentence and R is a reasoning scheme, R(p) is a sentence to be read as ‘p is a theorem of R’. Note that R(p) is a sentence that can feature as an axiom or theorem of a reasoning scheme; it is not how we will express a statement, made from our viewpoint as modellers, about the properties of R. When we assert that p is a theorem of R, we use the notation p ∈ T(R). Thus, for example, if R ₁ and R ₂ are (possibly distinct) reasoning schemes, R ₂(p) ∈ T(R ₁) denotes that the sentence R ₂(p), which expresses the proposition that p is a theorem of R ₂, is in fact a theorem of R ₁.

We take as given a non-empty set P ₀ of primitive sentences, such that no sentence in P ₀ contains any of the terms R*(.), R ₁(.), . . ., R_n(.). For each k ≥ 1, we define P_k to contain all of the following sentences (and no others): (i) every sentence which can be constructed from the elements of P_{k −1} using a finite number of connectives from the set {¬, ∧, ⇒}; (ii) every sentence of the form R*(p) where p ∈ P_{k −1}; and (iii) every sentence of the form R_i(p) where i ∈ {1, . . ., n} and p ∈ P_{k −1}. We define φ(P ₀) ≡ P ₀ ∪ P ₁ ∪. . . . For any given specification of P ₀, φ(P ₀) is the domain in which the reasoning schemes of our model operate.

We now define the following concept as a representation of the links between private and common reason. An interactive reasoning system among the population N = {1, . . ., n} is a triple < P ₀, R*, (R ₁, . . ., R_n)>, where P ₀ is a set of primitive sentences, R* is a reasoning scheme, and (R ₁, . . ., R_n) is a profile of reasoning schemes, such that each of the (n+1) reasoning schemes has the domain φ(P ₀) and the following conditions hold:

We will say that an interactive reasoning system <P ₀, R*, (R ₁, . . ., R_n)> is consistent if each of its component reasoning schemes is consistent.

The Awareness condition stipulates that each agent's private reasoning has access to the theorems of common reason. The Authority condition stipulates that, from the premise that some sentence p is a theorem of common reason, each agent's private reason infers p. The Attribution condition stipulates that common reason attributes its own theorems to the private reason of each agent. Awareness and Authority together imply that each theorem of common reason is also a theorem of the private reason of each agent i. Thus, we may think of common reason, not as involving any unexplained collective consciousness, but simply as a sub-routine that each agent can individually use to generate certain theorems. It is convenient to identify this sub-routine using the device of R*, and then to impose Attribution upon it, in order to capture the idea that certain conclusions are reached by all agents in the population in the same way and also attributed by each of them to each other.

We will say that there is iterated reason to believe p in population N if the proposition R_j(. . . R_k(p). . .) ∈ T(R_i) is true, for all finitely nested sentences R_j(. . . R_k(p). . .) and all i, j, . . ., k ∈ N. The following theorem establishes that, in an interactive reasoning system, there is iterated reason to believe all sentences that are theorems of R*:

Theorem 3. Consider any population N of agents and any interactive reasoning system <P ₀, R*, (R ₁, . . ., R_n)> among the population N. For every sentence p ∈ T(R*), there is iterated reason to believe p in population N.

Iterated reason to believe is the closest Lewisian analogue of Aumann's concept of common knowledge. Theorem 3 allows our analysis of CKR to be carried out almost entirely in terms of R*. Nevertheless, private reasoning schemes are an essential part of our modelling strategy, since it is only by virtue of the connections between private and common reason that we can claim that an analysis of R* is informative about iterated reason to believe.

There is a close affinity between Theorem 3 and Lewis's (Reference Lewis1969: 52–60) analysis of common knowledge. Lewis defines p to be ‘common knowledge’ in a population N if some ‘state of affairs’ A holds, such that (i) everyone in N has reason to believe that A holds, (ii) A ‘indicates’ to everyone in N that everyone in N has reason to believe that A holds, and (iii) A ‘indicates’ to everyone in N that p. He defines ‘A indicates to person i that p’ as ‘if i has reason to believe that A holds, i thereby has reason to believe that p’. He sketches a proof of the theorem that if p is common knowledge in this sense, and given (not fully specified) premises to the effect that individuals share, and have reason to believe that they share, certain principles of rationality, inductive standards and background information, there is iterated reason to believe p in N. Cubitt and Sugden (Reference Cubitt and Sugden2003) reconstruct the theorem and its proof, using an explicit specification of the properties of ‘indication’, motivated by interpreting ‘i has reason to believe that p’ as saying that p is treated as true in a mode of reasoning that i endorses and ‘A indicates to i that p’ as saying that, in that mode of reasoning, there is an inference from ‘A holds’ to p.

The same ideas can be represented more directly in an interactive reasoning system, in which ‘shared’ background information and principles of rationality are represented by axioms and inference rules of R*. For this purpose, we treat p and ‘A holds’ as sentences in the formal language of that interactive reasoning system. That some state of affairs A is such that, if it occurs, its occurrence is public and self-evident can be represented by the material implication that if (in fact) A holds, then ‘A holds’ ∈ A(R*). That there are common standards of inductive inference such that, if there is common reason to believe that A holds, there is thereby common reason to believe p can be represented by «from {‘A holds’}, infer p» ∈ I(R*). Given these conditions, it follows from Theorem 3 that if (in fact) A holds, then there is iterated reason to believe p in N.

Our method of modelling CKR in a given game will be to represent practical and game-theoretic rationality in terms of axioms and inference rules, and to attribute these to common reason in an interactive reasoning system among the population comprising the players of the game. By virtue of Theorem 3, any sentences that can be derived using those axioms and inference rules will be the object of iterated reason to believe among the players.

6. DECISION RULES: PRACTICAL RATIONALITY EXPRESSED BY SENTENCES

In this section, we develop a general method of representing principles of practical rationality for a game in the form of a particular kind of sentence, which we call a ‘decision rule’. This concept uses a purely formal notion of ‘permissibility’, to be interpreted as permissibility with respect to whatever principles of practical rationality are to be represented.

Here, and throughout Sections 6–9, we fix a given game in G. Our analysis applies to any such game but we suppress phrases of the form ‘for all games in G’ except in formal results. Differences between games become important again only in Sections 10 and 11.

We now add two further pieces of syntax to our formal language. For every player i and for every s_i ∈ S_i, p_i(s_i) is a sentence to be read as ‘s_i is permissible for i’ and interpreted as stating that, normatively, i may choose s_i (but not that he must, since two or more strategies might be permissible for him). For every player i and for every s_i ∈ S_i, m_i(s_i) is a sentence to be read as ‘s_i is possible for i’ and interpreted as stating that s_i might in fact be chosen by i. Sentences of the form p_i(s_i) or ¬p_i(s_i) (the latter read as ‘s_i is impermissible for i’) are permissibility sentences. For each permissibility sentence p_i(s_i) or ¬p_i(s_i), the corresponding possibility sentence m_i(s_i) or ¬m_i(s_i) (the latter read as ‘s_i is impossible for i’) is its correlate, and vice versa.

We will say of any conjunction of sentences that it states each of its conjuncts. A prediction about a player i is a conjunction of the elements of a consistent set of possibility sentences referring to the strategies available to i, satisfying the conditions that (i) not every strategy in S_i is stated to be impossible; and (ii) that, if every strategy but one in S_i is stated to be impossible, the remaining strategy is stated to be possible. Notice that (ii) is not redundant because, in general, a prediction about i may not refer to every strategy available to i. (Since we want to be able to represent how reasoning can proceed in successive steps from initial premises to progressively richer conclusions, we need to be able to represent predictions that are incomplete in this sense.) Conditions (i) and (ii) express the presumption that, as S_i exhausts the options available to i, it cannot be the case that all its elements are impossible; and, if every element but one is impossible, that establishes that the remaining one is possible (indeed, certain). A conjunction of the elements of a non-empty set of predictions about individual players other than i, where that set contains no more than one non-null prediction about any such player, is a collective prediction about N\{i}.

Analogously with the concept of prediction, a recommendation to a player i is a conjunction of the elements of a consistent set of permissibility sentences referring to the strategies available to i, satisfying analogues of conditions (i) and (ii); here the presumption that those conditions embody is that normative requirements must be logically capable of being satisfied.

The definition of a correlate is extended to recommendations and predictions, so that for each recommendation there is a unique correlate prediction and vice versa: the correlate of a recommendation (resp. prediction) is the conjunction of the correlates of its component permissibility (resp. possibility) sentences.Footnote ¹³

We also need to represent sentences that assert nothing substantive about what player i might or might not play or, correspondingly, about what would or would not be normatively permissible for i. Sentences of the first kind play a crucial initiation role in our model of reasoning as, intuitively put, we need to represent how a player ‘starts’ with no substantive conclusions about what another player may or may not play and only ‘reaches’ such conclusions though steps of reasoning. We therefore define a further piece of syntax. We use an arbitrary tautology, denoted by #, to represent a sentence with no content – the null sentence. Thus, # can be a null recommendation, a null prediction or null collective prediction. In view of this, the correlate of # is #.

Recommendations to a player i, collective predictions about the set of players N\{i}, and predictions about i are sentences that have special roles to play in what follows. To distinguish them from other sentences, we use the sentence variables y_i for recommendations to i, x _−i for collective predictions about N\{i}, and z_i for predictions about i. Using this notation, a maxim for player i is a material implication x _−i ⇒ y_i. The interpretation is that, conditional on the prediction x _−i about the behaviour of players other than i, the permissibility sentences stated by y_i are mandated by some conception of practical rationality. Note that the maxim # ⇒ y_i is logically equivalent to the recommendation y_i.

A decision rule for player i is a conjunction of all elements of a set F_i of maxims for i, such that F_i satisfies the following conditions: (i) (Distinct Antecedents) for all x _−i: F_i contains at most one maxim whose antecedent is logically equivalent to x _−i; and (ii) (Deductive Closure) for all x _−i′, for all non-null y_i′: if the material implication x _−i′ ⇒ y_i′ is logically entailed by a conjunction of all elements of F_i, then F_i contains a maxim x _−i″ ⇒ y_i″ such that x _−i″ is logically equivalent to x _−i′ and y_i″ logically entails y_i′. By virtue of Distinct Antecedents, a decision rule for i makes a set of recommendations to her that are conditional on logically distinct predictions about the other players. In view of this, the Deductive Closure condition implies that, for any collective prediction, all the permissibility sentences implied by the rule, given that prediction, are summarized by a single maxim of the rule. As the consequent of that maxim is a recommendation, this condition guarantees that the set F_i is consistent, and that F_i does not logically entail the falsity of any collective prediction. In this sense, a decision rule for player i is compatible with every possible collective prediction about the other players. However, it need not contain maxims covering all these possibilities. We use D_i as a sentence variable standing in for a decision rule for player i.

7. COMMON PRACTICAL REASONING IN A GAME

We now use the concepts of an interactive reasoning system and of a decision rule, developed in Sections 5 and 6 respectively, to model CKR in a given game. To do so, we first specify P ₀, the set of primitive sentences, so that it contains # and, for each i ∈ N and for each s_i ∈ S_i, the sentences m_i(s_i) and p_i(s_i) (and no other sentences). This specification implies that all decision rules are in φ(P ₀). Next, we specify a particular profile of decision rules D = (D ₁, . . ., D_n). We then construct reasoning schemes R* = <φ(P ₀), A(R*), I(R*)>, R ₁ = <φ(P ₀), A(R ₁), I(R ₁)>, . . ., R_n = <φ(P ₀), A(R_n), I(R_n)> in the following way. R* is constructed by using the rules:

1. (1) A(R*) = {#, D ₁, . . ., D_n};

2. (2) I(R*) contains the rules of valid inference and those specified below, and no other rules:

   1. (i) for all p ∈ φ(P ₀): «from {p}, infer R_i(p)» ∈ I(R*);

   2. (ii) for all i ∈ N, for all y_i, z_i ∈ φ(P ₀) such that y_i is a recommendation to i and z_i is the prediction about i that is the correlate of y_i: «from {R_i(y_i)}, infer z_i» ∈ I(R*).

For each i ∈ N, R_i is constructed by using the rules:

1. (3) A(R_i) = {p ∈ φ(P ₀) | p = R*(q) for some q ∈ T(R*)};

2. (4) I(R_i) contains the rules of valid inference and those specified below, and no other rules:

   for all p ∈ φ(P ₀): «from {R*(p)}, infer p» ∈ I(R_i).

By virtue of rules (2i), (3) and (4), which respectively ensure that the Attribution, Awareness and Authority requirements are satisfied, < P ₀, R*, (R ₁, . . ., R_n)> is an interactive reasoning system.Footnote ¹⁴ Rule (1) provides R* with substantive axioms, in the form of the decision rules in D. Rule (2ii) provides R* with an inference rule that is specific to our modelling of game-theoretic rationality. This inference rule embeds in common reason the following principle: from i's having reason to believe some recommendation that applies to him, it can be inferred that he will act on that recommendation. In this sense, common reason attributes practical rationality to each player.

An interactive reasoning system <P ₀, R*, (R ₁, . . ., R_n)> defined in relation to a profile D of decision rules and constructed according to rules (1) to (4), with P ₀ as specified at the start of this Section, is a common-reasoning model of the game; D is its common standard of practical rationality.

It is immediate that, for any profile D of decision rules for any game in G, a corresponding (and unique) common-reasoning model exists: the model is constructed by defining P ₀ as specified and following rules (1) to (4). What is not so obvious (since rules (1) to (4) attribute substantive axioms, as well as some inference rules besides those of logically valid inference, to the component reasoning schemes) is whether the model so constructed is consistent. The following theorem establishes this property:

Theorem 4. For every game in G, for every profile D of decision rules for that game, the common-reasoning model in which D is the common standard of practical rationality is consistent.

Theorem 4 shows that our framework can represent coherently common knowledge of any conception of practical rationality that can be formulated as a profile of decision rules. Together with Theorem 3, it establishes the credentials of our Lewisian modelling approach.

8. THE RECOMMENDATION ALGORITHM

We now focus on the content of common reason in the common-reasoning model defined by a given profile D of decision rules, in so far as that content relates to permissibility and impermissibility of strategies. (From rules (3) and (4) and Theorem 4, each R_i will have the same content as common reason in relation to permissibility and impermissibility of strategies.)

For each player i and each strategy s_i, we can ask whether, in the common-reasoning model, it is a theorem of R* that s_i is permissible for i (i.e. whether p_i(s_i) ∈ T(R*) is true). We can also ask whether it is such a theorem that s_i is impermissible for i (i.e. whether [¬p_i(s_i)] ∈ T(R*) is true). By virtue of Theorem 4, it cannot be the case that p_i(s_i) ∈ T(R*) and [¬p_i(s_i)] ∈ T(R*) are both true. But it can be the case that neither is true – that is, that common reason is silent about whether s_i is permissible or impermissible. Thus, in general, a common-reasoning model implies a trinary partition of each player's strategy set S _i, the three elements of which are {s_i ∈ S_i| p_i(s_i) ∈ T(R*)}, {s_i ∈ S_i| [¬p_i(s_i)] ∈ T(R*)}, and {s_i ∈ S_i| p_i(s_i) ∉ T(R*) and [¬p_i(s_i)] ∉ T(R*)}. We call this partition the common-reasoning partition for player i.

These arguments indicate that the common-reasoning model, for a given profile of decision rules, defines a ‘solution’ of the game that is interpretable as indicating which strategies are shown by common reason to be permissible (resp. impermissible).Footnote ¹⁵ In general, a solution may have the property that some s_i is neither shown by common reason to be permissible nor shown by common reason to be impermissible. The reader might want to ask what, in such a case, it is rational for the relevant player i to do. However, the Lewisian approach does not analyse what it is ‘really’ rational for a player to do, but rather what players have reason to believe, given that they endorse certain modes of reasoning. Within this approach, the closest analogue of the reader's question is ‘Which permissibility sentences does i have reason to believe to be true?’ The answer (in view of rules (3) and (4) for the construction of R_i in the common-reasoning model) is: ‘The set of permissibility sentences that are theorems of R*’. All that can be said (for the relevant case) is that, using only the reasoning resources that have been attributed to her by the model, i cannot determine whether s_i is permissible or impermissible.

Although Theorem 4 establishes the existence of a solution, it does not in itself show how we, as analysts, can discover that solution; nor does it indicate a specific line of reasoning whereby common reason can reach the conclusions about permissibility (resp. impermissibility) of strategies that are summarized by the profile of common-reasoning partitions. Each of these gaps can be filled by defining a particular algorithm, as we now explain.

For any profile D = (D ₁, . . ., D_n) of decision rules, we define the recommendation algorithm as follows. The algorithm has a succession of stages k = 0, 1, 2, . . ., at each of which, for each player i, it generates as its output a recommendation to i, denoted y_i^k. As an initiation rule, we set y_i ⁰ = #, for each i. Then, for each stage k > 0, and for each player i, y_i^k is obtained through three operations. Operation 1 generates, for each i, a prediction about i, denoted z_i^k, that is defined as the correlate of y_i^{k –1}. Operation 2 generates, for each i, a collective prediction about N\{i}, denoted x _−i^k, that is defined as z ₁^k ∧ . . . ∧ z_{i −1}^k ∧ z_{i +1}^k ∧ . . . ∧ z_n^k. Operation 3 determines y_i^k, for each i, as follows: if there is a component maxim of D_i that has as its antecedent a sentence logically equivalent to x_−i^k, then y_i^k is the consequent of that maxim; otherwise, y_i^k = #. The algorithm halts if a stage k* is reached at which y_i^k* = y_i^{k* –1}, for all i. If such a k* is reached, then, for each player i, y_i^k* is the final output of the algorithm. We can now state:

Theorem 5. Consider any game in G and any profile D of decision rules for the game.

1. (i) The recommendation algorithm for D halts at some finite stage k* > 0.

2. (ii) For each player i, let y_i^k* be the final output of the recommendation algorithm for D, and R* be common reason in the common-reasoning model with D as common standard of practical rationality. For each i ∈ N, and for each s_i ∈ S_i:

   1. (a) s_i is stated by y_i^k* to be permissible if, and only if, p_i(s_i) ∈ T(R*); and

   2. (b) s_i is stated by y_i^k* to be impermissible if, and only if, [¬p_i(s_i)] ∈ T(R*).

This theorem establishes that, for any profile D of decision rules, the corresponding recommendation algorithm halts and generates, as its final output for each player i, a recommendation for i that conjoins exactly those permissibility sentences for i that are theorems of R* in the common-reasoning model with D as common standard of practical rationality. Thus, the algorithm is a tool by which we, as analysts, can discover the common-reasoning partition for each player i.

The recommendation algorithm can also be interpreted as tracking a line of reasoning by which common reason can establish the conclusions captured by the players’ common-reasoning partitions. To see this, consider the algorithm for any given profile D of decision rules. With respect to each player i, the output of stage 0 (i.e. the null recommendation) is an axiom of, and therefore a theorem of, R* in the corresponding common-reasoning model, by virtue of rule (1) for construction of that model. The output of each subsequent stage can be derived by using, in a specific sequence, axioms and inference rules of R* together with the output of the previous stage. At each stage, Operation 1 tracks inferences from the output of the previous stage that are licensed by rules in I(R*) by virtue of rules (2i) and (2ii) for the construction of a common-reasoning model; Operation 2 tracks inferences from the conclusions of those tracked by Operation 1 that are licensed by rules of logically valid inference in I(R*), by virtue of rule (2); and, finally, Operation 3 tracks inferences licensed by rules of logically valid inference provided by rule (2), using as premises D, which is an axiom of R* by virtue of rule (1), and the conclusions of inferences tracked by Operation 2.

9. CATEGORIZATIONS

In Section 10 below, we compare our Lewisian common-reasoning approach with the Bayesian approach described in Sections 2 and 3. But first we define some concepts introduced by Cubitt and Sugden (Reference Cubitt and Sugden2011) that are useful in making the comparison, because they provide a convenient way to summarize binary and trinary partitions of sets of strategies.

For any player i, an ordered pair <S_i ⁺, S_i ⁻> of subsets of S_i is a categorization of S_i if it satisfies the following conditions: (i) S_i ⁺ and S_i ⁻ are disjoint; (ii) S_i ⁻ ⊂ S_i; and (iii) if S_i\S_i ⁻ = {s_i} for any s_i ∈ S_i, then S_i ⁺ = {s_i}. In general, a categorization of S_i defines a trinary partition of S_i, whose elements are the positive component S_i ⁺, the negative component S_i ⁻, and the residual set S_i\(S_i ⁺ ∪ S_i ⁻).

Now consider any non-empty set N′ ⊆ N of players. For each i ∈ N′, let <S_i ⁺, S_i ⁻> be any categorization of S_i. In order to allow us to aggregate across players, we define a ‘union’ relation ∪* between such categorizations such that ∪*_{i ∈N′} <S_i ⁺, S_i ⁻> ≡ <∪_i∈N′S_i ⁺, ∪_i∈N′S_i−>.Footnote ¹⁶ Each such ∪*_i∈N′ <S_i ⁺, S_i ⁻> is a categorization of ∪_i∈N′S_i; its positive component is ∪_i∈N′S_i ⁺; and its negative component is ∪_i∈N′S_i ⁻. For purposes of the main text, we need only the case where N′ = N. For this case, we use a shorthand notation in which $\mathbb {S}$ denotes ∪_{i ∈N}S_i and $\mathbb {S}$⁺ and $\mathbb {S}$⁻ denote, respectively, the positive and negative components of a typical categorization of $\mathbb {S}$. Such a categorization is exhaustive if $\mathbb {S}$⁺ ∪ $\mathbb {S}$⁻ = $\mathbb {S}$.

Consider any two categorizations C ′ = <$\mathbb {S}$⁺′, $\mathbb {S}$⁻′> and C ″ = <$\mathbb {S}$⁺″, $\mathbb {S}$⁻″> of $\mathbb {S}$. We define a binary relation ⊇* (read as has weakly more content than) between such categorizations such that C ″ ⊇* C ′ if and only if $\mathbb {S}$⁺″ ⊇ $\mathbb {S}$⁺′ and $\mathbb {S}$⁻″ ⊇ $\mathbb {S}$⁻′. If, in addition, either $\mathbb {S}$⁺″ ⊃ $\mathbb {S}$⁺′ or $\mathbb {S}$⁻″⊃ $\mathbb {S}$⁻′ holds, we will say that C ″ has strictly more content than C ′, denoted C ″ ⊃* C ′.

Consider any Bayesian model M of the game, as defined in Section 2. For each player i, the model specifies a set S_i*(M) ⊆ S_i of included strategies. Equivalently, for each i, M specifies a categorization C_i^M = <S_i ⁺(M), S_i ⁻(M)> of S_i, where S_i ⁺(M) = S_i*(M) and S_i ⁻(M) = S_i\S_i*(M).Footnote ¹⁷ Thus, aggregating across all players, M specifies a single categorization C^M = ∪*_{i ∈N} <S_i ⁺(M), S_i ⁻(M)> of $\mathbb {S}$. We will say that C^M is the inclusion categorization with respect to Bayesian model M. By construction, C^M is exhaustive.

Now consider the common-reasoning model of the game, for a given profile D of decision rules, as defined in Section 7. As Section 8 showed, for any player i, this model defines a trinary common-reasoning partition of S_i, two of whose elements are {s_i ∈ S_i| p_i(s_i) ∈ T(R*)} and {s_i ∈ S_i| [¬p_i(s_i)] ∈ T(R*)}, where R* is common reason. Thus, the model defines, for each i, a categorization <S_i ⁺(D), S_i ⁻(D)> of S_i, where S_i ⁺(D) = {s_i ∈ S_i| p_i(s_i) ∈ T(R*)} and S_i ⁻(D) = {s_i ∈ S_i| [¬p_i(s_i)] ∈ T(R*)}.Footnote ¹⁸ Again aggregating across players, the common-reasoning model for the profile D specifies a single categorization C^D = ∪*_{i ∈N} < S_i ⁺(D), S_i ⁻(D)> of $\mathbb {S}$. Unlike C^M, C^D may or may not be exhaustive, depending on the common-reasoning partitions resulting from profile D. As the positive (resp. negative) component of C^D is the set of strategies whose permissibility (resp. impermissibility) is established in common reason in the common-reasoning model, we will say that C^D is the common-reasoning solution of the game, with respect to the profile D of decision rules.

10. ICEU BAYESIAN MODELS REVISITED

In this Section, we compare our approach to modelling CKR, set out in Sections 4–8, to the canonical approach set out in Sections 2 and 3. We focus on the cases in which each approach is adapted to a conception of practical rationality provided by the ICEU standard.

We have already defined the concept of a Bayesian model which incorporates the ICEU standard – the ICEU Bayesian model. To explore the relationship between the two approaches, we need to specify a corresponding class of common-reasoning models in which the conception of practical rationality is ICEU. As explained in Section 7, the common-reasoning model is uniquely defined for any given game and any given profile of decision rules. So what we need to do is to define a profile of ICEU decision rules, for any given game. We do this in the following way.

For any player i and for any collective prediction x _−i about N\{i}, we define a probability distribution over S _−i as ICEU-consistent with x _−i if it satisfies the following conditions. First, probabilities are independent in that, for each s _−i ∈ S _−i, the probability of s _−i is the product of the marginal probabilities of the strategies appearing in s _−i. Second, every strategy that x _−i states to be impossible has zero marginal probability. Third, every strategy that x _−i states to be possible has strictly positive marginal probability. An ICEU maxim for i is a maxim x _−i ⇒ y_i such that (i) y_i states p_i(s _i) if, and only if, s_i maximizes i's expected utility relative to all probability distributions that are ICEU-consistent with x _−i, and (ii) y_i states ¬p_i(s _i) if, and only if, s_i does not maximize i's expected utility relative to any probability distribution that is ICEU-consistent with x _−i. Because the elements of a given set of sentences can be conjoined in different orders, there may be collective predictions (resp. recommendations) that are formally distinct from, but logically equivalent to x _−i (resp. y_i), and so there may be more than one ICEU maxim with the logical content of x _−i ⇒ y_i. By taking exactly one maxim from every set of logically equivalent ICEU maxims for i, we can construct a non-redundant set F_i of ICEU maxims for each player i.

Consider any two ICEU maxims x _−i′ ⇒ y_i′ and x _−i″ ⇒ y_i″. It follows from the definition of an ICEU maxim that if x _−i′ and x _−i″ are logically equivalent then so too are y_i′ and y_i″. Hence, given the definition of non-redundancy, F_i satisfies Distinct Antecedents. It also follows from the definition of an ICEU maxim that if x _−i′ logically entails x _−i″, then y_i′ logically entails y_i″.Footnote ¹⁹ Thus, F_i satisfies Deductive Closure. So any conjunction of the elements of a non-redundant set F_i of ICEU maxims for i is a decision rule for that player. Since all such conjunctions are logically equivalent, we can fix on any one of them as ‘the’ ICEU decision rule D_i for player i; and, in this way, construct ‘the’ profile D of ICEU decision rules for the game and ‘the’ ICEU common-reasoning model. This model implies a unique common-reasoning solution, which we may unambiguously take as the ICEU common-reasoning solution (as every profile of ICEU decision rules for the game yields the same such solution). We denote this solution by the categorization C*.

Assume for the moment that the game has at least one ICEU Bayesian model. Consider any such model M and let C^M be its inclusion categorization; necessarily, C^M is exhaustive. We now investigate the relationship between C^M and C*.

M can be interpreted as a model of a situation in which each player is rational in the ICEU sense, and in which it is common knowledge that each strategy in the positive component of C^M might be played, and that each strategy in the negative component will not be played. The Bayesian modelling strategy does not try to represent how players arrive at this common knowledge; but on the most natural interpretation, there is an implicit assumption that they do so by some process of reasoning whose premises include (but are not necessarily limited to) those of the ICEU standard of rationality (see Section 1). In contrast, the ICEU common-reasoning framework explicitly models what players have reason to believe about the game, given that the ICEU standard of rationality is axiomatic in common reason. Thus, given our interpretation of Bayesian models, it is natural to conjecture that if the permissibility (resp. impermissibility) of some strategy is a theorem of common reason, then every ICEU Bayesian model includes (resp. does not include) that strategy. It is also natural to conjecture that if the ICEU common-reasoning solution is exhaustive, then there exists an ICEU Bayesian model that includes (resp. does not include) every strategy whose permissibility (resp. impermissibility) is a theorem of common reason. The following theorems establish that these conjectures are indeed correct.

Theorem 6. Consider any game in G for which an ICEU Bayesian model exists. Consider any such model M of the game, and let its inclusion categorization be C^M. Let C* be the ICEU common-reasoning solution. Then C^M ⊇* C*.

Theorem 7. For every game in G: If the ICEU common-reasoning solution C* is exhaustive, then (i) there exists an ICEU Bayesian model of the game; and (ii) for every such model M, the inclusion categorization C^M is identical to C*.

Theorem 7 establishes that, in the special case in which the ICEU common-reasoning solution of the game is exhaustive, and with respect to the resulting categorizations, the Bayesian and common-reasoning approaches are equivalent. In this case, the ICEU common-reasoning model may be seen as justifying ICEU Bayesian models, in the sense that it describes explicit steps of reasoning whereby the players could establish the permissibility of those strategies included in each ICEU Bayesian model and the impermissibility of all other strategies.

But, now consider cases where the ICEU common-reasoning solution is not exhaustive. (As we will show in Section 11, this is a genuine possibility.) Nothing we have said in this Section excludes the possibility that such a game has an ICEU Bayesian model. However, we know from Theorem 6 that, for every such model M, C^M ⊃* C*. Thus, the unmodelled reasoning that players are implicitly assumed to use in any such M must be (assumed to) enable them to arrive at common knowledge, not captured in C*, about the possibility and impossibility of strategies. That reasoning would have to make use of axioms and inference rules in addition to (or stronger than) those of the ICEU common-reasoning model.Footnote ²⁰ But if an ICEU Bayesian model requires support of this kind, there seems no a priori reason to expect such a model to exist for all games. And, even if some such model does exist for a given game, the additional axioms and inference rules that it requires may not be appropriate for all contexts in which the game is played.

11. RESOLVING THE PARADOXES

We now turn to the paradoxes presented in Section 3. It is convenient to go straight to the most problematic case – the Tom, Dick and Harry Paradox of Game 3. The paradox is that this game has no ICEU Bayesian model.

Nevertheless, this game (like every other) has an ICEU common-reasoning solution, which can be identified by using the recommendation algorithm. The outputs of the algorithm are (for i = 1, 2, 3): y_i ⁰ = #; y_i ¹ = p_i(in_i); y_i ² = y_i ¹. The ICEU common-reasoning solution is <{in ₁, in ₂, in ₃}, ∅>. This categorization is non-exhaustive: for each player i, out_i is neither shown to be permissible nor shown to be impermissible.

It should not be surprising if, for some reasoning scheme R and some sentence p, neither p nor its negation is a theorem of R. The reasoning scheme R has, as resources, only axioms in A(R) and inference rules in I(R); thus, it is hardly surprising if we can find some p such that neither it nor its negation can be derived from those axioms using those inference rules. The significance of the previous paragraph is that it shows that this unsurprising general possibility applies specifically to the case where R is common reason in the ICEU common-reasoning model of Game 3 and p any permissibility sentence referring to the strategy out_i for any player i.

As the ICEU common-reasoning solution of Game 3 is not exhaustive, it is natural to ask whether there is any way in which the profile of decision rules in the ICEU common-reasoning model could be strengthened to yield an exhaustive categorization as the resulting solution. The answer is ‘No’: for each player in Game 3, there is no decision rule stronger than that player's ICEU decision rule.Footnote ²¹ More generally, one might ask whether it is possible to achieve an exhaustive categorization by augmenting the interactive reasoning system that constitutes the ICEU common-reasoning model with further axioms or inference rules. Obviously, Theorem 4 cannot vouch for the consistency of such a construction. But we can show that in any such interactive reasoning system that is consistent, neither the permissibility nor the impermissibility of any out_i is a theorem of common reason.Footnote ²²

If our modelling framework is accepted, the implication is this: given common knowledge of the ICEU standard of rationality, there is no way in which players could reason their way to the conclusion that the permissibility or impermissibility of out_i, for any player i, was common knowledge. However, an ICEU Bayesian model of Game 3, were it to exist, would rest on the implicit assumption that the players had reasoned their way to such a conclusion. Given this analysis, the non-existence of an ICEU Bayesian model is not surprising.

This analysis also provides the key to the other paradoxes. Consider the Proving Too Much Paradox of Game 1. For this game, as for all two-player games, a family of ICEU Bayesian models exists. The paradox, as we originally presented it, is that certain propositions about the choiceworthiness of strategies, namely those summarized by S ₁* = {first} and S ₂* = {left, right}, hold for all ICEU Bayesian models of Game 1, but there seems no way that the players could arrive at common knowledge of those propositions using only reasoning resources attributed to them by the assumption that the ICEU standard of rationality is common knowledge. Now that we have developed the concept of a common-reasoning model, we can tighten up the ‘there seems no way . . .’ clause. The ICEU common-reasoning solution tells us what conclusions about choiceworthiness (or permissibility) can be arrived at by players who have just those reasoning resources and how they can arrive at them.

In the case of Game 1, with the ICEU common standard of practical rationality, the outputs of the recommendation algorithm are: y ₁⁰ = y ₂⁰ = #; y ₁¹ = ¬p ₁(third), y ₂¹ = #; y ₁² = ¬p ₁(third), y ₂² = p ₂(left); y ₁³ = y ₁², y ₂³ = y ₂². The ICEU common-reasoning solution is <{left}, {third}>. As with Game 1, this solution is not exhaustive: first, second and right are neither shown to be permissible nor shown to be impermissible. So the intuition on which the initial statement of the paradox is based is correct.

Any ICEU Bayesian model of Game 1 must therefore rest on the implicit assumption that players have access to resources of reasoning beyond those arising from the ICEU standard of practical rationality being common knowledge among them, and that those resources enable them to arrive at common knowledge of an exhaustive categorization. We are not entitled to make this assumption merely by virtue of the existence of some Bayesian model of Game 1. Thus, the fact that first and right are choiceworthy in all Bayesian models of that game does not imply that the choiceworthiness of those strategies is an implication of common knowledge of ICEU rationality per se. Rather, it is an implication of the Bayesian representation of that common knowledge. That representation is premised on an implicit assumption to which the ICEU common-reasoning model lends no support.

The Three-Lane Road Paradox of Game 2 can be resolved in a similar way. Proposition 2 establishes that ICEU Bayesian models of Game 2 can be partitioned into two classes – those models M′ for which the inclusion categorization is C^M′ = <{in ₁, in ₂, out ₂}, {out ₁}>, and those models M″ for which the inclusion categorization is C^M″ = <{in ₁, out ₁, in ₂}, {out ₂}>. The paradox is that each of these models appears to represent a situation in which the players have common knowledge of an asymmetry between what rationality requires of one of them and what it requires of the other, even though the formal structure of the game is symmetric.

Again, we can use an ICEU common-reasoning model to discover what conclusions players can reach by reasoning that uses only those axioms and inference rules that represent common knowledge of the ICEU standard. For Game 2, the outputs of the corresponding recommendation algorithm are (for i = 1, 2): y_i ⁰ = #; y_i ¹ = p_i(in_i); y_i ² = y_i ¹. The ICEU common-reasoning solution is <{in ₁, in ₂}, ∅>; again, it is not exhaustive. There is no asymmetry between the players in this solution, but, for each player i, out_i is neither shown to be permissible nor shown to be impermissible. As in the other games, we are not generally entitled to assume that the players have access to additional (or stronger) axioms and inference rules that allow them to arrive at an exhaustive categorization of strategies by common reasoning, because that assumption can only hold if the additional axioms and inference rules induce an asymmetry between the players.Footnote ²³ If there is no source of asymmetry, the Bayesian modelling strategy must fail.

12. CONCLUSION

Our objective has been to investigate the relationship between two ways of understanding one of the core concepts of game theory – common knowledge of rationality (CKR). On one understanding, canonically expressed by Aumann (Reference Aumann1987), CKR is represented by a model of what players know about one another's strategy choices, conditional on every possible event. Each player's choices are required to be decision-theoretically rational with respect to her knowledge and beliefs, and the event represented by the model itself is required to be common knowledge. The strategies available to each player can then be partitioned into those that are and are not played in the model, and since the model is common knowledge, those binary partitions are common knowledge too. On the alternative understanding, inspired by Lewis (Reference Lewis1969) and developed here with our concept of a common-reasoning model, CKR is represented by the axioms and inference rules of a mode of ‘common reasoning’ that each player endorses and attributes to the others. This approach induces a trinary partition of each player's strategy set, since a given strategy can have one of three statuses in common reason – it can be shown to be permissible, shown to be impermissible, or neither.

By modelling reasoning, the Lewisian approach provides formal resources with which to assess the implicit assumption of the Bayesian approach, formalized in Section 2 and based on Aumann's canonical contribution, that, if CKR holds, then players can arrive at a binary partition of strategies into those that are rationally playable and those that are not. We have shown that, for a given standard of practical rationality, our common-reasoning model grounds this implicit assumption when, but only when, the resulting common-reasoning solution is exhaustive. Since this condition may fail, there can be conceptions of practical rationality common knowledge of which cannot be represented in a Bayesian model without creating the potential for puzzling or contradictory results such as those of Section 3. In contrast, we have shown that the Lewisian approach has no need to restrict the conception of practical rationality beyond a requirement of internal coherence.

To repeat what we said at the outset, we recognize that the paradoxes of Section 3 can be circumvented by various (non-Lewisian) adaptations of the Bayesian framework. The central contribution of this paper does not depend on our analysis of what we see as paradoxical implications of Aumann's approach, nor on whether we are right to see those implications as paradoxes, rather than as minor curiosities or as evidence against the particular conception of practical rationality that gives rise to them. Our core contribution is the provision of formal and general foundations for Lewisian solution concepts in game theory.

We have proposed, and derived from a model of reasoning, a very general solution concept – that of a ‘common-reasoning solution’ – which defines a trinary partition of strategies for any given normal-form game and any given coherent concept of practical rationality for that game. This proposal defines a path of possible research in which alternative conceptions of practical rationality are specified, properties of the resulting common-reasoning solutions analysed, and their relationships to existing concepts studied. Cubitt and Sugden (Reference Cubitt and Sugden2011) can be seen as taking initial steps along this path, in view of the relationship (described in Appendix 1) between their ‘categorization solutions’ and common-reasoning solutions. We expect that solution concepts that are generated in this way will often be quite distinct from those of standard game theory. The reason for that expectation is that few existing solution concepts are defined in terms of trinary partitions of strategies, but such partitions reflect the underlying structure of common-reasoning models. More basically, they stem from a fundamental feature of reasoning itself.