Abstract
This paper provides a conceptual exploration of the implication of Jeffrey’s rule of belief revision to account for rule-following behavior in a game-theoretic framework. Jeffrey’s rule reflects the fact that in many cases learning something new does not imply that one has full assurance about the true content of the information. In other words, the same information may be both perceived and interpreted in several different ways. I develop an account of rule-following behavior according to which, in the context of strategic interactions, following a rule is defined by two conditions. First, that agents must frame the interaction in a sufficiently similar way and be aware of the same salient properties, i.e. they must have the same partition of the event. Second, they must ascribe to others the same revised probabilities to what they take to be the common partition. In a game-theoretic framework, this also indicates that rule-following behavior cannot be identified merely to the existence of a common prior.
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Notes
The interested reader may consult Vineberg (2016).
In this paper, I will speak indifferently of propositions and events to refer to the object over which agents form beliefs. While propositions are expressed through a language constitutive of a given syntax, an event is a set of states of the world (or ‘possible worlds’) where a given proposition is true. Events are thus the semantic counterparts of propositions expressed in a given syntax, there is a one to one correspondence between propositions and sets of possible worlds.
As an event E is a set of possible worlds, a partition {Ei} is defined as a set of subsets Ei of E with empty intersections. In the language of propositions, the partition corresponds to a disjunction of mutually exclusive propositions.
Conditionalization in the case the rigidity condition does not hold but where beliefs over the elements of the partition {Ei} remain unchanged is sometimes referred to as ‘Adam’s rule’. See Bradley (2007) for a discussion and a formal analysis.
I thus defined an institution as a set of constitutive rules. There are debates in philosophy and in the social sciences regarding the relevance of the distinction between constitutive and regulative rules first proposed by Searle (1969). The following discussion does not depend in any way on one’s stance regarding them.
I am here essentially following Dennett’s (1989) functionalist account of intentionality.
Admittedly, the institutions-as-equilibria account does invoke preferences and beliefs, i.e. intentional attitudes. But it does so in such a way that it does not distinguish between intrinsic and derived intentionality. In this sense, it views rules as being summaries rather than as being constitutive of practices. This is equally relevant to study animal and human population but it arguably misses a distinctive aspect of rule-following behavior among humans. See also footnote 8 above.
The payoffs of the ‘Rule’ strategy are computed on the basis of the assumption that the player has a probability ½ of being the incumbent. For instance, against an Hawk player, a Rule player will be the incumbent and play Hawk half of the time (with a payoff of 0) and will be the challenger and play Dove the other half (with a payoff of 1), thus leading to an expected payoff of ½. Two Rule players will alternate between the [Hawk; Dove] and the [Dove; Hawk] outcomes, thus leading to an expected payoff of 7/2.
More formally, denote Γ a probabilistic space and {γi}i∈N a vector of private signals received by the players on the basis of some probabilistic distribution. Denote fi: Γ → Si the private signal function of each player i. Suppose that each individual signal corresponds to a strategy recommendation, i.e. fi(γi) = si. A function f: Γ → S then implements a correlated distribution of strategy profiles. This distribution is a correlated equilibrium if each player maximizes her expected utility conditional on her private signal and thus on her strategy, i.e. for any γi and all i there is no strategy si’ such that Eui(siʹ, s−i |γi) > Eui(si, s−i |γi) with E the expectation operator and s−i= (s1, …, si−1, si+1, …, sn).
Since Pi is a partition, we have for any pair w, w’ either Ii(w) = Ii(wʹ) or Ii(w) ∩ Ii(wʹ) = ∅. The use of information partitions has several implications regarding the players epistemic abilities. In particular, the players necessarily know that they know something (positive introspection) and know that they do not know something (negative introspection). It is well-known that epistemic models using information partitions are equivalent to the so-called S5 system of modal logic [see e.g. Bacharach (1993); Stalnaker (2006)]. We could easily relax several axioms and obtain a weaker system without altering anything that is said in the text.
Of course, as it should be obvious, this does not undermine the validity of Aumann’s theorem. Since Jeffrey’s rule is only an extension of Bayes’s rule, common knowledge of Bayesian rationality and the existence of a common prior are still together sufficient for playing a correlated equilibrium. What the example in the text establishes is that, for a given epistemic model with a common prior and where it is common knowledge that players are Bayesian rational if their posteriors are determined through Bayes’s rule, Bayesian rationality may no longer hold if posteriors are determined through Jeffrey’s rule. Relatedly, there is no claim here that—under Jeffrey’s rule—having the same partition of an event and ascribing the same probabilities to the elements of the partition are necessary conditions for the existence of a correlated equilibrium. Aumann’s theorem only establishes sufficient conditions and, as it remains valid, no additional necessary conditions can be established.
It should be noted however that by construction, players ‘know’ the epistemic model of the game they are participating in. Hence, if the model assumes a common prior, the players must know this. Nevertheless, the point is that this is not knowledge in the relevant sense, i.e. as defined by the knowledge operators of the model. As pointed out by Aumann and Brandenburger (1995), epistemic models are formal bookkeeping tools that allow the modeler to describe what the players know, believe and are doing. Hence, the players’ ‘meta-knowledge’ of the structural assumptions of the model implies nothing substantive.
A partition {Ei}* is strictly finer than a partition {Ei} if every cell in {Ei}* is an atom of {Ei}, i.e. if the cells of {Ei}* are strictly subsets of cells of {Ei}.
A similar point is made by Bacharach (2006) in his ‘variable frame theory’.
Of course, for any epistemic model using Jeffrey’s rule, a state space W and a pair \(\langle \left\{ {I_{i} ,C_{i} } \right\}_{i \in N} \rangle\), we can construct a formally equivalent epistemic model using Bayes’s rule with a unique set of partitions {Ii} of a state space W’. However, the latter will be far more complex than the space W. If the point is merely to provide a formal description of what is observed, both models are equivalent. But if the purpose is to account for the way individuals are actually reasoning in a strategic interaction, a model with a less rich state space is probably better.
Moreover, note that this only a necessary condition. Agents must also give the same probabilistic weights to the elements of the partition.
Signaling games are generally studied in economics within the context of principal-agent models where the player choosing first (the ‘agent’) knows her type but where the player choosing in second (the ‘principal’) ignores the agent’s type. In these models, the exogenous event corresponds to the agent’s type chosen by Nature. Nothing change at the formal level if the exogenous event refers to another thing than the agent’s type.
Indeed, equilibrium play is fairly common in nature.
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Acknowledgements
Different versions of this paper have been presented in front of several audiences, especially at a seminar of the Department of Philosophy and Centre for the Study of Social Action (University of Milan, Italy) and at the 4th international conference of economic philosophy (Lyon, France). I would like to thank the participants of these events for their comments, especially Francesco Guala and Ivan Moscati. I would like also to thank the two anonymous referees who have provided valuable comments on a previous version of this paper. All errors and omissions are mine.
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Hédoin, C. Practical reasoning, rule-following and belief revision: an account in terms of Jeffrey’s rule. Synthese 198, 7627–7645 (2021). https://doi.org/10.1007/s11229-020-02536-z
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DOI: https://doi.org/10.1007/s11229-020-02536-z