Game Theory, InsTITuTIons and The schellInG-Bacharach PrIncIPle: Toward an emPIrIcal socIal onToloGy

Documents de travail GREDEG GREDEG Working Papers Series

Cyril HédoinLauren Larrouy

GREDEG WP No. 2016-21http://www.gredeg.cnrs.fr/working-papers.html

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Game Theory, Institutions and the Schelling-Bacharach Principle: Toward an Empirical Social Ontology

Cyril Hédoinα University of Reims Champagne-Ardenne, France

Lauren Larrouyβ

University of Nice Sophia Antipolis, France

GREDEG Working Paper No. 2016-21

Version 1.1: May 2016 Do not quote without permission

Abstract: This article defends a methodological and theoretical claim according to which the combination of epistemic game theory with the recent developments in the so-called “theory of mind” is able to provide an empirically grounded and theoretically consistent perspective on the mechanisms through which institutions determine the individuals’ beliefs and choices. This move toward an empirical social ontology is captured through what we call the Schelling-Bacharach principle in game theory. According to it, game-theoretic analysis of coordination and cooperation should study how the players are actually reasoning in different game situations.

Keywords: Social ontology – Epistemic game theory – Institutions – Theory of Mind – Schelling-Bacharach principle

JEL codes: B41 – C72 – D02

α Full professor of economics, University of Reims Champagne-Ardenne, economics and management REGARDS research center (EA 6292). Contact: cyril.hedoin@univ-reims.fr β PhD student in economics, University of Nice Sophia-Antipolis, GREDEG/CNRS-UMR 7321. Contact: Lauren.Larrouy@gredeg.cnrs.fr Paper prepared for the 3rd International Conference Economic Philosophy, Aix-en-Provence, June 15th-16th 2016.

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Game Theory, Institutions and the Schelling-Bacharach Principle: Toward an Empirical Social Ontology

1. Introduction

During the 20th Century, Game Theory has introduced institutions in the realm of economic theory. Game theory can therefore offer significant contributions to social ontology, as institutions are fundamental components of social reality. More specifically, we argue in this paper that epistemic game theory can make an important contribution to the development of an empirical social ontology. Indeed, epistemic game theory provides a well-suited toolbox to analyze the nature and working of institutions. Our main claim is that the combination of an epistemic game-theoretic framework with the recent developments in the so-called “theory of mind” is able to provide an empirically grounded and theoretically consistent perspective on the mechanisms through which institutions determine the individuals’ beliefs and choices. This move toward an empirical social ontology is captured through what we call the Schelling-Bacharach principle in game theory. According to it, game-theoretic analysis of coordination and cooperation should study how the players are actually reasoning in different game situations.

We assume in this paper that institutions are defined as rules of behaviors and encompass regular patterns of behavior and convergent individual beliefs, i.e. individuals have common beliefs that everybody will follow these rules of behavior. With respect to this last dimension, epistemic game theory can grasp from a theoretical point of view a matter of social ontology. Epistemic game theory announces a shift in game theory that comes to adopt a bottom up methodology by investigating the players’ epistemic states, i.e. knowledge and beliefs, and modes of reasoning. Indeed, it attempts to analyze game situations from the perspective of the players and identify the players’ epistemic states compatible with various equilibrium concepts. As a consequence, epistemic game theory can bring scientific rigor in a domain of philosophy of social science mainly based until recently on metaphysical thinking and mere intuition.

Furthermore, the need to ground epistemic game theory on more solid empirical foundations and its current challenges overlaps with the ongoing shift from the standard social ontology to an empirical social ontology. As Perea (2013, 20) accurately outlines:

“[A]n important task for epistemic game theory in the future is to develop new game-theoretic concepts by first presenting new, natural collections of epistemic assumptions, and subsequently characterize the choices – or beliefs – that result from these. Some work has already been done in this direction, but there is still plenty of

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room for more. In particular, epistemic game theory could help to develop concepts that rely on elements of bounded rationality, or even irrationality, as to better accommodate the game-theoretic concepts to the observed behavior of people in laboratory experiments. The good news is that with epistemic game theory we finally have the tools that are necessary to successfully tackle these problems.”

In such a perspective, we introduce a discussion of Schelling and Bacharach’s respective works on strategic reasoning that, according to us, are directly relevant to the advancement of an empirical social ontology. They both exhibit the role of institutions in the players’ capacity to coordinate and identify these institutions as constitutive rules. For both of them, institutions rely on two intertwined dimensions: regular patterns of behaviors and common beliefs towards the fact that players believe in the capacity of institutions to induce coordination through these pattern of behaviors. More significantly, by identifying the players’ cognitive processes sustaining coordination, and more specifically through a descriptive theory of these processes grounded on cognitive psychology, they allow enhancing the empirical foundations of game theory. To support this claim we also introduce a discussion of a particular account of individuals’ capacity of mindreading – which is obviously a prerequisite of interactions: the Simulation Theory. This capacity of mindreading through simulation is very close to some of Schelling and Bacharach’s claims concerning individuals’ cognitive processes in games. As Simulation Theory is nowadays an important domain of investigation within Neurosciences, this reinforces the empirical foundations of their account of strategic interactions. We therefore argue that Schelling and Bacharach’s respective works examined in parallel with Simulation Theory offer a significant step in the naturalistic turn of social ontology.

The paper is organized as follows. The next section makes a few general statements about the relationship between different versions of game theory (evolutionary and epistemic) and social ontology. In particular, we suggest that the epistemic program in game theory constitutes a theoretical a methodological cornerstone in the advancement of an empirical social ontology. Correspondingly, building on epistemic game theory, we develop in section 3 a framework in which institutions are formalized as semantic epistemic models of games. We argue that such a framework is particularly useful to capture the working of institutions, i.e. how institutions determine people’s behavior in strategic contexts. Section 4 presents what we call the Schelling-Bacharach principle in game theory, i.e. the general idea that game-theoretic models have to capture the way the players’ reason in specific game situations. In section 5, we explore how the Schelling-Bacharach principle and the semantic epistemic models framework could be advantageously combined with insights from the theory of mind in philosophy of mind and cognitive sciences, and especially the so-called Simulation Theory. Our main claim is that the latter could be the decisive step towards an empirical social ontology. The last section briefly concludes the paper.

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2. Game Theory and the Notion of an “Empirical Social Ontology”

The study of institutions has led to an unexpected encounter between economics, game theory and social ontology over the last thirty years. If we agree over the definition of institutions as sets of rules, norms, conventions and organization regulating social interactions and that generate behavioral regularities,1 then the philosophers’ interest in the working and the nature of institutions is as old as philosophy itself. If we restrict ourselves to the philosophical thinking of the 20th century onward, Ludwig Wittgenstein’s seminal account of rule-following is of course an important instance of a philosophical inquiry into the nature and the working of some institutions, especially language. More recently, John Searle’s books The Construction of Social Reality (Searle 1995) and Making the Social World (Searle 2010) have generated a huge amount of comments from philosophers and social scientists, including economists, e.g. Smit, Buekens, and du Plessis (2011). David Lewis (1969) and Edna Ullmann-Margalit (1977) have been the first among philosophers and social scientists to make use of a game-theoretic framework to study some forms of institutions (conventions in the case of Lewis, social norms in the case of Ullmann-Margalit).

Though institutions have been a subject of interest for the various (heterodox) “institutionalist schools” in economics from the 19th century onward, it is only relatively recently that the topic has become an important one for economists. It is probably not by sheer luck that the explosion of works on institutions in economics coincides with the full inclusion of game theory in the economist’s toolbox. Pioneering works such as Schotter (1981) and Sugden (1986) have introduced the use of game-theoretic concepts and methods especially for the study of the emergence and the evolution of simple institutions fostering cooperation and permitting coordination. Game-theoretic models have also been tightly articulated with historical accounts to study the working of more complex institutions at the root for instance of impersonal exchanges (e.g. (Greif 2006); (Milgrom, North, and Weingast 1990)). The study of institutions is thus at the intersection of social ontology and economics, with game theory lying in the background. From this point of view, game theory and economics can be seen as making a contribution to social ontology, where the latter is defined as the study of the nature of the social reality. This is not surprising as institutions are clearly part of our social reality. The implication is potentially significant however: as a subfield of philosophy, social ontology primarily consists in metaphysical reflections. Quite the contrary, game theory (as a branch of mathematics) and economics are generally regarded as belonging to the realm of science. As a result, one may see the encounter between economics and social ontology as an opportunity to bring scientific rigor and accuracy to a field used to rely on mere metaphysical speculations.

1 For a similar but non-identical definition, see Greif (2006, 30). Greif includes as part of an institution the individuals’ beliefs about exogenous events and other’s choices and beliefs. Following Hédoin [(2012); (2016)], we rather take beliefs as being intentional states generated by rules, norms and conventions. In this sense, beliefs do not have the same status than rules, norms and conventions in the definition of what an institution is. Note that some economists restrict the term institutions to formal (legal) rules and prefer to refer to informal norms and conventions as the “culture” of a given population. See for example Alesina and Giuliano (2015).

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Things are not so simple however and at least two caveats are in order. First, the fact that economics contributes to social ontology does not mean that social ontology reduces to the use of a game-theoretic framework. This is clear in Searle’s work mentioned above where no use of economics and game theory is made altogether. More generally, economics and game theory remain peripheral to what Francesco Guala (2007) calls the “standard model of social ontology” (SMOSO). The latter is based on a mildly individualistic restriction, according to which “if all individuals were to disappear, all social institutions would disappear too” (Guala 2007, 961), and is constituted by three elements: reflexivity, performativity and collective intentionality. Reflexivity refers to the fact that “social entities are constituted by beliefs about beliefs”. That is, social facts depend on what individuals believe about them, acknowledging that social facts are themselves constituted by beliefs.2 Performativity underlines the fact that the social reality is literary continuously created by the individuals through their actions. The notion of performativity first originated in the philosophy of language where a performative speech act is defined as an act performing the reality through the uttering of a sentence in the right circumstances. More generally, performativity is tightly linked to the human ability to ascribe status to objects, situations and persons through speech acts and more generally on the basis of a language. Because these statuses give rise to deontic powers, they also have implications at the behavioral level.3 Finally, collective intentionality refers to a broad idea according to which the belief systems that constitute the social reality may be partially grounded on a “we-mode” of reasoning and thinking. There is a great diversity of points of view here. On one end of the spectrum, some scholars argue that social reality is grounded on nothing but individual beliefs of the form “I believe that…” plus an epistemic condition about their mutuality or commonality in the population [e.g. (Bratman 1993)]. At the other end, some defend the view that there are authentic collective intentional states that should be distinguished from individual ones and on which social reality depends [e.g. (Gilbert 1989); (Searle 2010)]. As we will argue below, the SMOSO can be usefully related to and sustained by game-theoretic reasoning. Reflexivity and performativity in particular are easily accounted for in a game-theoretic framework. Even collective intentionality can be given a game-theoretic meaning through the notion of “team reasoning” [e.g. (Hakli, Miller, and Tuomela 2010); (Sugden 2000)]. However, most of the accounts falling into the SMOSO are not grounded on game theory. They are not grounded on any empirical analysis either. Indeed, according to Guala (2007, 966) a strong feature of the methodology related to the SMOSO is that “ontological issues are to be resolved by means of linguistic analysis and intuition” (emphasis in original).

As a second point worth noting, the contribution of economics to social ontology has essentially taken place outside the SMOSO on the basis of a particular variant of game theory,

2 The existence of self-fulfilling prophecies in the social world is of course one of the most significant implications of reflexivity. Interestingly, reflexive situations are particularly well-fitted to the kind of equilibrium analysis favored by economists [(Grunberg and Modigliani 1954); (Simon 1954)]. They are also naturally accounted for in a game-theoretic framework (Guala 2013). This point will be essential below in our use of an epistemic game-theoretic framework. 3 This feature is a major one in Searle’s (2010) account, especially through his notion of status function.

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namely evolutionary game theory. First developed and used in biology,4 evolutionary game theory has been introduced in economics during the 1980’s and has since become a major tool for the study of institutions in both philosophy [e.g. (Skyrms 1996)] and economics [e.g. (Young 1998)]. Evolutionary game theory has particularly been used to shade light on the mechanisms of emergence and evolution of norms and conventions and as such has led to a range of contributions concerned with social ontology.5 Some evolutionary game-theoretic models used by economists build on a strong analogy with the models developed by biologists. In particular, the so-called “replicator dynamics” transition function has been used to account for the mechanisms through which some behavior or “cultural trait” disseminates in a population. Formally, the replicator dynamics applied to cultural evolution takes the following form:6

(1) 𝑑𝑑𝑑𝑑𝑠𝑠(𝑡𝑡)𝑑𝑑𝑡𝑡

= 𝛽𝛽𝛽𝛽𝑠𝑠(𝑡𝑡)[𝜋𝜋𝑠𝑠;𝒑𝒑(𝑡𝑡) − 𝜋𝜋𝒑𝒑(𝑡𝑡)].

According to expression (1), the growth rate of the proportion ps of a strategy s (corresponding to a behavior or any other phenotypic trait) in the population at time t is a linear function of the difference between the payoff 𝜋𝜋𝑠𝑠;𝒑𝒑(𝑡𝑡) yielded by s when the distribution of strategies at t is given by the vector p and the mean payoff 𝜋𝜋𝒑𝒑(𝑡𝑡) in the whole population. When applied to biological evolution, the replicator dynamics straightforwardly formalizes the process of natural selection as soon as the payoff functions π(.) are measured in terms of (expected) fitness. When used in the context of cultural evolution, the replicator dynamics is generally interpreted as representing an imitation mechanism. Here, the payoff functions are measured in terms of (expected) utility and it is assumed that the higher the utility yielded by a strategy, the more likely it is that the agents using it will be imitated by those using strategies yielding lower utility. In this case, β is a positive constant representing the propensity of the individuals to imitate more successful people.

The replicator dynamics is an aggregative transition function, i.e. a mathematical expression of an evolutionary process that takes place at the population level. The relevance of this function as an explanation of the mechanisms underlying this evolutionary process depends however on its ability to capture them from an empirical perspective. On this basis, several scholars have pointed out that the economists’ use of evolutionary game theory has mostly stayed at a pre-scientific, metaphysical stage until now [(Grüne-Yanoff 2011a); (Grüne-Yanoff 2011b); (Guala 2007); (Sugden 2002)]. To understand this point, one should again consider the interpretation of the replicator dynamics in a biological context. As indicated

4 See Smith (1982) for a synthesis. 5 Economists are generally not explicit regarding the nature of these contributions and most of them would probably not frame them as such. Note however that Smit et al.’s (2011) “incentivized-action view” can be seen as providing an explicit ontological rationale of the way many economists account for institutions. Interestingly, Smit et al. build their view in opposition with Searle’s account of the nature of the social reality. 6 For a derivation of the replicator dynamics for cultural evolution, see for instance Bowles (2006) or McElreath and Boyd (2008).

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above, the payoff functions are then measured in terms of expected fitness, i.e. the expected reproductive success of the bearers of a given strategy. Assuming perfect genetic or phenotypic transmission (i.e. children play exactly the same strategy than their parents), it follows that a strategy yielding more than the mean expected fitness will necessarily expand in the population as time unfolds. This is nothing but a mechanism of natural selection. It may eventually face counter-tendencies originating in other biological or genetic mechanisms (mutations, genetic drift). In any case, as fitness is an empirically grounded measure, expression (1) corresponds to a useful tautology providing an explanatory template to answer specific empirical issues. The same cannot be said however when the replicator dynamics is applied to cultural evolution. Two difficulties emerge in this context: firstly, while payoff functions are expressed in terms of utility, the measure of the latter has been a notoriously difficult issue in economics from the day of Marginalism. The definition of utility as a representation of choices favored by some revealed preference theorists [e.g. (Binmore 1998)] is especially problematical as expression (1) obviously requires a standard for making interpersonal comparisons of utility. Such a standard is hard to settle in a revealed preference framework and it is far from clear how utility comparisons could be empirically made. If instead we define utility as an independent quantity measuring “satisfaction”, “happiness” or “social success”, then expression (1) is obviously no longer a tautology: the assumption that more successful strategies are more likely to be imitated than less successful ones becomes a theoretical hypothesis that has to be tested against empirical evidence. This leads to a second difficulty about the mechanisms through which individuals imitate each other or learn the best strategy. Economists using evolutionary game theory have actually not undertaken any empirical study of these mechanisms. In other words, few things are known about how individuals actually imitate each other or learn from past experiences. There is clearly a great number of plausible mechanisms of imitation and individual learning that may sustain the replicator dynamic or other transition functions at the aggregate level.7

This last problem is the major reason why evolutionary game theory applied to the study of institutions is not more scientific than the SMOSO. The best this research program has led to are conjectural statements of the kind “if individuals learn this way, then such institution could have emerged this way”. In other words, it allows to answer questions of the kind “how institution X could have evolved?” but is unable to assess, among all the plausible candidates, which are the more likely to be actually responsible for the evolution of a specific institution. Moreover, this approach is mostly silent regarding the way institutions themselves affect the agent’s behavior. The story told by the replicator dynamics on this subject is quite thin: one gives up his strategy for another one because the latter is “better”. But since nothing is told about the precise mechanisms of imitation and learning, it is not clear if the change of strategy is due to direct interactions and observations or to some form of “collective” learning. In this latter case, it seems that institutions as social object are directly determining people’s

7 Young (1998) explores evolutionary models where individual learning occurs on the basis of a “fictitious play” mechanism. In a nutshell, each player has access to a sample of the history of past plays in the game and chooses on this basis his best reply strategy, i.e. the strategy that maximizes his expected utility given the probability distribution of strategies in the sample. This leads to a different transition function than the replicator dynamics at the aggregate level.

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behavior. This is not to say that explaining how institutions evolve is not important. Nor do we intend to mean that conjectural answers to this issue are devoid of any value. Our point however is that both the SMOSO and evolutionary game theory have not delivered what all that they have promised (i.e. a full account of the nature of the social reality) because they do not empirically investigate the mechanisms through which individuals are able to coordinate and to cooperate on the basis of institutional resources.

This claim is actually at the core of Guala’s (2007) discussion of the notion of “empirical social ontology”. At first sight, the very notion of empirical social ontology may strike one as being contradictory. As part of metaphysical thinking, ontology has generally been thought as being outside the domain of empirical sciences. Ontology is indeed associated to armchair theorizing and linguistic analysis, as it has been already pointed out concerning the SMOSO. However, the notion of empirical social ontology can be understood as a peculiar instance of the so-called naturalistic turn in philosophy, especially the philosophy of science. From the naturalistic point of view, philosophical thinking should no longer remain divorced from the empirical sciences but instead should build on them by using their methods as well as their theoretical and empirical results. Economics and game theory have actually already largely contributed to the naturalistic turn, especially in moral philosophy and social ontology.8 An interesting and recent trend has been initiated by Herbert Gintis (2009) on the basis of Robert Aumann’s (1987) work on Bayesian rationality and the solution concept of correlated equilibrium. Gintis’ suggestion is that our understanding of the nature of social norms can be greatly enhanced by the use of game theory, and especially epistemic game theory, i.e. the branch of game theory that explicitly formalizes the players’ knowledge, beliefs and reasoning modes. In the rest of the paper, we follow Gintis’ path by taking epistemic game theory as our basic theoretical framework to account for the nature of institutions.9 On this basis, we consider the possibility that the so-called theory of mind can provide an additional and more empirically grounded building block to the construction of a fully naturalistic social ontology, as recently suggested by Brandenburger (2014, xxii) though he does not explore the issue further.

2. Formalizing Institutions as Epistemic Games

What is known as the “epistemic program” in game theory has initiated a significant twist in the game theorists’ methodology (Brandenburger 2014). On the basis of the research program opened in the 1950’s by Nash and others, game theorists have traditionally been interested in working out the way different classes of games have to be played on the basis of various solution concepts. This approach is for instance at the core of the Nash equilibrium solution

8 Guala (2007) cites Sugden (1986) and Bicchieri (2005) as two significant examples of contributions to social ontology based on game theory and economics (even though Bicchieri is actually a philosopher). 9 Hédoin [(2014) ; (2015); (2016)] and Hindriks and Guala (2015), among others, can be seen as contributions pursuing the same goal.

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concept: considering a set of commonly known strategy profiles to be implemented, which are those that satisfy some criterion of rational play, in this case here simply the best reply play. A strategy profile thus corresponds to a Nash equilibrium if and only each player plays his best response given the strategy choice of others. The standard methodology in game theory is thus “top-down”: start with a well-defined solution concept and figure out which strategy profiles satisfy the required criteria.

The methodology behind the epistemic program is based on a rather different “bottom-up” logic. This logic is constitutive of what we call epistemic game theory.10 The initial step consists in specifying what the players know and believe and how they reason on this basis, and then to ask which are the strategy profiles that could be implemented. Unsurprisingly, this methodology has allowed the epistemic program to produce significant results regarding the sufficient conditions for various solution concepts to be implemented, included the Nash equilibrium one (Aumann and Brandenburger 1995). However, we will use the resources of the epistemic program in a quite different fashion here. Our purpose is to set up a formal framework providing a first building block for an empirical social ontology. More specifically, we need a formal framework that could in principle produce testable propositions about the way more or less rational individuals are reasoning on a basis of a set of epistemic resources to solve coordination problems. These epistemic resources are provided by institutions. From a methodological point of view, the latter are thus seen as constituting the explanation for the way people behave in some specific situations: if we are able to characterize adequately institutions and how they work, then we should be able to explain why individuals behave as they do. Our main tools in this endeavor are what we call semantic epistemic models (s.e.m.), i.e. formal descriptions of the way some given game is played on the basis of what the players know and believe. This section introduces the formal framework and briefly discusses how it can be used to capture rule-following phenomena that are essential to understand how institutions work.

We start with a standard game G describing some strategic interaction. A game is a tuple G: < N, Si, uii∈N > where as usual, N is the set of n ≥ 2 players, Si is the set of player i’s pure strategies with i = (1, …, n) and ui is i’s utility function. We denote S = S1 x … x Sn the set of strategy profiles and therefore ui: S ℜ. Throughout the paper, we assume that the utility functions ui are ordinal (they are unique up to any positive monotonic transformation). A s.e.m. of G is a complete description of how the game is played and would have been played given what the player know and believe. It corresponds to a tuple I: < Ω, w, Ci, Rii∈N >. Ω denotes the set of states of the world (or “possible worlds”) w. A state is a complete specification of what the players would do, know and believe if w was the actual state of the

10 The terminology is not completely settled in the literature. For instance, Guala (2007) speaks of “epistemic game theory” to refer to what we will call here traditional or classical game theory, i.e. game theory as conceived and practiced by Nash and his followers. To be clear, we thus assume that there are currently three major research programs in game theory: the Nash equilibrium refinement research program (Harsanyi and Selten 1988) associated to the classical approach, the evolutionary program discussed in the preceding section and the epistemic program. Note however that there are a significant number of other programs (e.g. the cooperative games approach, behavioral game theory, the global games approach…).

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world. The state w corresponds to the actual state and thus gives the description of how the game is actually played. Ci: Ω Si is player i’s decision function, i.e. it indicates player i’s strategy choice at each state w. Finally, Ri is an accessibility relation defined for each player i and where wRiw’ means that state w’ is epistemically accessible from state w for player i. It indicates which are the worlds w’ that player i considers as possible when the actual world is w. For convenience, we define Ri: Ω 2Ω as player i’s possibility operator, with Ri(w) = w’ | wRiw’ for any w ∈ Ω, i.e. the set of states w’ which are accessible from w. Combining game G with such a s.e.m. I gives an epistemic game G: < G, I >.

A s.e.m. and especially the accessibility relation is a powerful tool to investigate the properties of various kinds of epistemic states and relations as well as their behavioral implications. The kinds of epistemic (or doxastic) states and relations represented depend on the properties satisfied by the accessibility relations. A standard approach in economics initiated by Aumann [(1976); (1987)] consists in using s.e.m. with knowledge-belief hierarchies. This is done by assuming that each player has an information partition and a prior probability distribution over Ω. Arguably, these assumptions are relatively strong as they imply in particular that the players’ probability one beliefs are necessarily true. Discussing their plausibility is well beyond the scope of this paper and here we will restrain ourselves to weaker epistemic requirements. More specifically, we will frame the whole discussion in terms of non-probabilistic belief operators.11 We proceed the following way. First, we require the possibility operators to satisfy the following two properties:12

a) For all i ∈ N and all w ∈ Ω, Ri(w) ≠ ∅. b) For all i ∈ N and all w, w’ ∈ Ω, if Ri(w) ≠ Ri(w’), then Ri(w) ∩ Ri(w’) = ∅.

The first property is a consistency requirement which imposes, as we will indicate below, that one cannot have inconsistent beliefs. The second property indicates that the possibility operators partition the state space Ω into equivalence classes, i.e. if w’, w’’ ∈ Ri(w), then Ri(w) = Ri(w’) = Ri(w’’). Together, properties (a) and (b) imply that an agent’s beliefs may be false but are necessarily consistent and satisfy conditions of positive introspection (one believes that he believes something) and negative introspection (one believes that he does not believe something). The second step is to provide a formal definition of the notion of belief on the basis of the possibility operators. This is done by defining n belief operators Bi in the following way. Define an event E ⊆ Ω as any set of states. The event that player i believes that E holds is denoted BiE. Formally, this is expressed as

11 Our choice to use non-probabilistic belief operators is without loss of generality. Probabilistic belief operators are less convenient to use and given our purpose would have not brought any much additional insight. 12 Alternatively, we can express the epistemic requirements in terms of the accessibility relations Ri directly. In this case, for each player i, Ri must be serial (for all w, there is at least one w’ such that wRiw’), transitive (for all w, w’, w’’, wRiw’ and w’Riw’’ imply wRiw’’) and Euclidean (for all w, w’, w’’, wRiw’ and wRiw’’ imply w’Riw’’).

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(2) BiE = w | Ri(w) ⊆ E.

Expression (2) states that player i believes that E holds at w if and only if all the states he considers to be possible are members of E. It can be shown that if the accessibility relations satisfy properties (a) and (b), then the belief operators respond to the following list of axioms:

(B1) BiΩ = Ω.

(B2) Bi(E ∩ F) = BiE ∩ BiF.

(B3) BiE ⊆ ¬Bi¬E.

(B4) BiE = BiBiE.

(B5) ¬Bi¬BiE ⊆ BiE.

(B1) and (B2) are the semantic counterparts of standard axioms in modal logic that imply a weak form of logical omniscience, i.e. one believes what is necessarily true as well as the logical implications of what he believes.13 Axiom (B3) is a consistency requirement that directly results from property (a). Finally, axioms (B4) and (B5) correspond respectively to the properties of positive introspection and negative introspection referred to above. One last step remains as we need to define a common possibility operator R* and a corresponding common belief operator B*. These are required because they will allow us to characterize situations where the members of some population share some belief state up to an infinite level (each person believes that each person believes that… and so on ad infinitum). This corresponds to what is generally characterized as “common knowledge”, even though here we will speak of commonly believed events. Although the common belief operator can be directly defined as the infinite intersection of the individual belief operators, it is more useful to first characterize the common possibility operator as the transitive closure of the union of the individual possibility operators, that is

(3) R1*(w) = ∪i∈NRi(w).

(4) Rk*(w) = ∪R1*(w’) | w’ ∈ Rk-1*(w).

13 Note that (B2) implies that if E ⊆ F (‘E implies F’), then BiE ⊆ BiF.

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(5) R*(w) = ⋃ 𝑹𝑹𝒌𝒌∗ (𝑤𝑤)∞𝑘𝑘=1 .

Now, the common belief operator can be straightforwardly defined in the standard way:

(6) B*E = w | R*(w) ⊆ E.

Expression (6) states that an event E is commonly believed in some population if all the states that are commonly considered as possible are members of E.

Our primary postulate in this paper is that on the basis of this framework and in the perspective of contributing to an empirical social ontology, it is useful to characterize an institution as a s.e.m.. In other words, an institution is more than a behavioral pattern but captures and determines what the individuals believe and how they reason. An institution thus corresponds to a whole practice based on the existence of a constitutive rule (i.e. a rule that is constitutive of the practice). For instance, the institution of baseball corresponds to the general practice of “playing baseball”. What is it to play baseball? Arguably, it consists in nothing but following the rules of baseball. The knowledge of these rules leads the players to form specific expectations regarding the behavior of others. These rules act as focal points in the sense that they help the players to coordinate in spite of the fact that they may have partially conflicting interests. On this basis and following Hédoin (2016), we thus impose two conditions on a s.e.m. to formalize an institution:

C1 – Rule-following. An institution exists only if the players are following at least one rule in the following sense: the event E that some rule holds must be mutually believed (BiE for all i) and the event F that some strategy profile (s1*, …, sn*) must be commonly believed (B*F).

C2 – Practical rationality. Each player i maximizes is utility given his belief that the profile (s1*, …, si-1*, si+1*, …, sn) is implemented. As a result, the strategy profile (s1*, …, sn*) is a (Nash) equilibrium.

The motivation for setting these two conditions is relatively straightforward. Condition C2 only requires that the players choose the action they prefer the most given their belief about what others are doing. It naturally implies that any institutionalized practice must take the form of a stable pattern at the behavioral level. Condition C1 states however that the existence of a behavioral pattern is not sufficient for an institution to exist (otherwise insects would also

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have institutions for instance). The behavioral pattern must be the result of the fact that the players are following a rule. Rule-following is captured in the semantic model by the inclusion relation ∀i∈N: BiE ⊆ B*F, i.e. the mutual belief in the fact that some rule holds implies the common belief that each person will behave in a certain way. We discuss more fully this relation in section 5 below. In particular, it should be noted that the semantic approach hides an important feature regarding the kind of practical reasoning that allows the players to infer what to do from the fact that they believe that a rule holds. The next section explains why this issue is an essential one, on the basis of Thomas Schelling’s and Michael Bacharach’s writings on salience and focal points. It will settle the ground to explore more fully the nature of the inclusion relation with some insights coming from the theory of mind.

4. The “Schelling-Bacharach Principle”: Framing and Focal Points in Games

Schelling and Bacharach conceptual and analytical contributions in game theory integrate three intertwined dimensions: an historical dimension (i.e. individual personal and social experiences), a contextual dimension (objective and social) and the players’ empathic capacity. These interdependent aspects explain the players’ ability, thanks to the incorporation of psychological assumptions, to solve coordination and cooperation problems. In fact, Bacharach retakes Schelling’s methodological and conceptual innovations and goes deeper in the incorporation of psychological assumptions regarding the players’ decision-making. In this manner, he progresses in the building of a ‘theory of focal points’ and justifies its analytical content in a theory of games that respects Schelling’s methodological ground.

In Schelling’s conception of games, the solution of a game is explained in terms of ‘stabilized’ convergent expectations, i.e. a point of time in which there is a “meeting of minds” among the players (Schelling 1980[1960], 54, 115). Albeit each player initially builds her own subjective game and the fact that players are heterogeneous, at this point of the game each player perceives the same thing of the strategic situation they face (ibid., 54). Reaching this state of convergence of subjective perceptions and in turn of individual beliefs is for Schelling a dynamic and ‘discovery’ process (see also Innocenti 2007). “The players must jointly discover and mutually acquiesce in an outcome or in a mode of play that makes the outcome determinate. They must together find “rules of the game” or suffer the consequences” (Schelling 1980[1960], 107). In the dynamic process leading to the players’ meeting of minds, at a certain point of time one of the numerous potential suggestive details of the game stands out. As Schelling argues:

“Most situations – perhaps every situation ... – provide some clue for coordinating behavior, some focal point for each person’s expectation of what other expects him to expect to be expected to do. Finding the key, or rather finding a key – any key that is mutually recognized as the key become the key – may depend on imagination more than on logic” (ibid., 57; original emphasis)

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This key is what Schelling calls a focal point. In the identification of a focal point, two dimensions matter: the “perceptual and suggestive elements” and the “structural elements” of a game (Schelling 1980[1960], 83-84). The former refer to the psychology of the players, e.g their perceptions, while the latter refer to the context of the game, i.e. the social and objective – or physical – environment. These two dimensions are obviously intertwined, however it is important in Schelling’s conceptual framework to differentiate them. The suggestive details of a game depend on the players’ perceptions and are both context-dependent and socio-culturally determined. Schelling distinguishes those two aspects of the environment because both can matter in the identification of ‘clues’ in order to reach a ‘meeting of minds’. In case of the physical environment, “some of the objective details of the situation can exercise a controlling influence” (ibid., 71). In case of the social environment of a game, players can exploit “analogies,” “precedents,” “incidents” (ibid., 90), “clichés,” “conventions” (ibid., 84-85), “institutions,” “traditions” (ibid., 91). When interacting, players can rely on everything that can be perceived as a reliable coordination device (ibidem). These coordination devices must be reliable in the sense that everybody believes that everybody else will conform to it, etc. (ibidem). In other words, these coordination devices, like institutions, conventions, etc. must be associated with stable patterns of behavior, i.e. with rules of behavior, which in turn induce convergent expectations. In this respect, the players must ultimately know that these coordination devices belong to the ‘value system’ of the other players. Hence, players must ultimately share a common background.

This requirement may however not be satisfied. In some circumstances, the players’ knowledge of the others’ value system may be too thin, so that they may not be sure that they will rely on a common institutional device to coordinate or eventually cooperate. In this case the players have to interact as long as necessary to discover each other’s value system and to ultimately act upon a common understanding of their situation.

All of this implies that in Schelling’s contribution, focal points, as institutional facts, are created through human interactions. Focal points are first voluntarily generated through strategic interactions due to the individuals’ need to coordinate and to conciliate their subjective and potentially divergent will. They are then associated with rules of behaviors inducing consistent and convergent expectations. In Schelling’s contribution, focal points are conceived in a pragmatic way (Sugden and Zamarrón 2006). To be reproduced, they have to recurrently allow successful coordination. Therefore, focal points progressively generalize to broader contexts. The more an institutional fact will be perceived as reliable in similar contexts, the more it will generalize and will be anchored in individuals’ knowledge of sociality. For Schelling (2006[1978], 126), “[t]he solution depend on some kind of social organization, whether that organization is contrived or spontaneous, permanent or ad-hoc, voluntary or disciplined”. It means that the institutional system in which the players are embedded and the structure of the social reality shape the resolution process of a game. Subsequently, institutions causally affect the players’ decision-making.

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Schelling’s framework provides the ground for Bacharach’s contribution to the analysis of focal points – as institutional devices – in game theory. Bacharach introduces in a comparative way the players’ subjective perceptions of the game they are to play through the notion of framing. For that purpose, Bacharach (1991; 1993; 1997; 2001; 2006) builds the Variable Frame Theory (VFT). However, contrary to the standard use of framing in behavioral economics and the Prospect Theory, Bacharach (2001, 4) is interested in natural framing, i.e. in the absence of any theorist’s manipulation to orient the players’ decision making. Accordingly, each player shapes her own subjective game, which is called a Variable Universe Game (VU-Game) (Bacharach 1993).

“According to the way that normal agents tend to ‘cut-up the world’ as they come to terms with a problem, this problem gives rise to payoff-matrices that differ widely in terms of the number of options and the pattern of payoffs: the payoff structure generated by a problem is concept-dependent” (Bacharach 1991, 3).

In the VFT, Bacharach defines a frame as “a set of concepts or predicates an agent uses in thinking about the world” (Bacharach 2001, 1), i.e. “a players’ frame is, most simply, the set of variables she uses to conceptualize the game” (Bacharach 1997, 4). Therefore, in the VFT, he incorporates a descriptive theory of the process through which frames come to the players’ mind, “what determines players’ frames” (Bacharach and Bernasconi 1997, 5), their structure, and effects on players’ perceptions, first and second order beliefs and strategic reasoning (Bacharach 1993, 258). According to Bacharach, the set of options a player has, i.e. her available ‘subjective’ strategies and her set of beliefs are comprised within the frames she handles. The frame a player handle can in turn allow circumscribing her set of options – or strategies,14 and her belief space concerning her co-players’ frames and beliefs. The set of frames and beliefs a player ascribes to the others are defined within her own frames. As Bacharach claims, a VFT model “gives us probabilities for players’ having various belief-spaces. It allows us, too, to say something about the beliefs which a player with one belief-space has about the belief-spaces of others” (Bacharach 1993, 258). As soon as the players’ frames, subjective strategies and beliefs are specified, Bacharach applies a principle of practical rationality. Each player maximizes her expected utility according to her beliefs of others’ frames and subjective strategies.

In the VFT, players’ frames – or repertoires – are structured in families of concepts.15 Some concepts and the families to which they belong instantiate according to the context of the game. For Bacharach, concepts come to the players’ mind into bundles and are classified within families (like color family, shape family, etc.). The probability of a concept to instantiate – and again its family – is a matter of salience and noticeability: “highly conspicuous features are more likely to come to mind” (Bacharach 1991, 16). However, 14 “[A]n option of a player is a feasible action for her described as she herself describes. It follows that a player’s

options must be descriptions of possible actions which only use attributes [i.e. concepts] in her frame” (Bacharach and Bernasconi 1997, 6).

15 Bacharach draws the VFT from matching games. Concepts therefore refer to characteristics of the objects players have to choose.

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“frames may vary … across players” (Bacharach 1997, 4). They depend on the players’ personal history and cultural background. The probability of a concept to instantiate is partly “given by the culture to which the players belong” (Bacharach 1993, 267). Bacharach asserts

“[T]he outcome of a given interactive problem situation can vary dramatically with the conceptual resources of the players. We have observed that there are two levels of such variation: the stochastic variations of the “coming to mind” process parameterized by the availability function V, and the possibility of alternative values of the parameters of V itself. The latter variation models effects of culture. VU-games may therefore be seen as a tool for giving precise expression to the emerging view … that the rational solutions of some games depend on the culture of the players” (Bacharach 1993, 271).

Bacharach therefore gives an analytical content to what was previously identified by Schelling as the players’ “value system”. When Scazzieri (2008, p. 197) refers to Bacharach’s account of framing, he emphasizes that in the VFT “framing shows the interplay between structural principles and evolutionary (historical) principles”. For him, there are “natural associations” between frames and context which are explained by the players’ personal history and their experience of sociality. This specific structure of frames explains Bacharach’s “theory of focal point” (1991; 1993; 1997). Within his conceptual and formal framework, two dimensions appear in salience and focal points: one which is context dependent and explained by the structure of the players’ frames, and one which is socially or culturally dependent.

The first dimension relies on the fact that some of the objective characteristics of the context of a game may be conspicuous and more noticeable than others and, accordingly, the concept describing this characteristic will instantiate with a higher probability. Therefore, “a frame is said to be salient if it has a strong tendency to be operative; some frames are more salient than others; and the salience of a given frame depends on the context. The salience of a frame depends on the salience (similarly defined) of its constituent concepts, and in particular of any constituent classifiers” (Bacharach 2001a, 5). Besides, Bacharach claims that “frames can have more or less power or potency to influence decision” (ibid., 5-6). The last statement is explained by the fact that in the VFT, “players think about salience strategically: i asks herself how likely it is that j has noticed what she has and so has the same options as her” (Bacharach and Bernasconi 1997, 39). This strategic dimension of salience therefore refers to social and cultural aspects. Bacharach (1991, 34), claims that the two characteristics of focal points (according to Lewis’ (1969) definition, resting on Schelling’s (1960) account): (i) ‘conspicuousness’ or ‘noticeability’ and (ii) ‘uniqueness’, “are logically independent, although they frequently appear in amalgam in discussion of salience”. The conspicuousness relies for Bacharach on framing, i.e. on the unconscious process through which frame come to the players’ mind. A concept “is conspicuous just if a predicate denoting it is highly available” (Bacharach, 1991, p. 34). This is the “primary salience” (Bacharach, Bernasconi, 1997). However the uniqueness rests on the reasoning phase, i.e. when ‘players think about

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salience strategically’. It means that players must put themselves in the others’ shoes to think from this angle what concept may be salient for the others and accordingly influence their decision. This leads to what Bacharach calls the “secondary salience”. An option is of “secondary salience for a player if she thinks it has primary salience for her coplayer” (Bacharach and Bernasconi 1997, 37).

The second – socially and culturally grounded – dimension of salience refers to conventions or institutional devices that may allow players to successfully coordinate. Bacharach (in Bacharach and Hurley 1991, 3) refers to the existence of a “cultural common sense”. He postulates “that every real player has the general knowledge her cultures gives her, such as knowledge of which arrangements are salient or traditional in that culture” (ibidem). For Bacharach, players’ identity matter in their decision-making. He identifies three self-identities that may matter when players interact: (i) “personal”, (ii) “relational” and (iii) “collective” identities (Bacharach 2006, 74).16 Collective identities in particular are constitutive of an individual.

“Personhood is the resultant, to the extent that it is so constituted, of a set of group identities; more exactly, the person is defined by the intersection of her group identities. But it is only to some extent, since there are plenty of person-defining features which do not correspond to group memberships.” (ibid., p. 88-89)

Collective identity “produces certain judgements, attitudes and behaviour” (ibid., p. 76) and in particular it induces the “internalization of group norms” (ibid., p. 80). Bacharach also refers to the self-categorization theory to argue that self-identification and more specifically the identification to a collective, a community, is a matter of framing. Bacharach asserts:

“Which of my collective personae is activated depends on the current accessibility’ of the categories to which I belong the relative accessibility of a category depends upon many things, which include the perceiver’s current expectations, tasks and purposes. In human interactions, the accessibility of categories is a special case of the notion of availability of frames at the heart of the variable frame theory of games.” (ibidem)

A strategic context involving coordination (or eventually cooperation) is a context that tends to prompt a natural collective identification for Bacharach (1999; 2006). The players face a common fate – which is the need to solve the game. ‘Common fate’ is one of the characteristics identified by Gestalt and Post Gestalt Psychology, to which Bacharach (2006) refers, which induces ‘entification’. The players therefore tend to identify the set of players involved in game as a collective and in turn their collective identities instantiate. As a consequence, players may rely on “group norms” and underlying patterns of behavior to coordinate. Besides, according to Bacharach, within a common culture, e.g. in a community 16 A player’s personal identity is linked “to aspects of her representation of herself that differentiate herself from

others” (ibidem), while her relational identity concerns her “self conception in terms of relationships with other individuals with whom she interacts” (ibidem).

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language, individuals tend to represent situations in a comparative way (Bacharach 2001, 9). The players’ frames have a propensity to be shared. The players tend to hold common structural frames (i.e. common conceptual repertoires or common classifiers concepts, etc.). Schelling’s “meeting of minds” can therefore be reached through Bacharach’s conception of ‘social’ framing. Moreover, empirical results (in Bacharach and Bernasconi 1997, 9) reveal “that beliefs about framing propensities are also shared”.

“[W]ithin a same culture the parameters of a conceptual scheme – its membership, its clustering, and the readiness to mind of the clusters in a given situation – are essentially shared. Furthermore, they are shared in this strong sense: not only does everyone have certain conceptual competences by virtue of belonging to the community, but every member knows that every member has them” (Bacharach 1993, 259).

The partial share of individual frames and beliefs – within a given community – induce that the players can ground their reasoning on the fact that they can refer to coordination devices inherited from this culture; everybody believe that everybody else will believe, etc. that everybody will conform to it. Indeed, for Bacharach (2001, 7), “human framing propensities stand behind the well-known ability of people to solve coordination problems by exploiting ‘focal points’”. To sum up, by both integrating players’ subjective perceptions of games and acknowledging that they must be embedded in a social and cultural context when playing, Schelling and Bacharach provide a methodological ground to (i) escape the indeterminacy problem and (ii) to incorporate institutions in the realm of game theory and more specifically of epistemic game theory.

We are now ready to state what we call the Schelling-Bacharach principle in game theory more explicitly. According to this principle, game-theoretic analysis of coordination and cooperation should study how the players are actually reasoning in different game situations. The study of reasoning modes may be pursued along different conceptual and theoretical approaches, such as for instance assuming that the players are using different frames. In any case, the point is that according to Schelling’s and Bacharach’s methodological and theoretical views, the tacit assumption of a symmetry between the game theorist and the players should be given up. Obviously, this has some incidence at the empirical level, as it is now a requirement to investigate how the players are actually reasoning. Arguably, both Schelling and Bacharach enhance our understanding of the nature and working of institutions. For both, focal points as institutional facts induce rules of behaviors and common beliefs on the fact that everybody will conform to those rules. This is explained by the fact that individuals believe in the capacity of focal points to serve as practical coordination devices. That is to say that the two dimensions C1 and C2 formalizing an institution by an epistemic model hold. Besides, by investigating the players’ cognitive processes when analyzing their capacity to coordinate – thanks to focal points – and grounding this investigation on psychological assumptions, and in particular on cognitive psychology, they (i) improve the empirical foundations of strategic reasoning in games and (ii) provide testable hypotheses.

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This is even truer if we consider that in Schelling and Bacharach’s respective work, one of the main dimensions in the players’ strategic thinking is empathizing or mindreading: a topic that is nowadays largely studied within Neurosciences.

5. Theory of Mind and Game Theory: Toward an Empirical Social Ontology

In a bottom up perspective, epistemic game theory purports to define the players’ hierarchy of beliefs – consistent with the equilibrium defined by the theorist – on the basis of their initial epistemic states. However the status of these epistemic states is ambiguous, both from an ontological and a formal perspectives (see e.g. Kadane and Larkey, 1982 ; Morris, 1995 ; Mariotti, 1995, 1996). First, it is not clear whether the beliefs are truly subjective Indeed, a major assumption on which virtually all the theory of incomplete/imperfect information game theory is based is that the players hold a common prior over the state space. Second, the origins and the formation of these beliefs is left undefined. This explains why any relevant contribution from game theory to the advancement of an empirical social ontology requires to explain the origin and the content of the players’ beliefs:

“[I]n order to reason our way towards sensible predictions about the opponents’ choices, it may be helpful to also reason about the possible desires and beliefs of our opponents. This naturally leads to the emergence of belief hierarchies which do not only describe what one believes about the others’ choices and desires, but also what one believes about the beliefs that others have about their opponents’ choices and desires, and so on.” (Perea 2013, 2)

This is precisely the purpose of the Theory of Mind (ToM), namely, to explain how people can ascribe to themselves and others some mental states like perceptions, intentions and beliefs. ToM refers to human capacity of “minreading” or “empathizing”, and mainly offers a “scientific support for the intuitive idea that understanding others is mediated by putting ourselves in their (mental) shoes.” (Goldman 2006, vii) A ToM insures two main functions: to (i) “[c]omprehend and explain” and (ii) “predict” others’ behavior (Michlmayr 2002, 5). As Churchland (1991, 57) suggests, “if one cannot predict or anticipate the behavior of one’s fellows at all, then one can engage in no useful commerce with them whatever”. A ToM is therefore a prerequisite to human social life, i.e. to both social and strategic interactions.

The landscape of the Theory of Mind (ToM) is explained by progressive enrichments from the mid-20th century to nowadays. The hot topic of mentalizing, which is a ‘second order mental activity’ and therefore (initially) an unobservable, first falls within the metaphysical and epistemological branches of Philosophy of Mind. The mentalizing issue has then progressively been tackled by Psychology and in particular Developmental Psychology, which introduces the experimental methodology and more recently by Neurosciences, which at last makes the cognitive process of mentalizing observable. Therefore, by providing data on

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individuals’ interactive cognitive processes, Neurosciences conform to the ongoing naturalistic turn in philosophy of mind. The main consequence is the progressive shift from a first approach of mentalizing, the Theory Theory (TT), to a second approach, the Simulation Theory (ST) which is – again, according to neuroimaging data – empirically grounded, and no longer intuitively-driven like the TT approach of mentalizing.

The TT states that to understand others we build a “folk theory” of human psychology, i.e. we build a “naïve” theory of our own and other’s mind functioning (Mychlmayr 2002, 10; Goldman 2012, 3). In the TT, mental states are “theoretical states of a common sense psychological theory, a “folk psychology”” (Goldman 2006, 7). Defining those theoretical states therefore relies on psychological laws explaining the relationships between: (i) ‘observable inputs’ and mental states, (ii) the mental states themselves, and (iii) mental states and behaviors or utterances (ibidem). In brief, in the TT “mental states concepts are essentially underpinned by theories; as a corollary, all inferences to mental states must be species of theoretical inferences… the [TT] of mentalizing [is] a theory that accounts for the mentalizer’s possessing a naïve psychological theory” (Goldman 2002, 13). Despite some sensible differences among scholars in the TT, it is always claimed that this “naive psychology, at bottom, is driven by a science-like theory, where a theory is understood as a set of lawlike generalizations” (Goldman 2012, 3). In this perspective, everybody functions as a scientist, collects data on mind functioning, makes assumptions, experiments them, etc. and make generalizations provided a sufficient amount of confirming experiments. Individuals establish a theory, making ‘theoretical inferences’ through their experience of social life.

One of the main consequences of the TT account of mentalizing and the rational account, which is considered as a specific approach of the TT (see e.g. Michlmayr 2002), is the imposition of constraints on mental states that individuals ascribe to others; those theoretical states must be ‘rational’, ‘pertinent’, or ‘consistent’ (Goldman 2002). In Goldman’s words “how the folk conceptualize mental states places significant constraint on the methods and conditions of ascription” (Goldman 2002, 9).

However, the TT approach is more and more contradicted, in particular due to new empirical data from neurosciences and neuroimaging, so that nowadays the Simulation Theory (ST) is dominant in the landscape of the ToM. Neuroimaging technics and the ST offer a new look to the cognitive processes at stake in mind reading. Besides, mental states like perceptions or beliefs for instance, are mainly unconscious and involuntary instantiated, so that imposing a consistency or rational constraint on them can be questionable (Goldman 2002). This is even truer regarding the purpose of this paper and considering the Schelling-Bacharach principle. By integrating the players’ perceptions in the realm of game theory both Schelling and Bacharach’s methodologies allow to bypass, on the one hand, the consistency constraints on players’ attribution of mental states to the other players, and on the other hand the external perspective that these constraints impose. The rationality theory, as a TT version of the theory of mind, entails that a game remains analyzed in a third person perspective without any subjective and intersubjective perspective. As emphasized by Harsanyi (1967/68; 1995) when

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Bayesian rationality is applied to game theory, each player acts or analyzes the game they are to play like an informed observer and let say, no longer as a participant to the game. The game is objectivized. Therefore, when imposing Bayesian rationality and common knowledge of Bayesian rationality, as it is done in Harsanyi and Aumann’s framework (and despite the claims of epistemic game theorists), players actually never try to represent from their subjective perspective – and as gamers – what the other players are thinking. As if they were game theorists, the players are able to infer others’ actions and the equilibrium, from the objective characteristics of the game (from its rules and matrix). Players have a prior about the result of others’ reasoning. However, the process itself of formation of those beliefs is never tackled. In addition, before playing the game, players’ prior are consistent each other, which means that before playing the game those priors are at the equilibrium. Everybody believes that everybody is able to compute the equilibrium and therefore will play his part in the strategy profile that are at the equilibrium.For an empirical social ontology, we can consequently argue that the intersubjective dimension of strategic interactions is misrepresented (see also Schmidt and Livet 2014).

The ST however offers a methodological ground to provide endogenous beliefs to games from the players’ subjective perceptions of the game. Contrary to the TT, the ST states that to understand and predict others’ behavior there is no need to refer to a theory of others’ mind functioning but to put herself in the others’ shoes, i.e. to empathize. Individuals use “the resources of [their] own minds to simulate … others.” (Davies and Stone 1995, 3). Individuals integrate in their decision process ‘pretend’ perceptions, intentions, or beliefs in order to come to an explanation or a prediction of someone else behavior (Michlmayr 2002; Goldman 1995; 2012). They imaginatively put themselves in the other’s shoes, try to see the situation from the other’s perspective and then imagine what would be her mental states, e.g. perceptions, intentions, beliefs etc. to predict her decision or behavior according to these ‘pretend’ mental states.

“A fundamental idea of ST is that mindreaders capitalize on the fact that they themselves are decision makers, hence possessor of decision-making capacities. To read the mind of others, they need not consult a special chapter on human psychology, containing a theory about the human decision-making mechanism. Because they have one of those mechanisms themselves, they can simply run their mechanism on the pretend input appropriate to the target’s initial position. When the mechanism spits out a decisional output, they can use the output to predict the target’s decision.” (Goldman, 2006, 20)

This implies that the players’ predictions of others’ decisions or behaviors are dependent on their own cognitive scheme, let say, on their own perceptions. This is one of the main justifications to argue in favor of the ST as a ToM to integrate in game theory and in particular to extend both Schelling and Bacharach’s work. When players simulate others’ reasoning, their beliefs (of others’ perceptions and beliefs) are defined within their own perceptions or frames. The ST therefore allows us to define the players’ hierarchies of beliefs

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when the players’ subjective perceptions of the game are integrated. This is compatible with Schelling and Bacharach’s framework since they both consider that the players are cognitively bounded and non-omniscient. In fact as emphasized by Michlmayr (2002, 25, referring to Goldman), “[i]t is … very effective to use one’s own mind since one can achieve accurate predictions and explanations with few cognitive resources.” Besides individuals and especially when belonging to a common community, can count on the fact that they have “major psychological similarities to any prospective attributor” (Goldman 1995, 3). Again, as mentioned above Bacharach explicitly argues that community membership partly induces shared frames and even shared beliefs. He asserts that within a common cultural or language community, individuals tend to hold shared frames, shared structural classifiers, etc. and that they tend to have, in turn, common beliefs. This is of particular importance since simulation “does not involve the very same states in the attributor as those undergone by the target” (Goldman, 1995, p. 11). The major difficulty to come to accurate predictions is therefore: are the ‘pretend’ mental states “sufficiently similar – in psychological and perhaps neurological terms – to their genuine counterparts” (ibidem). When referring to focal points, conventions or any institutional facts, those conditions of similarity and in turn common beliefs tend to be met.

It is of course beyond the scope of this paper to delve into the details of the cognitive mechanisms lying behind the process through which individuals form their beliefs. Nor is our purpose to adjudicate the disagreement between ST and TT on these matters. More relevant for us given what has been said in section 2 and 3 however is the fact that the ToM and especially ST clearly fit with our s.e.m framework. To be more specific, what the ToM could provide is the empirical content of the inclusion relation ∀i∈N: BiE ⊆ B*F that we have called the rule-following condition and that has been singled out as constitutive of the working of institutions. Recall that the inclusion relation is deemed to represent some mode of practical reasoning on the basis of which each player infers what he should do from some game situation. In our framework, an institution is instantiated by the mutual belief that some rule holds in a population and in a given situation. Each player infers from this belief a common belief that a certain behavioral pattern will be implemented. What our framework does not explicitly formalized is the specific cognitive mechanism that leads from the first mutual belief to the second, common belief. Therefore, the inclusion relation is akin to a “black box”. Arguably, any plausible empirical social ontology should seek to open up this box. As a consequence, we now clearly see what kind of contribution the ToM could make to an empirical social ontology combined with an epistemic game-theoretic framework. More precisely, the insights from ST suggest that the players’ ability to infer from the mutual knowledge that some rule holds what everyone will do in a given situation corresponds to a property of symmetric reasoning [(Gintis 2009); (Lewis 1969)]. Formally, a player i is a symmetric reasoner with respect to a player j if i assumes that whatever he infers from a given state of affairs is also inferred by j. It has been suggested that the very nature of rule-following and thus of institutions entails symmetric reasoning [(Hédoin 2016); (Lewis 1969)]. Arguably, symmetric reasoning may be seen as the default assumption used by the players in their simulation of others’ reasoning. While the assumption will appear to be erroneous in

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some specific instances, it will generally be pragmatically efficient to solve coordination problems. In another respect, the value of such a game-theoretic framework stems from the fact that it could helpful to build experimental protocols which in turn would contribute to advance the debates within the ToM.

6. Conclusion

It is worth noting that the three constitutive principles of the SMOSO, i.e. reflexivity, performativity and collective intentionality (at least in a diminished account) hold both in Schelling and Bacharach’s contributions while at the same time the latter are complying with the naturalistic turn in social ontology. Indeed, Schelling and Bacharach point to some empirical foundations for social ontology through their assumptions regarding the players’ cognitive processes in strategic reasoning. As we have suggested, contemporary developments in the ToM that refer to cognitive psychology and more recently to neurosciences, combined with an epistemic game-theoretic framework, could fully settle the establishment of an empirical social ontology.

Another important aspect of Schelling and Bacharach’s respective contribution that is beyond the scope of this paper but nevertheless worth noticing is that they conciliate the evolutionary and epistemic dimensions of institutions. They can theoretically bring these intertwined dimensions of institutions again from the perspective of an empirical social ontology.

There are still very few attempts to test in experimental settings the cognitive dimension of the players’ strategic reasoning in games and in particular regarding the content and the formation of their beliefs in such strategic context. Nevertheless, on the one hand the attempt to build a “theory of focal points” in game theory and, on the other hand, the integration of players’ frames in coordination and cooperation games, induce some experiments. For the former, we can mention Schelling (1960), Metha Starmer and Sugden (1994a; 1994b), Colman (1997), while for the latter Bacharach and Bernasconi (1997). Experiments on the level-k theory also enter in such a perspective (Bacharach and Stahl 2000; Hargreaves Heap, Rojo Arjona and Sugden 2012).

Gathering game theory and the theory of mind can provide the context for such new and promising experimental settings.

References

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Documents De travail GreDeG parus en 2016GREDEG Working Papers Released in 2016

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