1The role of informal norms in constraining available equilibria in social dynamics has been a subject of steadily increasing interest among social scientists. There is as yet no general theory of norms to play the role of a “default model” against which contributions can be assessed. This is partly because there is not a consensus on whether it is better to model a general phenomenon of norm responsiveness, with specific normative contents treated as varying with circumstances, or whether emphasis should instead be given to exploring the functionality of specific norms—e.g., fairness, reciprocity, modesty, patriotism, etc.—as separate exercises, with general normativity being treated as an emergent construct. The latter approach is typically referred to as the “social preferences” approach, and as such has been criticised for lack of desirable generality (Binmore, 2010; Smith and Wilson, 2019). Among efforts to model norm responsiveness in general, some influential models from economics are those of Sugden (1998), Bernheim (1994), Brock and Durlauf (2001), and Michaeli and Spiro (2015; 2017), which have inspired some experimental applications (Andreoni and Bernheim, 2009; Andreoni et al., 2017).

2The present essay is about general responsiveness to norms. Its intended contribution is methodological: we aim to show how a specific modeling technology—conditional game theory (CGT)—can be used to represent relationships between the stabilisation of norms in groups, and strategic choices of individual agents. The latter kind of process is typically modelled by economists, using standard game theory. The former has less clear-cut disciplinary “ownership”. Norms arise and spread partly through cultural evolution, for which there is a rich literature linking formal anthropological models descended from Boyd and Richerson (1985) with evolutionary game theory (e.g., Gintis, 2009; 2016). On shorter-run scales, norms are promoted by exercises of interpersonal influence. Where such influence shapes people’s conceptions of their identities, often mediated through occupation of professional and other recognised social roles, economists typically yield scientific responsibility to sociologists. The main explanation for this is only partly a matter of disciplinary inertia. It also stems from the fact economists are most at home building models in which agents’ preferences are exogenously given and stable, whereas identity formulation, at least as standardly conceived, involves endogenous preferences change. In making the obvious point that these various processes influence one another, we intend no challenge to historically useful divisions of interdisciplinary labour. However, as a matter of empirical fact, the processes obviously constrain and causally influence one another. We understand our work here as a way of focusing formally on these linkages. We therefore see it as contributing to cross-disciplinary unification, without any implication that disciplines should merge or fuse.

3We revisit this general theme in the concluding discussion. We do so not through reflections on abstract philosophy of social science, but by reference to technical considerations that are our primary focus. These are illustrated by the core exhibit of the article, a sample simulation that compares our approach to a prior contribution to the literature on modeling strategic effects of norms. This should be understood as a formal proof-of-concept, not as a substantive general result. We do not here present an intended empirical theory of the dynamics of norms. Rather, we furnish a modelling approach that we hold to be potentially useful for operationalising such theories in the lab or the field.

4One feature of norms that is now emphasised in almost all models is their conditionality: agents’ choices to conform their actions to norms are often sensitive to the extent to which they observe corresponding behaviour in those with whom they interact. Bicchieri (2006; 2017) distinguishes unconditional norms sharply from conditional (“social”) ones, following the Kantian tradition in regarding the former as the special domain of morality. For reasons explored by Binmore (1994; 1998; 2005), we are not persuaded that the distinction is sufficiently behaviorally stable to bear the weight that Bicchieri assigns to it. That is, we suspect that only a very rare and unusual kind of agent would go on following a norm if this was costly to her and she expected literally no one else to also follow it. This skepticism, about the empirical extent of morality in Kant’s sense, only strengthens the idea that normative conditionality is central to understanding choice-sensitive sociality (as opposed to the hardwired sociality of ants, termites, and cells) and ethics.

5There is a weak sense in which conditionality is sown into the very fabric of game theory: whether a given choice is an element of an equilibrium strategy for an agent is a function of what other agents choose. Therefore, whether an action in a strategic setting maximizes an agent’s utility function in ex ante expectation depends on whether it is a best reply to other players’ expected choices. But this only captures what Bicchieri (2006; 2017) calls “empirical” expectations, expectations about what others will in fact do, and, in conditions of uncertainty about utility functions, what others expect others to do, and expect others to expect others to do, and so on recursively. These kinds of expectations are modeled using well developed game-theoretic solution concepts, including Bayes-Nash equilibrium (Harsanyi, 1967) and quantal response equilibrium (McKelvey and Palfrey, 1995; Goeree et al., 2016), and, for extensive-form games, sequential equilibrium (Kreps and Wilson, 1982). Conditional responses can therefore, in principle, be purely implicit in best-response patterns across ranges of games.

6The conception of norms of interest to us here, however, models them as exogenous social structures that agents encounter as elements of their environments. Agents can then have (conditional or unconditional) preferences and beliefs about whether they, and other agents with whom they interact, regulate their choices, or should regulate their choices, by reference to them. This conception reflects the generally accepted empirical fact that norms display intertemporal and cross-generational persistence on a scale that is longer than individual preferences. Furthermore, as discussed and modelled by Kuran (1995), agents can under some circumstances go on following norms that are no longer preferred by majorities, or even by anyone at all, if something (e.g., fear of shame, or of the secret police) systematically interferes with revelation of true preferences.

7On this “social facts” (Gilbert, 1989) conception of norms, we might suppose that they are subject to conditional preferences in a special sense that goes beyond equilibrium dependence (though must be compatible with whatever definition of equilibrium features in the analysis). This is at the heart of Bicchieri’s (2006; 2017) proposed special utility structure for application to games in which expectations about norms are relevant for choice. As reviewed in Ross, Stirling, and Tummolini (2023), Bicchieri and her various co-authors have, however, tended to fall back on formulations more consistent with the social preferences approach when analysing data from their own experimental lab.

8Here we focus attention on models for application for choice data estimation that remain consistent with a “social facts” conception of norms. Specifically, we mobilize the utility structure proposed by Kimbrough and Vostroknutov (2016), hereafter KV, for applications to public goods games, dictator, ultimatum, and trust games run in the laboratory. This work represents a significant advance with respect to marrying generality and identifiability in choice data. We aim to deepen understanding of it as a model of empirical normative dynamics by embedding it in a more general framework.

9For reasons we will indicate, KV’s model only accommodates norms for which social welfare increases in the number of followers. Nor does it allow for an agent to persistently follow a norm she would be better off abandoning. Therefore, it does not allow for representation of “Kuran cases”, that is, norms that survive due to strategic preferences falsification and trap societies in inferior states relative to all plausible welfare criteria until something breaks informational symmetry. As the KV (2016) model is constructed to describe the specific class of social dilemma games indicated above, and is not intended to be general, it might be objected that in doing this we patch a tyre that is not losing any air. However, as we describe in the Discussion section of the article, the subsequently developed general model of Kimbrough and Vostroknutov (2020a) also excludes Kuran cases.

10Kuran cases remain out of reach of KV’s models for a deep reason. Although KV represent norms as social facts in the sense that agents have preferences about them as norms, norms so understood lack the kind of causal force that most social theorists regard as distinctive to genuine norms. As Kuran (1995) recognizes, persistent norms shape preferences. A norm that endures for a time due to preferences falsification may cause agents’ preferences to shift to accommodate it, and may consequently cease to be a barrier to Pareto improvement because it alters the location of the Pareto frontier, even while leaving market-valued wealth or income unimproved. That is, there are social adaptive preferences. Individual adaptive preferences are often regarded as phenomena that undermine standard normative economics. This is because they seem inconsistent with the idea of consumer sovereignty that in turn supports welfarism, the doctrine that the best economic policy is the one that most efficiently satisfies subjective preferences (Sen, 1992). As we discuss in Section 2, the implications of social adaptive preferences for normative economics are more nuanced and arguably more interesting. As with individual adaptive preferences, they are sometimes defensive responses to oppression, of sub-cultures or sub-communities, but they are also the basis of often celebrated “expansions of the moral circle” (Singer, 1981).

11Such endogenous preferences change is generally excluded in standard game theory, though there are exceptions (e.g., Bisin and Verdier, 2001). Providing a generalised approach to representing strategic preferences adjustment is the primary objective of CGT (Stirling, 2012; 2016). This is an extension of standard game theory that is designed to bring the social phenomenon of mindshaping (Zawidzki, 2013) within the ambit of strategic choice. Mindshaping occurs when agents coordinate with one another by aligning ex ante uncertain preferences. One useful way to understand social norms is as outputs of mindshaping processes in networked groups.

12KV’s models retain fixed preferences, and in most applications of interest to economists there are good reasons for this, as we will discuss. The model we develop is thus not intended to improve or replace either the KV2016 special model or the KV2020 general model of choice under norms. It is, rather, intended to represent a strategic process by which the normative expectations that the KV models treat as exogenous exert special, endogenous influence as norms. The games we model and those modelled by KV mutually constrain one another according to a relationship described by Ross (2005) under the label of “game determination”. As we explain, this can be understood, following Ross (2014), as step along a path to unifying macrostructural sociological and microeconomic theory in the formalism of game theory.

13The article is structured as follows. Section 1 explains its motivations in more detail and relates these to the motivations underlying the KV2016 special model of norm-governed social dilemma games. Section 2 compares simulations of play of a representative such game, the public goods game, using, respectively, KV’s model based on application of standard game theory and our construction of a model using CGT. Section 3 discusses wider implications of this comparison. In particular, we sketch a conception of an account of norms in which macrosociological and microeconomic analyses play distinct but complementary roles. Their relationship is characterised as analogous to the Nash program for using cooperative and non-cooperative game theory in tandem to exploit the strengths and avoid the limitations of each. Section 4 concludes and looks toward follow-up work.

1. Modelling Norms with Game Theory: From Mindreading to Normative Mindshaping

1.1 The Importance of Mindshaping Processes for Norm Emergence

14If norms are generally conditional, then agents playing games in norm-governed social contexts are under pressure to form expectations by making inferences about the normative attitudes of other players. As KV wrote in a 2013 working paper that was the ancestor of KV2016, “one way of thinking about social cognition is that an important part of ‘theory of mind’ is the ability to infer social norms from context”. They here refer to the extensive literature on “mindreading” (Nichols and Stich, 2003). Where conditional norms are in play, mindreading goes beyond updating of priors about other players’ preferences based on observations that regard all of their actions as symmetrically informative, because normative preferences will be revealed by choices only when their activating conditions happen to be satisfied.

15Zawidzki (2013) argues that mindreading is, in general, much more difficult than theorists have tended to assume, given the evidence available to participants in the typical interactions that mindreading hypotheses are intended to explain. Zawidzki musters substantial evidence that most phenomena that other theorists have characterised as instances of mindreading are in fact manifestations of mindshaping processes, in which interacting parties exert influence on one another to generate alignment of mutually attributed preferences and beliefs. This interpretation of evidence is set within a wider externalist conception of the ontology of “propositional attitudes” (PAs), such as belief and desire, that has become the dominant view among philosophers (Dennett, 1987; McClamrock, 1995; Bogdan, 1997; Clark, 1997). According to externalists, PAs are not private psychological states of individuals that must be inferred from behaviour, but publicly negotiated interpretations of relatively consistent patterns in behaviour (including verbal behaviour) that people use to make sense of one another and construct their own images of themselves as relatively coherent selves with unfolding biographies that comprise meaningful narratives. Mindshaping is typically relatively effortless and implicit because it is simply equivalent to the normal interpersonal construction of shared social reality.

16Zawidzki does not deny that successful mindreading sometimes occurs. But it is necessary only in circumstances where a party to an interaction is thought to be concealing or misrepresenting their self-ascribed beliefs or preferences, perhaps for strategic reasons, or where there is asymmetry between parties in the sophistication with which contents of PAs are distinguished and articulated (e.g., a psychotherapist and a patient, a parent and a child, or a scientific analyst and a research subject). Zawidzki argues, based on review of experiments by cognitive scientists, that successful mindreading in such cases is far from assured; and that when mindreading is successful it depends on foundations of prior mindshaping with respect to background beliefs and preferences that provide contextual leverage for inference.

17Mindshaping processes are not amenable to easy representation in standard game-theoretic models because they involve changes in preferences (as well as in beliefs). In standard game theory, preferences are the basic arguments for the utilities associated with outcomes, and in that sense define and individuate players. Since games with imperfect information typically involve strategic signaling and screening of information about which players hold relevant beliefs, or about types of players where types are distinguished by their preferences profiles, it is natural to interpret games between people as models of mindreading rather than mindshaping.

19This idealised division of labour blurs where the modeling of response to norms is concerned. After developing their special model of norms for ultimatum, dictator, public goods, and trust games, KV (2016) write that “[t]he most important unanswered question … and the one that we hope this research will encourage others to ask, is ‘where do norms come from?’” (2016, 635) Their subsequent general model addresses this question in one sense: according to that model, different norms arise for different strategic contexts, but always as solutions to a general problem of minimizing aggregate dissatisfaction with actual outcomes by comparison with achievable counterfactual alternatives. Binmore (1994; 1998), on the other hand, extensively addresses the question in a more fundamental sense, appealing to evolutionary psychology to explain why people are sensitive to norms in the first place. And we might turn to evolutionary game theory to account for general features of human sensitivity to normative influence. In abstracting away from idiosyncratic preferences of individual agents, explanations that rely on evolutionary modeling like Binmore’s are relatively long-run models while KV’s models, in which agents with fixed preferences maximize utility, are short-run ones. Though there is not yet a standard account of the ways in which the long-run models constrain the short-run models, there is no shortage of formal work relating evolutionary and classical game-theoretic solution concepts (e.g., Weibull, 1995), or of reflection on the nature of their consilience (Binmore, 1994; 1998; Gintis, 2009; 2016).

20According to Ross (2005; 2006; 2008), however, a crucial middle layer of analysis between evolutionary and standard game-theoretic analysis is needed but has been largely neglected in the literature. Early evolutionary models of norm emergence were inspired by biology and abstracted away from individual agents to focus on the adaptive value of behavioural strategies viewed as traits (Sugden, 1986; Binmore, 1994; but see Young, 2015, for a different approach). On the other hand, in standard game-theoretic models agents appear fully formed, with stable utility functions (see also Davis, 2010). As discussed above, however, individuals are socialised into relative stability with respect to preferences (and beliefs) by mindshaping. Furthermore, mindshaping has a strategic dimension in which already socialised agents play essential roles, and less-than-fully socialised “patients” co-develop their own agency by strategically blending accommodation, resistance, and creativity. Ross was therefore motivated to seek modeling approaches that, unlike biologically-inspired evolutionary models, preserve agency, while also allowing for representation of preferences that shift under social pressure.

1.2 Socialisation as Game Determination Between Generations

21The basic device that Ross (2006; 2008) exploits to try to achieve this ambition is an overlapping generations model, lifted from its traditional macroeconomic setting (Samuelson, 1958) and applied in a microeconomic, game-theoretic environment. He models what he calls “game determination” using non-terminating sequences of extensive-form games across generations of agents (in which generations can, depending on the intended scale of analysis, include different life stages of a single biological individual). Each sequential triplet of games has the following structure. The first stage, the “determining” game \({\mathit G’}_1\), involves players who adhere to norms, understood following Binmore (1994; 1998) as equilibrium selection devices, along with “pre-socialised” players whose utility functions are generated by replicator dynamics that represent natural selection (including Baldwin effects) of human psychology. The outcomes of determining games are rules (extensive-form structures and strategy sets) of the “determined” game stage \({\mathit G}_2\) that follows. All players’ utility functions in all stages incorporate, to an exogenously parameterised degree, interest in the welfare of their descendants over a horizon of two subsequent periods. Pre-socialised players implicitly choose norms these successors will be endogenously motivated to follow given the structure of the determined game \({\mathit G}_2\) that is selected from among the equilibria of the determining game \({\mathit G’}_1\). In \({\mathit G}_2\) the pre-socialised agents are replaced by socialised successors, who might or might not have modified the norms of their “tutors”. Determined games, which involve only socialised players, represent bargaining over distributions of resources that generate relative bargaining power in the subsequent determining game \({\mathit G’}_3\). In \({\mathit G’}_3\) agents who played \({\mathit G’}_1\) as socialised agents have died, replaced by pre-socialised descendants. This pattern iterates indefinitely.

22Ross intended this game determination model to be a more or less literal implementation of the account given by Binmore (1994; 1998). It was thus acknowledged to be incomplete in the same respects that Binmore’s philosophical theory is incomplete. First, it allows no scope for altruism about the interests of non-descendants, so coalitions of interests can only arise implicitly in the underlying replicator dynamics. But clearly humans often choose to participate in coalitions. An implication of this is that, as in the special model of KV2016, there is no scope for representation of norms that prescribe differential treatment of in-group and out-group members. Third, normative influence, i.e., mindshaping, is exclusively intergenerational, and there is none between generational peers. Fourth, the model is not general, in that the modeler can freely choose the parametric structures specifying interest in descendants’ welfare; and the parametric structures governing these interests as between socialised and pre-socialised agents must be chosen independently because socialised agents live for two periods and pre-socialised agents live for only one period.

23The game determination model’s retention of a core restriction of standard game theory, that no agent’s utility function changes, is also the main source of the limitations identified above. These limitations are removed in CGT.

1.3 Socialisation as Cyclic Influence in Social Networks

24The primary motivation for CGT is the insight that, just as agents adjust uncertain beliefs on the basis of observing beliefs of others, so they may resolve ambivalent preferences by comparing them with preferences of others in cultural, commercial or political reference groups. CGT is characterised by Ross and Stirling (2021) as a formal theory of mindshaping. The core technical manoeuvre in the construction of the theory is to apply the syntax of epistemic probability to the practical domain, i.e., to incentivised choice. It is therefore equipped to model what might be regarded as cognitive dynamics using data compatible with revealed preference theory.

25There are three stages involved in representing and solving a conditional game: socialisation, diffusion, and deduction. Socialisation is achieved by expressing preferences via conditional payoffs that reflect modulation by agents of their utility structures as functions of the preferences of those who socially influence them. This form of conditional reasoning is formally analogous to the use of conditional probabilities to modulate beliefs as functions of statistical influence. Just as a set of statistically dependent random variables can be expressed as a directed graph (a Bayesian network) with random variables as the vertices and conditional probabilities as the edges, so a community of socially influenced agents can be expressed as a directed graph (a social influence network) with agents as the vertices and conditional payoffs as the edges. Then, by formal analogy to the way in which statistical dependency is diffused throughout a Bayesian network to create the joint probability of the random variables as the product of the individual conditional probabilities, social influence is diffused throughout a social influence network to create a coordination function as the product of the individual conditional payoffs. In epistemology individual probabilities of the random variables are deduced from the joint probability by marginalisation; so, analogously, in CGT socially influenced payoffs of the individual agents are deduced from the coordination function by marginalisation. Following marginalisation, games are solved by application of standard equilibrium concepts for normal-form games (e.g., Nash equilibrium, quantal response equilibrium). CGT is equivalent to standard non-cooperative game theory if no social influence exists (i.e., a social influence network has no edges), which might be because no agents are ex ante ambivalent about their preferences, and the conditional payoffs thus are identical to categorical payoffs.

27Philosophically, we view the relationship between game determination theory (GDT) and CGT as follows. We are convinced that, as a matter of empirical fact, processes of socialisation and diffusion set the conditions for human interactions that game theorists model and solve deductively. However the mechanisms that transmit social influence are modelled mathematically, we expect Ross’s (2005) philosophical account to apply to them. But for now we reserve “GDT” for what is actually on the table by way of real theory, namely, Ross’s specific (2006; 2008) overlapping-generations model. Then we can say that both GDT and CGT aim to represent mindshaping. Because GDT preserves a one-to-one mapping of agents to utility functions, it is more conservative in its way of extending standard non-cooperative game theory. But the price of this is high: it can only represent intergenerational mindshaping. CGT, in also capturing peer-to-peer mindshaping, is more general. This also comes with a cost: CGT cannot usefully be applied to extensive-form games in which solutions rely on identification of outcomes with fixed preferences. Of course, a normal-form game is formally a set of extensive-form games. When a normal-form game with categorical preferences is “relaxed” in CGT to allow for conditional preferences, the associated set of extensive-form games typically explodes. As we argue in Section 3 this apparent cost reflects the scale shift between macrostructural and microeconomic modeling—it is why macrostructural modeling is insufficient for identifying the phenomena that interest microeconomists. Following arguments in Ross (2014), we believe that a general philosophy of social science should aim to show how disciplines can make complementary contributions while remaining distinct in the formal analyses specified by their general theories. The kind of complementarity we have in mind is exemplified in the application in Section 2 below. But it is merely exemplified. The project of theoretically specifying general mechanisms by which processes of socialisation and deductions of solutions to strategic interactions of fully socialised agents constrain one another awaits future work. We will return to sketch the challenge less cryptically at the end of the article.

1.4 Normative Mindshaping: From Categorical to Conditional Norm-Dependent Utility

28With this context in place, we return to KV’s modeling strategy. Building on previous work (Kessler and Leider, 2012), KV assume that the human tendency to comply with social norms can be fruitfully described with a norm-dependent utility function like the following:

29\[ u_i(x) = x - \phi_i|\eta - x| \]

30Here \(x\) corresponds to the agent’s material payoff, \(\eta\) is the action that is most socially appropriate in a given context (the norm), \(\phi_i \geq 0\) is an individual parameter that specifies the individual sensitivity to norms, and the distance \(|\eta - x|\) between the socially appropriate action and the actual one captures the disutility from norm violation. Notice that, if defined in this way, norm-dependent utility does not amount to a kind of social preference (in the technical sense), since the norm-following agent only cares about the extent to which the considered action conforms to the norm \(\eta\) and not about the payoff other agents derive. A social preference model instead typically assumes that a norm is embodied in a social utility where a “taste for fairness” (or some other specific social value) modulates how much an individual agent values alternative outcomes (e.g., Fehr and Schmidt, 1999). In contrast with this approach, an agent who is simply sensitive to norms in general can consistently be “selfish” on some occasions and “generous” on others if the norm that is relevant demands different actions in different contexts. Consider, for instance, that the same norm that specifies a “fair” distribution of a windfall gain may also prescribe a “selfish” one when resources have instead been earned (Oxoby and Spraggon, 2008).

31Notice that this formulation of the norm-dependent utility does not theoretically constrain where the norm \(\eta\) comes from. Thus, especially for the purposes of experimental work, KV rely on the experimental task proposed by Krupka and Weber (2013) in which, by exploiting the incentive structure of a pure coordination game, the relevant norm underlying a given social situation is inferred by eliciting the action which is believed to be the most appropriate by a reference group of other people (i.e., second-order normative beliefs or normative expectations in the sense of Bicchieri, 2006). Finally, as a complement to this procedure, KV introduce an additional task to also estimate, albeit indirectly, \(\phi_i\), conceived as a general, idiosyncratic trait revealing a categorical desire to actually comply with normative expectations.

32To better understand how the KV model works consider, for instance, a standard Public Goods (PG) game. As the workhorse of the experimental study of cooperation, the PG is commonly proposed as a model of a situation in which a group has the opportunity to invest in a common project with the potential to benefit all of its members. This collective welfare-optimizing outcome however risks not to be achieved because of the temptation to free-ride on the effort of others and the fear of being unfairly exploited may induce widespread defection. Indeed, the decline of cooperation in repeated PG is one of the most robust and best replicated findings in experimental economics (Ledyard, 1995; Chaudhury, 2011). Starting with the seminal contributions by Yamagishi (1986) and Fehr and Gachter (2000), it has been repeatedly observed that existence of a punishment mechanism in game rules can indeed sustain cooperation in these difficult contexts. In real informal settings amongst people, the relevant mechanisms are often naturally interpreted as norms (Smith and Wilson, 2019). Consistently with this interpretation, KV have shown that groups composed of norm followers (i.e., members with high \(\phi_i\)) are indeed capable of sustaining cooperation at similar levels even if punishment is not available, which is something that is not possible for groups of norm-breakers (i.e., with members with low \(\phi_i\)) that were unable to resist the well-known pattern of cooperation decline.

33To rationalize these results, KV propose that the relevant norm in a repeated PG is a norm of conditional cooperation. In a repeated PG such a norm encourages the i-th player, denoted \(X_i\), to contribute \(\eta\) at the start of the interaction (e.g., in period 1) and to keep contributing similarly in the next rounds if the other members have contributed the appropriate amount too. Violation of the norm by others make \(X_i\)’s violation appropriate. Indeed, KV have shown that, under such a norm, cooperation can be sustained as a Perfect Bayesian Nash Equilibrium if players’ norm-sensitivity parameters \(\phi_i\) are sufficiently high as well as being believed to be sufficiently high by all. In other words, in keeping with standard philosophical interpretation of game theory applied to humans, norms influence behaviour via a mind-reading process by which norm-followers infer the action that others believe to be most appropriate as well as their reciprocal desires to act according to it.

34Adopting the mindshaping perspective, however, opens up a complementary possibility. Instead of defining a (categorical) norm-dependent preference to cooperate conditionally on the cooperation of others, i.e., a preference for reciprocity in action, we can begin by specifying a conditional norm-dependent preference towards cooperation as such, i.e., the norm emerges from convergence in the preferences for cooperation in a group. In this approach norm-dependent preferences form due to the fact that the utility that a norm follower derives from cooperating in a PG is affected by the possibility that cooperation is assumed to be the action that one’s reference group prefer the most too. Viewed in this way, conditional norm-dependent preferences are the product of a mind-shaping, interactive influencing dynamics that aim to create and stabilize behavioral patterns in interactive contexts.

35As we explain in the third section, we frame the technical work of the next two sections as follows. KV, in both their special and general models, provide an improved microeconomic analysis of the operation of norms. Such analysis applies under the idealisation of fixed utility functions. For the huge range of applications that matter to economists, who specialise in understanding the marginal effects of changes in incentives, typically under institutional constraints that are commonly known, this idealisation is powerful, the basis of elegant deductive solutions. But on less granular scales, the scales on which social dynamics forge utility functions through mindshaping, the idealisation must be relaxed.

36The work in the next two sections below should be understood as exemplifying this relaxation. The example is not intended to show the full potential for representing social structure using CGT, which would construct sub-networks with heterogeneous agents. Such construction is illustrated in Ross, Stirling and Tummolini (2023). The example here is offered as a formal proof-of-concept with the purpose of demonstrating the causal effects of preferences diffusion, that is, normative mindshaping, within a single community whose network connections are assumed to be symmetrical throughout.

2. Simulating a Public Goods Game Using Conditional Game Theory

2.1 Unconditional KV Utility Model

37We consider the PG scenario introduced by KV (2016), which involves a collective \(\{X_1, \ldots, X_{n}\}\) of players where the players possess a common action set \(A = \{x_1, \ldots, x_N\}\) and each player has the option of contributing part of her endowment to the community and receiving some fraction of the total contribution in return. The KV payoff model for agent \(X_i\) is

38\[\pi_i(a_{i},a_{-i}) = u_{i,i}(a_{i}) + u_{i,-i}(a_{i},a_{-i})\]

39where \(a_i \in A\) is an action by \(X_i\) and \(a_{-i} = \{a_{1}, \ldots, a_{n}\} \setminus \{a_{i}\}\) is the set of actions of all agents \(X_{-i} = \{X_1, \ldots, X_n\} \setminus \{X_i\}\), \(u_{i,i}(a_i)\) is the part of \(X_i\)’s payoff that she improves by choosing \(a_{i}\), and \(u_{i,-i}(a_{i},a_{-i})\) is the part of the payoff chosen for \(X_i\) by \(X_{-i}\).

40In the KV (2016) model set-up, the payoffs are defined over the unit interval. In order to conduct simulations, we need to replace the unit interval action set employed by KV with a finite action set. We consider a set of four agents \(\{X_i, i=1, \ldots, 4\}\) with common action \(A = \{x_{1}, x_{2}, x_{3}\}\) with \(x_{1} > x_{2} > x_{3}\) expressed in units of dollars. For our model we set \(A = \{50, 25, 0\}\). The resulting payoff function is

41\[\pi_i(a_i,a_{-i}) = x_{1} -a_i +\alpha\sum_{j=1}^{n}a_j\]

42where \(u_{i,i}(a_{i}) = x_{1} - (1-\alpha)a_{i}\) and \(u_{i,-i}(a_{i},a_{-i}) = \alpha \sum_{j \neq i}a_{j}\) with \(\alpha \in [0,1]\).

43KV (2016) extend this model to create a repeated-play game defined over a sequence of stages \(t=0, 1, 2, \ldots\) that includes a norm-response component \(\phi_i\, \rho(s_{t-1})\, g\left(\|\eta -a_{it}\|\right)\), yielding, for \(t= 1, \ldots,\)

44\[\tag{1}\pi_i(a_{it},a_{-it})=x_{1} - a_{it}+\alpha \sum_{j=1}^{n}a_{jt} - \phi_i\, \rho(s_{t-1})\, g\left(\|\eta -a_{it}\|\right),\]

45where

• \(\phi_i\geq 0\) is a dimensionless parameter indicating the sensitivity of player \(X_i\) to deviations from the norm \(\eta\);

• \(\rho(s_t)\) is the norm response function of the average contribution \(s_t= \frac{1}{4}\sum_{i=1}^{4} a_{it}^*\) of all players, defined as \(\rho(s_t) = \begin{cases}1 & \text{if } s_{t} = \eta\\ 0 & \text{otherwise}\end{cases}\) where \(a_{it}^*\) is \(X_i\)’s choice at stage \(t\);

• \(g(\cdot)\) is a strictly convex increasing function that represents the disutility of deviating from the norm and is defined as \(g(z) = x_{1}\frac{1-e^{\frac{z}{x_{1}} } } {1-e}\, ,\) hereafter labeled as (2), with \(\|\eta-a_{it}\|= |\eta-a_{it}|\), the absolute value of the difference (notice that \(g\) is scaled to units of dollars), yielding \(g\big(\|\eta -x_{1}\|\big) = g\big(\|\eta -x_{3}\|\big) = g(\eta) =18.877\) and \(g\big(\|\eta -x_{2}\|\big) = 0\);

• \(\eta = \$25\) is the norm that governs agents’ behavior.

46KV (2016, Appendix B.1) propose the following strategy. At stage \(t=0\), each player takes an action \(a_{i0}^*\) drawn from a distribution \(F\) that is common knowledge. Since the initial actions strongly influence subsequent behavior, it is imperative that these actions are randomized such that the average \(s_{0}\) is uniformly distributed. For each

47\(s \in \{\$0,\$25,\$50,\$75,\$100,\$125, \$150,\$175,\$200\}\),

48Table 1 lists the combinations of \(a_{1} +a_{2}+ a_{3} +a_{4} = s\) for \((a_{ 1},a_{2}, a_{3}, a_{4}) \in A\). To ensure that the actions drawn from these subsets are uniformly distributed, we must draw from the set of constant-sum subsets with probability proportioned to the number of elements in the subset, and then draw uniformly from that subset. For example, there are 10 ways to achieve a sum of \($150\) (i.e., \(301 \cup 220\), cf. the rows of Table 1) so this subset is selected with probability \(10/81\) (81 being the total number of partitions, namely the sum of the fourth column of Table 1), and an element is drawn from this set with probability \(1/10\). We also require that each element of the set \(\{\phi_{1}, \phi_{2},\phi_{3}, \phi_{4}\}\) be drawn via the independent and identically uniform distribution over the interval \([\phi^* -\epsilon, \phi^* + \epsilon]\), where \(\epsilon> 0\) and \(\phi^*\) is the threshold value such that the maximizing choice for payoff (1) switches from \(a_i^* = x_{3}\) to \(a_i^* = x_{2}\) when \(\phi_i \geq \phi^*\) (the value for \(\phi^*\) will be computed subsequently).

Table 1. Initial Action Partition Sets for Constant Sum

\( x_{1}\)	\( x_{2}\)	\( x_{3}\)	# Partitions	Sum
4	0	0	1	$200
3	1	0	4	$175
3	0	1	4	$150
2	2	0	6	$150
2	1	1	12	$125
2	0	2	6	$100
1	3	0	4	$125
1	2	1	12	$100
1	1	2	12	$75
1	0	3	4	$50
0	4	0	1	$100
0	3	1	4	$75
0	2	2	6	$50
0	1	3	4	$25
0	0	4	1	$0

49Table 2 lists the cumulative distribution function for constant-sum partitions for each contribution sum and the corresponding inverse distribution. The randomization of the initial action proceeds as follows: let \(a\) be drawn from a uniform distribution over \([0,1]\), apply \(F^{-1}(a)\) to identify the constant-sum partition, and draw the initial actions \((a_{10}, a_{20}, a_{30}, a_{40})\) from a uniform distribution over the selected constant-sum partition.

Table 2. Cumulative Distribution and Inverse Distribution for Choice Initiation

\(x\)	\(x\)≤0	\(x\)≤ 25	\(x\)≤50	\(x\)≤ 75	\(x\)≤ 100	\(x\)≤ 125	\(x\)≤ 150	\(x\)≤ 175	\(x\)≤ 200
\(F(x)\)	1/81	5/81	15/81	31/81	50/81	66/81	76/81	80/81	1
\(a\)	\(a\)≤ 1/81	\(a\)≤ 5/81	\(a\)≤ 15/81	\(a\)≤ 31/81	\(a\)≤ 50/81	\(a\)≤ 66/81	\(a\)≤ 76/81	\(a\)≤ 80/81	\(a\)≤ 1
\(F^{-1}(a)\)	0	25	50	75	100	125	150	175	200

50KV’s payoff model for \(t>0\) is

51\[a_{it}^* = \arg\max _{a \in A} \{-(1-\alpha)\,a - \phi_i \,\rho(s_{t-1}) \, g(\|\eta -a\|)\}.\]

52Let

53\(\pi_i(x_{1}) = \begin{cases} -(1-\alpha) x_{1} -\phi_i\, g(\|\eta -x_{1}\|) = -(1-\alpha)50 - 18.877\, \phi_i & \text{ if } s_{t-1} = \eta\\ -(1-\alpha) x_{1} = -(1-\alpha)50 & \text{ otherwise}\end{cases}\)

54\(\pi_i(x_{2}) = \begin{cases} -(1-\alpha) x_{2} -\phi_i\, g(\|\eta -x_{2}\|) = -(1-\alpha)25 & \text{ if } s_{t-1} = \eta\\ -(1-\alpha) x_{2} = -(1-\alpha)25 & \text{ otherwise}\end{cases}\)

55\(\pi_i(x_{3}) = \begin{cases} -(1-\alpha) x_{3} -\phi_i\, g(\|\eta -x_{3}\|) = - 18.877\, \phi_i & \text{ if } s_{t-1} = \eta\\ -(1-\alpha) x_{3} = 0 & \text{ otherwise}\end{cases}\)

56and the optimal choice is

57\(\max \{\pi_i(x_{1}), \pi_i(x_{2}),\pi_i(x_{3})\} = \begin{cases}-(1-\alpha)25 & \text{if } \phi_i > \phi^*\text{ and } s_{t-1} = \eta\\-18.877 & \text {otherwise}\end{cases}\)

58where

59\[\tag{3}\phi^* = \frac{(1-\alpha)25}{18.877},\]

60resulting in

61\[\tag{4}a_{it}^* = \begin{cases}x_{2} & \text{if } \phi_i > \phi^* \text{ and } s_{t-1} = \eta\\x_{3} & \text{otherwise.}\end{cases}\]

62Specifying behavior in this repeated-play game is straightforward. Each player is initialized with a randomly selected action. If the average of their initial actions is consistent with expectations under the collective welfare improving norm and the sensitivity parameter \(\phi_i\) is greater than \(\phi^*\) for all players, then they all conform to the norm and continue to do so at subsequent stages. However, if the average of the initial actions is not consistent with normative expectations, then each player seeks to maximize her payoff at subsequent stages, regardless of the existence of a norm. However, even if the average initial action is norm-consistent, if \(\phi_i < \phi^*\) for any \(X_i\), then she will ignore the norm, thereby causing the other players to do likewise.

2.2 Norm-Compliant Utility Model

63As defined by KV (2016), the only mechanism to induce compliance with the norm is the history provided by \(s_{t-1}\). The function \(\rho\) signals to the network at stage \(t\) that the agents at stage \(t-1\) did or did not comply with the norm and invokes a penalty if they did not, but that is not the only mechanism for exertion of social pressure. We will suppose that if a norm exists then the players are aware of it, and that this awareness is crucial for motivating punishment and given that knowledge, each player may, regardless of knowledge of the past, feel social pressure to comply with the norm. The purpose of our simulation is to demonstrate the effectiveness of this supposition in the CGT framework.

64We model social influence via conditional utility. The utilities of standard game theory involve deductive inferences, that is, the expression \(u(a) > u(a’)\) establishes that \(a\) is preferred to \(a’\). Deductive inferences are also familiar from probability theory. Let \(pr[\cdot]\) denote a subjective probability mass function. The expression \(pr[a] > pr[a’]\) establishes that \(a\) is more likely than \(a’\). But probability theory also supports a different kind of inference, namely conditional inferences. The conditional probability statement \(pr[a|b] > pr[a’|b]\) does not provide sufficient information to conclude that \(a\) is more likely than \(a’\) without a supporting statement regarding the likelihood of \(b\). Similarly, we may adopt the syntax of probability theory to define a conditional utility of the form \(u(a|b) > u(a’|b)\), meaning that if \(b\) is actualized, then \(a\) is preferred to \(a’\).

65Following this line of reasoning, let us define the conditional utility for the PG. Let \(\boldsymbol{\Sigma}\) denote the set of all possible summative arrangements for \(X_{-i} = \{X_1, \ldots, X_4\} \setminus \{X_i\}\), the subset of agents excluding \(X_i\). Ordered lexicographically, the elements of \(\boldsymbol{\Sigma}\) are

66\(\Sigma_{111} = x_{1}+ x_{1}+x_{1}\)

67\(\Sigma_{112} = x_{1}+x_{1}+x_{2}\)

68\(\Sigma_{113} = x_{1}+x_{1}+x_{3}\)

69\(\Sigma_{121} = x_{1}+x_{2}+x_{1}\)

70\(\vdots\)

71\(\Sigma_{133} = x_{1}+x_{3}+ x_{3}\)

72\(\Sigma_{233} = x_{2}+x_{3}+ x_{3}\)

73\(\Sigma_{333} = x_{3}+x_{3}+ x_{3}\,.\)

74For any profile \( [a_{it}, a_{jt}, a_{kt}, a_{lt}]\), we define the exclusion sum

75\[\sigma_{-it} = a_{jt}+a_{kt}+a_{lt},\]

76as the sum of conjectures of all players excluding \(a_{it}\), the conjectured contribution of \(X_i\). Thus, \(\sigma_{-it}\) constitutes a conditioning conjecture set by \(X_i\) for \(X_{-i}\). There are seven possible summative values for \(\sigma_{-it}\), comprising the set

77\[S = \{0,25,50,75,100,125,150\}.\]

78For \(r \in S\) let

79\[\mathcal{S}_{r} = \big\{(a_{jt}, a_{kt}, a_{lt}) \in A^3 : a_{jt} + a_{kt} + a_{lt} =r \big\}\,\]

80denote the set of conditioning subprofiles whose sum equals \(r\). Table 3 displays the sets of all possible combinations, expressed in lexicographical order, of the ways \(A^3\) can be partitioned into constant \(r\) subsets.

Table 3. Conditioning Conjecture Partition Sets for Constant \(s_r\)

\(x_{1}\)	\( x_{2}\)	\( x_{3}\)	# Partitions	Sum
3	0	0	1	$150
2	1	0	3	$125
2	0	1	3	$100
1	2	0	3	$100
1	1	1	6	$75
1	0	2	3	$50
0	3	0	1	$75
0	2	1	3	$50
0	1	2	3	$25
0	0	3	1	$0

81Given the norm \(\eta\), the norm-compliance partition, denoted \(\mathcal{S}_\eta\), is the subset of \(\boldsymbol{\Sigma}\) whose elements sum to \(3\eta\). For \(\eta = \$25\),

82\[\mathcal{S}_{\eta} = 030 \cup 111 = \{(x_{1},x_{2},x_{3}),(x_{1},x_{3},x_{2}),(x_{2},x_{1},x_{3}), (x_{2},x_{2},x_{2}),(x_{2},x_{3},x_{1}), (x_{3},x_{1},x_{2}), (x_{3},x_{2},x_{1})\}.\]

83A conjecture \(\sigma_{-it}\) is norm-compliant if \(\sigma_{-it} \in \mathcal{S}_\eta\). The norm-compliant payoff function is

84\[\tag{5}\pi_{i|-i}(a_{it}|\sigma_{-it}, s_{t-1}) = \begin{cases}x_{1} - a_{it}+\alpha (a_{it} + \sigma_{-it}) - \phi_i(a_{it}, s_{t-1})\big[ g\big(\|\eta -a_{it}\|\big) -x_{1}\big]& \text{if } \sigma_{-it} \in \mathcal{S}_\eta\\x_{1} - a_{it}+\alpha (a_{it} + \sigma_{-it}) & \text{if } \sigma_{-it} \not\in \mathcal{S}_\eta\end{cases}\]

85where, for \(t=0\), \(s_{t-1} = \varnothing\). Thus, if \(X_i\) conjectures that \(X_{-i}\) will be norm-compliant and she is favorably disposed toward the norm, then she would also favor others complying. If she conjectures that \(X_{-i}\) will not be norm-compliant (i.e., she conjectures that \(X_{-i}\) favors outcomes in \(\{x_{2}\}^{\mathcal{c}}\)), \(\mathcal{c}\) being the complementary set, then she will seek, unconstrained, to maximize her utility without regard for the existence of a norm. The norm, then, conditions actions in the sense of Bicchieri (2006; 2017). \(X_i\)’s norm-compliance sensitivity, \(\phi_i\), is now a function of her conjectured action \(a_{it}\) and the history \(s_{t-1}\).

2.3 Two-Agent Conditional Public Goods Game

86To define a conditional game we must transform \(\pi_{i|-i}\) to become a mass function, yielding

87\[\tag{6}u_{i|-i}(a_{it}|\sigma_{-it}, s_{t-1}) = \frac{\pi_{i|-i} (a_{it}|\sigma_{-it},s_{t-1})}{\sum_{q =1:4} \pi_{i|-i}(a_{qt}|\sigma_{-it},s_{t-1})}\]

88In the interest of clarity (and brevity) we first develop a one-shot public goods conditional game model for only two agents; extending it to four agents is then conceptually straightforward.

89The most significant difference between this model and the KV model is the presence of direct linkages between the two agents whereby each agent modulates her preferences as a function of conjectures regarding the preferences of the other. Furthermore, these preference relationships are reciprocal: \(X_1\) exerts influence on \(X_2\), who influences \(X_1\), and so forth, thus, creating a cyclic network of the form

90where the arrows indicate the direction of social influence from the influencer to the influencee and \(s_{t-1}\) represents the information available at time \(t-1\) that influences the agents at time \(t\) according to the utility model \(\{u_{{1|2}s}(a_{1}|a_{2}, s_{t-1}) , u_{{2|1}s}(a_{2}|a_{1}, s_{t-1}) , u_{s}(s_{t-1})\}\). One of the consequences of expressing conditional utility with the syntax of probability theory is that the mathematical machinery that has been developed in the probability context can be imported into conditional game theory. In particular, the Markov chain convergence theorem may be applied to characterize this mindshaping. Let \(\tau\) denotes an iteration index which may be an interval but may also be viewed as an iterated calculation index that represents an adjustment process, or tâtonnement, as agents respond to their social environment. Suppose, before the social engagement begins (i.e., \(\tau = 0\)), that each agent attaches a conditional utility of the form \(u_{i|s}(a_{it} |s_{t-1};0)\). Analogous to the way beliefs are combined via the chain rule according to the syntax of probability theory to create a joint model of belief, we apply this syntax to combining preferences to create a joint model of preference, yielding

91\[ w_{ij|s}(a_{it},a_{jt}|s_{t-1}; \tau) = u_{i|js}(a_{it}|a_{jt},s_{t-1}) w_{j|s}(a_{j}|s_{t-1}; \tau)\]

93\[w_{i|s}(a_{it}|s_{t-1};\tau) = \sum_{a_{jt}} w_{ij|s}(a_{it},a_{jt}|s_{t-1}; \tau) = \sum_{a_{jt}}u_{i|js}(a_{it}|a_{jt},s_{t-1}) w_{j|s}(a_j|s_{t-1}; \tau-1)\, \]

94where it takes one unit of iteration time to transition from time \(\tau-1\) to \(\tau\). Notice that we do not sum over \(s_{t-1}\) since the sum of contributions at stage \(t-1\) is common knowledge at stage \(t\). Expressing this relationship using matrix theory notation yields

95\[\mathbf{w}_{i|s} (\tau) = T_{i|js} \mathbf{w}_{j|s}(\tau -1) \, ,\]

96where

97\[\mathbf{w}_{i|s}(\tau) = \begin{bmatrix}w_{i|s}(x_{1}|s;\tau) \\ w_{i|s}(x_{2}|s;\tau) \\ w_{i|s}(x_{3}|s;\tau)\end{bmatrix}\]

98and

99\[T_{i|js}= \begin{bmatrix}u_{i|js}(x_{1}|x_{1},s_{t-1})&u_{i|js}(x_{1}|x_{2},s_{t-1})&u_{i|js}(x_{1}|x_{3},s_{t-1}) \\u_{i|js}(x_{2}|x_{1},s_{t-1})&u_{i|js}(x_{2}|x_{2},s_{t-1})&u_{i|js}(x_{2}|x_{3},s_{t-1}) \\u_{i|js}(x_{3}|x_{1},s_{t-1})&u_{i|js}(x_{3}|x_{2},s_{t- 1})&u_{i|js}(x_{3}|x_{3},s_{t-1})\end{bmatrix}\]

100is the transition matrix for the linkage \(X_i \to X_j\) as conditioned by \(s_{t-1}\). Applying this expression iteratively yields

101\[\mathbf{w}_{i|s} (\tau) = T_{i|js} \mathbf{w}_{j|s}(\tau -1) = T_{i|js} T_{j|is}\mathbf{w}_{i|s}(\tau -2) = \cdots =(T_{i|js} T_{j|is})^{\tau}\mathbf{w}_{i|s}(0)\]

102or, more compactly,

103\[ \mathbf{w}_{i|s} (\tau) = T_{i|s}^{\tau}\mathbf{w}_{i|s}(0) \, ,\]

104where \(T_{i|s} = T_{i|js}T_{j|is}\) is the closed-loop transition matrix.

105The Markov chain convergence theorem establishes that, if \(T_{i|s}\) satisfies the regularity condition that all entries of \(T_{is}^m\) must be greater than zero for some integer \(m\), then

106\[\lim_{\tau \to \infty} \mathbf{w}_{i|s} (\tau) = \lim_{\tau \to \infty}T_{i|s}^{\tau}\mathbf{w}_{i|s}(0) = \bar{\mathbf{w}}_{i|s}\]

107for all initial conditions \(\mathbf{w}_{i|s}(0)\), where \(\bar{\mathbf{w}}_{i|s}\) is the eigenvector corresponding to the unique unit eigenvalue of \(T_{i|s}\). Thus, one does not need to actually perform the iteration; one can simply compute the eigenvector.

2.4 Four-Agent Conditional Public Goods Game

108We now apply the above development to a four-agent conditional public goods game employing the conditional utilities defined in (6), yielding the graph

where the center node is \(s_{t-1}\).The multidirectional influence paths greatly complicate the analysis of this game; thus we are motivated to develop a Markov equivalent graph of the form4 where we have exchanged a network graph with single-agent vertices and multidirectional edges for a network graph with multi-agent vertices and unidirectional edges.5 The issue now is to achieve Markov equivalence by defining edges that preserve the conditional relationships. Applying the chain rule, the transition from \(X_jX_kX_l \to X_iX_jX_k\) conditioned on \(s_{t-1}\) is, suppressing the stage index \(t\),

109\[w_{ijk|s}(a_{i},a_{j},a_{k}|s)=\sum_{a’_{j},a’_{k},a’_{l}}w_{ijk|j’k’l’s}(a_{i},a_{j},a_{k}|a’_{j},a’_{j},a’_{k},s) \,w_{j’k’l’|s}(a’_{j},a’_{k},a’_{l}|s)\, ,\]

110where primes distinguish between the conditioning conjectures \((a’_{j},a’_{j},a’_{k})\) and the conditioned conjectures \((a_{i}, a_{j},a_{k})\). Suppressing arguments and applying the chain rule,

111\[w_{ijk|j’k’l’s} = w_{k|ijj’k’l’s}\, w_{ij|j’k’l’s}\]

112where, again applying the chain rule

113\[w_{ij|j’k’l’ s} = w_{j|ij’k’l’ s}\, w_{i|j’k’l’ s}\,\]

114yields

115\[\tag{7}w_{ijk|j’k’l’ s} (a_{i}, a_{j},a_{k}| a’_{j}, a’_{k}, a’_{l},s) =w_{k|ijj’k’l’ s}(a_k|a_i,a_j,a’_j,a’_k,a’_l,s) \\ w_{j|ij’k’l’ s}(a_j|a_i,a’_j,a’_k,a’_l ,s) \, w_{i|j’k’l’ s}(a_i |a’_j,a’_k,a’_l,s)\, . \]

116However, The conditional functions \(w_{k|ijj’k’l’s}\) and \(w_{j|ij’k’l’s}\) involve self-conditioning, that is, the conjectured actions for \(X_j\) and \(X_k\) appear as both conditioning actions (marked by primes) and conditioned actions. Thus, these mass functions are degenerate; hence,

117\[w_{k|ijj’k’l’s}(a_k|a_i,a_j,a’_j,a’_k,a’_l,s) = \begin{cases} 1 & \text{ if } a_k = a’_k\\ 0 & \text{ if } a_k \neq a’_k\end{cases}\]

118\[w_{j|ij’k’l’s}(a_j|a_i,a’_j,a’_k,a’_l ,s) = \begin{cases} 1 & \text{ if } a_j = a’_j \\ 0 & \text{ if } a_j \neq a’_j\, , \end{cases}\]

119and (7) becomes

120\[w_{ijk|j’k’l’s} (a_{i}, a_{j}, a_{k}| a’_{j}, a’_{k}, a’_{l},s) =\begin{cases}w_{i|j’k’l’s}(a_{i}|a’_j, a’_{k}, a’_{l}, s) & \text{if } a_{j} = a’_j, a_{k} = a’_k \\0 & \text{otherwise.}\end{cases}\]

121Markov equivalence will thus be assured if

122\[w_{i|j’k’l’ s}(a_{i}|a’_j, a’_{k}, a’_{l}, s) =u_{i|j’k’l’ s}(a_{i}|a’_j, a’_{k}, a’_{l}, s)\, , \]

123and it follows that (suppressing the stage index \(t\)),

124\[w_{ijk|j’k’l’s} (a_{i}, a_{j},a_{k}| a’_{j}, a’_{k}, a’_{l}, s) =\begin{cases}u_{i|j’k’l’s}(a_{i}|a’_j, a’_{k}, a’_{l}, s) & \text{if } a_{j} = a’_j, a_{k} = a’_k \\0 & \text{otherwise}\end{cases}\]

125for \(i|jkl \in \{1|234, 2|341, 3|412,4|123\}\).

126Given these conditional linkages between the subgroups, we may close the loop by defining the subgroup-to-subgroup conditional transition matrices \(T_{ijk|jkls}\) connecting subgroup vertex \(X_jX_kX_l\) to \(X_iX_jX_k\) given \(s_{t- 1}\), yielding

127\[\mathbf{w}_{ijk|s} = T_{ijk|jkls} \mathbf{w}_{jkl|s}\, , \]

128where

129\[\mathbf{w}_{ijk|s} = \begin{bmatrix}w_{ijk|s} (x_{ 1} , x_{ 1} , x_{ 1} |s)\\w_{ijk|s} (x_{ 1} , x_{ 1} , x_{ 2}|s)\\w_{ijk|s} (x_{ 1} , x_{ 1} , x_{ 3}|s)\\\vdots\\w_{ijk|s} (x_{ 1}, x_{ 3} , x_{ 3}|s)\\w_{ijk|s} (x_{ 2}, x_{ 3} , x_{ 3} |s)\\w_{ijk|s} (x_{ 3} , x_{ 3} , x_{ 3} |s)\\\end{bmatrix}\]

130\[\mathbf{w}_{jkl|s} = \begin{bmatrix}w_{jkl|s} (x_{ 1} , x_{ 1} , x_{ 1} |s)\\w_{jkl|s} (x_{ 1} , x_{ 1} , x_{ 2} |s)\\w_{jkl|s} (x_{ 1} , x_{ 1} , x_{ 3}| s)\\\vdots \\w_{jkl|s} (x_{ 1} , x_{ 3} , x_{ 3} |s)\\w_{jkl|s} (x_{ 2} , x_{ 3}, x_{ 3} |s)\\w_{jil|s} (x_{ 3} , x_{ 3} , x_{ 3} |s)\\\end{bmatrix}\]

131and

132\[T_{ ijk|jkls}=\]

133\[\begin{bmatrix}w_{ijk|jkls}[x_{ 1},x_{ 1},x_{ 1} |x_{ 1},x_{ 1},x_{ 1},s] & \cdots& w_{ijk|jkls}[x_{ 1},x_{ 1},x_{ 1} |x_{ 3},x_{ 3},x_{ 3},s] \\w_{ijk|jkls}[x_{ 1},x_{ 1},x_{ 2} |x_{ 1},x_{ 1},x_{ 1},s] & \cdots& w_{ijk|jkls}[x_{ 1},x_{ 1},x_{ 2} |x_{ 3},x_{ 3},x_{ 3},s] \\w_{ijk|jkls}[x_{ 1},x_{ 1},x_{ 3} |x_{ 1},x_{ 1},x_{ 1},s] & \cdots& w_{ijk|jkls}[x_{ 1},x_{ 1},x_{ 3} |x_{ 3},x_{ 3},x_{ 3},s] \\\vdots & & \vdots\\w_{ijk|jkls}[x_{ 1},x_{ 3},x_{ 3} |x_{ 1},x_{ 1},x_{ 1},s] & \cdots& w_{ijk|jkls}[x_{ 1},x_{ 3},x_{ 3} |x_{ 3},x_{ 3},x_{ 3},s] \\w_{ijk|jkls}[x_{ 2},x_{ 3},x_{ 3} |x_{ 1},x_{ 1},x_{ 1},s] & \cdots& w_{ijk|jkls}[x_{ 2},x_{ 3},x_{ 3} |x_{ 3},x_{ 3},x_{ 3},s] \\w_{ijk|jkls}[x_{ 3},x_{ 3},x_{ 3} |x_{ 1},x_{ 1},x_{ 1},s] & \cdots& w_{ijk|jkls}[x_{ 3},x_{ 3},x_{ 3} |x_{ 3},x_{ 3},x_{ 3},s] \\\end{bmatrix}\]

134The closed-loop transition matrices are computed as

135\[T_{ijk|s} = T_{ijk|jkls} T_{jkl|klis}T_{kli|lijs}T_{lij|ijks}\, . \]

136Once these closed-loop transition matrices are defined, we apply the Markov chain convergence theorem to compute the converged functions \(\bar{\mathbf{w}}_{ijk|s}\). These converged coordination vectors are used to compute the individual converged utilities via

137\[\bar{\mathbf{w}}_{i|s} = \begin{bmatrix} \bar{\mathbf{w}}_{i|s}(x_{ 1})\\\bar{\mathbf{w}}_{i|s}(x_{ 2})\\ \bar{\mathbf{w}}_{i|s}(x_{ 3})\end{bmatrix}= T_{ i|jkls}\bar{\mathbf{w}}_{jkl|s}\, , \]

138where

139\[T_{i|jkls}= \begin{bmatrix}u_{i|jils}(x_{ 1}|x_{ 1},x_{ 1},x_{ 1},s ) & \cdots& u_{i|jkls}(x_{ 1} |x_{ 3},x_{ 3},x_{ 3},s) \\u_{i|jils}(x_{ 2}|x_{ 1},x_{ 1},x_{ 1},s ) & \cdots& u_{i|jkls}(x_{ 2} |x_{ 3},x_{ 3},x_{ 3},s) \\u_{i|jils}(x_{ 3}|x_{ 1},x_{ 1},x_{ 1},s ) & \cdots& u_{i|jkls}(x_{ 3} |x_{ 3},x_{ 3},x_{ 3},s) \\\end{bmatrix} . \]

140The converged cyclic network is

141where the links defined by the symbol ~> indicate that the network has reached steady-state once convergence is achieved.

2.5 Simulation Design

142The simulation consists of two simulation experiments to compare two populations, both playing the repeated PG as defined in KV (2016). Each population comprises four agents \(\{X_1, X_2, X_3, X_4\}\) engaging in \(K\) trials, with each trial comprising \(L\) stages. Population A plays the straight KV PG, and Population B plays the same game except for intermittent mindshaping episodes. This is a modeling convenience. We suppose that in reality mindshaping would occur continuously and incrementally. However, such continuous preference modulation cannot be directly represented consistently with standard game theory. We in effect probe the effects of background preference diffusion using comparative statics. The variable of interest is the frequency of norm-compliant choices between population A and population B.

143Population A

144The members of population A are all independently initialized at stage \(t=0\) according to the distribution defined by Table 2, yielding initial random choices \((a_{ 10},a_{ 20},a_{ 30}, a_{ 40})\). The payoffs for stages \(t>0\) are given by (1), with \(\alpha = 1/2\), \(\eta = \$25\), and \(g\) defined by (2). The threshold sensitivity level is \(\phi^*\) as defined by (3), yielding \(\phi^* = 0.662\). The sensitivity parameters \(\phi_i\) for each trial are chosen independently according to the uniform distribution \(U[\phi^* -\epsilon, \ \phi^*+\epsilon] \). The optimal solutions for \(t>0\) are defined by (4). Thus, if \(\phi_i > \phi^*\) for all agents and \(s_{t- 1} =\eta\), then they all play the norm for all subsequent stages. However, if \(\phi_i < \phi^*\) for any \(X_i\), then she plays to maximize her payoff, which causes all agents to play selfishly for all subsequent stages. This is the critical feature that is explored via population B. Namely, if an agent comes to the social engagement with a low norm-sensitivity, she may be susceptible to the influence of others, and there may be reachable equilibria in which she is induced to adjust her norm sensitivity accordingly.

145Population B

146The members of population B are initialized with a mindshaping exercise using the norm-compliant utility defined by (5). The distinctive feature of this utility is that, rather than being exogenously set ex ante, it is susceptible to modulation as a function of both the previous states \(s_{t - 1}\) and \(\sigma_{-it}\), the conditional conjectures of \(X_{-i}\). In fact, we may view the product \(\phi_i \rho(s_{t-1})\) in (1) as a coarse modulation of \(\phi\) as a function of \(s_{t-1}\)—an on-off switch. Although we model \(\phi_i\) as a function of both \(\sigma_{\it}\) and \(s_{t- 1}\), in this section we focus on conditionalization via \(\sigma_{-it}\). Given her conjecture that \(X_{-i}\) will follow the norm, if she were inclined to be a contributing citizen of the community, she would have an incentive to also follow the norm rather than risk the disapproval of her fellow citizens. This would motivate her to increase her norm sensitivity. On the other hand, if she were not normatively compatible with \(X_{-i}\), she would not be so inclined.

147Our simulation model for population B proceeds as follows. Each agent is initialized with CGT using the norm-compliant payoff function defined by (5). In order to make comparisons with the basic KV model, we modulate the \(\phi_i\) as used by KV by multiplying it by \(\gamma(a_{it})\) to create the modulated parameter.

148\[\tag{8}\hat{\phi}_i(a_{it},s_{t-1})=\begin{cases}\gamma(a_{it})\,\phi_i & \text{ if } \sigma_{-it} \in \mathcal{S}_{\eta}\\\phi_i & \text{ otherwise,} \end{cases}\]

149with \(\gamma(a_{it}) = \gamma_{q}\geq 0\) if \(a_{it} = x_{q}\), \(q\in \{1,2,3\}\), where \(\gamma_{ 2}\) increases norm-compliance sensitivity by conjecturing increased norm sensitivity if \(X_{-i}\) conjectures norm-compliance and \(\gamma_{ 1}\) and \(\gamma_{ 3}\) reduce norm-compliance sensitivity if \(X_{-i}\) were to reject norm-compliance.

150Our simulation for Population B uses the outcome from the CGT first stage for subsequent stages using the basic KV model, which are interrupted at random stages by inserting the CGT model as a mindshaping reinforcement. The simulation consists of \(K= 100\) trials with repeated games of length \(L =50\) stages. Table 4 displays the number of instances of norm-compliance for both populations for several values of \(\gamma_i\).

Table 4. Simulation Results

\((\gamma_{1},\gamma_{2}, \gamma_{3})\)	Population A	Population B
(0.5, 2.5, 0.5)	194	392
(0.2, 2.5, 0.2)	185	2598
(0.7, 3.0, 0.7)	124	1568
(1.0, 3.0, 1.0)	105	147
(0.5, 3.0, 0.5)	126	4653
(0.5, 2.5, 0.2)	164	2303
(0.2, 2.5, 0.5)	152	368

151Our simulations establish that mindshaping via CGT does indeed increase the frequency of norm-compliance. Figure 1 displays simulation results as a function of \(\gamma_{2}\) for various combinations of \((\gamma_{1}, \gamma_{3})\). These plots demonstrate considerable mindshaping effectiveness sensitivity to \(\gamma_{2}\), with significantly less sensitivity to \(\gamma_{1}\), resulting in three clusters of graphs for \(\gamma_{2} \in \{0.2, 0.5, 0.7\}\). Mindshhaping effectiveness approaches zero as \(\gamma_2\) declines toward \(2.2\) for any values of the other parameters. Figure 2 displays norm-compliance frequency with \( (\gamma_{1}, \gamma_{2})\) fixed at \((0.5,2.3)\) as \(\gamma_{ 3}\) increases from zero, which indicates that mindshaping becomes essentially ineffective for \(\gamma_{ 3} > 0.15\). We did not attempt a specification search for the equation governing the parameters relationships due to the arbitrary nature of the parameters restrictions; such theory would serve no generalising purpose. The point simply is to show that for some parameter values mindshaping makes a dramatic difference.

152The threshold \(\gamma_{ 2} \phi^*\) is significantly higher than the threshold \(\phi^* =0.662\) for the basic KV game due to the interrelationships that exist among the agents. In population B, \(X_{-i}\) exerts direct social influence on \(X_i\), in contrast to the indirect social influence that the history from stage \(t-1\) exerts at stage \(t\) in population A. Thus, although setting \(\phi_i> \phi^*\) will certainly cause the payoff defined by (6) to favor \(x_{ 2}\) over \(x_{ 3}\) viewed in isolation from all other agents, when modeled as a part of the community where members are subjected to direct social influence, a more substantial increase in \(\phi_i\) is required for \(x_{ 2}\) to be favored over \(x_{ 3}\)—see (5) and (8).

153These results establish that as the multiplier increases for \(x_{ 2}\) and decreases for \(x_{1}\) and \(x_{ 3}\), the frequency of norm-compliance increases. It is important to appreciate the distinction between the PG as defined by the basic KV model and the CGT-based PG. With the basic model, although each agent is influenced by the history of past actions, there are no explicitly defined social relationships among them. In particular, their norm sensitivities are held constant over all of their possible actions. By contrast, the CGT model allows agents to directly modulate their norm sensitivities as a function of their possible actions. In fact, as modeled in the simulations, the two models conditionalize on different criteria. The KV model expresses norm sensitivity conditions on the past behavior via \(\rho_i\), but the CGT model expresses norm sensitivity by conditioning on current conjectures as functions of explicitly defined social relationships. Although the CGT approach certainly permits conditioning on the history, we have deliberately avoided doing so in order to emphasize conditioning via direct social influence.

3. Discussion

154In this section we sketch the wider theoretical framework that the modeling and simulation exercise of the previous two sections is intended to exemplify. We emphasise that this is a sketch: substantial technical work necessary to develop it into theory lies ahead.

155We understand population A in the simulation as a community in which the only strategic interactions occur between agents with fixed utility functions. Population B illustrates a community in which interactions occur under the shadow of a background process of social normative pressure that alters utility functions through mindshaping. Such processes do not necessarily tend toward homogeneity; two pairs of individuals that begin with conflicting preferences from one another might, in developing closer within-pair preference structures, become less aligned between pairs. Mindshaping can thus promote polarisation, or balkanisation in parts of networks, in addition to normative convergence in other parts. Ross, Stirling and Tummolini (2023) simulate such cases, along with cases of preference falsification following Kuran (1995), in a setting where outcomes differ from one another only in relative risk. Which of these general kinds of social situations mindshaping produces depends on the structure of the network, which our work to date demonstrates but for which we have yet to develop general theory. We conjecture that in relatively simple networks such as the one simulated above, in which individuals mainly play games in which Pareto-dominant coordinated equilibria are available, mindshaping will tend to accelerate the emergence of efficiency, as in the simulation.

156One potential application of CGT modeling of mindshaping can occur in the experimental lab when subjects encounter novel situations to which pre-established norms do not apply. Experimenters in such situations often run pre-play phases to help agents learn about the game (Fudenberg and Levine, 1998). Under a well-explored range of conditions, such learning allows Bayesian agents to identify correlated equilibria (Aumann, 1974; 1987). Ross and Stirling (2023) show that mindshaping as modeled by CGT is among these conditions.

157Our application of CGT and mindshaping to the KV experiment, however, illustrates a more novel and ambitious modeling vision for relating long-run and short-run strategic processes. The idea envisages two kinds of models, at different scales of representation, that constrain one another through shared use of non-refined non-cooperative game theory. By “non-refined” we refer to game theory that aims to describe patterns of strategic behavior rather than prescribe it according to idealisation of stringently rational choice. Of course, game theory cannot be applied to entirely irrational or arational behavior. Agents’ choices must be sensitive to changes in incentives that can be identified using utility functions. Over interesting explanatory domains of interactions, models must predict, and risk refutation by failure of, stochastic dominance. Equilibria no weaker in the constraints they impose on identification than quantal response equilibrium must be supported by analysis. Taken together, these constraints amount, in philosophical terms, to the idea that most accurate and comprehensive explanation and generalisation of agents’ behavior requires adoption by the modeler of the intentional stance (Dennett, 1987; Ross, 2005; 2014).

159Among models of the effects of social norms, both the general and special models of Kimbrough and Vostroknutov (2016 and 2020a, respectively) count as conservative according to the above restrictions. This also applies to many models in the social-preferences tradition, notwithstanding our endorsement of Binmore (2016)’s complaint that this approach eschews generality and therefore poorly serves the goal of accumulation of knowledge in theory. In holding utility functions fixed, these models are best interpreted as describing strategic interactions of agents for which the distribution in a population of preferences, on which norms supervene, are exogenous. KV, as noted, do not aspire to describe processes by which norms originate and change. In our setting, we interpret this as reflecting the idea that what turns some agents’ preferences into norms are diffusion (mindshaping) processes of the kind illustrated in our CGT simulation.

161Adaptive preferences have been presented by various authors following Sen (1992) as challenges to standard welfare theory (see also Elster, 1983). Part of the problem is mere intertemporal inconsistency. Where this is not a problem, perhaps because time-scales are short, we see no motivation to supplement the tool-kit we called “conservative”. But a more substantive basis for concern is that individual preference adaptation as an empirical phenomenon often involves less powerful agents accommodating the objectives of more powerful ones. In a tradition in which the centrality of welfare as a proxy for well-being is based on emphasizing consumer sovereignty, and that in turn rests on aversion to paternalism by a morally non-neutral state, preference adaptation that results from power imbalance strikes many analysts as sinister.

162With respect to these normative issues, we note that our simulation of the PG in a CGT setting is a reminder of the morally ambiguous nature of mindshaping. It seems to us to be an obvious historical fact that large-scale normative change has very frequently reflected cultural imperialism and coercive homogenisation. On the other hand, to the extent that such homogenisation expands the moral circle, in the sense of Singer (1981), by promoting the spread of normative sensitivity across sub-networks, it is likely the essential basis for values that aspire to universal scope. In the more modest context of a PG, as our simulation demonstrates, it can accelerate and stabilise cooperative choice. We furthermore suggest that by directly representing the strategic element of social preference adaptation, CGT offers a technique for correcting some accounts of political and cultural dynamics that neglect the agency of the oppressed.

163Our modeling exercise leaves the general, formal integration of the macrosociological and microeconomic dynamics of norms as a pending project. We argue, however, that it displays a promising technical tool for the project, and begins to specify its conditions. We close the discussion with a sketch of the technical agenda we have in mind.

164Our template for the proposed relationship between CGT and standard game theory is the relationship between cooperative and non-cooperative game theory as conceived in the Nash program. Binmore (1998) provides a rich informal discussion. Since we are here outlining a proposed methodology rather than yet building real theory, this informal exposition is the appropriate reference point.

165Nash (1950; 1951; 1953) identified the value of a practical strategy, for use in applied work, that could exploit the strengths of both cooperative and non-cooperative game theory while avoiding their respective weaknesses. Cooperative game theory can identify highly general solutions to bargaining games that are robust to many changes in details about players’ utility functions and strategy sets. However, none of the various cooperative solution concepts that have been studied have been found to be robust across all bargaining games. By contrast, solution sets for non-cooperative games can be identified with confidence, and it is typically clear from a game’s structure which solution concept is most appropriate for application. One might therefore imagine that when God applies game theory to the strategic affairs of her creatures she doesn’t bother with cooperative models: she writes down the full non-cooperative extensive-form of the bargaining game among the individuals who are conjectured into coalitions by the less powerful minds of mortal theorists. She then computes the equilibria of the game that folds both the “pre-play” and “play” stages into a single extensive-form model with stages. Unfortunately, theorists attempting to construct applied models of real-world situations without unbounded cognitive resources typically can have little confidence that they have reliably sorted relevant from irrelevant differences among players and strategy sets when they try to reconstruct the pre-play cooperative game in non-cooperative extensive-form. Such sorting is essential to avoid combinatorial explosions of branches in game trees.

166Nash’s method for coping with this problem—the Nash program—is to use cooperative analysis as the basic tool for representing bargaining games, but then to use non-cooperative analysis to test proposed cooperative solutions. The straightforward idea is that the modeler should be obliged, after identifying a cooperative solution, to find a plausible non-cooperative model for which the solution in question is an equilibrium in extensive-form. Nash illustrated the strategy in defending his solution to the general problem of bargaining over the distribution of a cooperatively generated surplus.

167We suggest a similar approach to reconciling macrostructural and microeconomic analysis as we have characterized them here. This is the approach we illustrated in the second section as applied to KV’s PG.

168Consider first a theorist whose primary interest is in the dynamics of normative stability and change in a society or sub-society. She can begin by building a conditional game to describe the network of relationships among types of agents that she hypothesises on empirical grounds. But actual interactions involve agents with situation-specific parameters in their utility functions. To ensure that she will be able to compare her model with results of this more granular and particular modeling, our social theorist should avoid various restrictions that would limit generality. For example, she should not impose homogenous risk attitudes on her population or assume that all agents are expected-utility maximisers. This she can do by assigning them response functions with flexible (e.g., Prelec) decision weights, following the approach in Ross, Stirling, and Tummolini (2023). But there is one key restriction the conservative game theorist makes that she drops: she allows agents’ utility functions (and their decision weights on risky prospects) to be influenced through conditionalisation and marginalisation as Markov processes.

169Though the empirical process that is the social theorist’s explanatory target is dynamical, her CGT model is analysed as comparative statics. Use of CGT allows her to restrict her solution set to concepts that are as narrow as her favoured philosophy of game theory recommends; we would urge that she follows the advice of Binmore (2007) in making this choice. The structural analyst can then test whether the social normative equilibrium she has identified is stable across standard game-theoretic specifications. By this we mean that the equilibrium is reachable in an extensive-form unconditional model of her game, played by agents with utility functions “frozen” in the underlying social dynamics.

170The importance of explicitly attending to norms in a unified social analysis emerges at this methodological juncture. The point of the CGT modeling is to identify them. If the extensive-form unconditional games against which they are tested does not allow for norm-responsiveness in players’ utility functions, then there is no reason to expect that the conditional and unconditional solutions should align. Thus the unconditional game used in the test that is analogous to that recommended by the Nash program should be defined using utility functions such as those developed by KV (2020a).

171Now let us examine this methodological program from the microeconomist’s point of view. Her aim is to predict and explain outcomes of interactions among individual agents who are identified by their utility functions. If these agents are people or human institutions (or even, perhaps, elephants or ravens), their assessments of outcomes will be (heterogeneously) influenced by prevailing social norms. These norms are exogenous to the microeconomist’s model: her agents did not choose them. She might try to account for them by conjecturing and experimenting with various social-preferences models. But this is likely to be extremely inefficient, and in any event is likely to generate over-fitted models even if she finds a social-preferences conjecture that happens to “work”, because her agents’ utility functions will be more restricted than those that characterise the “frozen” output of the corresponding CGT analysis. In the KV setting, what matters is not which precise social states different players idiosyncratically prefer; rather, what she seeks to take into account is the distribution of responsiveness among her agents to whichever norms prevail and must be factored into their empirical expectations about the strategies they will encounter.

172In the PG example we simulated, under conditions where enough economic agents are sufficiently sensitive to a norm of cooperation to find equilibria, mindshaping processes can accelerate the extent of entrenchment of the norm over time. However, the threshold frequency of norm-sensitivity sufficient for increasing the frequency of cooperative play is substantially higher than the threshold required for a stationary non-zero distribution of cooperative play among agents without mindshaping. This result is intuitively appealing, suggesting a general micro-mechanism for the kinds of cultural ratchet effects that promote the spread of political participation, rule of law, and more globalised markets described by theorists of the historical spread of democratic capitalism, and of contemporary countries’ relative levels and rates of success in what Fukuyama (2014) calls “getting to Denmark” (Lal, 1998; Grief, 2006; Acemoglu and Robinson, 2012; Henrich, 2020).

173Mindshaping also allows for representation of processes of a kind that standard game-theoretic models do not aim to explain. Preference falsification or pluralistic ignorance about norms are sustainable as equilibria among agents with fixed preferences (Smerdon et al., 2020). However, the fixation of preferences blocks endogenisation of the kind of process described by Kuran (1995) in which preference falsification ends not with general discovery of social error, but with adaptation of preferences to comply with observed choices.

174As discussed in the second section, this has somewhat vertiginous implications for welfare analysis. General discovery of preference falsification should tend to increase welfare, by bringing both individual behaviour and policy into closer alignment with preferences. Elimination of preference falsification through preference adaptation should also tend to promote welfare improvement, but by changing preferences rather than behaviour or policy. This does not necessarily carry societies in the direction away from Denmark—think of a society with an initial majority of private racists who over time adjust their preferences to conform with public normative shaming—but we see no prima facie basis for general optimism. For example, the Chinese government might well have succeeded over the past couple of decades in creating more sincerely militant nationalism as a means of shoring up the legitimacy of its authoritarian institutions. Theorists who insist on subjective preference satisfaction as the touchstone for welfare assessment might have to acknowledge that the recent Chinese repression of Hong Kong’s autonomy may thus have improved Chinese social welfare, as measured against actual preferences in the general population. We do not intend this comment as a kind of passive-aggressive attack on welfarism. Perhaps it is a reminder of the virtues of welfare criteria that are often criticised as being too conservative. No matter how many nationalistic Chinese people have enjoyed better satisfied preferences through the assault on rule of law in Hong Kong, it is clearly no Pareto improvement, though it might satisfy Kaldor-Scitovsky criteria. Recent government behaviour in Xinjiang can constitute neither sort of welfare improvement, as the losers cannot be compensated by any feasible lump-sum transfer.

175A final discussion issue involves conceptual sanitation. We earlier suggested that Ross’s (2006; 2008) overlapping-generations model of what he calls “game determination” should be displaced by the more encompassing strategy adopted here. However, the general phenomenon of strategically influenced social-psychological change that Ross (2005) describes informally but more richly remains the explanatory target. We therefore adopt the following semantic policy going forward: mindshaping is the basic mechanism for game determination, of which CGT is a more general formal theory than Ross’s model.

Conclusion

176Constructing a theory of social norms is an ambitious, developing enterprise that will necessarily be the work of many hands from multiple disciplines. We have sketched a methodology that aims to facilitate that interdisciplinarity without sacrificing the accumulation of analytic clarity that the traditional division of labour in the social sciences has fostered. We follow the advice of Gintis (2009) in using game theory for social-scientific unification. But we also follow Binmore (1994; 1998; 2005) in retaining the straight and narrow path of standard microeconomic theory in modeling shorter-run games among socialised individuals. Modeling longer-run dynamics is a task for social theory more generally. Then the crucial trick for the promoter of unification to turn is finding a technical meta-language in which mutual constraints from models at different scales of abstraction can be stated. We proposed conditional game theory as furnishing a possible such meta-language.

177Norms are produced by mindshaping processes that are the critical mechanism explaining human ecological dominance (Henrich, 2015). These norms are exogenous features of the contexts in which individual economic agents encounter one another as traders and bargainers. We see these as the basis for fundamental axioms of a future theory of norms.

The authors thank the editor and three anonymous reviewers for comments that significantly improved the article. Luca Tummolini acknowledges funding by PRIN2022 “Understanding What is Mine” grant n°2022LYRT8E and PRIN PNRR 2022 “Normativity of Joint Action” grant n°P2022YYRK3.