axiomatic models of injunctive norms and moral rules

Axiomatic Models of Injunctive Norms and Moral Rules∗

Erik O. Kimbrough† Alexander Vostroknutov‡§

April 2021

Abstract

We propose a framework for modeling the in�uence of normative considerations on decision-making and social interaction. Our key insight is that axiomatic models like those used in cooper-ative game theory can be reinterpreted as theories of moral reasoning and norm formation. Ratherthan providing evaluative criteria for external observers, such models can instead be understood asdescribing the principles agents use to reason about the normative properties of a set of feasible out-comes in a choice set or game. Axioms can be chosen to represent widely-known moral theoriesand then used to generate a ranking of feasible outcomes from most to least normatively appealingunder a given theory. We illustrate with an example rooted in moral psychology which can accountfor extensive evidence of context-dependence in normative decision-making. We then distinguishtheories that endogenize the ideal from those that do not, and we explore when and where the judg-ments from such models will cohere. In a companion paper (Kimbrough and Vostroknutov, 2021), weshow how to apply such models using a simple utility speci�cation in which agents care both aboutconsumption and normative considerations and trade these o� when they con�ict.

JEL classi�cations: C91, C92Keywords: norms, axiomatic preferences, rules.

∗We would like to thank Elias Tsakas for help with formulating the axioms and Valerio Capraro, Ma�hias Stefan, and aseminar audiences at Maastricht University and the University of Konstanz for helpful comments. All mistakes are our own.

†Smith Institute for Political Economy and Philosophy, Chapman University, One University Drive, Orange, CA 92866,USA. email: [email protected].

‡Department of Economics (MPE), Maastricht University, Tongersestraat 53, 6211LM Maastricht, �e Netherlands. e-mail:[email protected].

§Corresponding author.

1 Introduction

In economics, the normative and the positive are traditionally treated separately. �is is true for neo-classical theory as well as game theory. Within the la�er, one way of thinking about the distinctionbetween cooperative concepts (esp. axiomatic bargaining theory) and non-cooperative ones is that theformer provide tools for formalizing particular notions of “the good” and “the bad” which can be used toevaluate the outcomes produced by the (amoral) interactions analyzed by the la�er. �is separation ofspheres, while tidy, overlooks extensive evidence that normative considerations are not merely evalua-tive but also enter into peoples’ decisions as goals to be aimed at, however imperfectly.

We contend that it is possible to improve our understanding of social decision-making by deepeningthe integration of cooperative and non-cooperative theory. In this paper, we show how the axiomaticapproach, typical for cooperative game theory, can be used to model how people moralize, with di�erentaxioms implying di�erent normative rankings of feasible outcomes (herea�er norms). �en, in a com-panion paper, by using these rankings as an input to simple models of “norm-dependent preferences,”in which decisions re�ect trade-o�s between normative goals and own utility maximization, we show anew way to introduce normative goals into models of choice (see Kimbrough and Vostroknutov, 2021).

Existing theories in which normative considerations enter into decision-making o�en build a particu-lar moral goal directly into the utility function (e.g., inequality aversion, e�ciency-seeking, maximin) orintroduce particular norms as purely evaluative criteria (as in models of image motivation, e.g. Benabouand Tirole (2011), or psychological game theory, e.g. Ba�igalli et al. (2019a)). While such models haveclear domains of application, they treat normative judgments largely as given without a�empting to ac-count for the way in which norms and morals emerge from the psychology, preferences, and constraintsof individuals.

One consequence of this is that existing models tend to be unable to account for the rich—evenradical—sensitivity of decisions to context. A large body of work in behavioral economics reveals spec-tacular diversity in the criteria that apparently guide human social behavior. Depending on the choicecontext, existing theory and evidence suggest that people are variously motivated by Pareto improve-ments, e�ciency, equality, maximin, reciprocity, guilt aversion, lying aversion, anger, and so on.1 More-over, experimental evidence suggests that simple changes to the choice set are su�cient to cause peopleto switch from privileging one criterion to another (e.g., List, 2007; Engelmann and Strobel, 2004; Gale-o�i et al., 2018). What determines the mapping between context and the choice criterion? A promisingapproach to a unifying explanation argues that apparent context-dependence in motivation re�ects ad-herence to context-dependent injunctive social norms (Lopez-Perez, 2008; Cappelen et al., 2007; Kesslerand Leider, 2012; Krupka and Weber, 2013; Kimbrough and Vostroknutov, 2016), but the literature so farhas failed to provide a coherent account of why and how social norms vary across contexts.

In our view, axiomatic models drawn from cooperative game theory can be (re)interpreted as modelsof moral reasoning and norm formation and used to provide a foundation for the kinds of norms that are

1For example, Charness and Rabin (2002); Fehr and Schmidt (1999); Dufwenberg and Kirchsteiger (2004); Engelmann andStrobel (2004); Ba�igalli and Dufwenberg (2007); Abeler et al. (2019); Ba�igalli et al. (2019b).

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taken for granted in existing models, while also providing an account of why those norms are context-speci�c and why other norms apply in other se�ings. Starting from axioms on moral psychology whichyield individual normative judgments and a set of assumptions on how those judgments are aggregatedinto shared injunctive norms, we show how to take the objects typically analyzed in non-cooperativegame theory and construct normative evaluations thereof that depend on the entire set of feasible al-locations and the preferences of interested parties. A particular set of axioms will imply a particularnormative ranking of feasible outcomes, and the axioms can be chosen to represent widely shared the-ories of moral psychology. �e introduction of psychological elements produces sensitivity to context:when the set of feasible allocations changes, so can the nature of the normative ranking generated by aparticular set of axioms. �us, context-dependent norms can emerge naturally from a theory of moralpsychology.

We illustrate the potential of this method by building a particular model of normative evaluationdesigned to capture some basic aspects of morality. We begin with the assumption that individual nor-mative judgments are fundamentally comparative rather than objective. �at is, at any outcome, peopleare dissatis�ed with what they have because of what they might have had instead. In other words, eachperson’s normative evaluation of each outcome depends directly on how it compares to all other possi-ble outcomes.2 An outcome is more dissatisfying the more numerous (and be�er) are the other availableoptions that would be preferable to it. We then model aggregation of individual judgments into a sharednorm via another basic human psychological ability: the capacity for empathy, or the ability to imaginehow others would feel in similar situations. We assume that all people feel dissatisfaction of the kinddescribed above and that everyone knows that this is true of everyone else. We then assume that normsemerge that account for the judgments of everyone involved, with the most appropriate outcome beingthe one that minimizes the aggregated dissatisfaction of interested parties. �us, the resulting injunctivenorms are context-dependent in the sense that they depend critically on all feasible allocations.

In the companion paper (Kimbrough and Vostroknutov, 2021), we show that a functional represen-tation obtained from our axioms o�ers substantial explanatory power across a wide set of experimentalenvironments in which behavior is known to be context-dependent, with changes in behavior acrosscontexts tracking the predicted changes in injunctive norms. Introducing choice-set-dependence thusallows us to account for a number of otherwise puzzling observations, but it comes with a cost: as thecomplexity of the choice se�ing grows, normative evaluation involves increasingly strong assumptionsabout one’s knowledge of others’ preferences and increasing computational complexity. While choice-set-dependent norms of the kind modeled here seem to track how people’s normative intuitions and theirbehavior vary with context in laboratory se�ings, for practical purposes (e.g., in organizations, in thelaw) such norms are di�cult to identify, specify, and articulate.

2�is implies that normative rankings will rarely respect the IIA assumption, which is commonly made in axiomaticmodels of choice, but is commonly violated by human subjects. �e study by Cox et al. (2018), which acknowledges thisfact, represents an a�empt to preserve some form of IIA by introducing moral reference points and declaring that “irrelevantalternatives” are those that do not change these reference points and consequently the revealed choice. �is axiomatic modelis close in spirit to our model of “moral rules” below.

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�is complexity may help explain the appeal, both in theory and in practice, of “moral rules” whichsimplify moral reasoning by reference to abstract ideals. �us we show how to build a link between ourproposed framework and existing models that codify particular moral rules. Our approach here is tointroduce alternative axioms which specify an ideal reference outcome and then compute dissatisfactionrelative to that ideal, rather than relative to the full set of feasible outcomes.

Normative rankings of outcomes, when derived from a model of this kind, can usually be summa-rized via general rules like those mentioned above (e.g., maximin, payo� e�ciency). However, modelsthat include an ideal reference outcome sacri�ce the radical context-dependence that comes from thepsychological bo�om-up construction of injunctive norms. �ey are, thus, necessarily unable to explainsome observations in which di�erent normative principles seem to drive choices in di�erent circum-stances. We prove that the normative evaluation of outcomes implied by any such set of axioms cannotmatch those generated by our choice-set-dependent axioms. �e implication is that normative reasoningof the kind modeled by our preferred axioms cannot always be summarized by simple, general principles.

�at said, there are likely to be certain contexts in which the norms generated by choice-set-dependentaxioms do in fact correspond to a particular moral rule. We suggest that this correspondence (or lackthereof) provides a criterion by which to identify the kinds of moral rules that are likely (or unlikely)to emerge in a given context. �us, we a�empt to identify the kinds of se�ings in which a context-dependent model generates a normative ranking that corresponds to that implied by a particular moralrule. By imposing various restrictions on the set of possible outcomes (e.g., random payo� vectors,constant-sum payo� vectors, social dilemmas, coordination games, etc.) we can ask which moral ruleis most consistent with the norms implied by our choice-set-dependent axioms and thereby predict thekinds of rules people would be expected to articulate if explaining appropriate behavior to one another inse�ings of that kind. �at is, our framework provides a meta-theory of moral rules, along with an accountof their origins—as approximations of the norms that emerge from our treatment of moral psychology—and a procedure by which we can identify which moral rules are likely to emerge in which contexts.

2 Models of Moral Reasoning

Axiomatic models of bargaining and cooperative games o�en start from a set of normatively appealingaxioms (e.g., individual rationality, symmetry, IIA) that the “fair bargain” (Nash, 1950)—or the alloca-tion considered the most socially appropriate by all players—should satisfy. Nowadays, such models aretypically seen as existing outside or above the subject ma�er of non-cooperative game theory, and sowhen such models come into contact with strategic interactions, their role is typically comparative orevaluative. �at is, we tend to think of the axioms as providing the standpoint from which we (modelers)can criticize the outcomes or allocations that arise in non-cooperative models or in real bargaining sce-narios.3 We might argue that the “fair bargains” reached in some class of non-cooperative games violate

3�e literature on the design of mechanisms to implement various cooperative solutions is probably the clearest illustrationof this point since those models take the cooperative solutions as normative and ask how a designer can ensure they are

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IIA or generate asymmetric results in symmetric problems, and are therefore normatively undesirable.While we think this is a valuable use of axiomatic models, we argue that their role can be augmented bya shi� in perspective. If we recognize them as models of moral reasoning and norm formation, then thereis no reason that they need to remain outside or above the concerns of non-cooperative game theory;instead, we can move the evaluative locus of axiomatic theory from the minds of modelers to the mindsof agents themselves. �at is, we can use the tools of cooperative game theory and bargaining theoryto build models of how agents themselves normatively evaluate a set of feasible allocations. Paired witha model of norm-dependent utility (Kessler and Leider, 2012; Krupka and Weber, 2013), this providesan account of how normative considerations in�uence choice. Under this view, the axioms codify thestructure of the norms themselves and, thus, become a part of the positive research program aimed atuncovering how actual norms, followed by actual human beings, emerge.

Existing cooperative solution concepts provide a starting point for thinking about the problem inthis way. Consider, for example, the bargaining problem studied by Nash (1950). Let S be a set of pay-o� vectors for two players that is compact, convex, symmetric, and includes the origin, which servesas a disagreement point (Nash’s assumptions). Such S can represent the set of feasible allocations ofsome non-cooperative game, say, a Public Goods game with two players. Suppose also that the play-ers have norms entering their (otherwise sel�sh) utility function, and that these norms satisfy Nash’saxioms. Speci�cally, the players believe that 1) the allocations on the Pareto frontier of S are more so-cially appropriate than those not on it; 2) the most appropriate allocation satis�es IIA (on subsets ofS that include it); and that 3) the most appropriate allocation is symmetric. Using Nash’s famous re-sult that under these conditions the solution maximizes the product of consumption utilities, we canmodel the behavior of such norm-abiding agents with the norm-dependent utility function of the formwi(xi, x−i) = xi + φiη(x1, x2), where (x1, x2) ∈ S; xi represents agent i’s linear consumption utility;η(x1, x2) = x1x2 is the norm function de�ned by the axioms; and φi ≥ 0 is i’s general propensity tofollow norms (Kimbrough and Vostroknutov, 2016).

Next, we can take any non-cooperative game Γ that has S as the set of feasible allocations andaugment it with the norm-dependent utilities wi, thus obtaining a modi�ed game Γ′(φ1, φ2) that canbe analyzed using any non-cooperative equilibrium concepts. Structured in this way, Γ′(φ1, φ2) nowrepresents a positive model of Γ with norm-following agents whose morality is based on the assumptionsof the Nash Bargaining Solution (NBS). Notice that Γ′(0, 0) = Γ, so at one extreme, where players do notcare about following norms at all, we get the standard non-cooperative game that can be analyzed withsome version of Nash equilibrium, while at the other extreme, when φ1 and φ2 are very high, the playerswill choose the NBS in any game form that has S as the set of feasible allocations. �is la�er resultfollows from a simple fact that for high enough propensities to follow norms, the players’ incentivesbecome exactly aligned (since norms typically prescribe what is good for the society), so they both willstrive to reach the Nash Bargaining Solution. With measures of φi in hand that can be obtained fromvarious experimental tasks (e.g., Kimbrough and Vostroknutov, 2018), these predictions are testable.

achieved (e.g., Nash, 1953; Perry and Reny, 1994).

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�is example demonstrates our intuition that cooperative solution concepts can be used as falsi-�able models of morality. Our job in developing this line of research is, therefore, to come up withrealistic axiomatizations that overcome the constraints imposed by the known solution concepts. Forexample, many of them (e.g., NBS, Kalai-Smorodinsky solution) assume that S is convex to guaranteethe existence of the solution, or that the solution is Pareto optimal.4 However, life is full of surprisesand non-convexities. �us, in order to construct useful accounts of morality, we need solution conceptsthat work on any (compact) set of allocations, that always exist, and that are constructed on the basisof simple psychological assumptions. �ese requirements make sure that we build axiomatizations thatdescribe plausible accounts of norms that could realistically evolve in our species (Henrich, 2015).5 Judi-cious choice of axioms allows us to build models in which norms respond to context and can representthe universe of “as is” human social relationships. In the next section we present our version of suchaxioms.

3 Dissatisfaction Functions for Injunctive Norms

Our model begins with the assumption that an agent’s own normative evaluation of any particular out-come depends on how it compares to all other feasible outcomes. Put simply, agents prefer outcomesthat yield higher utilities and are dissatis�ed with outcomes that yield lower utilities because of the coun-terfactual availability of higher-utility outcomes. First we de�ne a function that captures this notion ofdissatisfaction.6

We start with a large set C of all possible consequences, N players, and a utility/payo� functionu : C→ RN , which for each consequence de�nes a vector of utilities (payo�s). Suppose that the imageof u is RN so that all payo�s are possible. We will work with �nite subsets of C (called contexts), witha typical one denoted by C ⊂ C. Analogously to Kimbrough and Vostroknutov (2021), we think of Cand the collection of associated payo� vectors u[C] as a set of all feasible allocations in the context of achoice problem. In addition, suppose that there are functions di : R2 → R+ that for any consequencesx, y ∈ C and player i de�ne i’s dissatisfaction with x because of y as di(ui(x), ui(y)). Assume that forall i ∈ N , di(ui(x), ui(y)) is weakly increasing in ui(y), which represents the idea that counterfactualoutcomes with higher utility increase dissatisfaction.

Given these ingredients, in the �rst step we de�ne personal dissatisfaction functions Di(x |C) foreach player i that, for each subset C and each consequence x ∈ C , give a non-negative real numberrepresenting the personal dissatisfaction of player i associated with x inC . �is provides us with a theoryof individuals’ normative evaluations of each outcome, as a function of all other possible outcomes - a

4A special e�ort has been exerted in the literature to �nd equivalent axiomatizations that do not assume Pareto optimality(see e.g., Karos et al., 2018).

5Alger and Weibull (2013) also present an account of the emergence of Kantian moral preferences that are evolutionarilystable.

6Conceptually, these normative evaluations are made prospectively, before an outcome has been realized. We suggest inKimbrough and Vostroknutov (2021) that an appropriate moniker for the sentiment being captured might be “pre-gret”.

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choice-set-dependent model of normative judgment.In the second step we de�ne an aggregate dissatisfaction function D(x |C) that creates a composite

of the dissatisfaction of all interested players. �is function then directly translates into an injunctivenormative valence associated with an element x given the set of possibilities C . We propose a set ofaxioms that describe the properties of dissatisfaction di, personal dissatisfaction Di, and aggregate dis-satisfactionD. We start with the axioms that de�ne the connection between dissatisfaction di and utilityof player i.

A1 (Relativity). ∀i ∈ N ∀t, r, a ∈ R di(t+ a, r + a) = di(t, r).

A1 states that adding a constant to the utilities being compared does not change the dissatisfactionwith one outcome because of the possibility of another. So, if player i gains or loses the same amountof utility in all consequences then this does not a�ect her dissatisfaction. A1 ensures that di can beexpressed as a function of the di�erence of utilities.

A2 (No Rejoice). ∀i ∈ N ∀t, r ∈ R with t ≥ r we have di(t, r) = 0.A2 says that players do not feel dissatisfaction with a superior outcome because of inferior conse-

quences. In other words, any positive sentiment that a player may feel because there exist some inferiorconsequences (as in, “hey, it could be worse”) does not in�uence the normative valence of the supe-rior consequence. It is important to note that we do not assume here that players do not feel any suchsentiment in these circumstances, only that this sentiment does not in�uence the normative valence ofsuperior outcomes.

A3 (Homogeneity). ∀i ∈ N ∀t, r, α ∈ R with α > 0 di(αt, αr) = αdi(t, r).

A3 states that if all utilities or payo�s are multiplied by a positive constant, then the dissatisfactionsare also multiplied by the same constant. �is ensures that dissatisfactions are proportional to utilities ina linear way, thus, connecting the two concepts. We could have assumed that there is some non-linear,say concave, relationship between dissatisfaction and utility. However, we already allow utility to be anon-linear function of payo�s. A3 re�ects an idea that all non-constant marginal e�ects of payo�s arealready encoded in functions ui.

Finally, we assume non-triviality and importantly equivalence of dissatisfactions across players.

A4 (Non-triviality and Anonymity). ∀i ∈ N di(0, 1) = 1.

A4 serves two purposes. First, it makes sure that players do feel non-zero dissatisfaction. Second,it postulates the “equivalence” of dissatisfactions of all players. �is means that all players are equallydissatis�ed if they are at a consequence that gives them 0 utils and there is another consequence thatgives them 1 util. �is assumption amounts to the claim that the dissatisfaction from the same amounts ofutils makes all players dissatis�ed in the same way. �is is how we operationalize the idea that normativeevaluations are built on empathy.

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�e following proposition establishes the functional form of di equivalent to axioms A1-A4.

Proposition 1. �e following two statements are equivalent:

1. di satis�es A1-A4;

2. di(t, r) = max{r − t, 0}.

Proof. See Appendix A.

Next, we provide axioms that de�ne the personal dissatisfaction Di of player i associated with asingle consequence x.

A5 (Singleton). ∀i ∈ N ∀x ∈ C Di(x | {x}) = 0.

Axiom A5 states that if only one consequence is available or possible, then there is nothing to bedissatis�ed about, so the dissatisfaction of each player i is zero. A5 may sound trivial, nevertheless itrules out any situations in which players are dissatis�ed due to speci�c properties of an allocation x

given the choice set {x}. �is makes our model incompatible with social preference utility speci�cationswhere the utility of a player may depend directly on the payo�s received by other players at the sameconsequence. A form of social preferences, in the common meaning of the term, could be introduced if inA5 we assumed that dissatisfaction is not zero, but rather depends on x. However, we deliberately eschewthis path, as we believe it is more economical to understand social preferences as an epiphenomenon ofnorm-dependent preferences (a view we also spell out in Kimbrough and Vostroknutov, 2016, 2021).Indeed, one of our goals is to show how particular kinds of social preferences (and predictable variationin social preferences across contexts) can be explained via the model presented here.

A6 (Dissatisfaction). ∀i ∈ N ∀C ⊂ C, x ∈ C, y ∈ C\C

Di(x |C ∪ {y}) = Di(x |C) + di(ui(x), ui(y)).

Axiom A6 de�nes the personal dissatisfaction function of player i. It says that given any set ofconsequences C , any consequence x in this set, and any consequence y outside C the dissatisfactionwith x in the augmented set C ∪ {y} equals that of x when y is not in the set plus some non-negativenumber di(ui(x), ui(y)) that depends only on i’s payo�s in x and the payo�s in the added consequencey. Moreover, the higher the payo�s in y the higher is the dissatisfaction, which is guaranteed by theassumptions on di made above. �e important implications of this de�nition are 1) that i’s dissatisfactionwith x in di�erent sets of consequences is connected and 2) that the amount by which it changes when aconsequence y is added only depends on the payo�s in x and y and does not depend on the characteristicsof C . We think of A6 as capturing another basic sentiment, namely that player i feels dissatisfactionwhenever a new possibility represented by y that could give i a higher payo� appears.

�e following result connects the axioms above and the representation that we test in Kimbrough

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and Vostroknutov (2021).


1. Di satis�es A5-A6;

2. Di can be expressed as Di(x |C) =∑

c∈C\{x} di(ui(x), ui(c)).


Next, we aggregate the dissatisfactions across players and de�ne D. �is aggregation proceduretransforms individual normative judgments into a shared injunctive social norm. We start by assum-ing that D(x |C) is a function of Di(x |C) for all i ∈ N . Speci�cally, we assume that D(x |C) =

G(D1(x |C), ..., DN(x |C)), where G : RN → R+ is increasing in all arguments. �e following axiomsdetermine how aggregation is done.

A7 (Aggregate Satisfaction). G(0, ..., 0) = 0.

A7 simply states that if each player feels the lowest dissatisfaction of zero, then the aggregate dissat-isfaction is also the lowest and equals to zero.

�e last axiom de�nes how changing dissatisfaction of one player changes the aggregate dissatisfac-tion. In order to incorporate social context as de�ned in Kimbrough and Vostroknutov (2021), we assumethat players have social weights (ωi)i∈N where ωi ∈ (0, 1]. �ese weights can represent social status,in/outgroup relationships, kinship, or their combination and they determine how much each player’sdissatisfaction counts in the computation of aggregate dissatisfaction.

A8 (Aggregate Change). ∀i ∈ N ∀t1, ..., tN ∈ R+ ∀ai ≥ −ti G(ti + ai; t−i) = G(ti; t−i) + ωiai.

�e notation G(ti; t−i) singles out the ith argument of G. A8 says that if player i’s personal dissat-isfaction changes by ai then the aggregate dissatisfaction changes by the same amount weighted by ωi.�is incorporates an idea that social norms are more sensitive to changing dissatisfactions of “important”players with high ωi as compared to “unimportant” ones with low ωi. �e following proposition puts allthe axioms together.


1. di satis�es A1-A4, Di satis�es A5-A6, D satis�es A7-A8.

2. D can be expressed as

D(x |C) =N∑i=1

ωiDi(x |C) =N∑i=1

∑c∈C

ωi max{ui(c)− ui(x), 0}

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Proof. See Appendix A.7,8

Figure 1 illustrates the representation result in Proposition 3 with all players’ dissatisfactions weightedequally (ω1 = ω2 = 1). �e le� panel shows dissatisfactions of players P1 and P2 in the context of theset of allocations {(3, 6); (5, 5); (6, 3)}. �e colored lines correspond to dissatisfactions at identically-colored allocations. So, the personal dissatisfaction of P1 at the light-grey allocation (3, 6) is equal to3 + 2 = 5, which is also the aggregate dissatisfaction since P2 does not feel dissatisfaction at (3, 6).�e same holds symmetrically for the dark-grey allocation (6, 3) with the names of players switched.At (5, 5) the personal dissatisfactions of both P1 and P2 are 1, and the aggregate dissatisfaction at (5, 5)

is 1 + 1 = 2. �us, in this context, the allocation (5, 5) has the lowest sum of dissatisfactions, and is,therefore, the most socially appropriate.

Payoff of P1

Pay

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63 54

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1

Payoff of P1

6

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Dissatisfaction of P1 at (5,5)




Figure 1: Le�. A-model dissatisfactions of the allocations in {(3, 6); (5, 5); (6, 3)}. Right. A-modeldissatisfactions of the allocations in {(3, 6); (5, 5); (6, 3); (4, 5)}.

In the right panel of Figure 1 we add one more consequence—(4, 5), denoted by the white dot—so the set of allocations here is {(3, 6); (5, 5); (6, 3); (4, 5)}. Notice that (4, 5) increases the aggregatedissatisfaction with (3, 6) by 1 due to the additional dissatisfaction of P1 (P2 at (3, 6) is not additionallydissatis�ed, because she gets 6 in this allocation, which is higher than her payo� of 5 in the newly

7�e representation in Proposition 3 contains the maximum of a utility di�erence and zero, which might remind manyreaders of the inequality-averse utility function introduced by Fehr and Schmidt (1999). However, our model is conceptuallydi�erent: inequality aversion considers utility di�erences at a given outcome across players; our model of dissatisfactionconsiders utility di�erences of the same player across outcomes. �en we aggregate these across players to de�ne the injunctivenorm.

8In the case with only one player (N = 1) and keeping in mind the result of Proposition 1, the normative ranking ofoutcomes implied by our axioms is consistent with regret avoidance similar in �avor to that considered by Fiore�i et al.(2021). In the canonical treatment of regret (e.g., Loomes and Sugden, 1982), it is de�ned as a negative feeling associated withthe unchosen alternative. �e form of “regret” considered here and in Fiore�i et al. (2021) can be felt about any counterfactuals,regardless of their current a�ainability with choice. However, our model does not reduce choice to the maximization of aregret averse utility function; rather, the aggregated injunctive norms that we construct can be thought of as preferences of aregret-averse player who cares also about the regrets of other players, and norm-following is traded o� against maximizationof consumption utility in choice.

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added allocation). It also increases the aggregate dissatisfaction of (6, 3) by 2, due to the additionaldissatisfaction of P2. �us in this context, allocations (3, 6) and (6, 3) do not have the same aggregatedissatisfaction as was the case in the le� panel: adding an additional allocation that is “closer” to (3, 6),makes the dissatisfaction with (6, 3) grow more than the dissatisfaction of (3, 6).9

�e representation in Proposition 3 provides a simple mathematical expression for computing dissat-isfactions of any allocation x in any context C . �e norm-dependent utility of player i associated withD(x |C) can be then de�ned as

wi(x |C) = ui(x) + φi[−D(x |C)],

where [·] stands for the linear normalization of the function within to the interval [−1, 1].10 �e mainpoint is simple: player i trades-o� personal consumption utility ui(x) and a linear function [−D(x |C)]

of the negative of aggregate dissatisfactions (since they ought to be minimized). In Kimbrough andVostroknutov (2021) we call η(x |C) = [−D(x |C)] a normative valence of x inC and η a norm function.

4 Dissatisfaction Functions for Moral Rules

In this section we describe an alternative set of axioms that incorporate some moral rule according towhich the dissatisfaction function is constructed. By moral rule we mean some abstract ideal criterionagainst which all allocations are compared; this might include concepts like payo� e�ciency, Pareto op-timality, equality, or some combination of these, or any other abstract principle that might be o�ered asa normative guide to behavior. Such moral rules are succinctly summarized, readily codi�ed and learnedand therefore salient both in our daily lives and to researchers who study social behavior. Most of theliterature that deals with social welfare and economic e�ciency is based on such abstractions. �eseaxioms serve several purposes. First, they show how to represent any choice-set-dependent moral rule,which de�nes the normatively best element of all choice sets C , with a dissatisfaction function. Sec-ond, contrasting the dissatisfaction functions representing these abstract moral rules with the radicallychoice-set-dependent dissatisfaction functions described above, we highlight the extent to which theformer can serve as approximations of the normative evaluations implied by the la�er.

�e key insight is that the normative evaluations captured in the dissatisfaction functions that re-sult from our �rst set of axioms cannot be replicated by a moral rule. �is is important because weo�en communicate our moral arguments in terms of such rules, and we o�en want to codify normativejudgments into laws or other forms that must be explicitly stated (e.g., religious texts). �e issue is thatabstract ideals are necessarily simpli�cations and thus they are unable to capture some of the richnessand complexity that arises from our model of injunctive norms. �at is, there will always be implicationsfrom the radically choice-set-dependent axioms that will not follow from an abstract moral rule. �is

9�is argument is used in the proof of Proposition 7 below.10In Kimbrough and Vostroknutov (2021) we provide more details about why we need this normalization.

10

result demonstrates the possibility that there may not exist an ideal-coded set of explicit rules that ex-actly capture intuitive normative perceptions, and may imply that situations in which these normativeperceptions are in con�ict with codi�ed normative ideals will always arise.

As before, we start with a large set C of all consequences, its �nite subsets C , and a utility functionu : C→ RN . �e di�erence from the previous analysis is that now we require that for any C there existsa special non-empty set C∗, which represents the set of ideal payo� vectors that in the context of C areconsidered the most socially appropriate, or having zero dissatisfaction. In addition, there is a functionu∗(v |S) that for each vector v ∈ RN and all subsets S ⊂ RN that are equal to u[C∗] for some C de�nesan element in S that is a “reference point” for v in S.11 �is function is used to assign to any element ofCthe ideal element in C∗ to which it is compared, or in reference to which the dissatisfaction is expressed.It has the following property: u∗(v |S) = v whenever v ∈ S, which makes sure that the reference pointfor any ideal element is the element itself. Notice that u∗ is only needed when there are sets C∗ that arenot singletons. If all C∗ are singleton sets, as is the case for most ideals, then there is no need to de�neu∗. Finally, there are functions fi : R2 → R+ that de�ne dissatisfactions for each player i ∈ N . Namely,fi(vi, u

∗i (v |S)) stands for the dissatisfaction that player i feels when her utility is vi and the utility that

she would receive in the ideal situation is u∗i (v |S).In what follows, we denote the second set of axioms as B1 . . .B7, for ease of exposition we distinguish

between the A-model; which captures our radically choice-set-dependent theory of normative evalua-tions presented above, and the B-model, which includes an ideal. As above, to describe the B-model westart with the axioms describing the properties of fi.

B1 (Relativity). ∀i ∈ N ∀t, r, a ∈ R fi(t+ a, r + a) = fi(t, r).

B1 is exactly the same as A1, and it states that adding a constant to the utilities being compared doesnot change the dissatisfaction. So, only the di�erence in utilities ma�ers.

B2 (No Rejoice). ∀i ∈ N ∃βi ∈ (0, 1] such that ∀t ≥ 0 we have fi(0,−t) = βifi(0, t).

B2 is di�erent from A2. It states that there might be an asymmetry in how dissatisfaction is perceivedwhen the ideal utility is above or below the received one. Speci�cally, if player i receives utility 0 whenthe ideal utility is −t, then her dissatisfaction could be lower than the dissatisfaction she gets whenthe ideal utility is t. Notice that we require βi > 0. �e reason for this is the following. If βi is zero,then player i is not dissatis�ed when her utility is greater than that in the ideal case. However, thenit would be possible that there are consequences for which aggregate dissatisfaction is zero (all playershave utility higher than than the ideal one), but which are not in the ideal set. We deliberately excludesuch possibilities since by construction C∗ should include all ideal elements. So the requirement βi > 0

11Speci�cally, there is no need to de�ne u∗(v |S) for all subsets S of RN , but only for those that can play a role of a set ofideal payo� vectors u[C∗]. �is becomes important when we de�ne choice-set-independent dissatisfaction with only one C∗for all C in Section 4.1. �ere we need to de�ne u∗ for only one set S.

11

implies that there are no payo� vectors that have zero dissatisfaction but are not included in C∗.

B3 (Homogeneity). ∀i ∈ N ∀t, r, α ∈ R with α > 0 fi(αt, αr) = αfi(t, r).

�is axiom is the same as A3. Finally, as above we assume non-triviality and equivalence of dissatis-factions across players.

B4 (Non-triviality and Anonymity). ∀i ∈ N fi(0, 1) = 1.

�e following proposition provides the representation.


1. fi satis�es B1-B4;

2. fi(t, r) = max{r − t, 0}+ βi max{t− r, 0} for some βi ∈ (0, 1].


�e next axiom de�nes the personal dissatisfaction Fi(x |C) of player i. �is is very straightforward.However, it is worth noticing before we state the axiom that unlike in the A-model, in the B-model we donot require that Fi is zero for singleton choice sets C = {x}. �is is because it is easy to imagine a situ-ation where there is only one allocation which nevertheless deviates from the ideal. For example, whenthe ideal for each player is the average payo� in x (see Section 4.1). �is highlights the sense in whichideals involve abstraction; in the A-model this is not possible since the normatively best consequence isde�ned endogenously from the available consequences in C .

B5 (Dissatisfaction). ∀i ∈ N ∀C ⊂ C, x ∈ C Fi(x |C) = fi(ui(x), u∗i (u(x) |u[C∗])).

In words, player i’s personal dissatisfaction at consequence x in C is equal to his dissatisfactionbecause of an ideal reference element in C∗. �is de�nition looks redundant. However, it creates avery important restriction on the dissatisfactions across di�erent sets of consequences. Speci�cally, B5asserts that for any two sets C1 and C2 that have u[C∗1 ] = u[C∗2 ], or have the same ideal payo� vectors,the i’s personal dissatisfaction of all elements x ∈ C1 ∩ C2 is the same. �is is the main feature thatdistinguishes the B-model from the A-model, where such connection is only a�ainable in speci�c rarecases. In general, the dissatisfactions of x ∈ C1 ∩ C2 will not be the same in C1 and in C2 because allelements of these sets are used to compute them. So, instead of providing a representation result, weformulate this implication as a proposition.

Proposition 5. B5 implies that for anyC1 andC2 with u[C∗1 ] = u[C∗2 ] it is true thatFi(x |C1) = Fi(x |C2)

for all x ∈ C1 ∩ C2.

Proof of Proposition 5. By the property of u∗. �

Finally, we de�ne aggregated dissatisfactionF (x |C). �is is done in the same way as in the A-model.We assume that F is a function of Fi’s: F (x |C) = E(F1(x |C), ..., FN(x |C). �e following two axioms

12

describe the properties of E.

B6 (Aggregate Satisfaction). E(0, ..., 0) = 0.

�is is the same as A7. �e last axiom is the same as A8 assuming the existence of social weights(ωi)i∈N .

B7 (Aggregate Change). ∀i ∈ N ∀t1, ..., tN ∈ R+ ∀ai ≥ −ti E(ti + ai; t−i) = E(ti; t−i) + ωiai.

�e following proposition puts all B-axioms together.


1. fi satis�es B1-B4, Fi satis�es B5, F satis�es B6-B7.

2. F can be expressed as

F (x |C) =N∑i=1

ωiFi(x |C) =N∑i=1

ωi(max{u∗i − ui(x), 0}+ βi max{ui(x)− u∗i , 0}),

where u∗i is short for u∗i (u(x) |u[C∗]) and βi ∈ (0, 1] are some coe�cients.

Proof of Proposition 6. Similar to the proof of Proposition 3. �

�e following examples show how to express some commonly studied moral rules using a dissatis-faction function F that satis�es the axioms above. In each case, we do not specify the exact F , but ratherits ingredients: C∗ for each C and the function u∗. For simplicity, we assume in all examples that ωi = 1

for all i ∈ N .

Pareto Optimality. For all C we have C∗ ⊆ C , and for all x, y ∈ C if u(x) ≥ u(y) with strict inequalityfor at least one component then y /∈ C∗. �us, C∗ is non-empty for all C . u∗(r |S) can be de�ned as anelement of S that is the closest to r in Euclidean metric.

Payo� E�ciency. For all C we have C∗ ⊆ C , and for all x, y ∈ C if∑

i∈N ui(x) >∑

i∈N ui(y) theny /∈ C∗. u∗(r |S) can be de�ned as an element of S that is the closest to r in Euclidean metric.

Maximin. For all C we have C∗ ⊆ C , and for all x, y ∈ C if mini∈N ui(x) > mini∈N ui(y) then y /∈ C∗.u∗(r |S) can be de�ned as an element of S that is the closest to r in Euclidean metric.

Choice-Set-Dependent Inequality Aversion. Take any C and compute the average payo� that eachplayer gets in all consequences: a∗i =

∑c∈C ui(c)/|C|. Let C∗ = {(a∗i )i∈N}. �is is a singleton ideal set,

which is sensitive to all payo�s that players can receive. In this case there is no need to de�ne u∗.12

Cooperative Solution Concepts. Many cooperative solution concepts can be expressed as moral rules.12A similar but less payo�-sensitive principle of inequality aversion can be speci�ed if we take into account only the highest

and the lowest payo�s that players receive in C . For each player compute m∗i = minc∈C ui(c) and m∗i = maxc∈C ui(c).Let C∗ = {(m∗

i +m∗i

2 )i∈N}. �is is again a singleton ideal set. However, now it is less choice-set-dependent: adding anyconsequences that do not change m∗i and m∗i does not change the ideal element and thus does not a�ect the dissatisfactiondistance of existing elements in C .

13

To illustrate one possible way this can be done, we take the general class of bargaining solutions inKaros et al. (2018) that includes NBS and Kalai-Smorodinsky solution. �ese solutions, characterized by0 ≤ p < ∞ and de�ned on convex sets S, pick an allocation that is weakly Pareto optimal and havethe ratio of payo�s for any two players i and j equal to the ratio of their maximal payo�s in S raisedto the power p and denoted by api (S)/apj(S) where ai(S) is the maximal payo� that i can get in S. �iscan be expressed in our framework by de�ning for each set C the ideal singleton set C∗ = s∗(C), wheres∗(C) is the point on the ray going through the origin and the point (ap1(C), ..., api (C)), and such thats∗i (C) = ai(C) for some i ∈ N . �is way, for p = 0 we get a moral rule for the Nash Bargaining solutionand for p = 1 we get a moral rule for the Kalai-Smorodinsky solution.

Next, we demonstrate how dissatisfactions are computed for moral rules via the same example as inFigure 1 to illustrate the contrast between A- and B-models. Suppose we take the above-de�ned Payo�E�ciency rule as the moral rule. �en, allocation (5, 5) is the ideal in both contexts depicted on Figure2, since its sum of payo�s is greater than in any other allocation. We set C∗ = {(5, 5)} for both contexts.

Payoff of P1

Pay

off

of P

2

6

3

5

4

63 54

2

2

Payoff of P1

6

3

5

4

63 54

2

1

2

1

1

1





C* C*

Figure 2: Le�. B-model dissatisfactions of the allocations in {(3, 6); (5, 5); (6, 3)}. Right. B-modeldissatisfactions of the allocations in {(3, 6); (5, 5); (6, 3); (4, 5)}.

According to the representation in Proposition 6, the dissatisfactions at (3, 6) are equal to 2 for P1and β2 · 1 for P2 (light-grey lines on the le� panel of Figure 2). �e aggregate dissatisfaction at (3, 6) is2 + β2. At (6, 3), the dissatisfaction of P1 is β1 · 1, and the dissatisfaction of P2 is 2. �us, the aggregatedissatisfaction at (6, 3) is 2 + β1. When we introduce an additional allocation (4, 5) in the right panelof Figure 2, the aggregate dissatisfactions of (3, 6) and (6, 3) do not change—unlike what happened inthe A-model—because dissatisfactions are measured with respect to an ideal, and as long as that idealis the same, the dissatisfactions do not change. �is demonstrates the less-context-dependent nature ofthe B-model as compared to the A-model.

Finally, to predict how a chosen moral rule will in�uence behavior in a particular se�ing, one needonly normalize the function −F (x|C) to the interval [−1, 1] and then plug it into a norm-dependentutility function as discussed at the end of Section 3.

14

4.1 Dissatisfaction Functions for Choice-Set-Independent Moral Rules

It is worth paying a�ention to the simplest class of ideal dissatisfaction functions for which the idealset C∗ is independent of C . We assume that for all C ∈ C there is only one C∗, and that the B-axiomshold as before. In this case, for any non-ideal x ∈ C the dissatisfaction from x is the same for allC ∈ C that include x. In other words, Fi(x |C) = Fi(x) = fi(ui(x), u∗(u(x) |u[C∗])). �is modelclosely approximates outcome-based social preference models, with the only di�erence being that in ourapproach all players have the same aggregate dissatisfaction F (c) a�ached to an outcome, whereas insocial preference models di�erent players can have di�erent social utility terms at the same outcome (e.g.,inequality aversion a la Fehr and Schmidt, 1999). One argument in favor of our way of modeling socialpreferences is that the construction of the norm from aggregate (and not only individual) dissatisfaction—since F (x) is interpreted as the aggregate dissatisfaction of all players—means that we prioritize thesocial component of social preferences. In our framework, each player cares about all others, since theirdissatisfaction enters her utility function through the norm, and not only about how much their ownpayo� diverges from that of others.

�e following example illustrates what a choice-set-independent dissatisfaction function may looklike.

Choice-Set-Independent Inequality Aversion. Let C∗ = {x ∈ C | ∀i, j ∈ N ui(x) = uj(x)}. �isis a ray in RN containing all allocations that give the same payo� to all players. For any x ∈ C andi ∈ N let a =

∑j∈N uj(x)/N and u∗i (u(x) |u[C∗]) = (a, ..., a) ∈ C∗ be the average payo� (same for all

players).

5 Non-Equivalence of Injunctive Norms and Moral Rules

It should be obvious that B-models will not, in general, be equivalent to the A-model, because idealsare speci�ed exogenously by the modeler. However, we also show that the A- and B-models are notequivalent in another sense. Speci�cally, we formalize the intuition provided by the examples above andshow that abstract moral rules of the kind captured in the B-model are less context-dependent than thenorms generated by the A-model. To do this we need a notion of equivalence between them. Giventhat the d and f functions in A- and B-models are not �xed, it might seem as though we could focus onwhether they have the same maximal element. However, the entire ranking of outcomes is relevant forchoice because interior solutions of the utility maximization problem with norm-dependent preferencesare sensitive to the relative normative valences of all the outcomes. �us, we compare the models byasking whether they induce the same ranking of consequences in terms of dissatisfaction.

Let <C be de�ned as the preference relation that represents the aggregated dissatisfactions in theA-model in some set C :

∀C ∈ C x <C y ⇔ D(x |C) ≥ D(y |C).

Similarly, for some B-model, de�ned by the collection of all C and C∗ together with a distance function

15

f , let 3C represent the dissatisfactions:

∀C ∈ C x 3C y ⇔ F (x |C) ≥ F (y |C).

Let us say that a B-model is equivalent to the A-model if <C and 3C are the same for all C ∈ C. Ourtask now is to show that there is no B-model that is equivalent to the A-model satisfying all A-axioms.To do that, let us try to construct a B-model that is as close as possible to the A-model. B-models arede�ned through ideal elementsC∗ and a distance function. So, for eachC de�neC∗ as the set of minimalelements of <C for each C and choose any dissatisfaction function f .

23

45

67

Dis

satis

fact

ion

(3, 6) (5, 5) (6, 3)

y = (4, 5)

y = (5, 4)

Figure 3: A-model dissatisfactions of the same allocations in Cy = {(3, 6); (5, 5); (6, 3); y}, where y =(4, 5) or y = (5, 4).

�e following example shows how no such model can be equivalent to the A-model. Consider setsof four allocations for two players Cy = {(3, 6); (5, 5); (6, 3); y} where y can take on di�erent values,say y = (4, 5) or y = (5, 4). In the A-model both C(4,5) and C(5,4) have minimal dissatisfaction at (5, 5),as was shown in Figure 1. So, when we construct a B-model as described above for both of them we setC∗ = {(5, 5)}. By Propositions 5 and 6, in any B-modelF ((3, 6) |C(4,5)) = F ((3, 6) |C(5,4)) and the sameholds for allocation (6, 3). �us, according to any B-model either (3, 6) or (6, 3) has larger dissatisfactionin both C(4,5) and C(5,4). However, Figure 3 shows that this is inconsistent with the A-model in which therelative dissatisfactions of (3, 6) and (6, 3) change depending on y. �e dissatisfaction of (3, 6) is smallerthan that of (6, 3) in C(4,5), but larger in C(5,4). �us, no B-model is equivalent to the A-model. We statethis result as a proposition.13

Proposition 7. �ere is no B-model that is equivalent to the A-model in the sense of equivalence of the

13It may seem that we could construct an equivalent B-model by creating an ideal consequence that gives each player thehighest payo� that they can receive at any element in C , but because the normative evaluation of each outcome depends onall other outcomes (and not just on the ideal), this will not generate an equivalent ranking.

16

preference orderings of dissatisfactions that the models imply.

Proof of Proposition 7. By example on Figure 3. �

�is proposition demonstrates that if it is the aggregate dissatisfaction that people care about—as theA-model postulates—then there is no B-model that re�ects the same criterion. We have shown that thisis the case with the class of B-models that are the closest to the A-model, namely those that have thesame minimal dissatisfaction as the A-model in all sets C . For more general B-models the discrepancyis even larger. If C∗ ( C , or there are elements of C∗ outside of C , then adding these elements toC does not change any dissatisfactions. However, in the A-model in general the dissatisfactions of allelements will change a�er such an addition, and the minimal dissatisfaction will not be assigned in thesame way by the A- and B-models. Moreover, the dissatisfaction-minimizing allocations in the A-modelare never Pareto-dominated (see Proposition 1 in Kimbrough and Vostroknutov, 2021). �us, if in aB-model some C∗ contains Pareto-dominated consequences, as for example in the Inequality Aversionexamples described above, then this will always contradict the predictions of the A-model except for somespecial cases. Finally, the discrepancies in induced preference orderings between A- and B-models arenot “rare” or measure zero in some sense: the switch in the dissatisfaction rankings of some allocationsas demonstrated in Figure 3 can be very easily constructed and are rather typical for the kind of context-dependence coming out of the A-model. �us, we should expect systematic di�erences between A- andB-models in terms of dissatisfaction rankings.

6 Why Moral Rules?

As shown above, the radical choice-set-dependence of the A-model of injunctive norms, in which theevaluation of each consequence depends on all other feasible consequences, makes it impossible to char-acterize its normative implications succinctly with an abstract ideal. If we take seriously the idea thatthe A-model provides an account of peoples’ normative intuitions, this poses a problem - it will not, ingeneral, be possible to specify abstract ideals whose implications will satisfy our normative intuitions inall cases. Dissatisfaction will not in general be minimized by following moral rules. What good, then,are abstract moral rules? We argue that one might nevertheless prefer an inexact moral rule to a preciseinjunctive norm because of the la�er’s relative complexity.

Consider a simple case with the social weights of all players equal to 1. Suppose that there are Nplayers and consider some set of consequences C that has k elements. �en, to compute normativevalences of all elements of C in the A-model one needs to compute Nk2 payo� di�erences (and decidewhether they are greater or equal to zero). So, we can say that the complexity of this problem isO(Nk2).For the similar B-model with all βi = 1 one needs to compute only Nk payo� di�erences and decidewhether they are greater or less than zero. So the complexity of this problem is O(Nk). So, for any�xed number of players, the normative valences in the A-model can be computed in quadratic time,whereas the valences in any B-model can be computed in linear time. As the number of consequences

17

grows, so does the appeal of an abstract rule for assigning normative valences. Of course, this ignoresthe computations involved in �nding C∗ for each C , but here too moral rules would seem to have anadvantage because they can be articulated in a way that generalizes across choice se�ings (e.g., “oneought to maximize e�ciency”), facilitating coordination and cooperation.

�is is an argument for why we might wish to have some clearly articulated abstract moral ruleswhich we use to guide behavior, but it leaves open the question of what those moral rules might look like.We suggest that the most appealing moral rules for a given class of choice se�ings would be those whichmost o�en approximate the normative intuitions captured in injunctive norms (based on the A-model)computed in those se�ings. We thus ask, for various classes of choice se�ings, what are the commonlyobserved properties of the injunctive norm generated by the A-model in those se�ings that might besummarized as a moral rule. First, we show that regardless of the choice se�ing, all dissatisfaction-minimizing outcomes under the A-model will be Pareto optimal, such that this moral rule is alwayssatis�ed by an injunctive norm. Second, for the special case of choice se�ings with exactly two possibleconsequences, we show that the injunctive norm implies payo� e�ciency maximization. More generally,using simulations in which we append random payo� vectors to various numbers of consequences underdi�erent restrictions (e.g., constant sum), we ask how frequently the injunctive norm, calculated usingthe A-model, corresponds to various other plausible (or commonly expressed) moral rules: “maximizee�ciency”, “minimize inequality”, etc. �en we consider the moral rules that might be derived frominjunctive norms for payo� vectors that correspond to broad classes of games that are of interest toeconomists, such as social dilemmas, coordination games and tournaments. As it turns out, the injunctivenorms implied by the A-axioms o�en cohere reasonably well with existing moral rules such as e�ciencymaximization, inequality aversion, etc, depending on the context. We thus show how our A-model canbe used to provide a meta-theory of moral rules, predicting which moral rules are likely to emerge inwhich choice se�ings and helping explain why there exists a diverse set of moral rules.

6.1 Moral Rules for All Choice Settings

As we have shown in Kimbrough and Vostroknutov (2021), the injunctive norms implied by the A-modelhave an important property: in any set of consequences, a Pareto dominated consequence is alwaysnormatively inferior to the one that Pareto dominates it (i.e., it always generates more aggregate dis-satisfaction). �is regularity is easy to spot from observation. �us injunctive norms derived from theA-model always satisfy the Pareto optimality criterion. �is is the only moral rule that is exactly repre-sented by the A-model; however, it is worth noting that our model goes further in that it also provides aranking across Pareto optimal elements of the choice set. �us, although all dissatisfaction-minimizingconsequences are Pareto optimal, the converse is not true, and the ability of the Pareto optimality crite-rion to predict the normatively best consequence under the A-model is limited.

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6.2 Moral Rules for Random Payo� Vectors

First, from Kimbrough and Vostroknutov (2021) we know that, as long as agents’ utility functions do notexhibit too much curvature, payo� e�cient allocations are dissatisfaction-minimizing in the A-modelfor all choice sets with exactly two consequences. For larger sets of consequences, this is not true ingeneral. Nonetheless, it is true quite frequently, enough so that it is plausibly reasonable to summarizethis tendency of the injunctive norm as a moral rule. To illustrate the point, we simulate random sets ofpayo� vectors of varying sizes and for di�erent numbers of players, and compute the percentage of casesin which the dissatisfaction-minimizing consequence under the A-model (herea�er, A-norm) would alsobe chosen by the moral rule “choose the most e�cient outcome.”14

0.1

.2.3

.4.5

.6.7

.8.9

1E

ffici

ency

= A

-no

rm (

%)

2 3 5 10 20 100 150Number of consequences

2 players

3 players

5 players

10 players

20 players

Chance

0.1

.2.3

.4.5

.6.7

.8.9

1E

ffici

ency

ran

king

= A

-no

rm r

anki

ng

(%)

2 3 5 10 20Number of consequences

2 players

3 players

5 players

10 players

20 players

Figure 4: Le�: the percentage of A-norms that are payo� e�cient for random sets of consequences.500,000 sets for each case. Right: the percentage of cases in which the A-dissatisfaction function ranksoutcomes according to their payo� e�ciency.

�e le� panel of Figure 4 shows the results. With 3 players and a large number of consequences,around 80% of A-norms are payo� e�cient. �is percentage tails o� as the number of players grows.With 10 or 20 players and large sets of consequences the number of payo� e�cient A-norms drops to 65%,but the correspondence remains striking, happening far more o�en than would be predicted by chancealone. A more stringent test asks whether the ranking of outcomes according to the A-model is identicalto the ranking according to payo� e�ciency, or what percent of the time the A-model is equivalent tothe moral rule in the sense de�ned in Section 5. �e right panel of Figure 4 shows that the number ofequivalent rankings goes to zero rather rapidly with the number of consequences.

�is simple analysis shows the value of our framework. For example, we can hypothesize that thepayo� e�ciency criterion may be a good approximation of the default moral rule for small sets of conse-quences and players. As these numbers grow, payo� e�ciency becomes progressively worse at predictingthe most normatively desirable outcome (A-norm). �is suggests that some other moral rule might begin

14For each given number of players and given number of consequences, we generate 500,000 sets of random payo� vectorswith payo�s independently drawn from the uniform distribution on [0, 1].

19

to look appealing in such cases. We summarize this as a conjecture.

Conjecture 1. For small sets of randomly chosen payo� vectors with 3 to 5 consequences, people will identifymaximization of payo� e�ciency as a moral rule since it coincides with the A-norm in 80-90% of cases. Asthe set of consequences and the set of players grow, the payo� e�ciency heuristic should be used less o�en.

Next we consider maximin criterion as a moral rule for random payo� vectors (see the end of Section4 for the de�nition). �e le� panel of Figure 5 shows the percentages of cases in which the A-normcoincides with the maximin ideal. Overall the maximin rule corresponds to the A-norm less o�en thanpayo� e�ciency. However, the di�erence becomes considerable only for large sets of players and largesets of consequences. �erefore, we cannot a priori rule out the possibility that maximin can be usedas a moral rule for small numbers of players and small sets of consequences. Indeed, Engelmann andStrobel (2004) and Baader and Vostroknutov (2017) show that around 40% of subjects choose accordingto maximin in 3-player mini-dictator games with 3 consequences. Interestingly, another 40% follow thepayo� e�ciency criterion, which also corresponds to the A-norm quite o�en in such situations: for 3players and 3 consequences the number of payo� e�cient A-norms is 90% and the number of maximinA-norms is around 75% (see Figures 4 and 5).

0.1

.2.3

.4.5

.6.7

.8.9

1M

axim

in =

A-n

orm

(%

)


0.1

.2.3

.4.5

.6.7

.8.9

1M

axim

in =

A-n

orm

, Log

Util

ity (

%)


2 players

3 players

5 players

10 players

20 players

Chance

2 players

3 players

5 players

10 players

20 players

Chance

Figure 5: Le�: the percentage of A-norms that are maximin for random sets of consequences. Right: thepercentage of A-norms with log utility that are maximin for random sets of consequences. 500,000 setsfor each case.

In Kimbrough and Vostroknutov (2021) we argue that maximin is more likely to be the A-norm whenplayers use log utility to compute dissatisfactions. �e reason is that log utility embellishes the dissat-isfactions of “poor” players, in the sense that the same logged-payo� di�erence generates much moredissatisfaction of poor than rich players. �us, we argue that people who are prone to compute dissatis-factions taking relative wealth of others into account are more likely to follow maximin as a moral rule(for some extreme examples of such behavior see MacFarquhar, 2016). �e right panel of Figure 5 showsthe percentages of cases in which the A-norm coincides with the maximin rule in randomly generatedlogged-payo� vectors. �e performance of the maximin rule increases—as compared to the le� panel of

20

Figure 5—especially for the large sets of players and consequences. �is provides support to our intuitionin Kimbrough and Vostroknutov (2021). We summarize our �ndings as a conjecture.

Conjecture 2. For small sets of players (2 to 5) and randomly chosen payo� vectors with any number ofconsequences, people will identify maximin as a moral rule since it coincides with the A-norm in 70-90%of cases under the assumption that people use log utility to compute dissatisfactions. �is also holds to alesser degree even without log utility. As the set of players grows, the maximin rule should be used less o�en.Given that both payo� e�ciency and maximin �t the A-norm rather well when the set of players is small,we should expect to observe both rules used in these circumstances.

0.1

.2.3

.4.5

.6.7

.8.9

1

2 3 5 10 20 100 150 2 3 5 10 20 100 150

Equ

ality

= A

-nor

m (

%)

Number of consequences

2 players

3 players

5 players

10 players

20 players

Chance

Choice-Set-Dependent Inequality Aversion Choice-Set-Independent Inequality Aversion

Figure 6: �e percentage of A-norms that coincide with predictions of Choice-Set-Dependent and Choice-Set-Independent Inequality Aversion for random sets of payo� vectors. 500,000 sets for each case.

By way of contrast, we can also ask whether other well-known moral rules correspond to the A-norm in these se�ings. For example, if we repeat the exact same exercise as above and ask how o�enthe A-norm makes the same prediction as inequality aversion in randomly chosen payo� vectors, we getrather di�erent results. Figure 6 shows that for general random sets of payo� vectors, the choice-set-dependent inequality aversion model corresponds to the A-norm at a rate no be�er than chance, whilethe choice-set-independent model does only slightly be�er than chance. �is is not surprising given thatthe A-norm is always Pareto optimal, whereas inequality aversion models tend to favor payo� vectorsclose to the center of any set C . However, it is plausible that inequality aversion models may be be�ersuited to situations like the Dictator game in which payo� e�ciency is constant and the payo�s of eachplayer are non-negative. We explore the properties of the A-norm in those se�ings next.

6.3 Moral Rules in Constant-Sum Settings

In the above example, we computed injunctive norms over randomly chosen payo� vectors. A randomset of payo� vectors has with probability 1 a unique highest e�ciency element, which is o�en favored by

21

the A-model. So, the 80% result above should be taken with care.15 Moreover, the sets of payo� vectorsreside in a multidimensional space, and it is easy to �nd a measure zero subspace which corresponds toa particular widely-studied game and for which the payo� e�ciency moral rule may not correspond tothe A-norm. For example, the set of payo� vectors corresponding to a Dictator game has measure zeroin the two-dimensional space of payo� vectors for 2 players. �e results in Figure 4 cannot possibly bein�uenced by the properties of such measure zero subspaces since the probability that the sets of payo�vectors from these subsets are randomly drawn is zero. �at is, if we randomly draw sets of payo� vectorsfrom R2 we will never draw one where all payo� vectors have constant payo� e�ciency. At the sametime, in this measure zero subset, the payo� e�ciency moral rule does not have any explanatory powerat all since all payo� vectors have the same payo� e�ciency. And, there is a voluminous literature thatreveals economists’ interest in behavior in such se�ings. �us, we apply our method to choice se�ingsof this kind and again ask whether the A-norm generally corresponds to another widely studied moralrule: inequality aversion.

Figure 7 shows the percentage of cases (out of 500,000 randomly generated sets of payo� vectorsfrom a simplex de�ned by the condition

∑i∈N ui(x) = 1 and by the condition that all payo�s are non-

negative) in which the choice-set-dependent and choice-set-independent inequality aversion models fa-vor the same allocation as the A-norm. A �rst interesting observation is that both models correspond tothe injunctive norm at a rate be�er than chance. However, the choice-set-dependent model does so morefrequently than the choice-set-independent model in se�ings with only a few consequences, though thegap shrinks as the choice set grows. �us, we might expect the more complex moral rule to be favoredwhen the choice set is small and the less complex moral rule to be favored as the choice set grows largesince it is about as likely to correspond to the injunctive norm and is simpler to apply.

0.1

.2.3

.4.5

.6.7

.8.9

1

2 3 5 10 20 100 150 2 3 5 10 20 100 150

Choice-Set-Dependent Inequality Aversion Choice-Set-Independent Inequality Aversion

Equ

ality

= A

-nor

m (

%)

Number of consequences

2 players

3 players

5 players

10 players

20 players

Chance

Figure 7: �e percentage of A-norms that coincide with predictions of Choice-Set-Dependent and Choice-Set-Independent Inequality Aversion for random sets of payo� vectors with constant payo� e�ciency.500,000 sets for each case.

15�is result also may depend on the distribution from which random draws are taken. We use a uniform distribution.However, it is not inconceivable that the results will change if one takes some other distribution, say truncated normal.

22

Conjecture 3. In choice sets with constant or li�le varying payo� e�ciency, inequality becomes a keyforce driving normative comparisons. Moral rules based on payo� equality are thus likely to be favored,with simpler (i.e. choice-set-independent) versions of such rules more likely as the choice set grows.

�us far, we have deliberately focused on choice problems for clarity of exposition and to emphasizethat the normative ranking of outcomes generated by our model depends only on the payo�s at eachoutcome (and the social weights). Nevertheless, it is useful to explore what injunctive norms look likein various widely studied strategic interactions (i.e., games), while cautioning that the model does notdirectly predict that people will choose the normatively most appealing outcome. It is worth reiteratingthat the injunctive norms (and moral rules) described herein should be thought of as inputs to a player’sutility-maximization problem; it tells players what they ought to do, but in models of norm-dependentpreferences, this is still traded o� against what they want to do if normative considerations were ir-relevant (i.e., against own-utility maximization). As we show in Kimbrough and Vostroknutov (2021),strategic considerations and heterogeneity in individual preferences for following norms come togetherto predict the actual decisions that people will make. With this caveat in mind, we next ask what kindsof moral rules we would expect to be salient on the basis of the A-model in social dilemmas, coordinationgames, and tournaments, under the assumption that social weights are equal for all players. For obviousreasons, heterogeneity in social weights may complicate ma�ers, but we will note a few cases in whicha li�le hierarchy would resolve an otherwise tricky normative ambiguity.

6.4 Moral Rules for Social Dilemmas

An obvious moral rule for social dilemmas is that players ought to cooperate, and indeed this is fre-quently, but not always, consistent with the injunctive norm as derived from the choice-set-dependentA-model. For intuition, consider a two-player Prisoner’s Dilemma game played by coequal strangers. Itis easy to see that the injunctive norm will generally favor the outcome cooperate/cooperate since theresulting payo�s are typically e�cient and relatively egalitarian in most prisoner’s dilemma games stud-ied by economists. �e exception arises if one player’s payo� from unilateral defection is so high thatthe injunctive norm favors the outcome cooperate/defect (or defect/cooperate); this is because e�ciencyconsiderations can dominate in such extreme cases. Nevertheless, in most social dilemmas the moralrule “players ought to cooperate” will neatly capture the injunctive norm.

We provide formal analysis that generalizes this point and consider a class of symmetric two-playersocial dilemmas in which the A-norm corresponds to the most e�cient outcome even though for bothplayers there are other consequences that bring them higher utility. Consider the set of payo� allocationsfor two players shown in Figure 8. Point (z, z) represents the most e�cient symmetric allocation, orcooperative outcome. Allocations (xi, yi) and (yi, xi) that are weakly less e�cient (xi + yi ≤ 2z) andsatisfy xi ≤ z, yi ≥ z for all i represent the possible unilateral defection outcomes that give more payo�

23

z yi

xi

yi

(a,b)

Payoff of P1

Pay

off

of P

2

z

x + y = 2z

xi

Figure 8: Symmetric two-player social dilemma.

to the defector than in the cooperative outcome and less to the other player. Finally, arbitrary points(ai, bi) with ai ≤ z and bi ≤ z for all i represent the possible mutual defection outcomes.

�is is a rather general class of games that includes most Prisoner’s Dilemmas (with e�ciency of thecooperative outcome restricted to be at least as high as the defect-cooperate outcome), the two-playerPublic Goods game, and even the Dictator game as a special case. �e following proposition shows thatthe allocation (z, z) is the A-norm.

Proposition 8. For 2 players consider the set of payo� vectors that consists of 1) point (z, z); 2) n pairs ofpoints (xi, yi) and (yi, xi) with xi + yi ≤ 2z and such that xi ≤ z for all i = 1..n and z ≤ y1 ≤ y2 ≤ ... ≤yn; 3) any �nite number of other points (ai, bi) with ai ≤ z and bi ≤ z. �en (z, z) is the A-norm (has theleast A-dissatisfaction).


To provide additional intuition about what kind of behavior is implied by the injunctive norm insocial dilemma games consider Figure 9 that illustrates the aggregate dissatisfaction functions in a 2-player Public Goods game (using parameters derived from Fehr and Gachter, 2000) and a Trust game(using parameters from Berg et al., 1995). On both graphs the points on the 2D plane are the payo�s thatplayers can obtain and the color codes the aggregate dissatisfaction, with dark red being the outcomewith the least dissatisfaction and dark blue the outcome with the most dissatisfaction.

�e most appropriate consequence in the PG is for both players to contribute the whole endowment,which follows from Proposition 8, and the most inappropriate consequence is to contribute nothing. �enormatively best outcome satis�es moral rules about maximizing e�ciency and equality.

In the TG the most appropriate consequence is for the �rst mover to send everything to the secondmover (1 token), and for the second mover to return slightly more than half of the resulting amount

24

Payoff of the first mover

Payoff

of

the s

eco

nd

mover

Payoff of Player 1

Payoff

of

Pla

yer

2

Figure 9: Le� graph: the choice-set-dependent injunctive norm derived from the A-model in a 2-player PublicGoods Game. Right graph: the choice-set-dependent injunctive norm derived from the A-model in a Trust Game.Dark red shows the most appropriate consequence and deep blue shows the least appropriate one.

(second mover returns 1.66 tokens and keeps 1.34 tokens). �is should not come as a surprise since thepayo�s in the Trust game are not symmetric and do not satisfy the assumptions of Proposition 8. It isalso worth noting that for any choice of the �rst mover, the A-norm prescribes that the second moverought to return around half the resulting money (which is equal to the amount sent times 3) back to the�rst mover. �at is, the norm favors what looks like positive reciprocity. We discuss the connection toreciprocity more thoroughly in Kimbrough and Vostroknutov (2021).

6.5 Moral Rules for Coordination Games

In coordination games, the ability of the injunctive norm derived from the A-model to generate a co-herent moral rule depends on what other assumptions we make about payo�s. In coordination gameswith multiple Pareto-ranked equilibria (minimum e�ort games, stag hunts), the injunctive norm willalways select the Pareto-optimal equilibrium as the normatively best outcome. By contrast, in gameswith symmetric, non-Pareto-ranked equilibria (e.g., matching pennies, ba�le of the sexes), the injunctivenorm provides li�le guidance if players are treated identically in aggregating dissatisfaction. However,in these kinds of se�ings, heterogeneity across players can help resolve normative ambiguity, as suchgames are more likely to have a unique normatively best outcome if the players’ utilities are not weightedidentically in computing the norm (e.g., if it is one of the two players’ birthday in the ba�le of the sexes).

6.6 Norm-Free Choice Settings

While injunctive norms apply to a broad array of choice se�ings, there are some se�ings in which the in-junctive norm simply provides no guidance about what one ought to do. �at is, there exist environmentsin which the normative evaluation implied by the A-model is constant across all feasible consequences.

25

Let the tuple 〈N,C, u〉 that consists of the set of players, some set of consequences C , and a utilityfunction u be called an environment and consider the following de�nition.

De�nition 1. Call an environment 〈N,C, u〉 norm-free if D(x) ≡ d, where d is some constant.

We do not (yet) have a characterization of the set of norm-free environments in terms of payo�s,and in fact, we suspect that there are no intuitively interpretable constraints that de�ne them. Anyenvironment with two consequences and constant e�ciency is norm-free, as well as, for example, theenvironment with two players de�ned by C = {a, b, c} and u(a) = (0, 3), u(b) = (3, 0), u(c) = (1, 1).Nevertheless, there is an important class of norm-free environments that we would like to describe.�ese are the environments that have a structure reminiscent of a tournament.

De�nition 2. Call 〈N,C, u〉 a tournament if there are prizes xi ∈ R for i = 1..N such that C is the setof all 1-to-1 functions c : N → {x1, x2, ..., xN} and u(c) = (c(i))i∈N .

In words, in a tournament N players “compete” for prizes in the set {x1, x2, ..., xN}. Each prize isassigned to some player, and the set of consequences consists of all such assignments, as in professionalsports or poker tournaments. Note that the prize could be winner-take-all such that an indivisible objectwill be assigned to one of the players. An important property of tournaments is that they are norm-free.We prove this result in a proposition.

Proposition 9. Any tournament is norm-free.


�us, built into our model of norms that account for the dissatisfaction of all interested parties, isthe existence of situations in which norms have no bite. Under our model of norm-dependent utility(Kimbrough and Vostroknutov, 2021), players in such se�ings are expected to behave as self-interestedutility maximizers, which seems a reasonable assumption about players’ motivations in tournaments.

7 Discussion

Our paper is motivated by the mounting evidence that economic decision-making re�ects not only self-interested payo� maximization but also normative considerations that allow us to cooperate, coordinate,and pursue common goals. �is evidence has led to fruitful modeling e�orts that capture these o�en-con�icting motivations in a norm-dependent utility function, which models people as trading o� whatthey ought to do against what they want to do from the point of view of consumption utility. �is callsfor formal theories of how agents determine what they ought to do in any situation. Our work addressesthis call by bridging a gap between cooperative and non-cooperative game theory.

Our innovation is to reinterpret axiomatic models of the kind developed in cooperative game theory

26

as theories of moral reasoning and norm formation. Typically such reasoning is conducted by modelerswho use the normative rankings of outcomes from such models to assess whether a normatively desirableoutcome (as derived from the axioms) has been achieved by agents who do not necessarily share this ideal.We argue that instead, it makes sense to put the moral reasoning into the minds of the decision-makersthemselves. �is approach allows us to combine the best elements of cooperative and non-cooperativegame theory, using the former to identify the kinds of allocations that norm-following agents will aimat and the la�er to identify conditions under which those allocations are likely to actually be a�ained.

We illustrate the potential of this method by developing an axiomatic model of injunctive normsbased on plausible assumptions about human moral psychology. We assume that individuals’ normativejudgments arise from their consideration of the entire set of feasible outcomes and their prospective dis-satisfaction with each outcome as compared to all counterfactual outcomes. To turn individual rankingsof dissatisfaction into an injunctive norm, we further assume that people employ empathy, in the sensethat they know and consider the fact that all other people feel dissatisfaction in an analogous manner.We thus assume that injunctive norms emerge that rank outcomes according to the aggregate dissatis-faction that they would produce, with the most appropriate outcome in the feasible set being the onethat minimizes aggregate dissatisfaction of interested parties.

We highlight that injunctive norms derived from this model are radically choice-set-dependent. Howappropriate a particular outcome is depends crucially on the other possible outcomes. In a companionpaper (Kimbrough and Vostroknutov, 2021), we show that a functional form derived from this model hassubstantial explanatory power in experiments and in particular can account for many observations ofradical context-dependence in choice that cannot be readily explained by standard theory or in modelsof outcome-based social preferences. However, in this paper we also provide a potential explanation forwhy social preference models have normative appeal (and why there are many cases in which they seemto �t the data well).

We introduce a second set of axioms in which dissatisfaction is computed not relative to all otherfeasible outcomes, but rather, relative to some ideal as speci�ed by a moral rule: for example, maxi-mize e�ciency or minimize inequality. We show how such ideals are able to closely approximate socialpreference models, and then we ask, for a variety of possible choice se�ings: how o�en does the choice-set-dependent injunctive norm derived from the �rst set of axioms correspond to the ranking implied bythe second set of axioms under a given moral rule? �is allows us to identify choice se�ings for whicha particular moral rule is likely to be salient, precisely because it accurately re�ects moral intuitions ascaptured in the injunctive norm.

�us, the model provides a meta-theory of moral rules, as approximations of radically choice-set-dependent injunctive norms. We argue that the complexity of computing injunctive norms and thedi�culty of summarizing their implications renders such approximations desirable so that normativegoals can be easily communicated. We then provide plausible moral rules for a variety of widely studiedchoice se�ings.

One �nal distinction between moral rules and our radically choice-set-dependent injunctive norms is

27

worth drawing out. Choice set dependence means that normative goals are always de�ned endogenouslyon the set of feasible allocations; moral rules, on the other hand, can de�ne the ideal in terms of possibil-ities that exist outside the feasible set. �us moral rules can have an aspirational quality; when the idealoutcome is outside the feasible set, knowledge of a moral rule may induce people to seek changes to thefeasible set.

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29

Appendix (for online publication)

A ProofsProof of Proposition 1. (1⇒ 2). By A1 di(t, r) = di(0, r−t). By A2, whenever r−t ≤ 0 we have di(0, r−t) = 0.When r − t > 0, by A3 it is true that di(0, r − t) = (r − t)di(0, 1). By A4 then di(0, r − t) = r − t. �us, we canwrite di(t, r) = max{r − t, 0}. N

(2⇒ 1). A1-A4 hold trivially. �

Proof of Proposition 2. (1⇒ 2). Take any �nite C with more than one element and take any x ∈ C . Enumeratethe elements of C :

C = {x1, x2, ..., xK} ∪ {x}.

By A5 Di(x | {x}) = 0 and by A6 Di(x | {x, x1}) = di(ui(x), ui(x1)). Add elements one by one and use A6repeatedly to get

Di(x |C) =K∑j=1

di(ui(x), ui(xj)) =∑

c∈C\{x}

di(ui(x), ui(c)).

as desired. N

(2⇒ 1). A5 and A6 hold trivially. �

Proof of Proposition 3. (1⇒ 2). For all t1, ..., tN ∈ R+ A2 implies G(t1, ..., tN ) = G(0, ..., 0) +∑

i∈N ωiti. ByA1, G(t1, ..., tN ) =

∑i∈N ωiti. �us, since Di satisfy A5-A6 and di satisfy A1-A4, we have

D(x |C) =N∑i=1

ωiDi(x |C) =N∑i=1

∑c∈C

ωi max{ui(c)− ui(x), 0}

as desired. N

(2⇒ 1). A7-A8 are trivial. We get A1-A6 from the proofs of Propositions 1 and 2. �

Proof of Proposition 4. (1 ⇒ 2). By B2, f(0, 0) = 0 and by B1, fi(t, r) = fi(0, r − t). So if r − t ≥ 0 thenfi(t, r) = fi(0, 1)(r − t) = r − t. �e last equality is by B4. When r − t < 0, by B2 and the above we havef(t, r) = βifi(0, t− r) = βi(t− r). So, together we can write

fi(t, r) = max{r − t, 0}+ βi max{t− r, 0}

as desired. N

(2⇒ 1). B1-B4 are trivial. �

Proof of Proposition 8. Let us begin with calculating the normative value of (z, z). �e points (ai, bi) areirrelevant for this since they are Pareto-dominated by (z, z) or equal to it. �us, they do not evoke dissatisfactionat (z, z). �e pairs of points (xi, yi) and (yi, xi) only in�uence dissatisfaction of (z, z) through yi’s and not xi’ssince they are less than or equal to z. �erefore, the dissatisfaction at (z, z) is

D(z, z) = 2n∑

i=1

(yi − z),

1

which is shown in Figure 10.

z y1yjy

3y

4

y1

xj

y3

y4

yj

(a,b)

Payoff of P1

Pay

off

of P

2

z-x j

y-zj

y-zk

z

x + y = 2z

y-z

i

Figure 10: Illustration of the dissatisfaction calculations.

Now, �x any index j and consider the point (yj , xj). �e dissatisfaction D(yj , xj) can be wri�en as

D(yj , xj) =

n∑i=1

(yi − z) + (n+ 1)(z − xj) +

n∑i=1

(yi − z)−∑k<j

(yk − z)−∑k>j

(yj − z) + δj .

Here (n + 1)(z − xj) is the additional dissatisfaction as compared to D(z, z) because of the points (xi, yi) and(z, z), the two sums with k is the additional dissatisfaction because of the points (yi, xi), and δj ≥ 0 is thedissatisfaction because of points (ai, bi) and the points (yi, xi) with xi ≥ xj . Figure 10 illustrates. �us, thedi�erence in dissatisfactions between D(yj , xj) and D(z, z) is equal to

∆ = (n+ 1)(z − xj)−∑k<j

(yk − z)−∑k>j

(yj − z) + δj = (n+ 1)(z − xj)−∑k<j

(yk − z)− (n− j)(yj − z) + δj .

Using the assumed condition xi + yi ≤ 2z, which is the same as z − xj ≥ yj − z, we get

∆ ≥ (n+ 1)(yj − z)−∑k<j

(yk − z)− (n− j)(yj − z) + δj

or∆ ≥ (j + 1)(yj − z)−

∑k<j

(yk − z) + δj .

�is can be rewri�en as∆ ≥ 2(yj − z) +

∑k<j

(yj − yk) + δj ≥ 0.

�e last inequality follows from the assumption that z ≤ y1 ≤ y2 ≤ ... ≤ yn. �erefore, the dissatisfaction ofany point (yj , xj) is weakly higher than that of D(z, z). Points (ai, bi) also have higher dissatisfaction than (z, z)because they are Pareto-dominated by it (see Proposition 1 of Kimbrough and Vostroknutov (2021)). �is makes(z, z) the A-norm. �

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Proof of Proposition 9. Let 〈N,C, u〉 be a tournament. Set C consists of N ! consequences corresponding toall possible assignments of the prizes, and in each consequence each payo� from {x1, x2, ..., xN} happens onlyonce. Assume without loss of generality that x1 ≤ x2 ≤ ... ≤ xN . If we look at the payo�s of player i in allconsequences, we �nd that she receives any payo� xj in (N − 1)! consequences. Slightly abusing notation, wecan express the dissatisfaction of the consequence that gives player i payo� xj as

Di(xj) = (N − 1)!N∑

`=j+1

(x` − xj).

�is amount is the same for all players. Since in each consequence each payo� happens exactly once, the aggregatedissatisfaction is the same for each consequence. �

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axiomatic models of injunctive norms and moral rules

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