Temptation and Guilt�

Jawwad Noor Linxia Ren

January 9, 2011

Abstract

This paper builds a model of guilt on the observation that guilt-avoiding behavior must itself be cause for guilt. The agent experiencesguilt when she submits to any temptation, but the anticipation of guiltin turn generates a temptation to avoid guilt. In an application tosocial preferences, where the literature explains behavior by a concernfor social image, the model uses temptation and guilt by sel�shnessto accommodate all key �ndings in experiments on dictator games.The model also yields welfare implications di¤erent from those in theliterature. Results are reported from an experiment that suggests theexistence of a temptation to be sel�sh.

Keywords: Guilt, normative preference, temptation, social prefer-ence, dictator games, welfare.

1 Introduction

Guilt is the painful experience that accompanies a �bad�choice. This de�ni-tion presumes the existence of a normative preference �the agent�s personal

�Noor (email: jnoor@bu.edu) and Ren (email: lxren@bu.edu) are at the Dept of Eco-nomics, Boston University, 270 Bay State Road, Boston MA 02215. We thank BartLipman, Philipp Sadowski, Mike Luca, Rob Munger, Sean Horan, seminar participants atBoston University and participants at the Far East Econometric Society Meeting (2009)for useful comments. Ren gratefully acknowledges �nancial assistance from the Instituteof Economic Development at BU. The usual disclaimer applies.

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view of what should be chosen �which is the standard against which a choiceis judged to be good or bad. The de�nition also presumes the existence offactors that may cause choice to deviate from normative preference. Thesefactors may be attributed to the agent�s desires �refer to these as her temp-tation preferences. The agent�s choices are thus a struggle and a search fora compromise between normative and temptation preferences. When nor-mative and temptation preferences con�ict, pain is inevitable: good choicesrequire the painful exertion of self-control, but bad choices entail the painfulexperience of guilt.

1.1 A Theory of Guilt

What behavior reveals that an agent experiences guilt? A response thatcomes naturally to economists is that the experience of any painful emo-tion gives rise to pain-avoiding behavior. Thus, it is natural to hypothesizethat the behavioral implication of guilt is guilt-avoidance. Evidence of guilt-avoidance is suggested by experiments in social psychology and in economics.Ehrich and Irwin [16] demonstrate that consumers avoid obtaining informa-tion on the ethicality of products, such as whether the wood used in a pieceof furniture comes from endangered rain forests, or whether a cell phone wasmade by a company with overseas factories that employ child labor. The re-searchers �nd that such �willful ignorance�manifested most strongly amongthose people who had claimed to care most about the ethical issue at hand.That is, those that are likely to experience guilt more intensely also exhibita greater tendency to avoid information that could cause guilt.While it may seem natural that a model of guilt should be built around

guilt-avoidance just as a model of any painful emotion should be built aroundpain-avoidance, this paper takes the view that guilt-avoidance is a di¤erentphenomenon compared to usual forms of pain-avoidance. An agent�s actionto avoid (say) risk can be taken at face value: the fact that risk is painful tobear is a complete explanation for risk-avoiding behavior and has the obviouswelfare implications. We hold that this does not necessarily hold true forguilt-avoidance. While pain-avoidance is typically never itself painful, weargue that guilt-avoidance should itself be cause for guilt. If guilt is causedby a betrayal of one�s resolve toward normative goal, then guilt-avoidance issuch a betrayal as well. In the context of the earlier example, if an agentconsiders it wrong to knowingly consume an unethical product, he must alsoconsider it wrong to willfully avoid information on the ethicality of a product.

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In a word, while it is natural (and also suggested by experimental evidence,such as Ehrich and Irwin [16]) that agents who experience guilt also desire toavoid guilt, we believe that a complete theory of guilt should also explain whyguilt-avoidance might itself generate guilt. In fact we believe it is necessarythat it does. Other than completing the conceptual framework, the answerwill suggest both the behavioral implications and welfare implications of thedesired theory.We make sense of �guilt by guilt-avoidance� in the following way: We

hypothesize that the desire to avoid guilt is not a normative goal, but ratheritself a temptation. We hypothesize that it arises because opportunities ofguiltless indulgence are more tempting than those of guilt-ridden indulgence.This induces a temptation to avoid guilt. It also completes the picture: Itexplains why a moral agent may try to avoid guilt, and it explains why guilt-avoiding behavior may itself give rise to guilt. (We discuss implications forbehavior and welfare below in the context of our application.)Based on this analysis, we model an agent whose choices are determined

by the maximization of an �aggregation�of normative and temptation pref-erences less guilt costs and self-control costs. This describes both how anagent chooses from a menu (choice problem) and also how he selects a menufor himself to face in the future. The normative preference over menus takesinto account the normative value of the choice in a menu and the associatedself-control costs �in particular, guilt-avoidance is not a motive. The temp-tation preference over menus takes into account the temptation value of thechoice in a menu and the associated guilt costs �guilt-avoidance is a motivehere.1

This paper contributes to the literature on self-control problems (Laib-son [29], Gul and Pesendorfer [22]) by studying what we consider to be anintegral part of the problem of dealing with temptation, and by identifying aconceptual nonlinearity associated with it.2 Our second contribution is the

1The formal framework for studying temptation introduced by Gul and Pesendorfer[22] is now su¢ ciently well-developed in the decision theory literature to accommodateour theory of guilt (Noor [34], Kopylov [26], Kopylov and Noor [28]). We outline ourformal model in Section 2 and provide behavioral foundations in Section 5.

2This paper has limited overlap with the work on guilt by Battigalli and Dufwenberg[5]. These authors de�ne guilt by the utility loss experienced when behavior falls shortof others�expectations. In contrast, in our model, guilt is the consequence of behaviorfalling short of an abstract normative preferences, which could in principle depend onexpectations, though we do not model this. (See Ellingsen et al [17] for experimentalresults that suggest insigni�cant a¤ect of others� expectations on behavior.) Another

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application detailed below. While much theoretical work has been done onmodels of agents with the propensity for self-control (see Gul and Pesendorfer[22] and subsequent literature, Fudenberg and Levine [19]) there have beenrelatively few demonstrations of the empirical relevance of the propensityof self-control (Fudenberg and Levine [19, 20, 21], Noor and Takeoka [35]).This is in contrast with the many applications of the dynamic inconsistencyassociated with temptation. We provide a demonstration of the relevance ofthe notion of self-control in the context of social preferences.

1.2 Application to Social Preferences

We apply our model to the literature on social preference. The applicationexploits the behavior generated by the model, which includes guilt-avoidance.�Guilt by guilt-avoidance�remains in the background, manifesting itself onlyin our stance on welfare.Various theories in social psychology try to explain why moral people be-

have immorally. It may be because moral values have not been taught wellenough for them to withstand temptation, or it may be because people�s ca-pacity for selective perception and rationalization enables moral disengage-ment in particular situations (Bandura [6, 7]). A relatively recent theoryadds a third possible explanation: the key motivation for moral behaviormay come from peoples�desire to appear moral. Indeed, studies �nd that ifagents can avoid having to behave morally and pursue self-interest instead,they do. This behavior is referred to as moral hypocrisy (Batson et al [4]).Some of the evidence is as follows:� In seminal work in psychology, Batson et al [4] conducted an experi-

ment where subjects were required to allocate two tasks between themselvesand a partner. The �desirable�task came with the opportunity to win moneywhile the �undesirable� task was dull and payed nothing. They were toldthat their partner would not be informed that they were allowed to assignthe tasks. Subjects were given the opportunity to privately �ip a fair coin inorder to help them make the allocation decision. The researchers found thatthe coin �ippers allocated the desirable task to themselves 90% of the time,even though they privately �ipped a fair coin. In a follow-up questionnaire,75% of subjects who assigned themselves the desirable task indicated that

di¤erence is that Battigalli and Dufwenberg [5] focus on strategic settings whereas wefocus on a single decision maker.

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they believed that the morally correct thing to do was to assign it to thepartner. The results have been replicated multiple times in the psychologyliterature.� In economics, moral hypocrisy has been demonstrated in dictator game

experiments. A �dictator�is asked to allocate an endowment between himselfand a passive �recipient�. Lazear et al [30] �nd that while (experienced)dictators share with recipients on average 20-30% of their $10 endowmentwhen playing the dictator game, 50% of them exit the game with the fullendowment when given the option. Dana et al [10] �nd that a signi�cantproportion exit the game even at a cost. Hammam et al [24] �nd that,when given the option, dictators delegate the endowment allocation decisionto agents who tend to be more favorable toward the dictator. Dana et al[11] demonstrate that agents exploit moral wiggle room (such as uncertaintyabout the outcome of an action for others) in order to behave sel�shly. Thus,dictators that share, thereby appearing to possess a sense of morals, alsoexploit opportunities that increases their share of the pie.� Anecdotal evidence includes the observation that people who con-

tribute to beggars when they encounter them may also cross the street toavoid encountering them.While initial research on fairness and social preferences suggested the

existence of an intrinsic concern for altruism and fairness, the newer researchundermines this conclusion. It suggests that observed moral behavior isnot indicative of the true extent of an agent�s moral imperative, but ratheroverstates it because it may be the result of social pressure. This view hasgained some favor in economics, particularly in the study of social preferencesand the dictator game (see also Neilson [32], Andreoni and Bernheim [2], andDella Vigna et al [12]). Our model of guilt suggests a di¤erent relationshipbetween observed behavior and unobserved moral imperative: Choices re�ectthe interaction of a personal moral imperative and the temptation to besel�sh. Behavior in the absence of social pressure understates the extent of anagent�s moral imperative because of the heightened in�uence of temptation.Social pressure may enable an agent to better align his choices with hismoral perspective. We show that the key comparative statics observed inexperiments on the dictator game are all natural implications of our model.Does it matter whether the world is viewed as populated by amoral moral

hypocrites rather than by temptation-stricken moralists? The answer is yes,on both positive and normative grounds. (The following discussion also high-lights the peculiar behavioral and welfare implications of our theory of guilt).

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On the positive side, we show that the moral hypocrisy and moral in-tegrity are not observationally equivalent. Speci�cally, we argue that to theextent that choices from a �distance�(by some appropriate measure of dis-tance) are less subject to temptation, temptation-stricken moralists wouldbehave more morally from a distance. In particular, their behavior maychange with distance. In contrast the behavior of moral hypocrites shouldremain unchanged, as the driving force behind their behavior is how theyappear to others, which is not a function of distance. We conduct a versionof the dictator game experiment and �nd that behavior indeed becomes moregenerous with temporal distance.On the normative side, there is the question of whether excuses or oppor-

tunities for immoral behavior should be provided. The welfare implication ofmoral hypocrisy is that agents may be better o¤ if they can avoid the costs ofbehaving morally. Thus, Batson et al�s subjects are better o¤ when they aregiven the coin, and dictators in dictator games are better o¤ when they aregiven the silent exit option. In contrast, temptation-stricken moralists areworse o¤, since the coin or the silent exit option puts them in the presenceof temptation. Indeed, social pressure may make the agent better-o¤, as itmay serve as a commitment device.

The remainder of the paper is organized as follows. Section 2 presents ourmodel of guilt. Section 3 discusses the application to dictator games. Sec-tion 4 presents the results of our experiment. Section 5 provides behavioralfoundations for our model and also de�nes and characterizes a comparativemeasure of proneness to guilt. Section 6 discusses related decision-theoreticliterature. Section 7 concludes. All proofs are contained in appendices.

2 A Model of Guilt

Denote the space of alternatives by �. A menu (choice problem) is a non-empty subset of �, and the set of menus is denoted byM. Generic elementsof � are x; y; z while those ofM are a; b; c.The agent has a personal view of how she should behave when faced with

a menu. This view may re�ect ethical concerns, but more generally maybe her perspective on what is the right balance between social norms andself-interest, for instance. We refer to this perspective on how she shouldbehave as her normative preference. Her actual choices are also in�uenced

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by desires �refer to these as her temptation preferences. Choices are thus astruggle and a search for a compromise between normative and temptationpreferences. When normative and temptation preferences con�ict, pain isinevitable: good choices require the painful exertion of self-control, but badchoices are accompanied by the painful experience of guilt.Observe that the notions of normative and temptation preferences are

necessary for any discussion of guilt. Since foundations for the notions ofnormative and temptation preferences have been provided in the work ofGul and Pesendorfer [22] (henceforth GP), we take inspiration from it andwrite down an extension of their model that accommodates guilt.

2.1 Choice

We provide here an outline of the model and specify how choice is deter-mined by it. For the theoretically-inclined readers: the formal model and itsbehavioral foundations are presented in Section 5.

Choice from a Menu. Normative and temptation preferences over alter-natives are represented by utilities u; v : � ! R respectively. When facedwith a menu a 2 M, the agent�s choice C(a) from the menu is the set ofmaximizers:

argmaxx2a

�u(x) + v(x)�

hmaxz2a

u(z)� u(x)i��maxy2a

v(y)� v(x)��(1)

= argmaxx2a

fu(x) + v(x)g : (2)

We �rst interpret (1) and then verify that it equals (2). Begin with some intialobservations: The normatively best choice in the menu a solves maxz2a u(z)whereas the most tempting choice solves maxy2a v(y). An alternative x 2 awill satisfy normative and temptation preferences to varying degrees. To theextent that x may not respect normative preferences, the agent su¤ers thecost of guilt [maxz2a u(z)� u(x)] � 0. If x does not satisfy desires, she feelsthe frustation measured by [maxy2a v(y)� v(x)] � 0. This can be interpretedas the cost of self-control, which by de�nition must be proportional to thedegree of frustration of desires.Given these observations, (1) is interpreted as follows: a choice x from

menu a yields normative utility u(x) less guilt costs, and temptation utilityv(x) less self-control costs, and the agent maximizes the sum of these costs.

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Observe that actual costs are not of any consequence in describing choicefrom a menu. Leaving aside what she �experiences�in the menu, the agent�schoice at the end of the day simply maximizes the sum of normative andtemptation utilites: since maxz2a u(z) and maxz2a v(z) are constants when ais given, they are sunk at the time of choice and thus do not determine choice.Thus, her choice from a menu seeks a compromise between her normativeperspective and her desires �this is re�ected in (2).3

The experience of guilt and self-control costs reveals itself in a di¤erentdomain, to which we now turn.

Choice between Menus. Our study will involve not only a discussionof how an agent would choose from menus but also how she would choosebetween menus. This will be required, for instance, to model the �exit option�in experiments on the dictator games (see the Introduction), and also tocapture the a¤ect of guilt and self-control costs. Below we will refer tochoice of menu as taking place in some period 1, and subsequent choice froma menu as taking place in period 2.

t=1�choose menu a

"temptation to avoid ex post guilt

� � �t=2�

choose alternative x2a"

temptation and guilt

The kind of considerations entering period 2 choice will exist also in period1. Thus, the agent will be presumed to have a normative preference andtemptation preference over menus, just as she does over alternatives as above,and her choice of menu will be subject to temptation and guilt as well. Thisis formalized as follows: given a menu of menus A, her choice is

C1(A) = argmaxa2A

�U(a) + V (a)�

�maxb2A

U(b)� U(a)���maxc2A

V (c)� V (a)��

= argmaxx2a

fU(a) + V (a)g : (3)

That is, her choice of menu seeks a compromise between her normative andtemptation preferences over menus, which are represented by U and V resp,and described next.

3Though still giving rise to (2), an alternative to (1) that is familiar from the literatureon temptation is considered at the end of Section 6.

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Normative and temptation preferences over menus will be derivative con-cepts. Her normative preference over menus is represented by a utilityU :M! R of the GP form

U(a) = maxx2a

[u(x)��maxy2a

v(y)� v(x)�]: (4)

The agent cares about three things: her time 2 choice from that menu, thenormative utility of that choice, and the self-control cost associated with it.4

Thus, she normatively prefers menus that have normatively better choicesand lower self-control costs. A key hypothesis here is that she does not careto avoid guilt costs. We justify this hypothesis in next Section.Her temptation preference over menus is represented by a utility V :

M! R of the form

V (a) = �maxx2a

[v(x)��maxy2a

u(y)� u(x)�]: (5)

The parameter � > 0 represents her relative time discounting (we normal-ize the normative discount factor to 1). Like the normative perspective, thetemptation value of a menu is sensitive to three things: her period 2 choicefrom that menu, the temptation utility of that choice, and the guilt costassociated with it.5 Thus, she is tempted by menus that lead to more tempt-ing choices and lower guilt. In particular, this expresses our key hypothesisthat guiltless indulgence is more tempting than guilt-ridden indulgence. In-tuitively, guilt dulls the pleasure from the consumption of tempting alterna-tives. For instance, it may be harder to enjoy an expensive dinner when abeggar (an option to be charitable) is visible from the window. Hence, inperiod 1 the agent is tempted to avoid guilt.

4It is worth observing the internal consistency of the model. Note that

argmaxx2a

[u(x)��maxy2a

v(y)� v(x)�] = argmax

x2a[u(x) + v(x)] = C(a):

Therefore, the maximizer x 2 a in the de�nition of U is indeed the period 2 choice from a.5Note again the the internal consistency of the model:

argmaxx2a

[v(x)��maxy2a

u(y)� u(x)�] = argmax

x2a[v(x) + u(x)] = C(a):

Therefore, the maximizer x 2 a in the de�nition of V is indeed the period 2 choice froma.

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This completes our description of the model. Observe that, analogous toperiod 2, the choice (3) is such that self-control and guilt costs associated withthe choice of menu do not express themselves in period 1, except through theimplication that the agent seeks a compromise. Note however, that period 2self-control and guilt costs express themselves in period 1 choice, since theyenter U and V . The agent�s normative preference underlying U strugglesto avoid temptation and self-control costs while the temptation preferenceunderlying V seeks temptation but avoids guilt costs.

2.2 Welfare and Guilt-Avoidance

In our model, the agent�s welfare is de�ned by her normative perspective.That is, her normative perspective is her personal welfare criterion, and itindicates what she feels the �best�thing to do. Her normative perspectiveis u in period 2 and U in period 1. Unlike traditional revealed preferencetheory, there is a wedge between choice and welfare �choice in any periodis �contaminated�with temptation. Unlike behavioral economics, the agent�s�experienced utility�is not her standard for welfare. For instance, guilt costsare not a consideration for our agent�s normative preference. This last claimrequires justi�cation, to which we now turn.For concreteness, suppose that there is a �good� option g and a �bad�

option b. Suppose further that the agent submits to the temptation by b inperiod 2 when faced with a choice from fg; bg:

C(fg; bg) = fbg:

In period 1, she anticipates her choice and recognizes that her choice will beaccompanied by pangs of guilt. Contrary to our hypothesis, suppose thather normative perspective takes the view that she should avoid guilt. Thenthe key observation to make is that in period 1, for some option B thatis �worse� than b, it will be the case that fBg is normatively preferred tofg; bg. The reason is that even if B is a worse option, its consumption isguiltless,6 whereas the bad option is chosen with guilt in fg; bg. Therefore, ifa guilt-avoidance motive exists in normative preference, then it would makeit normatively acceptable for to commit to a worse option. We �nd this

6Observe that in a singleton menu fBg, the only option is to consume B therefore thereis no room to feel guilty. Guilt would be experienced if a better option was a choice, as itwas in fg; bg.

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uncompelling as a feature of normative preference. If guilt is caused bya betrayal of one�s resolve towards normative goals, then guilt-avoidance�such as in this example �involves precisely such a betrayal. If the agentbelieves she should be moral/other-regarding/fair, then the act of committingto being immoral/sel�sh/unfair is itself immoral/sel�sh/unfair. In a moralcontext, if being immoral is cause for guilt, then it is also true that �to wantto be immoral is to be immoral�(Elster [18, pg 65]).7 Indeed, we believe thatguilt-avoidance should itself be cause for guilt, and thus that it cannot be afeature of normative preference.In our model, a guilt-avoidance motive does indeed exist, but its source

is temptation preferences, not normative preferences.

3 Application to Dictator Games

In this section we show that an appropriately specialized version of our modelcan unify the existing evidence on dictator games. A generic alternativeavailable to a dictator is an allocation x = (x1; x2) where x1 denotes herown consumption and x2 denotes the recipient�s consumption. Consumptionmay potentially be lotteries. A dictator playing the game faces a menu a ofpossible allocations. If the endowment is (M; 0) and the dictator can shareany part of her endowment, then we denote her menu by

dg = f(M � s; s) : 0 � s �Mg:

The option to exit the game with $M is the singleton menu:

e = f(M; 0)g:

If the dictator is o¤ered a choice of either playing the game or exiting, shefaces the menu of menus:

A = fdg; eg :Assume that a dictator�s normative preference is for an equal division

of the pie, and that the temptation preference is to maximize own materialpayo¤. Normative utility is u and temptation utility is �v, where � is a scalar

7An illustration in an interpersonal context also helps make the point: guilt-avoidanceis like a parent forcing his child to smoke so that the child does not feel guilty about beinga smoker.

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that parametrizes the intensity of temptation. Note that the dictator�s choicefrom any menu can be written as

C(a) = argmaxx2a

��u(x)�

hmaxz2a

u(z)� u(x)i�+ �

�v(x)�

�maxy2a

v(y)� v(x)���

= argmaxx2a

fu(x) + �v(x)g :

Evidently, greater intensity � of temptation is associated with lower self-control. Finally, the dictator�s choice over menus is given by

C1(A) = argmaxx2a

fU(a) + V (a)g :

where the parameter � in the model has been set to 1 for simplicity.We assume that u(M � s; s) is a hump-shaped function of s, with the

maxima at s = M2. That is, the agent normatively desires an even split.

Assume also that v(M � s; s) is just a function of own payo¤, and therefore,v is strictly decreasing in s. Both functions are twice di¤erentiable andconcave in s. An immediate observation is:

Proposition 1 For a dictator with temptation intensity �, denote the choiceby

C(dg) = f(M � s�; s�)g:Then s� is decreasing in �:

3.1 Evidence Accommodated by Model

We summarize evidence and show that our model can accommodate it.

Dictators tend to share though they share less than 50%: In ourmodel, since C(a) maximizes u + �v the agent�s choice is a compromise be-tween her normative desire to share 50% and her temptation to share nothing.Thus, the agent will tend to share but not as much as 50%.

Dictator exhibit a strict preference for exit: Dana et al [10] �nd thatover 25% of dictators exit at a cost (speci�cally, they exit with $9 when thepie is worth $10). Broberg et al [9] �nd that 64% of their dictators exhibitwillingness to exit the game for as little as 82% of the pie.

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In our model, since C1(A) maximizes U+V and since V maximizes temp-tation utility less guilt costs, the agent may exhibit a strict preference forcommitting to a sel�sh option even at a price �commitment to a sel�sh op-tion implies that there will be no ex post guilt from consuming that option.If today�s guilt from choosing this option is not overwhelming, the agent willcommit to the sel�sh option.

Proposition 2 There exists ��; �� s.t. for all �� � � � ��,

C(dg) = f(M � s; s)g, s > 0, and C1(fdg; eg) = feg

The proposition tells us that there are values of � for which the agentmay share when playing the dictator game but also strictly prefer to exit ifgiven the opportunity. At these values, there is enough self-control to sharewhen playing the dictator game, but not enough self-control for guilt to berelatively unimportant and thus for normative preference to overcome thetemptation desire to exit.

Dictators that share less also care less about exit: Dana et al [10] �ndthat of those subjects who (under anonymity) o¤er nothing to the receiver,only 1 of 24 subjects took the exit option. Broberg et al [9] �nd that subjectswho o¤er nothing also value exit less than those subjects who o¤er positiveamounts.In our model if C(a) o¤ers nothing then that is indicative of high �. That

is, the intensity of a temptation to be sel�sh is high. For the same agents,C1(A) would care less for commitment: observe that V maximizes temptationutility less guilt costs and that higher � implies a relatively lower importanceof guilt costs. Since the preference for commitment comes from guilt costs,this implies a lower desire to exit at a price.The reservation price c for the exit option is de�ned implicitly by

U(f(M � c; 0)g) + V (f(M � c; 0)g) = U(a) + V (a):

Higher the value of c, the less willing an agent is to exit at a given price.

Proposition 3 There exists ��� s.t. @c@�< 0 for all � � ���.

While dictators with lower self-control will share less with recipients, theproposition tells us that for su¢ ciently low self-control, dictators will notonly share nothing but will also not value commitment highly.

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The cut o¤ point ��� is strictly less that the �� in Proposition 2. Agentswith temptation intensity ��� � � � �� share with recipients, but the morethey share, the higher their temptation preference for exiting.

Dictators prefer less information: Dana et al [11] show that subjects be-have more sel�shly when they have �moral wiggle room�. In their experiment,subjects had the option of playing a random version of the dictator game:

a = f(6; `); (5; `)g;

where ` = (12; 5; 1

2; 1) is the lottery yielding $5 or $1 to the recipient with even

probabilities. Dictators also had the option of resolving the uncertainty, andbeing left with one of the following dictator games with 50% probability:

a1 = f(6; 5); (5; 1)g or a2 = f(6; 1); (5; 5)g:

Observe that in a1 the sel�sh choice is also fairest choice. Approximately half(44%) of the subjects preferred not to reveal the information, and of thesethe majority (86%) chose the sel�sh option.Our model would explain this as follows. When faced with a the agent

may guiltlessly choose to be sel�sh as her choice does not a¤ect the distribu-tion of the recipient�s outcome. Acting sel�shly in a1 would also be guiltlessbut not in a2. Therefore, when � is high (so that she expects to act sel�shlyin any menu), not revealing the information permits a sel�sh action withoutguilt, whereas revealing the information would lead to a sel�sh action withstrictly positive expected guilt. This will be re�ected in �rst period choiceif the �rst period guilt from such a choice is not overwhleming. Denote by(12; a1;

12; a2) then lottery over a1 and a2.

Proposition 4 There exists ���� s.t. C1(fa; (12 ; a1;12; a2)g) = fag for all

� � ����.

3.2 Accommodating the E¤ect of Anonymity

In order to accommodate the �nding that dictators�choices are di¤erent un-der anonymity, we consider an enriched version of our model by hypothesizingthat � is in fact a function:

� = �(o);

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where o is some measure of how observable the agent�s choice is to the re-cipient. We hypothesize that intensity of temptation is decreasing in observ-ability.8

Dictators share less under anonymity: Dana et al [10] �nd that whileonly 24% of dictators o¤ered nothing to receivers in the standard dictatorgame with no anonymity, 46% o¤ered nothing under anonymity. Similarly,Lazear et al [30] �nd that only 19% of dictators o¤ered nothing under noanonymity, but 33% o¤ered nothing under anonymity. For a related experi-ment that varies �social distance�, see Ho¤man et al [25].In our model, since C(a) maximizes u+�v and since � is hypothesized to

increase under anonymity, it follows that choices become more sel�sh withanonymity.

Dictators seek anonymity: In Andreoni and Bernheim [2], the playerswere given the endowment (20; 0). The dictator had to decide how much toshare. His choice was enforced with probability 1 � p, and with remainingprobability nature would determine the allocation. Speci�cally, nature wouldenforce each of (19; 1) or (1; 19) with probability p

2. Recipients were not

informed of the share o¤ered by the dictator. It was found that the fractionof subjects o¤ering (19; 1) increased with p and those o¤ering more fair splitsdecreased.Two observations are relevant here to see that our model can accom-

modate this. First, while unfair allocations yield guilt, as p increases theexpected guilt reduces. Thus, dictators would be more likely to make unfairallocations. Second, the observability o of the agent�s choices is a function ofthe share o¤ered. For instance, if the agent o¤ers (20; 0) and this is enforced,then recipients will deduce the agent�s o¤er, but if the agent o¤ers (19; 1)then the recipient is less sure whether this was due to a move by nature orbecause it was the dictator�s o¤er. Since � increases with lower o, the agent istempted to lower o. Indeed, as p reduces, attempts to maximize temptationstrengthen (due to lower expected guilt cost) and this includes attempts atlowering o.

8An axiomatization of this extended model would involve that the ex ante preference%0be indexed by observability o. Then a condition across %0�s is required that re�ects theidea that self-control reduces with o.

15

3.3 Accommodating Menu-Dependence

In order to accommodate evidence of menu-dependence of dictators�choices,we consider an enriched version of our model by hypothesizing that � is afunction of the form

� = �(maxy2a

v(y))

where maxy2a v(y) is the temptation value of the most tempting option avail-able in the menu a. Observe that this introduces menu-dependence into �.We hypothesize that intensity of temptation is increasing in �maxy2a v(y)�.9

This says that self-control becomes weaker in the presence of greater temp-tation.

Dictators give less when taking is also an option: List [31] and Bard-sley [3] �nd that giving is menu-dependent. For instance, List [31] �rstconsidered a standard dictator game where the endowment was ($10,$5) anddictators could share upto $5 of their endowment with the recipient:10

a1 = f(10� s; 5 + s) : 0 � s � 5g:

The mean o¤er among dictators was $1. However, when given also the optionof taking exactly $1 from the recipients,

a2 = a1 [ f(11; 4)g;

few dictators took the new option but the rate of giving substantially declined,and the mean o¤er fell to $0.Our model would explain this as follows: dictators are tempted to be

sel�sh, but this temptation becomes more intense when more sel�sh oppor-tunities are o¤ered. Thus, the introduction of the taking option is accom-panied by a weakened resolve toward normative goals, which induces themenu-dependence of choice.Formally, observe thatmaxx2a1 v(x) = v(10; 5) < v(11; 4) = maxx2a2 v(x).

Thus � is higher at a2 than at a1, and consequently the choice from a2 ismore sel�sh than that from a1.

9An axiomatization of this extended model would weaken Independence as in Noor andTakeoka [35] to accommodate such menu-dependent self-control.10More precisely, they could share only in $0.5 increments. We abstract away from this

in the description.

16

4 Temptation and Social Preferences: An Ex-periment

We have shown that our model can accommodate �ndings in social preferenceexperiments. Each of the �ndings have alternative explanations in terms ofa concern for social image (see the cited references). For instance, such aconcern can clearly rationalize the �nding that subjects give when playingthe dictator game but also prefer a silent exit. In this section we devise asimple experiment that separates the social image theory from our temptationtheory of social preference. To be clear, our goal is to show that temptationis relevant for social preference theory, not that social image is irrelevant.We seek an instrument which responds to temptation but is otherwise

unrelated to social image concerns. An idea prevalent in the temptation lit-erature is that immediate temptation is harder to resist than distant tempta-tion (Noor [34]). Put di¤erently, choices from a distance more closely revealnormative preferences. Thus, changes in behavior (�reversals�) that arisedue to distancing are attributable to temptation (depending on the context,of course). On the other hand, a concern for social image will not induce achange in behavior with distance as long as the degree of visibility of choicesto others is held constant. We use these ideas in the following way.

Design. Our experiment consists of two dictator game treatments. In eachtreatment, dictators must divide $10 between themselves and the AmericanRed Cross. Treatment A is a standard dictator game. The payments to bothparties are made immediately at the end of the experiment �the dictatorsare paid in cash and the charity receives payment electronically. In treatmentB, these payments are made one month later.Dictators�responses were anonymous (their sealed cash payments were

connected to their identities by a code). Thus, we tried to minimize theimpact of social image concerns. More importantly, however, we were carefulto equalize any residual impact across the two treatments. This was done byensuring that the protocols used in each treatment were identical in everyrespect except for the di¤erence in payment dates.The experiment was conducted with undergraduates at Boston Univer-

sity that were taking summer economics courses. There were 22 subjects intreatment A and 26 in treatment B.

Results. Consistent with what a temptation theory of social preference

17

would predict, we found that dictators on average gave less in treatment Athan in treatment B. That is, dictators behaved signi�cantly more sel�shlywhen immediate cash was at stake. From a (temporal) distance, dictatorswere substantially more generous, with the average contribution almost dou-ble that without distance: the average donation out of $10 by dictators was$3.875 (std 4.28) in treatment A and $6.529 (std 3.81) in treatment A. Thissuggests the existence of a temptation to be sel�sh and the existence of asubstantial normative preference to give. Observe that while the average giv-ing rate was less than 50% in treatment A (which is consistent with whatis typically observed in dictator game experiments), the average giving ratewas more than 50% in treatment B.

Table 1: Dictators (%) keeping $xDictators�payment x Group A Group B$0 18.2% 42.3%0 < x < $5 13.6% 11.5%x = $5 9.1% 23.1%5 < x < $10 22.7% 9.1%x = $10 36.4% 15.4%

Formally, we reject the null hypothesis that the distribution of dictators�behavior was independent of distance. We performed three tests for thedi¤erences in distributions of the two samples: the Kolmogorov-Smirnov test,Wilcoxon rank-sum (Mann-Whitney) test and Epps-Singleton test. The testresults are summarized in Table 1. All three test rejected the null hypothesisat 10% level, with the smallest p value being 0.025 (rank-sum) and the largestbeing 0.063 (Kolmogorov-Smirnov).

Table 2. Statistical AnalysisTests Test Statistic p valueKolmogorov-Smirnov test 0:3601 0:063Wilcoxon rank-sum test �2:241 0:0250Epps-Singleton test 10:669 0:03055

A noteworthy observation is that there are factors that may push sub-jects to be less generous in treatment B, and thus our results potentially

18

underestimate the extent of subjects�generosity from a distance. Considerthe following:(1) Charity payments are more valuable when made immediately than

later. Therefore, there is an incentive for concerned subjects in treatmentB to pay charity immediately out of their own pockets today, and simplydemand $10 after a month.(2) Subjects in treatment B could have saved on cognitive costs or retained

�exibility in the face of uncertainty by opting to receive $10 after a month,and then at that time deciding how to allocate it between themselves andcharity.A possible alternative explanation for our �nding is the existence of a

warm glow. In either treatment, the warm glow from giving may be im-mediate, in which case it would be relatively more important than delayedpersonal consumption in treatment B, thus leading to higher contributionrates. A simple experimental test of this possibility is to conduct two treat-ments, where the personal payment in both treatments is after a month, butin one treatment the charity payment is immediate whereas in the other itis after a month. If no di¤erence in contributions is found, then that wouldsupport a warm glow explanation.

5 Behavioral Foundations

We examine here the behavioral foundations of our model of guilt. This al-lows us to con�rm that the behaviors that characterize the model are peculiarto the existence of temptation and guilt, and this provides justi�cation forinterpreting our model as one of temptation and guilt. The purpose of thissection is to provide a complete presentation of our model.11

As noted in Section 2, the agent�s choice of menus contains informationabout normative and temptation preferences over alternatives, which in turncontains information about anticipated choice from menus, and the antici-pated costs of self-control and guilt. Therefore, assuming that the agent issophisticated in the sense of being able to foresee his future choices and expe-riences, the identi�cation of the normative and temptation preferences over

11We claim no technical novelty here. The formal framework for studying temptationintroduced by Gul and Pesendorfer [22] is now su¢ ciently well-developed in the decisiontheory literature to accommodate our theory of guilt (Noor [34], Kopylov [26], Kopylovand Noor [28]).

19

menus is su¢ cient to identify all the relevant components of our model. Weexploit ideas developed in the axiomatic literature following GP [22]. As inNoor [34], we hypothesize that choices from a distance re�ect the normativeperspective. Instead of formulating an in�nite horizon model as in Noor [34],we communicate the basic ideas in the simplest possible way here.12 Specif-ically, we simply assume that there is an ex ante stage �a �special period0��that precedes period 1, and that the agent�s choices in that period arefully normative. Thus guilt is absent in period 0, though anticipated guiltis not. While our terminology suggests that the �distance�between periods0 and 1 is temporal, other forms of distance may be invoked, such as if theagent was asked to dictate choices for others.While the agent chooses alternatives in period 2 and menus in period 1,

we take as a primitive his (normative) preferences % over menus of menusin period 0. To visualize these objects, consider the following examples. Ifa restaurant is a menu a, then a neighborhood of restaurants is a menuof menus A = fa1; a2; ::g. Similarly if the allocation choices available in adictator game is a menu a and the exit option is a menu b (yielding a singleallocation), then a choice of whether to exit or play the game is a menu ofmenus A = fa; bg.Our task is to (i) identify normative and temptation preferences over

menus, and (ii) behaviorally characterize the functional forms (4)-(5). This isdone through period 0 preference using ideas familiar from GP but extendedto a setting where preferences are de�ned over menus of menus. So, if theagent prefers to commit to menu a rather than b:

fag % fbg;

then this suggests that from her ex ante (normative) perspective a is better.Thus such �commitment preferences� identify the agent�s normative prefer-ence over menus. Moreover, if she exhibits a preference for commitment

fag � fa; bg;

then the preference for avoiding menu b reveals that b is tempting. These

12The ideas in [34] can be used to formulate an in�nite horizon version of our modelin which every period the agent picks current consumption and also a menu for the nextperiod. The resulting model is such that each period integrates both the period 1 andperiod 2 choices. That is, the agent experiences guilt from immediate bad consumption,but is tempted to avoid future guilt and thus chooses menus accordingly.

20

connection permits us to take the next step and identify the behavioral im-plications of our model for %.

5.1 Preliminaries

For any compact metric space Z, let �(Z) denote the set of probabilitymeasures on the Borel �-algebra of Z; endowed with the weak convergencetopology; �(Z) is compact and metrizable [1], and we often write it simply as� with generic elements x; y; z, and we often refer to these are alternatives.LetM1 = K(�) denote the set of all nonempty compact subsets of �, withgeneric elements a; b; c. When endowed with the Hausdor¤ topology,M1 isa compact metric space. An element a 2 M1 is referred to as an interimmenu. For � 2 [0; 1], �x + (1 � �)y 2 � is the �-mixture that assigns�x(S) + (1 � �)y(S) to each S in the Borel ��algebra of Z. Similarly,�a + (1 � �)b � f�x + (1 � �)y : x 2 a; y 2 bg 2 M1 is an �-mixture ofmenus a and b. LetM0 = K(M1) denote the set of all nonempty compactsubsets ofM1, endowed with the Hausdor¤ topology. An element A 2 M0

is referred to as an ex ante menu. Its generic elements are A;B;C and an�-mixture of menus A and B is given by �A+(1��)B � f�a+(1��)b : a 2A; b 2 Bg 2 M0. BothM0 andM1 are compact (Aliprantis and Border [1,Theorem 3.71]) and the mixture operations in these spaces are continuous.Our primitive is a preference relation % on M0.13 The interpretation

is that the agent chooses a menu of menus A 2 M0 in the ex ante stage,subsequently selects a menu a 2 A in the interim stage, and then picks �nalconsumption x 2 a in the ex post stage. Choice in the ex ante stage is priorto the experience of temptation, and choice in the remaining stages is subjectto temptation. In particular, guilt that may be experienced in the ex poststage a¤ects what menus tempt in the interim stage.

ex ante stage t=0�choose menu of menus A

"purely normative

� - - - � � -interm stage t=1�choose menu a2A

"temptation to avoid ex post guilt

� � �ex post stage t=2�

choose x2a"

temptation and guilt

Suitably adapted versions of GP�s axioms (behavioral assumptions) im-posed on % characterize the following basic representation theorem (see

13This choice domain is used in Kopylov and Noor [28]. An important di¤erence betweenthis domain and the domain of multi-period menus used in GP [23] is the absence oflotteries over menus. This feature is obtained by exploiting Kopylov [26].

21

Kopylov and Noor [28]). In the interest of brevity, we simply assert theexistence of this representation as an axiom. Say that a function u : X ! Ris linear if for all � 2 [0; 1] and x; y 2 X,

u(�x+ (1� �)y) = �u(x) + (1� �)u(y):

Let U be the set of all continuous linear functions u : X ! R. Similarly,de�ne linearity for functions onM1 and let U1 be the set of all continuouslinear functions V :M1 ! R.

Axiom 1 (Basic) The preference % has a utility representation U :M0 !R such that for all A 2M0 and a 2M1,

U(A) = maxa2A

�U(a)�

�maxb2A

V (b)� V (a)��

(6)

s.t. U(a) = maxx2a

�u(x)�

�maxy2a

v(y)� v(x)��; (7)

where u; v : X ! R and V :M1 ! R are continuous linear functions.

The ex ante preference % over menus of menus admits the representation(6), which in turn identi�es the agent�s normative and temptation preferencesover menus, represented by U and V respectively. The desired functionalform (7) for normative utility U is asserted. However, the desired functionalform for V , given by (5), has yet to be ascertained. We turn to this now.

5.2 Main Axioms

We augment the basic model with three behavioral assumptions. The follow-ing interpretation of behavior are needed to interpret the assumptions. Thereader should keep track of whether each is a statement about temptation orchoice in the interim or ex post period.� fag � fa; bg reveals that in the interim stage, menu b tempts menu a.� fag � fa [ bg reveals that in the ex post stage, an alternative in b is

a source of temptation in menu a [ b. Observe that interim choice is trivialin both fag and fa[ bg. Therefore the preference reveals information aboutwhat is experienced in the ex post period.� fa [ bg � fbg reveals that ex post choice from a [ b belongs to a.

Observe that if the anticipated choice from a [ b lay in b, then there wouldbe no reason to strictly prefer committing to the larger menu a [ b.Our �rst behavioral assumption is a consistency requirement.

22

Axiom 2 (Temptation Consistency) For singleton menus a; b 2M1,

fag � fa; bg () fag � fa [ bg:

Take any pair of singleton menus, a = fxg and b = fyg. TemptationConsistency states that committed consumption fyg is more tempting thanfxg in the interim stage if and only if y is more tempting than x ex post. Thisis an innocuous consistency requirement that ensures that there is no wedgebetween what �nal consumption the agent �nds tempting in the interimperiod and in the ex post period.The next behavioral assumption expresses the idea that temptation pref-

erence over menus is �forward-looking�in the sense of being sensitive to expost choice. Observe that correctly assessing ex post choice implies also acorrect assessment of ex post guilt.

Axiom 3 (Temptation Sophistication) For any menus a; b 2M1,

fa [ bg � fbg =) fag 6� fa; a [ bg

The axiom states that if the agent anticipates that the ex post choicefrom a [ b is not an element of b, then in the interim stage the agent cannever be tempted by the menu a[ b, relative to the menu a. The bite comesin cases where b contains something tempting. The axiom then says that thepresence of that temptation does not attract the agent toward a [ b in theinterim period, if it is the case that the temptation in a [ b is never chosen.The contrapositive of the axiom implies that if a[ b tempts a in the interimstage, then it must be because a [ b contains overwhelming temptation.The �nal axiom is the substantive one that relates temptation by menus

to guilt. De�ne the relation �0 overM1 by

a �0 b if there is x 2 a s.t. ffxgg � ffygg for all y 2 b:

That is, a �0 b if the �most virtuous�alternative in a is strictly better thanthat in b according to the ex ante normative perspective.

Axiom 4 (Guilt-Averse Temptation) For all interim menus a; b 2 M1

such that fag � fa [ bg,

fa [ bg � fa [ b; bg =) a �0 b:

23

Suppose b contains greater ex post temptation than a, in which case a[ band b o¤er the same ex post temptation. The axiom states that if b is moretempting than a[b in the interim stage, then it must be because consumptionfrom b is subject to less guilt. This is re�ected in the fact that b o¤ers strictlyless virtuous consumption than a [ b, that is, a �0 b.Finally, say that% is nondegenerate if there areA;B 2M0 and a; b 2M1

such that A � A [ B � B and fag � fa [ bg � fbg. That is, the agentanticipates resisting temptation in some a [ b 2M1 and A [B 2M0.

Theorem 1 A nondegenerate preference % satis�es Basic, Temptation So-phistication, Temptation Consistency and Guilt-Averse Temptation if andonly if % has a utility representation (6)�(7) such that for all a 2M1,

V (a) = �maxx2a

[v(x) + u(x)�maxy2a

u(y)]; (8)

where � > 0 and u; v are a¢ nely independent.

Thus, under the three behavioral assumptions in addition to Basic, we ob-tain a characterization of our model in terms of ex ante preference. The proofis presented in the appendix. We close by con�rming that our representationhas the desired uniqueness properties.

Theorem 2 If % is nondegenerate preference that admits a representation(6)�(8), then it has another representation with components �0 > 0 andu0; v0 2 U if and only if �0 = �, u0 = �u + �u and v

0 = �v + �v for some� > 0 and �u; �v 2 R.

5.3 Guilt-Proneness

In this section we seek to understand what guilt-proneness means in ourformal model. We proceed by de�ning a comparative behavioral notion ofguilt-proneness and then characterizing it. We �nd that guilt-proneness isequivalent to the tendency to exert self-control.Consider two agents, % and %�, both of whom satisfy our axioms, and

who are ex post similar in that they have identical normative and temptationpreferences over �nal consumption: for all x; y 2 �(Z);

ffxgg % ffygg () ffxgg %� ffygg; and ffxgg � ffx; ygg () ffxgg �� ffx; ygg:

24

Since our model identi�es guilt through temptation by menus, guilt-proneness is naturally identi�ed with the lower tendency to be tempted bymenus in which guilt-ridden indulgence is experienced. To formalize thisidea, say that

De�nition 1 (Guilt-Proneness) % is more guilt-prone than %� if for anymenu b 2M1 and singleton menu a 2M1,

fag � fa; a [ bg =) fag �� fa; a [ bg:

By Set Betweenness, fag � fa; a [ bg implies fag � fa [ bg. That is,the most tempting alternative in b is more tempting than the alternativein a (recall that a is a singleton). Note that adding the alternative in ato the menu b will then not change the maximum temptation, but it mayimprove on the normatively-best alternative. Suppose that the agent exhibitsfag � fa; a[ bg, which says that a[ b tempts a. The representation impliesthat the chosen alternative in a [ b cannot be the alternative in a.14 Thatis, adding the alternative in a to menu b does not change the choice in thelatter. However, the improvement in the normative content can then onlyreduce the temptation value of a [ b relative to b, as it leads the agent tobear a potentially greater degree of guilt. The de�nition says that when a[btempts the agent % despite this, then a[b will also tempt the less guilt-proneagent %�. The contrapositive states that if a [ b �normatively improves�onb to the extent that an agent %� ceases to be tempted by it, then the moreguilt-prone agent % will also cease to be tempted by it. The next theoremcharacterizes this de�nition.

Theorem 3 Let % and %� be an ex post similar pair of nondegenerate pref-erences with guilt representations (u; v; �) and (u�; v�; ��) respectively. Thenthe following statements are equivalent.(a) % is more guilt-prone than %� :(b) Without loss of generality, v = v� and u = �u� and � � 1.

Thus, in our model, to be guilt-prone is to place a higher weight on norma-tive preferences in ex post decisions. That is, guilt-proneness is equivalent to

14Suppose it is. Then the choice in a and a[b are the same, but the choice in a is guiltlessbecause the menu is a singleton, whereas the choice in a [ b is potentially guilt-ridden.Therefore, a [ b cannot possibly tempt a if the choice is the same in both menus.

25

a higher �intensity�of normative preference. This is intuitive: an agent whoexperiences guilt more strongly will place more importance on her normativepreference, and conversely, an agent who puts high value on her normativepreferences is also more prone to feel pangs of guilt if she deviates from them.Observe that agents who place higher weight on normative preferences

will also tend to exhibit a higher degree of self-control in ex post choices �recall that the agent maximizes u+ v, and so if u is �relatively more intensethan v�then choices will follow it more closely. Therefore, we see that greaterguilt-proneness is equivalent to exhibiting greater ex post self-control. This iscon�rmed in the next theorem. Note that the behavioral de�nition of �moreex post self-control�is familiar from GP, and states that whenever %� is ableto resist temptation, then so is %.

Theorem 4 Let % and %� be an ex post similar pair of nondegenerate pref-erences with guilt representations (u; v; �) and (u�; v�; ��) respectively. Thenthe following statements are equivalent.(a) % is more guilt-prone than %� :(b) % has more ex post self-control than %�, that is, for all x; y 2 �(Z);

ffxgg �� ffx; ygg �� ffygg =) ffxgg � ffx; ygg � ffygg:

A curious feature of our model is its lack of parameters �there is onlyone, namely, �. Our analysis reveals that in our model the notion of guilt-proneness is intimately tied with the notion of self-control, and in particularnot controlled by any separate parameter. Intuition strongly suggests thatit should not be any other way: it does not seem meaningful to expectto be able to change an agent�s sensitivity to guilt without also a¤ectingher tendency to resist temptation. While the previous section explored thetestable implications of sensitivity to guilt on an agent�s ranking of menusof menus, these results highlight a testable implication for ex post choice,namely, the existence of self-control.We conclude by noting that the parameter � can be interpreted in terms

of how the agent discounts, in the interim period, the temptation of a menurelative to its normative value. Smaller values of this parameter mean thattemptation by menus gets less weight in interim decisions, which thereforeimplies greater self-control in the interim stage.

26

6 Related Models

The GP [22] model has a two period time line such that in period 1 the agentranks menus and in period 2 chooses out of a menu. Their model adopts apreference %1 over menus as the primitive and axiomatize a representationof the form

U(a) = maxx2a

[u(x)��maxy2a

v(y)� v(x)�]:

While period 1 choice is unmodelled, there are results in the literature (Noor[34]) that show how their model can be augmented so as to provide a jointrepresentation for period 1 and period 2 choice.Notions closely related to guilt have been studied in GP-type models in

the decision theory literature. Dillenberger-Sadowski [15] specialize to thedictator game setting, and study the shame associated with behaving un-fairly when being observed by the recipient. The model is a discrete versionof the GP representation, where u is interpreted as the agent�s private pref-erences (that is, preferences when she is not observed by the recipient) andv re�ects the perceived social norm. If the agent does not behave exactly asdictated by v �and in particular behaves �sel�shly�by following u �then sheexperiences shame. That is, the analog of �self-control costs�here is �shame�.Unlike this paper, the authors�motivation is mainly to model the exit deci-sion by dictators, rather than other evidence from the literature. Moreover,the authors do not propose to explain social preferences in terms of tempta-tion, but rather reinterpret a temptation model in a way that behaviorallyre�ects the social image theory of social preferences that has been studied ineconomics.Kopylov [27] studies the negative emotions experienced by a perfectionist

who is unable to meet her perfectionist standards. A special case of his modelthat is relevant here is the extention of GP given by:

W (x) = max�2x

fu(�)��max�2x

v(�)� v(�)�+ k

�max�2x

u(�)� u(�)�g; x 2 Z:

Here, u represents the agent�s perfectionist standards and v represents de-sires, and the new term in the representation is the cost (guilt, anger, disap-pointment) of deviating from her perfectionist standards.Though not intended as such by the authors, the above models of painful

emotions can be readily reinterpreted as models of guilt �in the �rst model

27

shame would be reinterpreted as guilt and in the second it would be thecost of deviating from perfectionist standards. To see how these reinter-preted models are di¤erent from our model, observe that the preference %1over menus in both models exhibits a guilt-avoidance motive. In particular,the reinterpreted models would be consistent with the example discussed inSection 2.2. That is, they would exhibit

fBg �1 fg; bg;

whereB is a normatively worse alternative than b. This would be the peculiarimplication of guilt in these models: the agent would commit to somethingworse than b in order to avoid the pain of guilt. While these reinterpretedmodels exhibit guilt aversion, our model provides also a theory of guilt aver-sion. Speci�cally, it hypothesizes that guilt-aversion is a temptation and thatit is welfare-irrelevant.For completeness, we observe that choice of menus in our model maxi-

mizes the utility

(U + V )(a) = maxx2a

fu(x) + �v(x)� [maxy2a

v(y)� v(x)] + �[maxy2a

u(y)� u(x)]g:

This representation di¤ers from the two above. Most notably, while pref-erences over singleton menus in the above models is represented by u, inour model it is represented by u + �v. The interpretation is that tempta-tion acts on the agent even at the time of choice of menu, and in particular,even choice among singleton menus is a tug-of-war between normative andtemptation motivations. As in the special case of Kopylov [27], there is adesire to avoid self-control costs and also guilt costs (these reduce utility).However our model makes explicit that the desire to avoid self-control costsis a feature of normative motivations, whereas the desire to avoid guilt isthat of temptation. Formal justi�cation for this requires richer data than apreference over menus �see Section 5.

7 Concluding Remarks

In this paper we observe that a study of guilt calls for a consideration of�guilt by guilt-aversion�. We build a model that treats guilt-aversion as atemptation, rather than as a welfare-relevant feature of normative preference.The model�s foundations give behavioral support to our interpretation of

28

the constructs used in the model. An analysis of guilt-proneness revealshighly intuitive properties of the model. This paper applies the model tothe topic of social preferences and moral hypocrisy. The model is shown toaccommodate a wide range of evidence that seems to suggest that peopleare moral hypocrites, and suggests di¤erent welfare implications than thoseimplied by the view that people behave generously due to social pressurerather than a moral imperative. Our experiment suggests the existence ofa moral imperative to be generous that struggles with a temptation to besel�sh.An interesting next step in the study of guilt is the study of self-deception.

While this paper has shown that agents may take particular physical actionsto alleviate guilt, psychologists have argued that the mind can take actionsof its own to alleviate guilt. By selective memory and selective or biasedinterpretation, an agent may �nd ways to justify immoral behavior. Suchjusti�cation helps them view the immoral behavior as moral, and thus toavoid guilt, at least temporarily. Guilt may still be borne �deep inside�or atlater points in time, but self-deception places some wedge between immoralaction and the experience of guilt.15

A Appendix: Proofs of Propositions

Proof of Prop 1. Due to the hump-shape of normative utility u (M � s; s),

du

ds= u2 � u1

�< 0 M

2< s < 1

> 0 0 < s < M2

d2u

d2s= u22 � 2u12 + u11 < 0 s 2 [0;M ]

Similarly, for temptation utility v (M � s; s), for s 2 [0;M ],

dv

ds= v2 � v1 < 0

d2v

d2s= v22 � 2v12 + v11 < 0

15The reader is referred to Kopylov and Noor [28] for an axiomatic model of self-deception (albeit without guilt).

29

Dictators choose s� to maximize u (M � s; s)+v (M � s; s). Since duds; dvds<

0 for any s > M2, it must be that s� � M

2. That is, both normative and temp-

tation utilities are decreasing for s > M2, thus optimal choice will be at a level

of sharing less than M2.

Let �0 be the threshold temptation intensity such that,

du

ds+ �0

dv

ds

����s=0

= 0: (9)

That is, at �0 the agent chooses exactly not to share. Given d2ud2s+ �d

2vd2s< 0,

it must be that whenever � � �0, thenduds+ �dv

ds

��s=0

� 0=) du

ds+ �dv

ds

��s2(0;M2 )

< 0

=) s� = 0, that is, the agent continues not to share for any temptationintensity � higher than �0. For � strictly lower than �0, we have s� 2

�0; M

2

�,

which is determined by the �rst order condition,

FOC :du

ds+ �

dv

ds= 0:

Note that � = � du=dsdv=ds

���s=s�

. Di¤erentiating wrt s� and inverting leads to:

ds�d�

=(dv=ds)2�

d2v=ds2�| {z }

<0

(du=ds)| {z }>0

��d2u=ds2

�| {z }<0

(dv=ds)| {z }<0

���������s=s�

< 0

Therefore, we have that as � increases from 0 to �0, s� decreases from M2to

0. �

Proof of Prop 2. Recall the threshold �0 de�ned in (9). In order forC(dg) = f(M � s; s)g for s > 0, it must be that � < �0. Thus de�ne �� = �0.For C1(fdg; eg) = feg, we need to establish

U (dg) + V (dg) < U (e) + V (e) :

Observe that

U (dg)+V (dg) = 2 [u (M � s�; s�) + �v (M � s�; s�)]�u�M

2;M

2

���v (M; 0)

30

and U (e) + V (e) = u (M; 0) + �v (M; 0)

So the desired inequality requires

2 [u (M � s�; s�) + �v (M � s�; s�)]

< u

�M

2;M

2

�+ u (M; 0) + 2�v (M; 0)

De�ne the di¤erence

D (�) : = U (dg) + V (dg)� [U (e) + V (e)]

= 2u (M � s�; s�)� u�M

2;M

2

�� u (M; 0) + 2� [v (M � s�; s�)� v (M; 0)] :

ThenD (0) = u�M2; M2

��u (M; 0) > 0 andD (�0) = u (M; 0)�u

�M2; M2

�< 0.

Take the �rst derivative wrt �, and observe

dD (�)

d�=

2du

ds

ds�d�

+2�dv

ds

ds�d�

+ 2 [v (M � s�; s�)� v (M; 0)]

= 2

�du

ds+�dv

ds

�| {z }

=0

ds�d�

+ 2 [v (M � s�; s�)� v (M; 0)]

= 2 [v (M � s�; s�)� v (M; 0)] < 0

Therefore, D (�) decreases from D (0) > 0 to D (�0) < 0 when � increasesfrom 0 to �0. Conclude that there exists �� such that D (�) < 0 for �� <� < �0 = �

�. This completes the proof. �

Proof of Prop 3. Begin by noting that U (a)+V (a) = U (fM � c; 0g)+V (fM � c; 0g) implies2u (M � s�; s�)� u

�M2; M2

�� u (M � c; 0)

+� [2v (M � s�; s�)� v (M; 0)� v (M � c; 0)] = 0=) u1 (M � c; 0) dc+ �v1 (M � c; 0) dc

+ [2v (M � s�; s�)� v (M; 0)� v (M � c; 0)] d� = 0=)

dc

d�=v (M; 0) + v (M � c; 0)� 2v (M � s�; s�)

u1 (M � c; 0) + �v1 (M � c; 0)| {z }>0

:

Recall the quantities �0 and D (�) de�ned in the proofs of earlier proposition.Then s� = 0 whenever � � �0, and moreover,

31

D (�) > 0=) U (dg) + V (dg) < U (e) + V (e)=) c > 0.Thus, v (M; 0)+v (M � c; 0)�2v (M � s�; s�) = v (M � c; 0)�v (M; 0) <

0 for � � �0, and so dcd�

�����0

< 0. By continuity, there exists ��� < �0 such

that dcd�< 0 for all � � ���.�

Proof of Prop 4. Given our assumptions on u and v, we have u (5; 5) >u (6; 1) and v (6; 1) > v (5; 5). Note thatU(a1) + V (a1) = u(6; 5) + �v(6; 5)U(a2)+V (a2) = maxf2 [u(6; 1) + �v(6; 1)] ; 2 [u(5; 5) + �v(5; 5)]g�[u(5; 5) + �v(6; 1)]U(a)+V (a) = u(6; l)+�v(6; l) = 1

2[u(6; 5) + u(6; 1)]+1

2� [v(6; 5) + v(6; 1)]

Let ���� be the temptation intensity so that in game a2 the agent isindi¤erent between choosing the fair choice (5; 5) and the unfair choice (6; 1).That is, ���� solves u(6; 1)+����v(6; 1) = u(5; 5)+����v(5; 5). Then u(6; 1)+�v(6; 1) > u(5; 5) + �v(5; 5) for any � > ����. Therefore, (6; 1) is chosen ingame a2 when � > �

���. Moreover,

U (a) + V (a)� U(12; a1;

1

2; a2)� V (

1

2; a1;

1

2; a2)

=1

2

�u(6; 5) + u(6; 1) + �v(6; 5) + �v(6; 1) + u (5; 5)�u(6; 5)� �v(6; 5)� 2u (6; 1)� �v (6; 1)

�=

1

2fu (5; 5)� u (6; 1)g

> 0

Thus C1(fa; (12 ; a1;12; a2)g) = fag. �

B Appendix: Experiment Instructions

B.1 Treatment A

Please read the instructions carefully. Failure to follow them will invalidateyour responses, and no payments will be made.

Your Task. There are $10 available to you at the end of this class, andyou will divide this between yourself and the charity American Red Cross.You can keep it all or keep nothing, or keep part of it. The only restrictionis that the smallest denomination should be 25 cents.

32

You will receive your payment in cash at the end of this class, and thecharity will be sent its payment electronically over the internet within thehour following the class.

How. (a) You have up to 10 minutes to decide how to divide the $10between yourself and the charity. Write your share (in dollars) on the answersheet and inside the �ap of the envelope.16

(b) Write your study code on the answer sheet and on the envelope. Yourstudy code is the �rst two letters of mother�s �rst name, the �rst two lettersof father�s �rst name, the �rst two letters of month of birth and the last 2digits of your social security number.17

(c) Fold the answer sheet and put it in the envelope. Do not seal theenvelope. Hand in the envelope.

Protocol for your Payment.At the end of class, all subjects will be asked to approach the desk one

by one. Tell the cashier your study code, and you will receive your envelope,sealed with your payment inside.

Con�dentiality. It is crucial for this experiment that your individualallocation decision be con�dential. Do not write your name or student IDanywhere on the answer sheet. Keep your answer sheet hidden from everyonearound you. The only person who will ever see your allocation decision will bethe cashier, but s/he will not know your identity! The Principal Investigatorswill only see anonymous responses.

B.2 Treatment B

Please read the instructions carefully. Failure to follow them will invalidateyour responses, and no payments will be made.

Your Task. There are $10 available to you after a month, and todayyou will divide this future amount between yourself and the charity AmericanRed Cross. You can keep it all or keep nothing or keep part of it.

16If you write di¤erent amounts you will not receive any payment.17E.g. if your mother�s and father�s �rst names are Jane and John resp., your birthday

is in Oct and the last 2 digits are 22, then your study code is JAJOOC22.

33

You will receive your payment in cash at the end of class on 2nd Aug 2010,and the charity will be sent its payment electronically over the internet withinthe hour following that class.

How. (a) You have up to 10 minutes to decide how to divide the $10between yourself and the charity. Write your share (in dollars, smallestdenomination 25 cents) on the answer sheet.(b) To preserve complete con�dentiality, you need a code: it is constructed

by taking the �rst two letters of mother�s �rst name, the �rst two letters offather�s �rst name, the �rst two letters of month of birth and the last 2 digitsof your social security number.18 Write your code on the answer sheet andon the envelope.(c) Fold the answer sheet and put it in the envelope and hand it in. Do

not seal the envelope.

Protocol for your Payment.At the end of class on the 2nd of Aug 2010, you will tell the cashier your

code and subsequently receive your envelope, sealed with your payment in-side.19 The cashier will not have any way of knowing the amount of paymentyou are collecting.

Con�dentiality. It is crucial for this experiment that your individualallocation decision be con�dential. Do not write your name or student IDanywhere on the answer sheet. Keep your answer sheet hidden from everyonearound you. Your identity will not be known by the cashier when yourpayment is being placed in the envelope, and the amount of payment willnot be known by the cashier when you collect the sealed envelope. ThePrincipal Investigators will only see anonymous responses.

C Appendix: Proof of Theorem 1

Lemma 1 There exist �; k2; k3 2 R, such that for all a 2M1,

V (a) = �maxa(u+ v) + k2max

av + k3max

au (10)

18E.g. if your mother�s and father�s �rst names are Jane and John resp., your birthdayis in Oct and the last 2 digits are 22, then your study code is JAJOOC22.19If you do not collect your envelope that day we will help you get your payment as

soon as possible. The charity will be paid as scheduled.

34

Proof. We �rst show that

maxau = max

bu and max

a(u+v) = max

b(u+v) and max

av = max

bv =) V (a) = V (b) :

(11)This is proved as in Kopylov and Noor [28]. The argument is then completedby appealing to Harsanyi�s aggregation theorem.Suppose the above equalities hold. Note that U(a) = U(b). Suppose by

way of contradiction that V (a) < V (b). Then (U + V ) (a) < (U + V ) (b).Since% is nondegenerate, there exist x�; y� such that U (fx�g) > U (fx�; y�g) >U (fy�g). Let a� = " fx�g + (1� ") a and b� = " fy�g + (1� ") b and take" > 0 s.t.

(U + V ) (a�) < (U + V ) (b�) : (12)

Such " exists by continuity of U; V . Note also that by linearity of U; V andu,

U (a�) > U (a� [ b�) > U (b�) (13)

andmaxa�u < max

b�u: (14)

Given (12), there are two possibilities.

� (U + V ) (a�) < (U + V ) (a� [ b�). Combined with (13), V (a�) < V (a� [ b�).Therefore, fa�g � fa�; a� [ b�g, which contradicts Temptation Sophis-tication.

� (U + V ) (b�) > (U + V ) (a� [ b�). Combined with (13), V (b�) > V (a� [ b�).Therefore, fa� [ b�g � fa� [ b�; b�g. However, given (14), this contra-dicts Guilt-Averse Temptation.

This establishes (11). To prove the lemma, �rst restrict attention the setof convex interim menus:

Mc1 = fco(a) : a 2M1g;

where co(a) denotes the convex hull of a with respect to the mixture oper-ation. The set Mc

1 is a mixture space, and thus the pareto condition (11)yields the desired form for V onMc

1 by an application of Harsanyi�s aggre-gation theorem (see Border [8]).20 Extend the representation to all menus

20Harsanyi�s theorem delivers the desired form plus a constant, but the constant can beset to zero wlog given the uniqueness properties in Theorem 1.

35

inM1 by exploiting the fact that linearity of V implies the �Indi¤erence toTiming�condition V (a) = V (co(a)) for all a 2 M1 (see Dekel, Lipman andRusticini [13]).

Lemma 2 �+ k3 = 0 and �+ k2 > 0.

Proof. Suppose by way of contradiction that �+k3 6= 0. Consider two cases:Case i: �+ k3 > 0 or [�+ k3 < 0 and �+ k2 < 0]We show that Temptation Consistency must be violated. By the non-

degeneracy of % and the fact that u and v are nonconstant and a¢ nelyindependent, there exist x0; y0; x00; y00 such that:

u(x0) = u(y0); v(x0) < v(y0)

v(x00) = v(y00); u(x00) > u(y00)

By the linearity of u and v, for any � 2 (0; 1),

u(x0�x00)� u(y0�y00)v(y0�y00)� v(x0�x00) =

1� ��

u(x00)� u(y00)v(y0)� v(x0) := f(�):

Observe that f(�) ranges between 0 and in�nity.We �rst show that V (fx0�x00g) > V (fy0�y00g). If �+ k3 > 0, then we can

�nd � such that f(�) > �+k2�+k3

. But then,u(x0�x00)�u(y0�y00)v(y0�y00)�v(x0�x00) = f(�) >

�+k2�+k3

=) (�+ k3)u(x0�x00)+(�+ k2) v(x

0�x00) > (�+ k3)u(y0�y00)+(�+ k2) v(y

0�y00)=) V (fx0�x00g) > V (fy0�y00g), given the functional form for V established

in the previous lemma. On the other hand, if [� + k3 < 0 and � + k2 < 0]then we �nd � such that f(�) < �+k2

�+k3and use an analogous argument to show

that V (fx0�x00g) > V (fy0�y00g).Next, observe that the linearity of u and v implies

u(x0�x00) = �u(x0) + (1� �)u(x00) > �u(y0) + (1� �)u(y00) = u(y0�y00);

and similarly, v(x0�x00) < v(y0�y00). Letting x := x0�x00 and y := y0�y00, wetherefore have

u (x) > u (y) ; v (x) < v (y) and V (x) > V (y):

However, these inequalities imply ffxgg � ffx; ygg and ffxgg � ffxg; fygg,which contradicts Temptation Consistency, as desired.

36

Case ii: �+ k3 < 0 and �+ k2 � 0,By nondegeneracy, there exists x; y 2 4 (Z) such that

ffxgg � ffx; ygg :

By the representation, u (x) > u (y) and v (x) < v (y). Moreover, by theform for V ,

V (fxg) = (�+ k3)u (x) + (�+ k2) v (x)

< (�+ k3)u (y) + (�+ k2) v (y)

= V (fyg) :

But then ffxgg � ffxg; fygg, which contradicts Temptation Consistency.We have therefore shown that � + k3 = 0. Observe that V (fxg) =

(�+ k2) v(x). Suppose by way of contradiction that � + k2 � 0. By non-degeneracy, there is x�; y� such that u (x�) > u (y�) and v (x�) < v (y�).However, it would then follow that V (fx�g) > V (fy�g), and in particular,ffx�gg � ffx�; y�gg and ffx�gg � ffx�g; fy�gg, contradicting TemptationConsistency. Thus, �+ k2 > 0. This completes the proof.

Lemma 3 k2 = 0:

Proof. By nondegeneracy, there is x; y such that ffxgg � ffx; ygg � ffygg.By the representation, u (x) > u (y) ; v (x) < v (y) and u (x)+v (x) > u (y)+v (y). However, by Temptation Sophistication,ffx; ygg � ffygg=) ffxgg � ffxg ; fx; ygg=) V (fx; yg) � V (fxg)=) �v(x) + k2v(y) � �v(x) + k2v(x) by previous lemmas=) k2v(y) � k2v(x):Since v (x) < v (y), it follows that k2 = 0, as desired.

Lemma 4 The representation (u; v; �) is unique in the sense that (u0; v0; �0)is the representation for the same preference if and only if, u0 = �u + �1,v0 = �v + �2, and �

0 = �.

Proof. The �rst two requirement is given in GP. For the third part, if �0 6= �,then V 0 = �0 [max (u0 + v0)�maxu0] = ��0 [max (u+ v)�maxu] + ��0�2.According to theorem 1, V 0 = �V + �3. Combine these two equations,�3 = ��

0�2+� (�0 � �) [max (u+ v)�maxu], which is not a constant given

the nondegeneracy conditions, a contradiction.

37

D Appendix: Proof of Theorems 3 and 4

Suppose % and %� are a pair of nondegenerate preferences with guilt repre-sentations (u; v; �) and (u�; v�; ��) respectively.

Lemma 5 % and %� are ex post similar if and only if wlog v = v� andu = �u� for some � > 0:

Proof. The claim �(=�is trivial, so consider �=)�.The claim that u = �u�

for some � > 0 is obvious. Suppose by way of contradiction that v andv� are ordinally distinct. Then there is x; y s.t. wlog v (x) � u (y) andv� (x) � v� (y) with one strict inequality. Since nondegeneracy implies thatv and v� are nonconstant, we can assume v (x) > v (y) and v� (x) < v� (y)wlog. If u (x) < u (y) then ex post similarity implies ffygg �� ffy; xgg andthus v� (x) > v� (y), a contradiction. Thus u (x) � u (y).By nondegeneracy and ex post similarity,

ffwgg � ffw; zgg and ffwgg �� ffw; zgg

for some w; z. Observe that by the representation, for all �,

ff�w + (1� �)gxg �� ff�w + (1� �)x; �z + (1� �)ygg :

However, there exists � such that

ff�w + (1� �)gxg 6� ff�w + (1� �)x; �z + (1� �)ygg ;

contradicting ex post similarity. Therefore, v and v� are ordinally equivalent.Since both are linear, they are cardinally equivalent. By the uniqueness resultin Theorem 1, we can take v = v� wlog by rede�ning � if necessary.

Lemma 6 % is more guilt-prone than %� if and only if � � 1:

Proof. =): Suppose by way of contradiction that � < 1. By the non-degeneracy of % and the fact that u and v are nonconstant and a¢ nelyindependent, there exist x0; y0; x00; y00 such that:

u(x0) = u(y0); v(x0) < v(y0)

u(x00) > u(y00); v(x00) = v(y00)

38

By the linearity of u and v, for any � 2 (0; 1),

u(x0�x00)� u(y0�y00)v(y0�y00)� v(x0�x00) =

1� ��

u(x00)� u(y00)v(y0)� v(x0) := f(�)

Choose � such that� < f(�) < 1:

Let x = x0�x00, y = y0�y00 and z = z0�z00. Then,

f(�) < 1 =) u(x)� u(y)v(y)� v(x) < 1 =) u(y) + v(y) > u(x) + v(x); (15)

f(�) > � =) u(x)� u(y)v(y)� v(x) > � =)

u(y)

�+ v(y) <

u(x)

�+ v(x); (16)

u (x) > u (y) ; v (x) < v (y) (17)

Let a = fxg, b = fyg. Then a [ b = fx; yg. By (17),

fag � fa [ bg; fag �� fa [ bg (18)

By (15),

V (a [ b) = � [u (y) + v (y)� u (x)] > �v (x)V (a) = � [u (x) + v (x)� u (x)] = �v (x)

�=) V (a [ b) > V (a)

This implies, together with (18),fag � fa; a [ bg.On the other hand, by (16),

V � (a [ b) = ��hu(x)�+ v (x)� u(x)

�

i= ��v (x)

V � (a) = ��hu(x)�+ v (x)� u(x)

�

i= ��v (x)

=) V � (a [ b) = V � (a)

Together with equation (18), fag �� fa; a [ bg, which contradicts withthat % is more guilt-prone then %�.(=: If � = 1 then the proof is trivial. So suppose � > 1.The preference

fag � fa; a [ bg implies,

U (a) > U (a [ b) =) u (x) > maxa[b

hu+

�v �max

a[bv�i

V (a) < V (a [ b) =) v (x) < maxa[b

hv +

�u�max

a[bu�i

39

By these inequalities and by de�nition of V �,

V � (a [ b) = ��maxa[b

�v +

1

�

�u�max

a[bu��

� ��maxa[b

hv +

�u�max

a[bu�i

> ��v (x) = V � (a)

On the other hand, by the de�nition of U�,

U� (a [ b) = maxa[b

1

�

hu+ �

�v �max

a[bv�i

� maxa[b

1

�

hu+

�v �max

a[bv�i

<1

�u (x) = U� (a)

Therefore, fag �� fa; a [ bg.

Lemma 7 � � 1 if and only if % has more ex post self-control than %� :

Proof. =): If � = 1 then the proof is trivial. So suppose � > 1 and letffxgg �� ffx; ygg �� ffygg. Then u� (x) > u� (y), v� (x) < v� (y) andu� (x) + v� (x) > u (y) + v (y). By the previous lemma, u (x) > u (y) andv (x) < v (y). Moreover by � > 1,

u (x)+v (x) = (�� 1)u� (x)+u� (x)+v� (x) > (�� 1)u� (y)+u� (y)+v� (y) = u (y)+v (y) :

Therefore, ffxgg � ffx; ygg � ffygg.(=: Suppose by way of contradiction that � < 1. By the construction

used in the previous lemmas, we can �nd x; y such that u (x) > u (y), v (x) <v (y), u (x) + v (x) < u (y) + v (y), but u(x)

�+ v (x) > u(y)

�+ v (y) (let f (�) 2�

1; 1�

�). That is,

ffxgg � ffx; ygg � ffyggffxgg � � ffx; ygg �� ffygg ;

a contradiction.

40

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