On sympathy and gamesq - uni-frankfurt.de

Journal of Economic Behavior & OrganizationVol. 44 (2001) 1–30

On sympathy and gamesq

David Sally∗Johnson Graduate School of Management, Cornell University, 371 Sage Hall, Ithaca, NY 14853, USA

Received 11 August 1998; received in revised form 6 July 2000; accepted 7 July 2000

Abstract

Standard game theory turns a blind eye toward social interaction between the players in a game.Hence, a given game offers the same set of equilibria regardless of the identities of the players andthe specifics of the social context. The predictive value of game theory is severely limited as a result.This paper provides a formal treatment of games set in a context of social interaction. Based on theoriginal insights of Adam Smith and other political economists and social psychologists, the innatehuman quality of sympathy is examined. The effect of sympathy in a game may be to transformthe payoffs and lead to quite different choices in a setting of face-to-face play, play among friends,and play after finding interpersonal similarities. © 2001 Elsevier Science B.V. All rights reserved.

JEL classification:C70; C72; D00

Keywords:Sympathy; Game theory; Prisoners’ dilemma game; Cooperation; Social interaction

1. Introduction

What happens when friends or acquaintances, rather than anonymous strangers, play aparticular game? How quickly do strangers lose their anonymity? What happens if the play-ers are in the same room, as opposed to different buildings, when they choose? What happensif the players share a group identity or a certain cultural background? Standard game theory,in which only interactions between the strategies are allowed, provides no explanation of

q This paper was enriched through proximal interactions with Gary Becker, Colin Camerer, Jim Coleman, BobFrank, Bob Gibbons, Chip Heath, Vrinda Kadiyali, David Laibson, James Montgomery, Sendhil Mullainathan,Sam Peltzman, Lester Telser, Dick Thaler, Mike Waldman, and seminar audiences at the University of Chicago,Cornell University, and summer conferences sponsored by the Russell Sage Foundation. It was also moldedthrough interactions at a remove with two referees and Richard Day. The sympathetic understanding of thesevarious readers and audiences means that any remaining errors reside in the most dimly lit, cobwebbed corners ofmy own mind.

∗ Tel.: +1-607-255-5002.E-mail address:[email protected] (D. Sally).

0167-2681/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved.PII: S0167-2681(00)00153-0

2 D. Sally / J. of Economic Behavior & Org. 44 (2001) 1–30

changes in behavior caused by acquaintanceship and closeness. By contrast, Sally (1995)has shown that contact and discussion increase cooperation in prisoners’ dilemma exper-iments. It is also likely that the process of game playing itself may create changes in theplayers’ attitudes towards each other. This paper attempts to fill and span these lacunae byalloying the insights of earlier political economists, especially Adam Smith, with the morecurrent conclusions of social psychologists and the formal techniques of game theoristssuch as Rabin and Rotemberg.

The base metal in the bridging blend is sympathy, one of Adam Smith’s essential ideas.Refined inThe Theory of Moral Sentiments, which he wrote beforeThe Wealth of Nationsand returned to for major revisions shortly before he died, sympathy brackets the notionsof the invisible hand, division of labor, and extent of the market. In fact, sympathy, or“fellow-feeling”, is the cornerstone of Smith’s understanding of individual behavior. Here,Smith’s formulation of sympathy is discussed first and then formalized within the modernpsychological evidence. I prove the existence of a sympathetic equilibrium for all finitetwo-person games. Next, I analyze non-Nash outcomes in the prisoners’ dilemma and ex-plore changes in its social setting, and subsequently consider various alternative two-persongames. Finally, I compare the results due to sympathy to those arising from fairness.

2. Sympathy defined

2.1. The words of Smith and Mead

Adam Smith (Smith, 1790) perceived sympathy as an essential, ubiquitous presence insociety. He defined it as “our fellow-feeling with any passion whatever” (p. 10), not just amutual sense of loss or of tragedy or a paternalistic attempt to comfort. Sympathy “doesnot arise so much from the view of the passion as from that of the situation which excitesit” (p. 12). If we share someone’s anger, it is due to our understanding of the factors and thecontext that caused the person to get mad in the first place. We can reach a thorough compre-hension of that context only if we imaginatively transport ourselves into the other person’ssituation — sympathy arises from “changing places in fancy with the sufferer” (p. 10).1

Smith suggested a geometry of human relations: we perceive a space in which our self isthe origin and other people are arrayed at recognizable positions and at a calculable distancefrom the origin. Our ability to change places in fancy with another declines as the othermoves further away from the self; accordingly, sympathy is an inverse function of distance.Smith described how mutual sympathy declines within the extended family from the highlevel among brothers and sisters and parents and children to lesser levels among cousins,second cousins, etc. (pp. 219–220).

1 There is a controversy extant especially within the psychoanalytic literature over the distinction between“sympathy” and “empathy”. The former is equivalent to occupying the shoes of another, while the latter, toplacing our shoes in the other’s footprints. Wispe (1986) wrote, “To know what it would be like ifI were theother person is empathy. To know what it would be like tobe that other person is sympathy (p. 318)”. Thesetwo phenomena cannot be cleanly separated, yet it is clear that an interpersonal judgment that is sensitive to theidiosyncrasies of the other, i.e. sympathy, is more powerful within strategic social interactions.

D. Sally / J. of Economic Behavior & Org. 44 (2001) 1–30 3

Smith’s conception of distance has both psychological and physical components (Viner,1972), and we should not underestimate the importance of the latter. Smith believed thatphysical proximity is essential to the development of mutual sympathy: our strong fellow-feeling for our immediate family arises from living in the same house with them (p. 219);boarding-schools are an unspeakably pernicious institution, interfering with naturally ben-eficial domestic education (p. 222); and a cost of commercial society is the dispersal offamilies, which was proved by the “weaker family relations” in England than in Scotland(p. 223).2 Furthermore, we can form sympathetic attachments to those people with whomwe interact on a regular basis — “colleagues in office” or “neighbors” (p. 224) — evenif they are not related to us. Clearly, psychological affinities such as shared nationality,similar preferences, or common ancestors will reduce the distance between people. Butpsychological connections must be, in many cases, supported by physical relationships.Smith appeared to place great faith in the aphorism, “Out of sight, out of mind”.

However, it seems to me arbitrary to limit sympathy to the passions and to excludefellow-understanding of another’s mind or reason. George Herbert Mead (1934) linkedthe very existence of mind and intelligence with the ability to role-play and to take theattitude of the other. Within the philosophy of mind, simulation theory, as propoundedby Gordon (1986), Ripstein (1987), and Goldman (1989), asserts that we understand theintentions of another person and predict his behavior based on a sympathetic model ofhis mind. This simulation ability is fundamentally situated in our capacity for pretending:usually, it develops within young children, and its absence is pathologically coincident withan inability to pretend.3 Thus, when I speak of sympathy, I am referring to an essentialelement of human nature that is manifest in our capacity to recognize and anticipate eachother’s feelingsandthoughts.

We can now see the intimate relationship between sympathy and strategy. The notion of abest response entails that a player “change places in fancy” with his opponent.4 A rationalplayer must create a mental model of other participants in a game, and this act of modelingnaturally narrows psychological distance and creates fellow-feeling. The paradigmatic caseis that of the chess player:

“A good chess player has the response of the other person in his system. He can carryfour or five moves ahead in his mind. What he is doing is stimulating another person todo a thing while he stimulates himself to do the same thing. That enables him to analyzehis mode of attack into its different elements in terms of the responses coming from hisopponent and to then reconstruct his own activity on that basis (Mead, p. 243).”

In addition, if common knowledge is to be part of the structure of a game, then the creationof such mutual understanding will affect the fellow-feeling among the players and may altertheir basic actions.

2 See Rosenberg (1990) for a cost-benefit accounting of the overall effect of commercial society on the stock ofmoral capital according to Smith.

3 See Wimmer and Perner (1983) and Wellman (1992) on the normal child’s development of a theory of mind, andBaron-Cohen et al. (1985) on the autistic child’s inability to pretend and its consequences. Sally (2001) examineswhat autism proves about sympathy and the social mind.

4 There is even a linguistic connection between “response” and interpersonal identification. “Response” shares aLatin root with “sponsor” —sponsus, meaning “answerable person”. Thus, just as the sponsor is somehow presentin the one who is sponsored, so a “response” includes some form of the one who is responded to.


The logic of interpersonal strategy, then, necessitates that a player’s mental model ofan opponent not have the same effects upon him as the presence of the opponent himself.Simply put, we cannot plan to influence another if the mental model we are required to makeof the other, causes us to withdraw. What the mental model lacks, of course, is the physicalpresence emphasized by Smith, and thus, the anticipation or the memory of interacting witha person affects us differently than the physical presence of someone immediately beforeus. In the same way that the disjunction between the idea of gambling and the actual sightand feel of the green felt (Sartre, cited in Elster, 1984), makes self-control andintrapersonalstrategy possible, so the disjunction between the anticipation of an encounter and a person’spresence itself, and that between the anticipation of someone’s words and the actual, verbalutterance makesinterpersonal strategy practicable.5

2.2. Sympathy formalized

Let λ represent sympathy, andiλj , the fellow-feeling personi has for personj, 6 and letiλj ≡ `(ϕij , ψij ), whereϕ andψ stand for physical and psychological distance, respec-tively. I will presume thatϕ, ψ , ∈[0, δ], with δ being the maximal distance between twoindividuals.7 Hence, asϕ → 0, a person moves nearer, eventually becoming visible, iden-tifiable, then touchable. Asψ → δ, another person dissolves from friend or acquaintanceor co-national to a mythic Other, whose mind and tastes are unintelligible. To be faithful toSmith, the function, must have the following characteristics:

1. `(0,0) = 1. Everyone fully sympathizes with one who is identical to the self.2. (∂`/∂ϕ), (∂`/∂ψ) < 0. Sympathy declines as distance increases.8

3. `(δ, δ) = 0. There is no fellow-feeling for another who is far away and foreign.These three characteristics establish that

iλj ∈ L ≡ [0,1]. (1)

Furthermore, I will specify that

iλj = 1 − ωϕij + (1 − ω)ψij

δ(2)

with ω belonging to the unit interval and representing the weight placed by an individualon the physical distance from another person.

Social and interpersonal context will be manifest as a constraint on the range of possibledistances and the resulting levels of sympathy (this approach is one key difference betweenthis paper and other “altruistic” models such as Rotemberg (1994).) If I am making a decision

5 See Sally (2000a) for a thorough development of the relationship between immediacy and self-control asembodied by Odysseus in his encounter with the Sirens.

6 For iλj , read “i el j.” The similarity with the phrase “i likes j” is intentional.7 Presumably, each distance is measured by a different scale and has a distinct endpoint. Nevertheless, we can

transform one measure, such asϕ′, by lettingϕ = ϕ′(δψ/δϕ′ ). ϕ andψ would share the same interval.8 Marshall (1975) advanced a similar proposition: “Also if we take account of the fact that the total happiness will

be best promoted generally by each man’s taking special account of the happiness of those around him, and notdissipating his attentions over space, we may introduce another factor e−λr , whenr is the distance — measuredperhaps partly in terms of geography and partly in terms of kindred — of a man from the several units” (p. 318).


affecting another who is halfway around the globe (ϕ = δ), then my fellow-feeling has anupper bound of 1− ω, i.e. he has at best a ghostly presence for me. If I am considering afellow opera buff, then my sympathy has a lower bound above 0, i.e. I cannot completelyignore our shared taste. In a similar fashion, a history of repeated interaction, whether offriendly encounters or of hostile posturings, will circumscribe the feasible set of distancesand fellow-feelings. Thus, an individual will be unable to pull far away from his friend ordraw too near to his enemy.

2.3. Evidence on distance9

Proxemics is the branch of social psychology studying interpersonal positioning andphysical distance. Argyle and Dean (1965) proposed that there is an “intimacy equilibrium”between any pair of people, based on physical, visible factors: those who are more intimateor friendly will establish an equilibrium of greater physical closeness. They presentedevidence that people attempt to maintain this equilibrium if one factor is exogenouslyaltered. For example, in a laboratory, they observed that eye contact was reduced as thechairs of seated strangers were moved closer. Mehrabian (1969) found that the degree ofliking and the physical separation and gaze avoidance between two people were negativelyrelated. Burgess (1983) analyzed people as they walked through a mall and discovered thatcompanions were nearer to each other than to strangers, and that as the density of the crowdincreased, the companion groups compressed so as to maintain spacing from strangers.

For my purposes, then,ϕ represents more than the Euclidean distance between twoindividuals: it consists of other physical factors such as eye contact, shoulder orientation,posture, and facial expressions. The equilibrium affirmed by Argyle and Dean is manifest inthree ways: first, for any two people,i andj, ϕij may not equalϕji ; second, changes in theequilibrium distance may not be easily and unilaterally achieved, as by taking a step forward.Thirdly, in general, the more proximal someone is to us, the more readily we identify withhim and the more sympathy we have for him. Sympathy increases with propinquity.

The interpretation of a particular interaction, however, depends upon the expected con-figuration, in other words, the convention of proximity arising from a general social normor a history of previous interactions. Differences from the convention engender a processof interpretation: if the perceived intention of the partner is to manipulate, then sympathywill be dampened; if it is to identify, then sympathy will be reciprocated.10 This process of

9 For a complete review of the social psychological grounding of sympathy, see Sally (2000b).10 Robert Benchley wrote, “The most common of all antagonisms arises from a man’s taking a seat beside youon the train, a seat to which he is completely entitled.” I claim that the level of antagonism is related to the supplyof empty seats. Note that the spacing of seats on a subway, train, or bus is closer than the range of conventionaldistances for Americans (12′′–20′′). If there are a large number of unoccupied seats, the already-sitting views thechoice of the about-to-sit as a non-random assessment of each sitter’s personal space, with the conclusion thatthis already-sitting has the smallest personal space and would be least offended. This implied assessment is itselfoffending and creates the bad feeling. The more this outcome appears to be a choice, the greater the antagonism.Someone who wanders the aisle before sitting is hated, but the one who takes the last remaining seat producesminimal animosity. Of course, this scene is also influenced by psychological identification, which may diminishthe presumptive offense. It is through such minor scenes on the subway that racial, social, and physical differencesand similarities are reinforced.


interpretation and reciprocation will appear in Section 3 in my notion of effective sympathy.Hence,(∂iλj /∂ϕij ) < 0, yet the final effect of a turning away, or a frown, or a touch willdepend on the intentions and fellow-feeling attributed to the actor.

Priest and Sawyer (1967) tracked the friendships formed within a new dormitory at theUniversity of Chicago. They found that proximity was positively related to both recognitionand liking of other students: roommates were liked more frequently than neighbors, whowere liked more than floormates, etc. They maintained that the destrategized context ofproximal interactions allows sympathy to work unretarded: “It is easy for roommates tointeract without each encounter being anoccasion” (p. 635, emphasis added). Hence, theattempt to form a relationship over a long distance is more likely not to be reciprocated, allelse being equal.

A related stream of work within social psychology has concentrated on psychologicalsimilarity and liking. Since Aristotle (1991) wrote, “And we like those who resemble usand have the same tastes” (p. 197), and Darwin (1936) claimed, “If, indeed, such men areseparated from him by great differences in appearance or habits, experience unfortunatelyshows us how long it is, before we look at them as our fellow-creatures” (p. 492), manyother writers have affirmed the same proposition. For example, Byrne and Nelson (1965)proposed that attraction to another is a linear function of the proportion of shared attitudes,and Lydon et al. (1988) showed that shared preferences for activities such as shopping,reading, and camping, also increase the amount of sympathy. Psychologists have gath-ered empirical evidence in the field, finding positive correlations among college roommategroups (Newcomb, 1956; Byrne, 1971); the dysphoric (Locke and Horowitz, 1990); spouses(Richardson, 1939) and lovers (Hill et al., 1976).

Psychological distance is, nevertheless, an uncertain measure. It is less easily observed,and more prone to perceiver bias than is physical distance. Two frequently mentioned biasesin interpersonal perception are the fundamental attribution error (or correspondence bias)and the false consensus effect (Ross, 1977). The first flaw arises from an overeagernessto attribute a person’s actions to his personal characteristics or beliefs and to overlook thesituational constraints, e.g. ascribing Communist views to another subject who wasrequiredby an experimenter to write an essay espousing this position (Jones and Harris, 1967). As aresult, when we observe a consonant action, we are too likely to perceive similarity, and whenwe see a dissonant action, we are too likely to discern dissimilarity. The false consensus biasis a related effect in which the individual believes that his dispositions are more prevalentin society, and thus, more likely to be shared. Consequently, as Ross detailed, both sides ofan issue can believe that their position is more popular. Still, these tendencies do not vitiatemy conception of sympathy. First, they do implicitly support the existence and importanceof fellow-feelings: we expect to be able to identify with a broad set of strangers, and we useavailable data (possibly poorly) to assess interpersonal distances. Second, the pervasivenessof these phenomena limits the strategic vulnerability of an individual, since all sides maybe falsely attributing similarity. Finally, as Swann (1984) rejoined, every social interactioninvolves a process of identity negotiation among all participants. Hence, the target mayprovide clues to the perceiver to increase the latter’s accuracy, or the target may modify hischaracter to fall in line with the perceiver’s expectation.

One implication of my characterization ofψ andϕ is the overpowering role of chance inthe establishment of sympathy. At the extreme, one person just as easily as another would


be our friend if he had happened to be seated next to us in class, and one person just as easilyas another would be our friend if we had not said the wrong thing upon our introduction.So it is that a Democrat may remain close to his best friend if he learns that the latter votedRepublican, but he would distance himself from any new acquaintance who was Republican.It is possible that if we knew everything about someone at the moment we met, we wouldhave no friends at all.

Secondly,ψ andϕ may have deep evolutionary roots. Darwin thought of sympathy as acharacteristic of human nature: “[Man has] retained from an extremely remote period somedegree of instinctive love and sympathy for his fellows. We are indeed all conscious thatwe do possess such sympathetic feelings. . . ” (p. 480). Sympathy is the outcome of naturalselection among humans as social animals, but this outcome is stable only for discerningsympathy — “with all animals, sympathy is directed solely towards the members of thesame community” (p. 478). Modern evolutionary theorists support Darwin’s contentions.Thiessen and Gregg (1990) declared that mate choice based on propinquity and phenotypicsimilarity has a genetic basis. Peck (1993) demonstrated that the choice of interaction withina repeated prisoners’ dilemma is sufficient to allow cooperators and defectors to co-exist, theformer matched in long-term friendships and the latter churning through new acquaintances.

Thirdly, and most critically, the acknowledgment of the roles ofψ andϕ in the pro-duction of sympathy will permit a proper evaluation of the relationship between languageand rational behavior. The media of communication may be distinguished by their inherentphysical properties: corporeal presence decreases from face-to-face discussing to telephon-ing to emailing to writing. Moreover, the development of common ground between uttererand receiver, which is indispensable in the creation of meaning, reduces psychological dis-tance. Newcomb expressed this thought succinctly, “Communicators tend to become moresimilar to each other, at least momentarily, in one or more respects, than they were beforethe communication” (p. 578).

3. Utilizing sympathy

Edgeworth (1881) was the first economist to postulate a specific utility function incorpo-rating sympathy: “the object whichX (whose own utility isP) tends — in a calm effectivemoment — to maximize, is notP, butP+λΠ ; whereλ is a coefficient of effective sympathy”(p. 53n, quoted in Collard, 1975). The linear form he suggested has been adopted by thosesocial scientists who study altruism in two different areas: intergenerational consumptionand game theory. Examples of theorists studying consumption and using a linear combina-tion of utilities are Becker (1974), Bernheim and Stark (1988), Bruce and Waldman (1990),Montgomery (1994), and Mulligan (1997). Game theorists using a linear combination ofpayoffs include Rescher (1975), Collard (1978), and Rotemberg.11

One problem with the usual linear form, e.g.U1 = u1 + λu2, is that an altruistic personis subject to the Samaritan’s dilemma in a multi-period setting, as demonstrated by Bruce

11 Geanakoplos et al. (1989) and Palfrey and Rosenthal (1988) study sympathetic transformations of a givenpayoff matrix, but both papers assume a lump sum added to the payoffs that is independent of the other player’soutcome.


and Waldman. One way to distinguish sympathy from altruism is to suggest that a rationalindividual who is aware that he is being manipulated by the object of his sympathy will actquickly to reduce his fellow-feeling for the other, much as a person who has inconsistentpreferences will react harshly against one who is triangulating choices so as to drain himof income. Hence, our level of effective sympathy is a function not only of our own choiceof fellow-feeling, but also the sympathy directed toward us by all those with whom we areinteracting.

Therefore, letΛ be an effective sympathy function:Λ≡Λ(iλj ,j λi). I will retain thebasic linear form, so that personi’s overall utility when interacting with personj is Vi =vi +Λ(iλj ,j λi)vj , where thev’s are the immediate utilities of the participants. This formsupports Smith’s famous insight that total utility is positively related to the utility of others:“How selfish soever man may be supposed, there are evidently some principles in his nature,which interest him in the fortune of others, and render their happiness necessary to him,though he derives nothing from it except the pleasure of seeing it” (p. 9).

The first characteristic ofΛ concerns its maximum magnitude. Smith writes, “Every manfeels his own pleasures and his own pains more sensibly than those of other people. Theformer are the original sensations; the latter the reflected or sympathetic images of thosesensations. The former may be said to be the substance; the latter the shadow” (p. 219).Accordingly, the weight placed on another’s outcome must not be greater than the weightplaced on our own outcome.

Condition 1:

Λ(iλj , jλi) ≤ 1 for all iλj , j λi ∈ LThe feelings another has for us will change the feeling we have for that person. Smith

believed in the reciprocal nature of sympathy, and Mead was also explicit on this point:“There are persons with whom one finds it difficult to sympathize. In order to be insympathy with someone, there must be a response which answers to the attitude of theother. If there is not a response which so answers, then one cannot arouse sympathy inhimself (p. 299).”

Therefore, ifi regardsj positively, i.e.iλj > 0, theni’s effective sympathy moves positivelywith the regard shown byj.

Condition 2A:

∂Λ(iλj , jλi)

∂jλi> 0 for allj λi ∈ L if iλj > 0

Condition 2A reflects the notion that another’s sympathy may pull our feelings in onedirection or another. For instance, if someone becomes more distant to us, then we losefellow-feeling for that person, much as Smith described:

“What chiefly enrages us against the man who injures or insults us, is the little accountwhich he seems to make of us, the unreasonable preference which he gives to himselfabove us, and that absurd self-love, by which he seems to imagine, that other people maybe sacrificed at any time, to his conveniency or his humour (p. 96).”Condition 2B encapsulates the thought that another’s fellow-feeling can never make us

turn head over heels with affection:


Condition 2B:

Λ(iλj , jλi) ≤ max(iλj , j λi) for all iλj , j λi ∈ LIt is natural, when postulating that effective sympathy is affected by reciprocity, to sup-

pose, if two people are concordant with regard to their levels of fellow-feeling, that thislevel will be effectively expressed. Thus

Condition 2C:

Λ(λ, λ) = λ for all λ ∈ LIf we regard someone neutrally, then we take no account of what happens to him. Smith

wrote, “That we should be but little interested, therefore, in the fortune of those whom wecan neither serve nor hurt, and who are in every respect so very remote from us, seemswisely ordered by Nature. . . ” (p. 140). Hence

Condition 3:

Λ(0, j λi) ≤ 0 for all j λi ∈ LFinally, small changes in our sympathy for another or in his sympathy for us should resultin a minor modification in effective sympathy:

Condition 4:

Λ(iλj , jλi) is continuous for alliλj , j λi ∈ LThere are a variety of functions that satisfy all of these conditions (see Sally (1998) fordetails). In order to see the effects of social interaction on specific games, I will specify oneparticular sympathetic function:

Λ1(iλj , j λi) = iλj + iλj (jλi − iλj ). (3)

The sum on the right-hand-side of Eq. (3) has two parts: as in previous models, a rationalindividual enjoys the linear sharing of the utility of the other, but if his fellow-feeling is notrequited, then there is a loss associated with social closeness. It is this possible disutilitythat distinguishes my model from earlier ones, and that introduces a strategic element tosocial interaction.

4. Sympathetic equilibrium

I will now apply the concept of fellow-feeling and the existence of an effective sympathyfunction to two-person normal-form games. LetA1 andA2 be the finite set of actions players1 and 2 can take, respectively. In the usual manner, letS1 andS2 be the set of mixed strategiesfor each player. Let the payoff function for playeri be defined asvi : S1 × S2 → R. Thus,we have a well-defined game,G = {S1, S2; v1, v2}. Extend the game by embedding it ina social context. That is, we will suppose that the players may be anonymous, friendly,sullen, talkative, similar, different, distant, co-present, or some combination. They may bein different rooms, buildings, or countries; they may be in a laboratory, a conference room,


a café, or a living room. Furthermore, they may have the chance to meet or avoid each other,look into each other’s eyes or turn away, speak a kind word or disagree, touch or flinch, etc.The social context and social interaction precede the choice of strategy in the underlyinggame. I will assume, then, that the preferences of each player in this extended game maybe represented by the functionVi : L× S1 × L× S2 → R, where

Vi = vi(si, sj )+Λ1(iλj , j λi)vj (si, sj ). (4)

This payoff function is the same, simple, linear form discussed earlier. Hence, we haveconvertedG into the following game:

GS = {L× S1, L× S2;V1, V2} (5)

This conversion is related to the transformations of payoff matrices into effective matricesanalyzed by Collard (1975), Kelley and Thibaut (1978), and Raub (1990). However, noneof these authors explicitly included reciprocity in his formulation, so that a player’s matrixtransformation was independent of his belief about how the opponent was transmuting hisown payoffs.

The game specified here is closest to the model of Rotemberg, who allowed players tochoose an altruism parameter followed by a the choice of a strategy. He assumed that thechoice of altruism is unconstrained but that the objective of players is to maximize onlytheir immediate utilities, i.e.vi , in my formulation, rather thanVi . One of the weaknesses ofRotemberg’s approach is that it misrepresents the dynamics of social distance and interac-tion as detailed in the previous section: effective sympathy depends on the social moves ofboth players. In this sense, the model here is close in spirit to that of Rabin (1993) who mod-els fairness as depending on the players’ beliefs about each other’s actions. Rotemberg alsonever considered that the choice of altruism, or in my model, social distance, may be con-strained and determined by the social context: only rarely is it a completely unbound choice.

A very simple game may help clarify a number of the points made above. One child sitson a step while holding a popsicle, and another sits beside him. Fig. 1 shows the actionshe can take (A1) and the resulting payoffs(v1(ai), v2(ai) for ai ∈ A1).Based on his ownimmediate utility, the child will eat all of the popsicle himself. Sharing his popsicle is asympathetic equilibrium only if the child feels close to the otherandbelieves that the other

Fig. 1. A popsicle game.


will reciprocate his feelings.12 The belief in this type of outcome is the basis for the successof a pigeon drop scam, in which a con artist, claiming to have found a large sum of money,uses the recognized requirement of reciprocity to convince a “pigeon” to post a bond tosecure his share of the illusory cash. If the child was alone, he would not consider leaving thepopsicle for someone unknown to him to pick up. The eventual recipient’s lack of identityand the inability to reciprocate constrains the popsicle holder’s effective sympathy to verylow levels. Thus, identity-less charity, such as dropping dollar bills on a busy city streetcorner, is considered irrational.

I am claiming thatGS is the game perceived and played by rational, human actors whoare naturally sympathetic and who are affected by the social context of the underlyinggame. I want to prove that this extended game has a solution. To do that, I will follow thestructure of the proof that normal-form games with continuous payoff functions have a Nashequilibrium in mixed strategies.

Lemma. Vi is continuous overL× S1 × L× S2. 13

It is natural to establish the following analogue:

Definition 1. A sympathetic equilibrium is a Nash equilibrium ofGS.Playeri chooses(iλj , si) to maximizeVi , given his belief that playerj is selecting(jλi, sj ).

Does a sympathetic equilibrium exist? Yes, it does.

Proposition 1. For any finite game G, GS has at least one sympathetic equilibrium.

Proof. By the lemma,V1 andV2 are continuous overS1×L×S2×L, which is a nonemptycompact subset ofRn1+n2+2. Therefore, there exists a Nash equilibrium in mixed strategies(Glicksberg, 1952, cited in Fudenberg and Tirole, 1991). �

The original gameG can be recovered ifL is restricted to{0}, sinceVi(0, si ,0, sj ) =vi(si, sj ). Attempts at this restriction are manifest in the laboratory in the exertions ofexperimenters to shield subjects from any sight of or sign from potential partners, and inthe market from its inception as a traveling fair to its modernization in faceless e-trading.

The notion of a mixed strategy over sympathy levels is somewhat paradoxical since ran-domization seems a particularly unsympathetic way to choose an interpersonal connection.A refinement of our original definition above is the following:

Definition 2. A sympathetic equilibrium is pure if it contains determined levels of sympathyand either a pure or mixed strategy fromS1 × S2. 14

12 A related image in a sharing game with slightly altered payoffs is expressed in this couplet from the Rig-Veda:“Two birds, friends joined together, clutch the same tree.One eats the sweet fruit; the other looks on without eating.”

13 All longer proofs and calculations are found in Appendix A.14 The existence of a pure sympathetic equilibrium can be guaranteed ifVi(si , iλj , sj , j λi ) is quasi-concave in(si , iλj ). See Sally (1998) for details.


Fig. 2. Normalized prisoners’ dilemma.

By focusing the subsequent analysis on pure sympathetic equilibria, the sequential game,GS, can be presented in a normal form consisting of contingent strategies involving socialinteraction and a move in the focal game.

5. Sympathetic equilibria in the prisoners’ dilemma

The meta-analysis of Sally (1995) demonstrated that the degree of cooperation observedin experimental prisoners’ dilemmas could not be explained by models of strict self-interest.The Nash equilibrium of a PD is mutual defection. The amount of cooperation by laboratorysubjects and by individuals in other settings in which common interest and self-interestconflict must mean that people are either not maximizing their utilities or not playing theobservable game. I shall show that cooperation is incorporated in a number of sympatheticequilibria under varying social contexts.

Fig. 2 shows a normalized prisoners’ dilemma game, as formulated by Rotemberg. Thetemptation to defect from mutual cooperation is represented byt, and the sucker’s lossfrom being defected upon is−s; t and s are positive and may vary widely so long ast − s ≤ 1, assuring that the jointly dominant outcome is mutual cooperation. If the playersare completely anonymous to each other and the game is being played immediately, thenthe only equilibrium is (0, D, 0, D), regardless of the level of the payoffs or their ratios. Thisoutcome is, of course, the only Nash equilibrium in the underlying prisoners’ dilemma.

Suppose, however, that the PD is being played a year from now. The choice of sympathyis then effectively unconstrained, as the players can remain complete strangers or becomeintimately acquainted. In this instance, the number and identity of the sympathetic equilibriadepend on the relationship among the material payoffs in the game. Fig. 3 displays theextended, sympathetic prisoners’ dilemma in normal form.

Three pure sympathetic equilibria involving cooperation may exist for particular payoffvalues: (1, C, 1, C); (1/2, C, 0, D); (0, D, 1/2, C). Because the payoffs for mutual defec-tion are normalized to 0, there may be an equilibrium for everyλ ∈ [0,1] of the form(λ, D, λ, D). Appendix A contains the calculations of all the sympathetic equilibria for thenormalized prisoners’ dilemma. The computation of the sympathetic equilibria for a moregeneral prisoners’ dilemma are also included in Appendix A, and this analysis demonstratesthat (0, D, 0, D) and (1, D, 1, D) should receive most of our attention, since they exist for awider range of payoffs.

Fig. 4 is a graph relating the absolute values oft ands and their ratio to the identity ofthe sympathetic equilibria.


Fig. 3. Normal form representation of sympathetic prisoners’ dilemma.

There are a number of observations to make about this graph. First, within the constrainedparameter space determined byt − s ≤ 1, there is at least one pure sympathetic equilib-rium at each point. Second, mutual defection is neither the only sympathetic equilibrium,nor is it an equilibrium for all parameter values. Ifs/(1 + t) < 1/4, then the sole Nashequilibrium of the underlying game is not included in any sympathetic equilibrium. Third,if the temptation is limited, a cooperative equilibrium is always possible, and in this case,it is also always pareto optimal. In fact, ift ands are only moderately different from 0,

Fig. 4. Sympathetic equilibria for normalized prisoners’.


Fig. 5. Risk dominant, sympathetic equilibria for normalized PD.

then mutual sympathetic cooperation is the only equilibrium (area C). On the other hand,(0, D, 0, D) is a sympathetic equilibrium (area D) if the mutual cooperator is tempted by agreater than doubling of his immediate payoff, and if the sucker’s loss is quite larger than0. Fourth, mutual sympathetic defection can occur only whens > 1 + t . It is an almostwistful equilibrium: as though each player fully identifies with the fear and weakness ofthe other. Fifth, there are “sacrificial” equilibria (area S), in which the temptation to defectbecomes a gift from the cooperator to the defector. This gift is relatively large, compensatesthe cooperator for his loss through sympathetic sharing, and discourages the defector fromreciprocating.

Finally, there are a number of areas with multiple equilibria and the concomitant co-ordination problems. Although this multiplicity seems to conform to choices subjects inexperiments make when faced with this dilemma, and although this need for coordina-tion seems to reflect the uncertainty of participants about the outcome, it might be valu-able to distinguish among the various equilibria. It is often argued that pareto dominancemight serve as a means of focus, and if it is so in this case, then (1, C, 1, C) would befocal for all t ≤ 1, and areas S and D would remain indeterminate. Yet, Cooper et al.(1992) presented experimental evidence demonstrating that risk dominance rather thanpareto dominance serves as a focal mechanism in coordination problems.15 Fig. 5 displays

15 Although Harsanyi and Selten (1988) assert the primacy of payoff dominance over risk dominance in theirsolution concept, see Sally (2000c) for a review of the literature on the importance of risk dominance, and for anapplication to rational speech and linguistic choices.


the risk dominant equilibria across the parameter space (the calculations are contained inAppendix A):

For s > 1 andt < 1, (0, D, 0, D) is risk dominant jointly over (1, C, 1, C) and allequilibria configured as (λ, D, λ, D). Whens/(1 + t) > 1/4, (1, C, 1, C) is risk dominantonly if both the potential loss and the temptation are limited, i.e.s + t < 1. Furthermore,in its region of overlap with the sacrificial equilibria (1, C, 1, C) is risk dominant only ift <∼ 0.69. Thus, the spread of mutual sympathetic cooperation may be limited not by itslack of pareto dominance, but rather by its being strategically risky if the loss from beingsuckered is relatively large.

These results differ from those of Rotemberg, who found two different regions of equi-libria: when t > s, then mutual defection with no altruism; whens > t , then mutualcooperation with moderate levels of altruism. This pattern, however, is not borne out inthe empirical evidence (Sally, 1995). In some sense, Rotemberg’s model suggests that thepayoff matrix is just as deterministic in his extended game as it is in the core game, and so,he seems to misrepresent the complexity of the socially embedded PD. The outcomes hereshow that the unconstrained choice of social distance does complicate matters immensely,while often allowing players to “solve” the dilemma and cooperate.

6. Social context in the prisoners’ dilemma

Having considered the situation of either great sympathetic fluidity or extended pregameinteraction in the previous section, I turn now and face the more realistic game situationsin which sympathy, physical and psychological distances are circumscribed by the givensocial context and the details of the interpersonal setting. Such circumscription would cor-respond more closely to the psychological evidence presented earlier on the mechanics ofsocial distance.

Initially, let me examine sympathetic floors, i.e.λ ∈ [λ,1] with λ > 0. Such a floor mightbe create by both players seeing each other in the same room, or by being prior friends, or bycommunicating. The calculation of the general sympathetic equilibria within the normalizedprisoners’ dilemma are included in Appendix A, and Fig. 6 illustrates them forλ = 1/2.

As the floor increases(λ → 1), the following shifts occur: mutual sympathetic cooper-ation becomes more prevalent as its border pivots at (0, 1) up toward the edge of feasiblepayoffs; mutual defection occurs only for largersandt values, and is the sole sympatheticequilibria only if boths andt are very large; sacrificial equilibria fill the void once occu-pied by mutual defection for larger temptations, but also, are squeezed against the uppermargin,t − s = 1, by mutual cooperation. In addition, (1, C, 1, C) is risk dominant over alldefection equilibria across a greater portion of their overlap: in Fig. 6, everywhere but thenorthernmost piece of the common area.

As one might expect, the shifts in cooperation and defection are reversed when sympathyis bounded above, i.e.λ ∈ [0, λ] with λ < 1. Fellow-feeling might have a ceiling becausethe players are unacquainted, distant, and uncommunicative. As shown in Appendix A,(λ,C, λ,C) is a sympathetic equilibria when the temptation to defect is less than the ceiling(t < λ). Mutual, unsympathetic defection is unaffected by a falling ceiling untilλ < 1/2, atwhich point it expands towards thet-axis. The area of possible sacrificial equilibria expands


Fig. 6. Sympathetic equilibria with sympathy floor(λ = (1/2)).

slightly for 3/4 > λ ≥ 1/2. As the ceiling moves below a half, the sacrificial equilibriainvolve (λ,C) and exist when the sucker’s payoff is relatively small andt > λ. Fig. 7exhibits the equilibria forλ = 1/4.

As is clear from Section 2 above, constraints on sympathy result from limits on physicaland psychological distance. These latter limitations may be used to model various socialcontexts. Suppose that one player opens a door, walks in, and shakes the hand of the otherbefore each decides whether to cooperate. If this meeting has symmetric effects so thatϕij , ϕji ≤ ϕm < δ, then, from Eq. (2),λ = ω(1 − ϕm/δ). Suppose that two tourists fromthe same country are involved in a dilemma in a foreign land. The mutual knowledge andsalience of shared citizenship in this potentially xenophobic situation create a psychologicalcloseness so thatiψj , jψi ≤ ψξ < δ andλ = ω(1 − (ψξ/δ)). 16

Note that since experimental subjects often share a greeting upon arrival at the laboratoryor a co-citizenship as college sophomores, this analysis would lead us to expect positivelevels of cooperation even in a controlled experiment.

Different modes of communication can be distinguished by the floors to fellow-feelingthey form. Non-verbal communication, which requires that the players be able to see eachother, sets an upper bound to physical distance,ϕnv < δ. Any use of language requiresthat mutual knowledge be created and employed.17 The more complicated the signal is,the greater the common ground necessary for the coordination of meaning. Therefore,

16 Mead affirmed this phenomenon: “We meet the man in some distant country whom perhaps we would seek toavoid meeting at home, and we almost tear our arms off embracing him” (p. 218).17 See Lewis (1969), Schiffer (1972), Grice (1975) and Sally (2000d) for thorough discussions.


Fig. 7. Sympathetic equilibria with sympathy ceiling(λ = (1/4)).

this common knowledge constrains the psychological distance between communicationparticipants,ψck < δ. Finally, there is a psychological immediacy to the voice that evokesa different response in the listener than the memory or anticipation of an utterance does.18

Credit this vocal immediacy with a narrowing of social distance,ψv < δ. In ascending order,the sympathetic floors produced by non-verbal communication, writing, telephoning, andface-to-face discussion may be represented:

λnv = ω(1 − ϕnv

δ

); λw = (1 − ω)

(1 − ψck

δ

);

λt=(1 − ω)

(1 − ψck+ψv − δ

δ

); λf–f = 1 − ωϕnv + (1 − ω)(ψck + ψv − δ)

δ

Based on the analysis above, we should expect that cooperation would become morefrequent as language media with higher sympathetic floors were employed, and indeed, thisprediction coincides with the empirical findings of Sally (1995) and Wichman (1970).

Finally, the cultural background of the players may also be manifest as constraints onsocial distance. An egalitarian ideology may create a ceiling to social distance, while anindividualistic ethos may create a floor. Moreover, certain roles such as nurse, teacher, CEO,

18 Smith wrote, “The plaintive voice of misery, when heard at a distance, will not allow us to be indifferent aboutthe person from whom it comes. As soon as it strikes our ear, it interests us in his fortune, and, if continued, forcesus almostinvoluntarily to fly to his assistance” (p. 36, emphasis added).


Fig. 8. Nascent battle of the sexes.

or executioner, may be paired with explicit, strong sympathetic constraints.19 The existenceof social classes or distinct social roles may be reflected in asymmetric constraints. Givena high level of physical interaction, the servant is allowed to feel psychologically close tothe master, but the master is usually proscribed from feeling close to the servant. The sameskew may be present between a leader and his followers: Homans (1995) described justsuch a difference between the leader of the Bank Wiring Room at the Hawthorne worksand the other wiremen. So, the asymmetry of social class, status, or roles would facilitatethe observation of the sacrificial outcomes in the PD, that is, ift is large enough, we wouldexpect to see the servant cooperate and the master defect.

7. Sympathetic equilibria of other two-person games

Until now, I have considered only the sympathetic version of the prisoners’ dilemma. Thissection will examine the sympathetic equilibria ofGS whenG is another type of two-persongame.

7.1. Battle of the sexes

A given social context may create a battle of the sexes game where none existed before.A basic game in which there is one stable outcome (S, P), is displayed in Fig. 8; the playersorder a pizza with spinach and pepperoni and eat their favorite halves.

If p ≤ 4, this Nash outcome is reflected in pure sympathetic equilibria of the form(1λ2, S, 2λ1, P ) with 1λ2, 2λ1 ≤ (

2/√p) − 1. Suppose, however, that these players have

very similar tastes (except for pizza) and know that they have similar tastes. This psycholog-ical closeness may prevent either from feeling enough distance to sustain this equilibrium.Therefore, ifλ >

(2/

√p) − 1, the pure sympathetic equilibria entail coordination on only

one type of pizza. Along the same line, if both diners really love their favorite type of pizzaandp is large enough, thenGS (even with a full range of distances available) becomes abattle of the sexes with two equilibria: (0, S, 1/2, S) and (1/2, P, 0, P). There is an intuitiveappeal to these findings: a social situation that is trivially solved by acquaintances withmoderate predilections, may be confounded by the fellow-feelings of caring partners andtheir ability to take pleasure in the enjoyment of the other, and it may be jumbled by the

19 See Montgomery (1998) for a much more systematically developed and related argument.


Fig. 9. A zero-sum game.

presence of strong preferences. Friendship and sharp tastes can make some simple gamesquite messy.

7.2. Zero-sum

One of the assumed tenets of game theory is that every player knows the possible actionsand payoffs of each of the other players, or, at the very least, the identity of the sets fromwhich they are drawn. In the terms advanced by Knight (1957), there may be risk in a game,but not uncertainty: a player may have to resort to a subjective probability estimation, but heis certain about the identity of the support of this function. In some sense, this world is onewithout surprise.20 Any move, howsoever improbable, is simply an unlikely single-trialrealization; no one is able to plan an ambush, a trap, or a surprise attack, and expect greatersuccess. However, this assumption fails to capture the reality of many strategic situations, inwhich the possible creative moves of the opponent are unknown. This ignorance may ariseeither from an infinite, unbounded action set or from the cognitive limits of the players. Forexample, chess has a finite number of moves, yet the game does not appear to devolve intoposterior probability matching, rather, players may be both creative and surprising.

Sympathy has a strategic usefulness in this setting, a utility best illustrated in the case ofa pure zero-sum game. A very simple game is pictured in Fig. 9.

If this entire matrix is common knowledge to both players, then the mixed strategyoutcome is the following:p11 = (1/3), p12 = (2/3), p22 = (1/3), p23 = (2/3); andexpected payoffs,−(1/3) and (1/3) for 1 and 2, respectively. Suppose that either playermay be unaware of one of the moves accessible to the other.21 For instance, player 1 maynot realize thata23 is an action available to player 2, and thus, he may perceive thata11guarantees victory. If player 2 becomes familiar with player 1, he may become aware of thisgap in his knowledge and be able to exploit it by choosinga23 with certainty. On the otherhand, the first player may learn about this potential action by identifying with his opponent,and be able to counter it by mixing ina12.

Other awareness, or sympathy, is an undeniable input of strategy. Accordingly, there isroom for a dance of intimacy typical of such strategic settings, a minacious tango in which

20 An opponent choosing a move given a subjective probability of zero is also surprising. But if the probability wasassigned in a rational fashion, then such a move must be the initial foray of a sequenceunknownto the assessor.21 See Bacharach (1993) for a more formal treatment of this idea.


knowledge of the other is sought while secrecy of the self is guarded. By imaginativelyprojecting himself into the shoes of the other, a player may be able to assess what capa-bilities his opponent has. Thus, just before he went east to lead the Army of the Potomac,Grant claimed, “I know Lee as well as he knows himself. I know all his strong points, andall his weak ones. I intend to attack his weak points, and flank his strong ones” (Smith,1984, p. 182). Consequently, the achievement of common knowledge, often blandly as-sumed, necessarily entails a narrowing of psychological distance and an attendant increase insympathy.

Sympathizing with one’s opponent, and thereby, attaining mutual knowledge, in a zero-sum game is a maximin strategy. It leavens the pain of loss with the shared pleasure of theother’s victory, and it reduces vulnerability to an unknown reaction. Suppose thatψ < δ

is necessary in order to correctly assess the other’s strategic capabilities. This psycho-logical closeness creates a sympathetic floor,λ. The non-sympathetic mixed equilibriumre-emerges: player one bundles a sympathy level of(λ+ 1)/2 with a11 anda12, and playertwo links λ with his previous actions; the actions are mixed with the same frequencies asbefore; the expected payoffs move towards zero asλ increases, butV1 is always non-positiveandV2 is always non-negative.

I can draw a few conclusions from this application. First, sympathy does not changethe basic outcome of a zero-sum game: the “victor” does not suddenly lose because ofthe fellow-feeling arising from understanding the other’s strategy. However, in order toprotect against an unanticipated loss, sympathy does temper the joy of victory with thefellow-feeling for the defeated. Third, there may be non-zero-sum games in which a playermay rationally forego the strategic advantage of learning more about his opponent.22 Lastly,involvement in a zero-sum game may create an understanding and friendship although noneexisted before the contest began. Thus, Rommel (1953) in his papers presented a detailedmental model of Montgomery intermingled with high praise, and Jack Dempsey and GeneTunney exchanged personal favors throughout their retirements, with Tunney continuingto regard Dempsey as the top heavyweight boxer of all time (Dempsey, 1977; Heimer,1969).23

8. Comparison of fairness and sympathetic equilibria

Rabin (1993) has formalized the notion of fairness and has analyzed how fairness trans-forms a number of standard games. In Rabin’s model, fairness is an endogenously generatedlump sum added to the stated payoffs in a game, a sum whose quantity is determined by thechoices available to each player and their beliefs about each other’s actions. With respect tothe normalized prisoner’s dilemma, Fig. 10 shows how the fairness equilibria spread acrossthe parameter space.

22 Bob Frank pointed out this implication to me.23 Clearly, there are counter-examples, such as the enmity of Frazier for Ali, but in these cases the closenessnecessitated by strategy is overwhelmed by the conduct of the competitors within the game itself, e.g. Ali’shumiliating, pre-bout insults (see Hauser, 1991). My point simply is that the sympathy arising from strategy andthe sense of shared participation may out-weigh the inequality of outcomes in the game.


Fig. 10. Fairness equilibria of normalized prisoners’ dilemma.

The fairness extension of the prisoners’ dilemma is unlike the sympathetic extension ina number of ways. First, (D, D) is an equilibrium for every pair of temptations and sucker’slosses.24 Second, although (C, C) is a fairness outcome ift ≤ 1/2, it is never either the soleor the risk dominant equilibrium. Third, sacrifice cannot be a stable choice among Rabin’splayers. Finally, mutual cooperation is extinguished as the scale of the payoffs increases.Suppose that each of the payoffs in the normalized prisoners’ dilemma were multiplied bythe same factor,ζ > 1. For every positive temptation, there exists a maximum scale factorabove which the rational player is enticed to defect. In particular, fort > 0,ζ = (1/2t), andfor ζ > ζ , considerations of fairness pale besides the gains to defection. Notice that relativesympathetic preferences are not distorted by altering the scale of the payoffs. BecauseVi = ζvi(sj , sj ) + Λ(iλi,j λi)ζ vj (si, sj ) = ζVi , ζ does not affect the ranking of theutility arising from various sympathy and action pairs. From a qualitative perspective, thiscontrast between fairness and sympathy is sensible. We might consider fairness an objectivesocial standard that is vulnerable to the vagaries of personal reward, in the same way thatother social regulations are violated when they become too costly. On the other hand, ifsympathy is rooted in interpersonal identification, there is a subjective symmetry that isnaturally undisturbed by the height of the ground on which the individuals encounter eachother.25

24 Note that (D, D) is an abbreviation of the true psychological equilibrium ((D1, 1[D2], 1[2[D1]]), (D2, 2[D1],

2[1[D2]])) where i [x] means thati believes that “x”.25 Rabin proved that any strict Nash equilibrium of the underlying game must re-emerge as a fairness equilibriumat some finite multiple of the payoffs. This general proposition clearly does not hold for sympathetic equilibria.


Rabin advanced some general propositions about fairness equilibria based on the follow-ing two definitions: an outcome in the underlying game ismutual-maxif, for i = 1,2, j 6=i, si ∈ arg maxs∈Si vj (s, sj ); if both players are minimizing the immediate utility of theother, then the outcome ismutual-min. For example, joint defection in the simple prisoners’dilemma is mutual-min. Given the effective sympathy function assumed above, it is straight-forward to derive the following corollary propositions for sympathetic equilibria (proofsare found in Appendix A):

Proposition 2. If (s1, s2) is a Nash equilibrium of G and is mutual-max, then it is includedin a sympathetic equilibrium.

As shown by the analysis of the sympathetic prisoners’ dilemma, this proposition does nothold for mutual-min Nash equilibria: they are fairness equilibria, but not necessarily sym-pathetic. However, letλT ∈ (0,1) be a fellow-feeling threshold, and allowλ ≤ 1 andλ ≥ 0to be a ceiling and floor to sympathy, respectively, based on the identity of the participantsand the nature of the social interaction. Then, we have these two propositions:

Proposition 3. If (s1, s2) is a strict Nash equilibrium of G, then there existsλT such thatfor all λ ≤ λT, (s1, s2) is included in a sympathetic equilibrium.

Therefore, if people have no way of identifying above a certain extent with each other,then they may both sympathetically choose the best-response action of the underlyinggame.26 This proposition generalizes the finding of the effect of a sympathetic ceiling onthe presence of mutual defection within the prisoners’ dilemma.

Proposition 4. If (s1, s2) is a jointly dominant outcome,27 then there existsλT such thatfor all λ ≥ λT, (s1, s2) is included in a sympathetic equilibrium.

Therefore, if people have no way of distancing themselves across a certain cleft, thenthey may select the optimal solution for the group, even if this outcome was not stablein the underlying game. This conclusion is a generalization of the previous finding that alower bound to fellow-feeling increases the presence of mutual cooperation in the prisoners’dilemma.

There is one final point to be made about the relationship between fairness and sympathy,and it is that these two notions are by no means inimical, and in fact, occur concomitantlyin many social interactions. The lump-sum nature of the reward from fairness capturesan important aspect of sympathy I have not discussed to this point, an aspect outlined bySmith in this passage: “Sympathy, however, enlivens joy and alleviates grief. It enlivensjoy by presenting another source of satisfaction; and it alleviates grief by insinuating intothe heart almost the only agreeable sensation which it is at that time capable of receiving”(p. 14). Thus, the kindness of sharing the grief of a friend may make sympathizing a

26 The “may” in this last clause reflects the possibility that other sympathetic equilibria also exist.27 That is, actions (s1, s2) such that for anys′1 ∈ S1 ands′2 ∈ S2, (s′1, s

′2) 6= (s1, s2), v1(s1, s2) + v2(s1, s2) >

v1(s′1, s

′2)+ v2(s

′1, s

′2).


utility-maximizing interaction. On the other hand, the relativistic nature of sympathy leav-ens the strictness of viewing fairness as a purely objective phenomenon. Hence, friendsmight solve a dilemma concerning a million dollars as easily as they do one involving tendollars.

9. Conclusion

What do economists see when they look at a game? Certainly, a matrix or a tree, setsof strategy spaces, probabilities, equilibria, but maybe most importantly, interdependency.What do players see when they look at a game? Certainly, choices, payoffs, risks, but maybemost importantly, interaction. The simple point of this paper is that strategic interdependencyand social interaction are intimately linked. They are linked primarily because humans areendowed with the capacity for sympathy, and sympathy guides strategy formulation andis generated by social interaction. From this perspective, playing a game is but one formof social interaction, and game theory, then, should be part and parcel of a broader socialrelations theory.

Sympathy can be the basis for this broader theory. Our feelings for our fellow play-ers are quite predictably affected by their physical and psychological distance from us.Many games take place in a social context in which the players may be able to act toaffect these distances — I may call you for a meeting, wink, extend a hand, offer a com-pliment, disagree with your opinion, insult your parents, turn my back, close my eyes,etc.

The paper has demonstrated, using formal language, that familiar games embedded insympathy, social distance, and social relations may be transformed:• The prisoners’ dilemma has a variety of possible sympathetic outcomes depending on

the specifics of the payoff matrix and the players’ relationship. If I were a gamemaster,I could partially control the outcome of the PD by altering the identities of the players,their relationship and interaction, and the social environment.

• Mutual cooperation should be expected when the PD is played by players who havecommunicated, who sit across the table from each other, who belong to the same club,who are both surrounded by foreigners or outsiders, etc.

• Mutual defection should be expected with players who sit halfway around the globefrom each other, who are anonymous, who drink Coke and Pepsi, respectively,etc.

• Sacrifice in the PD may be expected when a manager plays an employee or a master,a servant; when only one player is informed of the identity and background of another;when players are seated across a one way mirror, etc.

• Simple games can be complicated by sympathy (the nascent Battle of the Sexes), and allgames may be affected by the social setting (propositions 2, 3, and 4).

• A sequence of zero sum games may create and constitute a relationship (e.g. Tunney andDempsey; Evert and Navratilova).

• In most cases, strategy requires taking the role of another, or stepping into the shoes ofanother, and thus, might affect the sympathy and behavior of the strategist.


Appendix A

Lemma. Vi is continuous overL× S1 × L× S2.

Proof. Let ni equal the number of elements inAi . L × S1 × L × S2 is a compact sub-set ofRn1+n2+2, since it is closed and bounded. Definev∗

i : L × S1 × L × S2 → R,such thatv∗

i (iλj , si , j λi, sj ) = vi(si , sj ) for all iλj , j λi, si , sj . Similarly, defineΛ∗:L×S1×L×S2 → R, such thatΛ∗(iλj , si , j λi, sj ) = Λ(iλj , jλi) for all iλj , j λi, si , sj . Setε and letΛ∗(iλj , si , j λi, sj )−Λ∗(iλ′

j , s′j , j λ

′i , s

′j ) < ε. Then, by definition,Λ(iλj , jλi)−

Λ(iλ′j , j λ

′i ) < ε. SinceΛ is continuous, there existsδ such that|(iλj , j λi)− (iλ′

j , j λ′i )| <

δ, where | | represents the Euclidean norm. Due to the bounded nature of simplexes,|(si, iλj , sj , j λi) − (s′i , iλ

′j , s

′j , j λ

′i )| <

√4 + δ2. Therefore,Λ∗ is continuous overL ×

S1 ×L× S2. A similar argument can be made forv∗i . Since its components are continuous,

v∗i +Λ∗v∗

j is continuous (Rudin, 1976), andVi is also continuous. �

A.1. Sympathetic equilibria for normalized PD

I have specified thatΛ = Λ1. The pure sympathetic equilibria will be found through abest response search.1. (1, C, 1, C). Clearly, a unilateral reduction of sympathy without changing to defection

lowers utility. A defector would chooseλ to maximize 1+ t − λ(1− λ+ 1)s, and thus,would setλ = 0. His payoff would be 1+ t , which is less than 1+ 1 × 1 for t < 1.

2. (λ, D, λ, D). A unilateral change in sympathy while still defecting would leave utilityunchanged. A cooperator would chooseλ′ to maximize−s + λ′(λ − λ′ + 1)(1 + t).This amount is maximized whenλ′ = ((λ + 1)/2). Substituting, the maximal sacri-ficer utility is −s + ((λ + 1)/2)((λ + 1)/2)(1 + t), which is greater than 0+ λ × 0if ((λ + 1)/2)2 > (s/1 + t). Note that if mutual defection is rewarded [punished]with an immediate payoff ofε [−ε], then (λ, D, λ, D) is dominated by (1, D,λ, D)[(0, D, λ, D)].

3. (1/2, C, 0, D) and (0, D, 1/2, C). The defector has no incentive to increase his sympathysince he would share in the loss of the cooperator. Moreover, his payoff from cooperatingwould be 1+ λ((1/2) − λ + 1). For λ = (3/4), best-response cooperative utility ismaximized at (25/16). Defection is more attractive if 1+t > (25/16). On the other hand,the player choosing ((1/2), C) has chosen an optimal sympathy level sinceλ(0− λ+ 1)peaks at (1/2). Defection is alluring if 0> −s + (1/2)(0 − (1/2)+ 1)(1 + t).

4. There are no other pure sympathetic equilibria. Note that, despite appearances from thenormal form in Fig. 3, (3/4, C, 3/4, C) is not an equilibrium, sinceV1(3/4,C,3/4,C) <V1(7/8,C,3/4,C).

A.2. Sympathetic equilibria for generalized PD

Fig. A-1 shows a generalized prisoners’ dilemma game. The definition of a prisoners’dilemma requires thata > c > d > b and 2c > a + b. GivenΛ1, the following are thesympathetic equilibria and the conditions limiting their existence:


(0,D,0,D) if d ≤ 0 and 4(d − b) ≥ a

(1,D,1,D) if d ≥ 0 and 2d − b ≥ a

(λ,D, λ,D) if d = 0 and−(

4

(λ+ 1)2

)b ≥ a

(12,C,0,D

)if d ≤ 0 and 4(d − b) < a and 16a > 25c

(0,D, 12,C) if d ≤ 0 and 4(d − b) < a and 16a > 25c

(1,C,1,C) if b ≥ 0, or if b < 0 and 2c > a

A.3. Risk dominant, sympathetic equilibria for normalized prisoners’ dilemma

Following Harsanyi and Selten, I can calculate the Nash products for each equilibriumin those regions where there are more than one. Whent ≤ 1 and 4s ≥ 1+ t , (1, C, 1, C) isrisk dominant over (0, D, 0, D) and any other (λ, D, λ, D) if(2 − (1 + t))2 > (0 − (−s))2 or, in other words, 1− t > s. 28

Whent ≤ 1 and 4s ≤ 1 + t , (1, C, 1, C) is risk dominant over either (0, D, 1/2, C) or(1/2, C, 0, D) if

1

4(1 − t) >

(1 + t

4

) (t − 1

2

).

This is satisfied fort < ((3 − √33)/− 4) ≈ 0.686.

A.4. Sympathetic equilibria with sympathy floor,λ > 0

Let GS once again be the normalized prisoners’ dilemma with both players preferencesgiven byΛ1.1. (1, C, 1, C). Now, the potential defector cannot fully distance himself from his part-

ner, so he remains a cooperator as long as 2> 1 + t − λ(1 − λ + 1)s. This in-equality is represented by an area below a line pivoting at the point (0, 1) in (s, t)

28 I am claiming that (0, D, 0, D) risk dominates (λ, D, λ, D) since its Nash product with respect to (1, C, 1, C) isgreater.


space. As the floor increases, the border swings from the horizontal toward the limit oft − s = 1.

2. (λ,D, λ,D). The minimally sympathetic defection equilibrium is stable against sacrificeif 0 > maxλ − s + λ(λ − λ + 1)(1 + t). The right-hand side is maximized whenλ =((λ + 1)/2). Therefore, this mutual choice is stable ifs > ((λ + 1)/2)2(1 + t). Thiscondition is similar to those we have seen earlier: the floor simply eliminates mutualdefection equilibria with lower degrees of fellow-feeling.

3. (λ,D, ((λ + 1)/2),C) and (((λ + 1)/2),C, λ,D). The inequality from the defectionequilibria in condition 2 above, must be reversed so that the gift-giver is not enticedto defect. The gift-receiver will feel no incentive to reciprocate if 1+ t − λ(((λ +1)/2)− λ+ 1)s > maxλ 1 + λ(((λ+ 1)/2)− λ+ 1). The reciprocator would do bestif he choseλ = ((λ + 3)/4). Therefore, the southeast boundary of sacrifice satisfiest > ((λ + 3)/4)2 + ((3λ − λ2)/2)s. As the floor increases, mutual cooperation beginsto crowd out sacrifice.

4. Risk dominance. The comparative Nash products for (1, C, 1, C) and(λ,D, λ,D)are (2 − (1 + t − λ(2 − λ)s))2 and (s − λ(1 + t))2, respectively. Whenλ = 1/2,the former is greater than the latter fors < 6 − 2t , in other words, for almost allthe overlapping region pictured in Fig. 6. Given this same floor, (1, C, 1, C) is riskdominant over the sacrificial equilibria only when 1− t + (3s/4) > (1 + t)(t −(5s/8) − (3/4)), representing a small sliver in the southwest corner of the commonregion.

A.5. Sympathetic equilibria with sympathy ceiling,λ < 1.

1. (λ,C, λ,C). Either cooperator is enticed by unsympathetic defection which yields utilityof 1 + t , compared to cooperative utility of 1+ λ. So, as long asλ > t , sympatheticcooperation is a possibility.

2. (0, D, 0, D). If social distance can still be reduced enough to achieve fellow-feelingnot less than a half, then the border of unsympathetic defection is the same as seenin part I of Appendix A,s > ((1 + t)/4). However, if λ < 1/2, then sacrificingaffords less utility and defection expands towards thet-axis according tos > λ(1 − λ)

(1 + t).3. Sacrificial equilibria. The identity of these equilibria varies with the height of the ceiling.

(1/2, C, 0, D) and (0, D, 1/2, C) are pure sympathetic equilibria ifλ > 1/2 ands <((1 + t)/4). For a ceiling above (3/4), the southern border of sacrifice is the same aswith no constraint,t > (9/16); for (3/4) > λ ≥ (1/2), sacrifice crowds out cooperationand the border ist > λ((3/2) − λ). When closeness becomes even more difficult andλ < (1/2), the sacrificial equilibria are(λ,C,0,D) and(0,D, λ, C) bounded byλ > t

ands < λ(1 − λ)(1 + t).

A.6. Proofs of additional propositions

Proposition 2. If (s1, s2) is a Nash equilibrium of G and is mutual-max, then it is includedin a sympathetic equilibrium.


Proof. The identity of the sympathetic equilibrium will depend on whether the immediateutilities are positive.1. vi(s1, s2) ≤ 0 for i = 1,2. Both players minimize fellow-feeling with the other in

order not to share in misery. Because (s1, s2) is a Nash equilibrium ofG, and since anyshift in action makes the other person worse off, (0,s1, 0, s2) is a Nash equilibrium ofGS.

2. vi(s1, s2) > 0 for i = 1,2. Both players maximize fellow-feeling with the other inorder to share in happiness. By assumption, antipathetic utility is unavailable. Choos-ing a different action would lower the immediate utilities of both players. Therefore,(1, s1,1, s2) is a Nash equilibrium ofGS.

3. v1(s1, s2) > 0 and v2(s1, s2) < 0. Applying analogous reasoning from parts 1 and 2above,(0, s1,1/2, s2) is a sympathetic equilibrium ofG.

�Proposition 3. If (s1, s2) is a strict Nash equilibrium of G, then there existsλT such thatfor all λ ≤ λT, (s1, s2) is included in a sympathetic equilibrium.

Proof. Proposition 3 covers the case when (s1,s2) is mutual-max, so, let it be not mutual-max.Then, without loss of generality, there exists a non-empty set of actionsS

′′1 ⊆ S1 such

that S′′1 = {s′′1 ∈ S1|v2(s

′′1, s2) > v2(s1, s2)}. Let s′′1

max be a maximal element ofS′′1. If

v2(s′′max1 , s2) ≤ 0, then player one has no incentive to shift from unsympathetically maxi-

mizing his own immediate utility. Assume, therefore, thatv2(s′′1

max, s2) is positive.Let S′

1 = {S1 − s1}. Then, lets′1max ∈ S′′

1 such thatv1(s′1

max, s2) ≥ v1(s′1, s2) for

all s′1 ∈ S′1. DefineλT = min(1, (v1(s1, s2) − v1(s

′1

max, s2))/v2(s′′max1 , s2) − v2(s1, s2)).

Because of the strictness requirement, the numerator is positive, and because (s1, s2) is notmutual-max, the denominator is also greater than zero. Therefore, 1≥ λT > 0.

The situation most threatening to the stability of (s1, s2) occurs whens′1max = s

′′max1 .

If v2(s1, s2) ≥ 0, thens1 is included in a positive sympathetic equilibrium ifv1(s1, s2) +λTv2(s1, s2) ≥ v1(s

′1

max, s2)+λTv2(s′1

max, s2). (This condition relies on the fact thatλT ≥λT(λ − λT + 1) for anyλ ∈ [0, λT]). Substituting forλT, this weak inequality must hold.On the other hand, ifv2(s1, s2) < 0, then an unsympathetic equilibrium withs1 is endan-gered only ifv1(s1, s2) < v1(s

′1

max, s2)+ λTv2(s′1

max, s2). Again, substituting forλT, thisinequality cannot hold sincev1(s1, s2) > v1(s

′1

max, s2).The analogous argument can be made from player 2’s perspective, which would establish

another lower bound onλT. The lesser of these two lower bounds would still be greaterthan zero, and would permit (s1, s2) to be incorporated in a sympathetic equilibrium.�

Proposition 4. If (s1, s2) is a jointly dominant outcome, then there existsλT such that forall λ ≥ λT, (s1, s2) is included in a sympathetic equilibrium.

Proof. Again, suppose (s1, s2) is not a mutual-max outcome. The structure of this proof issimilar to the previous one above. LetS∗

1 = {s∗1 ∈ S1| v1(s∗1, s2) > v1(s1, s2)+ v2(s1, s2)}.

If S∗1 is empty, then there are no appealing alternatives to s1 for player 1. AssumeS∗

1is not empty ands∗max

1 is a maximal element. Then, defineλT = ((v1(s∗max1 , s2) −

v1(s1, s2))/(v2(s1, s2) − v2(s∗max1 , s2))). Because (s1, s2) is jointly dominant,λT < 1.

It must be true thatv2(s∗max1 , s2) is negative, so the best alternative is to minimize sym-


pathy and chooses∗max1 . Note thatλT ≤ λT(λ − λT + 1) for anyλ ∈ [λT,1]. A sufficient

condition for (s1, s2) to be part of a sympathetic equilibrium isv1(s1, s2)+λT v2(s1, s2) ≥v1(s

∗1, s2)+λTv2(s

∗1, s2). Substituting forλT, this weak inequality must hold. This argument

carries through for player 2, creating another upper bound on the threshold. The larger ofthese two is still less than one, and would allow (s1, s2) to be incorporated in a sympatheticequilibrium. �

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