reciprocity, altruism, solidarity: a dynamic model

DOI 10.1007/s11238-006-7192-6Theory and Decision (2006) 60:371–394 © Springer 2006

FRIEDEL BOLLE and ALEXANDER KRITIKOS

RECIPROCITY, ALTRUISM, SOLIDARITY: ADYNAMIC MODEL

ABSTRACT. Reciprocity is a decisive behavioural rule resulting insuccessful co-operation or deterrence. In this paper, a dynamic modelis proposed, where reciprocity is described by changes in altruistic (ormalevolent) ties. Multiple steady states may exist in one of which theremay be general cooperation (solidarity) and the other being one of uni-versal malice (war of each individual against all other individuals). Weapply our theory to a number of examples, illustrating that the agents’initial preferences determine whether a steady state of solidarity, selfish-ness or malice will emerge.

KEY WORDS: Altruism, cooperation, dynamic model, reciprocity, soli-darity

1. INTRODUCTION

The willingness to behave altruistically and reciprocally isdeeply rooted in all human societies. The advantage of everyindividual in a society adapting altruistic behaviour is appar-ent: external effects of one’s behaviour are internalized and—in comparison to merely egoistic behaviour—more efficientstates are realized. Furthermore, the expectation of positive(negative) reciprocity may additionally enhance altruistic deci-sions. If we expect “friendly” behaviour to be rewarded and“unfriendly” behaviour to be punished then we may take intoaccount the consequences of our actions for others and, thus,act “as if” altruistic.

To formalize behaviour beyond mere self-interest, severalmodels introduced various kinds of social preferences. Withsome exceptions most of them either deal with choices resem-bling only altruistic preferences or they focus on reciprocalchoices alone. Most models have a static framework in the

372 FRIEDEL BOLLE AND ALEXANDER KRITIKOS

sense that these preferences for altruistic or reciprocal choicesare exogenously given. The major motivation of the presentpaper is now, in contrast to these models, to explicitly embedaltruistic and reciprocal preferences in one dynamic approach,where we then are able to analyse the long-term consequencesof the individuals’ attitudes towards the peers in their society.We will shortly discuss some of the existing models (and howthey relate to our work) in this section.

By now altruistic behaviour is mostly modelled as a pos-itive externality of the income of others on one’s utility(see inter alia Collard (1978)). General models on altru-ism are also able to describe the existence of spite (whenthe external effect is negative). Specific models of inter-per-sonal preferences were proposed by the “inequity aversion”approaches of Bolton and Ockenfels (2000) and Fehr andSchmidt (1999). They explain that the relative performanceof incomes between two related persons decides whether theexternality will be positive or negative, and where any non-selfish choice will be done in order to reduce the differencebetween one’s own and all others’ payoffs. An alternativemodel has been formulated by Arrow (1975) and by Andreoni(1995) who suggests a self-centered positive utility of giving.

In games with more than one stage, these models of altru-ism are to a certain extent capable of describing indirect rec-iprocity: When the income of a person A is increased andwhen, thus, the marginal utility of A’s income is decreasing(for instance, because A is inequity averse) then A has alarger propensity to act in favour of others. The reason forincreasing the income of A could be someone else’s altruis-tic act towards A, and the increased readiness of A to actaltruistically may but must not be (by definition) in favour ofthe benefactor of A. These models are, thus, able to capturereciprocal choices. However, they do not take into account theintentions of the fellow player, when he decided for a specificaction at the previous stage.

Different to such “material welfare based approaches”Becker’s (1981) model on altruism involves an interdepen-dency of utility functions,1 i.e. altruistic choices of a person

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A do not only depend on the parameters of his own utilityfunction but also on the parameters of his peer’s utility func-tion (Person B). In this model A takes into account B’s atti-tude about A.

While in Becker’s (1981) approach the relations between Aand B are implicit, Levine (1998) explicitly starts in his exten-sive form model with utility functions where specific “types”are defined.2 Under complete information about all parame-ters of A’s and B’s utility function, altruistic (or non-altruistic)decisions do not depend anymore on actual behaviour buton the given parameters in this type-based approach: Forinstance, A will show the more altruism for B, “the more Blikes A”. In Becker’s approach the direction of influence isambivalent (Bolle and Kritikos (2004)): person A may showmore or less3 altruism if a person B starts to like A more.Under incomplete information about these parameters in thefellow person’s utility function, however, there can be a linkbetween actual behaviour (for instance fair behaviour) of Btowards A—which may serve as a signal that B likes A—andthe kind of A’s reaction towards B.4 The models of Becker(1981) and Levine (1998) imply a different kind of indirectreciprocity than the approaches mentioned before. Choices ofreciprocal altruism may arise on the basis of mutual regard.

Parallel to these indirect explanations of reciprocal behav-iour, direct approaches to reciprocity were formulated. Theseapproaches are based on the argument that a person A isguided by the desire to increase or decrease his fellow playerB’s income depending on how fair B treated A on an earlierstage. Thus, these models analyse decisions in strategic set-tings where the intentions of the fellow players have a directimpact on the decision making process of A.

Since in this case intentions need explicit modelling, Rabin(1993) suggested in his model the following principles of rec-iprocity for normal form games: Person A will reward a per-son B if he perceived person B’s action as kind (B’s intentionsmust have been friendly) and A will punish B if he perceivedB’s action as unfair (B’s intentions must then have been hos-tile). To separate friendly from hostile actions, intentions were


defined depending on the reference point of equitable materialpay-off.

Dufwenberg and Kirchsteiger (2004) developed similar prin-ciples of reciprocity for extensive form games and introducedconditional cooperation of final players as an option lead-ing to a “sequential reciprocity equilibrium” (a concept notdefined in the approach of Rabin). Falk and Fischbacher(1998) extended the approach of Rabin (1993) insofar asthey combined his reciprocity approach with the inequalityaversion model of Fehr and Schmidt (1999). This modifi-cation enabled the relaxation of Rabin’s restrictive assump-tions about perceiving intentions as kind or unkind. Last butnot least, in an overview by Sethi and Somanathan (2003)it is analysed for various kinds of interactions under whatconditions reciprocal preferences may survive in an evolu-tionary context. They emphasize the importance of repeatedinteractions.

These modifications improve the description of behaviour instrategic games, but they all suffer from the main shortcomingof reciprocal approaches, namely—as Fehr and Schmidt (1999)point out—these models do not allow us to explain decisionsin isolated one-stage games.

Based on experimental evidence, Kritikos and Bolle (2001)therefore emphasized the necessity to combine the behaviouralvariables of altruism and reciprocity in a joint model. Theapproach pioneered by Charness and Rabin (2002) is in thisspirit, where so-called social welfare preferences are linked toreciprocal concerns. Their approach is specified so that socialwelfare preferences put emphasis on the positive side of altru-ism, while reciprocal motives appear according to this conceptonly when a person is confronted with mean actions by theirfellow players. The approach of Charness and Rabin is able toexplain decisions observed in various kinds of one-shot exper-iments as well as in experiments with a strategic context. Atthe same time, it has to be pointed out that their model isnot able to explain any behaviour in the domain of the oppo-site side of altruism (the spite component) and of positivereciprocity—both relevant fields of decision making as earlier

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experiments showed (for an overview see, e.g. Kritikos andBolle (2004)).

In this paper, we will formulate a synthesis of altruism andreciprocity theories, which differs in several ways from pre-vious research. While the papers mentioned before basicallycapture interpersonal reciprocal decisions we will focus on amore general approach, where we analyse a kind of social rec-iprocity, which does not only encompass interpersonal recip-rocal choices but which also spots on the consequences ofthese choices on a more “macro-level”. Second, we analysethose dynamics, which may arise, when altruism and reciproc-ity are combined in a common model. Van Dijk and vanWinden (1997) propose a similar dynamic model but withcrucial differences. We will compare their approach with ourmodel in Section 4.

We provide a simple linear, n-person model of preferencesthat assumes that person A’s propensity to make an altruis-tic action towards person B is characterized by an altruismparameter (= A’s marginal utility of B’s income) which hasan initial (positive, neutral or negative) value. This parameterchanges from period to period and is influenced by the bene-fits person A received from B in the previous period.5

As a main result of our model we will show how theaccount of past relations with a person leads to “friendly”or “unfriendly” behaviour towards this person. An act offriendly behaviour, regarded as an isolated action, is denotedas altruism. According to this model, an individual’s altruismis strengthened if others do this person a favour so that in thelong run reciprocal altruism can be induced. If the initial atti-tude towards others is friendly6 a self-enforcing process mayunfold where accounts of friendliness are increased step bystep until a steady state of general and strong altruism withina group is reached. We denote one’s attitude and behaviour inthis steady state as solidarity. The self-enforcing process maybe driven into the opposite direction, as well—either causedby tempting behavioural alternatives, by the initial behaviourof others, or by one’s own “initial values”. Accordingly, wewill prove this result with the so-called Janus7 proposition.


The remainder of the paper is organized as follows: Section2 is dedicated to the presentation of the basic model, and onthe technical questions of existence and uniqueness of steadystates. In addition, the Janus proposition is presented. Basedon the steady states characterized in Section 2, we will pro-vide in Section 3 three stylized examples illustrating the JanusProposition and we will discuss the separability of externali-ties, which is central for our approach (as well as for all othermodels of explicit reciprocity). Section 4 compares our modelwith similar approaches in the literature. Section 5 concludes.

2. THE MODEL AND ITS STEADY STATES

Assume a group of n persons. We describe and denote anaction of person i by the vector Ti = (Ti1, . . . , Tin) of exter-nal effects connected with this action. Ti is called transfer vec-tor. Tij describes changes of income of j caused by the actionTi of i. Ti can be chosen from a set of feasible actions Ri .Instead of describing a single action, it may also be assumedthat Ti comprises many actions and results from a certain“mode of behaviour” during a time interval. As Ti describeschanges of incomes, we have to define a benchmark, which iscalled “normal income”. In Section 3, we will show how “nor-mal income” is usually used in game-theoretic settings and wewill discuss the “decomposition” of the effects of joint actions.

Our basic model consists of two simple assumptions. First,an action in period t is selected so that it maximizes the linearutility function

Uti =

∑

j

atijT

tij (1)

with atii = ei > 0 and positive or negative coefficients at

ij .(Below, we will see that, for most purposes, we can set ei =1.)

Second, we assume that all individuals i can observe Tij ,and that j reacts to the transfer Tij , which he receives from i.The altruism parameter at

ij of i with respect to j may changefrom period to period and does so according to the followingrule.

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at+1ij = δia

tij +γiT

tji, 0<δi <1, γi >0, j �= i. (2)

The initial values of a0ij as well as of γi and of δi are exoge-

nously given and agent specific. aik decays relative to the con-stant aii = ei >0 which describes the “strength of egoism”.

Our simple models (1) and (2) describes a self-enforcingprocess. Altruistic preferences of i may lead to altruisticchoices of i with Tij > 0, hence eliciting the altruism of oth-ers and increasing the probability of altruistic choices of oth-ers with Tji > 0. Then, in return, i’s altruistic preferences arefurther enforced.

We assume that, in every period, every individual i willchoose a transfer vector from Ri . A steady state of thedynamic process, is defined by

atij =aij for all i, j, t. (3)

Because of (2) this requires T tij =Tij and

aij = γi

1− δi

Tji for j �= i. (4)

Maximizing the utility function (1) implies(Tij

)∈ arg maxRi

∑

j

aijTij . (5)

Thus, Equations (4) and (5) describe a steady state. If all (ormost) aij are positive, this state is called a Solidarity State.8

According to (4), in a steady state the altruism parameter aij

of i is influenced by transfers received from j, by the strength ofhis reaction yi to these transfers and by the decay parameter δi .

A steady state needs actions Ti , which are (in the languageof Game Theory) best replies to (T1, . . . , Ti−1, Ti+1, . . . , Tn)when the coefficients aij are as in (4). The analysis of the exis-tence of a steady state is reduced to the question whether agame-theoretic equilibrium exists with strategy spaces Ri andpay-off functions

Vi = eiTii +βi

∑

j �=i

TijTji, βi =γi/(1− δi). (6)


PROPOSITION 1 (Existence). If the action sets Ri are compactand convex, then there is always a steady state with aij =βiTji, Tj = (Tji)=optimal action ofj.

Proof Straightforward.9

Decisions remain the same if a utility function is multi-plied by a constant. Thus (6) has another implication: As longas we focus on steady states, only the relation βi/ei is deci-sive. Thus, we will simplify the approach beyond Section 2 byassuming ei =1.10

Steady states are sometimes unique and sometimes—as wewill show in the following examples—multiple steady statesmay exist. The crucial point of the following Janus Proposi-tion is now that under certain conditions, if there is a (steady)state with a certain pattern of friendliness or hostility betweenthe members of a group, there is also a steady state whereall relations change their sign, i.e. all hostile relations becomefriendly and vice versa. Thus, the configuration of initial val-ues decides whether a steady state of solidarity, egoism or uni-versal malice will emerge. We continue with

PROPOSITION 2 (Janus Proposition). Let us assume that allRi are symmetric with respect to the axis Tii , i.e. if (Tij )j=1,... ,n ∈Ri, then also (Tij )j=1,... ,n ∈Ri with Tii =Tii and Tij =−Tij for j �=i. Then, if there is a steady state with Tij =T ∗

ij , aij =βiT∗ij , j �= i,

we also have a steady state with aij =−aij , j �= i.

Proof If T ∗i maximises (6) with given Tj =T ∗

j for j �= i, thenTi = Ti maximizes

Vi = eiTii +βi

∑

j �=i

Tij Tj i (7)

with given Tj = Tj , j �= i.The Janus Proposition highlights the ambiguity in predict-

ing steady states. The proposition assumes that every set Ri

has to be symmetric while at the same time neither the Ri norγi and δi need to be equal. But even without complete symme-try of Ri , steady states of completely different characters maybe realized. There may be solidarity states, where everybodyhelps all others, where everybody is eager to carry out actions

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with large positive external effects. Yet, in the same group,with the same opportunity set of choices, steady states arepossible, where there is war of every individual against allother individuals, where the group or the society suffers fromenvy and spite.

3. APPLICATIONS OF THE JANUS PROPOSITION

3.1. A useful two-player game example

In Figure 1, we will provide a first example to explain theconsequences of the Janus proposition. In this two playergame, both players have to choose between three possibleactions shown in Figure 1. The set of actions, which aredenoted in terms of transfers are R1 ={(1,0), (−1,3), (−1,−3)}and R2 = {(0,1), (3,−1), (−3,−1)}. Thus, both players maychoose between an “egoistic act”, which does not affect theother one, a costly “altruistic act” (−1, 3) for player 1, anda costly “malevolent act” (−1,−3) for player 1.

If both players have egoistic preferences, the unique Nashequilibrium is ((1,0), (0,1)). The dynamic process of ourmodel as defined in (1) and (2) may support this equilibrium,if the initial values a0

ij are sufficiently small or zero. How-ever, if ei = 1, β1 = β2 = β > 2/9, then also the efficient statewith mutual altruistic acts ((−1,3), (3,−1)) can be reached, aswell as the highly inefficient state with mutual malevolent acts((−1,−3), (−3,−1)). Thus, which of the three different steadystates is the outcome of the dynamic process, depends on theinitial values.

3.2. Separable externalities or the decomposition of pay-offsin games

In this subsection, we apply our approach to the PrisonersDilemma. It will illustrate that the characteristics of theJanus Proposition do not depend on the complete symmetryof action spaces. More importantly, however, we will alsodemonstrate the necessity to decompose payoffs, i.e. to assign


Figure 1. A two-player Game illustrating the Janus Proposition.

specific transfers to specific agents. In the second part of thissubsection, we will discuss this procedure in order to clarifywhy payoff decompositions are necessary and what kind ofproblems might be solved in a better way if we are indeedable to describe actions by transfer vectors. As we will alsosee, our discussion might be relevant for other approaches ofreciprocity.

Figure 2 displays a typical two-person Prisoners’ DilemmaGame, where the unconditional symmetric decomposition ofthe matrix values leads to transfers Ci with Tii = 1, Tij = 1and Di with Tii = 2, Tij = −2. (Adding up these incomesfor all combinations of choices leads again to the Prisoners’Dilemma game of Figure 2.) In addition to the symmetricdecomposition there is an infinite number of unconditionalasymmetric decompositions for the two players of this game.One example is C = (T11, T12) = (0,0),D1 = (T11, T12) = (1,−3),

C2 = (T12, T22)= (2,2),D2 = (T12, T22)= (−1,3). Again, if we addup the transfers between both players, the Prisoners’ Dilemmagame of Figure 2 results.

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Figure 2. A typical Prisoners’ Dilemma game.

When we analyse the dynamic process as defined in Equa-tions (1) and (2) for the transfers of the symmetric decompo-sition of the game, the following outcomes may result.

(D1,D2) is always a steady state, while (C1,C2) is a steadystate if li = 1, βi > 1/3. The latter is accompanied by a12 =β1, a21 = β2. This exemplifies the Janus proposition in a casewithout complete symmetry of the Ri . If βi > 1/3 holds forboth persons the dynamic process defined in (1) and (2) canend, depending on the initial values a0

ij , either in a steadystate of mutual co-operation (C,C) or of mutual defection(D,D). If in contrast βi < 1/3, there is only one steady state,namely (D,D).

For asymmetric decompositions of the values in Figure 2,the above defined process derives different conditions forsteady states. This remark highlights a further major distinc-tion between the processes defined in (1) and (2) and the stan-dard game theoretic treatment of a strategic game, where thedecomposition of the game, has no impact on the Nash equi-librium strategies.

Thus, the example from the last section as well as thePrisoners’ Dilemma example illustrate that the interplaybetween altruism/spite and reciprocity may lead to steady statesconsisting of highly efficient outcomes, which would not berealized by players being fully rational and acting in a self-interested manner. At the same time, highly inefficient steady


states might be realized in games where rational players wouldchoose strategies always leading to the Pareto-efficient outcome.Reality provides ample evidence that such (in-)efficient steadystates are not an artefact but existent (see the list of examplesprovided in Frank (1988)).

In the Prisoners Dilemma we highlighted the importance ofbeing able to decompose payoffs and to assign specific trans-fers to specific agents. We have to discuss to what extent it isa realistic and practicable procedure to decompose payoffs insuch manner.

From a theoretical point of view, theories that explicitlymodel reciprocal choices need to interpret and to evaluateother player’s actions. A may reciprocate only if he knows (orbelieves that he knows) how B’s actions affected A or howit intended to affect A. Therefore, in a first step it is alwaysimportant to find out whether in a certain situation intentionsmatter, i.e. whether they are relevant for the resulting utilityfunctions of the players.

An obvious example is the criminal law, which deals forinstance with personal damages, in comparison to the civillaw, which deals with economic damages. Under criminal law,it plays a crucial role, whether a person A intended to injure aperson B or whether the injury of person B was accidentallyconducted by person A. In this case the differing intentionslead to tremendously different amounts of punishment. Underthe civil law, usually it is not investigated whether a producerintended to deliver a product of bad quality to a consumer orwhether the product was accidentally of bad quality. In bothcases, the product of bad quality has to be substituted by thefirm.

If intentions matter it is important to be able to decom-pose payoffs. Under what conditions is that possible? Fromthe game-theoretic perspective, in abstract games the resultingpay-offs usually cannot be divided in a part attributed to theaction of Person A and in another attributed to the actionof Person B; the pay-offs result from the joint action. In thiscase, reciprocity has to be based on presumed intentions [Falkand Fischbacher (1998), Rabin (1993)] or on an otherwise

A DYNAMIC MODEL 383

motivated “decomposition” of pay-offs. Although the strate-gic problem is not affected, the decomposition has an impor-tant influence on behaviour not only if—as shown above—thedecomposition affects steady state conditions. Decompositionsmay also become crucial if for instance an experiment explic-itly decomposes the resulting pay-offs in games (as was donein experiments by Pruitt (1970) and Antonides (1994)). There-fore, any specific decomposition of pay-offs needs its particu-lar motivation.

Several experiments were explicitly designed to analyseto what extent positive or negative reciprocity are observ-able phenomena. In these experiments payoffs were decom-posed and it was “obvious” what kind of behaviour could belabelled as fair or hostile, as in the Trust Games (see, e.g.Berg et al. (1995), Bolle (1998) or Mc Cabe et al. (2003)), theGift Exchange Game (Fehr et al. (2003)) or in the Sequen-tial Prisoners’ Dilemma Game (Andreoni and Miller (1993),Bolle and Ockenfels (1990), Cooper et al. (1996), Clark andSefton (2001)). Also Public Goods experiments, which are car-ried out in a large number and variety have an “obviousdecomposition”.

Contrary to game theoretic models not only many experi-mental situations but also “real situations” are often “decom-posed”. After formulating these situations as an extensive ornormal form game, however, the decompositions are “hidden”and it seems as if any arbitrary decomposition were possible.However, there also exist situations in reality, for example, inmarket games, where no observable decompositions are avail-able and where we necessarily have to choose a decompositionmethod if we want to apply reciprocity theories.

Thus, the recently developed theories aiming to capture theobserved reciprocal choices in formal models had to definewhat behaviour can be labelled as kind or hostile which againimplicitly defines intentions. Rabin (1993) evaluates kindnessin comparison with “equitable behaviour”: The kindness ofa person A towards B is positive (negative) if A’s actionprovides B with more than (roughly) the mean of the lowestand the highest income he could get. This kindness is con-


ditional on B’s action. For the further analysis of the pres-ent approach, we define Tij (conditional on the action of j )by comparing j ’s actual income with the income he wouldhave received under i’s best reply to j ’s action. Thus, Tij

also measures kind behaviour. Compared with Rabin’s (1993)approach, we have substituted the standard of “equitable” by“best reply” behaviour (similar to Levine (1998)).

3.3. A class of duopoly games

The basic idea of this subsection’s model is to extend theapplication of the previous example of the Prisoners’ DilemmaGame (with two discrete choices) to a game, where two per-sons may choose among a continuum of strategies. We willinvestigate the conditions under, which stable steady statesmay be realized. We will further demonstrate the possibility todefine the transfer vectors Tik.

As Bester and Guth (1998) in their analysis of evolutionarystability of altruism we use a simple duopoly model, which allowsa broad spectrum of interpretations. The formal structure of theduopoly game is able to describe for instance the external effectsof two hunters or two fishermen chasing in two separate pre-serves or in a common preserve. At the same time it is real-istic to assume in a duopoly of two firms competing againsteach other, that these firms do not only focus on straight profitmaximization but also consider “peaceful” or “hostile” strate-gies (where they take each other’s profits positively or negativelyinto account of their next period’s behaviour).

For reasons of simplicity, we will use in the following thenotation for firms. Assume there are two identical firms com-peting in a market where each firm faces a demand function

xi =1−pi +αpj , j �= i, α <1, pi =prices. (8)

Every linear demand function can be transformed to (8) bychoosing the quantity and the price unit. The goods producedare substitutes and strategic complements (Bulow et al. (1985))for α > 0 and complements and strategic substitutes for α < 0.

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Zero costs are assumed. Thus the profit of firm i is

Gi =xipi. (9)

If we now interpret this structure as the situation of twohunters or of two fishermen then we may translate the duop-oly game for α > 0 to a situation, where the hunters or thefishermen chase in separate but adjacent preserves. pi is thenthe number of hours per period which i spends hunting, xi

is the amount of prey per hour he catches. The more i hunts,the more of his potential prey leaves his preserve, a fractionof α moves to j ’s preserve, a fraction 1 −α moves elsewhere.Gi is the total amount of prey caught per period.11 Alterna-tively for α<0, we may assume that the two hunters share thesame preserve but specialize on different kind of prey. (α>−1)or hunt the same kind of prey (α =−1).

The unique Nash equilibrium in this market or huntinggame is

pi =pE = 12−α

(10)

and it is connected with payoffs

Gi(α)=[

12−α

]2

. (11)

From the firms’ (hunters’) point of view, for α > 0, pE is toosmall to be efficient and for α <0, it is too large.

Assuming now that Equation (1) describes the preferencesof the hunters (the firms), we have to first—as mentionedabove—define the transfers Tij . If j hunts pj hours (he sellsat price pj ), then i’s reaction function (without any altruisticpreferences) suggests

pi = 1+αpj

2. (12)

If i chooses pi , this level is interpreted as a zero transfer toj . If i selects another pi the transfers are

Tij =Gj(pi,pj )−Gj(pi, pj )<0 or >0, (13)Tii =Gi(pi,pj )−Gi(pi, pj )�0. (14)


Given this definition, let us consider how hunters (firms) willbehave. We will simply assume that they form rational expec-tations about each other’s choices, i.e. they will choose Nashequilibrium strategies based on their altruistic preferences. i’sbest reply to pj (with altruism and ei =1) is

pi = 12(1+ (1+aij )αpj ). (15)

For aij =aji =a, the following Nash equilibrium results:

p∗i =p∗ = 1

2−α(1+a), G1 =G∗

i = 1−αa

(2−α(1+a))2. (16)

According to this outcome, Pareto improvements can be real-ized with an increasing a. At a = 1 the two players realizeprofit (prey) maximization. In order to find out, which a aresteady states, we have to analyse the dynamic process of (1)and (2). We will further assume for reasons of simplicity thatthe process starts with a symmetric combination of altruismparameters. For at

ij =atij =at , δi = δ, γi =γ we find:

at+1 = δat +γ Tji(at )

= δat +γα −α2 +α2(at

/2)

(2−α(1+at))2 . (17)

Figure 3 describes the dynamic process for 0 <α < 1. Figure 4for α <0. This example—visualized in Figures 3 and 4—clearlyshows that there can be in both cases three steady state equilib-ria (labelled I–III). Basically, we will focus on steady state I forwhich we present in Table I an overview over the numerical val-ues a∞, which are realized under different α and β = γ/(1 − δ).The steady state II is not stable so that we will not analyse itany further. Steady State III is connected with altruism coeffi-cients larger than 1. Since both players having such extensivealtruism leads to certain paradoxical situations, we will alsodisregard the steady state III.

The most striking result of the analysis of steady state Iis that, irrespective of the value of β, in situations whereα > 0, i.e., when hunters (or fishermen) chase in separate

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Figure 3. Visualization of the dynamic process (16) for 0<α <1.

Figure 4. Visualization of the dynamic process of (16) for α <0.


TABLE I

Steady states a∞ for different (α,β) values

α − 12 − 1

414

12

β

12 −0.28 −0.13 0.13 0.331 −0.13 −0.06 0.06 0.132 −0.06 −0.03 0.03 0.06

preserves (firms offer substitutes/strategic complements), pos-itive steady state values (altruism) are realized, while for α <

0, i.e. when hunters chase in common preserves (comple-ments/strategic substitutes), we observe negative steady statevalues (spite). In both cases hunting hours (prices) increasecompared with the equilibrium under egoistic preferences. Forthe hunter example, we come to the conclusion that the pos-itive external effects of separate preserves results in “friendlyrelations” between the hunters who increase the intensity ofhunting so that both hunters profit. If they have a commonpreserve, hunting has negative external effects and they hunttoo much. This result also holds when the hunters had egoisticpreferences. The steady state preferences, however, are spiteful(a∞ <0) and the hunters hunt even more in order to decreasethe success of their fellow hunter.12

4. RELATED APPROACHES IN THE LITERATURE

There are two approaches, which are to some extent inthe same spirit as the one, which we presented here. Inthis section we will compare the outcomes of these twoapproaches with the results we found using our model. Theapproach developed by van Dijk and van Winden (1997) stud-ied already a model with dynamic and interdependent utilityfunctions. They apply their model to investigate steady statesin public good games. However, there are some crucial differ-

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ences with respect to the assumptions on which each modelis based. On the one hand, their assumption on the dynam-ics of the utility function is more general than in this model.For instance, van Dijk and van Winden (1997) propose a non-linear (Cobb–Douglas) interpersonal utility function under alinear restriction (a budget constraint) while we analyse lin-ear utility functions. On the other hand, their model requiresrestrictions on the space of parameters and variables, whichwe do not require. They investigate a two-person model; weprovide a n-person model. We allow the income of othersto have a larger weight in our utility function than our ownincome. By doing so, it is possible to capture a larger vari-ety of distributional and efficiency-oriented preferences thanin the model of van Dijk and van Winden.

More importantly, under some additional assumptionsabout the functional forms of their model, van Dijk and vanWinden (1997) prove the existence of a unique steady state.The present model emphasises the possibility (not the neces-sity) of multiple steady states. For instance, in the first exam-ple of section III, there is a unique steady state if β <2/9 andthere are three steady states if β �2/9. One important (but notthe only) reason for van Dijk and van Winden’s uniquenessresult is that in their public goods model only positive trans-fers are possible. Contrary to this, we take also into accountpossible negative external effects of actions. Insofar this paperextends the research of van Dijk and van Winden on thedynamics of social preferences.

Secondly, the results in the last part of Section 3 are sim-ilar to those of the model presented by Bester and Guth(1998) and supplemented by Bolle (2000). The difference isthat, under these earlier approaches, the evolution of altru-ism is modelled. Now we investigate the impact of reciprocalpreferences if human beings have a certain initial altruisticgood will capital. The present approach is, therefore, moreflexible and the process leading to the derived steady statesneeds in any case “less time” than an evolutionary process.We would argue that evolution might select the γi, δi(or βi).


With (1) and (2), individuals can instantly adjust to situationswith strategic complements or strategic substitutes.

5. CONCLUSION

This paper proposes a dynamic model of reciprocity betweena finite number of agents. In our simple linear approach,we assume that the agents’ preferences can be described byinterpersonal utility functions. Making use of these utilityfunctions, we analysed how the interplay between altruisticpreferences and reciprocity in a repeated exchange processmay lead to steady states of altruistic utility functions andaltruistic or malevolent acts. We provide in Section 3 sev-eral examples (such as the Prisoners’ Dilemma or the duopolygame) illustrating how an (in)efficient outcome may emerge.In the central Janus Proposition of our model, we show thatin the same situations highly efficient as well as inefficientstates may (but must not necessarily) exist. The examplesfurther show that such ambiguity of predicting steady statesmay exist even beyond the symmetry assumption of the JanusProposition.

A main insight of this paper is that for these simple gamesdifferent steady state patterns exist, all of which are consis-tent with certain types of social preferences. There is, hence,a structural uncertainty with respect to the question whichbehavioural pattern will emerge when a certain game is playedbetween two or more specific agents.

In its present form, our model has some obvious limitations.In our theory, we assume myopic decisions and separableexternalities. These assumptions explain only one relevantamong several possible cases. It would also be interestingto analyze how forward-looking individuals would try tomanipulate the future preferences of their fellow agents bybehaving altruistically (or spiteful) for strategic reasons in ear-lier periods. This analysis is left for future research.

The central aspect of our model is its dynamic structure.Therefore, in connection, with further experimental research

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the basic approach of Section 2 may enable us to cope withseveral further interesting questions, for example, the emer-gence of structured groups. Is it possible that under certain(even minimal) asymmetries of the Ri , societies start to sepa-rate into sub-groups, which are friendly (altruistic) within andhostile (spiteful) between the groups (or vice versa)? Is it pos-sible that, if one tries to model societies (primitive or mod-ern) that “too large” societies start to separate into subgroupsif scarcity or other kind of stress increases? Is it possible thatritual gift giving (Christmas, birthday, etc.) and ritual celebra-tions (family, working place, etc.) are meant to prevent such aseparation? The contrary problem is the integration of new-comers, of immigrants, and of religious minorities. For us itseems to be promising to analyse these problems of dynamicsocial interdependencies in the framework of the above model.

ACKNOWLEDGEMENTS

We are very grateful for the comments and suggestions ofJonathan Tan, Frans van Winden, and two anonymous ref-erees, as well as of the Seminar participants in Athens andHamburg. We would further like to thank for the financialsupport of the ‘Deutsche Forschungsgemeinschaft’ (Contractnumber BO 747/8-1).

NOTES

1. For the connection between Becker’s utility based externalities andthe otherwise assumed income based externalities see Bolle (1991).

2. Note however, that Levines linear approach cannot be derived fromBecker’s model (see Bolle and Kritikos (2004)).

3. A child may argue: because you, mother, participate so much in myhappiness, it is better I eat this piece of chocolate alone.

4. Levine (1998) as well as Kritikos and Bolle (2005) are using the the-oretical framework mainly to explain the results of economic exper-iments on altruistic behaviour.

5. It has to be emphasized that only the initial preferences are exog-enously given in our approach. For a model explaining how such


preferences might emerge in a process of education and of adaptingsocial norms, see Kritikos and Meran (1998).

6. Even in dictator experiments with anonymity conditions, where wecannot expect to initiate a lasting relationship, some subjects showthis kind of altruism. Remember that also Tit for Tat-strategies startwith friendly behaviour. To a certain extent, our model describes akind of “smoothed” Tit for Tat behaviour.

7. Janus, the Roman god of doors and gates, and also of beginnings(January), is usually portrayed with two faces of different character,looking into opposite directions.

8. For a more general discussion on the notion of Solidarity, cf.Kritikos et al. (in press).

9. Textbooks of Game Theory, e.g. Fudenberg and Tirole (1993, p. 34)or Holler and Illing (1996, p. 63) Note that Equation (4) describescontinuous functions, which are linear with respect to i’s actions.

10. In Bolle and Kritikos (2004), where necessary and sufficient condi-tions for the stability of steady states are derived, it is shown thatstability depends not only on the relation βi/ei .

11. For the sake of simplicity, at the beginning of each period the sameamount of prey is present.

12. This outcome has another important implication: whenever thenecessity of defining property rights is discussed, egoistic preferencesare assumed to exist. The latter result, however, shows that it is evenmore important to clearly define property rights when agents havenon-egoistic preferences as considered in the present paper.

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Address for correspondence: Alexander Kritikos, Europa-UnisersitatViardrina, Dept. of Economics, Grosse Scharrnstr. 59, Frankfurt 15230,Germany. E-mail: [email protected]

Friedel Bolle, Department of Economics, Europa-Unisersitat Viardrina,Frankfurt, Germany.

Alexander Kritikos, IMOP, Athens University of Economics andBusiness, Greece.

reciprocity, altruism, solidarity: a dynamic model

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