An Unintentional Theory of Reciprocity

Transcription

1 An Unintentional Theory of Reciprocity William Neilson* and Jill Stowe** March, 006 Abstract An intentions-free theory of reciprocal behavior is proposed. Different behavioral patterns involving reciprocity, both positive and negative, are distinguished and modeled based only on characteristics of games which are observable. In particular, reciprocal behavior is determined only by the payoffs in the game. After formally modeling reciprocal behavior in a general setting, the model is applied to the investment game and a modified investment game. The distinction between the two games allows us to make predictions concerning the amount of reciprocity expected in different situations. JEL CLASSIFICATION: A3, C70, D63 KEYWORDS: *Texas A&M University, TAMU 48, College Station, TX ; wsn@econ.tamu.edu **Fuqua School of Business, Duke University, Box 900, Durham, NC ; stowe@duke.edu

2 Introduction Reciprocity suggests that individuals respond kindly to kind actions (positive reciprocity) and unkindly to unkind ones (negative reciprocity). One specialized set of reciprocal behaviors are trust and trustworthiness. According to Camerer (003), Trust is risky. Trustworthiness must also go against the Trustee s self-interest, to test whether people are willing to sacrifice to satisfy moral obligation. Evidence of many types of reciprocal behavior, both positive and negative, and trust have been common in many experimental settings such as ultimatum games, public goods games, gift-exchange games 3, investment games 4, and trust games 5. This paper presents a theory of reciprocity which can be used to describe many types of reciprocal behavior, including altruism, spitefulness, and trustworthiness. Although a few papers already exist which provide preference-based theories of reciprocity, ours stands alone for two reasons. 6 First, as far as we know, this paper represents the first attempt to formally distinguish between different types of reciprocal behavior based solely on the payoffs of the game. 7 More specifically, as will be discussed in depth, we define player s choices relative to his favorite payoff combination and player s least favorite payoff combination, all given player s action. In addition, our paper is unique in that reciprocity is modeled using an observable feature of games, namely payoffs, rather than the unobservable intentions prevalent in other papers. This is a key feature of our paper because basing these behaviors on observable characteristics implies that different reciprocal responses should be directly testable. In contrast to the model presented here, there are two theories of reciprocity which depend primarily on beliefs or intentions. In Dufwenberg and Kirchsteiger (004), reciprocity is solely-intentions driven. In their model, individuals form beliefs about how kind or unkind an opponent s action was. The assumption is that individuals experience psychological payoffs (which depend on beliefs about kindness) as well as material payoffs. In the Falk and Fischbacher (006) theory of reciprocity, both outcomes and intentions matter. An individual s utility is the sum of material outcome and the reciprocity utility; reciprocity utility consists of the kindness of an outcome (whether a favorable or unfavorable outcome is intentional) and the response to experienced (un)kindness. In both of these papers, the central idea is that kindness hinges on the intentions, or motives, behind the outcome. 8 In contrast to these approaches, we choose to model reciprocity and trust based on a feature which is observable in games. Specifically, payoffs are observable, but intentions are not. 9 More to the point, individuals can observe the payoff combinations induced by another player s actions, but they can only conjecture at the intentions or motives behind that choice, and an outside observer can ascertain neither the first player s intentions nor the second player s beliefs about the first player s intentions. We contend that it is more useful to develop a model based on observable characteristics of games. 0 See Camerer (003) for a survey of ultimatum game results. Croson (988) 3 Fehr, Kirchsteiger, and Riedl (993); Charness and Haruvy (999) 4 Berg, Dickhaut, and McCabe (995) 5 McCabe, Rigdon, and Smith (003); Cox (004) 6 An alternative to the preference-based approach to modeling reciprocity is the repeated-games approach. Applications include informal insurance markets (Coate and Ravallion (993)), reciprocal exchange (Kranton (996)), and the trading of favors (Neilson (999), Mobius (00)). 7 Cox (004) experimentally differentiates trust from and reciprocity and altruistic distributional concerns, but does not distinguish between trustworthiness and reciprocity. 8 In addition, Cox and Friedman (00) introduce a parametric model of other-regarding preferences in which the emotional state of the decision maker determines the individual s marginal rate of substitution between her payoff and her counterpart s. The emotional state of the decision maker depends on the (un)kindness of her counterpart s choices. 9 Many experiments, such as Dufwenberg and Gneezy (000) and Charness and Dufwenbery (005), elicit players beliefs. While beliefs may be elicited ex post, in which sense beliefs may be observable (to the degree that they are elicitable) to the experimenter, our position is that these beliefs are not observable to players in the game. 0 Hey (003) shows that unless one is prepared to make certain heroic assumptions about preferences and dynamic

3 Our goal is to construct the simplest possible model based on the players preferences (which are assumed to be known, as is commonly assumed) and observable characteristics of the game that can capture different types of reciprocity. We first assume that, similar to Neilson and Stowe (005), the decision maker s preference function is a weighted sum of the utility from his own payoff and his opponent s payoff, using (possibly) different utility functions to evaluate each players payoffs. A single coefficient determines other-regarding attitudes. The observable feature of games that we focus on, the payoff combinations, represents the characteristic which leads to increases or decreases in the other-regarding coefficient. In this paper, we restrict our attention to two player, two-stage games. Suppose that when player chooses action s, player is left with a choice among the payoff combinations in the set A. Whenplayer choosess instead, player is left with a choice in A. If player s favorite payoff combination in A is preferred to his favorite payoff combination in A, then player views player as being kinder when she plays s instead of s. Reciprocity says that kindness begets kindness, and player s other-regarding coefficient rises, increasing the weight he places on player s payoff. Let us be specific about what we mean by reciprocity. In this paper, reciprocity refers to the response of one player s preferences to another player s action. This is different from the notion of reciprocity used in Rabin (993) and Dufwenberg and Kirchsteiger (004). In these papers, reciprocity is based on anticipated kindness, which depends on beliefs. Hence, reciprocity can occur in a simultaneous move game, such as when cooperation occurs in the Prisoner s Dilemma. In contrast, our definition of reciprocity implies that it can only arise in dynamic games. Section 6 applies the model to two games. First, we apply the model to the investment game (Berg, Dickhaut, and McCabe (995)). According to our definitions, the investment game is a game about trust and trustworthiness. Second, we modify the investment game in a way which removes the trust component but still allows for reciprocity. Analyzing the predictions from these two games results in an intuitive comparison in the amount of reciprocity shown by the second mover. The rest of the paper proceeds as follows. Section introduces the notation and definitions. Section 3 provides a description of individual preferences. Section 4 offers a general approach to reciprocity, and it also defines a specific kind of reciprocal behavior, namely trust and trustworthiness. Section 5 characterizes the effects of trust and reciprocity in a setting similar to that of Proposition 3 of Section 4. Section 6 applies the model to the investment game and a modified investment game. Section 7 compares our model with Rabin (993), Dufwenberg and Kirchsteiger (004), and Falk and Fischbacher (006) and then concludes. Notation and Definitions Letplayersandbethetwoplayersinthegame. Apayoff combination is an ordered pair x = (x,x ) X,whereX R is a closed set, and where player receives x and player receives x.let A be the set of compact subsets of X,andapayoff set A A is a set of payoff combinations. Both players begin the game with preference orderings over payoff combinations. Let % 0 i denote player i s weak preference ordering before any moves are made, and let Vi 0 : X R be the continuous utility function that represents these preferences. In our model, players attitudes toward their opponents can depend on the choice set they face. Accordingly, we introduce some terminology governing comparisons of choice sets. The first form of decision making processes, experiments capable of revealing intentions are impossible. In contrast, intentions-based models assume that beliefs about intentions behind actions alter other-regarding attitudes. In the Charness and Rabin (00) model of social preferences, reciprocity is modeled in a similar manner. 3

4 comparison involves the notion that one payoff set leaves a player potentially better off than another payoff set, and the second defines a concept which allows us to compare payoff sets according to the players worst outcomes. Definition The payoff set A is an improvement of A for player i if there exists some x A such that x % 0 i x for all x A. Accordingtothisdefinition, if payoff set A contains a payoff combination that player i likes better than any payoff combination in payoff set A, thena is an improvement of A for player i. For example, if player s initial preferences are completely self-oriented, so that V 0 (x,x )=v(x ),wherev is an increasing function, then the payoff set A is an improvement of A for player if her highest attainable payoff in A exceeds her highest attainable payoff in A. The notion opposing an improvement is called a worsening. Thepayoff set A is a worsening of A for player i if A is an improvement of A for player i. Definition The payoff set A puts player i more at risk than A if there exists some x A such that x ¹ 0 i x for all x A. If player i s least favorite payoff combination in A is worse than her least favorite payoff combination in A, thena can potentially make her worse off than A can. So, the choice of A puts her at risk of attaining a worse outcome than the choice of A does. For example, if player is completely self-oriented, A puts her more at risk than A does if her lowest possible payoff in A is below her worst possible payoff in A. We say that the payoff set A puts player i less at risk than A if A puts her more at risk than A. The relations an improvement of for player i and putsplayeri more at risk than form the basis for defining reciprocal behavior. Given the significance of these relations, it is worthwhile to explore their properties. Proposition The relation is an improvement of for player i is complete, transitive, and continuous, and can be represented by the function M i : A R given by M i (A) =supvi 0 (x)s.t.x A. x Proof. Let i denote the relation, so that A i A means that A is an improvement of A for player i. Let M i be as defined. By the maximum theorem (e.g. Green and Heller, 98), M i is continuous since Vi 0 is continuous. Furthermore, A i A if and only if M i (A ) i M i (A), so the function M i represents the relation i. It follows that i is complete, transitive, and continuous. Proposition shows that the relation on which the definition of reciprocity is based has several desirable properties. First, and most importantly, the relation is an improvement of for player i isa complete ordering, so that for any two sets A and A, eithera is an improvement of A for player i, A is a worsening of A for player i, or both. The comparison of the two sets is always well-defined. Second, the comparison is also transitive. Third, the comparison is continuous, which is the standard notion of well-behaved for a (preference) relation. Finally, the relation can be represented by a continuous real-valued function, which simplifies its use. The relation puts player i more at risk possesses the same properties as is an improvement of, and this is set forth by Proposition. Proposition The relation puts player i more at risk is complete, transitive, and continuous. Furthermore, A puts player i more at risk than A if and only if m i (A ) i m i (A), wherethefunction m i : A R given by m i (A) =inf V i 0 (x)s.t.x A. x 4

5 The proof is similar to that of Proposition. 3 Preferences Itisassumedthatattimet (t = 0,,or in a two-stage game) player i s preferences over payoff combinations can be represented by the function V it (x i,x j )=( c it )u i (x i )+c it v i (x j ) () for i, j =,, whereu i is player i s utility function and v i is the utility function player i uses to evaluate player j s outcome. Equation () says that the decision maker s utility from a payoff combination is a weighted sum of the utility she gets from her own outcome and the utility her opponent gets, using different utility functions for each player. This model in () is similar to the Neilson and Stowe (005) model of other-regarding preferences, but there is one major difference: it uses potentially different utility functions for players i and j. The coefficient c it is called an other-regarding coefficient, which can be either positive or negative, and it measures how much weight player i places on player j s outcome at time t. The following proposition demonstrates the usefulness of the preference specification, establishes that the coefficients really do reflect other-regarding behavior, and provides a result that can be exploited later in the paper. Proposition 3 Define W (z; c) =( c)u(f(z)) + cv(g(z)), whereu and v are strictly increasing and concave, f is strictly decreasing, g is strictly increasing, and W is a strictly concave function of z for all c. Letz maximize W.Thenz increases when c increases. Proof. The first-order condition for a maximum is W z (z; c) =0.Differentiate implicitly with respect to c to get W zz (z; c) dz /dc + W zc =0,ordz /dc = W zc /W zz. By assumption W zz < 0. Consequently, the sign of dz /dc is the same as the sign of W zc = u 0 f 0 + v 0 g 0 > 0. Proposition 3 captures the following scenario. Suppose that player can choose the value of some variable z, and higher values of z are better for her opponent but worse for herself. 3 Accordingtothe proposition, when player s other-regarding coefficient becomes higher, she chooses a higher value of z, thereby increasing her opponent s payoff at the expense of her own. The dictator game fits the scenario in Proposition 3. In the dictator game the proposer is given some amount of money k to split between himself and his opponent. In that case z would correspond to the amount given away, with g(z) =z and f(z) =k z. According to the proposition, a proposer with a higher other-regarding coefficient gives away more than a proposer with a lower other-regarding coefficient. Also, when c =0in this game, the dictator gives away nothing. Consequently, c =0 corresponds to complete selfishness, while a decision maker with c>(<)0 is called positively (negatively) other-regarding. A slightly different interpretation of a given value of the other-regarding coefficient in a static setting is that c defines the decision maker s distributional concerns. The other-regarding coefficients can change over time as the game proceeds. We contend that changes in payoff sets are what cause the other-regarding coefficients to change. The next section makes explicit the ways in which other-regarding coefficients change as a response to changes in payoffs. 3 It is sufficient but not necessary that f is strictly decreasing. The same result holds as long as v 0 g 0 >u 0 f 0. However, the interesting case in the one detailed above in which a player is willing to sacrifice her own payoff in order to increase the payoff of her opponent, and this entails f is strictly decreasing while g is strictly increasing. 5

6 4 Reciprocity Reciprocity means that individuals act according to how they are treated by others. For example, kindness is repaid with kindness, and unkindness is repaid with unkindness. In terms of the model, this suggests that an individual responding to a kind (unkind) action by his opponent will increase (decrease) the weight he places on his opponent s outcome. It remains to specify which actions are perceived as kind and which are perceived as unkind. Consider the following sequential game. Player chooses an action, which player observes, and then player chooses an action which determines the final payoffs. Player s choice determines the set of feasible payoff combinations among which player has to choose, which is player s payoff set. Now consider two possible actions for player resulting in either payoff set A or A forplayer. IfA is an improvement of A for player, then choosing the action that leads to A is kinder than choosing the action that leads to A. After all, according to player s initial preferences, º 0, the best payoff combination in A is preferred to the best payoff combination in A. If player chooses the kinder of two actions, reciprocity says that player should respond by being kinder to player, which corresponds to player having a higher other-regarding coefficient. To make this more explicit, let c be player s other-regarding coefficient when she makes her choice as the first mover of the game, and let c be player s other-regarding coefficient when he makes his choice as the second mover of the game. Since nothing has yet happened affecting player s attitudes toward player, it is natural to assume that c = c 0, so that her preferences when making her move are identical to her preferences before the game starts. Player s other-regarding coefficient depends on the payoff set induced by player s action, so c = c (M (A),m (A)), wherea is a payoff set, c : X R is a function, and M (A) and m (A) come from Propositions and. Given this functional form for c, the decision maker s preference function can be represented by V (x; A) =( c (M (A),m (A)))u (x )+c (M (A),m (A))v (x ). () Reciprocity and trust are determined by properties of the function c. Definition 3 Player exhibits reciprocity if c / M 0. This definition of reciprocity encompasses both positive and negative reciprocity. Positive (negative) reciprocity is the notion that a kind (unkind) action by player leads to a kind (unkind) response by player, and a kind (unkind) action by player is one that improves (worsens) the payoff set for player. According to the definition, such an improvement (worsening) increases (decreases) player s otherregarding coefficient, and by Proposition 3 player becomes kinder (less kind) when his other-regarding coefficient rises (falls). Player s choice also has an effect on her own payoff set. In particular, player sometimes sacrifices her own outcome when choosing an action. We propose that player responds favorably to this sacrifice. Definition 4 Player exhibits indebtedness if c / m 0. If player takes an action which potentially diminishes her least favorite payoff combination (i.e. she makes a choice which puts herself at risk), player observes that player made the sacrifice to either help or hurt him. Barring any actions by player prior to the start of the game, player has no reason to sacrifice her own payoff to hurt player. Hence we say that by making the sacrifice, player is trusting that player will reciprocate with trustworthiness (c increases). This is the subject of the next definition. Definition 5 If player exhibits both reciprocity and indebtedness, player is trustworthy. 6

7 5 Trust and Reciprocity in a General Setting In this section we characterize the effects of trust and reciprocity in a generalized setting similar to that in Proposition 3. Consider the following two player sequential game. In round player chooses action s from the set S. In round player observes s and then chooses action s from the set S (s ), where the dependence of S on s reflects the fact that player s action may constrict or expand player s action choices. Player s monetary payoff is determined by the function π (s,s ) and player s monetary payoff is determined by π (s,s ). Let M (s )=sup ζ ( c 0 )u (π (s,ζ)) + c 0 v (π (s,ζ)), sothatm (s ) is the highest utility player can receive from the game given that player has already chosen s. 4 Let m (s )=inf ζ ( c 0 )u (π (s,ζ))+c 0 v (π (s,ζ)), sothatm (s ) is the least utility player can receive from the game given that she has already chosen s. Assume that M ( ) and m ( ) are differentiable. Player s problem is to choose s to maximize V (s,s )=[ c (M (s ),m (s ))]u (π (s,s )) + c (M (s ),m (s ))v (π (s,s )). The first-order condition is [ c (M (s ),m (s ))]u 0 (π ) π + c (M (s ),m (s ))v s (π 0 ) π =0. (3) s Equation (3) defines player s best response given player s action, or s (s ). Differentiating the first-order condition with respect to s and rearranging yields: ds ds = D D " ( c ) u 0 µ π s µ µ u 00 π + v 0 π s s µ π s + u 0 Ã µ π + c v 00 π π!# + v 0 π s s s s s s dc (4) ds where D = V (ds ). We can assume that D is negative so that the first-order condition identifies a maximum. The first term in (4) exists whether or not c changes when s changes, so it is irrelevant for finding the impact of reciprocity and indebtedness. The second term captures the effects of c changing, and we are interested in the sign of this term. We can further decompose dc = c dm + c dm (5) ds M ds m ds The first term in (5) is the impact on player s other-regarding coefficient caused by reciprocity, and the second term is the impact caused by indebtedness. Proposition 4 Assume that u 0 > 0, v 0 > 0, andd<0. Reciprocity leads player to strictly increase s when s increases if one of the following conditions holds: (i) c / M > 0, dm /ds > 0, π / s < 0 and π / s > 0; or (ii) c / M > 0, dm /ds < 0, π / s > 0 and π / s < 0. If dm /ds =0then reciprocity has no effect on behavior. 4 Earlier in the paper, M was defined as functions of payoff sets. Since s uniquely defines a payoff set, we simplify the notation by writing M (s ) instead of M (A(s )). 7

8 Proof. From the above analysis, the portion of ds /ds governed by reciprocity is given by µ µ u 0 π + v 0 π dc dm. D s s dm ds This expression is positive if either (i) or (ii) holds. Furthermore, if dm /ds =0then the expression is zero. To understand Proposition 4, look at the first set of conditions. The requirement that c / M > 0 means that player exhibits strict reciprocity, and it is the only condition in the set that relates to preferences. The other two conditions are properties of the game. The condition dm /ds > 0 states that an increase in s leads to a worsening for player, which, when condition (i) holds, makes player more positively other-regarding. The third condition in (i), π / s < 0 and π / s > 0, statesthat the game is parameterized so that an increase in s is good for player (monetarily) and bad for player, such as when player returns money in the investment game. Hence, the conditions in (i) suffice for positive reciprocity. The conditions in (ii) represent conditions for negative reciprocity. The condition dm /ds < 0 states that an increase in s leads to a worsening for player, which, when condition dm /ds < 0 holds, makes player more negatively other-regarding. The conditions π / s > 0 and π / s < 0 state that the game is parameterized so that an increase in s is bad for player (monetarily) and good for player such as when player returns nothing in the investment game. One such parameterization might be the investment game in which strategies are defined as amounts kept rather than amounts sent (see footnote 6). Finally, note that if dm /ds =0, then the expression is zero. This suggests that an increase in player s action creates neither an improvement nor a worsening for player, or that player s favorite outcome does not change when player makes her choice. Given the requirement that c / M > 0, player s other-regarding coefficient remains the same. A similar result holds for indebtedness. Proposition 5 Assume that u 0 > 0, v 0 > 0, andd<0. Indebtedness leads player to strictly increase s when s increases if one of the following conditions holds: (i) c / m < 0, dm /ds < 0, π / s < 0 and π / s > 0; or (ii) c / m < 0, dm /ds > 0, π / s > 0 and π / s < 0. If dm /ds =0then indebtedness has no effect on behavior. Proof. From the above analysis, the portion of ds /ds governed by indebtedness is given by µ µ u 0 π + v 0 π dc dm. D s s dm ds This expression is positive if either (i) or (ii) holds. Furthermore, if dm /ds =0then the expression is zero. The conditions in (i) of Proposition 5 are as follows. First, the requirement that c / m < 0 means that player exhibits strict indebtedness. The condition dm /ds < 0 suggests that an increase in s puts player more at risk, which makes player more positively other-regarding. In conjunction with the last condition (which has the same interpretation as in Proposition 4), the conditions in (i) amount to player responding positively to player s sacrifice. In part (ii) of Proposition 5, the condition dm /ds > 0 states that an increase in s puts player less at risk, which makes player more negatively other-regarding. Player increases s in response to an increase in s because doing so improves his own payoff and reduced the payoff of his counterpart. 8

9 Finally, when dm /ds =0, the expression is zero. This suggests that an increase in player s action does not change her least favorite outcome. Given the requirement that c / m < 0, this results in no change in player s other-regarding attitudes. Player s optimal strategy will ultimately depend on player s best-response. Suppose that player initially begins the game self-interested so that c =0, and that she knows player s preferences. 5 Player chooses s to maximize V (s,s )=u (π (s,s (s )). The first derivative is u 0 [ π / s + π / s ds /ds ]. (6) The value of s which sets (6) equal to zero represents player s optimal strategy. However, it is possible that no interior maximum exists. In many games of interest, an increase in s moves π and π in different directions but at fixed rates, so that ( π / s )/( π / s ) is constant. An interior optimum exists if there is a value of s for which ds /ds = ( π / s )/( π / s ).Ifnot,playersetss either as high or as low as possible. Suppose that ds /ds > 0. Ifthefirst set of conditions from Propositions 4 or 5 hold, then π / s > 0, or an increase in player s strategy is good monetarily for player. If an increase in player s strategy is bad for herself ( π / s < 0), as long as π / s ds /ds is greater than (less than) π / s,the derivative in (6) is positive (negative), suggesting that no interior maximum exists and that player s optimal strategy is to choose s as high (low) as possible. If the second sets of conditions hold, then π / s < 0. This means that the second term of the bracketed expression in (6) is negative, and hence it is only optimal for player to choose s as high as possible if π / s > π / s ds /ds. 6 Applications to Two Games To demonstrate the applicability of the model of reciprocity and indebtedness, consider the following two-player sequential game, the familiar investment game. Player is endowed with k and chooses how much of it to give to player. The amount given to player is multiplied by λ>, and then player decides how much to give back to player. If player gives away y and player gives away y,player s payoff is π = k y + y, while player s payoff is π = λy y. To analyze this game, first assume that c 0 = c 0 =0. This assumption simplifies the analysis and ensures that all behavior is driven by reciprocity rather than fairness concerns. To further simplify the notation, since initial preferences are completely self-oriented, write c in equation directly as a function of the highest payoff for player and the lowest and highest payoffs for player, instead of their utility values. Finally, further assume that player knows the function c,sothatplayercan perfectly predict player s response. In the investment game, player s highest possible payoff is M (y )=λy and player s lowest possible payoff is m (y )=k y.consequently,dm /dy = λ>0 and dm /dy =. Furthermore, π / y =and π / y =. Thus,if c / M > 0 the game fits the requirements of condition (i) of Proposition 4, and reciprocity leads player to increase y when player increases y. Similarly, if c / m < 0 the game fits the requirements of condition (i) of Proposition 5, and indebtedness places 5 While this is unrealistic in laboratory settings, it is in line with the normal assumptions of game theory under which players know their opponents payoffs as measured by utility values. This assumption allows us to determine how player reacts to changes in player s degrees of reciprocity. 9

10 further upward pressure on y when player increases y. 6 Behavior in the investment game, then, reflects both reciprocity and indebtedness. Experimental results show that on average, first movers invest about 50% of their endowment, and second movers repay an average of 90% of what was invested (Berg, Dickhaut, and McCabe (995)), suggesting that trust is rewarded and that player s give back almost the amount initially invested. In addition, as our model predicts, an analysis of their data shows that, on average, the amount player s give back to player s is increasing in player s investment. However, player s investment and the amount player sends back are not strongly correlated (r =.478). Equation 4 becomes dy = λ( c )u 00 + c v 00 (u 0 + v 0 dy ( c )u 00 + c v 00 + ) ( c )u 00 + c v 00 dc (7) dy where dc = λ c c > 0. (8) dy M m Since λ> the first term in (7) is greater than, and by (8) the second term is positive. Consequently, not only is dy /dy positive, as argued above, but it is also greater than, and this is true even without the contribution from increases in y making player kinder. Player more than repays player s investment. This prediction of the model is not supported by the Berg et al. evidence, but neither is it contradicted. In the treatment with no history, their point estimate for dy /dy is 0.90, and the 95% confidence interval for the estimate is [0.83,.59]. In the treatment with history, their point estimate for dy /dy is.56, the 95% confidence interval is [0.659,.653]. Hence, in both treatments, their data do not reject the hypothesis that dy /dy. Now look at the decision made by player, who is assumed to be completely self-oriented with c 0 =0. Even though she is self-interested, she knows that once she makes an investment, her opponent will have an other-regarding coefficient that is different from its initial value of c 0 =0. From 5, the derivative of her payoff function is π 0 = u 0 (dy /dy ). Since u 0 > 0 and, as determined above, dy /dy >, the payoff function is monotone increasing and hence our model predicts that player should invest her entire endowment. This is not frequently seen experimentally, however, since by combining both treatments from Berg, Dickhaut, and McCabe (995), we find that only 0% of the subjects invest their entire endowments. 7 Since the investment game involves both trust and reciprocity, it becomes interesting to analyze a game which features only one of these to types of behavior. Consider a modified investment game in which player, when given the opportunity to send any amount back to player, is also allowed to choose a (0, 0) outcome. If player chooses the (0, 0) outcome, then player does not get to keep the remainder of the endowment that she did not send. This modification means that player s minimum payoff is always zero, and so an increase in y does not put player more at risk. Consequently, the trust component does not enter the game, and so the comparative statics derivative in (8) becomes dc dy = λ dc dm. 6 Alternatively, one could parameterize the investment game so that player keeps z and player keeps z. Then m (z )=z, M (z )=λ(k z ), π (z,z )=z + λ(k z ) z,andπ (z,z )=z. Accordingly, dm /dz < 0, dm /dz > 0, dπ /dz < 0, anddπ /dz > 0. With this parameterization, the game fits the requirements of condition (ii) of Propositions 4 and 5. 7 Bohnet and Zeckhauser (004) find that individuals dislike being betrayed, and this betrayal aversion may explain whymostplayer sdonotinvestalloftheirendowment. 0

11 . Our model suggests that player is less responsive to contributions by player in the modified investment game than in the original investment game; however, since it still holds that dy /dy >, our model predicts that player should invest her entire endowment. 7 Conclusion In this paper, we develop a theory of reciprocity and trust which is based only on observable features of the game, namely, payoffs. As Hey (003) discusses, heroic assumptions about preferences and dynamic decision making processes are necessary for the existence of experiments capable of revealing intentions. This unintentional model is more tractable than those based on intentions, and because it is based on payoffs, it should be directly testable. Two notions form the basis for defining reciprocity and trust. First, reciprocity (both positive and negative) is based on whether an individual s action is kind or not. We use the notion called improvement to define kindness. Any action by player which improves player s most favorite outcome (M ) is kind, whereas an action worsening player s outcome is unkind. Second, trust and trustworthiness depend not only on kindness but also on the sacrifice (lowering of the least favorite payoff, m ) that player makes to help player. For this idea, we introduce the notion called putting at risk. Player responds favorably to player when player puts her own welfare at risk by allowing the possibility of realizing a worse outcome. Given these definitions, there are two other natural factors to consider. First, player may care about his own least favorite payoff (m ) resulting from player s choice. However, given that player moves last, we find no compelling reason for player to choose his worst outcome, and hence his least favorite payoff is likely insignificant to his decision. It is more likely that player may care about player s most favorite payoff (M ). If player makes a choice which is an improvement for player but is also an improvement for herself, player may not perceive the action as kindly because the choice may be self-serving for player. The upshot of this extension is that player may also exhibit envy, which we would define as c / M < 0. Our contention is that given player s choice, player s other-regarding coefficient is increasing with respect to M, decreasing with respect to m, and decreasing with respect to M. The direction of the change in c is dominated by the M effect, but that effect may be magnified or mitigated due to the m and M effects. However, we believe that there are few situations in which player s concern over M is significant, and hence, we choose to model player s reciprocal responses based only on M and m. Our interpretation of reciprocity is different from other reciprocity approaches. Rabin (993) models reciprocity in a simultaneous move game, and he motivates his argument for using intentions with Game (Γ) in the appendix. He argues that whether player s choice of F is kind depends on player s beliefs about what player will do. The choice is kind if player believes that player will choose d but is unkind if player believes that player will choose f. Our model implies with certainty that the choice of F is kind because choosing F results in an improvement for player and because it puts player more at risk. Consequently, player s other-regarding coefficient increases, but it might not increase enough so that player chooses f over d to reciprocate player s kindness. This also illustrates that whileourmodelcanbeusedinfinite games, it is more clearly understood in games in where there are continuum of payoffs. In Dufwenberg and Kirchsteiger (004), reciprocity is solely intentions-driven. For illustration, they apply their model to the Sequential Prisoner s Dilemma (also in the appendix). When players are not motivated by reciprocity concerns, the subgame perfect equilibrium is (D; d, d). However, player may

12 want to choose c in either node if player believes that player is kind. Hence, cooperation may result. Our model suggests that player s choice of C is kind. Again, choosing C results in an improvement for player and puts player more at risk. In turn, player s other-regarding coefficient increases. If it increases enough, player will also choose to cooperate by choosing c. Unlike in Dufwenberg and Kirchsteiger s model, however, choosing D is never kind on the part of player. This makes sense, because player choosing D makes player strictly worse off. Falk and Fischbacher s (006) theory of reciprocity depends on both the material outcome as well as the intentions behind the outcome. Intentions are partially determined via interpersonal comparisons of outcomes. In contrast, our definitions of reciprocity and indebtedness depend solely on intrapersonal comparisons of realized outcomes and unchosen alternatives. Their theory predicts relevant stylized facts from the ultimatum game, the gift-exchange game, the reduced best-shot game, the dictator game, the prisoner s dilemma, and public goods games. It can explain the difference in behavior between intentional and random outcomes, and furthermore, their model can explain why outcomes tend to be fair in bilateral interactions but unfair in competitive markets. In Section 6, we analyze the investment game (a game with reciprocity and indebtedness, or trust) and the modified investment game (a game with only reciprocity) using the general setup in Section 5 as a guideline. Similarly, it is possible to extend the analysis to a game with negative reciprocity or no reciprocity. For example, according to our setup, a game with negative reciprocity is a game in which player s choice results in a worsening for player and leaves player s least favorite outcome unchanged. Our model predicts that player s other-regarding coefficient decreases following player s strategy. In addition, a game in which player makes a choice which neither improves or worsens player s favorite outcome nor puts herself more or less at risk represents a game with no reciprocity. Given that neither of the other-regarding coefficient s arguments change, the other-regarding coefficient remains constant and hence no reciprocity exists.

13 References [] Berg, Joyce, John Dickhaut, and Kevin McCabe (995). Trust, Reciprocity, and Social History. Games and Economic Behavior 0, -45. [] Bohnet, Iris and Richard Zeckhauser (004). Trust, Risk, and Betrayal. Journal of Economic Behavior and Organization 55(4), [3] Camerer, Colin F. (003). Behavioral Game Theory: Experiments in Strategic Interaction. Princeton University Press, Princeton, NJ. [4] Charness, Gary and Martin Dufwenberg (005). Promises and Partnership. Working paper, UC Santa Barbara and University of Arizona. [5] Charness, Gary and Ernan Haruvy (999). Altruism, Equity, and Reciprocity: An Encompassing Approach. Universitat Pompeu Fabra, Department of Economics Working Paper No [6] Charness, Gary and Matthew Rabin (00). Understanding Social Preferences with Simple Tests. Quarterly Journal of Economics 7, [7] Coate, Stephen and Martin Ravallion (993). Reciprocity without Commitment: Characterization and Performance of Informal Insurance Arrangements, Journal of Development Economics 40, - 4. [8] Cox, James C. (004). How to Identify Trust and Reciprocity. Games and Economic Behavior 46, [9] Cox, James C. and Daniel Friedman (00). A Tractable Model of Reciprocity and Fairness. Working paper, University of Arizona and UC Santa Cruz. [0] Croson, Rachel T.A. (998). Differentiating Altruism and Reciprocity, forthcoming in C.R. Plott andv.l.smith(eds.),handbook of Experimental Results. [] Dufwenberg, Martin and Uri Gneezy (000). Measuring Beliefs in an Experimental Lost Wallet Game. Games and Economic Behavior 30, [] Dufwenberg, Martin, and Georg Kirchsteiger (004). A Theory of Sequential Reciprocity. Games and Economic Behavior 47(), [3] Falk, Armin and Urs Fischbacher (006). A Theory of Reciprocity. Games and Economic Behavior 54(), [4] Fehr, Ernst, Georg Kirchsteiger, and Arno Riedl. (993) Does Fairness Prevent Market Clearing? An Experimental Investigation. Quarterly Journal of Economics 08, [5] Green, Jerry and Walter P. Heller (98), Mathematical Analysis and Convexity with Applications to Economics, in Kenneth J. Arrow and Michael D. Intriligator (eds.), Handbook of Mathematical Economics, vol., North-Holland. [6] Hey, John D. (003). Are Revealed Intentions Possible? Working paper, Universities of Bari and York. [7] Kranton, Rachel E. (996). Reciprocal Exchange: A Self-Sustaining System, American Economic Review 86,

14 [8] McCabe, Kevin, Mary Rigdon, and Vernon Smith (003). Positive Reciprocity and Intentions in Trust Games. Journal of Economic Behavior and Organization 5(), [9] Mobius, Markus M. (00). Trading Favors, working paper, Harvard University. [0] Neilson, William (999). "The Economics of Favors," Journal of Economic Behavior and Organization 39, [] Neilson, William and Jill Stowe (005). An Axiomatic Characterization of Other-Regarding Preferences. Working paper, Texas A&M University and Duke University. [] Rabin, Matthew (993). Incorporating Fairness into Game Theory and Economics. American Economic Review 83(5), [3] Sugden, Robert (984). Reciprocity: The Supply of Public Goods Through Voluntary Contributions. Economic Journal 94(376),

15 Appendix F f 0 D d 0 Game ' (Rabin (993)) C D c d c d Game Sequential Prisoner s Dilemma (Dufwenberg and Kirsteiger (004)) 9