Leadership, Reciprocity and Moral Motivation - lameta

Leadership, Reciprocity and Moral Motivation : Experimental Evidence from a Sequential Public Good Game By David Masclet(a), Marc Willinger(b) and Charles Figuières(c) This version : Abstract

We run a series of experiments in which subjects have to choose their level of contribution to a pure public good. The design differs from the standard public good game with respect to the decision procedure. Instead of deciding simultaneously in each round, subjects are randomly ordered in a sequence which differs from round to round. We compare sessions in which subjects can observe the exact contributions from earlier decisions ("sequential treatment with information") to sessions in which subjects decide sequentially but cannot observe earlier contributions ("sequential treatment without information"). The results indicate that sequentiality increases the level of contribution to the public good when subjects are informed about the contribution levels of lower ranked subjects while sequentiality alone has no effect on contributions. Moreover, we observe that earlier players try to influence positively the contributions of subsequent decision makers in the sequence, by making a large contribution. Such behaviour is motivated by the belief that subsequent players will reciprocate by also making a large contribution. We also discuss the effect of group size on aggregate contributions. Finally, we imagine a model of behaviours where agents’ preferences incorporate a “weak” moral motivation element. This model organizes consistently the patterns observed in the lab.

(a) CREM, Faculté des Sciences Economiques, 7 place Hoche, 35 065 Rennes, France e-mail : [email protected] (b) LAMETA , Faculté de Sciences Economiques, avenue de la mer, C.S. 79606 / 34960 - Montpellier cedex 2, France e-mail : [email protected] (c) LAMETA, 2 Place Viala. 34060 cedex 01, Montpellier, France. e-mail : [email protected]

1. Introduction Sequential contributions is a widespread practice for a large variety of public goods: individual efforts in housework, telethons, scientific research, ratification of international treaties about environmental issues, etc… In these examples contributions are not necessarily in monetary terms but they share two important features: the amount of public good provided depends on the aggregate level of contributions, and later contributors in the sequence can observe, to some extent, previous contributions.

Surprisingly, the economic literature on sequential contributions is rather scarce, while an overwhelming bulk of knowledge has been accumulated on simultaneous contributions environments. Other things being equal, should one expect a higher level of public good provision when contributions are sequential? Getting back to the telethon example, does the publicly released information about contributions affect donations during the campaign ?

Varian (1994) demonstrated that a sequential contribution environment exacerbates the incentives to free-ride, in comparison to a simultaneous contribution environment. In his model, self-centered players have different non-linear preferences for the public good. While his prediction was never tested, a few experiments have been performed for the case of the standard linear payoff functions in the public good. In sharp contrast with the zero contributions predictions, these experiments found a positive effect of sequentiality compared to simultaneous contributions (Moxnes and van der Heijden, 2003, List and Lucking-Reiley, 2002, Shang and Croson, 2003, Güth et al., 2004). In the present paper, we seek to contribute to previous literature by investigating the following question : “why do subjects contribute more under sequentiality ?”.

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The experimental literature suggests at least three reasons for larger contributions under sequentiality : reciprocity, leading by example, and pure ordering. They can be described briefly as follows. The reciprocity reason rests on the idea that some decisions are motivated by moral sentiments rather than by a rational choice to achieve an objective. A gift or a favor as a response to a gift or a favor enter into this category. In a public good context, ,a contributor reacts to the observed contributions of lower ranked players in order to maintain a balance or to avoid moral indebtness, even if he believes that he has no influence on subsequent contributors1 , hence on the level of public good he will enjoy. Second, according to the « leading by example” hypothesis, early contributors may feel responsible for setting a good example for later contributors. By making a large contribution, they expect later contributors to react positively, for instance in accordance with the reciprocity motive. Finally, even an agent who is unable to observe previous contributions and who believes that he cannot influence subsequent contributions, might still be influenced by the knowledge of his ranking in the sequence. Such an effect, called “pure-ordering” effect, was found in common pool resource (CPR) dilemmas experiments (Rapoport et al., 1993, Budescu et al., 1995, Suleiman et al., 1996, Rapoport, 1997), and is likely to be also present in a context where contributions to a public good are made sequentially. More generally, ordering effects may arise in contexts where subjects move sequentially, without being able to observe previous moves of other players.

Our objectives are both experimental and theoretical. Experimentally, the challenge is to isolate the effect due to leadership from the two other effects (reciprocity and ordering). We rely on an experimental setting where subjects have to choose sequentially their level of contribution to a pure public good. We compare two information conditions : a treatment without information and a treatment with information. In the treatment without information, subjects decide sequentially but without being 1

For a recent and comprehensive introduction to the economics of reciprocity, see Kolm (2006). 3

able to observe previous contributions. In the treatment with information, subjects observe the individual contributions of subjects who decided earlier in the sequence. The reference treatment is a simultaneous public good game (no sequential move and no information). Each treatment has the same number of rounds. In treatments where contributions are made sequentially, in each round subjects are randomly assigned to a rank. Our design allows us to separate the leadership and the reciprocity effects from the ordering effect. Furthermore, our analysis of the data allows one to identify the effect of the players’ rank on their level of contribution and to sort out the leadership effect and the reciprocity effect. Finally, we investigate whether population size affects the level of average contributions by considering two population sizes. Increasing population size might have mixed effects. On one hand, leaders’ influence might be stronger because more people observe early contributions. On the other hand, if later players reciprocate only partially because of a selfish bias (see Neugebauer et al., 2007), the decay of contributions might be stronger. Which one of these effects is strongest is an empirical matter.

The main experimental findings are as follows: i) in the sequential treatment with information the average contribution is larger than in the simultaneous treatment, and observed contributions decline with the order of play ii) sequentiality alone has no effect on contributions; iii) the size of the group has no significant impact on the average level of contributions; iv) the decline of the average contribution with the rank of the player is attributable to a vanishing leadership effect rather than to the erosion of reciprocity.

The theoretical objective is to propose of model of behavior that organizes consistently the data. The suggested theoretical explanation is a result from a back and forth process between theories and observations, both those already available in the literature and those produced during our experiment. 4

We end up with “morally motivated” agents who play a unique Stackelberg equilibrium for the sequential treatment with information. The model assumes that ex ante, each individual holds a morally ideal contribution, which she updates according to her observed contributions by earlier players. Assuming identical players, at the Stackelberg equilibrium all players in the society contribute a positive amount which declines with the player’s rank in the sequence. Furthermore, the model also predicts that the average contribution of the society’s members increases if players are more sensitive to observed contributions. If the population size increases the prediction of the model, as far as the average contribution is concerned, is ambiguous. The average contribution might be larger because the leadership effect gets stronger when the number of agents gets larger. On the contrary, the average contribution might fall if the additional contributors' preferences are such to induce lower contributions.

The paper is organized as follows. In section 2 we discuss the related literature on the possible effects of sequentiality. Section 3 presents our model of moral motivation. Section 4 describes the experimental design and section 5 presents our results. Finally, we conclude in section 6 by drawing the implications of our analysis.

2. Related literature

2.1. Sequentiality with observation: reciprocity and leadership effects

While most experimental research on voluntary contributions to a public good, has focused on simultaneous contribution environments, there are a few recent experiments that studied a sequential contribution environment (Weimann, 1994, Bardsley, 2000, Güth et al., 2004, Potters et al., 2004, Houser and Kurzban, 2005). While Weimann (1994) did not run directly a sequential public good 5

game, his experiment addresses the question of the effect of observing individual contribution. He investigates the effects of giving subjects false reports of the others’ contributions to the public good in the previous round, before being asked to choose their own contribution level in the current period. In the low condition, subjects were told that the others had contributed 15.75% of the endowment on average, and in the high condition 89.75%. Contributions were significantly lower in the low condition compared to the honest baseline, but where not affected in the high condition compared to the baseline. Using a one-shot public good game, Bardsley (2000) compares a “sequential” and a “simultaneous” public good task. In the sequential task subjects had to contribute after observing previous contributions of their group mates, being aware that the observed contributions were actually artefacts except one (which would be paid out at the end of the session). The results show that subjects are more cooperative in simultaneous-play than in sequential-play games and that contributions diminish with the rank in the sequence. In a recent paper, Houser and Kurzban (2005) proposed a novel design to study reciprocity in a sequential public goods experiment. In their experiment subjects simultaneously make an initial allocation to the group account. Subsequently, the current aggregate contribution to the group account is revealed to exactly one player who is given the opportunity to revise his contribution. The authors find that 63% of the subjects are conditional cooperators. In Güth et al. (2004), the leader’s contribution is announced publicly before the followers decide simultaneously about their contribution. They found that such an announcement significantly increases the average level of contributions to the public good, although followers contribute significantly less than the leader. According to the above experimental findings, the first contributor tries to set the “good example” by making a large contribution in order to lead followers towards high contributions. Güth et al. (2004)’s interpretation is that followers condition their contribution on the leader’s contribution like in a repeated simultaneous contributions game, where subjects reciprocate previous period average contribution ( Keser & Van Winden ( 2000) Fischbacher et al. (2001), Fischbacher & Gaechter (2006)). Using a public goods framing, Moxnes and 6

van der Heijden (2003), and Gächter and Renner (2004) have implemented leadership in the lab by letting a group leader decide, first. The other group members had to decide how much to contribute after being informed about the leader’s decision. The common finding of these papers is that players contribute more with a leader than “without.2

The positive leadership effect found in the experimental literature raises a central question about subjects’ motivation to act as a leader : why do subjects in the position of the first contributor choose to make a larger contribution than they would make otherwise ? Moreover does the leadership vanish with the rank in the sequence of contributions ?

2.2. Sequentiality without observation : “ordering effect”

If agents’ actions are unobservable, a game where players act sequentially is strategically equivalent to a game where they act simultaneously. In contrast, Rapoport and colleagues (Rapoport et al., 1993, Budescu et al., 1995, Suleiman et al., 1996, Rapoport, 1997) reported a pure positional order effect in common pool resource dilemma games. They observed that despite perfect informational symmetry among players (request disclosure), the first mover has a tendency to take a larger part, while later movers in the sequence, have a tendency to take less. In their resource dilemma game the overexploitation of the common pool leads to null profits for all players. It is therefore likely that subjects rely on the tacit cue, commonly accepted, that the first served should take a larger amount than the followers. Typically, such cues can be used as a coordination device to facilitate equilibrium selection. 2

Leadership effects were also sudied by Potters et al. (2004) but in a context of information asymmetry between players. They showed that players choose more frequently to contribute sequentially than simultaneously with significantly larger contributions under sequentiality. Several field studies also showed the importance of the leadership-effect in various contexts (List and Lucking-Reiley, 2002, Shang and Croson, 2003). . 7

In linear pure public good games with a unique Nash equilibrium the coordination issue is irrelevant. Nonetheless, we cannot exclude a priori that sequentiality as such might influence contributions for some unknown reason. For example, Abele and Ehrhardt (2005), observed larger contributions in a two-player simultaneous move game than in the equivalent sequential move game, i.e. without observability. According to the authors subjects are more likely to cooperate in simultaneous moves games, because they have a stronger “feeling of groupness” than in the sequential moves game.

Besides the leadership effect, this literature suggests that sequentiality as such can generate various effects, either by affecting the saliency of some equilibria3, or by affecting the players’ reasoning, or simply by affecting their perception of the game. However, all experiments which have studied order of play, with the exception of Güth et al. (1998), involved a coordination problem : common pool resource games, step-level public goods games, battle-of-sexes, … In contrast, in Güth et al.’s (1998) experiment, a first mover advantage could only be achieved by a deviation from the dominant strategy play. They found that the pure-ordering effect vanishes in this context.

However, in the light of the literature about the timing of moves, it is important to control for possible pure timing effects when investigating leadership and reciprocity effects induced by the observation of early contributions. In order to disentangle these effects, our experiment is based on a voluntary

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Cooper et al. (1993) found a first mover advantage in a two-player battle of sexes games. When the game is played sequentially without observability, the equilibrium which is more favourable to the first mover is played more frequently, probably because it becomes “more salient”. Weber and Camerer (2004) explored this “saliency” explanation by manipulating the timing of moves in a weakest link coordination game. Their aim was to distinguish between the “saliency of first mover advantage” explanation and the idea of “virtual observability” introduced by Amershi et al. (1989). According to the virtual observability theory in a sequential game without observability, players apply subgame perfection as if all moves were observable, provided that the selected equilibrium is also an equilibrium for the original game. The virtual observability prediction fits better their data on the sequential weakest-link coordination game, than the pure ordering prediction 8

contribution game with a dominant strategy equilibrium in order to avoid any coordination problem for the players.

3. Reciprocity, Leadership and Moral motivation in a sequential contribution environment

In this section, we present the theoretical predictions of our model. We present first the standard gametheoretic predictions of a sequential voluntary mechanism (VCM). We then propose a variant of this model, based on the assumption that players have a “weak moral motivation”. Our experimental data will be compared both to the predictions of this model and to the standard game-theoretic predictions. Let’s consider first the standard game-theoretic predictions of the sequential VCM. In the sequential version of the VCM introduced by Isaac et al. (1984), players choose their contributions sequentially. First the leader chooses her contribution, which is announced to the second player in the sequence. Then the second player chooses her level of contribution, etc…. Assume that each player’s utility function is given by : U i ( xi , x− i ) = w − xi + β ( xi + x− i ), i = 1,...,n

where xi is player i’s contribution to the public good, x-i the contribution of the other players, and β < 1 is the marginal utility from consuming the public good G = xi + x-i. Each contribution xi

must satisfy 0 ≤ xi ≤ e . Applying backward induction, it is easily seen from (1) that the theoretical prediction coincides with that for a standard simultaneous VCM. Due to β < 1 , the last subject in the sequence contributes zero to the public good. Since earlier players in the sequence will anticipate this, their dominant strategy is 9

to free ride as well. Therefore, if the game is played once, there is a dominant strategy to contribute zero. If the game is finitely repeated, the only subgame perfect equilibrium of the game is for all players to contribute zero in each period.

Let’s consider now a variant of this model, assuming that players have a weak moral motivation, which means that a player’s moral motivation to contribute is influenced by her observation of other players’ contributions. Under this particular assumption, leadership means trying to manipulate other players’ moral motivation in order to affect their level of contribution. The leader can only obtain such an influence on others’ contributions by manipulating her own contribution. The model exposed below is an adaptation to a sequential contribution context of the more general model developed by Figuières et al. (2007) for the standard repeated game of voluntary contributions.

The central idea is that preferences are structured into several layers (Harsanyi, 1955, Sen, 1977), some with moral reasoning, the interplay of which finally explains actual decisions. For instance, suppose that individuals have two layers of preferences: self-interested preferences and social preferences. Upon deciding how much to contribute to a particular public good, any individual has in mind a morally ideal level of contribution (Nyborg, 2000, Brekke et al, 2003), which is derived under the assumption that he

is able to judge matters from society’s point of view, following a combination of the utilitarian philosophy and a version of Kant’s Categorical Imperative: “how much should I contribute to maximize social welfare, given that everybody else would act like me?” The answer ends up in a Paretian level of contribution; and any discrepancy between this morally ideal course of action and the actual decision incurs a loss of utility. Finally, an agent’s actual contribution will be the result of a trade-off between his marginal utility of improving his self-image by making a large contribution, his

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marginal utility from enjoying more of the public good and his marginal utility of consuming more of the private good. In some sense people make intrapersonal comparisons between their utility layers.

Brekke et al (2003)’s approach of moral motivation, however interesting, also raises a few issues. Considering what should be the best plan for society, would individuals’ answers be independent of the society in which they live? For instance, would they take a Pareto optimal allocation as an ethically valuable goal regardless of the fact that all other people free-ride? Would they consider all societies equally deserving? Answering yes would be at odds with many experimental results supporting the idea that, for many people, reciprocity in public goods issues matters a lot, e.g. in a public good context, “ I free-ride, not because it is my self interest, but because everybody else does”. The answers to those questions are sufficiently unclear to call for an important qualification of Brekke et al (2003)’s theory of behaviors.

Our originality in this matter is to assume that individuals’ moral responsibility is not only the result of some autonomous ethical reflection, as would do impartial outside observers, but it is also partly determined by their actual social environment. It is therefore affected by the observations of others’ actions: an agent who observes other agents’ contributions to the public good, will eventually revise his initial morally ideal behavior, hence the expression “weak moral motivation”. Of course, in a more balanced perception of reality, some individuals might have a purely intrinsic motivation to contribute and therefore will always stick to it whatever the other agents contribute. Our intuition however, is that most individuals are amenable to reconsider their morally ideal point according to the current social morality partly revealed by other members of their group. We suggest that the reciprocity hypothesis, a widespread argument to explain individual decisions in public good experiments, could be rooted in the

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updating of one’s moral motivation. Such an updating arises naturally in a context where individual contributions to the public good are made sequentially. Turning to formal matters, the cardinal representation of agents’ preferences with moral motivation is given by: U i ( xi , x −i ) = w − xi + β ( xi + x −i ) − vi ( xi − xˆ i ), i = 1,..., n

The novel aspect of the above preferences comes from the moral motivation embodied in the function vi (.) . If xˆ i stands for agent i’s moral ideal, then his utility loss for any deviation from his moral ideal is vi ( xi − xˆ i ) . There are two important remarks about the function vi (.) . Firstly, in general the moral

motivation function is specific to each individual. We shall however assume that all individuals suffer the same loss from a deviation from their moral motivation. Secondly, and more importantly, the moral ideal xˆi is also specific to each individual even if all agents share the same loss function. In general xˆ i can take any value between 0 and wi . This seems consistent with the experimental findings about

over-contributions in public goods games. The average contribution in the first round is about half the endowment, and there is a large variance in individual contributions. Two natural assumptions about the loss function are as follows: Assumption 3.1. v(0) = 0, v( xi − xˆi ) > 0 iff x i ≠ xˆi . >    Assumption 3.2. v' (.)=0 ⇔ xi − xˆi <   

>    =0 <   

The first assumption is obvious. The second assumption means that, starting from a situation where agent i contributes less (more) than her moral ideal, a marginal increase of xi reduces (increases) her loss of utility.

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As previously announced, we shall conceptualize the moral ideal xˆ i of each agent as a function of two arguments. The first argument, noted xi*, captures an autonomous ethical logic, that we shall leave relatively unspecified for the purpose of this paper. It could be derived from a Kantian Categorical Imperative combined with a utilitarian philosophy, as in Brekke et al (2003), and would correspond in this case to a Pareto optimum level of contribution. If agents are symmetric agents all those individual Pareto optimum levels are equal, i.e. xi* = x* = w, ∀i . Note that from an experimental point of view xi* is not observable. The second argument/logic captures social influences on ethical thoughts via the

observation of the average contribution of the previous agents in the sequence xi −1 . The observation of others’ contribution leads agents to revise their own moral ideal. We assume that xˆi , called the effective moral ideal from now on, is a convex combination of the autonomous moral obligation xi* and the

observed average contribution of previous agents: xˆ i = (1 − θ i ) xi* + θ i xi −1 = xi* − θ i ( xi* − xi −1 ) , θ i ∈ [0,1] , i = 2,3,…,n.

The case of the first contributor is slightly different. Since he has no previous observation to update his ethical position, then his effective moral ideal is simply xˆ1 = x1*. Remember that actions are taken sequentially: agent 1 decides first, followed by agent 2, then agent 3 and so until agent n. In order to arrive at tractable solutions for the Stackelberg equilibrium, we restrict the model to the symmetric case4. We do this by introducing two simplifying assumptions. First, we assume symmetry, i.e. all players are assumed to have the same autonomous moral ideal, xi* = x* for all i, with 0 < x* < w. Second, we shall also assume that θ i = θ ∈ [0,1], ∀i . Note that by our first assumption, we exclude the uninteresting case of no autonomous moral ideal (x* = 0) - this would mean individuals have no personal ethical ideal - and the case where the autonomous moral ideal corresponds to the Pareto optimum contribution (x*= w). Even though the autonomous moral ideal 4

See Figuières et al. (2007) for a more general model, based on a Nash equilibrium. Also, at the end of this section we will close the discussion with the study of an asymmetric 3-agent economy. 13

could be arbitrarily close to w, this corner case is eliminated for two reasons. The possibility of corner decisions complicates substantially the analysis, because we would need to take into account of a sequence of discontinuous best replies functions to be anticipated by each leader. We discard this possibility to arrive at the phenomenon of interest as simply as possible. Secondly, there is no empirical reason to believe that subjects take the Pareto contribution as a reference for their autonomous moral ideal. Rather, the large available experimental evidence on voluntary contributions to a public good suggests that on average, in the first round, subjects have a moral motivation to contribute something to the public good, but far below the Pareto optimum level of contribution. Under the previous restrictions the interior Stackelberg equilibrium turns out to be:

x n = xˆ n + v' −1 (−1 + β ) , x n −1 = xˆ n −1 + v' −1 ( −1 + β +

βθ

) n −1 ,

… k   x n−k = xˆ n −k + v' −1  − 1 + β + βθ (n − h) −1    h =1  

∑

… n −1   x1 = x * + v' −1  − 1 + β + βθ (n − h) −1    h =1  

∑

Those equilibrium conditions can be understood as follows: at equilibrium, agent n-k chooses a level of contribution xn−k which equalizes his marginal utility from deviating from his effective moral ideal to his marginal utility of contributing an additional unit to the public good − 1 + β + βθ

βθ

k

∑ ( n − h) h =1

−1

k

∑ ( n − h) h =1

−1

, where

captures the influence of his marginal contribution on the remaining subsequent players.

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Note that, because the last expression is always positive, early players can contribute more than their moral ideal.

Let us define   v ' −1 (− 1 + β ) for   k  −1   v ' 1 β βθ ( n − h ) −1  , − + +   ∑ h =1   

g (k ) =

k = 0 , for

k

≥1 .

Notice that g’(k) > 0 for k ≥ 1. By construction, g(k) is agent k’s optimal departure from his ideal level of contribution. Using this definition, the Stackelberg equilibrium can be rewritten: x1 = x * + g (n − 1) x 2 = xˆ 2 + g (n − 2) = (1 − θ ) x * + θx1 + g (n − 2) = x * + θg (n − 1) + g (n − 2) x3 = xˆ 2 + g (n − 3) = (1 − θ ) x * + θx 2 + g ( n − 3) = (1 − θ ) x * + θx * + = x* +

θ 2

θ 2

[(1 + θ ) g (n − 1) + g (n − 2)] + g (n − 3)

[(1 + θ ) g (n − 1) + g (n − 2)] + g (n − 3)

....

xk = xˆk + g(n−k)= x* + Lk[g(n−1),g(n−2),...,g(n−k −1),g(n−k)] where Lk [g (n − 1), g (n − 2),..., g (n − k − 1), g (n − k )] is a linear combination, with positive coefficients, of g (n − 1), g (n − 2),..., g (n − k − 1), g (n − k ) .

Consider now the following assumption on the parameters: n −1

Assumption 3.3.

1−β

∑ n−1h ≤ βθ h =1

Assumption 3.3 is easy to satisfy; for instance as the marginal utility of the public good tends to zero, the right hand side of the inequality growths arbitrarily large. Note that Assumption 3.3 is equivalent to

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* assuming that the first contributor (agent 1) contributes less than his moral ideal, x1 < x . If the latter

* inequality is satisfied, by Assumption 3.2 we have v' ( x1 − x ) ≤ 0 which means

(or equivalently Assumption 3.3).

Proposition 3.1. Under Assumption (3.3), successive contributions to the public good are decreasing:

x1 > x2 > … > xk > …> xn . Proof : See Appendix

Without Assumption 3.3, some agents can contribute to the public good above their moral ideal, as )

ν

)

2 illustrated by the following example. Assume the moral obligation function is v( xi − xi ) = 2 ( xi − xi )

and let the parameters take the following values :

ν = 0.1, ω = 40 , x∗ = 20, β = θ = 0.9 and n = 3.

Computing the Stackelberg equilibrium, one finds:

x1 = 31.15 , x2 = 33.08 , x3 = 29.90.

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This example illustrates the property that although early contributions may increase, they will eventually decrease at some point in the sequence. Actually, without Assumption 3.3 we have the following weaker proposition :

Proposition 3.2. There exists an agent j ≤ n such that, starting from that agent successive contributions

to the public good are decreasing: xj > xj+1 > …> xn . Proof : See Appendix 2

Another interesting property obtains when a marginal increase of the first agent’s contribution has a positive impact on the ideal contribution of the second agent. Formally: ∂

∂

Assumption 3.4. ∂θ [θg (n − 1)] = g (n − 1) + θ ∂θ g (n − 1) > 0

Proposition 3.3. Under Assumption 3.4, the higher the importance of observed behaviours on agents’

moral obligation (higher θ), the higher the contributions at a Stackelberg equilibrium.

Proof : See Appendix.3

According to proposition 3.3 any average xh is an increasing function of θ. When θ tends to zero, the Stackelberg equilibrium converges to the Nash equilibrium of a public good game with morally motivated contributors. Proposition 3.3 implies therefore that Nash contributions are lower than Stackelberg contributions. This particular case of moral motivation function boils down to Brekke et al (2003). With their model of preferences, the Nash and the Stackelberg equilibrium of the game with payoffs given by (4) are identical. Moral motivation per se is not sufficient to explain the regularities 17

observed in the lab, except the tendency for agents to over-contribute, which is not predicted by standard Nash contributions without moral motivation.

Finally, the importance of a leadership effect can be partly measured through the following question: what is the effect on the average contribution of adding further agents? Intuitively, there can be two opposite effects. One may expect an increase of the average contribution because leaders have more followers to influence positively, that is adding more agents strengthens the leadership effect. But this effect might be counter-balanced by a free-rider effect if the added agents have a relatively lower taste for the public good hence, other things equal, lower contributions at the equilibrium. Of course, under the assumption of identical agents considered so far, this negative effect is supposed to disappear. This property can indeed be established:

Proposition 3.4 : With identical agents endowed with utility functions as in (4), the average

contribution xn is increasing with the number of agents n. Proof

:

Recall

that

n −k   g (n − k ) = v' −1  − 1 + β + βθ (n − h) −1  is   h =1  

∑

increasing

with

n-k.

Since

x k = xˆ k + g (n − k ) = x * + Lk [g ( n − 1), g (n − 2),..., g (n − k − 1), g ( n − k )] , i.e. contribution of player k is a linear

combination of g (n − 1), g (n − 2),..., g (n − k − 1), g (n − k ) with positive coefficients, increasing n will increase xk.

Proposition 3.4 is intuitive: the larger the number of followers who can be induced to contribute, the stronger the marginal incentive for any leader to increase his own contribution. An equivalent interpretation is as follows: when more followers are to be influenced, increasing the level of public good by one unit can be done (indirectly) at a lower cost.

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However, as mentioned above outside the symmetric framework the positive leadership effect can be counterbalanced, for instance when the added contributors attach less importance to their moral obligation (low vi). Without loss of generality, this can be shown by comparing the average contribution of a two-agent group with the average contribution in a three-agent group. )

ν

)

2 Assume again the moral obligation function is v( xi − xi ) = 2 ( xi − xi ) and let the parameters take the

following values :

ν = 0.7, ω = 40 , x∗ = 20, β = θ = 0.9 and n = 2.

Computing the Stackelberg equilibrium, one finds an average contribution xn = 0.578571. Adding a third agent, similar to the two previous ones but for the parameter of his moral obligation function, which is lower v3 = 0.04, the average contribution becomes xn = 0.45582. As we shall see in the next section, experimental evidence do not show a clear-cut decrease of the average contribution as the number of agents increases, suggesting that the added agents have had a weaker moral motivation.

While our model is primarily intended to capture the leadership effect, by assuming that early players can influence later players’ moral motivation, it also accounts for reciprocity. Indeed, the process by which a player’s moral motivation is updated takes into account the observed contributions of previous players. Therefore, if her revised moral motivation is increased (decreased) she will contribute more (less) than she would have contributed otherwise. This effect is independent from the leadership effect, since even the final player in the sequence updates his moral motivation and adjusts his contribution accordingly without being able to influence any further player. In a simultaneous contribution game, our model provides therefore a rationale for reciprocal behaviour (see Figuières et al., 2007).

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4. Experimental design

The experiment consisted of 16 sessions of 15 periods each. Experimental sessions were conducted both at the University of Rennes5 and at the university of Montpellier6 in France.7 252 subjects were recruited from undergraduate classes in business and economics at both sites. None of the subjects had previously participated in a public good experiment and none of them participated in more than one session. The experiment was computerized using the Ztree program. On average, a session lasted about an hour and 20 minutes8 including initial instructions and payment of subjects.

We set up an experimental design that allows us to investigate the effect of information accumulation on individual contributions in a sequential contribution environment. The reference treatment is a simultaneous voluntary contribution game. At the beginning of each period, each member of a group of n subjects is endowed with 10 tokens that he has to allocate between a private account and a group

account. Following previous notations, player i’s payoff is given by : n

u(xi, x−i) = 10 − xi + 0.5 × ∑ xh h =1

Subjects were instructed to indicate only their contribution to the group account, the remainder of their endowment being automatically invested in their private account. Any token invested in the group account generates the same payoff for each member of the group.

5 6 7 8

CREM (Centre de recherche en Economie et Management) , LAMETA (Laboratoire de Recherche en Economie Théorique et Appliquée) No significant differences were found in data between the two locations of the experiments. The sequential treatments took slightly more time in large groups. 20

In the reference treatment, players choose their contribution simultaneously (simultaneous treatment thereafter). Under standard behavioral assumptions, there exists a unique dominant strategy equilibrium for the one-shot game where each player i = 1, …,n, contributes xi = 0. Furthermore, there exists a unique subgame perfect equilibrium for the finitely repeated game where each player i contributes xi = 0 in every period. In the experiment, the constituent game was repeated exactly 15 periods. Note that the group optimum is achieved if each player contributes his endowment to the public good.

Besides our "simultaneous treatment", we have two treatments for which subjects take their decisions sequentially, called sequential treatment without information and sequential treatment with information. In the latter treatment previous contributions are observable while in the former they are not. In both treatments decisions are taken sequentially. This is done by assigning in the beginning of each round each subject to a rank in the decision sequence. Each subject is informed about his rank for the current round, and about the individual contributions of all lower ranked subjects in the sequential treatment with information. Therefore, the least informed subject is the subject who is ranked first in the sequence whereas and the most informed subject is the one who is ranked last in the sequence. In the sequential treatment with information there is a unique equilibrium where the first ranked player contributes zero, the second ranked player contributes zero, etc… The predictions are therefore the same for all three treatments – zero contribution in each round.

While the information condition is our main treatment variable, we also study the impact of group size on the level of contribution in the sequential contribution environment, by comparing the average contribution in groups of 4 subjects (small groups) with groups of 8 subjects (large groups). According to our model, an increase of the size of the group of contributors can have two opposite effects on the average level of contributions. A positive one by strengthening the leadership effect, since for any rank 21

they are more subsequent players which can be influenced. However if the size of the group becomes larger, there might also be a greater temptation to free ride for higher ranked subjects since the accumulated contribution becomes larger. We relied on the same presentation for all treatments9. The ranking of the subjects in the sequential treatments was randomly determined by the computer program, both within and across rounds. Therefore, subjects rank in the sequence varied from round to round. At the end of each round, the computer screen displayed the subject’s investment decision, the total group contribution and the earnings of the group account as well as the total earnings. Cumulated earnings since the beginning of the game, as well as the number of the round were also on display. After each round, subjects could see their detailed records since the beginning of the experiment. Contributions were anonymous. Table 1 provides a summary of the experimental design. The first four columns indicate the session number, the corresponding treatment, the number of groups and the number of subjects who participated in the session. The last column indicates the group size (4 or 8 members per group). A partner matching protocol was in effect for all sessions (i.e the groups remain unchanged during the entire session). [Table 1: About Here]

9

To control for the existence of a possible "framing effect", we also ran two sessions with a variant of the reference treatment, labelled " simultaneous treatment with framing". This control was required because the sequential version of the contribution game required a slight alteration of the usual presentation of the instructions. For this variant the investment in the group account is presented as an explicit addition of individual contributions which matches the presentation that was used for the sequential contribution treatments. The instructions pointed out that each subject's contribution would be identified by an index, e.g. subject i's contribution is noted Ii, and that the payoff of the group account would be given by 0.5×(I1+I2+...+IN). This point was described to the subjects in the following language : "I1 is member 1's investment to the collective account I2 is member 2's investment to the collective account I3 is member 3's investment to the collective account This presentation, by making explicit the summation of individual contributions, could have influenced the subjects decisions in a non predictable way. However, the results indicate no significant difference at any level of significance in average contribution between the simultaneous treatments with and without framing.

22

The comparison of average contributions in the benchmark and in the sequential treatment without information allows us to isolate a pure ordering effect if any. We hypothesize, based on the results of Güth et al. (1998), that only knowing their rank in the decision sequence should not affect subjects’ contributions. The comparison of the contributions between the two sequential treatments - with and without information - allows us to evaluate the combined effects of leadership and reciprocity resulting from the information asymmetry between players. Based on our model of weak moral motivation, we hypothesize that the combination of the two effects should affect positively the level of contributions for two reasons : 1) lower ranked subjects in the sequence will contribute more in order to influence later contributors (leadership effect) and 2) subsequent players will reciprocate the observed contributions of lower ranked subjects (reciprocity effect). Finally, the comparison between the sequential treatment with information and the benchmark treatment captures both the effect of sequentiality and of information asymmetry. Note that our experimental design does not allow to separate reciprocity from leadership. However, our data analysis will allow us to identify which of these two effects drives the decline in contributions according to the rank of the player in the sequential treatment with information. Furthermore, to some extent we are able to isolate ex post these two motives by comparing subjects’ contributions in extreme positions (first and last) with subjects’ contributions in intermediary positions Indeed, the first player’s contribution within a sequence cannot be motivated by “reciprocity" since no information about others’ contributions is available to him. However, he might be motivated by leading by example. At the other extreme, the last player in the sequence cannot lead by example, and therefore his contribution can only be motivated by reciprocity. Only players that have an intermediary position can be influenced both by "reciprocity" and "leadership".

5. Results

23

This section is organized as follows. Subsection 5.1 reports the observed patterns of average contributions for our three treatments . We analyze the treatments in relation to each other and to the benchmark treatment, and evaluate the hypotheses stated in sections 3 and 4. In subsections 5.2 we analyse the determinants of the contribution behaviour separately for each treatment.

5.1. Average individual contribution

Figures 1 and 2 illustrate the time path of average individual contributions by period for small and large groups respectively. The period number is shown on the horizontal axis and the average individual contribution on the vertical axis, where the maximum possible individual contribution is 10. These figures show the same pattern for all treatments : there is initially a positive level of contribution to the group account which declines with repetition (except for the sequential treatment with information in large groups, in which the average contribution level does not change appreciably as the game is repeated). This result is in line with several other experiments that have documented that the average contribution tends to decline with repetition (Isaac et al. 1984, Isaac and Walker, 1988, Andreoni, 1988, Weimann, 1994).

[Figures 1, 2 and table 2 about here]

Result 1 summarizes our findings both about the effects of sequential contributions with and without information.

24

Result 1 : Levels of contribution are higher under the sequential treatment with information than under

the sequential treatment without information. Sequentiality without observability has no significant impact on the level of average contribution.

Support for result 1 : Table 2 shows the average contribution for each treatment. The first three

columns of table 2 indicate the average individual contribution for each small group. The last three columns contain the same data for each large group. Comparison of treatments suggests that sequentiality with information positively affects average contributions. For both small and large groups, average contribution levels are higher in the sequential treatment with information than in the sequential treatment without information. A nonparametric Mann-Whitney rank-sum test10 for small groups shows that the difference in average contributions between the sequential treatments with and without information is significant at the p < .10 level, (z = -1. 7, two-tailed). A similar test of the difference between the sequential treatments with and without information for large groups also indicates a positive and significant effect of information (z = 2.082; two tailed test).

In order to isolate the pure effect of sequentiality, we compare the average level of contribution in the simultaneous treatment and in the sequential treatment without information. Our results indicate that for both small and large groups changing the timing of moves without changing the information condition has no significant effect on contributions (z = - 0.145 for small groups and z = 1.601 for large groups, respectively). The comparison of the simultaneous treatment to the sequential treatment with information indicates that the combined effect of sequentiality and observability of previous contributions in the sequence increases the average contribution level in small groups (z = - 1.843), but not in large groups (z = - 1.44; two tailed test). The insignificant difference between the baseline 10

In all statistical tests reported in this paper, the unit of observation is the group. 25

treatment and the sequential treatment with information for large groups suggests that the positive effect of information is partly offset by a negative effect induced by sequentiality alone, though this effect is not significant. Finally, Mann-Whitney tests of the difference of contributions between each treatment according to group size indicate no significant effect for the size. A Mann-Whitney test of the differences between the simultaneous treatment with size 4 and the simultaneous treatment with size 8 yields an insignificant z = 0.307. Similar results are obtained for the sequential treatment without information (z = - 1.486 ; two-tailed) and for the sequential treatment with information (z = - 0.480; two tailed).

5.2. Determinants of contribution

Table 3 provides formal evidence about the influence of sequentiality and observability on contributions. The dependent variable is the amount of tokens contributed in the tth round. The independent variables include several dummy variables including the variable "information", to control for the influence of information on contribution and the binary variable “sequentiality” to control for a pure order effect. The variable "information" takes value 1 if subjects are informed about previous contributions in the sequence and 0 otherwise. Finally, the variable "Sequentiality×information" takes value 1 if the treatment introduces both sequentiality and information and 0 otherwise.

We also included two variables that measure the influence of received information about previous players’ contributions (reciprocity effect) : the lagged average contribution of the other members of the t −1 r inf t group (x−i ) and the average contribution of lower ranked subjects in the current period (x−i ). In the

simultaneous treatment and the sequential treatment without information, only information about past periods is available. In contrast, in the sequential treatment with information, subjects may be 26

influenced both by the information received from previous periods and information from previous decisions in the sequence for the current period. In addition, we also introduced a dummy variable for the first rank in the group. This variable indicates whether the first player in the sequence contributes more than other group members ("leadership" effect). It takes value 1 if subjects are in the first position in the sequence and 0 otherwise. Finally we also introduced a dummy variable for the last round (i.e. period15) and a dummy that controls for the size effect. [Tables 3 about here] The estimates summarized in table 3 confirm our previous findings. The specifications of the second and third columns reveal that individuals increase their contribution when they are informed about the contributions of each lower ranked subject. Table 3 also indicates that the coefficient associated with "sequentiality alone" is not significant, confirming that sequentiality without observability has no significant impact on the level of average contribution.

We now turn to our main question : how does sequentiality and observability of others’ contributions affect a subject’s contribution ? Our answer is stated in results 2 and 3. Result 2 summarizes our findings about the relationship between observability of other players’ contribution and the level of own contribution. Remind that our conjecture is that a subject’s contribution is both affected by the observed contributions of the lower ranked subjects and by his trial to influence the contributions of higher ranked subjects. In result 3 we report our findings about the dynamics of the leadership effect and the reciprocity effect within a sequence.

Result 2: Consistent with a "reciprocity" effect, an individual's contribution in period t is higher (a) the

higher the contributions of the other group members in period t-1 for all treatments, (b) the higher the contributions of the lower ranked players in the sequence in period t (in the sequential treatment with 27

observability). Consistent with a "leadership" effect, individuals who decide first in the sequence, contribute significantly more than other group members in the sequential treatment with observability.

Support for result 2 :

Table 3 shows that in all treatments the variable “others’ lagged average contribution " has a positive and significant influence. High contributions on the part of the other group members are imitated or reciprocated by high individual contributions. Note that this effect is stronger for the simultaneous treatment and sequential treatment without information than for the sequential treatment with information. For the latter treatment, the variable "contribution of lowest ranked players" has a positive and highly significant impact on own contribution. This result suggests that in the sequential treatment with information subjects tend to rely less on information from the previous period than on the more relevant information from the current period. The variable "contribution of lower ranked subjects” has also a significant and positive impact on the last player’s contribution in the sequence (last two columns of table 3). This result suggests the existence of a pure reciprocity effect, since the last player in the sequence has no influence whatsoever. Taken together, the above results support the idea that individuals reciprocate previously observed contributions in the sequence. Noteworthy, the first ranked player who cannot rely on any information generated within the sequence relies more strongly on the previous period average contribution observed in the group (significant at 1%)11.

Consistent with earlier findings of the experimental literature, subjects who are ranked first in the sequence (variable "first position") contribute significantly more than subsequent players, indicating a strong leadership effect (significant at 1%). The coefficient associated to variable “first position” is not 11

The results are available on request. 28

significant in the sequential treatment without information, confirming the absence of a pure timing effect. Finally table 3 also reveals a significant end game effect in all treatments.

Result 3 indicates the dynamic in the sequence of both the leadership and reciprocity effects over time.

Result 3 : Contributions remain unaffected by the position in the sequential treatment without

information. In contrast, in the sequential treatment with information, the level of contribution declines with the position in the group, indicating a vanishing leadership effect consistent with the weak moral motivation hypothesis.

Support for result 3:

[Figures 3 and 4 about here] Figure 3 shows the average contribution of small groups, by position (rank) in the group, for the two sequential treatments. Figure 4 provides similar information for large groups. Both figures indicate that the average contribution in the sequential treatment with information decreases with the position in the game. In contrast, the average level of contribution in the sequential treatment without information does not vary across positions. Figures 3 and 4 also reveal that the average contribution of the early players in the sequence is higher than the baseline whereas the opposite is true for later players in the sequence. Indeed for small groups, the average contribution of the three first players in the sequence is larger than the average contribution in the simultaneous treatment, but the average contribution of the fourth player is lower than in the benchmark indicating a possible "end-of-sequence effect". Figure 4 shows a similar pattern for large groups: the average contribution of the first six players in the sequence is larger than the average contribution in the simultaneous treatment whereas the average contribution of the last two players is lower than in the benchmark treatment. 29

[Tables 4 and 5 about here]

Further evidence about this leadership effect can be found in tables 4 and 5 which display the average contribution levels by position respectively for small and large groups. In both tables, the second and the fifth columns indicate the overall average individual contribution for each group, respectively for the sequential treatments with and without information. The third and sixth columns give the average individual contribution for the first position in the group. Finally, the fourth and seventh columns provide similar information for the final position of the group. Our data clearly indicate that contributions in the sequential treatment with information are higher in the first position than in the last position. Our finding that the average contribution declines within a sequence according to the rank of the contributor, seems to mimic the general pattern of the decline of the average contribution in a repeated game where contributions are simultaneous. No such rank-effect appears in the sequential treatment without information.

Result 3, illustrated by figures 3-4 and tables 4-5, agrees with propositions 3.1 and 3.2 of our model of weak moral motivation, which predicts a vanishing leadership effect. However we cannot exclude the possibility that reciprocity declines with the rank in the sequence. Indeed, as argued previously, the temptation to free-ride might increase as the observed provision of the public good increases. In order to sort out the reciprocity effet and the leadership effect, we estimate a model of the determinants of contributions (see Table 6). [Table 6 : about here]

30

We take as dependent variable xi the contribution of subject i. The independent variables are subject i’s lagged contribution and the variable “contribution of lower ranked individuals”. We add dummy variables for each position in the group. The variable "position 2" is equal to 1 if subject i is in the second position and equal to 0 otherwise. The other position dummies are defined in a similar way. The results are interpreted in relation with the omitted category (position 1). We also take into account several interaction variables to capture the effect of observing previous contributions in different positions in the sequence. As could be expected from previous results, the position in the sequence has no influence on players’ contribution in the sequential treatment without information. In contrast, the position in the sequence has a negative impact in the sequential treatment with information, for both group sizes. Notice that the values of the coefficients are weaker for early positions than for later positions, indicating that the negative impact gets stronger as the position rank increases.

The above result is compatible both with a vanishing leadership effect and with a declining reciprocity effect, or both. To get some insight about which hypothesis fits best the data, we introduced two additional explanatory variables : "aggregate contribution of lower ranked subjects" and “aggregate contribution of lower ranked subjects × position”12. Introducing these variables does not affect our previous finding concerning the significant decline of contribution with position. The coefficient associated with the variable “contribution of lower ranked individuals” is positive and significant, indicating that subjects are positively influenced by the observation of previous contributions in the rank. However this effect remains constant across position since none of the interaction variables is significant. We conclude that the decrease of contributions with the position in the sequence is not due to a declining reciprocity effect but reflects a vanishing leadership effect, which agrees with our model of weak moral motivation. 12

The estimates, which are not reported here, are available upon request. 31

6. Conclusion

We studied an experimental game of voluntary contributions to a public good, in which players move sequentially. In our test treatment later players can observe the contributions of previous players, while in our control treatments, all players contribute without knowing the contributions of the other players. Our results show that sequentiality without observability does not significantly affect the average level of contribution, compared to the simultaneous contribution treatment, in accordance with Güth et al. (1998), who showed that in a game with a unique equilibrium, the positional order effect is seriously weakened.

Our main experimental results may be summarized as follows : i) the average level of contribution is significantly increased, when subjects contribute sequentially and have the opportunity to observe previous contributions, and ii) the average contribution declines with the position in the sequential contribution game. We showed that result ii) is due to a vanishing leadership effect..

Ou experimental findings are consistent with the predictions of our model of weak moral motivation. Contributions are not purely intrinsically motivated but are conditioned on observed contributions within each sequence and across sequences, in line with earlier findings on conditional cooperation in social dilemma games (Keser & van Winden, 2000). Our model of updating moral motivation provides a rational for conditional cooperation, based on mixing intrinsic motivation together with reciprocal behaviour. The model is also compatible with the so-called leadership effect, highlighted in several experiments on public goods provision. Subjects who decide earlier in the sequence know that their contribution will affect positively the morally ideal contributions of later decision makers. Accordingly 32

they try to encourage them by making a large contribution. As the decision sequence moves toward the last player, implying that fewer players are likely to be influenced, the leadership effect vanishes, and the average contribution declines in the higher ranks of the game. This outcome is both observed in our data and predicted by our model.

We also found that in the sequential treatment with observability, the level of contribution is only weakly influenced by the previous period average contribution in contrast to the treatments without observability. Instead, subjects rely more strongly on the contributions observed in the current sequence. Also, we found that the size of the group does not have a significant impact on contributions. One possible explanation, consistent with our model, might be that the increased leadership effect is crowed out by an increased free-rider effect.

The fact that later contributors are influenced by the observed contributions of early players in sequential games can also have important policy implications. Posting information on previous contributions might therefore be an efficient tool for increasing the level of contributions, suggesting that the design of public policies should take into account the leadership effect. For example, public announcements of previous efforts to reduce emissions might increase society’s overall abatement effort. Clearly, the leadership effect alone is not sufficient to solve the social dilemma arising in voluntary contribution games. Fehr & Gächter (2000) showed that the introduction of costly punishment opportunities provides strong enough incentives to overcome the dilemma. Our results suggest that the same outcome can be reached with less demanding punishment opportunities. Introducing asymmetric punishment opportunities, i.e. early players can only punish later players, might provide enough incentives to increase the level of contributions to the socially optimal level.

33

Appendix

1.Proof of Proposition 5.1 Under Assumption 3.3, one has −1+ β + βθ ∑(n−h ) ≤ 0 , ∀k ≥ 1. k

−1

h =1

(

)

As a result g(k )= v'−1 −1+ β + βθ ∑h =1(n−h ) ≤ 0 , due to the properties of function v(.) explicited in Assumption 3.2. −1

k

The proof will be established recursively. Note first that x1 = x*+g(n−1) ≤ x* because g(n−1) ≤ 0 . Then observe that x2 = xˆ2 + g(n−2) ≤ x1 because on one hand xˆ2 ≤ x* , as a convex combination of x* and xˆ1 ≤ x* , and on the other hand g(n−2) ≤ g(n−1) ≤ 0 since g(.) is an increasing function. Now assume that the property xh ≤ xh −1 holds for h = 3,…,k, for some k. To complete the proof, it must be established that xk +1 ≤ xk . If, as assumed, the property is true until h = k, then the sequence of averages xh decreases with h until h = k. It follows that xˆk +1 , a convex combination of x* and xk , is lower than xˆk , a convex combination of x* and xk −1 > xk . To conclude,

xk +1 = xˆk +1+ g(n−k −1) < xk = xˆk + g(n−k ), because xˆk +1 < xˆk and g(n−k −1) < g(n−k ). QED.

2. Proof of Proposition 3.2 Without Assumption 3.3, the sign of −1+ β + βθ ∑(n−h ) , ∀k ≥ 1. k

−1

h =1

is ambiguous. Therefore g(n-k) could be positive. However, as k increases it finally becomes negative, sing g(.) is a monotonic function and g(0) = v’-1(-1+β ). This means there is an agent j ≤ n such that g(nk) ≤ 0, for all k ≥ j. Starting with this agent j, there are two cases to be considered.

34

The first case is where the previous average is above the moral obligation, xk −1 > x* , k ≥ j. Then xk = xˆk + g(n−k ) < xk −1 , because on one hand xˆk ≤ x k −1 , as a convex combination of x* and x k −1 > x* , and on the other hand g(n−k ) ≤ 0 . To complete the proof, it is sufficient to observe that:

[

xˆk +1 = (1−θ )x*+θ 1 (k −1)x k −1 + xk k

]

= (1−θ )x*+θ xk −1 − θ 1 xk −1 +θ 1 xk k k

[

]

= xˆk + θ 1 xk − xk −1 < xˆk since xk −1 > xk , k as established above. As soon as the average contribution declines, one can reproduce the recursive demonstration of the previous proposition to show that subsequent contributions are lower. The second case, where x k −1 < x* , follows a similar logic. First, xk = xˆk + g(n−k ) < xk −1 , because on one hand xˆk ≤ x* , as a convex combination of x* and x k −1 < x* , and on the other hand g(n−k ) ≤ 0 . Again the average contribution declines, so that the previous reasoning applies to complete the proof. QED.

3. Proof of Proposition 3.3 Here also, the proof is established recursively. It is readily checked that x1 is an increasing function of θ . Under Assumption 3.4 this is also the case for x2. Assume this property hold for xh , h = 3,…,k and let us show it holds also for xk+1. Because

xh = x*+Lh(g(n−1), g(n−2),...,g(n−k )), ∀h ,

it must be true that each function Lh , h≤ k, is an increasing function of θ . Then, by construction

xk +1 = (1−θ )x*+θ x k + g(n−k −1) ,

[

]

= (1−θ )x*+θ 1 kx* + ∑h =1Lh + g(n−k −1) , k k

= x*+θ 1 ∑h=1Lh + g(n−k −1) . k Observing the last line, the result simply follows from the property that each function Lh (.), h≤ k, is an increasing function of θ and g(.) is also an increasing function of θ . QED. k

References 35

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37

Figure 1. Average contribution level per round for N=4 10 9 8

Average contribution

7 6 5 4 3 2 1 0 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Periods

Sequential treatment with information

sequential treatment without information

Simultaneous treatment

Figure 2. Average contribution level per round for N = 8 10 9 8

Average contribution

7 6 5 4 3 2 1 0 1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

Period

Sequential treatment with information

Simultaneous treatment

38

sequential treatment without information

Figure 3. Average Contribution for each position in the game (size N=4) 8

7

6

5

4

3

2

1

0 f ir s t pos .

s d pos .

thir d pos .

P osit ion

Sequenti al wi th i nf or mati on

Sequenti al wi thout i nf or mati on

f our th pos .

Si mul taneous

Figure 4 . Average Contribution for each position in the game (size N=8) 7

6

Average Contribution

5

4

3

2

1

0 first pos.

sd pos.

third pos.

fourth pos.

fifth pos.

sixth pos.

seventh pos.

eighth pos.

Position

Sequential with information Simultaneous

39

Sequential without information

Table 1: Number of independent observations per cell Session Treatment number 1 Simultaneous game 2 Simultaneous game # 3 Simultaneous game # 4 Simultaneous game 5 Simultaneous game 6 Simultaneous game 7 Sequential game with info 8 Sequential game with info 9 Sequential game with info 10 Sequential game with info 11 Sequential game with info 12 Sequential game without info 13 Sequential game without info 14 Sequential game without info 15 Sequential game without info 16 Sequential game without info # simultaneous game with framing

Number of Groups 5 4 4 2 2 2 3 3 2 2 2 4 4 2 2 2

Number of subjects 20 16 16 16 16 16 12 12 16 16 16 16 16 16 16 16

Size of the group 4 4 4 8 8 8 4 4 8 8 8 4 4 8 8 8

Table 2. Group Average Contribution Levels (standard deviation in brackets) Group

Sim 1 2 3 4 5 6 7 8 9 10 11 12 13

4.3# (2.94) 4.15# (2.83) 5.91# (2.33) 2.78# (3.26) 5.25# (2.93) 6.08# (2.52) 2.61# (2.96) 2.5# (2.48) 4.95 (1.92) 2.05 (1.99) 2.96 (3.17) 5.25 (2.47) 4.46 (3.13) 4.09 (2.68)

N=4 Sequ without Sequ with info info 5.4 4.58 (4.57) (3.67) 4.7 6.23 (3.17) (3.11) 4.28 5.15 (4.72) (2.99) 5.93 5.53 (3.06) (3.31) 4.21 5.35 (3.12) (3.30) 4.1 4.68 (3.05) (3.11) 2.28 (2.59) 3.05 (3.03)

4.24 (3.41)

5.25 (3.32)

Sim.

4.84 (3.44) 4.95 (3.38) 3.48 (3.44) 5.26 (3.34) 3.52 (3.61) 3.18 (3.35)

4.20 (3.42)

N=8 Sequ without Sequ info with info 2.65 3.85 (3.43) (3.08) 2.7 3.88 (2.5) (3.85) 3.22 6.025 (2.43) (3.08) 4.28 5.50 (2.91) (4.03) 4.05 4.24 (3.04) (2.22) 3.34 6.53 (2.72) (4.28)

3.37 (2.83)

5.03 (3.42)

# The results show no significant difference at any level of significance in average contribution between the simultaneous treatments with and without framing. All simultaneous treatments with large groups were conducted with framing.

40

Table 3: Random-effects GLS regression of Individual Contribution: Information and Order Effects13 All treatments

Others's average cbt (lagged) Information Sequentiality

0.299*** (0.030) 1.047*** (0.293) -0.272 (0.369)

Treat Sequ. treat. without (sequ with info. and without (simult and info) seq without info) 0.287*** 0.362*** (0.038) (0.035) 1.072*** (0.276) -0.243 (0.270)

Seq*Information

Simult treat and seq. treat with info

Sim. treat.

Seq. without info

Seq. with info.

Seq. with info. Final position N=4

Seq. with info. Final position N=8

0.249*** (0.036)

0.321*** (0.048)

0.404*** (0.053)

0.125* (0.059)

0.087 (0.160)

0.303 (0.1944)

0.214 (0.189)

1.486*** (0.270) 0.256*** (0.031) -0.414 (0.483) -0.927** (0.405) 2.611*** (0.437) 840 (0.12)

0.176* (0.110)

0.222*** (0.086)

-2.088 (1.419) 2.066** (1.042) 84 0.07

-0.700 (1.036) 0.822 (1.172) 84 0.06

0.798*** (0.299)

First position Contribution of lower ranked Size subjects N=4 Round 15 Constant Observations R squared

0.232 (0.230) -1.070*** (0.191) 2.751*** (0.249) 3528 0.08

0.354 (0.281) -0.890-** (0.256) 2.464*** (0.259) 2128 0.1

0.233 (0.269) -1.003*** (0.217) 2.479*** (0.272) 2520 0.1

0.110 (0.296) -1.291*** (0.231) 3.043*** (0.292) 2408 0.1

0.076 (0.395) -1.347*** (0.281) 2.760 (0.358) 1400 (0.1)

0.328 (0.362) -0.566* (0.337) 1.929*** (0.296) 1120 (0.07)

Standard errors in parentheses * significant at 10%; ** significant at 5%; *** significant at 1%

13

Similar results were found using Tobit models that control for censured data. 41

Table 4. Small groups average contribution by position in the game (standard deviation in brackets) Group

Sequential treatment with info All positions

1 2 3 4 5 6

First position

4.58 (3.67) 6.23 (3.11) 5.15 (2.99) 5.53 (3.31) 5.35 (3.30) 4.68 (3.11)

6.33 (3.24) 7.33 (1.87) 6.86 (2.16) 6.33 (2.60) 6.93 (2.93) 4.4 (2.13)

Last position 2.26 (3.43) 4.33 (3.90) 3.4 (3.37) 3.13 (3.15) 2.86 (2.58) 4.13 (3.92)

5.25 (3.32)

6.36 (2.64)

3.35 (3.40)

Sequential treatment without info All positions First position

7 8

5.4 (4.57) 4.7 (3.17) 4.28 (4.72) 5.93 (3.06) 4.21 (3.12) 4.1 (3.05) 2.28 (2.59) 3.05 (3.03) 4.24 (3.41)

5.13 (4.77) 3.8 (3.21) 4.33 (4.77) 5.46 (2.97) 3 (2.92) 5.33 (3.08) 1.53 (1.84) 3 (3.42) 3.94 (3.37)

Last position 6.2 (4.49) 4 (3.11) 4.73 (4.90) 4.93 (3.69) 3.66 (2.46) 4.13 (2.97) 1.8 (3.12) 3.8 (2.73) 4.15 (3.43)

Table 5. Large groups average contribution by position in the game (standard deviation in brackets) Sequential treatment with info

Sequential treatment without info

Groups All positions First position 3.85 4.26 (3.08) (3.41) 2 3.88 6.33 (3.85) (3.19) 3 6.025 8.26 (3.08) (1.83) 4 5.50 7.86 (4.03) (3.02) 5 4.24 4.6 (2.22) (2.55) 6 6.53 6.6 (4.28) (4.70) Average 5.03 6.31 (3.42) (3.11) 1

Last position 2 (2.23) .93 (2.63) 6.2 (3.91) 2.66 (4.23) 2.8 (2.54) 5.53 (4.71) 3.35 (3.37)

42

All position First position Last position 2.65 3.13 2.93 (3.43) (3.96) (3.55) 2.7 2.8 2.53 (2.5) (2.30) (2.79) 3.22 3.46 3.2 (2.43) (2.41) (1.89) 4.28 4.2 4.06 (2.91) (3.12) (3.08) 4.05 2.46 5 (3.04) (2.19) (3.31) 3.34 2.4 2.86 (2.72) (2.5) (2.26) 3.37 3.07 3.43 (2.83) (2.74) (2.81)

Table 6: Random-effects GLS regression of contribution by position in the game N=4 Seq. without Information (1) Others' average cbt (lagged) 0.457*** (0.068) Position 2 0.717 (0.435) Position 3 -0.07 (0.435) Position 4 0.286 (0.435) Position 5

Sequential with Information (3) 0.172*** (0.070) -2.303*** (0.577) -3.319*** (0.569) -4.873*** (0527)

Position 6 Position 7 Position 8 0.298***

Agregate contr. of lower ranks round15 Constant Observations

N=8 Sequential without Information (5) 0.236** (0.093) 0.409 (0.406) -0.254 (0.405) -0.203 (0.407) 0.182 (0.407) -0.387 (0.412) 0.342 (0.405) -0.119 (0.395)

-1.766*** (0.609) 1.971*** (0.471) 448

(0.061) -1.634*** (0.582) 5.744*** (0.533) 336

0.139 (0.395) 2.216*** (0.250) 672

Sequential with Information (7) 0.0700 (0.074) -1.710*** (0.514) -2.148*** (0.503) -1.874*** (0.497) -2.806*** (0.511) -3.081*** (0.486) -4.417*** (0.482) -4.339*** (0.472) 0.160*** (0.036) -0.583 (0.501) 4.938*** (0.311) 672

Standard errors in parentheses, * significant at 10%; ** significant at 5%; *** significant at 1%

43