1 Introduction In standard public goods experiments, a subject participating in a multiple round decision making environment, has to allocate a fixed

Transcription

1 Strength of the Social Dilemma in a Public Goods Experiment: an Exploration of the Error Hypothesis Marc Willinger Λ Anthony Ziegelmeyer Institut Universitaire de France BETA-Theme Université Louis Pasteur BETA-Theme Université Louis Pasteur 61, avenue de la For^et Noire 61, avenue de la For^et Noire F Strasbourg Cedex F Strasbourg Cedex September 29, 2000 Abstract We present the results of an experiment onvoluntary contributions to a public good with a unique dominant strategy equilibrium in the interior of the strategy space. The treatment variable is the equilibrium contribution level. By increasing the equilibrium contribution level, we reduce the strength" of the social dilemma. Though we observe that the average level of contribution rises with the equilibrium contribution level, the average rate of overcontribution is not affected in a systematic way. Over-contribution is statistically significant only at the lower level of equilibrium contribution but not at the higher levels. We showthat the Anderson, Goeree, and Holt (1998) logit equilibrium model which combines altruism and decision errors fits quite well our laboratory data. Keywords: error, logit equilibrium experimental economics, public goods, interior equilibria, altruism, decision JEL Classification: C72, C92, H41 Λ Corresponding author. Tel. : , fax: , willma@cournot.u-strasbg.fr. 1

2 1 Introduction In standard public goods experiments, a subject participating in a multiple round decision making environment, has to allocate a fixed number of tokens between a private and a public activity, and gets a payoff which increases linearly in each activity. Experiments on linear public goods games have shown that subjects tend to allocate a significant number of their tokens to the public activity. This fact contradicts the induced incentives to free-ride, i.e. aself-interested subject has a single stage dominant strategy whereby he allocates all his tokens to the private activity. Given that the number of decision rounds is finite, contributing nothing to the public activity inall periods is the unique subgame perfect equilibrium. This prediction is based on the assumption of common knowledge that players are rational payoff maximizers. Unless otherwise specified, when we refer to dominant strategy", equilibrium contribution/solution" or Nash equilibrium", we shall systematically rely on such an assumption. Observed cooperation in linear public goods games is robust to various treatment variables. Even if some of them (repetition, low marginal values of the public good) reduce sharply the contribution level, cooperation is not completely eliminated (see Ledyard (1995) for a survey). If the single-shot dominant strategy is to contribute no token to the public activity, any individual deviation entails a positive contribution (if negative contributions are not allowed). This design may therefore lead to a bias toward over-provision of the public good. One way to overcome the bias consists in modifying the payoff structure in such away that the equilibrium solution corresponds to partial contribution. Interior equilibrium solutions are obtained by defining a non-linear payoff function, either in private and/or public good. In contrast to experiments based on linear public goods games, over-contribution is less systematic in experiments which have used quadratic payoff functions (for a survey on the evidence from interior- Nash public goods experiments see Laury and Holt (forthcoming)). While Keser (1996), and van Dijk, Sonnemans, and van Winden (1997) found significant rates of over-contribution for payoff functions which are quadratic in the private good, Isaac and Walker (1998) found significant over-contribution only for low (but not for high) levels of equilibrium contribution by relying on payoff functions quadratic in the public good. It should be noted that introducing a diminishing marginal value of the public good, while maintaining the marginal value of the 2

3 private good constant, induces a unique Nash equilibrium for aggregate contribution, but multiple individual equilibria. In other words, there is no longer a single-shot dominant strategy. Sefton and Steinberg (1996) explored the coordination problem, arising from the multiplicityof individual equilibria, by comparing the unique dominant strategy equilibrium treatment with the non-dominant strategy environment. While they found that contributions have a higher variance in the non-dominant strategy environment, they showed that there is no significant difference in terms of levels of contribution between the two treatments. Our experiment has two primary objectives. First, we wanted to test whether overcontribution in the case of an interior dominant strategy equilibrium was robust to parametric changes. We compare the data of four equilibrium solutions: low" equilibrium contribution (L), medium" equilibrium contribution (M), high" equilibrium contribution (H) and very high" equilibrium contribution (VH). We observe significant over-contribution at the low level of equilibrium contribution but not at the higher levels of equilibrium contribution. Our results confirm Isaac and Walker's (1998) observation that moving the equilibrium level of contribution closer to the Pareto optimum, leads to a decrease in average over-contribution. 1 However, in contrast to the results of Isaac and Walker (1998), we did not find, for high levels of equilibrium contribution, that subjects on average tend to under-contribute. Second, we look for a plausible explanation for the diverse behavior patterns observed in our experiment. Though many competing hypotheses have been up to now suggested by the literature in order to explain the lack of free-riding as a dominant strategy in the laboratory (see Holt and Laury (forthcoming) for a survey), we restrict attention on two of them: altruism, and decision errors. 2 Altruism posits that contributions are an expression of concern for the payoff of other people. Decision errors can be provoked by subjects' confusion about the payoff structure or by unobserved preference shocks. Regardless of the source of the decision errors, we shall assume that relatively costly errors are less likely. With respect to our second objective, we show that the Anderson, Goeree, and Holt (1998) logit equilibrium model assuming linear altruism predicts fairly accurately the final levels of contributions observed in our experiment. The paper is organized as follows. Section 2 presents the experimental design and procedures. Experimental results are presented in section 3. In section 4 we investigate whether the error hypothesis is a good candidate for explaining the diverse behavior patterns observed in 3

4 our experiment. We first consider an exogenous formulation of the errors and then we test the predictive ability of Anderson, Goeree, and Holt's (1998) logit equilibrium model. We conclude the paper in section 5. 2 Experimental design We implement an experimental design in order to study the impact of the strength of the social dilemma on the contribution behaviors. As in Keser (1996), our design is based on a quadratic payoff function for the private good, i.e. the game has a dominant strategy solution which lies in the interior of the strategy space. 2.1 The four treatments Each participantwas randomly assigned to a group of four people (including himself) for the 25 periods of the experiment. At the beginning of each period, each subject was endowed with 20 tokens, which he had to allocate between a private and a public activity. 3 The constituent game has a dominant strategy equilibrium where each player contributes a positive number of tokens to the public activity, noted c Λ, which is also the per-period level of contribution predicted by the unique subgame-perfect equilibrium for the 25-fold repetition of the constituent game. Let x i be the number of tokens invested by player i (i = 1; 2; 3; 4) in his private activity and c i =20 x i the number of tokens invested in the public activity. The payoff of player i is given by Y i 0 1 xi ; c j j=1 A =41xi x i 2 + 4X c j ; (1) j=1 where 2 N is the marginal return from investing in the public good. The term 41x i x i 2 is the payoff from the private activity and P 4 j=1 c j is the payoff from the public activity. Each player i chooses the number of tokens to invest in the public good so as to maximize Q i subject to the budget constraint x i + c i =20,given other players' strategies. The payoff function of player i can be rewritten as Y i X xi ; j6=i A = (41 )xi x i X c j 1 4 j6=i c j ; (2)

5 P where c j6=i j is the (positive) externality generated by the other players on the payoff of player i. The level of the externality depends on the value of. There is a unique Nash equilibrium where each player invests x Λ ( ) =(41 )=2 in the private investment and contributes c Λ ( ) =( 1)=2 to the public good. Thesocialoptimum requires that each member contributes 20 tokens to the public activity. 4 In our experiment we compare the level of contribution for four different values of the marginal return from investing in the public good. The corresponding equilibrium contributions to the public good and the relative payoff differences between the social optimum and the equilibrium solutions are given in Table 1. The relative payoff difference reported in column 4 measures the percentage increase in individual payoff by moving from the equilibrium play to the social optimum. Table 1: Experimental design Marginal return Equilibrium Relative payoff difference Equilibrium to the level of between the optimum condition public good ( ) contribution (c Λ ) and the equilibrium Low (L) % Medium (M) % High (H) % Very High (VH) % Our experimental design allows us to compare contribution behaviors in a public goods game where the strength of the social dilemma is modified. Indeed, by moving the equilibrium level of contribution closer towards the Pareto optimum, the strength of the social dilemma is reduced. More precisely, a given deviation from the equilibrium contribution is less costly if the equilibrium condition is high than if the equilibrium condition is low (see Section 4.1 for more details). Incentives to free ride therefore become stronger as one moves towards a lower equilibrium condition. If, on the contrary, rational individuals cooperate to pursue their common interests, reaching the Pareto dominant outcome is easier in the low equilibrium condition than in the higher ones. This is because the relative payoff difference between the Pareto and the equilibrium level of contribution is stronger at lower levels of equilibrium contribution (see column 4 of Table 1). Since the conflict between individual and collective rationality becomes 5

6 weaker when the equilibrium is shifted towards the Pareto optimum, there is no clear prediction about the evolution of cooperation in our setting. 2.2 Practical procedures The experimentwas run on a computer network 5 in winter 1998 using 64 inexperienced students at the BETA laboratory of experimental economics (LEES) at the University of Strasbourg. The subjects were recruited by phone from a pool of 800 students. They were a mixture of economics (about 20% of the pool) and other majors. Four sessions were organized, with 4 groups of 4 subjects per session. A total of 4 independent observations per treatment was collected. Subjects were randomly assigned to a group of four players, to play a 25-fold repetition of the one-shot game, on a computer terminal, which was physically isolated from other terminals. Communication, other than through the decisions made, was not allowed. The subjects were instructed about the rules of the game and the use of the computer program through written instructions (available upon request), whichwere read aloud by a research assistant. Group and individual returns from the private and public activity were presented in tabular form to the subjects. A short questionnaire and two practice rounds followed. Subjects earned points that were converted into French Francs (FF) at the end of the session (at the time of the experiment 1$was slightly less than 6 FF). The number of points accumulated since the beginning of the experiment was on permanent display onthe computer screen and each subject could see his personal token investment and total investments in the public activity for all previous rounds. At the end of a session, each subject was paid privately the total amount earned during the session. Subjects earned a 30 FF show-up fee along with their earnings in the experiment. Table 2 gives the summary of the subjects' payoffs and the conversion rates used in our experiment. Note that with a constant conversion rate, for any given strategy profile such thatthesum of the contributions is strictly positive, subjects would earn more money in the high equilibrium condition than in the other treatments. Therefore, we adjusted the conversion rate in order to keep constant the monetary payoff from the equilibrium strategy at about 2.5 FF per round. Subjects on average earned more money in the low equilibrium condition than in any other of the treatments. The actual playing of the 25 repetitions of the one-shot game took less than 1 hour in each treatment. Note that the payoffs are quite high relative to such a short time span. 6

7 Table 2: Summary of the subjects' payoffs (show-up fee not included) Equilibrium Maximum Mean Minimum Conversion rate condition payoff payoff payoff (for 1000 points) Low (L) Medium (M) High (H) Very High (VH) Experimental results In order to compare the results of the various treatments, we shall use the rate of overcontribution" rather than the level of contribution. The over-contribution rate is defined as the contribution rate if the equilibrium level of contribution had been the minimum contribution (see Section 4.1 for a formal definition). Table 3 contains average over-contribution rates to the public good for period 1, period 25 and for all 25 periods (standard deviations are given in brackets). Table 3: Average over-contribution rates Equilibrium Average over-contribution rate condition in the first period in the last period over all 25 periods Low (L) b % (0.43) % (0.22) a % (0.36) Medium (M) % (0.55) % (0.49) % (0.46) High (H) 5.36 % (0.55) % (0.63) 2.07 % (0.53) Very High (VH) % (0.95) % (0.67) % (0.85) a Significantly different from zero at 5 % level. b Significantly different from zero at 1 % level. We observe that the largest average rate of over-contribution over all 25 periods obtains for the low equilibrium condition. Applying a χ 2 test, we can reject the null hypothesis of no difference between the average level of contribution and the equilibrium solution at the 5 percent level (p < 0:05). In treatments M, H and VH the average level of contribution is not significantly different from the equilibrium solution. This result contrasts with previous experiments on public goods games that have induced an interior equilibrium by specifying a 7

8 declining marginal value for the private good (see Keser (1996), Sefton and Steinberg (1996), and van Dijk, Sonnemans, and van Winden (1997)). Moreover, according to χ 2 tests the initial level of contribution differs significantly from the equilibrium solution only for treatment L. Finally, in none of the four treatments is the last period average contribution significantly different from the equilibrium solution. Our data show that over-contribution is not systematic when the equilibrium solution is to contribute a positive amount of the initial endowment, even if the equilibrium level of contribution is pretty far from the upper boundary. There is especially a sharp contrast between treatments L versus H. While the average rate of over-contribution declines when moving from L to M and from M to H, it increases when moving from H to VH. This is partly due to the very high level of contribution of one group which started in period 1 at almost full contribution (average group level of contribution of tokens) and remained at a very high level afterwards. Furthermore, any small positive deviation from the equilibrium at c Λ =17has avery strong impact on the rate of over-contribution. 6 This particularity of the VH treatment is further illustrated by the frequency distributions of the levels of contributions (see figure 1). In the L, M and H treatments we observe more than two peaks. For the VH treatment we only observe peaks at the equilibrium level and at the social optimum (together, the two levels of contribution represent almost 80 % of all contributions), which entails a higher variance of the rate of over-contribution than for other treatments (see Table 3). Using F-tests, we reject the hypothesis that the variance in the VH equilibrium treatment is the same than in any of the other treatments, at the 5 % level. Over-contribution rates in the VH equilibrium treatment exhibit significantly more variance than in the other treatments. 7 8

9 25% Equilibrium contribution is 7 25% Equilibrium contribution is 10 20% 20% Frequency 15% 10% Frequency 15% 10% 5% 5% 0% Level of contribution 0% Level of contribution (a) Low equilibrium condition (L) (b) Medium equilibrium condition (M) 35% Equilibrium contribution is 13 45% Equilibrium contribution is 17 Frequency 30% 25% 20% 15% 10% 5% Frequency 40% 35% 30% 25% 20% 15% 10% 5% 0% Level of contribution 0% Level of contribution (c) High equilibrium condition (H) (d) Very High equilibrium condition (VH) Figure 1: Observed frequencies of contribution Figure 2 shows the time paths of the average over-contribution rate for the four treatments. For treatments L and M, we observe that the average rate of over-contribution increases in the four initial periods, followed by an un-regular decline for the remaining periods. For the H treatment there is a tendency for the rate of over-contribution to move around the equilibrium level of contribution. We observe a higher variability forthe VH treatment than for the three other treatments. We also observe that the rate of over-contribution curve at the VH equilibrium is not located below the curves that correspond to the other treatments, but intersects each of them at several points. This can partly be explained by the higher variance at the VH equilibrium. As we already mentioned, by restricting the set of over-contribution levels when moving the equilibrium level upwards, deviations from the equilibrium level of contribution affect much more the variance of the over-contribution rate. 9

10 Over-contribution rate 60% 50% 40% 30% 20% 10% 0% -10% -20% Over-contribution rate 60% 50% 40% 30% 20% 10% 0% -10% -20% -30% P e r i o d -30% P e r i o d (a) Low equilibrium condition (L) (b) Medium equilibrium condition (M) Over-contribution rate 60% 50% 40% 30% 20% 10% 0% -10% -20% Over-contribution rate 60% 50% 40% 30% 20% 10% 0% -10% -20% -30% P e r i o d -30% P e r i o d (c) High equilibrium condition (H) (d) Very High equilibrium condition (VH) Figure 2: Time paths of the average over-contribution rate to the public activity Isaac and Walker (1998) observed significant under-contribution at high levels of equilibrium contribution. The question is therefore whether the curve of the average rate of overcontribution for the VH treatment should lie below the corresponding curve for the H treatment. As we already pointed out in the introduction, Isaac and Walker (1998) have a single-shot Nash equilibrium in total contribution. Their equilibrium is compatible with many combinations of individual levels of contributions in a group. Therefore, subjects probably had more difficulties to coordinate their contributions in their design than in ours, which has a dominant strategy equilibrium for the stage game. Taken together, our results clearly indicate that the strength of the social dilemma underlying the public goods game affects the apparent cooperation. Indeed, the average rate of over-contribution is significantly different fromzeroonly in treatment L where the strength of 10

11 the dilemma is the strongest. While an argument based on collective rationality seems to be roughly in accordance with our observed data, hard-nosed game theory cannot explain them. It should be noted that the economic/game theoretic prediction for free-riding is particularly sharp. In the next section, we explore the potential of theories which allow for both altruism and decision errors to explain our diverse observed behaviors. 4 The error hypothesis As already mentioned, incentives to free ride become stronger as one moves towards a lower equilibrium condition. This implies that a theoretical explanation based on exogenous costly errors would predict higher over-contribution rates for the high equilibrium conditions than for the low ones. This argument in combination with altruism is developed in the first part of this section. As our results are in contradiction with an exogenous formulation of errors, we shall consider a model of correlated errors. In this respect, Anderson, Goeree, and Holt (1998) proposed of model of endogenous errors, which allows also for altruism, that nicely fits the data of Keser (1996) and Isaac and Walker (1998). Since our results differ from those of previous public goods experiments with interior equilibria, our data allow an additional test of the predictive ability ofanderson, Goeree, and Holt's (1998) model. We shall show in the second part of this section that the logit equilibrium model of Anderson, Goeree, and Holt (1998) predicts fairly accurately the final levels of contributions of our laboratory data. 4.1 Altruism and exogenous errors In this section we assume that a subject deviates from the equilibrium contribution (c Λ ) only due to a decision error. By assuming exogenous decision errors, we look at the incentives to switch back to the equilibrium contribution level. Under the exogenous error assumption, we derive a na ve prediction" about the relation between the rate of over-contribution and the equilibrium level of contribution. 8 At the equilibrium level of contribution c Λ =( 1)=2, the marginal return from the private activity is equal to the marginal return from the public activity. An equivalent statement is that the marginal per capita return" from investing in the public activity is equal to one. The 11

12 marginal per capita return for subject i (MPCR i ) is defined as the ratio of the marginal return from the public activity ( ) to the marginal return from the private activity (2c i +1) MPCR i = 1+2c i : (3) For a selfish subject, the MPCR measures the strength of the incentives to reallocate tokens from the private to the public activity (and vice-versa) when the allocation is sub-optimal. If his MPCR is smaller than one, the selfish subject has an incentive toinvest more in the private activity and less in the public activity. The smaller the MPCR, the larger the gain from moving one token from the public to the private activity. Let i be the rate of over-contribution of subject i i = c i c Λ ( ) 20 c Λ ( ) : (4) The contribution of subject i can then be written as c i = c Λ ( )(1 i )+20 i and the MPCR i can be rewritten as MPCR i = + i (41 ) : (5) At the equilibrium level of contribution we have i = 0. Any increase of i lowers the MPCR and the incentive tocontribute more to the public good. Furthermore, for any positive 0, i.e. for a given rate of over-contribution, incentives to reallocate tokens from the public to the private activity are lower for larger values of. Since this is true for any positive value of i, under the na ve prediction we expect to observe larger rates of overcontribution for larger values of. This is due to the fact that a given deviation, in terms of over-contribution rate, is less costly if is high than if is low. The relation between the MPCR and the rate of over-contribution is illustrated in figure 3. How would altruism affects the na ve prediction? Considering pure altruism as an alternative explanation for over-contribution does not change qualitatively the prediction. If we introduce a parameter ff>0 that represents an individual's utility weight for the payoff of the other players of his group, the equilibrium contribution becomes c Λ ( ; ff) =( (3ff +1) 1)=2. 9 We can measure the increase in the equilibrium level due to altruism, in relative terms, by 12

13 1 0,9 0,8 0,7 0,6 Low equilibrium condition High equilibrium condition Medium equilibrium condition Very High equilibrium condition MPCR 0,5 0,4 0,3 0,2 0,1 0 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Overcontribution rate Figure 3: Relation between the MPCR and the over-contribution rate c Λ ( ; ff) c Λ ( ) : (6) 20 c Λ ( ) As one can see this ratio increases with which implies that pure altruism reinforces the na ve prediction that a higher level of induces higher over-contribution rates. The predictions of the na ve error model, with or without altruism, stand in obvious opposition with the experimental data reported in Section Altruism and endogenous errors: Anderson, Goere, and Holt's (1998) model Anderson, Goeree, and Holt (1998) (hereafter, AGH) consider an equilibrium error model which can simultaneously account for (pure) altruism. Each individual contribution level is chosen with a probability that depends on its expected payoff; low expected payoff contribution levels are less likely to be chosen than high expected payoff contributions, which means also that large errors with respect to the equilibrium level of contribution are less likely than small 13

14 errors. Altruism is introduced in the model through a weighting parameter ff. If ff = 0 the agent's utility does not depend on the other agent's payoff, while if ff > 0 he takes into account others' payoffs in his utility. The logit equilibrium for the quadratic public goods game predicts that the mean contribution lies between half the endowment and the (altruisminclusive) Nash equilibrium. Anderson, Goeree and Holt note: `The logit equilibrium model is not explicitly dynamic, and therefore, it does not explain features of laboratory data that arise because of interactions across periods, :::' (1998, p. 300). In this respect, they conjecture that their predictions should be more consistent with laboratory data coming from strangers" experiments or from the last period of a partners" experiment. We therefore restrict the comparison of our laboratory data with the predictions of AGH's model to the last period (period 25) for each treatment. The predictive abilityofagh's theory is first measured by considering the case without pure altruism. Setting ff = 0,the interval predicted by the logit equilibrium model contains 75%, 25%, 50% and 0% of our experimental data respectively in the low, medium, high and very high equilibrium condition. Although the model with ff = 0 seems acceptable for treatment L, itis less accurate for the other treatments. Therefore we consider strictly positive values of ff that can fit our data. Incorporating pure altruism would improve significantly Anderson, Goeree, and Holt's (1998) model predictive ability in treatment VH because adding altruism would move upward the upper level of the predicted interval (the average contributions in period 25 are equal to 17.5, 18.5, 17.5 and 18.5 for the four groups in the VH treatment). Nevertheless, at the same time as the predictive ability increases, the parsimony ofagh's theory decreases as well. We therefore apply Selten and Krischker's (1983) measure of predictive success, which trades off the predictive parsimony of a theory against its descriptive power (see also Selten (1991)). The measure of predictive success (S) is the difference between the hit rate h and the area a (S = h a), where the hit rate is the relative frequency of correct predictions and the area is the relative size of the predicted subset compared to the set of all possible outcomes. We also compare AGH's model with a model of pure altruism which does not take into account decision errors. At the individual level, AGH's theory is a distribution theory, i.e. a theory which determines a probability distribution over the space of contributions. If we consider the mean contribution 14

15 then it becomes an area theory, i.e. a theory which specifies a subset of predicted contributions. Thus, in each treatment, the measure of predictive success of AGH's theory is the difference between the relative frequency of correct predictions of the mean contribution for each group, i.e. the hit rate, and the size of the interval in which such meancontributions should lie, i.e. the area. We compare the predictions of AGH's theory with the predictions of a theory which does not account for decision errors. More precisely, we consider a very simple model which predicts either the Nash equilibrium if no altruism is present (ff = 0) or a higher equilibrium level if there is positive altruism (ff >0). Such a theory is a point theory, i.e. it predicts a single level of contribution. Thus, for each treatment, the measure of predictive success of this theory is the difference between the relative frequency of correct predictions of the individual contributions (the hit rate), and 1/21 which is the size of the interval" where such contributions should lie (the area). The results for both theories are summarized in Table 4. Table 4: Calculation of Selten and Krischker's measure of predictive success for the Anderson, Goeree, and Holt (1998) model and a model without decision errors Equilibrium Highest measure of predictive Highest measure of predictive success condition success for AGH's model for a model without decision errors Low (L) S =0: = 0:56 S =0: :26 (ff =0) (ff =0:1333) Medium (M) S =0: = 0:65 S =0:44 = :39 (ff =0:0317) (ff =0) High (H) S =0: = 0:51 S =0:31 = :26 (ff =0:0247) (ff =0) Very High (VH) S =1: = 0:52 S =0:44 = :39 (ff =0:0381) (ff =0orff =0:0571) In order to compare the predictive abilities of both theories, AGH's equilibrium error model and a model without decision errors, we computed for each theory the highest measure of predictive success for each treatment. This means that we took, foreach treatment, the value of the weighting parameter ff which maximizes Selten and Krischker's measure of predictive success. Table 4 shows that AGH's theory always has a better predictive success than the point theory. For example, in the very high equilibrium condition, the standard equilibrium solution 15

16 (ff = 0) has a measure of predictive success equal to The same measure is obtained with positive altruism equal to ff =0:0571 in which case the equilibrium level is equal to 20. This is due to the fact that the majority oftheindividual contributions were in the last period of the VH treatment either equal to 17 or 20. AGH's theory has a larger measure of predictive success for ff = 0:0381, because, with such alevel of altruism, all last period's average contributions lie in the predicted interval ([10, 19]). Thus, even by subtracting an area of 10/21 from the hit rate equal to 1, the measure of predictive success of AGH's theory remains greater. Our comparison of the two theories' predictive success makes us sensitive to the fact that `::: endogenous errors do not cancel out, but rather lead to a systematic bias as compared to the Nash equilibrium. This underscores the importance of formal modeling of errors.' (Anderson, Goeree, and Holt (1998), p. 317). 5 Conclusion We designed a public good experiment tostudy the impact of the equilibrium level of contribution on the subjects' contributions to the public good. The major finding of our experiment is that over-contribution is not a systematic outcome when partial contribution to the public good is the single-shot dominant strategy. We observe significant over-contribution at the low level of equilibrium contribution but not at the higher levels of equilibrium contribution. We also looked for a plausible explanation for the diverse behavior patterns observed in our experiment. Several attempts have been made recently in order to distinguish between different theoretical explanations for observed over-contribution in public goods experiments (Palfrey and Prisbrey (1997), Levine (1998), Fehr and Schmidt (1999), Bolton and Ockenfels (2000), Brandts and Schram (forthcoming), among others). While all these models predict overcontribution, in general they do not provide a precise prediction about the level of contribution, especially for quadratic individual payoff functions. 10 By contrast, the model proposed by Anderson, Goeree, and Holt (1998) provides fairly precise predictions, which fit nicely the data of Keser (1996) and Isaac and Walker (1998). Since our results differ from those of previous public goods experiments with interior equilibria, our data allowed an additional test of the predictive ability of Anderson, Goeree, and Holt's (1998) model. We showed that the logit 16

17 equilibrium model of Anderson, Goeree, and Holt (1998) predicts fairly accurately the final levels of contributions of our laboratory data. Holt and Laury (forthcoming) survey several dynamic models of evolution and adjustment, which were proposed by theorists to explain the inconsistencies between public goods experiments results and standard game theory. They conclude that key elements like altruism, error, learning, endogenous preferences, and signaling are certainly present in the laboratory and that a good model will select the most important of these factors and ignore the others. It seems that the Anderson, Goeree, and Holt (1998) model has managed such a task. Nevertheless, the computation of Selten and Krischker's measure of predictive success for each treatment showed that different values of ff had to be considered in order to maximize this measure. AGH's theory does not account for such a result, which suggests that (pure altruism) is equilibrium level-dependent". Moreover, as previously mentioned, the logit equilibrium model only helps understanding the end behavior of the public goods game. We therefore definitely need a dynamic theory that accounts for the evolution of individual contributions over time. 11 Acknowledgments We are grateful to Simon Anderson, Guillaume Haeringer, Jully Jeunet, Nicolas Jonard, Frédéric Koessler, two anonymous referees and one editor for helpful comments on preliminary versions of the paper. Also gratefully acknowledged are numerous discussions with the participants of the following conferences, at which earlier versions of this paper were presented: the Economic Science Association annual meeting, Lake Tahoe, May 1999; the Ninth International Conference on the Foundations and Applications of Utility, Risk and Decision Theory, Marrakech, June 1999; and the 16 emes Journées de Micro-Economie Appliquée, Lyon, June Our special thanks to Gary Charness who suggested that we add the VH treatment in our experimental work. 17

18 Notes 1 Andreoni (1993) and Chan, Godby, Mestelman, and Muller (1998) also showed that by moving the lower boundary towards the Nash equilibrium level significantly decreases the rate of over-contribution. 2 It should be noted that some recent papers have tried to disentangle both explanations (see, e.g., Andreoni (1995) and Palfrey and Prisbrey (1997)). 3 Only entire tokens could be handled. 4 To obtain this result must be strictly larger than Based on an application developed by BounMy (1998) designed for Visual Basic. 6 Indeed, in the VH treatment, the rate of over-contribution can only take three different values, 1/3, 2/3 and 1, corresponding to levels of contribution 18, 19 and 20 respectively. A unit of deviation above the VH equilibrium increases the rate of over-contribution by 1/3, while the same deviation leads only to an increase of 1/13 at the L equilibrium, 1/10 at the M equilibrium and 1/7 at the H equilibrium. 7 Note that while the VH equilibrium treatment exhibits a higher variance in terms of rates of over-contribution it has a lower variance in terms of levels of contribution. 8 Although exogenous errors may also induce under-contributions, we do not address this issue because of the low frequency of under-contributions (see figure 1). 9 We assume, for the sake ofcomparing the na ve model" with the model of Anderson, Goeree, and Holt (1998), that altruism is captured by aunique parameter. So, the utility of a risk-neutral player is given by U i = Q P Q i + ff j6=i j. 10 In fact, in the latter case, the qualitative predictions can be in contradiction with our observed data. For example, in our framework, the Fehr and Schmidt model predicts that it is more likely to observe larger rates of over-contribution when the marginal return from investing in the public good increases (proof available upon request). This prediction contradicts our data, which show a decrease in the rate of over-contribution when the marginal return from investing in the public good increases from 15 to 21 and from 21 to For an attempt in this direction see Willinger and Ziegelmeyer (1999). References Anderson, S., J. Goeree, and C. Holt (1998): A Theoretical Analysis of Altruism and Decision Error in Public Goods Games," Journal of Public Economics, 70, Andreoni, J. (1993): An Experimental Test of the Public Goods Crowding-Out Hypothesis," American Economic Review, 83,

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