Relative performance of two simple incentive mechanisms in a public goods experiment

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Relative performance of two simple incentive mechanisms in a public goods experiment

Juergen Bracht, Charles Figuieres, Marisa Ratto

To cite this version:

Juergen Bracht, Charles Figuieres, Marisa Ratto. Relative performance of two simple incentive mechanisms in a public goods experiment. Journal of Public Economics, Elsevier, 2008, 92 (1-2), pp.54-90.

�10.1016/j.jpubeco.2007.04.005�. �hal-02658682�

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Relative Performance of Two Simple Incentive Mechanisms in a Public Goods Experiment

^∗

Juergen Bracht^a, Charles Figuières^b and Marisa Ratto^c

aDepartment of Economics, University of Aberdeen Business School, U.K. juergen.bracht@abdn.ac.uk bCorresponding author. INRA(UMR-LAMETA), 2 place Viala 34060 Montpellier Cx 1, France.

Þguiere@ensam.inra.fr Phone : +33 4 99 61 22 09 Fax : +33 4 67 54 58 05 cPolicy Studies Institute, London, U.K. m.ratto@psi.org.uk

April 2007

Abstract

We compare the performance of two incentive mechanisms in public goods experiments. One mechanism, the Falkinger mechanism, rewards and penalizes agents for deviations from the average contributions to the public good (Falkinger mechanism). The other, thecompensation mechanism, allows agents to subsidize the other agents’ contributions (compensation mechanism). It is found that both mechanisms lead to an increase in the level of contributions to the public goods. However, the Falkinger mechanism predicts the average level of contributions more reliably than the compensation mechanism.

Keywords: public goods, voluntary provision, incentive mechanisms.

JEL: H42, D62

∗We are grateful to the Leverhulme Trust for funding this project through CMPO. We thank Nick Feltovich, Simon Gächter, and two anonymous referees for valuable comments.

We would also like to thank Ken Binmore, Leslie Einhorn, Rosemarie Nagel, and participants at the CMPO seminar, the LAMETA seminar, the Universitat Pompeau Fabra microeco- nomics seminar, the ESRC research seminar in game theory, the International Meeting of Economic Science Association in Amsterdam, the 16th International Conference on Game Theory, and the Annual Meeting of the European Economic Association in Amsterdam for helpful discussions.

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1 Introduction

How does society design incentive systems for the production of public goods?

In such settings, the total provision generally falls short of the level required by eﬃciency because people attempt to free ride on each other’s contribution. To overcome this, theorists have designed suitable mechanisms to align individual incentives to collective interest. At the same time, laboratory experiments have continuously helped both to improve our understanding of free rider behavior and to test the eﬀectiveness of the theoretical solutions.

With a view towards actual implementation, the goal of this paper is to evaluate the relative strengths and weaknesses of two mechanisms proposed in the theoretical literature to resolve the free rider problem. One mechanism rewards and penalizes agents for deviations from the average contributions to the public good. We refer to thistax-subsidy mechanism proposed by Falkinger (1996) as theFalkinger mechanism. The other mechanism introduces a pre-play stage in which agents have the option of subsidizing each other’s contributions.

We refer to this two-stage game as thecompensation mechanism.¹

Both mechanisms are capable of increasing the level of contributions towards the eﬃcient level of the public good and share the property of simplicity.

Therefore, both could probably be applied successfully to real-life situations. In addition, neither mechanism maintains that the designer has perfect knowledge of agents’ preferences.

The important next step is to evaluate how they perform empirically. This assessment is the paper’s objective. We put the two types of mechanisms in the very same setting, and we conduct laboratory experiments to compare their effectiveness. Each mechanism has already been tested independently.² Andreoni and Varian (1999) assess the compensation mechanism in a prisoners’ dilemma.

Without the mechanism, only one-third of the subjects choose to cooperate.

The introduction of a subsidy stage increases cooperation from one-third to two-thirds. Falkinger et al. (2000) study the impact of the Falkinger mechanism in a small universe of public goods environments, varying both group size and payoﬀfunctions. The mechanism brings about an immediate and large shift towards the eﬃcient level of the public good. Equilibrium is a good predictor of behavior under the mechanism.

The originality of our paper is twofold. First, we test experimentally the perhaps simplest version of the compensation mechanism in a public goods framework.³ This game is a bit more complicated than the prisoners’ dilemma

1The term was introduced by Varian (1994a, 1994b) who builds on earlier works by Guttman (1978, 1985, 1987), Moore and Repullo (1988), and Danziger and Schnytzer (1991).

2The results of both tests are encouraging in light of the disappointing performances reported by some experimental papers for the Groves-Clarke, the Groves-Ledyard, and the Walker mechanisms (see respectively Arifovic and Ledyard (2004), Attiyeh et al. (2000), Harstad and Marrese (1981), and Harstad and Marrese (1982)). However, Chen and Plott (1996), Chen and Tang (1998), and Healy (2006) show that the performance of the Groves- Ledyard mechanism is very good when the punishment parameter is above the supermodu- larity threshold.

3See Danziger and Schnytzer (1991).

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used by Andreoni and Varian (1999). But our contribution is also (modestly) theoretical: our framework does not fulÞll the suﬃcient conditions for the existence of an equilibrium set forth by Danziger and Schnytzer (1991). Hence, we establish existence. Next, we run the two mechanisms in the same framework, choosing the simplest experimental game that allows comparison of the two mechanisms, a two-player game similar to the one studied in Falkingeret al.

(2000). Unlike Falkingeret al., we vary the value of the tax-subsidy parameter, with a view towards testing how the performance of the mechanism depends on this value. We adopt a value required by efficiency (that gives rises to multiple equilibria), a value very close to the efficient level (that guarantees uniqueness of equilibrium), and a value well below the efficient level (that assures uniqueness of equilibrium, too).⁴

Our mainÞndings are as follows.

(i) Both mechanisms provide more public goods than voluntary contribution games do.

(ii) The Falkinger mechanism with a low-value tax-subsidy parameter provides the same amount of contributions as the compensation mechanism, while the Falkinger mechanism with high-value tax-subsidy parameter provides a greater amount of contributions than the compensation mechanism.

The rest of the paper is organized as follows. Section 2 describes the public goods economy we have reproduced in the laboratory, and introduces the two types of incentive mechanisms. Section 3 presents the experimental design.

Section 4 describes the results. Section 5 concludes.

2 A public goods experiment with quadratic util- ity

The economic situation reproduced in the laboratory is as follows. Two agents i= 1,2 are endowed with an exogenous incomeyi, which they can divide between the consumptionci of a composite private good and a contributiongi to the production of a public good G. The production technology for the public good takes the simplest form : G=g1+g2. Whileciis enjoyed by agentionly, the public goods nature of G means that both agents beneÞt from it. Thus agenti’s preferences are typically represented by utility functions of the form:

Uⁱ(c_i, G), i= 1,2. (1)

In the laboratory, we have endowed the subjects with quasi-linear quadratic reward functions:

Uⁱ(ci, G) =Mici−1

2Nic²_i +G , Mi, Ni>0, i= 1,2.

4Note that the design allows us to compare our results to Chen and Gazzale (2004).

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Note that the functionsUⁱare concave, increasing in public goods consumption and, forc_i∈[0, M_i/N_i],non-decreasing in private goods consumption.

The reason for this particular choice of utility functions is twofold. First, there is a need to keep the framework as simple as possible to ensure that subjects come to a good understanding of the link between the proÞle of decisions and their monetary earnings. Second, the framework has to be relevant to the theoretical properties of the two mechanisms that we want to test experimentally, namely, their ability to provide the subjects with the correct incentives to take eﬃcient decisions, without requiring any knowledge of their preferences.

Since the compensation mechanism can be applied to a large set of situations, including those for which the Falkinger mechanism was designed, logically we are limited only by the requirements of the Falkinger mechanism. Most of the public goods games used in experiments are games with linear payoﬀs, i.e., games with the simplest conceivable framework, in which corner decisions are dominant strategies. Falkinger’s mechanism can be applied to such games.⁵ But that would require the designer to know agents’ preferences. If the am- bition is to challenge asymmetric information issues where the designer does not know agents’ preferences, then the mechanism does not have any advantage over more traditional Pigovian tax-subsidy schemes. By contrast, in public goods frameworks, where agents undertake interior decisions, i.e., strictly positive contributions, the mechanism can Nash-implement an allocation arbitrarily close to eﬃciency, with the requirement that the regulator knows only the number of agents involved in the problem, and not their preferences. We shall come back to this in Section 2.2.1.

With quasi-linear quadratic utility functions, the Nash equilibrium consists of interior contributions. Quasi-linear quadratic functions have the additional advantage that the marginal utility from public goods consumption is constant.

Hence, generally, the Nash equilibrium involves dominant strategies. Therefore, subjects will easily be able to deduce the equilibrium strategies (before the introduction of any mechanism).⁶

2.1 Free rider problem

How much income will agents contribute voluntarily to the public good? To answer this question, theorists suggest focusing on non-cooperative decisions that form a Nash equilibrium, whereby each agent optimizes her utility, taking as given the other agents’ decisions. Under this behavioral assumption, each agent’s problem takes the form of

ci≥max0, gi≥0 Uⁱ(c_i, G)

5Actually, it was applied to public goods games with linear payoﬀs in Falkinger et al.

(2000), in particular in theM1, M2andM3treatment sessions.

6To our best knowledge, only four published articles use this speciÞc quadratic framework with interior dominant strategies for the purpose of experimentation. They are Sefton and Steinberg (1996), Keser (1996), Willinger and Ziegelmeyer (1999), and Falkinger et al. (2000).

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s.t.

½ ci+gi =yi

G=gi+gj, gj given,

which, for an interior solution, leads to theÞrst-order condition: MRSⁱ= ^U_U^Gⁱi

c =

1. For our quadratic example, theÞrst-order conditions of the two agents are

½ −M1+N1(y1−g1) + 1 = 0,

−M2+N2(y2−g2) + 1 = 0.

Solving these equations, the Nash equilibrium quantities are g_i^N= 1−M_i

Ni

+y_i, c^N_i = M_i−1 Ni

, G^N=y₁+y₂+1−M₁ N1

+1−M₂ N2

.

For later use, the equilibrium utilities are Uⁱ¡

c^N_i , G^N¢

=M_iM_i−1 Ni −1

2N_i

µM_i−1 Ni

¶2

+y₁+y₂+1−M₁ N1

+1−M₂ N2

. As far as Pareto eﬃciency is concerned, Samuelson’s condition requires MRS¹+M RS²= 1. Overall, this last condition and the feasibility condition

form the system: ½ P

i 1

Mi−Nici = 1, P

icⁱ+G=P

iyⁱ, or, equivalently,

½ M1 − N1c1+M2 − N2c2= (M1 − N1c1) (M2 − N2c2) , c1+c2+G=y1+y2.

Samuelson’s condition differs from the Nash equilibrium condition, since the Nash equilibrium is generally not Pareto-efficient. The Nash equilibrium provides less public goods than the Pareto-efficient level.

2.2 Nash implementation with well-informed agents

This section describes two simple theoretical mechanisms designed to Nash- implement Pareto optimal decisions in situations where agents know each other’s preferences, but, of course, the designer does not.

2.2.1 Falkinger mechanism

This mechanism modiÞes each agent’s budget constraint by rewarding (penal- izing) contributions over (under) the mean of the other agents’ contributions.

To do so, the designer needs to choose a single parameter: a tax-subsidy rate β ∈ [0,1[⁷. Under this new institutional framework, the problem of agent i becomes

max

cⁱ≥0, gi≥0 Uⁱ(c_i, G)

7Withβ= 1there cannot be interior decisions; besides, ruling out this value also guarantees the second order condition.

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s.t.





ci+gi=yi+β(gi−gj) , β≥0 G=gi+gj

gj given.

Note that this mechanism is balanced, whatever the contributions, since the amount that an agent receives corresponds exactly to the amount that the other agent pays. TheÞrst-order conditions for interior solutions are

½ (β−1)M₁−(β−1)N₁[y₁−g₁+β(g₁−g₂)] + 1 = 0, (β−1)M2−(β−1)N2[y2−g2+β(g2−g1)] + 1 = 0.

Solving this system, oneÞnds a solution forGandc_i, conÞgured byβ. The nice feature of this mechanism is that in a public goods game involving n agents, with a value for β arbitrarily close to ⁿ⁻_n¹ (here β ' 1/2 since n = 2), the resulting equilibrium will be arbitrarily close to eﬃciency.⁸

Indeed, lettingβ→1/2these solutions tend to G^f= 2−M₁

N1

+2−M₂ N2

+y₁+y₂

c^f_i = Mi−2 Ni

, i= 1,2.

The equilibrium quantities fulÞll Samuelson’s condition for eﬃciency. The resulting limit utilities are

Uⁱ³ c^f_i, G^f´

=M_iMi−2 N_i −1

2N_i

µMi−2 N_i

¶2

+y₁+y₂+2−M1

N₁ +2−M2

N₂ , i= 1,2.

Clearly, the only piece of information needed by the regulator to compute the critical β is the number of agents. No information about the preferences is required. By contrast, in a linear environment, the payoﬀs would be

Uⁱ(c_i, G) =M_ic_i+G , M_i >0, i= 1,2.

Let the parametersM_i be such that1< M_i<2, i= 1,2. Under these parameters, the voluntary contribution equilibrium is characterized by complete free

8In our quadratic environment with two players, with β = 1−1/n = 1/2, the Nash equilibrium is not unique. For instance in the symmetric case, whereMi=M, Ni=N, yi= y, β= 1/2, any pair(g1, g2)such that

−1 2M+1

2N

! y−1

2(g1+g2)

"

+ 1 = 0

is a Nash equilibrium. See Falkinger (1996) for a discussion of uniqueness in a quasi-concave environment with strictly normal goods.

In a quasi-concave environment with strictly normal goods, the eﬃcientβ^∗= 1/(1−n)can be implemented as an interior equilibrium if the population is partitioned into subgroups (see Falkinger 1996, proposition 2). Uniqueness is discussed in Kirchsteiger and Puppe (1997).

In the commonly used quasi-linear environment in which the private good is the numeraire, the existence of a Nash equilibrium under the Falkinger scheme in which both decisons are interior is problematic (see Kirchsteiger and Puppe 1997, pp. 497-498).

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riding, i.e., g^N_i = 0, i= 1,2, while Pareto eﬃciency requires that both agents contribute their full endowment to the public good,g_i^P =y_i, i= 1,2. With the introduction of the tax-subsidy scheme, the Þrst-order conditions would lead the agents to eﬃciency if

½ (β−1)M₁+ 1>0, (β−1)M₂+ 1>0,

or, equivalently, ifβ >(M_i−1)/M_i, i= 1,2. In other words, the optimalβis given by any value such that

β >max

½M1−1

M₁ ,M2−1 M₂

¾ .

It follows that the regulator now needs to know the preference parameters, and the mechanism is less attractive.

2.2.2 Compensation mechanism

Several versions of the compensation mechanism exist in the literature. We have chosen the simplest.⁹

First, agents voluntarily subsidize the other agents’ contributions, i.e. agent ioﬀerss_ig_j to agent j (withs_i ∈[0,1]). Then, given this proÞle of subsidies, agents contribute to the public good.

It is very similar to the mechanism introduced by Danziger and Schnytzer (1991), except that we have ignored a difficulty that may arise when, given the subsidy rate and the contribution chosen by the other agent, agenti’s own choice would entail that she spends more than her endowment on the public good. This happens, for instance, when agent 1 sets her subsidy rate close to one and agent 2 chooses a contribution larger than agent 1’s income. To overcome this difficulty, Danziger and Schnytzer (1991) suggested completing their mechanism with any rule that, in such situations, generates a feasible proÞle of choices (see Danziger and Schnytzer 1991, p. 58). In the laboratory, one mayi)use such a rule to avoid theoretical drawbacks, but at the risk of confusing the subjects when they play an already quite complicated mechanism, orii)ignore this difficulty altogether, and delete afterwards those observations when one agent went bankrupt; this would make the subjects’ decision context easier to understand, but at the risk of collecting many worthless observations. We were content with the second possibility. It turned out that, out of 600 observations under the compensation mechanism, there were only 7 incidences of bankruptcy. In the experiments, whenever agents went bankrupt, we subtracted their losses from their show-up fee, and allowed them to play on.

Under this simple compensation mechanism, a subgame-perfect Nash equilibrium is found by solving this game backward. Formally:

9More sophisticated forms of this mechanism add penalization terms (see Varian 1994a).

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• Each agent’s maximization program in the second stage is

ci≥max0, gi≥0 Uⁱ(c_i, G)

s.t.





ci+ (1−sj)gi=yi−sigj

G=gi+gj

s_i, s_j, andg_jgiven.

It is worth noting that adding up the individual budget constraints, for any combination of contributions and subsidies, one hasci+cj+gi+gj = y_i+y_j; this version of the mechanism meets the balanced aggregate budget requirement both in and out oﬀequilibrium. Lets= (s₁, s₂)stand for the proÞle of subsidies and letc_i(s), G(s)denote the solutions to the above program; indirect utilities are

Vⁱ(s) =Uⁱ(c_i(s), G(s)).

• Moving backward to the Þrst stage, the subsidy decisions solve for each agent

simax∈[0,1] Vⁱ(s), sj given.

Danziger and Schnytzer (1991) and Varian (1994b) have established that, under reasonable assumptions, a subgame-perfect Nash equilibrium for public goods games with a subsidy stage exists and replicates a Lindahl equilibrium.

In symmetric games, this means thatsi =sj = 1/2. In our symmetric game, this would further imply thatcⁱ=c^j= 30, gⁱ=g^j = 20. A Lindahl equilibrium is Pareto optimal.¹⁰

However Varian’s mechanism is slightly diﬀerent from the one described above,¹¹and Danziger and Schnytzer’s assumptions are not met by our quadratic economy: they require that limci→0∂Uⁱ/∂G

∂Uⁱ/∂ci = 0,∀i. Therefore, the existence of a subgame-perfect Nash equilibrium should be ascertained for the game we have reproduced in the laboratory. Even in this particular example, such an exercise is of some theoretical interest. Indeed, the exercise helps to clarify that Danziger and Schnytzer’s assumptions are suﬃcient. But they are not necessary to guarantee their results.¹²

Lemma 1 Let Mi = 5, Ni = 1/10 and yⁱ = 50. With these numerical values, any subgame-perfect equilibria of the compensation game involves a strictly positive level of public goodG >0andsi+sj = 1.

1 0A Lindahl equilibrium is also individually rational. Falkinger’s mechanism does not generally possess this property, though in our symmetric context both agents are better-oﬀafter introducing this mechanism.

1 1In Varian’s mechanism there is an additional penalty term and both the subsidy and the tax that each agent faces are chosen by the other agent.

1 2For a related discussion concerning the existence of subgame perfect equilibria in this public good economy see also Althammer and Buchholz (1993).

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Proof. See Appendix A.

Proposition 2 For the above public goods game with the compensation mechanism, the Lindahl allocation is implemented as a subgame-perfect Nash equilibrium; it is made of a unique pair of subsidy rates,si=sj= 1/2, a unique level of public good, G = 40, but a continuum of individual contributions such that gi+gj = 40.

Proof. See Appendix B.

The intuition for multiple individual contributions outlined in Danziger and Schnytzer (1991) is illuminating. When 1−sj = si, the prices of direct contributions to the public goods viagi and indirect contributions via gj are the same. Agentiis thus choosing eﬀectively whether the aggregate level of public good to consume should be higher than, or equal togj. When the Lindahl level is aﬀordable, it is optimal to choose it. Formally, agenti’s reaction function is

gi= max (40−gj,0). (2) The graphs of the agents best-response functions coincide forg₁+g₂= 40.

Clearly, the existence of a continuum of equilibria is a weakness of this implementation concept, for a coordination problem might prevent the agents from playing their equilibrium strategies. On the other hand, in this symmetric game, the symmetric equilibrium whereg1=g2= 20is a focal point.

3 Experimental design

The aim of our experiments is twofold. First, we test the behavioral response of subjects who have experienced free riding to the introduction of both the Falkinger mechanism and the compensation mechanism. Second, we compare how behavior differs between the two mechanisms. We run one experiment for each mechanism. Participants play two games: a voluntary contribution game for twenty rounds (without knowledge of the form of the game played thereafter), and a second game with either mechanism for other twenty rounds. We refer to the Þrst game as ”control” and the second game as ”treatment”.¹³ For the Falkinger mechanism, we adopt three values of the tax-subsidy parameter to test how the performance of the mechanism is affected by the value. We consider the efficient value, β^∗ = 1/2, the closest value to the optimalβ^∗ that gives an integer solution,β= 9/19, and a lower value,β= 1/3.

Thirty subjects took part in the experiment with the compensation mechanism and eighty-four subjects participated in the experiment with the Falkinger mechanism (thirty subjects withβ= 1/2in three sessions, thirty subjects with

1 3Subjects behaved in the same strategic situation before taking part in the treatment game.

Hence, differences in contributions between the mechanisms can not be attributed to different types of situations in the control phase of the experiments. In addition, we check whether actual contributions in the control associated which each experiment are different. We do not Þnd substantial differences (see Section 4.1). Hence, we could assume that subjects responded to the control in a similar way.

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β= 9/19 in three sessions and twenty-four subjects with β = 1/3in two sessions). Subjects were recruited from the student population of the University of Bristol and the University of Aberdeen, mostly from the faculties of human- ities and social sciences. Subjects were divided into pairs and informed that they would always play against the same opponent in the experiment but would never learn her identity. The instructions were read aloud. Subjects had toÞll in a questionnaire to check their understanding of the rules before the start of play.¹⁴ At the end of a session, points earned were converted to pounds (at the rate100P oints=£0.125) and added to the show-up fee of£2.50. The average payment was£10.00. Each session lasted somewhat less than two hours.¹⁵

At the beginning of each round, each participant receives50points as endowment and has to allocate the endowment between two activities, one beneÞcial to both players (activity A), and the other one beneÞcial only to the donor (activity B). Subjects are asked to decide how many points they want to contribute to activity A. Their choices automatically determine the contribution to B. The subjects’ contributions determine the points (income) earned from both activities. Total income is the sum of the income from activity A and B. At the end of each round, subjects are informed about the contribution of the other group member to activity A and their own income from activity A and B. Income from activity A results from the sum of both group members’ contributions to A, whereas income from activity B results from the following formula:

Income from B= 5×contribution to B− µ 1

20

¶

×(contribution to B)² (3) This formula corresponds to the utility function presented in Section 2:

Uⁱ(ci, G) =Mici−1

2Nic²_i +G (4) withMi=M= 5, andNi=N= 0.1.

Subjects do not need to do any calculations as they are provided with a table listing the level of income from activity B corresponding to the feasible values of the contribution to B that the subject could make, from0to50. The table also shows both the income change when activity A increases by one unit and the change when activity B increases by one unit.

From a theoretical point of view, subjects in the control participate in a repeated game of complete information in which the stage game has a unique Nash equilibrium, i.e. (gi, ci) = (10,40). The subgame-perfect Nash equilibrium of the repeated game comprises repetitions of the static Nash equilibrium.

The diﬀerence between the control game and the mechanism game is the way in which the total contribution to activity B is calculated. Both the Falkinger mechanism and the compensation mechanism act on the budget constraint.

Recall that in the Falkinger mechanism the budget constraint is

ci=yi−gi+β(gi−gj), (5)

1 4The instructions for the compensation mechanism are included in Appendix D. The complete set of instructions is available from the authors upon request.

1 5We programmed and ran the experiment with the z-tree software (Fischbacher (1999)).

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and in the compensation game the equation is

c_i=y_i−g_i+s_jg_i−s_ig_j. (6) This common feature makes the design of the experiments quite straightfor- ward. We merely need to apply changes to the way the contributions to activity B are calculated. Otherwise, we can use the same instructions. The feature facilitates both communication to subjects and enhances their understanding of the consequences of their choices in the treatments.

Like Falkingeret al. (2000), we test the effectiveness of the Falkinger mechanism in quadratic games, but using a different group size and different values of the tax-subsidy parameter. Table 1 describes experimental games of Falkinger et al. (referred to as C4 and M4 in their study), and our modiÞcations. We consider the same parameter values of the payofffunction, and a smaller group size of two subjects. Our use of three values of the tax-subsidy parameter,β, is the important difference. Theoretically, we consider repeated games of complete information in which the subgame-perfect Nash equilibrium is the repetition of the unique static Nash equilibrium, i.e., (g_i, c_i) = (15,35) for β equal to 1/3 and (g_i, c_i) = (19,31) for β equal to 9/19. When β equals 1/2, a subgame- perfect Nash equilibrium over the twenty rounds entails the repetition of the stage equilibrium strategy pair(gi, ci) = (20,30).

In the compensation mechanism, subjects have the option to set a rate at which to support the other group member in order to encourage her to contribute to activity A. Numerical examples provided in the instructions show how a subject’s choice of a support rate translates into a payment to the other subject and how the opponent’s rate translates into a payment to herself. The instructions state that a payment made to the opponent decreases the subject’s contribution to activity B, while a payment from the opponent increases that contribution. After both subjects have decided on the support rate, they learn about their opponent’s rate, and then they decide how many points to allocate to activity A. In a subgame-perfect equilibrium of the game, the contribution to activity A is20and to activity B 30, i.e. (g_i, c_i) = (20,30).

4 Experimental results

We examine the eﬀectiveness of the incentive mechanisms by analyzing contributions towards the public good by pairs of subjects. Section 4.1 considers contributions in the voluntary contribution (VC) game. This game serves as a control, and, hence, we compare behavior in the VC game across experiments.

We test whether contributions are the same in the control phase of the experiments. When weÞnd that we can assume that behavior is the same, we interpret the result as showing that we were able to sample subjects randomly from the same population of individuals. Section 4.2 analyzes the behavioral response to the introduction of a mechanism. Using data from each experiment, we compare contributions under the treatment with contributions under the control.

We test whether the level of contributions is raised, and investigate whether

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Table 1: Experimental design Falkinger 2000 BFR 2006

Group size n 4 2

ParameterM 5

ParameterN .1

Endowmenty 50

Benchmark: Voluntary Contribution game

c^N 40

g^N 10

G^N 40 20

U^N 160.00 140

Public Good with Falkinger mechanism

β^∗= (1−n¹) 3/4 1/2

β 2/3 1/3 9/19 1/2

c^F 20 35 31 30

g^F 30 15 19 20

G^F 120 30 38 40

U^F 200.00 143.75 144.95 145.00

Eﬃciency? N o Y es

Public Good with compensation mechanism

c^{COM P} − 30

g^COMP − 20

G^COMP − 40

U^{COM P} − 145.00

Eﬃciency? − Y es

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behavior moves towards equilibrium. Section 4.3 compares the contributions in the incentive mechanisms. We test whether the two mechanisms perform diﬀer- ently. We refer to the sessions of the experiment with the Falkinger mechanism with β = 1/3 as F30¹⁶, to the sessions with the Falkinger mechanism with β= 9/19asF38¹⁷, to the sessions with the Falkinger mechanism withβ= 1/2 asF40¹⁸, and to the sessions with the compensation mechanism asCOM P. Fi- nally, Section 4.4 compares the level and speed of convergence to equilibrium in the Falkinger mechanism under particular values of the tax-subsidy parameter and takes a closer look at individual behavior in the compensation mechanism.

4.1 Contributions to the public goods in the control games

We consider whether the distributions of contributions to the public goods are signiÞcantly diﬀerent in the control game of the experiments.

Table 2, second column, reports the statistic mean of the sum of contributions, on data from all sessions of control games, for each experiment. Data from play of the game by the pairs of subjects are organized into four blocks of Þve rounds.

In the control sessions of each experiment, subjects start at a very high level of contributions, close to the Pareto eﬃcient level of40. Play moves towards the Nash equilibrium level. Nevertheless, there is still substantial overcontribution after a few rounds of play. In the lastÞve rounds, the observed average level is about12.78%higher than the predicted level.¹⁹

Table 2, second column, replicates a generalÞnding of voluntary contribution experiments (Ledyard 1995): initially, contributions are well above equilibrium.

However, contributions sharply decline towards the end. This remarkably robust Þnding has been replicated previously in voluntary contribution experiments in a nonlinear environment (Keser 1996; Sefton and Steinberg 1996; Falkinger et al. 2000). We conÞrm the validity of theÞnding in the nonlinear environment for a group size of two players.

Result 1 There is no signiÞcant diﬀerence between the contributions to the public goods across the controls of the Falkinger mechanism experiment and across the controls of the compensation mechanism experiment.

Support The mean of the sum of contributions in F30, F38, F40, and COMP is29.35, 28.08, 32.52 and29.92, respectively. The standard deviation of the sum of contributions on data forF30, F38, F40, andCOM P is 11.73, 10.06,16.90, and6.21, respectively. The median of the sum of contributions on data forF30is28.42,27.03forF38,30.00forF40, and29.40forCOMP.

We perform a conventional Wilcoxon rank-sum test of the hypothesis that two independent samples are from populations with the same distribution. We

1 6The total Nash contribution to the public good is30.

1 7The total Nash contribution is38.

1 8The total Nash contribution is40.

1 9We report the deviation in percentage from the total endowment to render the statistic comparable to earlier studies. In the lastÞve rounds, the average level of contributions is 25.55%higher than the predicted equilibrium level.

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compare the distributions of contributions in sessions forF30andF38andÞnd P rob >|z|= 0.3296. For the distributions of contributions in sessions for F30 and F40, we ÞndP rob >|z| = 0.3539. For the distributions of contributions in F30 and COMP, we Þnd P rob > |z| = 0.3797. For the distributions in F38 and F40, we Þnd P rob > |z| = 0.2290. For F38 and COMP, we Þnd P rob >|z|= 0.7697. ForF40andCOM P,P rob >|z|= 0.3296.

Hence, for each pair of independent samples, the results suggest that the medians are not statistically diﬀerent at any reasonable signiÞcance level.²⁰ We conclude that behavior in the control sessions across experiments may be considered the same. We now assess the impact of each treatment.

4.2 Comparing contributions to the public goods in the control and in the treatment games

This section examines the impact of the mechanisms by comparing the contributions under the control with the contributions under the corresponding treatment in each experiment. For each experiment, we perform a Wilcoxon signed-ranks test on matched pairs of observations, and a paired two sample t- test on the equality of means. We evaluate whether each treatment signiÞcantly increases the level of contributions.

4.2.1 Falkinger mechanism with β= 1/3

Figure 1 reports the time series of the mean of the sum of contributions to the public goods using data on both the control phase and the treatment phase with β= 1/3. In addition to the mean of the sum of contributions, theÞgure shows the 95% conÞdence interval around the mean.²¹ One observes a movement towards Nash equilibrium in the control game, a jump in the contributions after the introduction of the mechanism, and fast convergence to equilibrium.

Figure 1 here

Result 2 (initial average eﬀect) After the Falkinger mechanism with β= 1/3is introduced, there is an immediate and large shift towards the eﬃcient level of the public goods.

Result 3 (deviation over time)After a few rounds of play of the Falkinger mechanism with β= 1/3, contributions decline and move closer to the equilibrium.

2 0To check the robustness of our conclusion, we also evaluate the general statement of equality on unmatched data in a two-group randomized design by performing a mean comparison test. We test the speciÞc hypothesis that the means in two independent samples are equal.

We compare the means of contributions in sessionsF30andF38andÞnd that the p-value of the two-sided t-test with unequal variance is0.7647. For the pair(F30,F40), the p-value is0.4363. For(F30,COMP), the p-value is0.8655. For(F38,F40), the p-value is0.2875.

For(F38,COM P), the p-value is0.5970. For(F40,COM P), the p-value is0.4232. For each pair of independent samples, the results indicate that the means are not statistically diﬀerent at any reasonable signiÞcance level.

2 1We use the method of bootstrapping to calculate the conÞdence bounds (with a conventional number of 1000 replications).

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Support The top panel in table 2 reports the mean of the sum of contributions, on data from all sessions of the Falkinger mechanism with β = 1/3.

We calculate the statistic from observations on repeated play of the incentive mechanism by pairs of subjects. We display the statistic for four blocks ofÞve rounds, and for the entire length of play.

For the value of the tax-subsidy parameterβ= 1/3, theory predicts a level of the public goods equal to30. The introduction of the Falkinger mechanism causes a jump from a25.13in the lastÞve rounds of the control game to around 38.37in theÞrstÞve rounds of the incentive mechanism. Thereafter, contributions decline steadily over time. In the lastÞve rounds, the mean of the sum of contributions is29.00and the median of the sum of contributions (not reported in the table) is 29.33. Hence, the center of the distribution of the observed contributions shifts to equilibrium.

Result 4 (overall treatment eﬀect) The introduction of the Falkinger mechanism with β= 1/3increases the level of contributions towards the public goods. Overall, the increase is weakly signiÞcant. Both on data from the Þrst Þve rounds and on data from the lastÞve rounds of the mechanism, the increase is weakly signiÞcant.

SupportIt is reasonable to conjecture that the introduction of an incentive mechanism increases contributions. This hypothesis implies a one-sided test.

To test the hypothesis that the mechanism increases the level of contributions, we use a conventional Wilcoxon signed-ranks test. Table 2 reports the p-value of the test. The median of the sum of contributions of all pairs of subjects and twenty rounds is30.58.

In addition, the table reports the p-value of a t-test. The sum of contributions averaged over all pairs of subjects and twenty rounds is 32.15. The standard deviation of the sum of contributions is8.52.

WeÞnd that the null hypothesis of no difference between control and treatment can be rejected at the10%level of signiÞcance. We conclude that there is a weakly signiÞcant difference between control and treatment over all rounds.²² We also perform tests on data from rounds 1-5, 6-10, 11-15, and 16-20. The null hypothesis of no difference can be rejected on data from rounds 1-5 and data from rounds 16-20 at the10%level of signiÞcance. The null hypothesis of no difference is not rejected at conventional error levels on data from either rounds 6-10 or rounds 11-15.

We also note that the introduction of the Falkinger mechanism withβ= 1/3 decreases the variation in the contributions. More speciÞcally, it decreases the average width of the conÞdence intervals from 14.04 on data from the control to11.55on data from the treatment (see Figure 1). This is a35.47%decrease (in percentage from endowment).

2 2We follow the convention that a signiÞcance level of 1% or less is highly signiÞcant, a signiÞcance level between 1% and 5% is signiÞcant, while a signiÞcance level between 5% and 10% is weakly signiÞcant.

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Table 2: (1) Comparison of contributions in the control sessions, unmatched data; (2) comparision of contributions in the control (before) and treatment sessions (after), matched data, signiÞcant at: * 10% level, ** 5% level, *** 1%

level, one-sided test; (3) comparison of contributions in the treatment sessions, matched data.

Experiment with Falkinger mechanism,β= ¹₃ (F30) Sum of contributions p-value, test of equality Control Treatment of median of mean

Nash/Pareto 20/40 30/40

Rounds 1-5 35.50 38.37 0.0912* 0.0954*

Rounds 6-10 29.58 30.88 0.2048 0.2612

Rounds 11-15 27.18 30.37 0.1047 0.1283

Rounds 16-20 25.13 29.00 0.0731* 0.0750*

Overall 29.35 32.15 0.0422** 0.0676*

Experiment with Falkinger mechanism,β= ₁₉⁹ (F38) Sum of contributions p-value, test of equality Control Treatment of median of mean

Rounds 1-5 36.48 52.21 0.0030*** 0.0028***

Rounds 6-10 27.99 47.05 0.0008*** 0.0012***

Rounds 11-15 24.83 42.75 0.0023*** 0.0008***

Rounds 16-20 23.03 38.97 0.0011*** 0.0001***

Overall 28.08 45.29 0.0006*** 0.0002***

Experiment with Falkinger mechanism,β= ¹₂ (F40) Sum of contributions p-value, test of equality Control Treatment of median of mean

Rounds 1-5 39.12 47.00 0.0134** 0.0085***

Rounds 6-10 34.04 49.53 0.0013*** 0.0003***

Rounds 11-15 29.00 45.55 0.0006*** 0.0001***

Rounds 16-20 27.93 45.71 0.0004*** 0.0000***

Overall 32.52 46.95 0.0005*** 0.0000***

Experiment with compensation mechanism (COMP) Sum of contributions p-value, test of equality Control Treatment of median of mean

Rounds 1-5 35.65 37.07 0.4181 0.3702

Rounds 6-10 30.51 34.13 0.1822 0.1285

Rounds 11-15 29.16 31.75 0.3991 0.2661

Rounds 16-20 24.36 31.65 0.0498** 0.0326**

Overall 29.92 33.65 0.1533 0.1211