Claude Meidinger
email: claude.meidinger@univ-paris1.fr

Jean-Louis Rullière
email: rulliere@gate.cnrs.fr

Marie-Claire Villeval
email: villeval@gate.cnrs.fr

Free-Riding and Fairness in Principal -Multi-Agent Relationships: Experimental Evidence

Keywords: Fairness, Experimental Economics, Principal-Agent Relationship, Team Production. JEL Classification: J33, C91, C92, D63

-2 -Résumé : De quelle manière des motivations intrinsèques telles que la bienveillance et la réciprocité peuvent-elles influencées l'efficacité d'une relation "principal multi-agent" dans un cadre de production jointe par équipe ? Centré sur des problèmes de hasard moral en équipe [START_REF] Holmström | Moral Hazard in Teams[END_REF], ce papier rend compte d'une expérience cherchant à déterminer si la bienveillance d'un principal permet de réduire les comportements de passager clandestin au sein d'une équipe. Deux traitements sont effectués, l'un avec réassortiment (traitement "étranger"), l'autre sans réassortiment (traitement "partenaire") des membres des groupes de sujets. Les résultats de l'expérience montrent que i) des offres de contrats bienveillants favorisent la coopération au sein de l'équipe dans le traitement "étranger" et que ii) des interactions répétées n'améliorent pas nécessairement la ccopération au sein de l'équipe. Les résultats du traitement "partenaire" montrent la difficulté d'établir une coopération fructueuse entre des principaux et des membres d'équipe inégalement motivés par des considérations de bienveillance.

Mots clefs : bienveillance, économie expérimentale, relation principalagent, production en équipe.

Abstract: How do intrinsic motivations such as fairness and reciprocity influence the efficiency of a principal -multi-agent relationship when joint production in a team is considered? Focusing on moral hazard in teams [START_REF] Holmström | Moral Hazard in Teams[END_REF], this paper reports the results of an experiment designed to determine whether principal's fairness helps in reducing free-riding amongst team members. Two treatments were run, with reshuffling (stranger treatment) and without reshuffling (partner treatment). Experimental evidence shows that i) offers of fair contracts favor team cooperation in the stranger treatment, whereas ii) repeated interactions do not necessarily improve team cooperation. All the results of the partner treatment point to the difficulty of establishing a fruitful cooperation between principals and team members unequally motivated by fairness considerations.

INTRODUCTION

What is the impact of intrinsic motivations such as fairness and reciprocity on the efficiency of the principal -multi-agent relationship ? The importance of vertical fairness in the interactions between a principal and an agent has been documented by many experimental studies; the principal pays higher wages than those required by the incentive constraint and the agent delivers a level of effort greater than the one theoretically predicted [START_REF] Anderhub | Efficient Contracting and Fair Play in a Simple Principal-Agent Experiment[END_REF][START_REF] Fehr | Reciprocity as a Contract Enforcement Device: Experimental Evidence[END_REF][START_REF] Fehr | Gift Exchange and Reciprocity in Competitive Experimental Markets[END_REF][START_REF] Güth | An Experimental Study of a Dynamic Principal-Agent Relationship[END_REF][START_REF] Keser | Principal's Principles when Agents' Actions are Hidden[END_REF]. These experimental results have been obtained in designing principalagent games as a one-to-one relationship in which one principal faces one agent.

However, in real life, most people are working within teams. Do these results hold when considering a principal -multi-agent relationship? As pointed out by Holmström in his seminal paper on moral hazard in teams (1982), a multi-agent setting is characterized both by an incentive for agents to free-ride in the context of joint production and by competition in controlling incentives, as agents play a noncooperative game. An optimal allocation cannot be achieved as a Nash equilibrium with a binding budget balancing constraint (where the outcome is fully allocated to team members). A budget breaking rule, realized by introducing a principal as a residual claimant, can be efficiency-enhancing. The rule stipulates that the joint outcome will accrue to the principal in the event that the actual outcome has not reached the expected target level. Introducing a budget breaker would be more efficient than an active monitoring.

However, this punishment mechanism faces serious drawbacks. A problem of moral hazard on the part of the principal can be pointed out [START_REF] Eswaran | The Moral Hazard of Budget-Breaking[END_REF]. With such a contractual arrangement, the third party has an incentive to bribe one team member to shirk and this clandestine deal results in a sub-optimal outcome. [START_REF] Rasmusen | Moral Hazard in Risk-Averse Teams[END_REF] also shows that an all agent contract can still be efficient if workers are risk averse. Free-riding in a team can be eliminated by introducing a lottery in a "massacre contract" in which one randomly selected agent is rewarded and all the other agents are punished when an out-ofequilibrium outcome is achieved by the team. In contrast, Andolfatto and Nosal Cahiers de la MSE -2000.48 -4 -(1997) extend Holmström's perspective in designing a contract in which delivering a payment to the principal even though the desired outcome has been achieved helps reducing the agents' incentive to free-ride.

As far as principal-multi-agent relationships are considered, this paper intends to pursue this analysis by asking whether fairness and reciprocity help in reducing free-riding within a team. Our aim is not to design the more efficient payment scheme but to identify the influence of intrinsic motivations on the result of a principal -multi-agent relationship, beyond the respect of participation and incentive compatibility constraints. How do the proportion of the total revenue kept by the principal for himself (index of vertical fairness) and the threat of punishment (index of negative reciprocity) affect effort decisions within a team? Does vertical fairness favor solidarity amongst team members?

It should be noted that recent supporting evidence for horizontal fairness between agents has also been reported from experiments which consider either an all agent framework [START_REF] Nalbantian | Productivity under Group Incentives: An Experimental Study[END_REF][START_REF] Königstein | Profit Sharing in an Asymmetric Bargaining Game[END_REF][START_REF] Carpenter | Mutual Monitoring in Teams: Theory and Experimental Evidence[END_REF] or a principal -multi-agent framework (Rossi and Warglien, 1999;[START_REF] Güth | Discriminating Incentive Schemes in one Principal-Multiple Agents-Situations: A Theoretical and Experimental Analysis[END_REF]. In Güth and alii however, the authors do not really design a team but rather a collection of separate contracts between a unique principal and many agents. Experimental evidence of horizontal fairness is here delivered very indirectly, through the existence of less asymmetric contracts offered by the principal and through a lower dispersion of efforts when contracts are observable by all agents. Rossi and Warglien's approach is closer to ours inasmuch as it investigates how principal's fairness influences cooperation between two interdependent agents performing a production game. Our approach is however different. We consider heterogeneous agents and a threat of punishment by the principal.

Distinctively, this paper reports an experiment designed as a principalmulti-agent game in which a principal faces a team composed of two players with different productivity levels.

The principal first offers a contract that, in exchange of a non-binding desired outcome, specifies a team payment scheme (TPS-1) under which each agent receives an equal share. In such a scheme, because unequal production skills (i.e.

Cahiers de la MSE -2000.48 -5 -unequal productivity) amongst agents depend on natural endowments such as talents, the principal asks each agent to provide a same effort. Despite their differences in talents, she pays them according to a scheme based on a collective ownership in talents (see [START_REF] Fleurbaey | Cooperative Production with Unequal Skills: The Solidarity Approach to Compensation[END_REF]. Agents with different productivity levels but developing the same effort suggested by the principal will receive the same payoff.

The offer can be more or less generous, depending on both the share that the principal keeps for herself (either 25% or 50% of the outcome) and on the effort level suggested to achieve the desired outcome.

If the actual outcome is less than the desired one, the principal can decide whether she applies the previous TPS-1 or a second team payment scheme (TPS-2) under which wages are individualized according to productivity levels. In that case, each player bears a share of the TPS-2 implementation cost. This approach based on a private ownership of talents is substituted to the other one based on a collective ownership of talents.

With players only motivated by pecuniary incentives, it is obvious that the threat to apply TPS-2 is not credible. And because both agents develop a dominant free riding strategy under TPS-1, the issue will be a non cooperative one. Things can nevertheless be different if players take into account fair behavior.

In real life as in many experiments, the willingness to be kind to others seems highly contingent on the behavior of others. People help those who are helping them and hurt those who are hurting them [START_REF] Rabin | Incorporating Fairness into Game Theory and Economics[END_REF]. Clearly, such a behavior can entail some economic consequences both on vertical relationships (between the principal and the team) and on horizontal relationships (between team members). A generous contract offered by the principal in a framework of skill solidarity may, by reciprocity, induce cooperation within a team, as well as free riding in such a situation can trigger punishment. Intrinsic motivations within team (solidarity, peer pressure or mutual monitoring) may have something to do with the success of cooperation [START_REF] Kandel | Peer Pressure and Partnerships[END_REF]. This paper reports the results of two repeated game experiments with and without reshuffling that allow to study the impact of reciprocity and reputation in principal-multi-agent sequential move games. Two series of experiments were run, Cahiers de la MSE -2000.48 a stranger treatment (with reshuffling) and a partner treatment (without reshuffling). The stranger treatment serves as a benchmark for measuring the extent of genuinely reciprocal fairness (leading to non-strategic conditional behavior to reward kind acts and to punish unkind ones even if costly). The partner treatment leaves room for reputation building.

Experimental evidence shows that, in the stranger treatment, offers of fair contracts, some of them being equity-based, induce team cooperation, whereas offers of unfair contracts always induce team defection. This result is similar to those reported by Rossi and Warglien (1999). Despite the prediction that we should observe neither principal's fairness nor team cooperation if subjects are only motivated by pecuniary considerations, team cooperation occurs in half of fair contracts. Therefore, vertical behavior influences horizontal behavior within a team. However, in opposition to many other experimental results, repeated interactions do not necessarily improve team cooperation. All the results of the partner treatment point to the difficulty to establish a fruitful cooperation between principals and team members unequally motivated by fairness considerations. In the partner treatment, vertical behavior also influences horizontal behavior, through notably the agents' interpretation of intentions.

The remainder of this article is organized as follows. Section 2 develops the model and presents the general frame of the game, the four stages of the game, the standard strategic analysis of the game and its strategic analysis when reciprocal fairness is taken into account. Section 3 gives a brief presentation of the experimental design. Section 4 analyses the experimental results. A brief conclusion is given.

THE PRINCIPAL -MULTI-AGENT GAME

The game involves three players, a principal (player X ) and a team composed with two agents (players Y 1 and Y 2 ). These matched players have to share an outcome whose amount depends on their decisions. with p = 12 , the price of the output, and ρ 1 1 = , ρ 2 2 = . That outcome has to be divided according to a sharing scheme decided by the principal and which depends on both a parameter θ (that determines the principal's share) and a team payment scheme (that determines the allocation of ( ) 1-θ py between the agents).

Two team payment schemes are available in the game.

q The first one, TPS-1, consists of a team payment scheme according to which each agent receives an equal share of the outcome whatever his productivity level. This solidarity scheme is based on the idea of a collective ownership in talents. The players' shares are θpy for the principal and 2 / ) 1 ( py θ for each agent.

q The second one, TPS-2, is a productivity-based team payment scheme. This second scheme is based on the idea of a private ownership in talents. In that case, the implementation of individualization entails a cost C 0 (with C 0 =48) that is subtracted from py before the sharing. The players' shares are θ ( py C -0 ) for the principal and

j i i C py ρ ρ ρ θ + - - ) )( 1 ( 0 for agent i .
In both TPS-1 and TPS-2, the principal's payoff is equal to her outcome The game therefore consists of the following elements.

The principal designs a labor contract and makes a "take it or leave it " offer to the agents. The contract specifies a desired outcome that can be realized through a common non-binding level of effort e E * ∈ = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24} provided by each agent. It also specifies a value for θ , either equal to 0.5 or 0.25 which, once chosen, will be retained for both TPS. When the actual outcome, resulting from the agents' efforts, equals at least the desired outcome, TPS-1 is applied. When that expected outcome has not been achieved, the principal can freely decide to apply either TPS-1 or TPS-2.

In spite of the fact that e * is not binding, that level of effort is nevertheless indicated by the principal to her agents, as a kind of cheap talk.

The four stages of the game

q In the first stage, the principal indicates to the agents a common nonbinding level of effort e E * ∈ that determines a desired outcome amounting to

) ( * * 2 1 ρ ρ + = pe py
and chooses a θ either equal to 0.25 or 0.5.

q In the second stage, informed about the principal's offer, each agent independently accepts or declines the offer. If at least one agent refuses the contract, the round is over and the players' payoffs are null. If both agents accept, the third stage starts up.

q In the third stage, each agent Y freely and independently chooses in E a level of effort. This determines an outcome py p e e = + ( ) ρ ρ

1 1 2 2
to be shared.

q In the fourth stage, the principal is informed about the actual outcome py .

If py py ≥ * , she applies TPS-1. Otherwise, she can choose between enforcing TPS-1 or applying TPS-2, keeping the same value of θ in both cases.

The four stages of the game are common knowledge for all players at the start of the game. Tables 1 to 12 in Appendix show the payoffs' matrices for the two agents and the principal under TPS-1 and TPS-2.

The standard strategic analysis of the game

Because we avoid negative earnings, the participation constraint of a nonnegative payoff for the agents is clearly satisfied. And if the two agents cooperate and follow the principal's suggestion, a maximum common level of e* = 24 has to Cahiers de la MSE -2000.48 be required to maximize the principal's payoff. Under TPS-1, with θ = 0 25 . , that level of effort will also maximize the agents' cooperation payoffs π i coop , but not with θ = 0 5 . . In that sense, if the principal wants the agents to cooperate, she has to take into account an incentive compatibility constraint given by:

e Arg coop e i * ( ) max θ π = (1)
We therefore have e * ( . ) 0 25 24 = and e * ( . ) 0 5 18 = .

But clearly, that will not be enough to ensure the agents' cooperation. The principal must also take into account the possibility of using TPS-2 as a credible threat.

From the agents' point of view, regardless of the TPS adopted, both of them have a dominant free riding strategy. These strategies are under TPS-1:

e a p ifr i = - ( ) 1 4 θ ρ (2)
and under TPS-2:

e a p ifr i i j = - + ( ) 1 2 2 θ ρ ρ ρ (3)
In Tables 1 to 12 in Appendix, these strategies lead to the following effort values:

q With θ = 0 25 . , under TPS-1: e fr 1 =8 or 10 and e fr In one-shot games or in repeated games with reshuffling, with players only motivated by pecuniary incentives, the threat to apply TPS-2 is not credible. It follows that both agents have an incentive to free ride whatever the value of θ and will play their dominant free riding strategy under TPS-1. The predicted issue will Cahiers de la MSE -2000.48

-10 -then be a free riding solution. With θ = 0 25

. , payoffs will be 282 for Y1 and 217 or 226 for Y2. With θ = 0 5

. , payoffs will be 181 for Y1 and 154 for Y2. The principal's payoff π p will be equal to 232 or 238 with θ = 0 25 . and equal to 280 with θ = 0 5 . . Therefore, by backward induction, the principal's best choice is to select θ = 0 5 . in the first stage of the game and not to apply TPS-2 after a deviation in the fourth stage.

In repeated games without reshuffling, is there any interest for the principal to build a reputation of toughness ?

For θ = 0 25 . , both agents get a payoff of 280 if they cooperate by achieving the effort level of e* = 24 suggested by the principal. When TPS-2 is considered as a credible threat, an agent that defects has to play his free riding strategy under TPS-2. With such a strategy, Y1 cannot hope to get more than a payoff of 241 when free-riding under TPS-2 and therefore has an interest to cooperate if he expects Y2 will cooperate. But confronted with a cooperative Y1, Y2 has an interest to trigger TPS-2. By playing e2=22, he will get a payoff of 363. Knowing that, Y1 will not cooperate and the issue will then be the free riding solution under TPS-2 with a principal's payoff of 250.

For θ = 0 5 . , and therefore e* = 18 , both agents get a payoff of 181 if they achieve the required effort level. But here too, it is clear that agent Y 2 has an incentive to trigger TPS-2 if he conjectures that Y 1 cooperates. By playing his free riding strategy 16 2 = e under TPS-2, he will get a payoff of 220 instead of 181.

Knowing that, agent Y 1 will not cooperate and the issue will be the free riding solution under TPS-2, Therefore, in a repeated game without reshuffling, the principal's best choice is to select θ = 0 5

. in the first stage of the game and to apply TPS-2 after a deviation in the fourth stage.

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-11 -

The introduction of fairness considerations

It is clear that some contracts can be considered as being fair. With θ = 0 25 .

and e* = 24 for example, the principal is going to accept a reduced payoff of 316, instead of a payoff of 424 she can expect by asking for e* = 18 with θ = 0 5 . . She also offers the agents the opportunity to get an increased payoff of 280 if they cooperate, to be compared to the payoff of 181 they could get with θ = 0 5

. and e* = 18 . It is also clear that among fair contracts ( 25 . 0 = θ

), some of them can also be considered as being equity-based. With a suggested level of effort 18 ≤ e , each employee's payoff is equal to or greater than the principal's payoff. Such a vertical fairness may induce agents to cooperate. It may also motivate the principal to apply the TPS-2 threat (negative reciprocity) in order to punish the agents for not responding positively to her kindness even in a one-shot game.

One should also note that with θ = 0 5

. , because the principal's payoff is always greater than the agent's payoffs for any suggested level of effort, there can be no equity-based contracts. But there can be unfair contracts with a level of suggested effort above the one determined by the incentive compatibility constraint. For 18 > e , the principal tries to reinforce her first mover advantage by using cheap talk to suggest a high level of effort.

From that point of view, it seems that vertical fairness may lead to a more fruitful interpretation of observed behavior.

In one-shot games or repeated games with reshuffling, because reputation is ruled out by design, any contract with θ = 0 25

. may be interpreted as a contract offered by a principal motivated by fairness considerations. A look at the principal's strategy may also allow to distinguish principals playing fair only for strategic reasons from principals genuinely motivated by fairness. Only the latter will punish agents for not responding kindly to a fair offer.

In repeated games without reshuffling, where there is scope for reputation building, two different types of principals must be distinguished.

q On the one hand, greedy principals trying to maximize their payoffs with θ = 0 5 . through a toughness reputation that destroys team cooperation.

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-12 -With TPS-2 credible, Y2 will defect, the punishment will be applied and 292 = p π (better than 280 obtained without punishment for θ = 0 5 . ).

q On the other hand, genuinely kind principals trying to enforce cooperation through fair offers with a maximum payoff of 316 = p π . It should be noted that vertical fairness will usually not work without something that could be related to fair behavior among team members.

Without punishment, Y1 has a strong incentive to free ride (under TPS-1, he is offered half of the team's payoff despite his lower productivity and he will get a free riding payoff greater than Y 2 's payoff). With punishment, Y2 has the same incentive. It is also worth noting here that greedy principals have interest to mimic genuinely kind principals if they consider that fairness can be an efficient means to increase their payoff.

EXPERIMENTAL DESIGN

The experiments were performed in November 1999 at the University of Lyon. They were run in a computerized way with Regate as experimental software.

Subjects were drawn from the undergraduate population at the Management School of Lyon and no subject was experienced. Before the experiment started, written instructions were distributed to the participants and read aloud. A preexperimental questionnaire was distributed in order to check whether the rules of the game were fully understood. The payoffs functions and the productivity distribution are common knowledge. Tables were distributed to participants indicating the principal's and each employee's payoffs according to the individual effort levels, for each sharing rule (TPS-1 and TPS-2) and each value of the θ parameter (see Appendix).

12 subjects participated in a session (4 groups of 3 subjects for each period). 4 sessions were organized (3 sessions with 4 groups and 1 session with 3 groups). It gives 15 groups of observations. Each subject was randomly assigned either the role of a principal (player X ), or the role of an employee with a low (player Y 1 ) or a Cahiers de la MSE -2000.48

-13 -high productivity (player Y 2 ). In each period, each principal is matched with two heterogeneous employees. Two treatments were run.

q A first series of sessions consisted of a "stranger treatment" (7 groups).

Each subject played 15 rounds of the game in the same role but he could be matched with different players all along the session. The matching of a principal with two agents was determined randomly each period. This treatment excludes repeated game effects.

q A second series of sessions consisted of a "partner treatment" (8 groups) without reshuffling. In this case, each subject played 15 rounds of the game, staying in the same role and matched with the same two other players all along the session. This treatment enables to observe reputation effects and the principal's will to build a toughness reputation.

Each session started with 5 trial periods (which the results were not taken into account for analyses and payments) in order to help players to get accustomed with the rules of the game.

In each period, employees are informed of the level of effort suggested by the principal, * e , of the outcome share θ the principal intends to keep for herself, of the amount of the actual outcome, py, and of the team payment scheme which will be actually implemented (TPS-1 or TPS-2). In each period, the principal is informed of the acceptance or rejection of the contract, and of the size of the actual outcome. Each player in each period is also informed of the payoffs received by the two other players with which he is playing. This ended the period. During the experiment, each player could see on the upper half of his or her computer screen a table showing his or her decisions and results in all previous rounds of the game.

At the end of the session, each player was paid his or her average net payoff in the 15 rounds in French Francs according to the conversion rate 1 ECU= 0,4 FF.

A showing up fee of 20 FF was added.

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IV -EXPERIMENTAL EVIDENCE

The total number of contracts offered by the principals is 105 in the stranger treatment (ST) and 120 in the partner treatment (PT). Let us consider successively these two treatments. -15 -With θ =0.5, 14% of the contracts respect the incentive compatibility constraint (e=18). 42% of offers suggest a level of effort higher than the incentive compatibility constraint (unfair offers).

Team cooperation: defined as the respect by both agents of the suggested effort level, it should be noted that team cooperation is weak (21% of accepted contracts) (see Table 1). Table 1. Team cooperation in the stranger treatment Team cooperation is observed in 50% of accepted contracts with θ =0.25 but only in 11% of contracts with θ =0.5. Unfair contracts (θ =0.5 and e>18) never induce team cooperation. By contrast, equity-based contracts (θ =0.25 and e ≤ 18) usually support team cooperation. Therefore, with a broader fairness concept including equity considerations, it seems here that principal's fairness helps in reducing free-riding within the team (positive reciprocity).

Suggested effort (e) # cases

If one looks now at the individual agent behavior, one should note that the rates of defection increase with the value of e and are generally greater for θ =0.5 than for θ =0.25. When θ =0.25, the less (more) productive agent defects in 38.5% (42.3%) of the contracts. When θ =0.5, the less (more) productive agent defects in 73.3% (70.7%) of the contracts. For both values of θ , a 2 χ test (at a level of significance of 0.60) leads to the acceptation of equal frequencies of defection for agents Y1 and Y2 ( 2 χ = 0.3252 for θ =0.25 and 0.2198 for θ =0.5).

Cahiers de la MSE -2000.48 Punishment: in the stranger treatment, a principal motivated only by pecuniary considerations should never apply punishment (TPS-2). The experimental evidence shows that punishment has been applied in 13 cases out of 80 group defections (16.2%) and just once with θ =0.25 (negative reciprocity). One should point out that punishment, applied in 6 of the unfair contracts, reinforces in those cases the principal's meanness. The evidence therefore mostly points to principals either playing fair for strategic reasons or trying to maximize their payoffs through a first mover advantage with θ =0.5.

Payoffs: As displayed by

Table 2. Average payoffs in the stranger treatment

Fair contracts (including equity-based ones) produce a quasi-equal pie sharing among the three players, with a small advantage to the less productive agent. As a matter of fact, the principal gets her worse payoff when she offers equity-based contracts and her best payoff when she offers unfair contracts. But for both agents, the payoff gap between these two kinds of contracts is larger by far.

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-17 -

The partner treatment

Contracts: 8 groups have played the partner treatment, giving 8 series of independent observations (120 contracts offered).

39 contracts (33%) are fair (θ =0.25) and among them, 22 contracts are equity-based (e ≤ 18).Only 38% of the 39 contracts respect the incentive compatibility constraint.

In the 81 remaining contracts with In spite of the fact that the proportion of contracts offered with θ =0.25 is greater in the partner treatment (33%) than in the stranger treatment (25%), it should be noted here that a 2 χ (corrected for continuity) leads to the acceptation of equal frequencies of both types of contracts in both populations ( 58 , 1 2 = χ with a probability of 0.20). This may point to the fact that greedy principals do not really see any interest to mimic fair principals in order to build a reputation of fairness. A similar comparison made respectively for the first round of the repeated game (t=1) and the last round (t=15) shows no significant difference (at a level of 0.70) between the two populations.

However, it seems that most contracts with θ =0.25 cannot be interpreted as a sign of principal's fairness. Unlike the stranger treatment, in the partner treatment one should look at the evolution of proposed contracts over time to appreciate fairness intentions.

A look at Figures 2 and3 below shows that only three principals out of eight start the repeated game by offering fair contracts and that most of the fair contracts are proposed at rounds 10 and 11 (Figure 2), just after a sharp decline of the average size of the pie at round 9 (Figure 3). These observations therefore seem to point to the fact that most principals do not care about fairness. They try to enforce cooperation with θ =0.5 and worry about the shrinking size of the pie, just around the middle of the game, due to the offer of unfair contracts (from rounds 6 to 9, unfair contracts represent 46% of contracts with θ =0.5, and more than one third of the total contracts offered). But even though fair contracts offered at round 10 help to increase the average size of the pie, they are nevertheless proposed too late to be considered as a sign of trustful cooperation. Consequently, at the end of the game (rounds 12 to 15), the Cahiers de la MSE -2000.48 proportion of fair contracts collapses and the value of e increases, a classical effect of end game.

Finally, a total of 108 contracts have then been accepted. But just note that if, in the partner treatment as in the stranger treatment, all contracts offered with θ =0.25 are accepted, a smaller proportion of contracts offered with θ =0.5 are here accepted (81.5% versus 95% in the stranger treatment). This has something to do with the strategies of some agents that use contract rejection not only as a punishment device but also as an incentive one.

Team cooperation: Team cooperation is globally less developed in the partner treatment (16% of accepted contracts) than in the stranger treatment (see Table 3). A 2 χ test rejects the null hypothesis at a level of α=0.01 for θ =0.25 and at a level of α=0.15 for θ =0.5. Only associated with low levels of suggested effort (e<10 with θ =0.25 and e<12 with θ =0.5), team cooperation is indeed never achieved with unfair contracts. Table 3. The extent of team cooperation in the partner treatment Such a lack of team cooperation is explained by most of the principals' behavior, trying to enforce cooperation with θ =0.5 at the beginning of the game.

Suggested effort (e) # cases

Generally not considered as a sign of trustful cooperation, only 3 out of the 39 proposed fair contracts lead to team cooperation.

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Coordination failures and payoffs: Table 4 displays the distribution of groups' payoffs. A look at subjects' behavior in some groups illustrates the difficulties to achieve coordination. Two particular groups are here chosen : Group 1 which exhibits a failure of coordination with a genuinely kind principal and Group 3 that exhibit the same failure with a greedy principal.

Table 4. Distribution of payoffs in groups

q Let us first consider Group 1, the worst but one in terms of average pie. Figure 4 displays subjects' behavior.

Over the five first rounds, two out of the three subjects (the principal and agent Y1) try to set up a coordination on θ =0.25 and e=24. But agent Y2 defects by playing a low level of effort. Such a behavior triggers a retaliation war. It is interesting here to note that such a war takes place precisely between those two subjects that first agree on coordination. At round 5, partly for inciting the other agent to raise his contribution, Y1 defects (horizontal punishment). The principal retaliates by proposing an unfair contract in the next two rounds (vertical punishment) and this leads Y1 to reject both offers. Afterwards, fair offers cannot prevent Y1 to defect, then to reject the proposed contracts with θ =0.5 at rounds 13 and 14. Finally, the principal enforces punishment in the last round. By contrast, let us consider Group 3, which achieved the best average pie (Figure 5).

Efficiency can be here explained by a generalized lack of reciprocity concern.

First of all, except at round 10, it should be noted that the greedy principal ever uses her first mover advantage by choosing θ =0.5 and suggesting on the average a high level of effort (9 unfair contracts over 15 rounds).

Evolution of suggested and individual effort levels -Group 3 In addition, she mostly enforces punishment (at rounds 3, 4, 6, 11 and 15) with unfair contracts (4 out of 5 cases of punishment enforcement). One should note more generally that this strategy is also observed in other groups. In the aggregate, punishment is applied in 37% of unfair contracts, in 18% of other contracts with θ =0.5. In that group, agents do not seem to bother about reciprocation. There is no contract rejection despite the number of unfair contracts;

agent Y1 free rides (and he continues to do so even after a punishment) on the high productivity of agent Y2 who, in turn, tries not to get a reduced payoff by choosing on the average a level of effort greater than Y1's one.

V. CONCLUSION

Much experimental evidence has established, on the one hand, the importance of incentive to free-ride in groups and, on the other hand, the extent of fairness and reciprocity in one principal -one agent relationships. Distinctively, this paper considers a principal -multi-agent relationship and tries to determine whether principal's fairness helps in limiting the extent of free-riding within a team.

Two repeated game experiments have been run that help to study how team cooperation and free-riding develop in principal -multi-agent sequential move games. Experimental evidence shows that vertical fairness favors horizontal fairness between agents and thus efficiency in the stranger treatment. However, a repeated interaction between same subjects need not improve team cooperation.

One should first note that the results of our partner treatment are close to the evidence displayed by public goods experiments on individual behavior within groups. In such experiments, the level of individual contributions to the funding of a public goods declines over time to the free-riding equilibrium, mainly due to the coexistence of free-riders and conditional cooperators. However, when a punishment device is introduced in such experiments, individual contributions raise over time and converge progressively to the greatest possible level of contribution [START_REF] Fehr | Cooperation and Punishment in Public Goods Experiments[END_REF]. Distinctively, in our experiment, the punishment mechanism does not succeed in raising team cooperation, because of Cahiers de la MSE -2000.48

-23 -its asymmetric nature. There is no explicit horizontal punishment mechanism enabling a mutual monitoring within the team and the vertical mechanism that should favor team cooperation is not efficient in a repeated interaction because of its asymmetry: When the threat is credible, the more productive agent has an interest in free-riding in order to trigger the punishment and, thus, to increase his payoff.

One should also note that less cooperation is observed here than in a one principal -one agent framework because of the incentives to free-ride in a team production process. One of our main results is that a fruitful and long-lasting cooperation between principal and team-mates is all the more difficult to achieve with people differently motivated by fairness. Team cooperation occurs in our experiment if and only if two conditions are met. First, team cooperation requires that the principal offers a fair contract and does not apply the punishment that could trigger the more productive agent's free-riding. Second, team cooperation implies that the more productive agent should accept the equal sharing TPS in spite of his higher contribution to the outcome. In such a way, cooperation is more demanding since fairness must develop between agents and not only between one principal and one agent. In such an asymmetric principal -multi-agent relationship, the lack of cooperation in repeated interactions is due to the fact that selfish considerations easily dominate fairness.

  share and the agents' payoffs are determined by subtracting from their outcome shares a personal cost of effort C 4 . When running the experiments, we added 100 to these theoretical payoffs in order to avoid negative earnings.

  of 160 for Y1 , 164 for Y2 and 292 for the principal.

4. 1 .Figure 1 .

 11 Figure 1. Distribution of offers according to the value of theta and the suggested effort level (e) in the stranger treatment With θ =0.25, e should be equal to 24 but in fact, only 19% of the contracts respect this incentive compatibility constraint. 81% of the contracts suggest a lower level of effort.

Figure 2 .Figure 3 .

 23 Figure 2. Evolution of the value of theta over time in the partner treatment

Figure 4 .

 4 Figure 4. Evolution of subjects' behavior in Group 1

Figure 5 .

 5 Figure 5. Evolution of subjects' behavior in Group 3

  Let us first define ρ i and e i as player Y i 's productivity and effort levels,

	2.1. The general frame of the game
	respectively, with	{ } i ∈ 1 2 , . The resulting outcome is then py p e = ( ρ 1 1	+	e 2 2 , ) ρ
		-7 -
		Cahiers de la MSE -2000.48

  Table2, fair contracts produce more efficient results. The average size of the pie is greater for θ =0.25 than for θ =0.5. Equitybased contracts are not however more efficient than this average. By contrast, by suggesting high efforts, unfair contracts somehow succeed in achieving an amount to be shared greater than the average one obtained when θ =0.5.

	Nature of	Average	Principal's	Less	More
	contracts	size of the	payoff	productive	productive
		pie		agent's payoff	agent's payoff
	All contracts	708.2	265.0	173.0	170.2
	Contracts withθ =0.25	824.3	230.6	242.9	228.6
	Equity-based contracts	812.0	228.0	240.9	226.1
	Contracts withθ =0.5	670.0	276.3	149.9	151.1
	Unfair contracts	713.1	296.1	159.9	150.2

Table 1 -

 1 Cahiers de laMSE -2000.48 Rossi, Alessandro, and Massimo Warglien, "Fairness in Hierarchical Triangles: An Experimental Study", University of Trento, Rock Working paper, 1999. Player Y1's payoffs

										APPENDIX
									SHARING RULE : TPS-1
	A1-	θ	=	0	25 .
	Effort			Y2	2	4	6	8	10	12	14	16	18	20	22	24
	Y1
		2				126	144	162	180	198	216	234	252	270	288	306	324
		4				132	150	168	186	204	222	240	258	276	294	312	330
		6				136	154	172	190	208	226	244	262	280	298	316	334
		8				138	156	174	192	210	228	246	264	282	300	318	336
		10			138	156	174	192	210	228	246	264	282	300	318	336
		12			136	154	172	190	208	226	244	262	280	298	316	334
		14			132	150	168	186	204	222	240	258	276	294	312	330
		16			126	144	162	180	198	216	234	252	270	288	306	324
		18			118	136	154	172	190	208	226	244	262	280	298	316
		20			108	126	144	162	180	198	216	234	252	270	288	306
		22			96	114	132	150	168	186	204	222	240	258	276	294
		24			82	100	118	136	154	172	190	208	226	244	262	280

Table 2 -

 2 Player Y2's payoffs

	Effort	Y2	2	4	6	8	10	12	14	16	18	20	22	24
	Y1
	2		126	141	154	165	174	181	186	189	190	189	186	181
	4		135	150	163	174	183	190	195	198	199	198	195	190
	6		144	159	172	183	192	199	204	207	208	207	204	199
	8		153	168	181	192	201	208	213	216	217	216	213	208
	10		162	177	190	201	210	217	222	225	226	225	222	217
	12		171	186	199	210	219	226	231	234	235	234	231	226
	14		180	195	208	219	228	235	240	243	244	243	240	235
	16		189	204	217	228	237	244	249	252	253	252	249	244
	18		198	213	226	237	246	253	258	261	262	261	258	253
	20		207	222	235	246	255	262	267	270	271	270	267	262
	22		216	231	244	255	264	271	276	279	280	279	276	271
	24		225	240	253	264	273	280	285	288	289	288	285	280
						Cahiers de la MSE -2000.48

Table 3 -

 3 Player X's payoffs

	Effort			Y2	2	4	6	8	10	12	14	16	18	20	22	24
	Y1
		2				118	130	142	154	166	178	190	202	214	226	238	250
		4				124	136	148	160	172	184	196	208	220	232	244	256
		6				130	142	154	166	178	190	202	214	226	238	250	262
		8				136	148	160	172	184	196	208	220	232	244	256	268
		10			142	154	166	178	190	202	214	226	238	250	262	274
		12			148	160	172	184	196	208	220	232	244	256	268	280
		14			154	166	178	190	202	214	226	238	250	262	274	286
		16			160	172	184	196	208	220	232	244	256	268	280	292
		18			166	178	190	202	214	226	238	250	262	274	286	298
		20			172	184	196	208	220	232	244	256	268	280	292	304
		22			178	190	202	214	226	238	250	262	274	286	298	310
		24			184	196	208	220	232	244	256	268	280	292	304	316
	A2-	θ	=	0	5 .

Table 4 -

 4 Player Y1's payoffs

	Effort	Y2	2	4	6	8	10	12	14	16	18	20	22	24
	Y1
	2		117	129	141	153	165	177	189	201	213	225	237	249
	4		120	132	144	156	168	180	192	204	216	228	240	252
	6		121	133	145	157	169	181	193	205	217	229	241	253
	8		120	132	144	156	168	180	192	204	216	228	240	252
	10		117	129	141	153	165	177	189	201	213	225	237	249
	12		112	124	136	148	160	172	184	196	208	220	232	244
	14		105	117	129	141	153	165	177	189	201	213	225	237
	16		96	108	120	132	144	156	168	180	192	204	216	228
	18		85	97	109	121	133	145	157	169	181	193	205	217
	20		72	84	96	108	120	132	144	156	168	180	192	204
	22		57	69	81	93	105	117	129	141	153	165	177	189
	24		40	52	64	76	88	100	112	124	136	148	160	172
						Cahiers de la MSE -2000.48

Table 5 -

 5 Player Y2's payoffs

	Effort	Y2	2	4	6	8	10	12	14	16	18	20	22	24
	Y1
	2		117	126	133	138	141	142	141	138	133	126	117	106
	4		123	132	139	144	147	148	147	144	139	132	123	112
	6		129	138	145	150	153	154	153	150	145	138	129	118
	8		135	144	151	156	159	160	159	156	151	144	135	124
	10		141	150	157	162	165	166	165	162	157	150	141	130
	12		147	156	163	168	171	172	171	168	163	156	147	136
	14		153	162	169	174	177	178	177	174	169	162	153	142
	16		159	168	175	180	183	184	183	180	175	168	159	148
	18		165	174	181	186	189	190	189	186	181	174	165	154
	20		171	180	187	192	195	196	195	192	187	180	171	160
	22		177	186	193	198	201	202	201	198	193	186	177	166
	24		183	192	199	204	207	208	207	204	199	192	183	172

Table 6 -

 6 Player X's payoffs

	Effort	Y2	2	4	6	8	10	12	14	16	18	20	22	24
	Y1
	2		136	160	184	208	232	256	280	304	328	352	376	400
	4		148	172	196	220	244	268	292	316	340	364	388	412
	6		160	184	208	232	256	280	304	328	352	376	400	424
	8		172	196	220	244	268	292	316	340	364	388	412	436
	10		184	208	232	256	280	304	328	352	376	400	424	448
	12		196	220	244	268	292	316	340	364	388	412	436	460
	14		208	232	256	280	304	328	352	376	400	424	448	472
	16		220	244	268	292	316	340	364	388	412	436	460	484
	18		232	256	280	304	328	352	376	400	424	448	472	496
	20		244	268	292	316	340	364	388	412	436	460	484	508
	22		256	280	304	328	352	376	400	424	448	472	496	520
	24		268	292	316	340	364	388	412	436	460	484	508	532

Table 7 -

 7 Player Y1's payoffs

	Effort	Y2	2	4	6	8	10	12	14	16	18	20	22	24
	Y1
	2		105	117	129	141	153	165	177	189	201	213	225	237
	4		108	120	132	144	156	168	180	192	204	216	228	240
	6		109	121	133	145	157	169	181	193	205	217	229	241
	8		108	120	132	144	156	168	180	192	204	216	228	240
	10		105	117	129	141	153	165	177	189	201	213	225	237
	12		100	112	124	136	148	160	172	184	196	208	220	232
	14		93	105	117	129	141	153	165	177	189	201	213	225
	16		84	96	108	120	132	144	156	168	180	192	204	216
	18		73	85	97	109	121	133	145	157	169	181	193	205
	20		60	72	84	96	108	120	132	144	156	168	180	192
	22		45	57	69	81	93	105	117	129	141	153	165	177
	24		28	40	52	64	76	88	100	112	124	136	148	160

Table 8 -

 8 Player Y2's payoffs

	Effort	Y2	2	4	6	8	10	12	14	16	18	20	22	24
	Y1
	2		111	132	151	168	183	196	207	216	223	228	231	232
	4		123	144	163	180	195	208	219	228	235	240	243	244
	6		135	156	175	192	207	220	231	240	247	252	255	256
	8		147	168	187	204	219	232	243	252	259	264	267	268
	10		159	180	199	216	231	244	255	264	271	276	279	280
	12		171	192	211	228	243	256	267	276	283	288	291	292
	14		183	204	223	240	255	268	279	288	295	300	303	304
	16		195	216	235	252	267	280	291	300	307	312	315	316
	18		207	228	247	264	279	292	303	312	319	324	327	328
	20		219	240	259	276	291	304	315	324	331	336	339	340
	22		231	252	271	288	303	316	327	336	343	348	351	352
	24		243	264	283	300	315	328	339	348	355	360	363	364
						Cahiers de la MSE -2000.48

Table 9 -

 9 Player X's payoffs

	Effort			Y2	2	4	6	8	10	12	14	16	18	20	22	24
	Y1
		2				106	118	130	142	154	166	178	190	202	214	226	238
		4				112	124	136	148	160	172	184	196	208	220	232	244
		6				118	130	142	154	166	178	190	202	214	226	238	250
		8				124	136	148	160	172	184	196	208	220	232	244	256
		10			130	142	154	166	178	190	202	214	226	238	250	262
		12			136	148	160	172	184	196	208	220	232	244	256	268
		14			142	154	166	178	190	202	214	226	238	250	262	274
		16			148	160	172	184	196	208	220	232	244	256	268	280
		18			154	166	178	190	202	214	226	238	250	262	274	286
		20			160	172	184	196	208	220	232	244	256	268	280	292
		22			166	178	190	202	214	226	238	250	262	274	286	298
		24			172	184	196	208	220	232	244	256	268	280	292	304
	A4-	θ	=	0	5 .

Table 10 -

 10 Player Y1's payoffs

	Effort	Y2	2	4	6	8	10	12	14	16	18	20	22	24
	Y1
	2		103	111	119	127	135	143	151	159	167	175	183	191
	4		104	112	120	128	136	144	152	160	168	176	184	192
	6		103	111	119	127	135	143	151	159	167	175	183	191
	8		100	108	116	124	132	140	148	156	164	172	180	188
	10		95	103	111	119	127	135	143	151	159	167	175	183
	12		88	96	104	112	120	128	136	144	152	160	168	176
	14		79	87	95	103	111	119	127	135	143	151	159	167
	16		68	76	84	92	100	108	116	124	132	140	148	156
	18		55	63	71	79	87	95	103	111	119	127	135	143
	20		40	48	56	64	72	80	88	96	104	112	120	128
	22		23	31	39	47	55	63	71	79	87	95	103	111
	24		4	12	20	28	36	44	52	60	68	76	84	92

Table 11 -

 11 Player Y2's payoffs

	Effort	Y2	2	4	6	8	10	12	14	16	18	20	22	24
	Y1
	2		107	120	131	140	147	152	155	156	155	152	147	140
	4		115	128	139	148	155	160	163	164	163	160	155	148
	6		123	136	147	156	163	168	171	172	171	168	163	156
	8		131	144	155	164	171	176	179	180	179	176	171	164
	10		139	152	163	172	179	184	187	188	187	184	179	172
	12		147	160	171	180	187	192	195	196	195	192	187	180
	14		155	168	179	188	195	200	203	204	203	200	195	188
	16		163	176	187	196	203	208	211	212	211	208	203	196
	18		171	184	195	204	211	216	219	220	219	216	211	204
	20		179	192	203	212	219	224	227	228	227	224	219	212
	22		187	200	211	220	227	232	235	236	235	232	227	220
	24		195	208	219	228	235	240	243	244	243	240	235	228

Table 12 -

 12 Player X's payoffs

	Effort	Y2	2	4	6	8	10	12	14	16	18	20	22	24
	Y1
	2		112	136	160	184	208	232	256	280	304	328	352	376
	4		124	148	172	196	220	244	268	292	316	340	364	388
	6		136	160	184	208	232	256	280	304	328	352	376	400
	8		148	172	196	220	244	268	292	316	340	364	388	412
	10		160	184	208	232	256	280	304	328	352	376	400	424
	12		172	196	220	244	268	292	316	340	364	388	412	436
	14		184	208	232	256	280	304	328	352	376	400	424	448
	16		196	220	244	268	292	316	340	364	388	412	436	460
	18		208	232	256	280	304	328	352	376	400	424	448	472
	20		220	244	268	292	316	340	364	388	412	436	460	484
	22		232	256	280	304	328	352	376	400	424	448	472	496
	24		244	268	292	316	340	364	388	412	436	460	484	508

Cahiers de laMSE -2000.48