Social Preferences and Strategic Uncertainty: An Experiment on Markets and Contracts by Antonio Cabrales , Rafaele Miniaci ** Marco Piovesan ^***and Giovanni Ponti DOCUMENTO DE TRABAJO 2009-09

Serie Talento, Esfuerzo y Movilidad Social CÁTEDRA Fedea – Banc Sabadell Serie Capital Humano y Empleo CÁTEDRA Fedea – Santander

February 2009

* Universidad Carlos III de Madrid, CEPR and FEDEA.

** Università di Brescia.

*** University of Copenaghen.

*** Universidad de Alicante and Università di Ferrara.

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Social Preferences and Strategic Uncertainty: an Experiment on Markets and Contracts^!

Antonio Cabrales

Raffaele Miniaci

Universidad Carlos III de Madrid

Università di Brescia

Marco Piovesan

University of Copenaghen

Giovanni Ponti^†

Universidad de Alicante

and Università di Ferrara

KEYWORDS: Social Preferences, Team Incentives, Mechanism Design, Experimental Economics

JEL CLASSIFICATION: C90, D86

Social Preferences and Strategic Uncertainty: An Experiment on Markets and Contracts

Abstract

This paper reports a 3-phase experiment on a stylized labor market. In the Þrst two phases, agents face simple games, which we use to estimate subjects’ social and reciprocity concerns, together with their beliefs. In the last phase, four principals, who face four teams of two agents, compete by o ering agents a contract from a Þxed menu. Then, each agent selects one of the available contracts (i.e. he “chooses to work” for a principal). Production is determined by the outcome of a simple e ort game induced by the chosen contract. We Þnd that (heterogeneous) social preferences are signiÞcant determinants of choices in all phases of the experiment. Since the available contracts display a trade-o between fairness and strategic uncertainty, we observe that the latter is a much stronger determinant of choices, for both principals and agents. Finally, we also see that social preferences explain, to a large extent, matching between principals and agents, since agents display a marked propensity to work for principals with similar social preferences.

KEYWORDS: Social Preferences, Team Incentives, Mechanism Design, Experimental Economics

JEL CLASSIFICATION: C90, D86

When it comes to assess the distribution of rewards within and among organizations, economists have, by and large, taken the view that inequality is a natural consequence of disparities in ability, or simply of asymmetric information.¹ In marked contrast, social psychologists emphasize the deleterious e ects of inequality on workers’ motivations and social relations within the organization.² More recently, these two stylized perspectives have moved towards less extreme viewpoints. On the one hand, many economists now accept that workers have social (i.e. interdependent) and/or reciprocal preferences, with a strong taste against inequality.³ On the other hand, social psychologists recognize that there are situations in which inequality may be beneÞcial for an organization. For example, Bloom (1999) claims that, since “... greater dispersion is negatively related to the performance of those lower in the dispersion [...] and positively related of those higher in the dispersion, [...] it may be beneÞcial for a law Þrm to pay a relatively high salary to attract a top attorney or for a university to o er an endowed chair to a particularly productive scholar. In other types of organizations [...] the situation is quite di erent because the poor performance of a particular worker cannot be compensated for by the better performance of the other workers”.

The interesting part of this latter observation, from our point of view, is that the beneÞts of inequality seem to be directly linked to the existence of activities which display strategic complementarities, which often lead to multiple equilibria.⁴ This goes along the lines of Winter’s (2004) model of moral hazard in teams, which shows that complementarities are not only su cient, but also necessary for the optimal contract -the one which implements the high-e ort proÞle as the unique equilibrium of the game- to yield inequality in rewards. This is because, “if agents’ exertion of e ort induces a positive externality on the e ectiveness of other agents’ e ort, it is optimal to promise high rewards to some agents so as to make the others conÞdently believe that these highly paid agents will contribute, hence allowing the planner to save resources by o ering other agents substantially less”. Even though Winter’s (2004) result abstracts from the existence of social preferences, it adds an additional ingredient to the debate on inequality by showing that the principal faces a trade-o between robustness and fairness considerations: fairness can be obtained only at the expense of robustness to strategic uncertainty.⁵ In this respect, one can only expect this trade-o to be exacerbated by the presence of (inequality-averse) distributional preferences.

¹ A simple model with hidden actions would predict that agents with the same level of ability receive unequal pay even if in equilibrium they make the same amount of e ort, due to purely random variations in output. See e.g. Macho-Stadler and Pérez-Castrillo (1997), chapter 3.

² “Decision makers are likely to use the equity principle in employment contexts and to use the equality principle to allocate resources in social contexts in which maintaining harmony and positive relationships are the primary goals.” (Jawahar, 2005).

³ Bewley (1999) is the seminal reference o ering survey evidence on the importance of equity concerns in organizations. Other papers on this topic are Blinder and Choi (1990), Bewley (1995), Agell and Lundborg (1995, 2003) and Campbell and Kamlani (1997).

⁴ Many payo functions display strategic complementarities. A well-known one is the so-called “O-ring” production function, originally proposed by Kremer (1993), which has been applied in a large number of empirical and theoretical works. Heywood and Jirjahn (2004), or Mirza and Nicoletti (2004), are recent examples of papers in very di erent Þelds which take this production function as the basis of their empirical work.

The aim of this paper is precisely to test experimentally the idea that workers’ (heterogeneous) social preferences are crucial in determining the contracts they are o ered and choose.⁶ We are also interested in the way our experimental subjects resolve the trade-o between robustness and inequality, as they can choose either i) contracts in which -following Winter (2004)- the all-e ort proÞle is the unique equilibrium, but inequality is enhanced; or ii) contracts in which the all-e ort proÞle is not the unique equilibrium, but inequality is mitigated. In this respect, subjects more concerned with equity (and less worried about coordination failure) may Þnd convenient to opt for the latter alternative. Finally, since another solution to the trade-o is sorting (agents with similar distributional concerns work for the same Þrms), this will also be an important element of our experimental design.⁷

With these goals in mind, we design and perform an experiment with three phases.

1. In the Þrst phase , subjects are matched for 24 rounds with a di erent partner and have to choose among four possible options involving a payo pair -one for them, one for their matched partner- in a Dictator Game-type protocol. We use to estimate subjects’ purely distributional preference parameters within the realm of Charness and Rabin’s (2002, C&R hereafter) model.

2. In the second phase , subjects are again matched in pairs for 24 rounds and asked to choose among the same payo pairs. However, this time options correspond to “contracts”, as they yield a 2×2 e ort game induced by Winter’s (2004) technology, which subjects then have to play at a second stage. In reciprocity may play a role, since agents may condition their second-stage e ort decision on their teammate’s contract choice. Thus, we use to estimate subjects’ C&R reciprocity parameters, together with their beliefs in the e ort game.

3. Finally, in the third phase ! there are 4 principals and 4 pairs (“teams”) of agents.

⁵ Van Huyck, Battalio and Beil (1990, 1991) are probably the best known experimental works on the e ects of strategic uncertainty in coordination games. Crawford (1995) and Crawford and Haller (1990) are theoretical papers partly inspired by these experimental results. Heinemann, Nagel and Ockenfels (2008) experiments measure the extent and importance of strategic uncertainty in coordination games. López-Pintado, Ponti and Winter (2008) test directly Winter’s (2004) model in the lab.

⁶ Fershtman, Hvide and Weiss (2005), Rey-Biel (2008) and Kosfeld and von Siemens (2006) explore theoretically the e ects of social preferences on e ort and cooperation. On the experimental side, Charness (2004) shows that volition in choosing a wage has a signiÞcant e ect on subsequent costly e ort provision. Fehr, Klein and Schmidt (2007) show (theoretically and experimentally) that even a minority of people with concerns for fairness can alter the kind of contracts that are efficient.

⁷ Cabrales, Calvó-Armengol and Pavoni (2008), Cabrales and Calvó-Armengol (2008) and Teyssier (2008) show that social preferences lead to more productive workers sorting themselves into di erent Þrms than the remaining workers. In a controlled laboratory experiment, Dohmen and Falk (2006) Þnd that more productive workers self-select into Þrms with variable pay schemes. Krueger and Schkade (2007) and Bellemare and Shearer (2006) provide Þeld evidence suggesting that sorting by preference traits is an important determinant of contract choices.

Principals o er a contract game, such as those played in selected from a given set. The presence of several competing principals acts as a kind of menu of contracts, among which agents may sort themselves.

This three-stage experimental design (and the associated estimation strategy) is novel, and it is especially designed to solve the identiÞcation problem discussed by Manski (2002), as we use it to disentangle preference and beliefs parameters. Since in beliefs do not play any role, we use data from to estimate subjects’ distributional preference parameters. Under the assumption that the latter are constant across phases, we then use data from to estimate subjects’ reciprocity concerns and beliefs.⁸ Our estimation of distributional parameters is carried out at the level of each individual subject participating in the experiment.⁹

Let us summarize the main results of our study.

1. Subjects display a signiÞcant degree of heterogeneity in their decisions, and thus, in estimated preferences and beliefs.

2. This heterogeneity explains, to a large extent, agents’ behavior. That is, preferences and beliefs which best explain agents’ behavior in and ! also explain well the contracts they choose among those o ered by the di erent principals in ! together with their subsequent e ort decision.

3. We also observe that equality is a less important consideration than robustness, for both principals and agents, since the egalitarian (but not robust) contract is rarely selected and, when it is selected, it very often yields the (inefficient) low e ort outcome. This, in turn, implies lower proÞts, for both principals and agents.

4. Finally, we Þnd that principals and agents sort themselves according with their social preferences. An agent’s probability of selecting a contract in decreases with the distance between her estimated preferences and those of the principal for whom she ends up working for. Moreover, principals also end up o ering contracts in tune with their own estimated distributional preferences.

The remainder of this paper is arranged as follows. Section 1 presents the experimental design, while in Section 2 we develop an econometric model to estimate distributional preferences and beliefs. Section 3 discusses our testable hypotheses and reviews the relevant literature. Final remarks are placed in Section 4. Three Appendices provide proofs, additional statistical evidence and experimental instructions.

⁸ In comparison with the approach recently proposed by Bellemare et al. (2008), we do not rely upon hypothetical subjective probability questions to estimate beliefs. More generally, as Nyarko and Schotter (2002) acknowledge, belief elicitation has its own problems “As is true of all scoring functions, while payo s are maximized by truthful revelation of beliefs, there are other beliefs that could be stated that are more secure [...] If subjects were risk averse, such an action might be desirable.” We opted for our design because it allows us to identify cleanly the distributional preferences, separating them from belief identiÞcation, withou distracting the subjects with new tasks. Given the complication of the overall design, this seemed to us a sensible strategy.

⁹ Many other papers in the “social preferences” literature, such as Fehr and Schmit (1999,2003), Costa-Gomes and Zauner (2001), or Charness and Rabin (2002) only provide pooled estimates. One noticeable exception is the paper of Fisman, Kariv and Markovits (2007), which provides individual distributional preference parameters for those subjects whose behavior is sufficiently “consistent”, in the sense of Afriat (1972).

1 Experimental design

In what follows, we introduce the features of our experimental environment.

1.1 Sessions

Three experimental sessions were conducted at the Laboratory of Theoretical and Experimental Economics (LaTEx), of the Universidad de Alicante. A total of 72 students (24 per session) were recruited among the undergraduate population of the Universidad de Alicante -mainly, students from the Economics Department with no (or very little) prior exposure to game theory. The experimental sessions were computerized. Instructions were read aloud and we let subjects ask about any doubt they may have had.¹⁰ In all sessions, subjects were divided into two matching groups of 12. Subjects from di erent matching groups never interact with each other throughout the session. Given this design feature, we shall read the data under the assumption that the history of each matching group (6 in total) corresponds to an independent observation.

1.2 Choice sets

Our experiment involves, for each one of the 24 rounds # constituting each phase, two subjects, 1 and 2, deciding over a choice set of four options ! where each option constitutes a pair , with by construction. Each pair determines the payo matrix of a simple e ort game . The rules of are as follows. Each agent ! has to decide, simultaneously and independently, whether to make a costly e ort. We denote by agent )’s e ort decision, where if agent ) does (does not) make e ort. Let also denote the agents’ action proÞle. The monetary payo of agent ) is described by

\[\pi_ {i t} ^ {k} (\delta) = B + P (\delta) b _ {i t} ^ {k} - \delta_ {i} c.\tag{1}\]

where

\[P (\delta) = \left\{ \begin{array}{l} 0 \text {if} \delta_ {1} + \delta_ {2} = 0, \\ \gamma \text {if} \delta_ {1} + \delta_ {2} = 1, \\ 1 \text {if} \delta_ {1} + \delta_ {2} = 2, \end{array} \right.\tag{2}\]

, is a beneÞt not depending on e ort decision, and - is the cost of e ort. Players receive their payo in full if they both (independently and simultaneously) coordinate on the e ort decision. In our experiment we and

^{1 0} The experiment was programmed and conducted with the software z-Tree (Fischbacher, 2007). The complete set of instructions, translated into English, can be found in Appendix C.

The pairs were drawn at random in the positive orthant, but not uniformly.

Figure 1' The experimental contract sets

In Figure 1 we report all pairs used in the experiment. As Figure 1 clearly shows, these pairs are concentrated in two “clouds”, which di er one another from the fact that, for some pair, Player 1 (the “advantaged” player within the 2-member team) receives substantially more. As we explain in detail in Appendix A, the two clouds includes pairs which are the solutions of two di erent mechanisms design problems aimed at inducing both players to provide e ort. The two mechanism design problems di er in that

1. under the “weak e ort inducing” solution (wing hereafter) players have a strict incentive to put e ort only if the other does;

2. under the “strong e ort inducing” solution (sting hereafter) player 1’s payo is suf-Þciently high to provide a strict incentive to put e ort independently on what player 2 does, while player 2, like in the wing solution, has a strict incentive to put e ort only if player 1 does. This implies that, under the sting solution, the all-e ort proÞle is the unique equilibrium of the induced game, while under the wing solution also the all-no-e ort proÞle is an equilibrium.

Unlike Winter (2004), who focuses on Egoistic (i.e. non distributional) Preferences (EP), we solve the two mechanism design problems under a wide variety of distributional preferences analyzed by the literature. This explains the additional payo variability within each cloud (where the larger points in each cloud identiÞes the corresponding EP solution).

The interested reader can Þnd in Appendix A all the details. What is important to stress here is that our choice set provides sufficient variability in payo s to estimate individual social preferences in Section 2.1, and that the speciÞc variability we created (essentially, payo s of similar magnitude for player 2, while a substantial di erence in monetary prizes for player 1, depending on whether a wing or a sting solution is applied) gives some formal dress to the discussion on the trade-o between equality and robustness we proposed earlier.¹¹

Depending on the round #, the choice set could be made by i) 4 wing contracts generated from 4 di erent preference proÞles; ii) 4 sting contracts generated from 4 di erent preference proÞles; or iii) 2 wing and 2 sting generated by two di erent preference proÞles.

We grouped rounds into time intervals. A time interval is deÞned as a group of three consecutive rounds (starting at 1), and indexed by / so that round ! is part of time interval . Within each time interval subjects experienced each and every possible situation, i) to )))) The particular sequence of three situations within each time interval was randomly generated. We did so to keep under control the time distance between two rounds characterized by the same situation. Player position (either player 1 or player 2) was also chosen randomly, for each team and round.

^{1 1} The fact that monetary payo s are derived from a speciÞc theoretical exercise -instead of simply randomly generated- has no further impact on the experimental design. Subjects were not acknowledged, at any time, on where those numbers came from: they simply had to choose, at each round, one out of four di erent options, with no further explanation.

1.3 Phases

Subjects played three phases, to of increasing complexity, for a total of 72 rounds (24 rounds per phase).

' Dictator Game (24 rounds) In this phase we use a variant of the classic protocol of the Dictator Game. The timing for each round # and matching group is as follows:

1. At the beginning of the round, six pairs are formed at random. Within each pair, another (independent and uniformly distributed) random device determines player position (i.e. the identity of the best paid agent).

2. After each agent is informed of her player position in the pair (common to all options in ! she has to select her preferred choice. The monetary payo associated to each choice corresponds to the all-e ort proÞle payo ,

3. Once choices are made, another independent draw Þxes the identity of the Dictator.

4. The Dictator’s choice, &! determines monetary payo s for that pair and round.

: E ort Game (24 rounds) Stages 1 to 3 are identical to those of ' Instead of stage 4, we have

4 Subjects are asked to play the e ort game described above. Subjects’ action proÞle determines their Þnancial reward (1).

: The Market (24 rounds) At the beginning of ! within each matching group, 4 subjects are randomly chosen to act as “Principals”. Then, in each round #! these 4 principals have to select one contract within the choice set to be o ered to the 4 teams of agents in their matching group. We denoted by the set of contracts o ered by at least one Principal (this set may be a singleton, since contracts o ered by Principals may all coincide, as it often happened in the experiment). Agents have then to choose within this subset Stages 2-4 are then identical to those of ' The payo for the Principal is calculated as the di erence between total output, ! and total costs:

\[\pi_ {0} ^ {k} (\delta) = P (\delta) (V - b _ {1} ^ {k} - b _ {2} ^ {k}),\]

with and in the experiment (i.e. )'

1.4 Monetary payo s

All monetary payo s in the experiment were expressed in Spanish Pesetas (1 euro is approx. 166 ptas.).¹² Subjects received 1'000 ptas. just to show up, to which they summed up all their cumulative earnings throughout the 24×3 = 72 rounds of the experiment.¹³ Average earnings were about 21 euros, for an experimental session lasting for approximately 90 minutes.

1.5 Three (testable) questions

We are now in the position to specify the main objectives of our experiment.

Q1. Is it inequality aversion or strategic uncertainty aversion? Contracts have been calculated using two di erent mechanism design strategies, with rather di erent distributional characteristics. Two kinds of questions arise here.

Q1.1. Which contract type (sting or wing) is chosen more often by principals and agents? Evidence for this in Remark 1

Q1.2. What is the role of strategic uncertainty? That is, to which extent the (non) existence of multiple equilibria in wing (sting) a ects agents’ behavior in the e ort game. Evidence for this in Remarks 2 and 3.

Q2 Does separation emerge? That is, is the market able to sort (principals and) agents according to their distributional and reciprocity preferences? Evidence for this in Remarks 4 and 5.

Q3. Do models of social preferences work? That is, does a model with distributional and reciprocity preferences provide a reliable framework to predict principals and agents behavior? Evidence for this in Remark 6.

2 Identifying preferences and beliefs

In what follows, and ! identify our subjects matched in pairs. We assume that our subjects’ preferences follow C&R, as we explain in the following

^{1 2} It is standard practice, for all experiments run in Alicante, to use Spanish ptas. as experimental currency. The reason for this design choice is twofold. First, it mitigates integer problems, compared with other currencies (USD or Euros, for example). On the other hand, although Spanish pesetas are no longer in use (substituted by the Euro in the year 2002), Spanish people still use Pesetas to express monetary values in their everyday life. In this respect, by using a “real” (as a opposed to an artiÞcial) currency, we avoid the problem of framing the incentive structure of the experiment using a scale (e.g. “Experimental Currency”) with no cognitive content.

^{1 3} In other papers in this area subjects are paid according to the outcome of a randomly chosen period (instead of the accumulated payo s, as we do). Our design choice was dictated by our focus on “strategic” uncertainty, which led us to reduce other sources of uncertainty (notice that we also replaced the uncertain payo s of the theoretical benchmark by their certainty equivalent).

DeÞnition 1 (C&R Preferences))

\[\begin{array}{r c l} u _ {i} (\delta) & = & \pi_ {i} (\delta) \\ & & - (\alpha_ {i} - \theta_ {i} \phi_ {j}) \max \left\{\pi_ {j} (\delta) - \pi_ {i} (\delta), 0 \right\} - (\beta_ {i} + \theta_ {i} \phi_ {j}) \max \left\{\pi_ {i} (\delta) - \pi_ {j} (\delta), 0 \right\}, \end{array}\tag{3}\]

where “has misbehaved”, and otherwise (we provide an operational deÞnition of misbehavior a little later in this Section). In words, if player ! has “misbehaved”, player increases her “envy” parameter (or lowers her “guilt” parameter by an amount equal to . Thus, can be interpreted as player ’s sensitivity to negative reciprocity. Model (3) has the useful feature that it subsumes parameters which account for subjects’ distributional tastes a’ la Fehr and Schmidt (1999, F&S), and as well as for their tastes for reciprocity,

Our experimental setup seems particularly well suited to estimate both distributional and reciprocity concerns. With respect to the former, there are four relevant subsets of parameters, which we now describe. All these speciÞcations do not consider reciprocal motives (i. , it is always assumed ) and, in this sense, deÞne purely “distributional” preferences.

\[\mathrm{EgoisticPreferences(EP)} \colon \alpha_ {i} = \beta_ {i} = 0.\tag{4}\]

\[\text { Inequality Averse Preferences (IAP): } 0 \leq \beta_ {i} < 1, \alpha_ {i} \geq \beta_ {i}.\tag{5}\]

\[\text { Status Seeking Preferences (SSP): } \alpha_ {i} \in [ 0, 1), \beta_ {i} \in (- 1, 0 ], | \alpha_ {i} | \geq | \beta_ {i} |\tag{6}\]

\[\text { Efficiency Seeking Preferences(ESP):} \alpha_ {i} \in (- \frac {1}{2}, 0 ], \beta_ {i} \in [ 0, \frac {1}{2}), | \beta_ {i} | \geq | \alpha_ {i} |\tag{7}\]

Inequality averse preferences (5) were Þrst proposed by F&S. The literature has also focused upon two alternative subsets of parameters for (3), namely SSP (Frank, 1984) and ESP (Engelmann and Strobel, 2004). The former assumes that an increase in the other player’s monetary payo is always disliked, independently of relative positions. The latter, that a reduction in her own payo is acceptable only if it accompanied by an increase (at least of the same amount) in the other player’s payo . Even though C&R follow F&S in only considering IAP, we jointly call -with a slight abuse of notation- “C&R distributional preferences” the four types of preferences (4)-(7).

2.1 Estimating distributional preferences using

In each round -, let be a dummy variable which is equal to 1 if subject is the lower paid agent- and zero otherwise. Assuming that each subject is characterized by her own parameters and , her utility from choosing option / at round - can be written as

\[u _ {i t} ^ {k} = (1 - L _ {i t}) \left[ \pi_ {1 t} ^ {k} - \beta_ {i} (\pi_ {1 t} ^ {k} - \pi_ {2 t} ^ {k}) \right] + L _ {i t} [ \pi_ {2 t} ^ {k} - \alpha_ {i} (\pi_ {1 t} ^ {k} - \pi_ {2 t} ^ {k}) ] + \varepsilon_ {i t} ^ {k}.\]

According to this notation, subject chooses option / at round - if

\[u _ {i t} ^ {k} = \max \left(u _ {i t} ^ {1}, \dots , u _ {i t} ^ {4}\right).\]

Under the assumption that the stochastic term is iid with an extreme value distribution, the probability that individual chooses option / at round - is therefore

\[\begin{array}{l} \operatorname * {P r} \left(y _ {i t} = k | \pi_ {1} (.), \pi_ {2} (.)\right) = \\ \frac {\exp \left((1 - L _ {i t}) \left[ \pi_ {1 t} ^ {k} - \beta_ {i} \left(\pi_ {1 t} ^ {k} - \pi_ {2 t} ^ {k}\right) \right] + L _ {i t} \left[ \pi_ {2 t} ^ {k} - \alpha_ {i} \left(\pi_ {1 t} ^ {k} - \pi_ {2 t} ^ {k}\right) \right]\right)}{\sum_ {k = 1} ^ {4} \exp \left((1 - L _ {i t}) \left[ \pi_ {1 t} ^ {k} - \beta_ {i} \left(\pi_ {1 t} ^ {k} - \pi_ {2 t} ^ {k}\right) \right] + L _ {i t} \left[ \pi_ {2 t} ^ {k} - \alpha_ {i} \left(\pi_ {1 t ^ {\prime}} ^ {k} - \pi_ {2 t ^ {\prime}} ^ {k}\right) \right]\right)}. \end{array}\tag{8}\]

Notice that (8) allows for parameter heterogeneity across subjects. Thus, the iid assumption does not stem from neglected individual unobserved heterogeneity, and it is consistent with the random order of the four contracts in the choice set

In our estimates we do not constrain the parameters to adhere to any of the preference types (4)-(7). Our estimated couples can therefore potentially cover all the space. Figure A1 (in Appendix B) plots the estimated and of each subject participating to the experiment. In Table 1 we summarize this information by partitioning our subject pool, assigning each subject to the quadrant to of the space in which her estimated parameters are most likely to fall. At the same time, we group in an additional category those subjects whose estimated and are jointly not signiÞcantly di erent from zero (at the 10% conÞdence level). Subjects with IAP preferences are a subset of those included in the Þrst quadrant 19.4% of all the subjects), the pool in 22.2%) includes agents with SSP preferences, while those with ESP preferences fall in . For 19.4% of the subjects we cannot reject the null hypothesis of EP.

Table 1+ Preference types of agents and principals

2.2 Estimating reciprocity and beliefs using .

In ) after selecting their favorite option (now to be interpreted as a proper “contract”, that is, a beneÞt proÞle conditional on the joint e ort decision), agents are asked to play the induced e ort game, , in which they may condition their e ort decision upon the (publicly known) contract choice of their teammate.

This, in turn, implies that we can apply the full-ßedged behavioral model (3) to estimate our subjects’ reciprocal concerns. To do this, we need Þrst to operationally identify what “misbehavior” means in the context of our experimental setup. In this respect, we shall use contract choice decisions by ! and in Stage 1) deÞned as and ) respectively:

\[\phi_ {j} = \left\{ \begin{array}{c} - 1 \text { if } b _ {i} ^ {k _ {j}} < b _ {i} ^ {k _ {i}}, \\ 0 \text { otherwise }. \end{array} \right.\tag{9}\]

By (9), ! misbehaves by choosing a contract which assigns a strictly lower beneÞt than what would have guaranteed herself with

We can now look at agents’ e ort decisions in as the result of a process of expected utility maximization. Individual will choose to make e ort in Stage 2 if

\[E _ {\lambda_ {i} ^ {k}} \left[ u _ {i} ^ {k} \left(1, \delta_ {j} ^ {k}\right) - u _ {i} ^ {k} \left(0, \delta_ {j} ^ {k}\right) \right] > 0,\tag{10}\]

where indicates the expected value taken with respect to player ’s beliefs on !’s e ort decision, . We parametrize as a logistic function of the distributional features of contract and ) and on player ’s own misbehavior in Stage 1, :

\[\lambda_ {i} ^ {k} = \frac {\exp \left(\psi_ {1} \phi_ {i} + \psi_ {2} b _ {j} ^ {k} + \psi_ {3} (b _ {i} ^ {k} - b _ {j} ^ {k})\right)}{1 + \exp \left(\psi_ {1} \phi_ {i} + \psi_ {2} b _ {j} ^ {k} + \psi_ {3} (b _ {i} ^ {k} - b _ {j} ^ {k})\right)}.\tag{11}\]

Our belief speciÞcation (11) allows player to anticipate that her own behavior in Stage 1 may a ect ’s willingness to put e ort. In addition, and proxy the e ect associated with absolute and relative payo s. Our speciÞcation for the reciprocity parameter in (3) allows behavior to a ect e ort decision di erently, according to ⁰s player position (1 vs. 2) and to the Dictator role. Letting if individual ! is the Dictator, and zero otherwise, we have.

\[\theta_ {i} = \theta_ {1} D _ {i} (1 - L _ {i}) + \theta_ {2} (1 - D _ {i}) (1 - L _ {i}) + \theta_ {3} D _ {i} L _ {i} + \theta_ {4} (1 - D _ {i}) L _ {i}.\tag{12}\]

Assuming that the latent index on the LHS of (10) has an extreme value distribution, the probability to observe the subject ! making e ort given the plan & is given by

\[\begin{array}{l} \operatorname * {P r} \left(\delta_ {i} ^ {k} = 1 | (\alpha_ {i}, \beta_ {i}, \theta_ {i}), L _ {i}, D _ {i}, \left(b _ {1} ^ {k}, b _ {2} ^ {k}\right)\right) \\ = \frac {\exp \left(E _ {\lambda_ {i} ^ {k}} \left[ u _ {i} ^ {k} (1 , \delta_ {j} ^ {k}) \right]\right)}{\exp \left(E _ {\lambda_ {i} ^ {k}} \left[ u _ {i} ^ {k} (1 , \delta_ {j} ^ {k}) \right]\right) + \exp \left(E _ {\lambda_ {i} ^ {k}} \left[ u _ {i} ^ {k} (0 , \delta_ {j} ^ {k}) \right]\right)}. \end{array}\tag{13}\]

Since we posit that distributional preferences estimated in are constant across phases, the e ort decision taken in Stage 2 of reveals individuals’ subjective belief over their teammates’ e ort decision and their own sensitivity to reciprocity

Consistently, our estimation strategy is a two step procedure:

1. in we get estimates of distributional parameters, and from (8);

2. in we estimate - via partial maximum likelihood- the parameters of and replacing and in (13).

Given the two-step nature of the procedure, we use (8) to obtain bootstrap estimates of for each of the 72 subjects, and we use them to obtain a bootstrap distribution of Step 2 estimates.

Table 2 reports the estimation results, where the estimated standard errors of the parameters of and take into account matching group clustering.

Table 2% Estimated parameters of belief function and reciprocity.

As for our belief speciÞcation (11), we see that both coefficients associated with (relative) payo s, and are signiÞcant, indicating that player ! is expecting more e ort the higher payo and lower e ort if her teammates is Player and As for our account for reciprocity in ´5’s beliefs, ) we Þnd a positive coefficient, although not statistically signiÞcant. Similar considerations hold when we look at the estimates of the four coefficients for in (12) conditional on Player and Dictator positions.¹⁴ None of them is signiÞcant, and those associated with Player 2 (1) are positive (negative).

To summarize, our estimations do not yield statistically signiÞcant reciprocity parameters for subjects’ beliefs and behavior, at least conditional on the speciÞc functional forms (11)- (13). Conditional on the estimated distributional preferences we carry from , only (absolute and relative) payo s seem to have a signiÞcant e ect on how subjects form their beliefs and make their e ort decisions.¹⁵

3 Discussion

We devote this section to provide answers to our conjectural hypotheses and discuss several methodological (as well as empirical) issues raised by our novel theoretical and experimental setting.

3.1 1" Is it inequality aversion or strategic uncertainty aversion?

We Þrst analyze subjects’ revealed preferences over the type of contract, wing or sting, to see how subjects resolved the tension between fairness and strategic uncertainty we discussed earlier, and how this depends on their individual social preferences. As explained in Section 1, in 8 out of 24 rounds of the experiment, the choice set was compound by 2 wing and 2 sting contracts, built upon two pairs of distributional preferences (5)-(7). Table 3 reports the relative frequency of subjects’ choices of a sting contract in the 8 rounds in which both types of contracts were available.

θi.

^{1 4} In principle we could provide individual estimates for _!! In fact, we obtain estimates of _! which are θi signiÞcantly di erent from zero for only 20 out of 72 individuals, with 10 of them positive. But these results are difficult to interpret because given the number of observations available for each individual we are forced to impose that the reciprocity e ect does not vary according to the player position of the individuals (that is, Dictator vs. Non Dictator and Player 1 vs. Player 2). Table 1B in Appendix B provides prima facie evidence against these assumptions, thus we prefer to present a pooled estimate for !

^{1 5} It may be worth noticing at these stage that these qualitative results are robust across alternative functional speciÞcations for both beliefs and tastes for reciprocity. Results are not reported here, but are available upon request.

Table 3% Relative frequencies of the sting choice in the “mixed” rounds.

Remark 1 sting is the most frequent choice for all players and phases.

As Table 3 shows, in all phases, sting is by far the most popular choice, and this is particularly true for Player 1 (who, in goes for wing only 7 out of 288 times). Principals also display a higher preference for sting, even though choice frequencies are much closer to those of the less advantaged Players 2. To assess the extent to which social preferences a ect the probability of choosing a sting contract, we need to control for the inequality in the available choice set ) which varies substantially from round to round. In Appendix B we run a logit regression, whose main conclusions are:

1. The more “unequal” is the wing choice (i.e., the bigger are the payo di erences of the 2 wing contracts, relative to those of the 2 sting contracts in 6_%), the more likely is the choice of a sting contract, whatever the player position. On average, a 1% increase of a “relative inequality index” between wing and sting contracts we build for this purpose induces an increase of the 29% of the probability of choosing 78!9: for Player 2, and of 14% for the principals in

2. For principals, distributional parameters are not signiÞcant to explain the choice of contract type, while for Players 2 in , both ( and are signiÞcant, with opposite sign.

We now discuss agents’ e ort decisions in and . Table 6 shows that individual !’s willingness to put e ort is higher when she faces a 78!9: contract: when we focus on we see that, with a 78!9: contract, Player 1 puts e ort in 92% of the cases, while the same statistic drops to 51% in the ;!9: contracts. For Player 2 the corresponding Þgures are much lower (62% and 43%, respectively).¹⁶ If we compare the e ort decisions in and we observe that only for Player 1 in the ;!9: case there is an overall reduction of the e ort in . (51% vs 44%).

Table 4% Relative frequencies of positive e ort decisions in and

Remark 2 E ort is much higher in sting that in wing.

We now look at the extent to which contract choices are able to solve the coordination problems agents face in the e ort game. Table 5 shows that the relative frequencies of the all-e ort efficient equilibrium are about twice larger in sting than in wing (about 60% vs 30%).

2,

^{1 6} This consideration notwithstanding, it may be worth to remember -see Figure 1 - that di erence in e ort on behalf of Player 1 could be imputed to the higher beneÞts she enjoys under a sting contract. On the other hand, for Player absolute rewards do not vary much across contract types and higher e orts may be due, following Winter’s (2004) argument, to the reduced strategic uncertainty.

Remark 3 In wing, the inefficient all-no-e ort equilibrium pools more than of total observations, and it is played more frequently than the efficient all-e ort equilibrium.

While this frequency stays basically constant over phases and mechanisms, in sting the relative frequency of outcomes in which only Player 2 puts e ort never exceeds 4% while, in wing, this frequency is 3 times bigger. Also notice that about 30% of total observations correspond to a (non-equilibrium) strategy proÞle in which only one agent puts e ort.

Finally, if we look at the evolution of outcomes over time, we see that, for both wing and sting, the relative frequency of efficient equilibria is falling, although, in wing, this e ect is much stronger. In addition, the frequency of the inefficient no-e ort equilibria almost doubles, when we compare the Þrst and the last 12 repetitions of each phase.

Table 5% Outcome dynamics in the e ort game.

If we look at the mechanism design problem from the principal’s viewpoint, our evidence yields a clear preference for the “sting program”. Despite its being more expensive (since the sum of beneÞts to be distributed is higher), the di erence in average team e ort is sufficient to compensate the di erence in cost. In addition, in the “mixed” rounds of , principals o ering sting contracts were selected by agents with a much higher frequency. This, in turn, implies that average proÞts for a principal when o ering a sting contract in the “mixed” rounds was substantially higher, three times as much as the corresponding proÞts when o ering a wing contract (95.4 ptas. vs. 30.1).

3.2 2" Does separation emerge?

One way to interpret the results of the previous section is that distributional preferences play a role to resolve the trade-o implicit in the wing-sting choice only for Player 2. Matters change when is composed of the same contract type, either sting or wing, and therefore, di erences across contracts in are less pronounced. We refer to periods characterized by an homogeneous contract choice set as “non-mixed”. In this case, the wing-sting trade-o is not an issue, and principals and agents may Þne-tune their contract decisions to their individual distributional tastes. In Appendix B we show that when we focus our attention to relative inequality and relative total cost of chosen contracts by Principals and Agents (compared with the other available options in ) individual social preferences matter, and in the expected direction: more inequality averse Principals and Agents choose, on average, contracts in which inequality is reduced. By the same token, more inequality averse Principals go for “more expensive” contracts (i.e. contracts in which Agents’ beneÞts are higher).

Remark 4 Distributional preferences parameters estimated in account well for agents’ and principals’ and observed contract choices in and

This last remark could be interpreted as an indirect evidence on sorting: for both principals and agents, distributional concerns matter when it comes to decide which contract to o er and to choose. A more direct evidence on sorting would come from the direct inspection, in , of how distributional parameters explain the matching process. In other words, to properly understand sorting we need to look at the extent to which principals and agents of similar distributional tastes tend to form matches.

To do this, we estimate the probability that a principal is “chosen” by an agent in each period as a (logit) function of the (euclidean) distance -in the space- between agents’ and principals estimated distributional preferences:

\[\operatorname * {P r} \left(\textit {a g e n t i c h o o s e s p r i n c i p a l j} | (\alpha_ {i}, \beta_ {i}), (\alpha_ {j}, \beta_ {j}), D _ {c}\right) = \frac {\exp (\psi \sigma_ {i j} + \gamma^ {\prime} D _ {c})}{1 + \exp (\psi \sigma_ {i j} + \gamma^ {\prime} D _ {c})},\]

where and is a full set of matching group dummies. We estimate the model using only those periods in which not all the principals o er the same contract to the pool of agents. The estimated coefficient is -0.336, (bootstrap and cluster adjusted std. err. 0.099), for of 0.001. This evidence justiÞes the following

Remark 5 Agents are more likely to choose a contract o ered by a principal with more similar distributional preferences to her own.

3.3 3" Does the social preference model work?

To answer this question, we use data from to check whether our structural model is able to explain (and predict out-of-sample) agents’ e ort choices in Once we provide agents with parameters on tastes for distribution (estimated in , and reciprocity and beliefs about their teammate’s action in the e ort game (estimated in , we can fully characterize the agents’ e ort decision at the individual level in

Using the evidence from , each cell of Table 6 reports relative frequencies of actual positive e ort decisions, relative frequencies of predicted positive e ort decisions and relative frequencies of instances in which actual and predicted behavior coincide. Predicted behavior is identiÞed by subjects’ e ort decision which maximizes expected utility (3) in the e ort game, subject to their estimated preference parameters and their subjective beliefs,

Table 6% Actual and predicted behavior in Stage 2 of Phase

Overall (more details in Appendix B), the model seems to frame subjects’ decisions accurately, which justiÞes the following

^{1 7} Our behavioral model (3) clearly provides a suitable framework to predict agents’ e ort decisions. To also predict contract choices, it would be necessary to deal seriously with several additional problems. Three of those are particularly noteworthy. First, we would need to model agents’ beliefs on the probability of teammates “misbehavior” in the contract decision (and, in consequence, principals’ beliefs over those beliefs). Second, we would also need a robust model of competition among principals. And Þnally, we would have to deal with the incomplete information about agents’ (and other competing principals) preferences.

Remark 6 Estimated preferences and beliefs predict about 80% of observed agents’ e ort decisions.

A more indirect, but still useful, way to check for the ability of the model to account for the subjects’ behavior is to look at the robustness of estimates across alternative design speciÞcations. In this respect, two features of our experimental design looked, ex ante, particularly likely to have a ected our inferences from the data.¹⁸

1. In our experiment, player position assignment is the outcome of an i.i.d. draw. We did this to be able to obtain individual estimates of both distributional parameters, and . On the other hand, one might argue that, Þxing player position across the entire experiment, may yield di erent estimates for distributional and reciprocity parameters.¹⁹

2. Players were choosing their favorite contract before being acknowledged of the identity of the Dictator. The reason why we used this procedure (also known as the strategy method) was to collect observations on contract decisions for all subjects and rounds (not only in cases where a particular subject turned out to be the Dictator). However, one might argue that when using this procedure, fairness can be achieved in two ways: either by playing the “fair” equilibrium in each single round, or by playing the “unfair” equilibrium in each round (letting the random Dictator role allocation provide overall fairness). Thus, the uncertainty of not knowing whether the agent decision was binding could change agents’ behavior in di erent directions.²⁰

For these reasons, in May 2007, we run three extra sessions (i.e. 6 additional independent observations) to investigate these issues. In these new sessions we made only two modiÞcations of the original design:

(i) We Þxed the player position throughout the experience (i.e. across all 72 rounds).

(ii) We made public the identity of the Dictator before the contract choice (i.e. we only have observations on contract decisions on behalf of Dictators).

In what follows, we shall denote by , evidence coming from the original (alternative) treatment conditions. Clearly, in of & we can only estimate one distributional parameter per subject, either (Player 2) or (Player 1). Figure 2 shows the distributions of and estimated in of and

^{1 8} We thank two anonymous referees to point out these two possible drawbacks of our original design.

^{1 9} For example, inequality might be perceived as less important for the the “richest” Player 1, since she never experiences a position at the lower end of the stick (or, by the same token, inequality may be perceived as more important by the less favored Player 2).

^{2 0} For instance, it is possible that by Þxing the role of the Dictator before the choice of the contract we would observe that agents choose less often “fair” contracts (or would have a less pronounced concern for “fair" reciprocity).

Figure 2. Comparison of the distribution of and in and

As Figure 2 shows, social preference parameters display very similar distributions across treatments. For the empirical distributions depicted in Figure 2, the hypothesis of equality of the means is not rejected (with '-statistics equal to 0.24, and 1.07, respectively), and the Kolmogorov-Smirnov statistics do not reject the hypotheses of equality of the distributions (with KS statistics of 0.11 and 0.15, respectively).

By contrast, e ort decisions in and appear to be sensitive to treatment conditions (see Table 5B in Appendix B). Average e ort levels are higher in # $1, when subjects interchange player positions across rounds. This evidence points to a dynamic aspect of social preferences, which our static model cannot account for.

4 Conclusion

Our experimental results show that strategic uncertainty should be an important concern for those in charge of designing organizational incentives. In our context, where strategic uncertainty conßicts with social preferences in terms of their respective recommendations on contract design, the former seems to be the primary consideration. However, we also provide evidence showing that distributional preferences are a key determinant of contracts o ered and accepted, on e ort levels, as well as on how markets sort di erent attitudes towards distributional issues into di erent organizations.

Our experimental environment is certainly ad-hoc in some respects.²¹ Nevertheless, our results are encouraging, because a parsimonious model of individual decision making is capable of organizing consistently the evidence from a complex experimental environment. The stability of social preferences (and beliefs) across quite di erent environments is a positive piece of news for the research program in interdependent preferences.²²

We conclude by discussing three possible avenues for future research.

From a theoretical standpoint, it would be interesting to solve completely the mechanism design problem under incomplete information about the social preferences of the agent. From an empirical point of view, it would be interesting to observe the e ect of having agents of di erent productivities, which are also private information. In this way we could see how Þnely and in which ways “corporate culture” partitions the agents. Also, notice that, in our setup, the numbers of principals and agents exactly balance one another. Thus, the e ect of more intense competition on the side of either principals or agents is an empirically interesting extension.

^{2 1} Take, for example, our decision to give to only one agent the monopolistic power to decide the ruling contract for the entire team.

^{2 2} It is true that the literature has already discussed the ability of di erent models to explain quite diverse data sets. But this discussion has been done by showing that the same distribution of parameters that explains behavior in one experiment also explains behavior in a di erent one. Our experiments provide a more deÞnitive test, by following subjects’ choices, and showing their consistency with social preferences, across rather di erent tasks.

Finally, we also would like to check the extent to which agents’ decisions (and, consequently, the estimated distributional preferences which derive from these decisions) depend on whether the choice of the optimal contract is made before or after agents’ are told about their player position in the game. If agents choose the contract before knowing their relative position within the team (i.e. “under the veil of ignorance”), their decisions may also reßect individuals’ attitude to risk, as well as distributional considerations. This exercise would require to collect additional information about our experimental subjects on these two complementary dimensions, measuring how these dimensions interact in the solution of the decision problem facing them in the experiment.

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Table 1: Preference types of agents and principals

Table 2: Estimated parameters of beliefs function and reciprocity. Bootstrap and matching group adjusted standard errors

Table 3: Relative frequencies of the 0':;< choice in the “mixed” rounds

Table 4: Relative frequencies of positive e ort decisions in and . Number of cases for each player type in parenthesis

Fig. 1

Table 5: Outcome dynamics in the e ort game. Absolute values and row percentages

Fig. 2

Table 6: Actual and predicted behavior in Stage 2 of For each case we report relative frequencies of actual positive e ort decisions, relative frequencies of predicted positive e ort decisions, and the fraction of cases for which actual and predicted e ort behavior coincides. Number of cases in parenthesis.

NOT FOR PUBLICATION

1 Two mechanism design problems

1.1 Production technology

Technology closely follows Winter’s (2004) model of moral hazard in teams. Let deÞne the game-form associated with a given beneÞt proÞle, " The rules rules of the gameform are the following. Each agent ! has to decide, simultaneously and independently, whether to make a costly e ort. We denote by agent # ’s e ort decision, where if agent # does (does not) make e ort. Let also denote the agents’ action proÞle. The cost of e ort % is assumed to be constant across agents. Team activity results in either success or failure. Let deÞne production as the probability of success as a function of the number of agents in the team who have put e ort:

\[P (\delta) = \left\{ \begin{array}{l l} 0 \text { if } \delta_ {1} + \delta_ {2} = 0, \\ \gamma \text { if } \delta_ {1} + \delta_ {2} = 1, \\ 1 \text { if } \delta_ {1} + \delta_ {2} = 2, \end{array} \right.\tag{1}\]

with

If the project fails, then all (principal and agents) receive a payo of zero. If the project succeeds, then agent # receives a beneÞt, " Agent #⁰s expected monetary proÞt associated to contract ) is given by

\[\pi_ {i} ^ {k} (\delta) = P (\delta) b _ {i} ^ {k} - \delta_ {i} c.\tag{2}\]

The expected monetary payo for the principal is the di erence between expected revenues, for a given (randomly generated) value for the project ! and expected costs:

\[\pi_ {0} ^ {k} (\delta) = P (\delta) (V - b _ {1} ^ {k} - b _ {2} ^ {k}).\]

¹ This is how Winter (2004) models moral hazard: agents’ e ort a ect the overall probability of success of the project. However, since risk neutrality is assumed on agents’ behalf, the fact that technology follows a random -as opposed to deterministic, as in our design- process has no impact in the solution of the mechanism design problem.

Assume a principal who wishes to design a mechanism that induces all agents to exert e ort in (some) equilibrium of the game induced by 0( )! which we denote by ( ). A mechanism is an allocation of beneÞts in case of success, i.e., a vector that satisÞes this property at the minimal cost for the principal. Following Winter (2004), the principal may consider mechanisms that strongly or weakly implement the desired solution, depending of how concerned he is about equilibrium multiplicity. More precisely:

DeÞnition 1 (sting contracts) The contract is 12345678 e ort-inducing (sting) if all Nash Equilibria (NE) of ( ) entail e ort by all agents with minimal beneÞt distribution,

DeÞnition 2 (wing contracts) The contract is 9:;)78 e ort-inducing (wing) if there exists at least one NE of ( ) such that , with minimal beneÞt distribution.

2 The solutions

By analogy with our experimental conditions (and without loss of generality), we assume " In what follows, we shall assume that both agents hold either EP (as in Winter, 2004), or IAP, SSP and ESP, respectively. We allow for heterogeneous preferences, provided they belong to the same preference class.

2.1 Solution of the mechanism design problem under the wing program

In the case of wing, the search of the optimal mechanism corresponds to the following linear program:

\[b ^ {*} \equiv (b _ {1} ^ {*}, b _ {2} ^ {*}) \in \arg \min _ {\{b _ {1}, b _ {2} \}} [ b _ {1} + b _ {2} ] \text { sub }\tag{3}\]

\[u _ {1} (1, 1) \geq u _ {1} (0, 1)\tag{4}\]

\[u _ {2} (1, 1) \geq u _ {2} (1, 0)\tag{5}\]

\[b _ {1} \geq b _ {2} \geq 0\tag{6}\]

Assumption (6) is wlog. To solve the problem (3)-(6), we begin by partitioning the beneÞt space in two regions, which specify the payo ranking of each strategy proÞles in " This partition is relevant for our problem, since it determines whether in (1,0) - player 1 exerts e ort and player 2 does not - whether it is player 1 or 2 the one who experiences envy (guilt):

\[R _ {1} = \left\{b \in B: b _ {2} \leq b _ {1} - \frac {c}{\gamma} \right\};\]

\[R _ {2} = \left\{b \in B: b _ {1} - \frac {c}{\gamma} \leq b _ {2} \leq b _ {1} \right\}.\]

Let deÞne the two linear constraints upon which our partition is built. The strategy proof is as follows. We shall solve the linear program (3)-(6) in the two regions independently (since, within each region, social utility parameters are constant for each agent and strategy proÞle), checking which of the two solutions minimizes the overall beneÞt sum , and determining the constraints on preferences which determine the identity of the best-paid player 1.

2.1.1 Wing under EP

As for the solution of wing under EP (i.e. with , the linear program (3)-(6) simpliÞes to the following:

\[\min b _ {1} + b _ {2}\]

subject to:

\[\begin{array}{r c l} b _ {1} - c & \geq & \gamma b _ {1} \\ b _ {2} - c & \geq & \gamma b _ {2} \\ b _ {i} & \geq & 0; \quad \text {with} i = 1, 2 \end{array}\]

In this case, the solution of the problem is problem is trivial:

\[b _ {1} ^ {*} = b _ {2} ^ {*} = \frac {c}{1 - \gamma}.\]

2.1.2 Wing under IAP

As for the solution of wing under we need to add to the basic linear program (3)-(6) the IAP constraint.

Proposition 3 (winiIAP) The optimal wing mechanism under is as follows:

\[b _ {1} ^ {*} = \binom{\frac {c (- 1 + \alpha_ {2} (- 1 + \beta_ {1}) + 2 \beta_ {1} + \gamma (- 1 + 2 \beta_ {1}) (- 1 + \beta_ {2}) - \beta_ {1} \beta_ {2}}{(- 1 + \gamma) (1 + \alpha_ {2} - \beta_ {1} + \gamma (- 1 + \beta_ {1} + \beta_ {2})},}{\frac {c (- 1 + \beta_ {1}) (- 1 + \alpha_ {2} - \beta_ {2} + \gamma (- 1 + 2 \beta_ {2}))}{(- 1 + \gamma) (1 + \alpha_ {2} - \beta_ {1} + \gamma (- 1 + \beta_ {1} + \beta_ {2})}} i f \beta_ {1} < \frac {1}{2};\tag{7}\]

\[b _ {2} ^ {*} = \left(\frac {c (1 - \beta_ {1})}{1 - \gamma}, \frac {c (1 - \beta_ {1})}{1 - \gamma}\right) i f \beta_ {1} \geq \frac {1}{2},\tag{8}\]

with

To prove Proposition 3, some preliminary lemmas are required. Let deÞne the solution of the linear program (3-6) in

Lemma 4

\[\hat {b} ^ {1} = \left(\frac {c (1 + \alpha_ {2})}{(1 - \gamma) \gamma}, \frac {c (\gamma + \alpha_ {2})}{(1 - \gamma) \gamma}\right)\tag{9}\]

Proof. In ! agent 1’s monetary payo , as determined by ! is always higher (i.e. . This, in turn, implies that constraints (4)-(5) correspond to

\[b _ {1} \geq f _ {1} ^ {1} (b _ {1}) \equiv \frac {c (1 - \beta_ {1})}{(1 - \gamma) \beta_ {1}} - \frac {1 - \beta_ {1}}{\beta_ {1}} b _ {1};\tag{10}\]

\[b _ {2} \geq f _ {2} ^ {1} (b _ {1}) \equiv \frac {c}{1 - \gamma} + \frac {\alpha_ {2}}{1 + \alpha_ {2}} b _ {1}.\tag{11}\]

Let deÞne the value of such that . By the same token, let denote the intercept of ! i.e. ' Finally, let denote the slope of ' We then have and ' Also notice that and ' This implies that and intersect in the Þrst quadrant of the space. On the other hand, is never binding in this case, since and since ' This implies that is minimized where and intersect, i.e. when and ■

Lemma 5 In ! the optimal wing contract under IAP is (7) when , and (8) when , with

Proof. In the case of , constraints (4)-(5) correspond to

\[b _ {1} \geq f _ {1} ^ {2} (b _ {1}) \equiv \frac {c (1 - \beta_ {1})}{(1 - \gamma) \beta_ {1}} - \frac {1 - \beta_ {1}}{\beta_ {1}} b _ {1};\tag{12}\]

\[b _ {2} \geq f _ {2} ^ {2} (b _ {1}) \equiv \frac {c (1 - \beta_ {2})}{1 + \alpha_ {2} - \gamma (1 - \beta_ {2})} + \frac {\alpha_ {2} + \gamma \beta_ {2}}{1 + \alpha_ {2} - \gamma (1 - \beta_ {2})} b _ {1}.\tag{13}\]

This implies that (i.e. the Nash equilibrium condition for player 1 remains unchanged in both and ! and

We Þrst show that ' Let . If ! then the optimal solution in would be ' On the other hand, if then ' More precisely, if ! the optimal solution is (7), that is, the intersection between and ! the solution is (8), that is, the intersection between and

We are in the position to prove Proposition 3.

Proof. [Proof of Proposition 1]. To prove the proposition, it is sufficient to show that ' To see this, remember that . Also remember that is (not) binding for both and solves ! then ， which, in turn, implies

\[\hat {b} _ {1} ^ {1} = \frac {c (1 + \alpha_ {2})}{\gamma (1 - \gamma)} > x _ {1} ^ {2 1} = \frac {c (1 - \beta_ {1})}{1 - \gamma} \geq \hat {b} _ {1} ^ {2} \text { and }\]

\[\hat {b} _ {2} ^ {1} = \frac {c (\gamma + \alpha_ {2})}{\gamma (1 - \gamma)} > x _ {1} ^ {2 1} = \frac {c (1 - \beta_ {1})}{1 - \gamma} \geq \hat {b} _ {2} ^ {2}.\]

2.1.3 Wing with SSP

As for the solution of wing under SSP, we need to add to the basic linear program (3)-(6) the SSP constraint.

Proposition 6 (winiSSP) The optimal wing mechanism under SSP is (7), with

Proof. We begin by showing that, as in the case of IAP, the optimal wing contract in is (9). This is because, also in this case, is not binding, since and

On the other hand, the optimal wing contract in is (7), independently of the value of ' This is because, given , both and are positive. Since ; (i.e., as before, and intersect in the Þrst quadrant. Also notice that, given ' Two are the relevant cases:

1. If ! then and intersect outside ! and the optimal solution would be

2. If ! then the solution is (7) which overall cost is never greater than

We complete the proof by noticing, by analogy with the Proof of Proposition 3, that the optimal solution lies in ! rather than in '

2.1.4 Wing with ESP

In the case of wing with ESP, we need to add to the basic linear program (3)-(6) the ESP constraint.

Proposition 7 (winiESP) The optimal wing mechanism under ESP is (7), with

Proof. We begin by showing that here the optimal wing contract in is (9) if and ' This is because, like in the previous cases, is never binding, since and . On the other hand, given that and is binding if and only if (i.e. if

As for ! we begin to notice that (since and that ' This implies, like before, that and intersect in the Þrst quadrant. The rest of the proof is identical of that of Proposition 6.

2.2 Solution of the mechanism design problem under the sting program

In the case of sting, the search of the optimal mechanism corresponds to the wing linear program (3)-(6) with an additional constraint (implementation with a unique equilibrium):

\[u _ {1} (1, 0) \geq u _ {1} (0, 0).\tag{14}\]

The constraint (14) makes, on behalf of player 1, the choice of putting e ort a weakly dominant strategy.

2.2.1 Sting under EP

The solution of sting under EP is as follows (see Winter, 2004):

\[\begin{array}{l} {b _ {1} ^ {*} = \frac {c}{\gamma},} \\ {b _ {2} ^ {*} = \frac {c}{1 - \gamma}.} \end{array}\]

2.2.2 Sting under IAP

Proposition 8 The optimal sting mechanism under IAP is

\[\left\{ \begin{array}{l l} b _ {1} ^ {*} = \frac {c ((1 + \alpha_ {1}) (1 + \alpha_ {2}) - \gamma (1 - \beta_ {2}))}{\gamma (1 + \alpha_ {1} + \alpha_ {2} - \gamma (1 + \alpha_ {1} - \beta_ {2}))}, \\ b _ {2} ^ {*} = \frac {c (1 + \alpha_ {1}) (\gamma + \alpha_ {2})}{\gamma (1 + \alpha_ {1} + \alpha_ {2} - \gamma (1 + \alpha_ {1} - \beta_ {2}))}. \end{array} \right.\tag{15}\]

To prove Proposition 8, we follow the same strategy as before.

Lemma 9

Proof. In ! the constraints for agent 1 and 2 correspond to:

\[b _ {1} \geq f _ {1} ^ {1} (b _ {1}) \equiv \frac {c (1 - \beta_ {1})}{(1 - \gamma) \beta_ {1}} - \frac {1 - \beta_ {1}}{\beta_ {1}} b _ {1},\tag{16}\]

\[b _ {1} \geq f _ {3} ^ {1} (b _ {1}) \equiv \frac {c (1 - \beta_ {1})}{\gamma (1 - \gamma) \beta_ {1}} - \frac {1 - \beta_ {1}}{\beta_ {1}} b _ {1},\tag{17}\]

\[b _ {2} \geq f _ {2} ^ {1} (b _ {1}) \equiv \frac {c}{1 - \gamma} + \frac {\alpha_ {2}}{1 + \alpha_ {2}} b _ {1},\tag{18}\]

Let solves " We Þrst notice that (16) is not binding. This is because (16) deÞnes a constraint which is parallel to (17), but with a smaller intercept ! since " Also notice that, in this case, (17) is not binding either. This is because, , and

This implies that, in is minimized (like in wing) where and intersect, i.e. when and

Lemma 10 The optimal sting contract in is (15).

Proof. , the relevant constraints are as follows:

\[b _ {1} \geq f _ {1} ^ {2} (b _ {1}) \equiv \frac {c (1 - \beta_ {1})}{(1 - \gamma) \beta_ {1}} - \frac {1 - \beta_ {1}}{\beta_ {1}} b _ {1}\tag{19}\]

\[b _ {1} \geq f _ {3} ^ {2} (b _ {1}) \equiv - \frac {c (1 + \alpha_ {1})}{\gamma \alpha_ {1}} + \frac {1 + \alpha_ {1}}{\alpha_ {1}} b _ {1}\tag{20}\]

\[b _ {2} \geq f _ {2} ^ {2} (b _ {1}) \equiv \frac {c (1 - \beta_ {2})}{1 + \alpha_ {2} - \gamma (1 - \beta_ {2})} - \frac {\alpha_ {2} + \gamma \beta_ {2}}{1 + \alpha_ {2} - \gamma (1 - \beta_ {2})} b _ {1}.\tag{21}\]

Notice that, by analogy with ! condition (19) is not binding since and . Also notice that and " This, in turn, implies that, and always intersect in the interior of , which implies the solution.²

We are in the position to prove Proposition 8.

Proof. To close the proposition, it is sufficient to show that " To see this, notice that and cross exactly at the intersection with " Since and is interior to ! the result follows.

2.2.3 Sting under SSP

Proposition 11 The optimal sting mechanism under is

Proof. By analogy with the IAP case, in is not binding. Also notice that " Two are the relevant cases:

1. if ! then (20) is not binding, and the optimal solution is the intersection between and ! that is, ;

2. if ! then the optimal solution is the intersection between and that is, "

\[\hat {b} ^ {1} = \left(\frac {c (1 + \alpha_ {2}) (1 - \beta_ {1} (1 + \gamma))}{\gamma (1 - \gamma) (1 + \alpha_ {2} - \beta_ {1})}, \frac {c (\alpha_ {2} + \gamma (1 + \alpha_ {2})) (1 - \beta_ {1})}{\gamma (1 - \gamma) (1 + \alpha_ {2} - \beta_ {1})}\right).\]

As for ! the optimal sting contract is, again, (15 ). This is because, by analogy with the IAP case, conditions (19) and are not binding. Also notice that and " This, in turn, implies that, in is minimized where and intersect, which implies the solution.

² As it turns out, unlike the wini case, the search for the appropriate conditions on preferences to identify player 1 has no (algebraically manageable) closed-form solution, but it has to be evaluated numerically (as we did in the calibration of our experimental conditions).

2.2.4 Sting under ESP

Proposition 12 The optimal sting mechanism under ESP is (15).

Proof. By analogy with the previous cases, in ! (16) is not binding. Also notice that, in this case, (19) is not binding either, since and " Since ESP imply ! the unique solution in this case is " As for ! we Þrst notice that, given that " Since ESP also imply ! then the optimal solution is the intersection between and ! that is, (15).

NOT FOR PUBLICATION

1 Distribution of the social preference parameters and

In Figure 1 we plot the estimated and of each member of our subject pool.

Figure 1" Estimating individual social preferences

Figure 1 is composed of two graphs:

1. In Figure 1a) each subject corresponds to a point in the space, where we highlight the regions corresponding to the taxonomy. As Figure 1a) makes clear, our subjects display signiÞcant heterogeneity in their distributional preferences. Moreover, in many cases, the constraints on absolute values (in particular, in the case of IAP) are violated. This is the reason why, in what follows, we shall refer to the corresponding quadrant in Figure to identify each distributional preference type. In this respect, the majority of subjects falls in the Þrst quadrant (i.e. in the IAP case), followed by SSP and ESP. Finally, 10% of agents in our subject pool display both and negative (a case not covered by the theoretical literature on these matters).

2. Figure 1b) reports, together with each estimated pair (as in Figure 1a), the corresponding 95% conÞdence intervals associated to each individual estimated parameter. As Figure 1b) shows, we have now many subjects whose estimated distributional preferences fall, with nonnegligible probability, in more than one region. Moreover, for some of them (about 20% of our subject pool), we cannot reject (at the 5% conÞdence level) the null hypothesis of egoistic preferences.

2 Reciprocity and contract choice

Table 1 reports the relative frequencies of positive e ort decisions in , conditional on subjects’ behavior in Stage 1.

Table 1" Relative frequencies of positive e ort decisions in

Table 1 shows that in about 35% of the cases the contracts chosen by Player 1 are less favorable to Player 2 than the one actually chosen by Player 2, that is in 35% of the cases Player 1 misbehaves . This percentage is almost constant across contract types. On the other side, Player 1 observes her teammate to misbehave in 30% (=193/339) of the cases if the played plan is a '()* and in 38% (=202/525) of the cases if it is a +,()*" Actions following misbehavior are heterogeneous: in a '()* plan (’s e ort following misbehavior is typically lower than after correct behavior.¹ In a +,()* contract, however, only the non Dictator Player 2 e ort is signiÞcantly lower after misbehavior. In Table 1 we also track (’s willingness to make e ort following their own (mis)behavior# " Also in this case, misbehavior yields lower e ort proÞles with the '()* plans and - as before - with the +,()* plans a reaction appears only for Player 2 when she is not the Dictator.

It would be wrong to draw immediate conclusions from the descriptive statistics in the previous paragraph. In table 1 di erences in e ort are only related to misbehavior, but this does not control for other factors -such as absolute and relative payo s of the contract being played, or the contingent choice set available in the particular round, . A more comprehensive analysis can only be done by estimating a model in which we can control for all those factors (and their subtle interplay).

3 Determinants of sting/wing choice

We construct a measure of inequality associated to each contract / in # which measures the relative inequality induced by contract in comparison with the other available options in

\[\sigma_ {\overline {{k}}} = \frac {\left(b _ {1} ^ {\overline {{k}}} - b _ {2} ^ {\overline {{k}}}\right) - \min _ {k} \left[ b _ {1} ^ {k} - b _ {2} ^ {k} \right]}{\max _ {k} \left[ b _ {1} ^ {k} - b _ {2} ^ {k} \right] - \min _ {k} \left[ b _ {1} ^ {k} - b _ {2} ^ {k} \right]}, k = 1, \dots , 4.\tag{1}\]

By (1), # i.e. we normalize the inequality each contract implies with respect to the choice set . We thus deÞne a “relative inequality index” associated with the choice of '()* vs. a +,()* contract in "We are now in the position to estimate the following logit function:

\[\operatorname * {P r} \left(k _ {i t} \in s t i n g | \alpha_ {i}, \beta_ {i}, \omega_ {t}\right) = \frac {\exp \left(\psi_ {0} + \psi_ {1} \alpha_ {i} + \psi_ {2} \beta_ {i} + \psi_ {3} \omega_ {t}\right)}{1 + \exp \left(\psi_ {0} + \psi_ {1} \alpha_ {i} + \psi_ {2} \beta_ {i} + \psi_ {3} \omega_ {t}\right)},\]

where identiÞes the contract choice of individual ( at round ,. For Players 2 (Principals), we use observations from We do so to frame the contract choice problem over the same choice sets, # since in agents’ choice sets are determined by principals’ decisions. In Table 2 we report the partial maximum likelihood estimates of to with bootstrap standard errors.

!₂

α = 5%

¹ Formal tests of mean equality conditional on player position always reject the null at = 5%.

² Player 1 chose wing in only 7 times out of 288, so that the predicted probability is basically one.

Table 2 Sting vs. wing choice in the “mixed” rounds, logit regression Notice that:

1. Estimated are always positive and signiÞcant: the more unequal is the wing choice the more likely is the choice of a sting contract, whatever the player role: on average, a increase of the relative inequality index induces an increase of the 29% of the probability of choosing #$&'( for Player 2, and of 14% for the principals in . These results are maintained (both in sign and magnitude) if we use a Þxed-e ects logit model.

2. For principals, distributional parameters are not signiÞcant to explain the choice of contract type, while for Players 2 in , both * and are signiÞcant, with opposite sign.

4 Distributional Preferences and contract choice

We look at how principals’ and agents’ estimated preferences explain their contract decision, with respect to the two dimensions which are more natural for the problem at stake: a) the total cost of the contract and, b) its induced inequality . By analogy with we deÞne, for each choice set , the following two variables:

\[\begin{array}{r c l} \tau_ {\overline {{k}}} & = & \frac {\left(b _ {1} ^ {\overline {{k}}} + b _ {2} ^ {\overline {{k}}}\right) - \min _ {k} \left[ b _ {1} ^ {k} + b _ {2} ^ {k} \right]}{\max _ {k} \left[ b _ {1} ^ {k} + b _ {2} ^ {k} \right] - \min _ {k} \left[ b _ {1} ^ {k} + b _ {2} ^ {k} \right]}, \overline {{k}} = 1,..., 4, \text { and } \\ \rho_ {\overline {{k}}} & = & \frac {1 + \sigma_ {\overline {{k}}}}{1 + \tau_ {\overline {{k}}}}. \end{array}\tag{2}\]

We interpret as a measure of relative e ciency (or relative cost, from the principal’s viewpoint). Consequently, proxies the trade-o agents (principals) face between inequality and efficiency (total costs).

We study principals’ contract decisions by regressing and in . against subjects’ distributional parameters, and Given that, in both cases, the dependent variable is bounded both from above and from below (with upper and lower limits which are period dependent), we estimate the equations using a double censored tobit model:

\[y _ {i t} = \psi_ {1} \alpha_ {i} + \psi_ {2} \beta_ {i} + \psi_ {3} V _ {i t} + \psi_ {4} ^ {\prime} D _ {t} + v _ {i t},\tag{3}\]

where the dependent variable refers, alternatively, to the corresponding and induced by the contract choice made by individual & at time is the randomly generated value for the principal, and is a full set of period dummy variables. In Table 3 we report the partial maximum likelihood estimates of the parameters with bootstrap and cluster adjusted standard errors. We estimate the parameters separating the periods in which the contract menu includes both #$&'( and 6&'( contracts (“mixed” periods) from the others (“non mixed”).

Table 3 Relative cost choice (0 ) and inequality-total costs trade-o (2) for principals in

First notice that Principals opt for the most expensive contract available more than 50% of the cases (the latter corresponds to the right-censored observations), and more that of the cases in the non-mixed periods. By contrast, less than 10% go for the cheapest one. We explain this evidence by the e ects of competitions among principals, and the fear of having their o ered contract not chosen by any agent. Consistently with the evidence of Table also notice that, in the mixed periods, principals’ distributional parameters are only marginally signiÞcant in explaining the choice of and this is another indirect evidence of the predominance of the search for robustness we already observed for the 6&'(7#$&'( choice. By contrast, in the non-mixed periods, we see that both principals’ distributional parameters signiÞcantly explain their preferred In the natural direction: the highest the (inequality-averse) distributional concerns, the lowest the relative inequality, and the highest the relative cost for the principal.

Table 4 Inequality-inefficiency trade o for players in

As for the agents we use an equation similar to (3) - here plays no role - to study their choice about the inequality - inefficiency trade-o in . Estimation results, conditional on Player positions, are shown in Table 9: we generally Þnd -as intuition would suggest- a (negative and signiÞcant) relation between distributional concerns and relative inequality.

5 Predicted and actual e ort choices

We begin by looking at actual behavior. As Table 6 in the main text shows, the overall level of e ort in is similar to the level of e ort observed in (see Table 1): Player 1 puts e ort in 91% of the cases of sting contracts where this percentage for her teammate drops to 64%; both player types put e ort about 43% of the time when they face a 6&'( contract. The only di erence with respect to can be noticed for Player 1 with 6&'( contracts (51% of e ort decisions in vs 44% in . There is instead a noteworthy di erence about misbehavior. In fact, due to the competition among the principals who, in 80 cases out of 144 (6 matching groups ×24 rounds) converged to a single contract o er, the possibility to misbehave is severely reduced. The agent & observed agent misbehaving less than 10% of the times in 6&'( contracts (it was 30% in and less than 20% in #$&'( contracts (it was at least 34% in . Conditional on misbehavior (either there are some discrepancies between the e ort rates in and , but these are difficult to interpret as robust evidence against the hypothesis of consistent behavior between and because of the small number of observations available.³ As for the comparison between actual and predicted behavior, our behavioral model correctly anticipates subjects’ e ort decisions in in 894 out of 1152 cases (about 78%), with a slightly better predicted power in #$&'( rather than 6&'( (80% against 74%, respectively). As for the latter, the most likely forecast mistake (for both player positions) is to predict no e ort when agents decided otherwise.

6 Additional treatments

We now compare subjects’ e ort decisions in in the two di erent treatments. By analogy with Table 6, in Table 11 we report relative frequency of e ort decisions, disaggregated for contract type (wing or sting), player and dictator position.

Table 5. Relative frequencies of positive e ort decisions in and of

First notice that, in , both players, ceteris paribus, work less (on average, almost 25% less). This e ect is stronger for Player 1 in wing and Player 2 in sting. We also see that, for wing, there is a decrease in e ort frequencies, which about of the corresponding levels of , while for sting the di erences in e ort levels across treatments are smaller. As for reciprocity, remember that, in . we cannot measure misbehavior, given that only dictators are asked to elicit their favorite contracts (and, therefore, relative comparisons cannot be performed). This implies that we are not in the position to estimate beliefs and reciprocity parameters as we did for

#₂

(0/7=) 0,

#₃,

³ Take, for example, the case of of the non Dictator Player 2, whose average e ort, when ! = 1" is (7/37=) 0.19 and in and respectively.

Table 1: Relative frequency of positive e ort decisions in . Number of cases in Parenthesis

Table 2: ?$&'( vs @ &'( choice in the “mixed” rounds, logit regression

Table 3: Relative cost choice (0 ) and inequality - total costs trade o for principals in . All speciÞcations include a full set of period dummies. Bootstrap and cluster adjusted standard errors

Table 4: Inequality - inefficiency trade o for agents in . All speciÞcations include a full set of period dummies. Bootstrap and cluster adjusted standard errors

Table 5: Relative frequencies of positive e ort decisions in and of . Number of cases for each player type in parenthesis

Figure 1: Distribution of estimated * and in

NOTE: In the experiment, the instruction for each PHASE were given only after subjects had played the previous phases.

WELCOME TO THE EXPERIMENT!

This is an experiment to study how people make decisions. We are only interested in what people do on average.

Please, do not think we expect a particular behavior from you. On the other hand, keep in mind that your behavior will affect the amount of money you can win.

• In what follows you will find the instructions explaining how this experiment runs and how to use the computer during the experiment.

• Please do not bother the other participants during the experiment. If you need help, raise your hand and wait in silence. We will help you as soon as possible.

THE EXPERIMENT

In this experiment, you will play for 72 subsequent rounds. These 72 rounds are divided in 3 PHASES, and every PHASE has 24 rounds.

PHASE 1

• In each of the 24 rounds of PHASE 1, you will play with ANOTHER PLAYER in this room.

• The identity of this person will change from one round to the next. You will never know if you interacted with the OTHER PLAYER in the past, nor the OTHER PLAYER will ever know if he has interacted with you. This means your choices will always remain anonymous.

• At each round of PHASE 1, the computer will first randomly choose 4 different OPTIONS, that is, four monetary payoff pairs, one for you and one for the OTHER PLAYER. Every OPTION will always appear on the left of the screen.

Then, you and the OTHER PLAYER have to choose, simultaneously, your favourite OPTION.

Once you and the OTHER PLAYER have made your decision, the computer will randomly determine who (either you or the OTHER PLAYER) will decide the OPTION for the pair.

• We will call this player the CHOOSER of the game.

• The identity of the CHOOSER will be randomly determined in each round.

On average half of the times you will be the CHOOSER and half of the time the OTHER PLAYER will be the CHOOSER.

Thus, in each round, the monetary payoffs that both players receive will be determined by the choice of the CHOOSER.

PHASE 2

• In the following 24 rounds of PHASE 2, you will participate in a game similar to the previous one, with some modifications.

• In STAGE 1 of PHASE 2, a payoff matrix will be chosen, and in STAGE 2 of PHASE 2, each pair will face this payoff matrix, which will appear on the left of the screen.

What does this matrix mean?

In each round, you and the OTHER PLAYER will receive an initial endowment of 40 pesetas.

In each round, you and the OTHER PLAYER have to choose, simultaneously, whether to BID or NOT TO BID.

• Bidding costs 10 pesetas, not bidding does not cost anything.

• You choose the ROW, the OTHER PLAYER chooses the COLUMN.

• Every cell of the matrix (which depends on the monetary payoffs b1 and b2 and your decisions on whether or not to bid) contains two numbers.

• The first number (on the left) is what you win in this round. The second (on the right) is what the OTHER PLAYER wins in this round. There are four possibilities:

1. If both players bid, both add to their initial endowment their ENTIRE MONETARY PAYOFF b1 or b2 (to which the 10 pesetas cost of bidding will be subtracted).

2. If you bid, and the OTHER PLAYER does not, both players add to their endowment ONE FOURTH of the monetary payoff b1 or b2 (and the cost of bidding will be subtracted from you only);

3. If the OTHER PLAYER bids, and you don’t, both players add to their endowment ONE FOURTH of their monetary payoff b1 or b2 (and the cost of bidding will be subtracted from the OTHER PLAYER only);

4. If nobody bids, you and the OTHER PLAYER will only obtain the 40 pesetas endowment.

PHASE 2 is composed of 2 STAGES:

• In STAGE 1, you and the OTHER PLAYER have to choose your favorite OPTION, that is, the game that you would like to play in STAGE 2.

• After you and the OTHER PLAYER have made your decision, the computer will randomly determine who (either you or the OTHER PLAYER) will be the CHOOSER of the game. That is, the OPTION selected by the CHOOSER in STAGE 1 is the one played in STAGE 2.

• Like in PHASE 1, the identity of the CHOOSER, will be randomly determined in each round.

On average, half of times you will be the CHOOSER and half of times the OTHER PLAYER will be the CHOOSER.

Once the CHOOSER has determined the option that will be played in this round, you and the other player have to choose whether TO BID or NOT TO BID and the monetary consequences of your decisions are exactly those we just explained.

SUMMING UP

• In each of the 24 rounds of PHASE 2, you will play with ANOTHER PLAYER of this room.

• In STAGE 1, you and the other player, like in PHASE 1, have to choose simultaneously your favorite OPTION.

After you and the OTHER PLAYER have made your decisions on the OPTION, the computer will randomly determine which one of those OPTIONS is the game that you will play in STAGE 2. That is, the computer designs a CHOOSER.

In STAGE 2 you and the OTHER PLAYER have to simultaneously DECIDE whether to bid or not to bid. The payoffs of each round depend on your initial endowment of 40 pesetas, on both your choices (to bid or not to bid), on the OPTION chosen by the CHOOSER and on the cost of bidding of 10 pesetas.

• The PAYOFF MATRIX (which will always appear on the left of your screen) sums up, in a compact form, the monetary consequences of your choices.

PHASE 3

In the last 24 rounds of PHASE 3, you will play in a game similar to the one in PHASE 2 but with some differences.

• Within the 24 persons in this room, the computer will randomly choose two groups of 12.

In each group of 12 people, the computer will randomly determine 8 PLAYERS and 4 REFEREES.

The identity of PLAYERS and REFEREES is randomly determined at the beginning of PHASE 2 and it will remain the same for the rest of the experiment.

PHASE 3 has 3 STAGES.

• Like in the previous PHASES, in STAGE 1 the computer randomly selects 4 OPTIONS, (that is, 4 pairs of monetary payoffs (b1, b2) for the players.

In addition, in STAGE 1, each REFEREE picks an OPTION within the 4 available for that round (which may be the same or different among them).

• Thus, the 4 OPTIONS selected by the four REFEREES will be proposed to the 8 PLAYERS of their group.

In STAGE 2, the 8 PLAYERS will be randomly paired. PLAYERS will be rematched at every round.

• Then, just like in PHASE 2, each player has to select one among the 4 OPTIONS proposed by the 4 REFEREES.

• Just like in PHASE 2, the computer randomly determines which of the two OPTIONS chosen by the PLAYERS is played by the pair. That is, the computer designs a CHOOSER.

• Just like in PHASE 2, in the game, both PLAYERS have to choose simultaneously, whether TO BID or NOT TO BID.

!he monetary consequences for the players of their decision are exactly the same as in PHASE 2.

REFEREES’ PAYOFF

The REFEREES’ payoffs depend on

1. the OPTION they offer,

2. how many REFEREES in their group offer the same OPTION

3. how many CHOOSERS choose the same OPTION

4. Players’ actions in the game.

We shall make this clearer with some examples.

CASE 1

• First, suppose that the REFEREE offered an OPTION with payoffs (b1, b2) and that only one CHOOSER has chosen this option.

The payoff of each REFEREE depends on the positive VALUE randomly generated by the computer and that each REFEREE (and only her) knows, and, in addition, on the sum of the payoffs b1+b2 in the following way:

if both players bid, the REFEREE wins the difference between his VALUE and the sum of the payoffs; that is, V-(b1+b2);

• if one player bids and the other does not, the REFEREE wins ONE FOURTH of the difference between his VALUE and the sum of the payoffs; that is,

• if nobody bids, the REFEREE does not win anything.

In this case, the PAYOFF MATRIX for the REFEREE would be as follows:

CASE 2

Suppose now that more than one CHOOSER chose the option that the REFEREE offered. Moreover, suppose moreover that this REFEREE is the only one that picked this OPTION.

• In this case, the REFEREE gets the sum of the payoffs obtained with each couple that chose her OPTION.

• The payoff with each couple will be determined as in CASE 1, taking into account if they bid, if only one bids or nobody bids.

CASE 3

Suppose now that one or more CHOOSERS chose an option that the REFEREE offered. Moreover, suppose that more than one REFEREE picked the same OPTION. In this case, every single REFEREE that chose the same OPTION gets a payoff with the same structure as in CASE 2, but now, sharing this payoff with the other REFEREES that picked the same option.

CASE 4

Suppose now that no couple chose the option that the REFEREE offered. In this case, her payoff for this round will be 0.

References

1. 2009-09: “Social Preferences and Strategic Uncertainty: An Experiment on Markets and Contracts”, Antonio Cabrales, Rafaele Miniaci,Marco Piovesan y Giovanni Ponti.

References

1. 2009-08: “Hidden Information, Bargaining Power and Efficiency: An Experiment”, Antonio Cabrales, Gary Charness y Marie Claire Villeval.

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1. 2009-07: “Democracy and the curse of natural resources”, Antonio Cabrales y Esther Hauk.

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1. 2009-06: “Social Interactions and Spillovers: Incentives,Segregation and Topology”, Antonio Cabrales, Antoni Calvó-Armengol e Yves Zenou.

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1. 2009-05: “Chance Constrained Programming with one Discrete Random Variable in Each Constraint”,Emilio Cerdá Tena y Julio Moreno Lorente.

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1. 2009-04: “Economic Value of Weather Forecasting Systems Information: A Risk Aversion Approach”, Emilio Cerdá Tena y Sonia Quiroga Gómez.

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1. 2009-03: “Population Ageing, Inequality and the Political Economy of Public Education”, Francisco Martínez-Mora.

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1. 2009-02: “Real Wages over the Business Cycle: OECD Evidence from the Time and Frequency Domains”, Julian Messina, Chiara Strozzi y Jarkko Turunen.

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1. 2009-01: “The Determinants Of Misreporting Weight And Height: The Role Of Social Norms”, Joan Gil y Toni Mora.

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1. 2008-42: “Social Security incentives, exit from the workforce and entry of the young”, Michele Boldrin, Pilar García-Gómez y Sergi Jiménez-Martín.

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1. 2008-41: “The evolution and main determinants of productivity in Brazilian electricity distribution 1998- 2005: an empirical analysis”, Francisco Javier Ramos-Real, Beatriz Tovar, Mariana Iootty, Edmar Fagundes de Almeida y Helder Queiroz Pinto Jr..

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1. 2008-40: “Immigration and Housing Prices in Spain”, Simón Sosvilla.

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1. 2008-39: “Modeling the Immigration Shock”, Ana Montes y Michele Boldrin.

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1. 2008-38: “Immigration and the Demand for Health in Spain”, Sergi Jiménez, Natalia Jorgensen y José María Labeaga.

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1. 2008-37: “Immigration and Students' Achievement in Spain”, Natalia Zinovyeva, Florentino Felgueroso y Pablo Vázquez.

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1. 2008-36: “Immigration and Social Security in Spain”, Clara Isabel González, J. Ignacio Conde-Ruiz y Michele Boldrin.

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1. 2008-35: “Complements or Substitutes? Immigrant and Native Task Specialization in Spain”, Catalina Amuedo-Dorantes y Sara de la Rica.

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1. 2008-34: “Immigration and Crime in Spain, 1999-2006”, Cesar Alonso, Nuno Garoupa, Marcelo Perera y Pablo Vázquez.

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1. 2008-33: “A Social Network Approach to Spanish Immigration: An Analysis of Immigration into Spain 1998- 2006”, Rickard Sandell.

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1. 2008-32: “The Consequences on Job Satisfaction of Job-Worker Educational and Skill Mismatches in the Spanish Labour Market: a Panel Analysis”, Lourdes Badillo Amador, Ángel López Nicolás y Luis E. Vila.

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1. 2008-31: “Students’assessment of higher education in Spain”, César Alonso-Borrego, Antonio Romero-Medina.

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1. 2008-30: “Body image and food disorders: Evidence from a sample of European women”, Joan Costa-Font y Mireia Jofre-Bonet.

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1. 2008-29: “Aggregation and Dissemination of Information in Experimental Asset Markets in the Presence of a Manipulator”, HelenaVeiga y Marc Vorsatz.

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1. 2008-28: “The Measurement of Consensus: An Axiomatic Analysis”, Jorge Alcalde-Unzu y Marc Vorsatz.

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1. 2008-27: “Macroeconomic Consequences of International Commodity Price Shocks”, Claudia S. Gómez-López y Luis A. Puch.

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1. 2008-26: “The Effect of Short–Selling on the Aggregation of Information in an Experimental Asset Market”, Helena Veiga y Marc Vorsatz.

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1. 2008-25: “Adult height and childhood disease”, Carlos Bozzoli, Angus Deaton y Climent Quintana-Domeque.

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1. 2008-24: “On Gender Gaps and Self-Fulfilling Expectations: Theory, Policies and Some Empirical Evidence” Sara de la Rica, Juan J. Dolado y Cecilia García-Peñalosa.

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1. 2008-23: “Fuel Consumption, Economic Determinants and Policy Implications for Road Transport in Spain”, Rosa M. González-Marrero, Rosa M. Lorenzo-Alegría y Gustavo A. Marrero.

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1. 2008-22: “Trade-off between formal and informal care in Spain”, Sergi Jiménez-Martín y Cristina Vilaplana Prieto.