David L Dickinson
email: dickinsondl@appstate.edu

David Masclet
email: david.masclet@univ-rennes1.fr.

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

Norm Enforcement in Social Dilemmas An Experiment with Police Commissioners

Keywords: Norm enforcement, Common Pool Resources, Voluntary Contribution Mechanism, Police officers, Experiment JEL Classifications: C92, H41, D63

Do individuals trained in law enforcement punish or reward differently from typical student subjects? We analyze norm enforcement behavior of newly appointed police commissioners in both a Voluntary Contribution Mechanism game and a Common Pool Resource game. Our experimental design includes treatments where a reward or sanction institution is exogenously imposed, as well as treatments with endogenous selection of the norm enforcement institution. Compared to a standard student-subject pool, police commissioners cooperate significantly more in both games. With exogenous institutions, police commissioners bear a higher burden of punishment costs than non-police subjects. When the norm enforcement institution is endogenous, all subjects vote more in favor of rewards over sanctions, but police subjects with some work experience are more likely to vote for sanctions. Police subjects also reward and sanction more than the others when the institution results from a majority vote.

Introduction

When a police officer pulls you over, it is usually bad news. However, that was not the case for some drivers recently stopped in Sandy, Utah. The Sandy Police Department has implemented a new program that consists of handing out movie tickets to virtuous motorists who drive safely. A similar program has been recently tested in the south of France during the 2014 New Year's celebrations. In this case, police could reward drivers found to be below the blood alcohol driving limit by offering them a 20 euro gasoline ticket. These two examples provide insightful illustrations of how rewards could be used by police officers to encourage good behavior. 1 But such examples go against the old paradigm of the corrective policing model [START_REF] Becker | Crime and Punishment: An Economie Approach[END_REF].

Police almost exclusively resort to sanctions. While both sanctions and rewards are commonly used in several other fields, it is not the case in the realm of law enforcement. 2 An obvious reason why police almost exclusively use sanctions is that it is not natural to reward those who simply comply with the law-it makes more sense to reward those who outperform a norm. 3 Apart from this, there may be three additional reasons in the specific context 1 Other similar programs such as the "Pops from Cops" program implemented by the Decatur Police Department, Illinois are targeted toward youth. These reward programs consist of giving out coupons for free food, movie tickets or entrance to something fun to reward young people for doing something good-wearing a helmet while biking or skateboarding, using the crosswalks, skateboarding in designated areas, getting to school on time, speaking respectfully, etc. According to a law enforcement officer involved in such program, there may exist several positive externalities associated with such programs. According to him: "What's most important is that the ticket is a positive event, and when the youth sees the officer the next time, it will start off on a positive note. The ticket is the gateway to the relationship. If you reward good behavior, your return on investment will be more good behavior." (http://herald-review.com/news/local/pops-from-cops-coupons-allow-police-to-rewardyouthsgood/article_b19d94ec-2b1a-11e3-90eb-0019bb2963f4.html).

2 For instance, several kinds of rewards including financial rewards or promotions have long been used in organizations to motivate workers towards greater productivity (see [START_REF] Lazear | Personnel economics[END_REF]. Firms also commonly use models that combine both rewards and sanctions such as the "up or out" model that implies that employees who succeed in achieving a certain level of performance are promoted while those who fail to do so must leave the organization.

of norm enforcement to explain why the use of sanctions may be preferred by the police to the use of sanctions. Firstly, there may exist a pure framing effect: "destructive" contexts may favor sanctions compared to "constructive" contexts. The intuition is that norm violations involving destruction may trigger more negative emotions than norm violations that are apparently more passive. 4 Secondly, sanctions may be more efficient than rewards at norm enforcement when norm violation involves the destruction of wealth. Finally, one may also conjecture the existence of a pure police-specific effect. Specifically, police officers are more exposed to destructive contexts and disorderly elements of society. As such, police may have a stronger preference for sanctions compared to other individuals because such exposure may induce a bias in favor of sanctions [START_REF] Skolnick | Community-Oriented Policing Would Prevent Police Brutality[END_REF]. 5In this paper, we study norm enforcement (rather than law enforcement) by means of social dilemma games, and we analyze whether a population trained in law enforcement behaves differently in these decision environments. In particular, we study whether social dilemmas framed in a common resource, rather than public goods, context bias individuals towards punishment of norm violators, rather than rewarding of non-violators. Social dilemmas are popular for studying cooperation and social norms because group welfare is at odds with the dominant strategy of selfish free riding behavior in these environments. Early laboratory experiments have shown that initial contributions in Voluntary Contribution Mechanism (VCM) games are substantially above the Nash prediction, but decline steadily as the game is repeated [START_REF] Isaac | Divergent Evidence on Free Riding: An Experimental Examination of Possible Explanation[END_REF][START_REF] Andreoni | Why Free Ride? Strategies and Learning in Public Goods Experiments[END_REF][START_REF] Ledyard | Public Goods: a Survey of Experimental Research[END_REF]. However, cooperation can be sustained in the long run when punishment is available [START_REF] Yamagishi | The Provision of a Sanctioning System as a Public Good[END_REF][START_REF] Fehr | Cooperation and Punishment[END_REF]. 6 This finding is robust to various environmental conditions [START_REF] Masclet | Monetary and Non-Monetary Punishment in the Voluntary Contributions Mechanism[END_REF][START_REF] Bochet | Communication and Punishment in Voluntary Contribution Experiments[END_REF][START_REF] Anderson | Do Non-Strategic Sanctions Obey the Law of Demand? The Demand for Punishment in the Voluntary Contribution Mechanism[END_REF][START_REF] Carpenter | Punishing Free Riders: How Group Size Affects Mutual Monitoring and the Provision of Public Goods[END_REF]. Other studies have attempted to investigated the effectiveness of reward mechanisms to enforce the norm of cooperation [START_REF] Dickinson | The Carrot vs. the Stick in Work Team Motivation[END_REF][START_REF] Andreoni | The Carrot or the Stick: Rewards, Punishments and Cooperation[END_REF][START_REF] Walker | Rewards and Sanctions and the Provision of Public Goods in One Shot Settings[END_REF][START_REF] Sefton | The Effects of Rewards and Sanctions in Provision of Public Goods[END_REF][START_REF] Rand | Positive interactions promote public cooperation[END_REF][START_REF] Sutter | Choosing the carrot or the stick? Endogenous institutional choice in social dilemma situations[END_REF][START_REF] Dugar | Non-Monetary Incentives and Opportunistic Behavior: Evidence from a Laboratory Public Good Game[END_REF]. Most of these results show that rewards are somewhat less effective than sanctions in enforcing cooperation.

The originality of our paper is threefold. First, we investigate the effectiveness of both punishment and reward institutions not only in the VCM game, but also in the more destructive context of a Common Pool Resource game (CPR). In the CPR game, non-cooperation involves active withdrawals from the common pool, whereas in the VCM game non-cooperation involves failure to contribute to the public good (Ostrom et al., 1992). While the effect of sanctions and rewards is well documented in the VCM game, their effectiveness is less well-established when the social dilemma takes the form of a CPR. Some find that sanctions improve cooperation (Ostrom et al., 1992;Casari and Plott, 2003;[START_REF] Van Soest | Peer enforcement in CPR experiments: the relative effectiveness of sanctions and transfer rewards, and the role of behavioural types[END_REF], but others find the opposite result [START_REF] Janssen | Lab Experiments for the Study of Social-Ecological Systems[END_REF][START_REF] Cason | Swords without covenants do not lead to self-governance[END_REF]. In the same vein, the use of rewards to enforce cooperation in a CPR context has received less attention (exceptions include [START_REF] Vyrastekova | On the (in)effectiveness of rewards in sustaining cooperation[END_REF][START_REF] Stoop | A tale of two carrots: the effectiveness of multiple reward stages in a common pool resource game[END_REF], even though its relevance to the real world is clear. In our design, we run all of our norm enforcement treatments in both CPR and VCM contexts to test this framing effect.

The second originality of our paper is that we use a nonstandard subject-pool by enrolling a representative sample of new French police commissioners to complement participants from a standard subject-pool, mainly composed of engineering and business students. Indeed, our aim is to analyze whether police commissioners behave differently, in terms of institutional choices and norm enforcement. This population is perfectly suited for our study because police commissioners have self-selected in a 'mission-oriented' occupation in a particular destructive context of deterrence of crime and because their training and core function is in law enforcement [START_REF] Besley | Competition and incentives with motivated agents[END_REF]. Additionally, some of our police commissioners had completed their training two years prior to our experiments, while others had just completed their training. This allows us to measure whether some experience with crime and enforcement affects behavior in our games with norm enforcement options. 7 Our intuition is that police commissioners may have a stronger preference for sanctions because they self-select into an occupation that is exposed to destructive contexts and/or once enrolled they are explicitly trained to favor sanctions over rewards (Raganella and White, 2004; Wu et al., 2004). 8 Finally a third contribution of our experiment is to vary the way the enforcement institutions are implemented (either exogenously or endogenously) so that we can test whether, as government agents, police commissioners are more willing to utilize an institution when it results from a democratic choice. Specifically, in some treatments the enforcement institution is implemented by the computer, while in other treatments it results from a majority voting procedure.

Our experiment consists of four treatments: Baseline, Reward, Sanction, and Vote. The Baseline treatment is a standard public good game without any enforcement institution. In the Sanction (Reward) treatment, a new stage is added after the contribution stage. After being informed of each other group member's contribution, subjects can sanction (reward) group members at a personal cost. Finally, in the Vote treatment, a preliminary stage is added in which subjects have to vote for their preferred institution (reward or sanction), and the majority vote determines the institution that will be implemented.

To preview our main results, we find that socially desired behavior (i.e., VCM cooperation or CPR non-extraction) is higher in the VCM compared to CPR context, the existence of norm enforcement increases socially desired behavior, and police subjects contribute more (extract less) than non-police subjects. We also find that sanctions are used more frequently in the CPR context, and police subjects enforce norms more than non-police subjects. Though all subjects favor the reward institution when allowed to vote, we also find that police subjects with more experience in the police are more likely to vote for sanctions than other subjects. And finally, we find efficiency is highest with the reward institution and in the VCM context. While police subjects do not have higher payoffs than other subjects, their presence in mixed social dilemma groups raises efficiency (i.e., total group payoff) in that group.

The remainder of this paper is organized as follows. In section 2 we present the experimental design. Section 3 develops the predictions. Section 4 displays the results and section 5 discusses these results.

Experimental design

Our design consists of four treatments for each subject: Baseline, Reward, Sanction, and Vote.

The Baseline treatment allows us to compare behavior against results established in the literature.

The Reward and Sanction treatments add the possibility of assigning costly reward or punishment points, respectively, as a way of enforcing norms of cooperation. Finally, the Vote treatment implements an endogenous enforcement institution by allowing subjects to vote as to whether the reward or sanction institution should be used. These treatments are administered in both VCM and CPR environments, which means we have a 4x2 mixed design (4 treatments within-subjects, and CPR or VCM as between-subjects factors). Subjects were matched using an anonymous partner matching protocol, such that groups were kept constant.

Treatment parameters

Baseline treatments: In the Baseline VCM treatment, each of five homogenously endowed subjects allocate 20 ECU (Experimental Currency Units) to a private or a group account. Payoffs, as a function of the contribution of subject i, x i , are defined as follows:

! i VCM = 20 ! x i + 0.3 x j j=1 5 " (1)
In the Baseline CPR treatment, each of the five group members may each withdraw w i ∈ [0,20] from the group account. The payoff function of subject i is the following: 9

(2) 9 In contrast with [START_REF] Fehr | Cooperation and Punishment[END_REF], our Baseline treatments provide subjects with feedback regarding each other group member's contributions at the end of each period (thus keeping feedback constant across treatments).

It can be easily seen from eq. (1), and (2) that the two games are equivalent with Sanction treatments: The sanction institution is implemented exogenously. Each period now consists of two stages. The first stage is similar to the Baseline treatments. In the second stage, the subjects have the opportunity to assign costly sanction points, P, to each other group member. The payoff function, shown in the case of the CPR game, becomes:

(3) In eq. ( 3), cP ji j!i " is the total cost of sanctions assigned by subjects j to subject i, with c indicating the per-unit cost of each point received (c=2 ECU). 10 kP ij i! j

" is the total cost to subject i of the sanctions she imposes on all subjects j, with k=1 ECU being the cost to i of each sanction point assigned to any other player. A subject can assign a maximum of 10 sanction points to each other player. The total cost of points received cannot exceed the subject's earnings from the first stage.

Reward treatments: These treatments are similar to the Sanction treatments except that instead of punishing others, each subject can reward them by assigning reward points, R. Each reward point assigned costs k=1 ECU and each reward point received increases one's payoff by c=2 ECU. The payoff function, again shown in the case of the CPR game, is:

(4) 10 In [START_REF] Fehr | Cooperation and Punishment[END_REF] and in most following studies on sanctions in VCM games, the cost ratio is 1:3. We chose a weaker 1:2 ratio because we wanted to hold this ratio constant for reward and sanction points. We feared that using a higher ratio for reward points would lead subjects to assign points to others not to enforce the cooperation norm, but to create reciprocity and increase payoffs. A lower ratio should, we thought, limit this motivation.

where cR ji j!i " is the total gain of the rewards assigned by players j to player i and

kR ij i! j
" is the total cost to player i of all points assigned to others. A maximum of 10 points can be assigned to each other player and the total gain of points received cannot exceed the subject's earnings from the first stage.

Vote treatments:

Here, the sanction or reward institution is implemented endogenously by adding a preliminary stage where subjects vote once for whether the Reward or Sanction institution should apply to the subsequent rounds in the treatment. Voting entails no monetary cost, the majority vote institution is implemented, and there is no feedback on individual votes of others.11

Experimental procedures

The experiment was programmed using the Z-Tree software [START_REF] Fischbacher | z-Tree: Zurich Toolbox for Ready-made Economic Experiments[END_REF]. All sessions were conducted in the experimental laboratory of GATE (Groupe d'Analyse et de Théorie Economique) in Lyon, France. The experiment consists of nine sessions in total (five CPR and four VCM sessions). Within a session, the Baseline (B) treatment was always played first. The Reward (R) and Sanction (S) treatments were always adjacent (but counterbalanced in order).

The Vote (V) treatment was counterbalanced to be either before or after the Reward and Sanction treatments, which allows us to identify how inclinations for either rewards or sanctions were modified by the experience of these institutions. For groups where the Vote treatment occurred prior to the Reward and Sanction treatments, the two institutions were first described to each subject. Table 1 displays the details of the sessions, including treatment orderings administered.

[Insert Table 1 about here]

In a session, subjects played 10 periods in the Baseline and seven periods in each of the three other treatments.12 It was common knowledge that group member ID numbers were re-matched at the beginning of each period so that it was impossible to reciprocate the action of a specific group member in the previous period. 13 The final earnings in these tasks were the sum of payoffs from all 31 periods.

Each session involved 20 subjects, 8 to 10 of whom were police commissioners. Participants

were not aware of the characteristics of the other participants. This means that, for instance, business and engineering students did not know that they were interacting with police subjects.

Upon arrival, and after informed consent was obtained, subjects drew a ticket from an opaque bag assigning them to a specific terminal in the laboratory. The instructions were distributed for each part after completion of the previous part (see Appendix). 14 After reading the instructions aloud, we verified each subject's understanding of the instructions by means of a questionnaire and any questions were answered in private. 15 Sessions lasted two hours on average. The average subject earnings were €26.19 (S.D.=4.39). Subjects were paid their earnings in private in a separate room.

Subject pools

In total, 180 subjects from two different pools participated in this experiment. The regular pool includes 93 subjects who were recruited via the ORSEE software [START_REF] Greiner | An Online Recruiting System for Economic Experiments[END_REF]. Most were undergraduate students from the selective local engineering and business schools; the minority are older participants, either employees on the campus or retirees.

The other subject-pool consists of 87 individuals who had recently passed the very competitive national exam of police commissioners. We have two main types of subjects within this pool. The majority (73 of 87) were still studying in the French national school of police commissioners (Ecole Nationale Supérieure de la Police, or ENSP), the unique school for police commissioners in France located in the suburbs of Lyon. A minority (14 subjects) had finished training at ENSP two years prior and either had a permanent position as police commissioners or were in internships at the time of our experiments. These subjects participated in our experiment during a post-training return to ENSP, and participation was voluntary. The participants had to commit not to reveal the experimental protocol to other subjects before completion of all the sessions.

Our police sample is representative of the population of French newly appointed police commissioners since each promotion includes 40 students who spend two years at ENSP (meaning that the participation rate for the students at ENSP is above 90%). On average, police subjects are older (mean: 32.82 ± 7.37 year old, min=23, max=48) than the subjects from the other pool (mean: 24.82 ± 9.22 year old, min=18, max=64). 16 Additionally, police subjects are more typically male (79.66% compared to 44.26% males in the other subject pool).

12 3. Theoretical predictions

Standard predictions

We first derive predictions based on rational and selfish money maximizing agents, in which case the predictions are similar for police and non-police subjects. The equilibrium in a negative setting (CPR) is the same as in a positive one (VCM). Thus, we will merely describe predictions as a function of the enforcement institution and not the framing of the social dilemma.

In the Baseline treatments, it can easily be seen from equations ( 1) and ( 2) that the dominant strategy Nash equilibrium is for all subjects to place all 20 ECU in their private account (x i = 0) contributing nothing to the public good or, alternatively, extracting all from the common resource (w i =20). In equilibrium, each subject earns 20 ECU and total group earnings are 100 ECU. In contrast, the strategy leading to the Pareto equilibrium is x i = 20 (or, w i = 0), which generates total group earnings of 150 ECU (or 30 ECU per subject).

In both the Sanction and the Reward treatments, the only subgame perfect equilibrium, whether played once or finitely repeated, is also x i = 0 or w i =20 for all i and no use of sanctions or rewards because both are costly. In the Vote treatment, purely self-interested individuals should be indifferent between available institutions. Contribution or withdrawal behavior should not be affected by the choice of enforcement institution since sanctions/rewards are not credible.

In sum, with money-maximizing self-interested play, the equilibrium prediction is the same across all enforcement institutions, games, and subject pools. Observed differences must therefore be attributed to behavioral factors resulting from a desire to enforce behavioral norms and/or the psychological impact of a positive vs. negatively framed decision environment.

Behavioral predictions

Our first set of conjectures concerns the norm of cooperation in the Baseline. Introducing otherregarding preferences might take the form of advantageous inequity aversion [START_REF] Fehr | A Theory of Fairness, Competition and Cooperation[END_REF], social preferences for fairness and efficiency [START_REF] Charness | Understanding social preferences with simple test[END_REF], or imperfect conditional cooperation [START_REF] Fischbacher | Are People Conditionally Cooperative? Evidence from a Public Goods Experiment[END_REF][START_REF] Fischbacher | Social Preferences, Beliefs, and the Dynamics of Free Riding in Public Goods Experiments[END_REF]. Such theories predict that cooperation may result with sufficient numbers of cooperators in a group, though contributions may still decline over time. One may therefore conjecture that people with otherregarding preferences should contribute positive amounts of their endowment in all treatments.

Regarding subject pool effects, we predict that police commissioners will be more cooperative than others, given their stronger norm of civic cooperation and because they selfselected into a public good oriented occupation where helping others is key motivation [START_REF] Raganella | Race, gender, and motivation for becoming a police officer Implications for building a representative police department[END_REF]. Our conjecture is that civil servants, whose job is to serve public interests, might be more cooperative than a standard subject-pool because those in missionoriented jobs are usually intrinsically motivated agents [START_REF] Besley | Competition and incentives with motivated agents[END_REF] who place more value on the output of public organizations [START_REF] François | Public service motivation' as an argument for government provision[END_REF]. 17 This prediction is also based on evidence from cross-cultural studies showing that cooperation in VCM games is lower in countries with weak norms of civic cooperation and weak laws [START_REF] Herrmann | Antisocial punishment across societies[END_REF].

Regarding framing effects, based on the previous literature, we also expect to observe framing effects leading to higher cooperation in the VCM game than in the CPR game, despite similar monetary incentives [START_REF] Andreoni | Warm-glow versus cold-prickle: the effects of positive and negative framing on cooperation in experiments[END_REF][START_REF] Park | Warm-glow versus cold-prickle: a further experimental study of framing effects on free-riding[END_REF][START_REF] Messer | New hope for the voluntary contributions mechanism: The effects of context[END_REF].

We summarize our first set of conjectures as C1:

C1: a) Police subjects will cooperate more than non-police subjects. b) Cooperation is more likely in the VCM than in the CPR game.

Our second set of conjectures is related to exogenous norm enforcement institutions. [START_REF] Sutter | Choosing the carrot or the stick? Endogenous institutional choice in social dilemma situations[END_REF] demonstrate that the use of rewards in equilibrium is predicted by the Charness and Rabin model of social preferences only if there are enough subjects who value social welfare as strongly as their own payoff. Punishment in equilibrium is predicted by the inequity aversion model only under very restrictive conditions. Based on previous evidence, we conjecture that a fraction of the subjects will sacrifice resources to sanction and/or reward and that rewards will be preferred to sanctions because they are more efficient [START_REF] Sefton | The Effects of Rewards and Sanctions in Provision of Public Goods[END_REF][START_REF] Rand | Positive interactions promote public cooperation[END_REF][START_REF] Sutter | Choosing the carrot or the stick? Endogenous institutional choice in social dilemma situations[END_REF]. Because their occupation is mission-oriented [START_REF] Besley | Competition and incentives with motivated agents[END_REF] and also because it requires expertise in law enforcement to fight deviant behavior [START_REF] Wu | A study of Chinese cadets' motivation to become police officers[END_REF], we conjecture that police subjects will be willing to bear a disproportionate burden of the enforcement costs in punishing norm violators. Finally, regarding framing effects, since less cooperation might be observed in the CPR context, we expect to observe relatively more sanctions in the CPR. In addition, we also expect more sanctions in the CPR because norm violation involves active destruction of the resource, which may trigger more negative emotions.

We summarize our second set of conjectures as C2:

C2: a) Police subjects will enforce norms more than non-police, particularly with sanctions and in the CPR context. b) More sanctions (less rewards) will be assigned in the CPR context.

Our last set of conjectures is related to the endogenous norm enforcement institutions.

Because police fight crime with threats and sanctions, we conjecture that police commissioners are more inclined to vote in favor of sanctions than the other subjects. We also predict that, as government agents, they use the institution more intensely when it results from a democratic vote. Finally, endogenously selected institutions are expected to better sustain cooperation than exogenously implemented institutions (see [START_REF] Tyran | Achieving Compliance when Legal Sanctions are Non-deterrent[END_REF][START_REF] Kosfeld | Institution Formation in Public Goods Games[END_REF][START_REF] Sutter | Choosing the carrot or the stick? Endogenous institutional choice in social dilemma situations[END_REF].

We summarize our third set of conjectures as C3:

C3: a) Police subjects will vote more in favor of the sanction institution than others. b) Subjects, especially police subjects, will more intensively use enforcement when the institution results from a democratic vote. c) Endogenous institutions will increase cooperation relative to exogenously implemented institutions.

Results

Cooperation

In what follows, 'contribution' refers to the amount not extracted from the group account in the CPR game, or the amount placed in the group account in the VCM game. Figures 1 and2 display mean contributions over time per treatment in the VCM and CPR games, respectively.

[Insert Figures 1 and 2 about here]

Both figures exhibit the standard decay of contributions over time in the Baseline treatments.

Focusing first on the exogenous enforcement institutions, we find that mean contributions are significantly higher in the Sanction treatments compared to Baseline treatments (Mann-Whitney test, p=0.0004 and p=0.0247 in the VCM and the CPR games, respectively). Mean contributions are also higher in the Reward treatments compared to Baseline treatments (p=0.0288 and 0.0514).

Sanctions tend to outperform Rewards, but the difference is not significant (p=0.2913 and 0.2035). These findings are consistent with previous literature. Figures 1 and2 also indicate that when a vote resulted in using the Sanction institution, cooperation increases dramatically compared to Baseline (p=0.007 and 0.001, in the VCM and CPR games, respectively).

Endogenous sanctions also increase contributions compared to exogenous sanctions in the VCM game (p=0.065), but not in the CPR game (p=0.221). Compared to Baseline, the increase in contributions from endogenous rewards is not statistically significant (p=0.148 and 0.484 in the VCM and the CPR games, respectively), nor is the increase seen with exogenous rewards institutions (p=0.511 and 0.526). Regarding framing effects, our data show that contributions are effects. Model (2) adds a dummy for police subjects (Police) and other demographic controls.

Model (3) adds a dummy for police commissioners being in an active position after training (Police in activity). In models ( 2) and (3), we also control for trust and political orientation. 18[Insert Table 2 about here]

Table 2 shows that cooperation is higher when norm enforcement is possible, even more so under the Sanction treatments than under the Reward treatments relative to the Baseline. The coefficient of the Sanction treatment is significantly higher than the coefficient of the Reward treatment. We also find that endogenous sanctions lead to higher contributions than exogenous sanctions. In contrast, the coefficient of the Vote Reward treatment is not different than the coefficient of the Reward treatment. This could result from the fact that a majority vote in favor of sanctions is a clearer signal against free-riders than a vote in favor of rewards. These estimated effects are robust across models. Table 2 also indicates that, once we control for other variables, contributions are significantly higher in the VCM than in the CPR game, which supports C1b.

Table 2 also shows that the police-subjects contribute significantly more than the non-police subjects. 19 This finding supports conjecture C1a. Some experience in police work does not make any significant difference. In an additional estimate (not reported here but available upon request), we also tested whether the framing effect was different across subject pools by including an interaction term "police subject*VCM". This variable is not statistically significant.

These findings are summarized in Result 1.

Result 1. a) Cooperation is higher when norm enforcement is possible. b) Sanctions are more effective than rewards. c) Endogenous sanctions lead to higher contributions than exogenous sanctions. d) contributions are significantly higher in the VCM than in the CPR game. e) Police subjects contribute significantly more than non-police subjects.

Sanctions and Rewards

Figures 4 and 5 show the evolution of mean enforcement points assigned across periods in the VCM and CPR contexts, respectively. These figures show that subjects are willing to use both rewards and sanctions to enforce norms, even at personal cost. The use of the enforcement institutions declines over time in both games. It may be the case that enforcement is less necessary once it has been previously used within a group. A notable exception is that endogenous sanctions tend to increase in last 1-2 periods. These figures also show that there is no systematic tendency to more intensively use sanctions relative to rewards, except when the institution is endogenous. In this case, subjects are more willing to reward than punish in the VCM game (Mann-Whitney test, p=0.093), but not in the CPR game (p= 0.513).

[Insert Figures 4 and 5 about here]

Figure 6 suggests that police subjects enforce norms more intensively than non-police subjects in all treatments, and this difference seems larger in the CPR game. It also shows that the mean number of both sanction and reward points is lower in the VCM than in the CPR game.

[Insert Figure 6 about here]

Table 3 analyzes the determinants of norm enforcement by means of random-effects Tobit models to control for censored observations. We removed from this analysis the Baseline treatment. The dependent variable is the number of assigned points (both punishment and reward points) by each player i to another player j. In model (1) of Table 3, we include controls for the VCM game and for each enforcement institution (Reward treatment is the reference category).

Controls are included for points received in the previous period, period of the game, a dummy for last period, dummies for police-subjects, and other demographics. Model (2) includes several interaction variables to check whether police subjects' decisions differ across treatments.

[Insert Table 3 about here] Table 3 shows that subjects assign significantly less punishment points than rewards and that the number of points assigned declines over time. It indicates that police subjects are more likely to engage in norm enforcement than the other subjects. Model (2) shows that police subjects are more willing to enforce norms using sanctions. Police subjects also use rewards significantly more as long as the reward institution is endogenously implemented (i.e., by vote).

Tables 4 and5 analyze the determinants of the number of punishment and reward points a subject assigns to each other group member by estimating random-effects Tobit models. Model

(1) includes controls for the VCM game, whether the enforcement institution was endogenous or exogenous, points received in the previous period, and other variables mentioned before.

Compared to model (1), model (2) also includes a variable to measure the positive deviation of the contribution of the subject being assigned points from the average of the rest of the group (Pos Dev Avg), as well as a variable to measure the absolute value of a negative deviation from the group average (|Neg Dev| Avg). Note that one's own contribution is included in the group average contribution levels being considered in Pos Dev Avg and |Neg Dev| Avg, and so these models estimate effects while controlling for the subject's contribution level. Avg Contr Others measures the effect of others' contributions (excluding the target's contribution) on one's own decision to assign points. In addition, we control for trust and political orientation. The last two models are similar to (2) but estimate the model separately for exogenous (model ( 3)) versus endogenous (model ( 4)) enforcement institutions.

[Insert Tables 4 and5 about here]

Table 4 shows that subjects are significantly more willing to sanction group members in the CPR than in the VCM game when the sanction institution is implemented exogenously (model ( 3)). Subjects contributing more (less) than the group average are uniformly less (more) sanctioned, and subjects assign fewer points the higher others in one's group contributed. There is also evidence of blind negative reciprocity and of a declining time trend for norm enforcement via sanctions. Police subjects are more willing to sanction than non-police subjects. The magnitude of this effect is particularly large for newly minted commissioners with no job experience (model ( 4)). Two-stage Heckit estimations (not reported here but available upon request) show that the difference between subject-pools is due to the extensive margin decision to punish or not, and not to the intensive margin decision to assign more or less punishment points.

Table 5 show that an endogenous reward institution marginally increases the likelihood of rewarding group members (model ( 2)). Subjects assign more reward points to those contributing more than the group average, and more reward points in general when others contribute more.

There is also evidence of blind positive reciprocity and a negative time trend. Police commissioners assign more reward points than other subjects but only when the reward institution results from a majority vote (models (3) and ( 4)).

To sum up, Tables 3 to 5 reveal three results of particular interest. First, subjects are significantly less willing to punish than reward others and they are less inclined to use sanctions in the VCM than in the CPR game, independent of the lower contribution levels in the CPR context. This may reflect the fact that the negative frame of the CPR context is more likely to generate negative emotions and induce the use of punishment. Secondly, police subjects, especially those in activity, enforce norms with punishment significantly more than non-police subjects. This supports our conjecture C2a. There is also some evidence that novice police commissioners enforce norms with rewards significantly more than others. We attribute these police subject results to the fact that police work is one where law enforcement is almost entirely by use of sanctions, and experience with punishment and crime in their occupation may train their preferences towards punishment. Third, police commissioners tend to use the institutional norm enforcement more than the non-police subjects when the institution results from a majority vote. This police-specific "democracy effect" might result from the civic values in the occupation of police officers. For the whole sample, an enforcement institution resulting from a majority vote leads to higher norm enforcement, but only when the norm enforcement institution is reward. This is also a novel result in the literature.

Our main findings regarding norm enforcement are summarized in Result 2.

Result 2. a) Police subjects enforce norms significantly more than non-police, especially with sanctions. b) Police enforce norms significantly more when the enforcement institution results from a majority vote, while for the whole subject pool, this democracy effect is only observed for the reward institution. c) More sanctions are used under the CPR than under the VCM.

Voting on norm enforcement Institutions

Table 6 displays the distribution of votes for the reward and sanction institutions for each pool of subjects in the VCM and CPR games.

[Insert Table 6 about here]

Table 6 indicates that subjects are much more likely to choose the Reward over the Sanction institution. The proportion preferring rewards is significantly higher than the proportion preferring sanctions in both VCM and CPR games (binomial tests; p<0.01 for both), and there is no difference between the two games (Χ 2 test, p = 0.941). Although the majority of police subjects still voted for Rewards, they were more inclined to vote for Sanctions compared to nonpolice (32% vs. 24% for CPR and 35.1% vs. 20.9% for VCM). A Χ 2 -test indicates that the difference is however not statistically significant (p= 0.156 and p=0.373 for VCM and CPR, respectively).

Table 7 investigates the individual determinants of the vote in favor of the sanction institution by estimating Probit models. In model (1), the independent variables include a dummy variable for the VCM game and a control indicating whether the vote occurs after the subjects have experienced both enforcement institutions. Model (2) adds a dummy variable to account for whether the subject is a police commissioner and demographic controls. The last model also controls for whether the police subject already has some experience (i.e., "in activity").

[Insert Table 7 about here]

Supporting conjecture C3a, Table 7 confirms that police subjects exhibit a higher preference for sanctions compared to others. This preference is significant only for commissioners with more experience with the police, suggesting that a longer exposure to crime and punishment reinforces confidence in sanctions. In parallel, a longer experience to possible free riding and with each institution increases the preference for sanctions as shown by the significant coefficient associated to the variable "vote with experience of the game". Sanctions become more appealing after observing the decay of cooperation and the higher impact of sanctions relative to rewards.

This leads to our next result regarding the police subjects:

Result 3. Police subjects are more likely to vote for the sanction institution compared to others. This effect is driven by those with work experience as police commissioners.

Efficiency

Lastly, we investigate the efficiency, as measured by the sum of final payoffs. 8 also indicates that in all treatments the mean first-stage and final payoffs are higher in the VCM than in the CPR game, with the highest efficiency achieved when using rewards in both games and the highest efficiency losses associated with the sanction enforcement institution in the CPR game.

Finally, Table 9 reports the estimations of random-effects Generalized Least Squared models, in which the dependent variable is either the first-stage payoff (models (1) and ( 2)), or the final payoff (models (3) and ( 4)). Independent variables are those used in previous regressions, with the addition in models ( 2) and ( 4) of a variable indicating the number of police subjects in the group.

[Insert Tables 8 and9 about here]

Both models (1) and ( 2) show significant treatment coefficients, indicating that introducing a norm enforcement institution increases first-stage payoffs by encouraging cooperation. The coefficients are particularly high in the Vote Sanction treatments. Models (3) and (4) show that the Reward institution, whether implemented by vote or exogenously, increases efficiency. In contrast, the Sanction institution decreases efficiency, though the magnitude of the loss is small when sanctions are implemented by vote. Interestingly, those in groups with more police subjects make higher payoffs because police subjects tend to cooperate more and bear a higher share of the enforcement costs. This leads to our final result:

Result 4. a) Sanctions increase cooperation the most, but rewards increase overall efficiency. b) Efficiency is systematically higher in the VCM than in the CPR game. c) Police subjects do not earn more than the other subjects, but their presence in a group increases efficiency in the group.

Discussion

In this study, we examined norm enforcement using either carrots or sticks in two distinct social dilemmas (VCM and CPR). A field-experiment element in our design was the use of a sample of French police commissioners along with a standard subject pool. We have three main sets of findings.

First, experience with a "destructive" social dilemma context, either in the experiment (i.e., the CPR context versus VCM) or in real life through longer exposure to crime in police work, leads to a stronger use of sanctions. We also find that police subjects with some work experience show more preference for sanctions compared to non-police. These results contribute to our understanding of how police officers behave in social dilemmas. More sanctions are used in the CPR game than in the VCM game, even after controlling for lower contributions in the CPR context. This is possibly because a negative frame may triggers more negative emotions than a positive frame for the social dilemma. Rewards are not used any more frequently in the VCM than in the CPR game and they become less effective than sanctions over time in both games. Our results are also important because they lend some support to the external validity of laboratory games: despite our artificial environment, norm enforcement behavior in the laboratory is correlated with some experience in norm enforcement in real settings such as law enforcement. Finally, a longer exposure to free riding over time in the experimental game also leads subjects to favor sanctions over rewards for norm enforcement.

Secondly, we show that individuals in a mission-oriented occupation responsible for norm enforcement, like police commissioners, are both more cooperative in social dilemmas and they bear more of the costs of the norm enforcement. This is especially true when the enforcement institution uses punishment. This translates into higher earnings for groups populated with more police subjects.

Thirdly, when the choice of the institution depends on voting, a large majority of individuals prefer Rewards over Sanctions in both the CPR and VCM games. This result extends that of [START_REF] Sutter | Choosing the carrot or the stick? Endogenous institutional choice in social dilemma situations[END_REF] to a wider set of conditions. This preference may be due to a willingness to avoid the loss of efficiency associated with sanctions and/or the perspective of mutual benefits through positive reciprocity within groups. Interestingly, individuals are more likely to enforce norms, in general, when the majority has selected the enforcement institution compared to when it is imposed. This effect is stronger in police commissioners who are particularly sensitive to the democratic implementation of both types of enforcement institutions, possibly because they are more reactive to civic virtues and to intrinsic motivation in general.

Finally, we find that the endogenous sanction institution leads to higher contributions compared to the corresponding exogenously imposed institution.

We acknowledge a number of limitations to our study. It would be desirable to recruit more experienced police commissioners. Indeed, this would allow us to better identify whether the police-subject effect is due to intrinsic motivation and self-selection, or whether it is due to the role of experience in law enforcement. Moreover, the low number of periods in our experiment penalizes sanction efficiency because the benefits of sanctions take more time to develop than the benefits of rewards. Despite these limitations, we believe that it is important to expose norm enforcement mechanisms to a large variety of environmental and institutional conditions so as to evaluate their robustness and derive policy implications on institutional design.

Figure 1 :Figure 2 :Figure 3 :Figure 4 .Figure 5 .Figure 6 :

 123456 Figure 1: Average contributions per treatment in the VCM games

  Table 8 reports the first-stage and final payoffs in all treatments and games. Pooling the data from the VCM and CPR games, the first column indicates that the mean payoffs after the contribution stage (i.e., before enforcement points are subtracted or added) rank as follows: 22.96 ECU in the Baseline, 24.13 in Vote Reward, 24.74 in Reward, 25.94 in Sanction, and 28.91 in Vote Sanction.However, mainly due to the destruction of resources by sanctions, the final payoff ranking differs: 18.84 ECU inSanction, 22.96 in Baseline, 24.10 Vote Sanction, 26.65 in Vote Reward, and 26.88 in Reward. Table

Table 1 .

 1 Description of sessionsNote: B: Baseline treatment; R: Reward treatment; S: Sanction treatment; V: Vote treatment. CPR: Common Pool Resources; VCM: Voluntary Contribution Mechanism.

	Session	Sequence	Game	Number of subjects
	1	BRSV	CPR	20
	2	BRSV	VCM	20
	3	BSRV	CPR	20
	4	BSRV	CPR	20
	5	BVRS	CPR	20
	6	BVSR	CPR	20
	7	BSRV	VCM	20
	8	BVRS	VCM	20
	9	BVSR	VCM	20

Table 2 .

 2 Determinants of contribution levels Standard errors are in parentheses. *, **, *** indicate significance at the 0.10, 0.05, and 0.01 levels, respectively for the 2-tailed test. The 5580 observations correspond to the 180 subjects observed in 31 periods of game.

	Dependent variable:	Random-effects Tobit models
	Contribution	(1)	(2)	(3)
	Sanction treatment	12.31***	12.31***	12.32***
		(0.602)	(0.602)	(0.602)
	Reward treatment	6.101***	6.103***	6.104***
		(0.590)	(0.590)	(0.590)
	Vote Sanction treatment	29.64***	29.63***	29.63***
		(1.516)	(1.516)	(1.515)
	Vote Reward treatment	4.379***	4.383***	4.383***
		(0.639)	(0.639)	(0.639)
	VCM game	5.465***	5.634***	5.207**
		(2.114)	(2.088)	(2.136)
	Period	-1.369*** -1.368***	-1.368***
		(0.101)	(0.101)	(0.101)
	Last period	-1.458**	-1.456**	-1.456**
		(0.697)	(0.697)	(0.697)
	Police	----	4.270*	4.837**
			(2.389)	(2.466)
	Police in activity	----	----	-3.726
				(4.148)
	Age	----	-0.0815	-0.0763
			(0.130)	(0.129)
	Male	----	0.689	0.703
			(2.243)	(2.238)
	Trust		0.649	0.631
			(0.571)	(0.570)
	Political orientation		0.285	0.251
			(0.496)	(0.496)
	Constant	5.903***	2.435	2.725
		(1.531)	(4.651)	(4.651)
	N	5580	5580	5580
	Left-cens. obs.	1903	1903	1903
	Right cens. obs.	1643	1643	1643
	Log-likelihood	-10538.99	-10535.99	-10535.59

Note:

Table 3 .

 3 Determinants of sanction or reward decisions

	Dependent variable	Random-Effects Tobit models
	Number of
	reward or sanction points		(1)	(2)
	Sanction treatment		-0.385***	-0.610***
			(0.116)	(0.168)
	Vote Sanction treatment		-0.830***	-1.384***
			(0.245)	(0.434)
	Vote Reward treatment		-0.0397	-0.262
			(0.121)	(0.172)
	VCM game		-0.372	-0.366
			(0.402)	(0.402)
	Period		-0.310***	-0.310***
			(0.0359)	(0.0359)
	Last period		-0.0653	-0.0653
			(0.177)	(0.176)
	Police		0.944**
			(0.462)
	Police in activity		0.253	0.240
			(0.777)	(0.777)
	Police*Sanction treatment			1.075**
				(0.482)
	Police*Reward treatment			0.646
				(0.478)
	Police*VoteSanction treatment			1.511**
				(0.677)
	Police*VoteReward treatment			1.078** (0.488)
	Points received in previous period	0.290***	0.290***
			(0.0119)	(0.0119)
	Trust		0.0949	0.0948
			(0.107)	(0.107)
	Political orientation		-0.0110	-0.0117
			(0.0933)	(0.0932)
	Age		0.0294	0.0293
			(0.0243)	(0.0243)
	Male		-0.0168	-0.0225
			(0.421)	(0.420)
	Constant		-3.996*** (0.893)	-3.831*** (0.895)
	N		12960	12960
	Left censored obs.		9954	9954

Table 4 .

 4 Determinants of sanction decisions

	Dependent variable	Random-Effects Tobit models
	Number of	Exogenous and endogenous	Exo.	Endo.
	sanction points	institution	institution	institution
		(1)	(2)	(3)	(4)
	Vote Sanction treatment	0.150 (0.331)	0.372 (0.327)	-	-
	VCM game	-0.443	-1.548***	-1.809***	1.345
		(0.645)	(0.600)	(0.605)	(2.069)
	Received points in t-1	0.138***	0.094***	0.067**	0.252***
		(0.029)	(0.025)	(0.027)	(0.096)
	Pos Dev Avg	---	-0.116***	-0.109***	-0.639**
			(0.031)	(0.030)	(0.269)
	|Neg Dev| Avg	---	0.499***	0.493***	0.474***
			(0.022)	(0.026)	(0.060)
	Avg Contr Others	---	-0.197***	-0.208***	-0.383
			(0.029)	(0.031)	(0.271)
	Period	-0.357***	-0.331***	-0.298***	-0.309
		(0.074)	(0.065)	(0.066)	(0.263)
	Last period	1.044***	0.287	0.172	1.241
		(0.343)	(0.308)	(0.314)	(1.227)
	Police	-0.538 (0.736)	-0.031 (0.680)	-0.144 (0.681)	6.266** (2.466)
	Police in activity	2.519**	1.882*	1.814*	2.219
		(1.203)	(1.100)	(1.099)	(2.590)
	Age	0.143***	0.0911***	0.0941***	-0.0146
		(0.038)	(0.035)	(0.035)	(0.139)
	Male	0.352	0.214	0.482	-4.419**
		(0.665)	(0.621)	(0.623)	(2.170)
	Trust	---	0.099	0.049	-0.409
			(0.155)	(0.156)	(0.421)
	Political orientation	---	-0.033	-0.063	0.176
			(0.137)	(0.138)	(0.366)
	Constant	-8.245***	-4.159***	-3.630***	-3.713
		(1.212)	(1.413)	(1.409)	(6.736)
	N	5280	5280	4320	960
	Left censored obs.	4364	4364	3506	858
	Log-likelihood	-3800.05	-3410.88	-2943.10	-426.05

Note: Standard errors are in parentheses. *, **, *** indicate significance at the 0.10, 0.05, and 0.01 levels, respectively for the 2-tailed test. The 4320 observations correspond to the decisions of the 180 subjects observed in

Table 5 .

 5 Determinants of reward decisions Standard errors are in parentheses. *, **, *** indicate significance at the 0.10, 0.05, and 0.01 levels, respectively for the 2-tailed test. The 4320 observations correspond to the decisions of the 180 subjects observed in 6 periods of game (one variable is lagged) for each of their other 4 group members. The 3360 observations correspond to 140 subjects' decisions for each of their 4 group members in 6 periods in groups who voted for reward.

	Dependent variable:	Random-Effects Tobit models
	Number of	Exogenous and endogenous	Exo	Endo
	reward points	institution		institution	institution
		(1)	(2)	(3)	(4)
	Vote Reward treatment	0.041	0.181*	-	-
		(0.099)	(0.093)
	VCM game	-0.224	0.232	0.114	-0.432
		(0.465)	(0.438)	(0.494)	(0.525)
	Received points in t-1	0.193***	0.150***	0.133***	0.113***
		(0.014)	(0.013)	(0.019)	(0.019)
	Pos Dev Avg	----	0.183***	0.182***	0.162***
			(0.013)	(0.019)	(0.018)
	|Neg Dev| Avg	----	-0.339***	-0.334***	-0.328***
			(0.016)	(0.023)	(0.022)
	Avg Contr Others	----	0.199***	0.130***	0.182***
			(0.015)	(0.023)	(0.024)
	Period	-0.341***	-0.244***	-0.344***	-0.281***
		(0.037)	(0.034)	(0.048)	(0.049)
	Last period	-0.555***	-0.222	-0.443*	0.035
		(0.185)	(0.177)	(0.238)	(0.240)
	Police	1.135**	0.657	0.436	1.337**
		(0.532)	(0.502)	(0.564)	(0.607)
	Police in activity	-0.762	-0.400	-0.0493	0.686
		(0.909)	(0.852)	(0.946)	(1.098)
	Age	-0.003	0.033	0.0161	0.0124
		(0.028)	(0.026)	(0.030)	(0.032)
	Male	-0.130	-0.158	-0.011	-0.622
		(0.480)	(0.456)	(0.512)	(0.547)
	Trust	---	0.234**	0.217	0.462***
			(0.118)	(0.132)	(0.149)
	Political orientation	---	-0.0448	-0.078	-0.049
			(0.101)	(0.113)	(0.128)
	Constant	-2.019** (0.859)	-5.428*** (1.007)	-3.651*** (1.159)	-4.203*** (1.228)
	N	7680	7680	4320	3360
	Left censored obs.	5590	5590	3170	2420
	Log-likelihood	-7087.22	-6442.95	-3508.05	-2798.25

Note:

Table 8 .

 8 First-stage and final payoffs per treatmentNote: Standard errors in parentheses.

	Treatments	First-stage	Final	First-stage payoffs	Final payoffs
		payoffs	payoffs
		all games	all games	CPR	VCM	CPR	VCM
	Baseline	22.96	22.96	22.73	23.24	22.73	23.24
	treatment	(5.96)	(5.96)	(6.18)	(5.66)	(6.18)	(5.66)
	Sanction	25.94	18.84	25.45	26.55	17.33	20.72
	treatment	(5.61)	(11.62)	(6.54)	(4.09)	(12.95)	(9.39)
	Reward	24.74	26.88	24.16	25.47	26.23	27.70
	treatment	(6.79)	(8.95)	(7.24)	(6.11)	(9.61)	(7.99)
	Vote Sanction	28.91	24.10	28.53	29.54	22.18	27.31
	treatment	(5.27)	(7.41)	(6.01)	(3.71)	(7.85)	(5.24)
	Vote Reward	24.13	26.65	23.68	24.64	26.28	27.08
	treatment	(6.38)	(9.56)	(6.79)	(5.85)	(10.79)	(7.91)

Table 9 .

 9 Determinants of payoffs Baseline treatment in the CPR game is the reference category. Standard errors are in parentheses. *, **, *** indicate significance at the 0.10, 0.05, and 0.01 levels, respectively for the 2-tailed test. The 5580 observations correspond to the 180 subjects observed in 31 periods of game.

		GLS models
		First-stage	First-stage
	Dependent variables	payoff	payoff	Final payoff	Final payoff
		(1)	(2)	(3)	(4)
	Sanction treatment	2.587***	2.587***	-4.493***	-4.493***
		(0.195)	(0.195)	(0.287)	(0.287)
	Reward treatment	1.386***	1.386***	3.549***	3.549***
		(0.195)	(0.195)	(0.287)	(0.287)
	Vote Sanction treatment	4.687***	4.669***	-0.986*	-1.005*
		(0.360)	(0.360)	(0.529)	(0.529)
	Vote Reward treatment	1.022***	1.028***	3.820***	3.826***
		(0.213)	(0.213)	(0.314)	(0.314)
	VCM game	0.907*	0.908*	1.628**	1.489**
		(0.501)	(0.510)	(0.685)	(0.696)
	Period	-0.272***	-0.272***	-0.271***	-0.271***
		(0.033)	(0.033)	(0.048)	(0.048)
	Last period	-0.249	-0.249	-0.726**	-0.726**
		(0.227)	(0.227)	(0.334)	(0.334)
	Police	0.485	-0.001	0.371	-0.018
		(0.575)	(0.642)	(0.786)	(0.876)
	Police in activity		-0.733		-2.114
			(0.990)		(1.350)
	Number of police subjects		0.638**		0.758**
	in the group		(0.266)		(0.363)
	Age	-0.117***	-0.118***	-0.197***	-0.197***
		(0.031)	(0.031)	(0.042)	(0.042)
	Male	0.016	0.223	0.056	0.307
		(0.538)	(0.539)	(0.735)	(0.736)
	Trust	-0.175	-0.185	-0.032	-0.050
		(0.136)	(0.135)	(0.187)	(0.184)
	Political orientation	0.039	0.051	0.268	0.270
		(0.119)	(0.119)	(0.163)	(0.162)
	Constant	27.398***	26.018***	27.784***	27.406***
		(1.125)	(1.270)	(1.541)	(1.473)
	N	5580	5580	5580	5580
	Wald Chi 2	496.27	501.40	1057.57	1062.47
	R 2	0.10	0.10	0.15	0.15

Note:

While it may be natural to consider as virtuous those employees who outperform the norm (i.e. exert more effort than the norm), it is unclear what it means to reward individuals who outperform the law (or help enforce it) in the public domain.

For example, if a social norm is that public goods be maintained, violation of that norm may be viewed differently if it involves destroying an existing public good than if it involves the more passive failure to assist in its provision.

This idea is well illustrated by the comments of a police officer who was recently involved in the "Pops for Cops" program. He said: "Like many professional law enforcement officers, I brought a certain mentality to the job. I wanted to hunt down criminals -chase bad guys, kick in doors, get the bust. It was hunter v hunted… I can't escape the realities of my job -I have to hunt down criminals. But could I also work on the other end of the spectrum? Could I build positive relationships strong enough to keep youth out of trouble?"

Punishment is typically directed at those individuals who violate the norm of cooperation given by the mean group contribution level.

After conducting our experiments, we became aware of another experimental study of trust and norm enforcement conducted with applicants to the German police by G.[START_REF] Friebel | Sorting of Motivated Agents : Evidence from Applicants to the German Police[END_REF]. The two studies differ in several respects. First, their study focuses on how individuals self-select into an occupation based on their behavioral characteristics. Instead, we focus on comparing the use and efficacy of norm enforcement institutions in various environments given that subjects are police or non-police. We do not try to determine whether the behavior of police subjects is due to behavioral self-selection into the occupation or whether it results from the training in law enforcement they receive. Second, their subject-pool consists of students in the final year of the high school, who are may apply to the police. Instead, French police commissioners have higher education and are expected to manage large teams. Third, their design is based on a trust game with a third-party and individuals can use both rewards and sanctions in the same periods (in ours it is one or the other), and their study does not include endogenous institutions. Our studies are therefore complementary.

[START_REF] Prendergast | The motivation and bias of bureaucrats[END_REF] shows that among public employees, if social workers are more likely to be biased in favor of their clients, police officers are more likely biased against their clients, i.e. those who break the law. If behavior in law enforcement transposes to norm enforcement, commissioners may be more inclined to sanction than reward.

Our procedure differs from that used in[START_REF] Sutter | Choosing the carrot or the stick? Endogenous institutional choice in social dilemma situations[END_REF]. In their design, subjects could vote between standard VCM, reward, and punishment by approval voting; voting was costly and voluntary, and voting was repeated until unanimity was achieved.

Initially, we planned to run 10 periods in each of the four treatments. Unfortunately, and despite a pilot session with our usual subject-pool, we realized during the first session that some subjects were very slow to make decisions. Therefore, we decided to reduce the number of periods to seven after the Baseline treatment. We kept the same structure for the remaining sessions. For comparison, we chose to restrain our data analysis to periods 1 to 7 in the session where we ran 10 periods for all treatments.

This was done to avoid counter-punishment[START_REF] Denant-Boemont | Punishment, counterpunishment and sanction enforcement in a social dilemma experiment[END_REF][START_REF] Nikiforakis | Punishment and counter-punishment in public-good games: Can we really govern ourselves[END_REF] or counterrewarding[START_REF] Noussair | Punishment, reward, and cooperation in a framed field experiment[END_REF], as it has been shown that this possibility affects greatly the dynamics of groups.

 14 The instructions use the terms « contribution » in the VCM and « withdrawal » in the CPR. Sanctions (rewards) were labeled as "points that decrease (increase) others' payoffs

".15 At the beginning of the session we elicited risk attitudes using the[START_REF] Holt | Risk Aversion and Incentive Effects[END_REF] procedure. We found no significant difference between subject pools (the mean numbers of safe choices out of 10 were 5.83 (1.78) and 5.82 (1.71) for police and non-police subjects, respectively (Mann-Whitney test, two tailed, p=0.910). After risk elicitation, participants played a trust game(Berg et al., 1995) in both roles under the strategy method. We found no significant difference between subject pools in trusting behaviors (the mean amounts sent by player A to player B out of 10 were 2.69 (S.D.177) and 2.41 (SD 1.88) for police and non-police subjects, respectively (Mann-Whitney test, two tailed, p=0.448). Results from these two tasks were only given at the end of the session.

Note that we have variation in age across both subject-pools, and we systematically control for age in the analysis.

Investigations on the motivation for entering the police emphasize the importance of non-monetary dimensions, such as the willingness to help people and social status[START_REF] Hageman | Who joins the force for what reasons: An argument for 'the new breed[END_REF][START_REF] Moon | The reasons for choosing a career in policing among South Korean police cadets[END_REF][START_REF] Raganella | Race, gender, and motivation for becoming a police officer Implications for building a representative police department[END_REF] Tarng et al., 2008;[START_REF] Wu | A study of Chinese cadets' motivation to become police officers[END_REF]. Social contribution is also ranked as the first determinant of job satisfaction by police officers[START_REF] Carlan | The Search for Job Satisfaction: A Survey of Alabama Policing[END_REF].

"Trust" corresponds to the amount sent by player A to player B in the trust game. The "political orientation" variable corresponds to the response to the following question in the post-experimental questionnaire: «In politics one usually speaks of right and left. Where do you situate yourself on a scale from 1 on the left to 10 on the right? ».

[START_REF] Gächter | Trust, voluntary cooperation, and socio-economic background: Survey and experimental evidence[END_REF] also report that non-students contribute more than students but their results are from a oneshot VCM environment as opposed to our finding in a multi-period environment. In what follows, our additional results regarding norm enforcement and preference for punishments over rewards helps us attribute the subject pool effects we report to the special attributes of police officers, as opposed to non-students in general.

periods of game (one variable is lagged) for each of their other 4 group members. The 960 observations correspond to 40 subjects' decisions for each of their 4 group members in 6 periods in groups who voted for sanction.

We are particularly grateful to the Ecole Nationale Supérieure de la Police (ENSP) and in particular H. Martini and D. Lambert who helped make these experiments possible. We thank E. Priour for programming the experiment and E. Derigo and Z. Dai for valuable research assistance. The financial support of ANR (National Research Agency) for the CONFLICT program is gratefully acknowledged (ANR_jc25-02-08). This research was performed within the framework of the LABEX CORTEX (ANR-11-LABX-0042) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-007) operated by the French National Research Agency (ANR). The usual disclaimer applies and the authors are solely responsible for the content of this analysis.

higher in VCM compared to CPR, but the differences are not statistically significant. Baseline mean contributions, in terms of ECUs, are 6.49 in VCM and 5.46 in the CPR game (Mann Whitney,p=0.324). In the Sanction treatment they are 13.10 (VCM) compared to 10.91 (CPR) (p=0.347), while in the Reward treatment they are 10.94 (VCM) and 8.32 (CPR) (p=0.156).

We turn next to comparisons across subject-pools. Figure 3 displays average contribution levels for police and non-police subjects, by treatment in the VCM and the CPR games.

[Insert Figure 3 about here]

As shown in Figure 3, in all treatments, police subjects cooperate more on average than the others but the differences are not significant. The lack of a statistically significant subject pool effect is true in all treatments examined (Mann-Whitney, p>0.10 in all cases), except for the Vote-Sanction treatment, where police subjects contribute significantly more than non-police (18.42 vs 17.03 ECU, p=0.088). The fact that most non-parametric tests fail to identify significant effects here is not that surprising since our subject-pools differ on several characteristics not controlled for in the non-parametric data analysis. We therefore appeal to parametric data analysis to further investigate differences across subject pools.

Table 2 reports three random-effects Tobit regressions analyzing the determinants of the contribution decision. Random effects control for the lack of independence of the contributions of a given subject across decision rounds and Tobit models are justified by the fact that contributions are censured both at 0 and 20 ECU. We pool the data from all treatments and we standardize contributions of individual i in period t as the dependent variable by defining contributions in the CPR game as the amount (20-w it ) that is not withdrawn from the common resource. In models (1) to (3) of Table 2, independent variables include a dummy variable for the VCM game and treatment dummies, with the Baseline omitted as the reference treatment. We also include controls for period (1-7) and a dummy for the last period to control for end-game


Mark Dukes

Yvan Le Borgne

The sandpile model on a bipartite graph, parallelogram polyominoes, and a q, t-Narayana polynomial

Keywords: sandpile model, bipartite graph, q, t-Catalan

de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Résumé. Pour le modèle du tas de sable sur un graphe Km,n biparti complet, on donne une description des configurations récurrentes à l'aide d'une bijection avec des polyominos parallélogrammes décorés de rectangle englobant m × n. D'autres classes combinatoires apparaissent comme des cas particuliers de cette construction: par exemple les matrices de bicomposition et les ordres partiels évitant le motif (2+2).

Un processus d'éboulement canonique des configurations récurrentes se traduit par un chemin bondissant dans le polyomino parallélogramme associé. Nous définissons une famille de polynômes, baptisée de q, t-Narayana, à travers la distribution d'une paire de statistique (aire, poidscheminbondissant) sur les polyominos parallélogrammes similaire à celle de Haglund définissant les polynômes de q, t-Catalan sur les chemins de Dyck. Ainsi nous étendons une paire de statistique de Egge et d'autres à l'ensemble des polynominos parallélogrammes. Cela réponds à l'une de leur question sur des généralistations à d'autres objets combinatoires.

Nous conjecturons que les polynômes de q, t-Narayana sont symétriques et discutons des preuves de plusieurs cas particuliers. Nous montrons également une relation avec les polynômes de q, t-Catalan en restreignant notre paire de statistique à un sous-ensemble des polyominos parallélogrammes.

Introduction

The abelian sandpile model [START_REF] Dhar | Theoretical studies of self-organized criticality[END_REF] is a discrete diffusion model whose states are distributions of grains on the vertices of a non-oriented graph. A vertex of a graph is stable if the number of grains at the vertex is strictly smaller than its degree, otherwise it is called unstable. In addition to a randomised addition of grains, the dynamics of this model requires that an unstable vertex donate a grain via each of its incident edges to its neighbours. This is the so-called toppling or avalanche process of the model, whose terminaison is guaranteed by a distinguished vertex, called the sink, that never topples collecting grains going out of the system. This process defines a Markov chain on stable states. In the particular case of graphs, there exist many bijections from recurrent states to spanning trees of the same graph [START_REF] Dhar | Theoretical studies of self-organized criticality[END_REF][START_REF] Cori | The sandpile model and Tutte polynomials[END_REF]. Another rich result concerning the model shows that the recurrent states, together with a certain binary operation, form an abelian group called the sandpile group.

In this extended abstract we outline some results from our paper [START_REF] Dukes | Parallelogram polyominoes, the sandpile model on a bipartite graph, and a q, t-Narayana polynomial[END_REF] in which we classified recurrent configurations of the sandpile model on the complete bipartite graph K m,n in which a single designated vertex is the sink. Cori & Poulalhon [START_REF] Cori | Enumeration of (p, q)-parking functions[END_REF] classified recurrent configurations of the complete (k + 1)-partite graph K p1,...,p k ,1 where the sink is the single vertex that is connected to all other vertices. Our work complements [START_REF] Cori | Enumeration of (p, q)-parking functions[END_REF] by breaking the symmetry of K p,q,1 and showing how recurrent configurations of K m,n that have one sink can be interpreted as parallelogram (or staircase) polyominoes.

Minimal recurrent configurations (those whose number of grains is minimal) correspond to ribbon parallelogram polyominoes (those having minimal area). In the original paper [START_REF] Dukes | Parallelogram polyominoes, the sandpile model on a bipartite graph, and a q, t-Narayana polynomial[END_REF] we showed links between some special minimal configurations and other combinatorial structures. We will outline some of these rich connections at the end of Section 2 which are linked to a matrix analogue of set partitions. Such matrices where the subject of several recent papers that presented surprising connections between five seemingly disparate combinatorial objects: (2+2)-free posets, Stoimenow matchings, ascent sequences, permutations avoiding a length-3 bivincular pattern, and a class of upper triangular matrices (see [START_REF] Dukes | Composition matrices, (2+2)-free posets and their specializations[END_REF][START_REF] Bousquet-Mélou | 2+2)-free posets, ascent sequences and pattern avoiding permutations[END_REF][START_REF] Dukes | Ascent sequences and upper triangular matrices containing nonnegative integers[END_REF][START_REF] Claesson | Partition and composition matrices[END_REF]). It was through these structures that we first noticed a connection to the sandpile model and the initial motivation behind this work.

During the last two decades, a series of papers have examined a power series C n (q, t) that has become known as the q, t-Catalan function (or polynomial/number). This power series was introduced by Garsia and Haiman [START_REF] Garsia | A remarkable q, t-Catalan sequence and q-Lagrange inversion[END_REF] and has important links to algebraic geometry and representation theory. In the original paper they showed that two special cases of this polynomial had combinatorial significance. The first was that C n (q, 1) was the generating function of the area statistic over Dyck paths having semi-length n. The second was that q ( n 2 ) C n (q, 1/q) is the nth q-Catalan number. Haglund introduced a new statistic 'bounce' and conjectured that C n (q, t) was the generating function of the bistatistic (area, hagbounce) on the set of all Dyck path of semi-length n. Garsia and Haglund [START_REF] Garsia | A positivity result in the theory of Macdonald polynomials[END_REF][START_REF] Garsia | A proof of the q, t-Catalan positivity conjecture[END_REF] proved this conjecture using methods from the theory of symmetric functions. (See Haglund [START_REF] Haglund | The q, t-Catalan numbers and the space of diagonal harmonics[END_REF] for a concise overview of these results.) Egge, Haglund, Kremer & Killpatrick [START_REF] Egge | A Schröder generalization of Haglund's statistic on Catalan paths[END_REF] asked if the lattice path statistics for C n (q, t) can be extended, in a way which preserves the rich combinatorial structure, to related combinatorial objects.

We present a pair of statistics (area, parabounce) on parallelogram polyominoes which is one answer to their question. We call the resulting polynomials q, t-Narayana polynomials since they specialise to the Narayana numbers for the case q = 1 = t. The bivariate generating function for this pair of statistics N m,n (q, t) appears to be symmetric in both q, t and m, n. We conjecture and discuss the symmetry of this bistatistic and prove it for numerous special cases.

A parallelogram polyomino whose bounding rectangle is [0, m] × [0, n] is a polyomino uniquely described by a pair (upper(P), lower(P)) of paths which begin at (0, 0), end at (m, n), take steps in the set {n = (0, 1), e = (1, 0)}, and only touch at their endpoints. Let us write Para m,n for this collection of parallelogram polyominoes. Let Ribbon m,n be the set of polyominoes P in Para m,n whose area is m + n -1, the minimal value, which we will call ribbon polyominoes. (See Example 1 for two examples of these polyominoes.)

Given P ∈ Para m,n , let Bounce(P) be the unique path from (m -1, n) to (0, 0) which is defined as follows: Starting from (m -1, n), the path goes south until it encounters a vertex on the lower path of P. The path then turns to the west and continues straight until it encounters a vertex on the upper path of P. The path turns again to the south until it encounters a vertex on the lower path of P, and so on, until it reaches (0, 0). Let bounce(P) = (c 1 , c 2 , . . .) be the sequence of numbers where c 1 is the number of initial south s steps in Bounce(P), c 2 is the number of contiguous west steps that follow the initial run of south steps in Bounce(P), and so forth. For P 1 we have Bounce(P 1 ) = (s, s, w, w, w, s, w, s, s) and bounce(P 1 ) = (2, 3, 1, 1, 2). For P 2 we have Bounce(P 2 ) = (s, w, w, w, s, s, w, w, w) and bounce(P 2 ) = (1, 3, 2, 3).

The sandpile model on D m,n and recurrent configurations

In this section we show how recurrent configurations of the sandpile model [START_REF] Dhar | Theoretical studies of self-organized criticality[END_REF] on K m,n which has a designated vertex that acts as a sink can be classified in terms of parallelogram polyominoes. In what follows we will refer to the the directed bipartite graph K m,n that has a designated vertex, which we will call the sink, as D m,n . The purpose in doing so is to avoid confusion about two non-equivalent choices for the sink in K m,n (in general if we have many sinks then we merge them into a single one).

For a general (stable) configuration u on D m,n we define a collection of cells

f m,n (u) ⊆ [0, m] × [0, n] ⊆ R 2 .
A canonical toppling process for checking recurrent configurations is defined (using Dhar's criterion) for the graph D m,n , and it is then shown that a configuration being recurrent is equivalent to f m,n (u) being a parallelogram polyomino in Para m,n . During this, we show the canonical toppling process of a recurrent configuration to be intimately linked with the bounce path of the corresponding polyomino f m,n (u).

Let D m,n be the graph D m,n (V, E) with vertex set V = {v 0 , . . . , v m+n-1 } and edge set This is illustrated in Figure 1. We call {v 0 , . . . , v m-1 } and {v m , . . . , v m+n-1 } the top vertices and bottom vertices of D m,n , respectively. A configuration, or state, x on D m,n is an assignment x : {1, . . . , m + n -1} → N 0 = N ∪ {0}, the collection of non-negative integers. The value x i = x(i) represents the number of grains of sand on vertex v i . Vertex v 0 is treated as the sink and the number of grains of sand on this vertex is generally ignored. There is, however, one benefit to considering the number of grains at the sink and this will be mentioned in the Dhar criterion.

E = {(v i , v j ) : i ∈ {0, . . . , m -1} and j ∈ {m, . . . , m + n -1}}. v 0 v 1 v 2 v m-1 v m v m+1 v m+n-1 • • • • • • • • • • • •
A vertex is said to be stable if the number of grains at that vertex is strictly less than the number of edges exiting that vertex. Otherwise it is called unstable. If all vertices {v 1 , . . . , v m+n-1 } are stable then the configuration is said to be stable. Suppose x = (x 1 , . . . , x m+n-1 ) is a state on D m,n . If vertex v i is unstable then it topples: it donates one grain to each of its neighbours. Let Stable(D m,n ) be the set of stable states on D m,n .

Let us call a configuration Let σ(x) be the stable state the results from initial state x. A stable state is recurrent if it results as the stable outcome of an state where all vertices were unstable. Let Rec(D m,n ) be the set of recurrent states on D m,n and let Rec ⋆ (D m,n ) be the set of increasing recurrent states. Recurrent states may be classified for general graphs as follows:

u ∈ Stable(D m,n ) increasing if u 1 ≤ • • • ≤ u m-
Dhar's Criterion: a stable state u on a graph G is recurrent if and only if the stable state that results from an initial toppling of the sink results once again in u (see [START_REF] Cori | Enumeration of (p, q)-parking functions[END_REF][START_REF] Dhar | Theoretical studies of self-organized criticality[END_REF]).

Applying this to the graph D m,n gives us that a stable configuration u ∈ Stable(D m,n ) is recurrent iff σ(u 1 , . . . , u m-1 , 1 + u m , . . . , 1 + u m+n-1 ) = (u 1 , . . . , u m-1 , u m , . . . , u m+n-1 ).

(

) 1
The order of topplings is unimportant when checking that the Dhar criterion (1) holds. However, for our purposes it will prove useful to fix a canonical toppling process as follows: Let Q 1 be the set of all unstable vertices in the bottom row of D m,n (as a result of adding 1 to the height of each vertex in this bottom row). Topple all vertices in Q 1 and then let P 1 be the set of all unstable vertices in the top row of D m,n . Topple all vertices in P 1 and then let Q 2 be the set of all unstable vertices in the bottom row of D m,n . Continue this process until all vertices of D m,n are stable. Let

CanonTop(u) = (Q 1 , P 1 , Q 2 , P 2 , . . .) and canontop(u) = (|Q 1 |, |P 1 |, |Q 2 |, |P 2 |, . . .).
We note of course that it is possible for CanonTop(u) = canontop(u) = (), the empty sequence.

Example 3 Consider u = (0, 2, 1, 2, 1, 2) ∈ Stable(D 3,4 ). Adding one grain to every vertex in the bottom row gives w = (0, 2, 2, 3, 2, 3), which topples as follows: (0, 2, 2, 3, 2, 3) → (2, 4, 2, 0, 2, 0) → (2, 0, 3, 1, 3, 1) → (4, 2, 0, 1, 0, 1) → (0, 2, 1, 2, 1, 2). Since σ(w) = u we have that u ∈ Rec(D 3,4 ). The toppling process is CanonTop(u) = ({4, 6}, {2}, {3, 5}, {1}).

Given u = (u 1 , . . . , u m+n-1 ) ∈ Stable(D m,n ) with CanonTop(u) = (Q 1 , P 1 , Q 2 , P 2 , .
. .) and inc m,n (u) = (a 1 , . . . , a m-1 , b 1 , . . . , b n ) we define two collections of cells in the plane:

Bottom(u) = n i=1 [0, 1 + b i ] × [i -1, i]
and

Top(u) = [m -1, m] × [0, n] ∪ m-1 i=1 [i -1, i] × [0, 1 + a i ] Let f m,n (u) = Top(u) ∩ Bottom(u).
Example 4 Given u = (2, 1, 2, 0, 0, 2, 6, 1, 5, 1) ∈ Stable(D 7,4 ), we have inc 7,4 (u) = (a 1 , . . . , a 6 , b 1 , . . . , b 4 ) = (0, 0, 1, 2, 2, 2, 1, 1, 5, 6).

The collection of cells

f 7,4 (u) is ∩ =
The next theorem is the main classification theorem of [START_REF] Dukes | Parallelogram polyominoes, the sandpile model on a bipartite graph, and a q, t-Narayana polynomial[END_REF]. Part (a) of its proof relies on a careful analysis of the topplings in Dhar's criterion, and the canonical toppling or a recurrent configuration leads one to consider the bounce path within the corresponding polyomino.

Theorem 5 Let u = (u 1 , . . . , u m+n-1 ) ∈ Stable(D m,n ). (a) u ∈ Rec(D m,n ) iff f m,n (u) ∈ Para m,n . (b) If u ∈ Rec(D m,n ) then bounce(f m,n (u)) = canontop(u).
One may give a bijection from Rec(D m,n ) to the set of all parallelogram polyominoes whose bounce paths have been decorated with an ordered set partition X. There are two restrictions on the set partitions (i) the size of the ith set in this partition must correspond to the length of the ith segment of the bounce path, and (ii) the sets associated with the horizontal steps must partition {1, . . . , m -1} and the sets associated with vertical steps must partition {m, . . . , m + n -1}. Let SetPara m,n be the set of all pairs (P, X) where P ∈ Para m,n and X satisfies conditions (i) and (ii).

Example 6 Let P be P 2 in Example 1. Let X = ({8}, {1, 4, 5}, {7, 9}, {2, 3, 6}). Then (P, X) ∈ SetPara 7,3 and may be represented in the following way:

(P, X) = {8} {7, 9} {1, 4, 5} {2, 3, 6} Given u ∈ Stable(D m,n ) let Ω(u) = (f m,n (u), CanonTop(u))
. This construction, combined with parts (a) and (b) of Theorem 5, gives:

Corollary 7 Ω : Rec(D m,n ) → SetPara m,n is a bijection.
It is well-known from the literature on the sandpile model that recurrent configurations on a graph G are in one-to-one correspondence with spanning trees of G. Thus the number of recurrent configurations on D m,n is the number of spanning trees of K m,n , which is m n-1 n m-1 . Furthermore, Narayana [START_REF] Narayana | Sur les treillis formés par les partitions d'un entier et leurs applications à la théorie des probabilités[END_REF] enumerated the class of parallelogram polyominoes according to width and height. Thus The correspondence between recurrent configurations and parallelogram polyominoes is shown to have several special cases of interest. The details can be found in the main paper [START_REF] Dukes | Parallelogram polyominoes, the sandpile model on a bipartite graph, and a q, t-Narayana polynomial[END_REF] but we summarise some of the points here.

• Minimal recurrent configurations correspond to ribbon polyominoes. Configurations which are non-zero everywhere are a natural class to ask questions about. However there are no recurrent configurations that are both minimal and non-zero. The next natural class to look at are those configurations that are almost non-zero: call a configuration

u ∈ Rec(D m,n ) almost non-zero if u 1 , . . . , u m-1 , u m+1 , . . . , u m+n-1 > 0.
Configurations that are both minimal and almost non-zero are called minanz configurations.

• We call a configuration u ∈ Rec(D m,n ) square if m = n. Square minanz configurations are in one-to-one correspondence with square matrices which have no rows or columns of empty sets, and whose entries form a set partition of {1, 2, . . . n -1}, which are termed bicomposition matrices of size n -1.

• Let u ∈ Rec(D n,n ) be minanz and suppose that CanonTop(u) = (Q 1 , P 1 , . . . , Q k+1 = {n}). If x ∈ P i ∪ Q i then we say that vertex v x is in the ith wave of u, and write this as wave u (v x ) = i. The collection of minanz configurations u ∈ Rec(D n,n ) which satisfy the following wave property:

wave u (v n+x ) ≤ wave u (v x ) for all 1 ≤ x < n
are in one-to-one correspondence with upper-triangular bicomposition matrices of size n-1. Uppertriangular bicomposition matrices on n -1 have recently been shown to be in 1-1 correspondence with (2+2)-free partially ordered sets on {1, . . . , n -1}. We show the composition of these two correspondences, and how to read a recurrent configuration from the corresponding (2+2)-free poset.

3 q, t-Narayana polynomials and their symmetry

In this section we will introduce a polynomial that we call the q, t-Narayana polynomial. The polynomial is the generating function for the bistatistic (area, parabounce) on the set of parallelogram polyominoes.

In terms of recurrent configurations of the sandpile model, the area statistic is up to a constant the level of the configuration:

level(u) = u 1 + . . . + u m+n-1 -n(m -1),
and the hagbounce statistic is related to the canonical toppling process. When viewed in the parallelogram polyomino, the bounce path is almost identical to the bounce path that Haglund defined for Dyck paths.

Our polynomial is a natural extension of the area and bounce path statistics to the class of parallelogram polyominoes. More will be discussed about this in the subsections that follow.

For any polyomino P ∈ Para m,n , define its bounce weight to be

parabounce(P) = k i=1 ⌈i/2⌉c i ,
where bounce(P) = (c 1 , . . . c k ). This weight may also be described by summing weights on each step of the bounce path. The initial step has a weight of 1, and the weight of subsequent steps is incremented by 1 after each turn on the upper boundary path of the polyomino. Consider the following two polyominoes:

P 1 =
∈ Para 9,7 ; The distribution of the bistatistic (area, parabounce) on polyominoes in Para m,n is represented by the generating function N m,n (q, t) = P∈Param,n q area(P) t parabounce(P) .

We call these polynomials q, t-Narayana polynomials because {N m,n (1, 1)} m,n≥1 are the Narayana numbers that were mentioned in Corollary 8. The following conjecture has been verified for all pairs (m, n) with m ≤ 11 and n ∈ N by using generating functions resulting from the classical transfer-matrix method.

Conjecture 9 For all positive integers m and n, the distribution of the bistatistic (area, parabounce) on polyominoes in Para m,n is symmetric: N m,n (q, t) = N m,n (t, q).

Further to this we posit another symmetry (which has been checked for all pairs (m, n) with max(m, n) ≤ 9).

Conjecture 10 For all positive integers m and n we have N m,n (q, t) = N n,m (q, t).

Symmetry along the main diagonal of parallelogram polyominoes provides a bijection from polyominoes of Para m,n to Para n,m that preserves the area statistic. This mapping gives the following special case of Conjecture 10:

Theorem 11 For all positive integers m and n we have N m,n (q, 1) = N n,m (q, 1).

Define the generating function for the bistatistic of all these polyominoes, adding a variable z to record the height n of polyominoes:

N k,⋆ (q, t; z) = n≥1 N k,n (q, t)z n .
Representing polyominoes as regular expressions is useful in evaluating N k,⋆ (q, t; z). For example, one finds

N 2,⋆ (q, t; z) = t 2 q 2 z (1 -qtz)(1 -q 2 tz)(1 -qt 2 z) = N ⋆,2 (q, t; z) N 3,⋆ (q, t; z) = q 3 t 3 z 1 + q 3 tz 1 -q 3 tz + qt 3 z 1 -qt 3 z = N ⋆,3 (q, t; z).
Conjecture 9 also holds for minimal values of one of the two statistics:

Theorem 12 In Para m,n , for any i, there are as many polyominoes whose bistatistic (area, parabounce) is (m + n -1, i) as there are polyominoes whose bistatistic (area, parabounce) is (i, m + n -1).

This theorem is proved by showing that there exists a bijection Theorem 17

P∈LowerParan,n-1 q area(P) t parabounce(P) = (qt) 2n H n-1 (q, t 2 ).

One is led to Conjecture 9 by considering two configuration statistics pertaining to (sorted) recurrent configurations on D n . For the sandpile model on D n , the area of a Dyck path D is related to the number of grains in the mapped sorted recurrent configuration u since

area(D) + n-2 i=0 i = n-1 i=1 u i .
Also, the Haglund bounce statistic of a Dyck path D may be interpreted as the sum over all vertices of the number of times a vertex is observed stable before it topples during the parallel toppling process CanonTop(u). More precisely, all vertices in Q i are observed i times as stable: during the initial toppling of the sink and the toppling of each set Q 1 , . . . Q i-1 . Conjecture 9 is the result of translating these configuration statistics to the sandpile model on the graph D m,n .

Example 1

 1 The bounce path is indicated by a thick directed line in the following diagrams:

Fig. 1 :

 1 Fig. 1: The graph Dm,n.

1 Example 2

 12 and u m ≤ • • • ≤ u m+n-1 . Every configuration u ∈ Stable(D m,n ) may be written uniquely as a pair decomp m,n (u) = (inc m,n (u), sperm m,n (u)), where inc m,n (u) = u ′ is an increasing configuration, sperm m,n (u) = π is the lexicographically smallest permutation such that π : {1, . . . , m -1} → {1, . . . , m -1}, π : {m, . . . , m + n -1} → {m, . . . , m + n -1}, and u i = u ′ π(i) for all i ∈ {1, . . . , m + n -1}. Consider u = (5, 2, 1, 2, 4, 2, 1, 3, 2, 2, 1)∈ Stable(D 5,7 ). We have inc 5,7 (u) = (1, 2, 2, 5, 1, 1, 2, 2, 2, 3, 4) and sperm 5,7 (u) = (4, 2, 1,[START_REF] Cori | The sandpile model and Tutte polynomials[END_REF][START_REF] Garsia | A positivity result in the theory of Macdonald polynomials[END_REF][START_REF] Dukes | Composition matrices, (2+2)-free posets and their specializations[END_REF][START_REF] Cori | On the sandpile group of dual graphs[END_REF][START_REF] Egge | A Schröder generalization of Haglund's statistic on Catalan paths[END_REF][START_REF] Dukes | Parallelogram polyominoes, the sandpile model on a bipartite graph, and a q, t-Narayana polynomial[END_REF][START_REF] Dukes | Ascent sequences and upper triangular matrices containing nonnegative integers[END_REF][START_REF] Dhar | Theoretical studies of self-organized criticality[END_REF].

Corollary 8

 8 For all m, n ≥ 1, |Rec(D m,n )| = m n-1 n m-1 and |Rec ⋆ (D m,n )| = Nara(m + n -1, m), where Nara(a, b) = 1 are the Narayana numbers.

Υ:Fig. 2 :

 2 Fig.2: On the left is shown P = diag(x) where x = (5, 5, 3, 2, 2, 1) ∈ Rec † (K7). The Dyck path dyck(P ) = s 2 w 2 sws 2 wsw 2 is shown on the right.

(

  iii) HagBounce(D) = (|Q 1 |, |Q 2 |, . . . , |Q k |).Furthermore: (iv) area(P ) = (x 1 + . . . + x n ) -(n -1)(n -6)/2 = area(D) + 2n, and (v) parabounce(P ) = 2(hagbounce(D) + n).

† Supported by grant no. 090038013 from the Icelandic Research Fund. ‡ Thanks the Laboratoire d'informatique Gaspard-Monge, Marne-la-Vallée for their kind hospitality where part of this work was carried out. subm.

Example 13 An example of the bijection from Theorem 12. In this case m = n = 6. The sequence of diagonal lengths (d 1 , . . . , d m+n-1 ) = (1, 2, 3, 3, 3, 4, 3, 3, 3, 2, 1) give (x 0 , . . . , x 4 ) = [START_REF] Bousquet-Mélou | 2+2)-free posets, ascent sequences and pattern avoiding permutations[END_REF][START_REF] Bousquet-Mélou | 2+2)-free posets, ascent sequences and pattern avoiding permutations[END_REF][START_REF] Cori | Enumeration of (p, q)-parking functions[END_REF][START_REF] Cori | On the sandpile group of dual graphs[END_REF][START_REF] Dhar | Theoretical studies of self-organized criticality[END_REF] and (y 0 , . . . , y 4 ) = (0, [START_REF] Bousquet-Mélou | 2+2)-free posets, ascent sequences and pattern avoiding permutations[END_REF][START_REF] Cori | Enumeration of (p, q)-parking functions[END_REF][START_REF] Cori | On the sandpile group of dual graphs[END_REF][START_REF] Dhar | Theoretical studies of self-organized criticality[END_REF]. The ribbon Υ(P) is characterized among ribbons by the fact that its horizontal bounce steps from left to right are weighted by (d 1 , . . . d m-1 ) while its vertical bounce steps from bottom to top are weighted by (d m , . . . d n+m-1 ). In this subsection we will show a connection (Theorem 17) between a class of polyominoes, the sandpile model on the complete graph K n having one sink, and Haglund's bistatistic (area, hagbounce) on the set of all Dyck paths. We first need to introduce some notation relevant to Haglund's statistics. A Dyck path of length n is a path from (n, n) to (0, 0) that does not go above the main diagonal and takes steps in {s = (0, -1), w = (-1, 0)}. Let Dyck n be the set of all such Dyck paths of semi-length n. A general D ∈ Dyck n may be represented as a word D = d 1 d 2 . . . d n where d i ∈ {s, w}. Given D ∈ Dyck n , let area(D) be the number of complete unit squares contained between D and the diagonal line x = y. (The shaded triangular regions adjacent to the diagonal are not counted.) If Haglund's bounce path on D is s a(1) w a(1) s a(2) w a(2) • • • s a(k) w a ( k) then we will write HagBounce(D) = (a(1), a(2), . . . , a(k)). Haglund's bounce statistic, in this setup, is hagbounce(D) = a(2) + 2a(3) + . . .

Recurrent configurations of the sandpile model on the graph K n were studied in Cori & Rossin [START_REF] Cori | On the sandpile group of dual graphs[END_REF] and are classified in terms of parking functions. Let D n be the graph on the vertices V = {v 0 , . . . , v n-1 } with edges E = {(v i , v j ) : 0 ≤ i, j ≤ n -1 and i = j} where v 0 is a sink. The set of all stable configurations is

An integer sequence (t 1 , . . . , t n ) is a parking function if there exists a permutation π of {1, . . . , n} such that i ≥ t π(i) for all 1 ≤ i ≤ n. Cori and Rossin [5, Prop. 2.8] proved that a stable configuration (x 1 , . . . , x n-1 ) is recurrent iff (n -1 -x 1 , . . . , n -1 -x n-1 ) is a parking function. Let Park n be the set of all parking functions of length n. The recurrent configurations are

Let us call a configuration x ∈ Stable(D n ) sorted if it is weakly decreasing, and let

x is sorted}. For example, Stable(D 3 ) = {(0, 0), (0, 1), (1, 0), (1, 1)}, Rec(D 3 ) = {(0, 1), (1, 0), (1, 1)}, and Rec † (D 3 ) = {(1, 0), (1, 1)}. Using Cori & Rossin's [START_REF] Cori | On the sandpile group of dual graphs[END_REF] classification we have:

Section 2 defined a canonical toppling process for every recurrent configuration. This canonical toppling process was an ordered set partition of the vertices and recorded the order in which vertices toppled in parallel. We now extend the same definition to recurrent configurations of A dot above the number denotes an unstable vertex that will topple. Thus CanonTop(x) = ({1, 2}, {3, 4}, {5}).

Diagrams of sorted recurrent configurations in K n are precisely those diagrams which are polyominoes in Para

Every element P ∈ LowerPara n,n-1 is uniquely described by a Dyck path dyck(P ) of semi-length n -1 which is the path of the boundary of P from (n, n -1) to (1, 0) since its upper path is a word of (ne) * e. See Figure 2 for an example of this simple transformation.