Georg Kirchsteiger
email: georg.kirchsteiger@ulb.be

Tom Lenaerts
email: tom.lenaerts@ulb.be

Rémi Suchon
email: -remi.suchon@univ-catholille.fr

Does voluntary information disclosure lead to less cooperation than mandatory disclosure ? Evidence from a sequential prisoner's dilemma experiment. *

In sequential social dilemmas with stranger matching, initiating cooperation is inherently risky for the first mover. The disclosure of the second mover's past actions may be necessary to instigate cooperation. We experimentally compare the effect of mandatory and voluntary disclosure with non disclosure in a sequential prisoner's dilemma situation. Our results confirm the positive effects of disclosure on cooperation. We also find that voluntary disclosure is as effective as mandatory one, which is surprising given the results of existing literature on this topic. With voluntary disclosure, second movers with a good track record decided to disclose because they expect that not disclosing signals non-cooperativeness. First movers interpret nondisclosure correctly as a signal of non-cooperativeness. Therefore, they cooperate less than half as often when the second mover does not disclose.

Introduction

Cooperation is one of the most important reasons for the success of the human species. However, humans do not always succeed to cooperate. This is especially true in dilemma situations, where narrow self-interest induces agents to refrain from cooperation although mutual cooperation would be beneficial for everyone involved.

Such dilemma situations have been studied extensively during the last decades, using theoretical and empirical methods (for an overview, see e.g. [START_REF] Camerer | Behavioral game theory: Experiments in strategic interaction[END_REF]). It has been suggested that one can overcome such dilemmas by repeated interactions between the agents involved (see the literature on repeated games and on relational contracts, e.g. [START_REF] Brown | Relational contracts and the nature of market interactions[END_REF]; [START_REF] Mailath | Repeated games and reputations: longrun relationships[END_REF]; Dal [START_REF] Dal Bó | On the determinants of cooperation in infinitely repeated games: A survey[END_REF]). The problem can be also solved if agents are social-minded, i.e. when they care not only about their self-interest, but also about the welfare of everyone involved (see the literature on gift-exchange games [START_REF] Fehr | Does fairness prevent market clearing? an experimental investigation[END_REF], public good games, etc). Typically, such a solution to the dilemma problem requires trust. If a social-minded agent trusts that the partners "do the right thing" even if it is not in their narrow self-interest, a social-minded agent will cooperate. But if a social-minded agent does not trust his partners, he will refrain from cooperation to avoid being abused (see e.g. the literature on conditional cooperation in public good games, originated by [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]).

A lot of economic interactions, in markets or elsewhere, require trust between strangers, where trust is particularly hard to establish. In this case, a mechanism of information disclosure might help. If one interacts for the first time with another agent, but one is informed that this agent has been trustworthy most of the times in the past in similar situations, one has more trust in this new partner than if the new guy has a reputation for abusing trust.

In theory, this intuition can be supported by at least two complementary mechanisms. First, disclosing one's past record may help sustain indirect reciprocity [START_REF] Nowak | Evolution of indirect reciprocity[END_REF]: A stranger might punish an individual with a record of defection by restraining trust/cooperation, or by refusing to interact with such a person at all. This in turn gives a strong incentive to cooperate. Second, a good record might change the trustor's beliefs about the social-preference type of the trustee and as a consequence help cooperation [START_REF] Kreps | Reputation and imperfect information[END_REF]. A lot of electronic market places have implemented mechanisms of information disclosure to improve trust (of the buyer or the employer) and trustworthiness (of the seller or the job candidate). A number of lab experiments have confirmed that information disclosure helps sustaining cooperation in sequential social dilemma [START_REF] Bolton | How effective are electronic reputation mechanisms? an experimental investigation[END_REF][START_REF] Bohnet | Repetition and reputation: Implications for trust and trustworthiness when institutions change[END_REF][START_REF] Charness | Trust and trustworthiness reputations in an investment game[END_REF][START_REF] Duffy | Social norms, information, and trust among strangers: Theory and evidence[END_REF]. 1In this paper, we investigate experimentally how information on past behavior influences the willingness to trust and to cooperate in dilemma situations with partner choice. In particular, we study how properties of the information revelation mech-anism impact its effectiveness on achieving cooperative outcomes. We compare mandatory with voluntary information disclosure mechanisms. Mandatory disclosure of past behavior is quite intrusive and might be at odds with some ethical considerations. In addition, individuals may value privacy and control over their data.2 For these reasons reputation systems based on voluntary information disclosure may be more desirable. Moreover, most mechanisms have some discretion about revelation of past information. For instance, in online market with information about past behavior, it is often possible for participants to manipulate what to disclose from their past records, for instance by creating a new alias after a history of uncooperative behavior.

In theory, systems with voluntary disclosure should be as effective as mandatory disclosure due to the unravelling principle [START_REF] Milgrom | Good news and bad news: Representation theorems and applications[END_REF]. Those who choose not to disclose their record will all be treated the same, so there is an incentive for those with a good record to disclose their records. This leads to unravelling: only those with bad records will withhold information. Since this is anticipated, "nondisclosers" will be treated with skepticism, i.e. one does not cooperate with them or refrain from interacting with them at all. This mechanism provides an incentive to build a good record, i.e. to cooperate. However, evidence from sender-receiver experiments have demonstrated the limits of the unravelling principle. A common observation is that senders with a bad private information exploit receivers who are not skeptical enough (see e.g. [START_REF] Jin | Is no news (perceived as) bad news? an experimental investigation of information disclosure[END_REF][START_REF] Montero | Naivety about hidden information: An experimental investigation[END_REF][START_REF] Sheth | Disclosure of information under competition: An experimental study[END_REF] for recent examples). 3 Jin et al. (2021) show that repeated feedback is necessary to reduce the gap between actual behavior and the theoretical prediction of unravelling. In this paper we investigate whether these observed limits of the unravelling principle makes voluntary information disclosure less effective than mandatory information disclosure in promoting cooperation in a sequential dilemma situation with partner choice.

To investigate these questions, we set up a sequential prisoner's dilemma experiment (SPD), where first player 1 (P1 -by convention female) decides whether to cooperate (trust) or defect. After being informed about P1's decision, player 2 (P2 -by convention male) takes his cooperation decision (reciprocate trust or renege). Before the SPD takes place, P1 decides whether the game should be played or whether she takes an outside option instead. The outside option gives both players higher payments than what they get in the SPD if both defect, i.e. if they play the unique Nash equilibrium, but less than what they get when both players cooperate. Hence, if P1 does not trust that P2 would cooperate, she should take the outside option, while if she trusts him, she should opt into the SPD and choose cooperation.

Each subject plays this game repeatedly with fixed roles, but with changing partners with exogenous matching (stranger matching). Before P1 decides whether to pick the outside option or the SPD, she might get informed about her prospective P2's past cooperation choices. If information about P2's past choices is disclosed, it is correct and complete. The disclosure of information is either mandatory ("Mand" treatments), or it is voluntary, ie only disclosed if P2 wants so ("Vol" treatments).

The information might be disclosed with 100% probability, (no noise, "NN" treatments), or it is disclosed only with 90% probability (low noise, "LN" treatments).

This implies four treatments: MandNN, MandLN, VolNN, and VolLN. Besides, in a baseline treatment no information gets disclosed. In this treatment P1 has to choose between the SPD and the outside option without having any information about the previous choices of her prospective P2. The noise treatments allow us to measure the robustness of voluntary disclosure when seeing no information might be plausibly blamed on bad luck. They also allow us to explore skepticism, by comparing the behavior of P1 when lack of information is due to P2's choice (in the VolNN treatment) with the behavior of P1 when the lack of information might be due to chance (in the LN treatments).

As expected, we find that disclosure of the record of P2s' past choices increases P1s' trust levels as measured by their cooperation choices. As a result, full cooperation by both players is significantly more often observed when such an informationdisclosure system is in place. The interplay between information-disclosure and partner's choice can mitigate the problem of dilemma situations. In contrast to the existing literature identifying limits of unravelling, voluntary disclosure is just as effective as mandatory disclosure. When given the choice, P2s reveal their records most of the time, and the probability of information disclosure increases with the quality of P2's record. P1s anticipate this, and they are skeptical: In the voluntary disclosure treatments, they do not trust P2s of whom they do not see the records.

As a placebo test, we compare trust by P1s when seeing the records and when not seeing it in the MandLN treatment. In this treatment the impact of information on P1s' cooperation rates is much smaller than in the VolLN treatment. Importantly, we find that the level of disclosure and skepticism are already high in the first periods of the game, suggesting that learning plays only a limited role. Clearly, these results are in contrast with the limits of unravelling found in sender-receiver experiments. In addition, our results also reveal that information disclosure is much less effective when the information-system is not perfectly reliable, ie in treatments with noise. Even the rather low noise levels of the LN-treatments are enough to reduce the overall cooperation level substantially.

We are aware of only two papers comparing automatic and voluntary disclosure in a social dilemma [START_REF] Kamei | Endogenous reputation formation under the shadow of the future[END_REF][START_REF] Kamei | Voluntary disclosure of information and cooperation in simultaneous-move economic interactions[END_REF]. Both focus on simultaneous games, while we use a sequential prisoner's dilemma. Sequentiality is common in real life economic interactions, including online transactions (the buyer must pay before the sellers sends the product) or employer-employee relations. In contrast to simultaneous games, in a sequential game all the strategic uncertainty weights on the first mover (as noted by e.g. [START_REF] Ghidoni | The effect of sequentiality on cooperation in repeated games[END_REF]. In SPD, P2 does not have to form a belief about P1's choice, because P1's choice is already known to P2 when P2 makes his own choice. Hence, any cooperation choice of previous P1s should be irrelevant for P2's choice in a particular round. This in turn implies that in our game, P2's record reveals all the information P1 might need, while in the simultaneous game P1 would also have to know the cooperation choices of P2's prior partner to interpret P2's record correctly. Therefore, in both [START_REF] Kamei | Endogenous reputation formation under the shadow of the future[END_REF] and [START_REF] Kamei | Voluntary disclosure of information and cooperation in simultaneous-move economic interactions[END_REF], the information about one's opponent is more complex to interpret. Both papers find that, under random matching, the possibility to freely hide one's identity [START_REF] Kamei | Endogenous reputation formation under the shadow of the future[END_REF] or one's last action [START_REF] Kamei | Voluntary disclosure of information and cooperation in simultaneous-move economic interactions[END_REF] undermines the effect of information disclosure on cooperation. In contrast, our results indicate that voluntary disclosure may be just as effective as mandatory one, in a context in which the interpretation of the disclosed information is relatively straightforward.

The remainder of the paper is organized as follows. Section 2 details the experimental design, Section 3 presents our results and Section 4 discusses these results and concludes.

Design

The stage game

The stage game is a modified sequential prisoner's dilemma. Compared to a regular sequential prisoner's dilemma the main modification is that in each round P1 can choose whether she enters the game or not. If she decides not to enter, both players get an outside option with a fixed and equal payoff. The modified sequential prisoner's dilemma is shown in Figure 1.

P1 P2 plays the strategy method: he takes one decision for the node after P1 decided to enter and defect, and one decision for the node after P1 decided to enter and cooperate. These two choices are made before P2 gets informed about P1's actual choice. The use of the strategy method allows to gather more data and to insure that the history of play of a P2 does not depend on her past partners' behavior. The use of strategy method to elicit P2s' choices is common and does not affect behavior (Brandts andCharness, 2000, 2011).

π 1 = 5 ; π 2 = 5 P2 P2 π 1 = 10 ; π 2 = 10 π 1 = 1 ; π 2 = 20 π 1 = 20 ; π 2 = 1 π 1 = 3 ; π 2 = 3 O U T E n t e
The parameters were chosen to make cooperation hard to sustain: the temptation to defect for P2 is high, which is expected to deter entry and cooperation of P1 (e.g. [START_REF] Mengel | Risk and temptation: A meta-study on prisoner's dilemma games[END_REF][START_REF] Gaechter | The variability of conditional cooperation in sequential prisoner' s dilemmas[END_REF], study the impact of game parameters on cooperation in the prisoner's dilemma).

Matching

The stage game is played for 30 periods, which is common knowledge. Roles are fixed: each player is randomly assigned to be P1 or P2, once and for all before the first round. At every round, each P1 is matched with a randomly selected P2 (stranger matching). 4 Participants are payed for every period.

Experimental treatments

There are 5 experimental treatments. The Baseline treatment is fully described by the stage game and the matching introduced earlier. Beside the Baseline treatment, we have 4 treatments in which information about P2 may be revealed to P1 before she decides to enter. We call this information the record of P2. The record contains two pieces of information about P2. First, it reports the portion of times that P2 chose "Cooperate" at the Enter Cooperate node. Second, it reports the portion of times that P2 chose "Cooperate" at the Enter Defect node. Remember that P2s' choices are elicited using the strategy method. Therefore, the observed decision of a P2 in a particular period is independent of the actual choice of her P1 in this period and the record of a P2 is independent of the actual choice of the P1 he was matched with in past periods. The 4 information treatments differ in the way the record is disclosed to P1. There are two dimensions: First, record disclosure might be mandatory (the "Mand" treatments), or it might be P2's choice whether to disclose his record or not (the "Vol" treatments). Second, we vary whether disclosure is noisy: in the LN treatments, there is a 10% chance that the record is not transmitted to P1 when information transmission is mandatory (in the MandLN treatment), or when P2 chose to reveal his record (in the VolLN treatment). In the NN treatments, no such noise is introduced. These variations result in 5 treatments: Baseline, MandNN, MandLN, VolNN, VolLN. Table 1 summarizes the information treatments.

Procedure

We recruited a total of 386 participants for 25 sessions.5 14 sessions were run at the BEEL lab in Brussels in winter 2019/2020 (before the pandemic), while 11 sessions were run at the Anthropo-lab in Lille in fall 2021 (after the pandemic).

Results

Table 3 reports the descriptive statistics by treatment.6

The cooperative outcome is relatively rarely observed. This was expected because our parameters make defection tempting. P1s enter in the vast majority of the The time dynamic of the cooperative outcome is not impacted by the presence of a reputation system. There is a decline in the likelihood of the cooperative outcome, but it is small, and similar with or without record disclosure. See Appendix D.2 for more details.

Cooperation

Result 1: The existence of a record system increases the likelihood of the cooperative outcome. Voluntary record disclosure is as effective as mandatory disclosure.

On the other hand, introducing a small noise decreases the effectiveness of record disclosure substantially.

Support for Result 1: The cooperative outcome occurs in 11% of cases in the baseline, and in 21.7% of cases in the other treatments (p < 0.01, see Table C.2 and C.3 in Appendix C for details). To disentangle the respective effect of voluntary disclosure and noise, we run logit regressions explaining the occurrence of the cooperative outcome by dummy variables indicating a voluntary disclosure system and a noisy disclosure system, and the interaction of these two dummies. We control for the presence of a record system in every models. The respective marginal effects of voluntary and noisy disclosure are reported in Table 4.

( Since we used the strategy method, we observe the four pure strategies of P2s: i) Co-

Result 2:

When there is no noise, the existence of a record system increases cooperation levels of the P1s and P2s irrespective of whether disclosure is mandatory or voluntary. With noise the record system has no significant effect on cooperation.

The effect of a record system on the enter decision is small and at best weakly significant.

Support for Result 2: Table 3 provides strong first evidence in favor of Result 2. To disentangle the respective effects of noise and voluntary disclosure we run random-effect logit regressions explaining the occurrence of the entering, and cooperation of P1 and P2 by dummy variables indicating a voluntary disclosure system and a noisy disclosure system and the interaction of these two dummies. The marginal effects of "noisy" and "voluntary" are reported in Table 5. We control for the presence of a record system in every models. The effect of voluntary disclosure is never significant. The effect of noise on entry is not significant, but it is significant on P1s' and P2s' cooperation levels: noise reduces P1s' cooperation by more than 7 percentage points (p=0.018) and P2s' cooperation by more than 11 pp (p=0.011).

(1)

(

Disclosure, Record and skepticism

Noise has no effect on the disclosure decisions of the P2s. As one can see from Table 3 the disclosure rates are very similar in both voluntary disclosure treatments. When disclosure is voluntary, the disclosure decision depends mainly on P2's record.

Result 3: P2s with better records are more likely to disclose it. This is true for both the VolLN and the VolNN treatment.

Support for Result 3: We estimated random-effect logit models explaining the decision of a P2 to disclose his record by the rate at which he cooperated following cooperation by P1 and following defection in the past. Obviously, we use only the data from the Vol treatments. Table 6 reports the outcome of these regressions. Column (1) reports the marginal effects pooling the data from both Vol treatments. Column (2) contrasts the marginal effects of the VolNN (Noisy=0) and VolLN (Noisy=1) treatments. The probability of disclosure obviously increases in the level of cooperation following cooperation of P1 in the past. The effect of past cooperation following a defection of P1 is very small and not significant. Again, we do not see any significant impact of noise on the disclosure decision.

( This result can be also seen from Figure 2 that shows the predicted probability of disclosure depending on portion of P2's cooperation choices in all the past rounds. We run logit regressions on the subset of the data in which the record of P2 is displayed to P1. We explain the decision of P1 to cooperate depending on the record of P2, i.e. on the portion of times P2 chose cooperation in the past. We present the outcome of several models in Table 7, including random and fixed effects models.

For all specifications, the estimated parameters for P2s' previous cooperation are significant at the 1% level. Note that, in contrast with all other parameters, the sign for the effect of previous cooperation following defection on cooperation by P1 is negative. This make intuitive sense: the more likely P2 is to cooperate even after P1s' defection, the more likely P1 is to enter and defect.

(1)

( To illustrate this result, Figure 4 reports the predicted probability of cooperation by P1, depending on the record of P2, for each treatment with information disclosure (confidence interval are omitted for readability). It shows that the better the record, the higher the probability of cooperation by P1. This does not change across treatments.

Result 5: P1s show skepticism: In the Vol treatments P1s cooperate less when they do not see the record of P1 than when they do see it. This is not the case in the MandLN treatment. Skepticism is observed already in the early rounds of the game, and at best limited learning takes place.

Support for Result 5: We already checked that P1s are more likely to cooperate when they see the report of P2. We expect that the impact of non-disclosure on P1s' cooperation is weaker when non-disclosure cannot be attributed to a deliberate decision of P2, since in this case non-disclosure cannot serve as a signal of a poor record.

As one can see from Figure 5 non-disclosure reduces the P1s' cooperation rates by 10 percentage points (from 37.3 to 27.3%) in the Mand treatments where non-disclosure cannot be voluntary. In the Vol treatments, where non-disclosure is mainly due to deliberate decisions of the P2s, the non-disclosure induced a drop in P1s' cooperation rates more than twice as large, namely 26.2 percentage points (from 45.6 to 19.%). This is a first indication of the validity of Result 5. We run a random-effect logit model explaining P1's decision to cooperate by a dummy variable indicating voluntary disclosure and a dummy variable indicating that P1 saw the record, and an interaction of these two dummies. We test whether the marginal effect of seeing the record is different when disclosure is voluntary and when it's not. Table 8 shows the outcomes and confirms the result suggested by Figure 5: the effect of (not)

seeing the record is much weaker when disclosure is for sure not voluntary (p=0.01)

. In addition, in Column 2, we interact our variables of interest with a categorical variable indicating the period bracket. The values of the interactions suggest that the effect of not seeing the record is already present early in the game, and increases moderately.

Discussion and Conclusion

We studied the effect of different information-disclosure mechanisms on cooperation in a sequential social dilemma. We found that such institutions increase the likelihood of reaching the cooperative outcome, as both the first and second movers act

(1)

( more cooperatively. In addition, we found that an institution in which disclosure is voluntary is exactly as effective as one in which it is mandatory. In contrast, we found that a relatively small noise significantly undermines the effect of information disclosure.

A striking result is that voluntary disclosure is exactly as effective as mandatory disclosure. This result is consistent with unravelling, but it is in contrast with two previous experiment comparing voluntary and automatic disclosure of information in social dilemma [START_REF] Kamei | Endogenous reputation formation under the shadow of the future[END_REF][START_REF] Kamei | Voluntary disclosure of information and cooperation in simultaneous-move economic interactions[END_REF]. There are several differences between these papers and ours that may explain the different results. First, the underlying social dilemma are simultaneous in these papers, while it is sequential in ours. This changes the distribution of strategic uncertainty between the first and the second mover [START_REF] Ghidoni | The effect of sequentiality on cooperation in repeated games[END_REF], which may impact the effectiveness of information disclosure. Second, and more importantly, the information disclosed in the work of Kamei is more complex. In the simultaneous games, the second order information is not strictly controlled for, so the interpretation of one's counterpart's record depends on the belief one has on the records of the individuals with whom the counterparts interacted in the past. This may induce the (maybe mistaken) beliefs that a bad record will not be sanctioned with (a high likelihood of) defection.

In our experiment, sequentiality, combined with the strategy method, makes the interpretation of one's record much more straightforward, which might foster the incentives to behave as a cooperative type.

More generally, our results also contrast with the literature on the failure of unravelling in sender-receiver games. In this literature, senders typically send favorable information, but withhold unfavorable ones, exploiting the naivety of (some of) the receivers. This failure of unravelling is corrected only after some learning has occurred. This tends to suggest that voluntary disclosure should be significantly less effective than mandatory one, if effective at all. In our experiment, when given the choice, a large share of the subjects choose to maintain a good record and disclose it (see Figure D.7 in Appendix D.4.). First movers are skeptical: they generally avoid exposing themselves to being exploited by a second mover who does not disclose his record. This is observed already early in the game, and learning is limited. What can explain this difference between our results and that of sender-receiver game experiments? A first possibility relates to the nature of the information disclosed. In sender-receiver games, participants disclose a random variable which is exogenously determined and contains no signal about the senders' type. In contrast, building and sustaining a good reputation may have an intrinsic value for a participant in social dilemma game, because a good record signals to oneself as well as to others that one is a cooperative, "nice" type (see e.g. [START_REF] Bénabou | Incentives and prosocial behavior[END_REF], and the large subsequent literature on identity management). Note also that P2s' reputation in our experiment depends on all his previous choices. Hence, a P2 cannot hope to get a "clean" record after he has refused to cooperate with a cooperating P1. In contrast, in the sender-receiver game, the game starts from "scratch" in every round. Because of the reputational spillover between rounds participants might experiment less in our SPD than in a sender-receiver game. Irrespective of the fundamental reasons, our results suggest that the limits of unravelling identified in past sender-receiver experiments might depend on the nature of the strategic situation.

Another striking result is the effect of noise on the effectiveness of information disclosure. We designed the treatment with noise with two goals in mind: assessing the robustness of the effect of information disclosure, and providing a placebo test for skepticism (comparing the behavior of P1 when not seeing the record can vs cannot be blamed on luck). The strong effect of the relatively small noise may be explained by the often observed tendency of individuals to overweight small probabilities [START_REF] Tversky | Advances in prospect theory: Cumulative representation of uncertainty[END_REF].

It would be interesting to see whether agents prefer mandatory disclosure or voluntary disclosure mechanisms in dilemma situations. On the one hand, the experimental results suggest that mandatory disclosure is never worse and in simultaneous games even better than voluntary disclosure. This should induce agents to opt for a mandatory disclosure mechanism for general dilemma situations. On the other hand, it is well know that humans like (the illusion of) control, and this motivation is of course more in line with voluntary disclosure.

B Power analysis

We followed a simulation approach for our power analysis and sample size determination. The advantage of simulation based power analysis is that we can adapt it to the test we will use on the actual data. We focused on the likelihood of the cooperative outcome as our variable of interest, and proceeded as follows:

• We randomly created samples fixing a number of parameters:

-The number of observations per session.

-The number of sessions.

-The baseline probability of the cooperative outcome.

-The effect of the existence of an information disclosure mechanism (i.e.

an increase in the probability of the cooperative outcome, measured in percentage point).

-The session effect size (also measured in percentage point).

In each sample, half the observations were allocated to the baseline, and half to a treatment with information disclosure. We generated 100 samples for each parameter space, and for each sample, we ran a logit model explaining the probability of the cooperative outcome by the treatment dummy, with errors clustered at the session level. For each parameter space, we recorded the percentage of the times the treatment dummy was significant at the 5% level. This gives us the power of our experiment for the parameter space. We reproduced it for numerous parameter spaces. In Table B.1, we report the power analysis for some parameters. This suggests that, if we expect an effect size of 10 pp with a moderate session effect, and session of, on average, 14 participants, we have appropriate power to detect effect size of 10 or 15 percentage points.

Power C Support for the results (omitted in the text).

C.1 Test for Table 3 ( Table C.1: Regressions for the significance levels reported in Table 3 C.2 Support for Result 1. We regress a dummy variable indicating the cooperative outcome on a dummy variable indicating that there is a disclosure system (all treatments pooled). We report the marginal effect from a logit models.

D.3 Analysis of the 4 possibles strategies of P2

Figure D.6 reports the distribution of pure strategies of P2s across treatments. In Table D.1, we report the marginal effect of a multinomial logit regression explaining the strategy choice of P2s by the factorial interaction of "noisy" and "Voluntary".

The results suggest that, while "Voluntary" has no effect on strategy choice at all, noise reduces the likelihood of conditional cooperation by 10 pp and increases the likelihood of defection by the same amount. 7 Both these strategies make sense: if a P2 believes that P1s are not very skeptical, the first strategy makes sense [START_REF] Jin | Is no news (perceived as) bad news? an experimental investigation of information disclosure[END_REF] find evidence of such beliefs in a sender-receiver game). Conversely, the second strategy makes sense for a P2 who anticipates that the P1s are skeptical.

Figure D.7 represent the 2 clusters. We checked which strategy is the most beneficial for a P2, and we found that participants in the second cluster had higher average stage-game profits (7.42 vs 6.48, p = 0.0017 in a Mann-Whitney test).

7

The same message holds if we choose to identify 3 clusters. E (Seemingly) irrational behavior.

Some behavior are hard to rationalize at first sight: In a lot of cases, P1s enter but do not cooperate (overentry) and P2s cooperate even if P1 defected. In this section, we discuss whether such behavior is indeed an indication for irrationality.

Overentry by P1. First, "over"entry by P1 might actually be motivated by social curiosity (one learns the corresponding conditional choice of P2 when entering). In addition, P1s might believe that (there is a small probability that) P2s cooperate in response to defection. Even if this probability is small, this can explain the aforementioned choices: Let's say that P1 believes that P2s cooperate when P1 cooperates with probability q. Let's say that P1 believes that P2s cooperate when P1 defects with probability p. Given our parameters, two conditions must be met so that P1 enters AND defects:

(1): 20p-3(1-p) ≥ 5

(2) : 20p + 3(1 -p) ≥ +(1 -q)

(1) is the condition to "entering conditional on defecting" It imposes p > 2 17 ∼ 0.11.

(2) is the condition "defecting conditional on entering". It imposes p > (9q -2)/17

There is a large set of (p, q) satisfying ( 1) & ( 2). Some are à-priori unlikely (p > q), others are more plausible (e.g. p ≥ .15 & q ≤ .3 or p ≥ .2 & q ≤ .5 ). [START_REF] Miettinen | Revealed preferences in a sequential prisoners' dilemma: A horse-race between six utility functions[END_REF] for instance elicit the beliefs of participants in a sequential prisoner dilemma and find that first movers on average expect second movers to cooperate 50% of the time when P1 cooperates, and 20% of the time when P1 defects. This might be due to P1s misunderstanding the game, to P1s expecting errors by P2s or to P1s expecting that P2 might be concerned by social welfare.

Cooperation after defection. "cooperate when P1 defected" can be due to unconditional cooperation, i.e. P2s who cooperate irrespective of the decision of P1, motivated by e.g. altruism or social welfare. By choosing to cooperate to a defector, one increases the social welfare 3.5 folds (from 6 to 21) or by 15 ECUs (75cents).

Such concern for welfare is well documented in the literature (see e.g. [START_REF] Engelmann | Inequality aversion, efficiency, and maximin preferences in simple distribution experiments[END_REF]. [START_REF] Miettinen | Revealed preferences in a sequential prisoners' dilemma: A horse-race between six utility functions[END_REF] find that "concern for social welfare" is a good explanation (along with reciprocity, among others) of behaviors in the sequential prisoner dilemma. Interestingly enough, they also find that more or less 15% of

F Instructions

[Baseline]

Thank you for participating in this experiment on decision making. You are not allowed to communicate with other participants during the entire session. Please turn off your cell phone. If you have a question, please raise your hand and wait until an experimenter comes to you to answer your question in private.

For showing up on time, you receive a €3 show-up fee. In addition, your decisions during the experiment earn you money. During the session, earnings will be expressed in terms of Experimental Currency Units, ECUs. ECUs convert to euro at the following rate: 20 ECUs = €1. You will be paid by bank transfer shortly after the experiment. Your decisions in the experiment will remain anonymous. At the beginning of the session, each participant is assigned the role of PLAYER 1 or PLAYER 2, once and for all.

The experiment consists of 30 rounds. For each round, the computer program randomly forms pairs composed of one PLAYER 1 and one PLAYER 2, who will have the possibility to participate in an interaction.

In this interaction, PLAYER 1 and PLAYER 2 will have simple decisions to make: choosing either "UP" or "DOWN" according to a procedure described hereafter. These decisions will impact both their own earnings as well as the other PLAYER's earnings as summarized in Table 1. PLAYER 2 chooses "DOWN" PLAYER 2 choses "UP" PLAYER 1 chooses ''DOWN''

• Player 1 earns 3 ECUs • Player 2 earns 3 ECUs

• Player 1 earns 20 ECUs • Player 2 earns 1 ECUs PLAYER 1 chooses ''UP''

• Player 1 earns 1 ECUs • Player 2 earns 20 ECUs

• Player 1 earns 10 ECUs • Player 2 earns 10 ECUs However, the interaction might not take place in every round: at the beginning of each round, PLAYER 1 has to decide whether to participate or not, using the interface shown in Screenshot 1.

If PLAYER 1 decides to participate, he then chooses between "UP" and "DOWN".

If PLAYER 1 decides not to participate, he has no further decision to make in this round, and the decisions of PLAYER 2 have no impact on earnings: PLAYER 1 and PLAYER 2 both earn 5 ECUs.

PLAYER 2 is not informed of the decisions of PLAYER 1 at this point and as a consequence makes two decisions. One decision is made ''as if" PLAYER 1 had decided to participate, and had subsequently chosen "UP". The other decision is made "as if" PLAYER 1 had decided to participate, and had subsequently chosen "DOWN". PLAYER 2 makes these decisions using the interface shown in Screenshot 2.

To compute earnings, only the decision of PLAYER 2 corresponding to the actual decision of PLAYER 1 is used. In other words:

• If PLAYER 1 chose not to participate, none of the decisions of PLAYER 2 impact earnings and both PLAYERS earn 5 ECUs. • If PLAYER 1 chose to participate and "DOWN", only the decision of PLAYER 2 made "as if" PLAYER1 had chosen "DOWN" matters. In this case, the consequence of PLAYER 2's decision on earnings is found in the first line of Table 1. • If PLAYER 1 chose to participate and "UP", only the decision of PLAYER 2 made "as if" PLAYER1 had chosen "UP" matters. In this case, the consequence of PLAYER 2's decision on earnings is found in the second line of Table 1. Before the first round of the experiment, you will be asked to answer several questions aimed at checking your understanding of the decisions you will have to make.

After the last round, you will have to answer a demographic questionnaire. This questionnaire does not threaten the anonymity of your decisions.

Please read these instructions again before clicking OK to proceed to the experiment. Should you have any question, silently raise your hand and an experimenter will come to answer privately. Thank you for participating in this experiment on decision making. You are not allowed to communicate with other participants during the entire session. Please turn off your cell phone. If you have a question, please raise your hand and wait until an experimenter comes to you to answer your question in private.

For showing up on time, you receive a €3 show-up fee. In addition, your decisions during the experiment earn you money. During the session, earnings will be expressed in terms of Experimental Currency Units, ECU. ECU converts to euro at the following rate: 20 ECUs = €1. You will be paid by bank transfer shortly after the experiment. Your decisions in the experiment will remain anonymous. At the beginning of the session, each participant is assigned to the role of PLAYER 1 or PLAYER 2, once and for all.

The experiment consists of 30 rounds. For each round, the computer program randomly forms pairs composed of one PLAYER 1 and one PLAYER 2, who will have the possibility to participate in an interaction.

In this interaction, PLAYER 1 and PLAYER 2 will have simple decisions to make: choosing either "UP" or "DOWN" according to a procedure described hereafter. These decisions will impact both their own earnings as well as the other PLAYER's earnings as summarized in Table 1. PLAYER 2 chooses "DOWN" PLAYER 2 choses "UP" PLAYER 1 chooses ''down''

• Player 1 earns 3 ECUs • Player 2 earns 3 ECUs

• Player 1 earns 20 ECUs • Player 2 earns 1 ECUs PLAYER 1 chooses ''up''

• Player 1 earns 1 ECUs • Player 2 earns 20 ECUs

• Player 1 earns 10 ECUs • Player 2 earns 10 ECUs However, the interaction might not take place in every round: at the beginning of each round, PLAYER 1 has to decide whether to participate or not, using the interface shown in Screenshot 1.

To help PLAYER 1's participation decision, from round 2 on, the past decisions of PLAYER 2 will be disclosed on the left panel of PLAYER 1's decision screen, as shown in screenshot 2.

If PLAYER 1 decides to participate, he then chooses between "UP" and "DOWN".

If PLAYER 1 decides not to participate, he has no further decision to make in this round, and the decisions of PLAYER 2 have no impact on earnings: PLAYER 1 and PLAYER 2 both earn 5 ECUs.

PLAYER 2 is not informed of the decisions of PLAYER 1 at this point and as a consequence makes two decisions. One decision is made ''as if" PLAYER 1 had decided to participate, and had subsequently chosen "UP". The other decision is made "as if" PLAYER 1 had decided to participate, and had subsequently chosen "DOWN". PLAYER 2 makes these decisions using the interface shown in Screenshot 3.

To compute earnings, only the decision of PLAYER 2 corresponding to the actual decisions of PLAYER 1 is used. In other words:

• If PLAYER 1 chose not to participate, none of the decisions of PLAYER 2 impact earnings and both PLAYERS earn 5 ECUs. • If PLAYER 1 chose to participate and "DOWN", only the decision of PLAYER 2 taken "as if" PLAYER1 had chosen "DOWN" matters. In this case, the consequence of PLAYER 2's decision on earnings is found in the first line of Table 1. • If PLAYER 1 chose to participate and "UP", only the decision of PLAYER 2 taken "as if" PLAYER1 had chosen "UP" matters. In this case, the consequence of PLAYER 2's decision on earnings is found in the second line of Table 1. Before the first round of the experiment, you will be asked to answer several questions aimed at checking your understanding of the decisions you will have to make.

After the last round, you will have to answer a demographic questionnaire. This questionnaire does not threaten the anonymity of your decisions.

Please read these instructions again before clicking OK to proceed to the experiment. Should you have any question, silently raise your hand and an experimenter will come to answer privately. At the beginning of the session, each participant is assigned to the role of PLAYER 1 or PLAYER 2, once and for all.

The experiment consists of 30 rounds. For each round, the computer program randomly forms pairs composed of one PLAYER 1 and one PLAYER 2, who will have the possibility to participate in an interaction.

In this interaction, PLAYER 1 and PLAYER 2 will have simple decisions to make: choosing either "UP" or "DOWN" according to a procedure described hereafter. These decisions will impact both their own earnings as well as the other PLAYER's earnings as summarized in Table 1. PLAYER 2 chooses "DOWN" PLAYER 2 choses "UP" PLAYER 1 chooses ''down''

• Player 1 earns 3 ECUs • Player 2 earns 3 ECUs

• Player 1 earns 20 ECUs • Player 2 earns 1 ECUs PLAYER 1 chooses ''up''

• Player 1 earns 1 ECUs • Player 2 earns 20 ECUs

• Player 1 earns 10 ECUs • Player 2 earns 10 ECUs However, the interaction might not take place in every round: at the beginning of each round, PLAYER 1 has to decide whether to participate or not, using the interface shown in Screenshot 1.

To help PLAYER 1's participation decision, from round 2 on, there are 9 chances out of 10 that the past decisions of PLAYER 2 get disclosed on the left panel of PLAYER 1's decision screen, as shown in screenshot 2. There is 1 chance out of 10 than nothing gets disclosed, as in Screenshot 1.

If PLAYER 1 decides to participate, he then chooses between "UP" and "DOWN".

If PLAYER 1 decides not to participate, he has no further decision to make in this round, and the decisions of PLAYER 2 have no impact on earnings: PLAYER 1 and PLAYER 2 both earn 5 ECUs.

PLAYER 2 is not informed of the decisions of PLAYER 1 at this point and as a consequence makes two decisions. One decision is made ''as if" PLAYER 1 had decided to participate, and had subsequently chosen "UP". The other decision is made "as if" PLAYER 1 had decided to participate, and had subsequently chosen "DOWN". PLAYER 2 makes these decisions using the interface shown in Screenshot 3.

To compute earnings, only the decision of PLAYER 2 corresponding to the actual decisions of PLAYER 1 is used. In other words:

• If PLAYER 1 chose not to participate, none of the decisions of PLAYER 2 impact earnings and both PLAYERS earn 5 ECUs. • If PLAYER 1 chose to participate and "DOWN", only the decision of PLAYER 2 taken "as if" PLAYER1 had chosen "DOWN" matters. In this case, the consequence of PLAYER 2's decision on earnings is found in the first line of Table 1. • If PLAYER 1 chose to participate and "UP", only the decision of PLAYER 2 taken "as if" PLAYER1 had chosen "UP" matters. In this case, the consequence of PLAYER 2's decision on earnings is found in the second line of Table 1. Before the first round of the experiment, you will be asked to answer several questions aimed at checking your understanding of the decisions you will have to make.

After the last round, you will have to answer a demographic questionnaire. This questionnaire does not threaten the anonymity of your decisions.

Please read these instructions again before clicking OK to proceed to the experiment. Should you have any question, silently raise your hand and an experimenter will come to answer privately.

[Voluntary No Noise]

Thank you for participating in this experiment on decision making. You are not allowed to communicate with other participants during the entire session. Please turn off your cell phone. If you have a question, please raise your hand and wait until an experimenter comes to you to answer your question in private.

For showing up on time, you receive a €3 show-up fee. In addition, your decisions during the experiment earn you money. During the session, earnings will be expressed in terms of Experimental Currency Units, ECU. ECU converts to euro at the following rate: 20 ECUs = €1. You will be paid by bank transfer shortly after the experiment. Your decisions in the experiment will remain anonymous. At the beginning of the session, each participant is assigned to the role of PLAYER 1 or PLAYER 2, once and for all.

The experiment consists of 30 rounds. For each round, the computer program randomly forms pairs composed of one PLAYER 1 and one PLAYER 2, who will have the possibility to participate in an interaction.

In this interaction, PLAYER 1 and PLAYER 2 will have simple decisions to make: choosing either "UP" or "DOWN" according to a procedure described hereafter. These decisions will impact both their own earnings as well as the other PLAYER's earnings as summarized in Table 1. PLAYER 2 chooses "DOWN" PLAYER 2 choses "UP" PLAYER 1 chooses ''down''

• Player 1 earns 3 ECUs • Player 2 earns 3 ECUs

• Player 1 earns 20 ECUs • Player 2 earns 1 ECUs PLAYER 1 chooses ''up''

• Player 1 earns 1 ECUs • Player 2 earns 20 ECUs

• Player 1 earns 10 ECUs • Player 2 earns 10 ECUs However, the interaction might not take place in every round: at the beginning of each round, PLAYER 1 has to decide whether to participate or not, using the interface shown in Screenshot 1.

To help PLAYER 1's participation decision, from round 2 on PLAYER 2 can choose to disclose his past decisions (see Screenshot 2). If so, the past decisions of PLAYER 2 will be disclosed on the left part of PLAYER 1's decision screen, as shown in Screenshot 3.

If PLAYER 1 decides to participate, he then chooses between "UP" and "DOWN".

If PLAYER 1 decides not to participate, he has no further decision to make in this round, and the decisions of PLAYER 2 have no impact on earnings: PLAYER 1 and PLAYER 2 both earn 5 ECUs.

PLAYER 2 is not informed of the decisions of PLAYER 1 at this point and as a consequence makes two decisions. One decision is made ''as if" PLAYER 1 had decided to participate, and had subsequently chosen "UP". The other decision is made "as if" PLAYER 1 had decided to participate, and had subsequently chosen "DOWN". PLAYER 2 makes these decisions using the interface shown in Screenshot 4.

To compute earnings, only the decision of PLAYER 2 corresponding to the actual decisions of PLAYER 1 is used. In other words:

• If PLAYER 1 chose not to participate, none of the decision of PLAYER 2 impacts earnings and both PLAYERS earn 5 ECUs. • If PLAYER 1 chose to participate and "DOWN", only the decision of PLAYER 2 taken "as if" PLAYER1 had chosen "DOWN" matters. In this case, the consequence of PLAYER 2's decision on earnings is found in the first line of Table 1. • If PLAYER 1 chose to participate and "UP", only the decision of PLAYER 2 taken "as if" PLAYER1 had chosen "UP" matters. In this case, the consequence of PLAYER 2's decision on earnings is found in the second line of Table 1. Before the first round of the experiment, you will be asked to answer several questions aimed at checking your understanding of the decisions you will have to make.

After the last round, you will have to answer a demographic questionnaire. This questionnaire does not threaten the anonymity of your decisions.

Please read these instructions again before clicking OK to proceed to the experiment. Should you have any question, silently raise your hand and an experimenter will come to answer privately. At the beginning of the session, each participant is assigned to the role of PLAYER 1 or PLAYER 2, once and for all.

The experiment consists of 30 rounds. For each round, the computer program randomly forms pairs composed of one PLAYER 1 and one PLAYER 2, who will have the possibility to participate in an interaction.

In this interaction, PLAYER 1 and PLAYER 2 will have simple decisions to make: choosing either "UP" or "DOWN" according to a procedure described hereafter. These decisions will impact both their own earnings as well as the other PLAYER's earnings as summarized in Table 1. PLAYER 2 chooses "DOWN" PLAYER 2 choses "UP" PLAYER 1 chooses ''down''

• Player 1 earns 3 ECUs • Player 2 earns 3 ECUs

• Player 1 earns 20 ECUs • Player 2 earns 1 ECUs PLAYER 1 chooses ''up''

• Player 1 earns 1 ECUs • Player 2 earns 20 ECUs

• Player 1 earns 10 ECUs • Player 2 earns 10 ECUs However, the interaction might not take place in every round: at the beginning of each round, PLAYER 1 has to decide whether to participate or not, using the interface shown in Screenshot 1.

To help PLAYER 1's participation decision, from round 2 on PLAYER 2 can choose to disclose his past decisions (see Screenshot 2). If Player 2 chooses to disclose, there is 9 chances out of 10, that the past decisions of PLAYER 2 get disclosed on the left part of PLAYER 1's decision screen, as shown in Screenshot 3, and 1 chance out of 10 that nothing gets disclosed.

If PLAYER 1 decides to participate, he then chooses between "UP" and "DOWN".

If PLAYER 1 decides not to participate, he has no further decision to make in this round, and the decisions of PLAYER 2 have no impact on earnings: PLAYER 1 and PLAYER 2 both earn 5 ECUs.

PLAYER 2 is not informed of the decisions of PLAYER 1 at this point and as a consequence makes two decisions. One decision is made ''as if" PLAYER 1 had decided to participate, and had subsequently chosen "UP". The other decision is made "as if" PLAYER 1 had decided to participate, and had subsequently chosen "DOWN". PLAYER 2 makes these decisions using the interface shown in Screenshot 4.

To compute earnings, only the decision of PLAYER 2 corresponding to the actual decisions of PLAYER 1 is used. In other words:

• If PLAYER 1 chose not to participate, none of the decision of PLAYER 2 impacts earnings and both PLAYERS earn 5 ECUs. • If PLAYER 1 chose to participate and "DOWN", only the decision of PLAYER 2 taken "as if" PLAYER1 had chosen "DOWN" matters. In this case, the consequence of PLAYER 2's decision on earnings is found in the first line of Table 1. • If PLAYER 1 chose to participate and "UP", only the decision of PLAYER 2 taken "as if" PLAYER1 had chosen "UP" matters. In this case, the consequence of PLAYER 2's decision on earnings is found in the second line of Table 1. Before the first round of the experiment, you will be asked to answer several questions aimed at checking your understanding of the decisions you will have to make.

After the last round, you will have to answer a demographic questionnaire. This questionnaire does not threaten the anonymity of your decisions.

Please read these instructions again before clicking OK to proceed to the experiment. Should you have any question, silently raise your hand so that an experimenter will come to answer privately.

Figure 1 :

 1 Figure 1: The stage game.

  Figure A.1 in Appendix A gives an example of a record as disclosed to P1.

  operate when P1 cooperated (conditional cooperation), ii) Cooperate irrespective of the decision of P1 (unconditional cooperation), iii) Never cooperate (unconditional defection) and iv) Cooperate only when P1 defected (mismatch). In what follows, to study cooperation choices of P2, we pool together conditional and unconditional cooperation. Focusing strictly on conditional cooperation does not change our results. More details about the distribution of strategies across treatments are given in Appendix D.3.

Figure 2 :

 2 Figure 2: Predicted probability to disclose depending on one's past record.

Figure 3 :

 3 Figure 3: Cooperation rate depending on whether P1 saw the record, separate by voluntary / mandatory disclosure.

Figure 4 :

 4 Figure 4: The predicted probability of cooperation by P1 depending on the record of P2.

Figure 5 :

 5 Figure 5: Cooperation of P1 depending on whether P1 saw the record, contrasted by voluntary / mandatory disclosure.

D. 2

 2 Figure D.2: The dynamic of the cooperative outcome (period averages)

Figure D. 6 :

 6 Figure D.6: The distribution of the 4 pure strategies of P2, by Treatment

Figure D. 7 :

 7 Figure D.7: Cluster of P2s according to disclosure and cooperation.

*

  ********************************************************************************* Description of the decisions.

  At the end of each round, you are informed of your earnings for the present round and of the earnings you have accumulated up to this round. You are not informed of the decision of the other PLAYER. ****************************************************************************

Screenshot 1 :

 1 Participation decision (PLAYER 1) Screenshot 2: UP or DOWN decisions (PLAYER 2) [Mandatory No Noise]

*

  ********************************************************************************* Description of the decisions.

  At the end of each round, you are informed of your earnings for the present round and of the earnings you have accumulated up to this round. You are not informed of the decision of the other PLAYER. ****************************************************************************

Screenshot 1 :

 1 Participation decision (PLAYER 1)Screenshot 2: Participation decision with information disclosure(PLAYER 1) participating in this experiment on decision making. You are not allowed to communicate with other participants during the entire session. Please turn off your cell phone. If you have a question, please raise your hand and wait until an experimenter comes to you to answer your question in private.For showing up on time, you receive a €3 show-up fee. In addition, your decisions during the experiment earn you money. During the session, earnings will be expressed in terms of Experimental Currency Units, ECU. ECU converts to euro at the following rate: 20 ECUs = €1. You will be paid by bank transfer shortly after the experiment. Your decisions in the experiment will remain anonymous. ********************************************************************************** Description of the decisions.

  At the end of each round, you are informed of your earnings for the present round and of the earnings you have accumulated up to this round. You are not informed of the decision of the other PLAYER. ****************************************************************************

*

  ********************************************************************************* Description of the decisions.

  At the end of each round, you are informed of your earnings for the present round and of the earnings you have accumulated up to this round. You are not informed of the decision of the other PLAYER. ****************************************************************************

Screenshot 1 :

 1 Participation decision (PLAYER 1) Screenshot 2: Disclosure decision (PLAYER 2) XXX XX X Screenshot 3: Participation decision with information disclosure (PLAYER 1) Screenshot 4: UP or DOWN decisions (PLAYER 2) participating in this experiment on decision making. You are not allowed to communicate with other participants during the entire session. Please turn off your cell phone. If you have a question, please raise your hand and wait until an experimenter comes to you to answer your question in private.For showing up on time, you receive a €3 show-up fee. In addition, your decisions during the experiment earn you money. During the session, earnings will be expressed in terms of Experimental Currency Units, ECU. ECU converts to euro at the following rate: 20 ECUs = €1. You will be paid by bank transfer shortly after the experiment. Your decisions in the experiment will remain anonymous. ********************************************************************************** Description of the decisions.

  At the end of each round, you are informed of your earnings for the present round and your cumulated earnings, but not of the decision of the other PLAYER. ****************************************************************************

Screenshot 1 :

 1 Participation decision (PLAYER 1) Screenshot 2: Disclosure decision (PLAYER 2) XXX XX X

Table 1 :

 1 Summary of the treatments with record disclosure.

	Record disclosure is: Mandatory Voluntary
	Certain	MandNN	VolNN
	Noisy	MandLN	VolLN

Table 2

 2

	sums up the distribution of participants and sessions across treatments and
	cities. Our results are robust to the inclusion of a dummy variable indicating the

Table 2 :

 2 Summary of the sessions.

	city in which the session was run.
	On average a session lasted about 1 hour. The average earnings were e12.87
	(S.D.:2.37). Before starting the experiment, participants had to successfully pass
	a non-incentivized understanding questionnaire. At the end of the experiment, par-
	ticipants had to fulfill a demographic questionnaire. The full set of instructions can
	be found in Appendix F.

Table 4 :

 4 The marginal effects of voluntary and noisy disclosure.

  Standard errors in parentheses are clustered at the session level.

			2)	(3)
		Enter=1 P1's Coop=1	P2's Cond Coop=1
	Marginal effect of:
	Voluntary	-0.014	-0.036	0.031
		(0.038)	(0.031)	(0.055)
	Noisy	-0.056	-0.073 * *	-0.112 * *
		(0.044)	(0.031)	(0.044)
	Observations	5780	5780	5780
	Session characteristics	Yes	Yes	Yes
	Individual characteristics	Yes	Yes	Yes
	Period FE	Yes	Yes	Yes
	Sessions	25	25	25

* p<0.10, ** p<0.05, *** p<0.01, **** p<0.001 Session characteristics: City dummy and size of the session. Individual characteristics: Gender, age, occupational status, experience with experiments.

Table 5 :

 5 The effects of voluntary disclosure and of noise

Table 6 :

 6 The marginal effect of one's record on the probability of disclosure

Table 7 :

 7 The effect of the record of P2 on the decision to enter and cooperate by P1

  City dummy and size of the session. Individual characteristics: Gender, age, occupational status, experience with experiments.

			2)	(3)
	Coop.=1 Coop.=1	Coop.=1
	Marginal effect of seeing the record at:
	Voluntary=0	0.137 * * * *
		(0.035)
	Voluntary=1	0.256 * * * *
		(0.030)
	p-value diff.	0.010
	Voluntary=0 × Period≤ 10		0.074 * *
			(0.036)
	Voluntary=0 × 10 <Period≤ 20		0.197 * * * *
			(0.043)
	Voluntary=0 × Period> 20		0.138 * * *
			(0.048)
	Voluntary=1 × Period≤ 10		0.202 * * * *
			(0.035)
	Voluntary=1 × 10 <Period≤ 20		0.282 * * * *
			(0.028)
	Voluntary=1 × Period> 20		0.284 * * * *
			(0.049)
	VolNN			0.279 * * * *
				(0.043)
	MandLN			0.144 * * *
				(0.046)
	VolLN			0.231 * * * *
				(0.038)
	Observations	5587	5587	5587
	RE/FE	RE	RE	RE
	SE	Cluster	Cluster	Cluster
	Session Char.	Yes	Yes	Yes
	Indiv. Char.	Yes	Yes	Yes
	Standard errors in parentheses are clustered at the session level.
	* p<0.10, ** p<0.05, *** p<0.01, **** p<0.001
	Session characteristics:

Table 8 :

 8 The differential effect of (not) seeing the record, depending on the treatment.

  Standard errors are clustered at the session level. Results are in Table C.3.

		(1)	(2)	(3)
		Coop. outcome=1 Coop. outcome=1 Coop. outcome=1
	Disclosure	0.126 * * * (0.048)	0.126 * * * (0.047)	0.122 * * * (0.046)
	Observations	5780	5780	5780
	Session characteristics	No	No	Yes
	Period FE	No	Yes	Yes
	Sessions	25	25	25
	Standard errors in parentheses are clustered at the session level.
	* p<0.10, ** p<0.05, *** p<0.01, **** p<0.001.
	Session characteristics: City dummy and size of the session.
	Individual characteristics: Gender, age, occupational status, experience with experiments.

  TableC.3: The effect of the existence of a disclosure system on the likelihood of the cooperative outcome .

		(1)	(2)	(3)	(4)	(5)	(6)
		Enter=1 Enter=1 Enter=1 Coop.=1 Coop.=1 Coop.=1
	Marginal effect of:
	Information disclosed 0.140 * * * * 0.211 * * * *		0.220 * * * * 0.262 * * * *
		(0.020)	(0.026)		(0.025)	(0.021)
	VolNN			0.148 * * * *			0.279 * * * *
				(0.025)			(0.043)
	MandLN			0.054			0.144 * * *
				(0.036)			(0.046)
	VolLN			0.196 * * * *			0.231 * * * *
				(0.028)			(0.038)
	Observations	5587	4804	5587	5587	5385	5587
	R 2
	RE/FE	RE	FE	RE	RE	FE	RE
	SE	Cluster Bootstrap Cluster	Cluster Bootstrap Cluster
	Session Char.	Yes	Yes	Yes	Yes	Yes	Yes
	Indiv. Char.	Yes	Yes	Yes	Yes	Yes	Yes
	Standard errors in parentheses are clustered at the session level.
	* p<0.10, ** p<0.05, *** p<0.01, **** p<0.001.
	Session characteristics: City dummy and size of the session.
	Individual characteristics: Gender, age, occupational status, experience with experiments.

C.3 Support for Result 4.

Table C

 C

	.4: The effect of seeing the information about P2's past choices on P1s'
	choices. Marginal effects from Logit models.

Table 1 :

 1 Earning consequences of Player 1 and Player 2's decisions in the interaction.

Table 1 :

 1 Earning consequences of Player 1 and Player 2's decisions in the interaction.

Table 1 :

 1 Earning consequences of Player 1 and Player 2's decisions in the interaction.

Table 1 :

 1 Earning consequences of Player 1 and Player 2's decisions in the interaction.

Table 1 :

 1 Earning consequences of Player 1 and Player 2's decisions in the interaction.

A related literature study the effect of information about choices on cooperation in simultaneous interactions. See for instance[START_REF] Duffy | Cooperative behavior and the frequency of social interaction[END_REF];[START_REF] Camera | Cooperation among strangers under the shadow of the future[END_REF];[START_REF] Kamei | Play It Again: Partner Choice, Reputation Building and Learning From Finitely Repeated Dilemma Games[END_REF].

See e.g.[START_REF] Varian | Economic Aspects of Personal Privacy[END_REF];[START_REF] Acquisti | The economics of privacy[END_REF] for reviews on the economics of privacy and e.g.[START_REF] Benndorf | Privacy concerns, voluntary disclosure of information, and unraveling: An experiment[END_REF];[START_REF] Benndorf | The willingness to sell personal data[END_REF];[START_REF] Schudy | you must not know about me'-on the willingness to share personal data[END_REF] for recent experimental evidence on privacy preferences.

There is also evidence of failures of the unravelling principle outside the lab. For instance,[START_REF] Brown | To review or not to review? limited strategic thinking at the movie box office[END_REF] show that film studios exploit the fact that movie goers fail to anticipate that movies which are not reviewed before release tend to be of low quality.

Given the size of our sessions, participants meet more than once on average. This could in principle affect the evolution of cooperation through contagion.[START_REF] Ghidoni | Perfect and imperfect strangers in social dilemmas[END_REF] show that random matching in groups of 6 participants give similar results to perfect stranger's matching. In all our sessions we had more than 6 participants. In addition, all our results hold when we control for the number of participants in the respective session.

A preregistration of the experiment can be found at https://doi.org/10.1257/rct.4937. Note that, due to constraints related to the pandemic, we had to drop two treatments in which a larger noise was introduced. We have a power of 80% to detect an effect of 10 percentage points at the 5% level (more details are in Appendix B). This effect size is less than half the size of the effect of disclosing the second movers' history reported in[START_REF] Charness | Trust and trustworthiness reputations in an investment game[END_REF].

In TableD.1 in Appendix D.1, we report the dynamics of the Cooperative outcome for each session separately.

We thank Elias Fernandez and Antoine Deplancke for helping with the experiment in Brussels and Lille. This project received financial support from Fonds

A Screenshots

(1)

D.4 Individuals and behavioral types.

Does it pay to be skeptical? Our results suggest some skepticism among P1s.

Here, we check whether being skeptical is beneficial in terms of earnings. To do so, we compute for each individual P1 the extent to which her cooperation rate depends on whether she received the information. This dependence of cooperation on information serves as measure of skepticism. We regress P1s' average stage-game earnings on this measure of skepticism, and we find that more skeptical individuals earn more (OLS with standard errors clustered at the session level: b = 1.225, p = 0.03).

Does it pay to reveal? P2 chose to reveal in the majority of the cases where he actually had a choice. We saw that disclosure is significantly linked to the quality of the record. Now we want to investigate individuals strategies. To do so, we XXX XXX


A Ha

T Sim

M Ta

Robust motion planning for rough terrain navigation

This paper deals with motion planning for a mobile robot on rough terrain. We proposed in 9] a geometrical planner for articulated robots which can take into account uncertainty on the terrain and on the position of the robot. This paper aims at improving the robustness of the trajectory using landmark based approach.

We consider regions of the terrain where natural landmarks are visible. We propose a two-step planning approach taking advantage of these regions to reduce position uncertainty. First a path is determined between the initial and goal con gurations, based on simpli ed models of the constraints. Then a trajectory is planned along this path, verifying the validity and visibility constraints.

Introduction

Recent success of Sojourner rover mission on Mars gave a new fervor to planetary exploration with robots. Future missions may include complex tasks : pick up samples, long range displacements,. . .A higher level of autonomy may be required to perform such missions, Autonomous navigation on natural terrain remains however a complex and challenging problem.

Several systems for outdoor navigation have been developed. The Rocky robots 15] from Jet Propulsion Laboratory can reach speci ed positions, using a behavior control approach 12, 7]. Nomad 1] from Carnegie Mellon University operated more than 200 kilometers using various navigation modes (principally teleoperation). Another system from CMU, Navlab also performed autonomous navigation on natural terrain 17].

The adaptative navigation approach developed at LAAS within the framework of the EDEN experiment 3] demonstrated autonomous short-range navigation in a natural environment gradually discovered by the robot. The approach combines various navigation modes (re ex, 2D and 3D) in order to adapt the robot behavior to the complexity of the environment.

This paper addresses the problem of motion planning for the 3D navigation mode 14], selected when very rugged terrain has to be crossed by the robot. On uneven or highly cluttered areas, the obstacle notion is closely linked with the constraints on the robot attitude. Planning a trajectory on such areas requires a detailed modeling of the terrain and also of the robot's locomotion system. Several contributions recently addressed motion planning [START_REF] Shirley | Mars path nder microrover[END_REF][START_REF] Matthies | Mars microrover navigation : performance evaluation and enhancement[END_REF][START_REF] Ha | Motion planning on rough terrain for an articulated vehicle in presence of uncertainties[END_REF][START_REF] Gat | Path planning and execution monitoring for a planetary rover[END_REF][START_REF] Farritor | Physicsbased planning for planetary exploration[END_REF][START_REF] Deng | Landmark selection for path execution[END_REF]. In the navigation experiments 14] performed using the planning algorithms of 16] , motion control was limited to executing the paths returned by this planner by relying on odometry and inertial data. The unpredictable and cumulative errors generated by these sensors, especially signi cant on uneven and slippery areas, often caused important deviations leading to a lack of robustness at execution. To overcome this problem, the robot may be equipped with environment sensors (e.g. cameras) that can provide additional information by identifying appropriate features of the terrain. Localization on these features allow to overcome the problem of cumulative errors. It however introduces new constraints; in particular the visibility of the terrain features has to be checked along the trajectory. Some contributions addressed this problem (e.g. [START_REF] Gat | Behavior control for robotic exploration of planetary surfaces[END_REF][START_REF] Kubota | Path planning for planetary rover based on traversability probability[END_REF][START_REF] Chatila | Planetary exploration by a mobile robot: mission teleprogramming and autonomous navigation[END_REF]).

In this paper we extend the motion planner proposed in [START_REF] Shirley | Mars path nder microrover[END_REF][START_REF] Gat | Path planning and execution monitoring for a planetary rover[END_REF] in order to consider a set of given landmarks (e.g. terrain peaks). The landmarks partition the terrain into regions where they remain visible from the robot's sensor. The approach allows to produce trajectories that remain, if necessary, inside these regions where the robot can navigate with respect to the given landmarks using closed-loop primitives relying onto the sensor's data.

2 The planning approach

Problem statement

We consider a geometric model of an articulated robot shown on Figure 1, right. The robot is composed of three axles linked by passive joints allowing to adapt its shape to the terrain relief, like Marsokhod robot LAMA (see Fig. 1, left). The terrain is described by surface patches de ned from an elevation map in z associated with a regular grid in (x; y). A con guration is a vector q = (x; y; ) specifying the horizontal position/heading of the robot in the global frame. Additional parameters r(q) describing the angles of the articulations are determined from con guration q.

Figure 1: Example of placement -The mobile robot LAMA The terrain model also contains a set of point landmarks corresponding to major terrain features that the robot should be able to track at execution. Whenever possible, landmarks are used to localize the robot; otherwise odometry is used, involving a growing position uncertainty along the trajectory. Consequently, the motion planner must take into account: validity constraints related to the safety of the motion (e.g. stability of the vehicle, collision avoidance with the terrain and mechanical limits); visibility constraints which represent the ability of the robot to detect one or a set of landmarks from a given con guration; position uncertainty along the trajectory.

The planned trajectory is a sequence of valid con gurations between initial and goal con gurations q init and q goal . It also indicates the landmarks the robot should track during execution.

A two-step approach

In order to limit the complexity of the problem, the constraints are not considered simultaneously. We propose a motion planner based on the following twostep approach: the rst step is based on simpli ed models of the constraints in order to obtain a 2D trajectory called path. This path is composed of portions where landmarks are visible and others where no landmark can be used during execution; in the second step, a 3D trajectory is planned along the path. This trajectory must verify the validity constraints, and along portions for which landmarks are visible, the visibility constraints for these landmarks must also be veri ed. The nal trajectory is thus a sequence of trajectories alternately verifying or not the visibility of landmarks. The main advantage of this approach is that the selection of the landmarks is done in the rst step, with simpli ed models. The visibility constraints are then tested with these landmarks when it is necessary in the second step. Another advantage is that trajectory planning is fast because search is only done along the path and not all over the terrain. The following sections describe the two steps, called path planning and trajectory planning.

Path planning

This step consists in determining a path, i.e. a sequence of points (x i ; y j ) on a grid with the same discretization as the terrain model. Initial point P init of the path is the nearest point on the grid from initial con guration q init (the same for P goal ). A numerical propagation on the grid, initiated at P init , is used to determine the path. It is based on simpli ed 2D models of the constraints.

Constraint models

In order to check the constraints without computing the complete robot placement (q; r(q)), 2D models are presented, relying on position P(x P ; y P ) of the robot. To guarantee a correct link with the trajectory planning step, it is important that these 2D constraints re ect well the 3D ones.

Validity

A cost bitmap is computed by evaluating for each point of the terrain model, the slope and the roughness of a circular domain CD centered at this position, with a radius related to the size of the robot. It represents the di culty for the robot to cross the domain (see Fig. 2).

Landmark visibility

The sensor position is estimated from the circular domain CD representing the robot (see Fig. 3): S is eld of view constraint : the distance between sensor S and landmark L is within some limit values; non-occultation constraint : L is not hidden by the terrain i.e. segment SL] does not intersect the terrain. For each point of the terrain respecting the rst constraint, the second one is tested, using the e cient collision checker developped for the trajectory planner 9]. A visibility region is thus associated to each landmark, de ned by the set of points respecting both constraints (see Fig. 4). The value of I depends on the path followed and the localizations on landmarks (see Fig. 5).

Cost value P at P with uncertainty I is the maximal value of the cost bitmap on the domain centered at P and with a radius I.

Out of visibility regions uncertainty grows proportionally to distance and terrain di culty:

I B = I A + f(d AB ; B )
In visibility regions uncertainty is computed in two steps (see Fig. 5). First uncertainty without localization is evaluated as presented above (e.g. on Fig. 5 I 0 E = I D + f(d DE ; E )). Then a localization model is used :

I E = F(fL i ; d i g)I 0 E with F(fLi; d i g) = n Y i=1 (1 r 1 d i D max )
where fL i g i=1::n is the set of visible landmarks from the current position, d i is the distance between the sensor and landmark L i , and D max is the higher distance allowing to detect a landmark. The propagation starts form the rst node with init = 0 and stops when the wavefront reaches the goal P goal . The solution path is then obtained by chaining the nodes back to the rst one.

Propagation out of visibility regions

In this case, no localization on landmark is possible. Consequently potential B and uncertainty I B increase the same way, depending on the di culty of the terrain. For each node, cost, potential and uncertainty are evaluated. The node is then placed in a list sorted by cost and potential values. The wavefront expansion thus simultaneously considers the local terrain diculty and the global cost of the followed path. At each iteration, the rst node of the list is expanded, using a 8-neighbor propagation. A new node is created if there is no other node corresponding to the same point (x i ; y j ) of the grid. Node creation is also forbidden if uncertainty exceeds a bound I max , in order to limit path search to realist values of uncertainty. The propagation stops when the list is empty (failure), or when the wavefront reaches the node corresponding to P goal . If uncertainty I goal is lower than the expected limit I max goal (6 I max ), a solution is found.

Otherwise, the goal cannot be reached by the robot within the given uncertainty limits I max goal and I max .

Localization on landmarks

The position uncertainty decreases when a localization is performed in a landmark visibility region. Path planning takes into account localization with the uncertainty model presented in x3.1.3. Consequently, for a given uncertainty limit I max , the wavefront propagation covers a wider area than previously. Modi cations to the propagation algorithm concern the node expansion. Node creation now depends on the following conditions: uncertainty is lower than I max ; there is no node corresponding to the same position, or : after a localization, the uncertainty of the new node is lower than the old node's one. Consequently, for each visibility region reached by the wavefront, a new wavefront propagation is performed, overlapping the previous one. Various wavefronts can successively reach the goal, with di erent uncertainty values I goal . The propagation goes on while I goal > I max goal . The case where no wavefront reach the goal with the expected uncertainty is considered as a failure. The solution path is composed of portions in and out

Example

Figure 6 shows an example of path planning with one landmark visibility region. Fig. 6-b and Fig. 6-c present the wavefront propagations. Grey levels correspond to uncertainty value (from light to dark). When no localization is possible (b), uncertainty value is to high and the propagation does not reach P goal . After a localization onto the landmark (c), the propagation reaches P goal . Fig. 6-d presents the solution path onto the terrain. One may note that thanks to the cost bitmap, the path avoids uneven terrain, so ensuring consistency with the 3D validity constraints in the trajectory planning step.

Trajectory planning

The path planner provides a sequence of portions between P init and P goal , obtained with simpli ed models of the constraints. The trajectory planner then searches for a trajectory (i.e. a sequence of con gurations q(x; y; )) following the path and respecting the validity constraints (cf. Fig. 7) and also the visibility constraints. This planner is based on the algorithms described in 9].

Planning areas

The trajectory is searched on a domain called planning area around the portion of path. The area size depends locally on the evaluated uncertainty along the path. Trajectory planning is performed in two steps on this area, as in 9] : a preprocessing step to compute robot placements on the area, and then the planning step itself.

Graph search

The planning step incrementally builds a graph of discrete con gurations that can be reached from the initial con guration by applying sequences of discrete controls during a short time interval. The arcs of the graph correspond to feasible portions of trajectory, computed for a given control. Only the arcs verifying the validity constraints are considered during the search. A numerical potential development is performed to guide the search along the path (heuristic function). When the corresponding portion of path belongs to a landmark visibility region, the visibility constraint of this landmark is also tested along the arc. When the landmark is not visible at some points, a penalty is given to the associated arc for the search (if the landmark is not visible all along the arc, this one is not considered).

Complete trajectory

For the rst portion of trajectory, the search starts at con guration q init . It stops when the current conguration is in a domain (dx; dy; d ) around the nal point of the portion of path. This domain prevents from unexpected maneuvers near the goal. If the next portion belongs to a visibility region, the last con guration must also verify the visibility of the landmark. The next portion of trajectory begins at the last conguration of the previous portion, and so on until the last portion, where the goal con guration q goal must be considered. The complete trajectory is then the sequence of portions between q init and q goal . It contains information about landmark visibility, and the relative position between the robot and the landmarks in order to move the sensor in the right direction.

Simulation results

The complete approach is illustrated onto a terrain model containing two landmarks. Figure 8 shows the corresponding visibility regions, the path found at the path planning level, and also the uncertainty evolution along the path.

Conclusion

In this paper, we considered the problem of robust motion planning for a robot on a rough terrain. In order to improve the trajectory robustness, planning takes into account the presence of natural terrain features that may allow to localize precisely the robot. We proposed a two-step approach based on various models of the constraints. The rst step relies on a numerical propagation to determine a 2D path. The second step searches for a trajectory along this path, respecting the validity and landmark visibility constraints. Tests on various terrain models demonstrated the ability of the planner to solve e ciently di cult problems. p

Figure 2 :Figure 3 :

 23 Figure 2: A terrain model and the associated cost bitmap

Figure 4 :

 4 Figure 4: Landmark visibility region 3.1.3 Position uncertainty For a position P, the uncertainty is represented by a circular domain centered in P, and with a radius I.

Figure 5 :

 5 Figure 5: Uncertainty along a path on the grid

Figure 6 :

 6 Figure 6: Path planning of the visibility regions. It is a sequence of positions between P init and P goal for which an evaluation of the uncertainty and the visible landmark are available.

Figure 7 :

 7 Figure 7: Examples of placement's constraints (collision, mechanical limits, stability)

Figure 8 :

 8 Figure 8: Path planning

Figure 9 :

 9 Figure 9: Trajectory planning