Anne-Lise Fevre-Gautier

Philippe Wavrer

Pascale Michel

Municipal solid waste organic fraction management: A multicriteria analysis for societal performance

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

ORBIT2012

1 Municipal solid waste organic fraction management: A multicriteria analysis for societal performance A.-L. Gautier, BRGM, Ph. Wavrer, BRGM.

Abstract

The results presented here derive from the work conducted for an ongoing two-year research project funded by the French Ministry of Environment and started in 2011. In 2007, 14% of French municipal waste collected by public services were going through organic valorization, when more than 60% of municipal residual waste (representing 153 kg of household residual waste per inhabitant and per year) could still undergo an organic treatment. To fulfill European and national commitments though, French biological treatment capacities for municipal solid waste should be doubled by 2015. This can be achieved via home composting, municipal waste organic fraction biological treatment (composting or anaerobic digestion) or mechanical biological treatment (MBT). The deployment of these treatment capacities should however take into account local and territorial specificities in order to reach a sufficient level of waste management global performance, regarding economic, environmental and societal sustainability French and European commitments.

This project aims at performing a comparative assessment of scenarios including different biological treatment processes for municipal solid waste management in France. This assessment is focused on the global performance of the scenarios, in order to understand which organic fraction management mode, under which local or global conditions, can achieve the best performance for society. The objective is therefore to assess scenarios performance regarding environmental, economic, social and societal aspects of waste management sustainability challenges. This analysis is based on a set of waste management scenarios for municipal residual solid waste, including biological treatment for the organic fraction. This set of scenarios also includes a reference scenario representing a usual way of managing municipal solid waste in France without any type of biological treatment. These scenarios are designed according to representative French waste management modes and include: private individual or collective composting, separate collection and biological treatment of organic fraction (composting or anaerobic digestion) and mechanical biological treatment of mixed wastes (MBT). They include a set of parameters used to perform sensitivity analysis in order to estimate the scenarios' global performance dependency on technical and contextual conditions, regarding for example: municipal waste composition, sorting efficiency, biological process performance, opportunities and constraints for process products valorization, treatment installations capacity, public services organization, etc. The comparative assessment of scenarios is conducted via a multicriteria analysis built on a set of specific sustainability and performance indicators for waste management scenarios. These indicators aim at covering subjects such as energy efficiency (treatment processes, transports), health and ecology (natural resources consumption, emissions, effects of organic matter supplies to soils, etc.), economic equilibrium, quality of life, territory development, etc.

Expected results of this ongoing project should highlight the fact that the global performance of biological treatment doesn't only rely on technical process performances, but also on the adequacy level between waste composition, territorial organization for municipal solid waste treatment and local specificities regarding societal aspects of households waste production, sorting, collection and treatment.Therefore one of the conclusions should be that waste management scenarios including biological treatment should be designed according to territorial specificities in order to optimize its global performance, and national waste policy should consider global performance targets recommandations instead of process implementation requirements.


Christoph March
email: march@pse.ens.fr

Sebastian Krügel
email: kruegel@econ.mpg.de.

Anthony Ziegelmeyer
email: ziegelmeyer@econ.mpg.de.

Do We Follow Private Information when We Should? Laboratory Evidence on Naïve Herding *

We investigate whether experimental participants follow their private information and contradict herds in situations where it is empirically optimal to do so. We consider two sequences of players, an observed and an unobserved sequence. Observed players sequentially predict which of two options has been randomly chosen with the help of a medium quality private signal. Unobserved players predict which of the two options has been randomly chosen knowing previous choices of observed and with the help of a low, medium or high quality signal. We use preprogrammed computers as observed players in half the experimental sessions. Our new evidence suggests that participants are prone to a 'social-confirmation' bias and it gives support to the argument that they naïvely believe that each observable choice reveals a substantial amount of that person's private information. Though both the 'overweighting-of-private-information' and the 'social-confirmation' bias coexist in our data, participants forgo much larger parts of earnings when herding naïvely than when relying too much on their private information. Unobserved participants make the empirically optimal choice in 77 and 84 percent of the cases in the human-human and computerhuman treatment which suggests that social learning improves in the presence of lower behavioral uncertainty.

Introduction

With the help of a large meta-dataset covering 13 experiments on social learning games, [START_REF] Weizsäcker | Do We Follow Others when We Should? A Simple Test of Rational Expectations[END_REF] investigates whether participants follow others and contradict their private information in situations where it is empirically optimal to do so. Weizsäcker finds that participants are quite unsuccessful in learning from others. The average participant follows others only in situations where the evidence conveyed by their observable choices is so strong that the private information is wrong more than twice as often as it is correct. Economic experiments on social learning games have repeatedly concluded that Bayesian rationality organizes well most of participants' choices except for an inflated tendency to follow private information (among others, [START_REF] Nöth | Information Aggregation with Random Ordering: Cascades and Overconfidence[END_REF][START_REF] Goeree | Self-Correcting Information Cascades[END_REF]. By estimating the value of the available actions, the metastudy additionally shows that participants forgo substantial parts of earnings when falling prey to the 'overweighting-of-private-information' bias.

The bulk of Weizsäcker's meta-dataset consists of experimental treatments that implement the stripped-down model of information cascades developed by Bikhchandani, Hirshleifer, and Welch (1992, henceforth BHW). In this simple social learning environment, a sequence of participants each in turn choose one of two options with each participant observing all of her predecessors' choices. Induced preferences over the two equally likely options are common, and participants receive independent and equally strong private binary signals about the correct option. According to Bayesian rationality, once the pattern of signals leads to two identical choices not canceled out by previous ones, all subsequent participants should ignore their signals and follow the herd. Though of interest, the experimental evidence on social learning behavior provided by the existing literature is too restrictive. Of particular concern is the coarseness of the social learning environment which favors the emergence of the 'overweighting-of-private-information' bias. 1 In this paper, we investigate whether participants follow their private information and contradict herds in situations where it is empirically optimal to do so. To address this complementary issue, our social learning game relies on a richer information structure than BHW's stripped-down model. Following [START_REF] Ziegelmeyer | Fragility of Information Cascades: An Experimental Study Using Elicited Beliefs[END_REF], we consider two sequences of players, an observed and an unobserved sequence. Observed players sequentially predict which of two options has been randomly chosen with the help of a medium quality private signal (quality equals 14/21). At the end of each decision period, the choice of one observed is made public knowledge. In a matched pairs design, unobserved players guess which of the two options has been randomly chosen knowing previous public choices and with the help of a low, medium or high quality signal (quality equals 12/21, 14/21 or 18/21 respectively). Their choices remain private.

Our laboratory experiment uses an expanding strategy method-like procedure that allows us to detect herding behavior directly, allows participants to gain experience with many decision nodes, and generates a large dataset (see also [START_REF] Cipriani | Herd Behavior in Financial Markets: An Experiment with Financial Market Professionals[END_REF]. In the first part of each session, the signal's quality for the unobserved is fixed at the beginning of each of the three rounds, each player observes only one signal realization and makes only one choice. Each participant earns 0.4 (0.1) Euro for each correct (wrong) guess. The second part of each session is identical to the first part except that i) all unobserved make one choice in each decision period (8 choices in total); ii) all seven observed make one choice in decision period 1 and one choice is randomly selected to be made public, the remaining six observed make one choice in decision period 2 and one choice is randomly selected to be made public, and so on till decision period 7 where the remaining observed makes a last choice; and iii) for each participant, only one randomly selected choice is paid in each round. The third part of each session is identical to the second part except that each choice is made for both realizations of the private signal. Players are informed of the payoff-relevant realization of their private signal after having made their last choice. Finally, the fourth part of each session is identical to the third part except that i) there are six rounds; ii) unobserved make their choices for each quality of the private signal and they are informed of the payoff-relevant quality of their private signal at the end of each round; and iii) for each participant, only one randomly selected choice is paid and each participant earns 12 (3) Euro for a correct (wrong) guess.

A second novelty of our design is the use of preprogrammed computers as observed players in three out of the six experimental sessions. Unobserved players, on the other hand, are always embodied by 1 In situations where predecessors' choices do not point in any direction or point in the same direction as private information, following private information seems the only reasonable choice. In these situations, the few experimental choices not in line with private information have been understood as resulting from confusion. Moreover, in situations where an option is favored by exactly one choice over the other option and private information points in the opposite direction, Bayesian rationality is silent about the optimal choice. Note that about one third of the data in Weizsäcker's meta-dataset stem from [START_REF] Nöth | Information Aggregation with Random Ordering: Cascades and Overconfidence[END_REF] which considers two privately known signal precisions. However, the predictions in this variant of BHW's stripped-down model closely match the original ones.

human participants. Though the computers' strategy is not revealed to the unobserved participants, the latter face lower behavioral uncertainty in the computer-human treatment than in the humanhuman treatment and, no matter how big the contradicting herd is, it is always beneficial for them to follow their high quality signal. Our dataset contains 1,827 choices from 21 observed participants, 8,712 choices from 24 unobserved participants in the human-human treatment, and 8,712 choices from 24 unobserved participants in the computer-human treatment. Given the experimental choices of the observed participants, the estimation of the empirical value of actions leads to the conclusion that following the high quality signal is also the empirically optimal action for the unobserved participants in the human-human treatment no matter how big the contradicting herd is.

The richness of our dataset enables us to measure the success of social learning both in situations where it is empirically optimal to follow others (and contradict private information) and in situations where it is empirically optimal to follow private information (and contradict the herd). 2 We infer that, conditional on being endowed with a low or medium quality signal and observing a contradicting herd of size at least 2, participants make the empirically optimal choice in 75 percent of the cases. In contrast, conditional on being endowed with a high quality signal and observing a contradicting herd of size at least 2, unobserved participants choose optimally in only 56 percent of the cases. In the latter situations, the evidence conveyed by the observable choices is so weak that the private information is correct more than twice as often as it is wrong. Our new evidence therefore suggests that participants are prone to a 'social-confirmation' bias and it gives support to the argument that they naïvely believe that each observable choice reveals a substantial amount of that person's private information [START_REF] Eyster | Naïve Herding in Rich-Information Settings[END_REF]. Though both the 'overweighting-of-private-information' and the 'social-confirmation' bias coexist in our data, participants forgo much larger parts of earnings when herding naïvely than when relying too much on their private information. Finally, compared to the human-human treatment, we observe slightly less naïve herding and slightly more overweightingof-private-information in the computer-human treatment. Overall, unobserved participants make the empirically optimal choice in 77 and 84 percent of the cases in the human-human and computer-human treatment which suggests that social learning improves in the presence of lower behavioral uncertainty.

The next section describes the experimental design and practical procedures. Section 3 derives the relevant theoretical predictions. Section 4 presents our experimental results. Section 5 concludes. The supplementary material contains a translated version of our instructions.

The Experiment

In our information cascade experiment participants make binary decisions in sequence encumbered solely by state-of-Nature uncertainty, and they may condition their decisions both on private signals about the state of Nature and on some earlier decisions. Participants make informational inferences in many analogous situations distinguished by either the history of previous choices, the quality or the realization of the private signal. The experimental setting therefore allows participants to gain extensive experience with the combination of private and public information while offering at the same time the unique chance to carefully study social learning behavior at the individual level.

Our setting builds upon three main ingredients.

Observed and Unobserved Players

The experimental social learning game involves observed and unobserved players. Each repetition of the game begins with the random selection of one of two options which remains hidden to the players. The latter obtain independent private signals that reveal information about which of the two options has been randomly selected. Binary private signals for observed are of medium quality whereas the signal quality for unobserved is either low, medium or high. Players choose in sequence one of the two options, and the monetary payoff is larger for a correct prediction than for an incorrect prediction. Once all choices have been submitted in a given decision period (but the last one), the choice of one observed is made public knowledge. The choices of unobserved remain private. Participant keep the same role of observed or unobserved during the entire experimental session.

Increasing Reliance on the Strategy Method

Our experiment consists of four parts with later parts relying more on the strategy method than earlier ones. In each repetition of the game, 7 observed and 8 unobserved make choices over 8 decision periods.

In the first part of the experiment, participants gain direct-response experience with the social learning game. Each player is endowed with only one realization of the private signal and makes exactly one choice in each of the three repetitions of the game. Concretely, observed obtain a single draw from an urn containing 14 balls indicative of the randomly selected option and 7 balls indicative of the other option (hereafter, simply correct and incorrect balls). Unobserved obtain a single draw from an urn containing 14 (18 and 12) correct balls and 7 (3 and 9) incorrect balls in the first (second and third) repetition, respectively. In each of the first seven decision periods, one observed and one unobserved chooses one of the two options. In the last decision period, only the remaining unobserved makes a choice. Assignments to decision periods are random. From the second decision period on, players may condition their choices on the choices made by observed in previous decision periods. Participants receive 0.4 Euro for a correct prediction and 0.1 Euro otherwise.

The second part of the experiment is identical to the first one except that each unobserved makes 8 choices and each observed makes between 1 and 7 choices in each of the three repetitions of the game. In the first decision period, all 15 players choose one of the two options. The choice of one observed is randomly selected to be made public at the beginning of the next period and this player stops from making predictions. In the second decision period, all remaining 14 players choose one of the two options. The choice of one observed is randomly selected to be made public at the beginning of the next period and this player stops from making predictions. And so on, until the last decision period where all unobserved choose one of the two options. For each participant, only one randomly selected prediction is paid in each repetition.

The third part of the experiment is identical to the second one except that each choice is made for both realizations of the private signal. Players are informed of the payoff-relevant realization of their private signal after having made their last choice.

Finally, the fourth part of the experiment collects the largest number of contingent choices per repetition of the game. Though observed make on average the same number of contingent choices as in the previous part, unobserved choose one of the two options for each quality and realization of the private signal in each decision period. They are informed of the payoff-relevant quality and realization of their private signal after having made their last choice. Other differences with the previous part include the six repetitions of the game and the fact that participants receive 12 Euro for a correct prediction and 3 Euro otherwise.3

Observed as Computers or Humans

We use preprogrammed computers as observed players in half the experimental sessions. Unobserved, on the other hand, are always embodied by human participants. In sessions where unobserved sequentially learn from computers, they do so without knowing the computers' strategy. The third main ingredient of our experiment serves two purposes. First, we facilitate social learning for the unobserved in sessions where they observe the choices of computers since the latter behave in a simple deterministic way as they adopt the Bayesian rational strategy (see Section 3.2.1). This exogenous variation in behavioral uncertainty enables us to check one of [START_REF] Weizsäcker | Do We Follow Others when We Should? A Simple Test of Rational Expectations[END_REF] conclusion according to which participants make worse informational inferences in situations where public information is less clear. Second, in sessions where computers act as observed players, we can perfectly identify the empirically optimal action for unobserved at each decision node.

Treatments and Procedures

The experiment consists of the Computer-Human and Human-Human treatments. The two options from which one was randomly selected at the beginning of a repetition were labeled 'BLUE' and 'GREEN'. Option 'BLUE' had a 11/20 probability to be selected and option 'GREEN' had a 9/20 probability to be selected. We conducted three sessions in each treatment. Most participants were students at the Friedrich Schiller University of Jena, and a few were students at the University of Applied Sciences Jena. The experimental sessions took place at the Experimental Laboratory of the Max Planck Institute of Economics (ELMPIE) in Jena, and participants were invited using the ORSEE recruitment system [START_REF] Greiner | An Online Recruitment System for Economic Experiments[END_REF]. Each session in the Computer-Human and Human-Human treatment involved 9 and 16 participants respectively, one participant being randomly selected to serve as the experimental assistant.

At the start of each session in the Human-Human treatment, experimenters demonstrated the optionselection procedure to small groups of participants. An experimenter shuffled a deck of 20 cards and laid them down on a table with the back of the cards facing the assistant. 11 cards had a blue front and 9 cards had a green front. The assistant then picked 1 card out of the 20 cards, the front color of the picked card determining the randomly selected option. Experimenters also showed to participants how the order of predictions was randomly determined in the first part of the experiment.

After the two demonstrations, paper instructions for part 1 were distributed and participants were given time to read them once at their own pace. Instructions were then read aloud, participants learned about their role (observed or unobserved ), and they answered a few control questions. Experimenters checked participants' answers, and they explained mistakes privately to participants whenever needed. After that, part 1 was ran following the "balls and urns" procedure of [START_REF] Anderson | Information Cascades in the Laboratory[END_REF]. In each of the three repetitions of the social learning game, participants were asked to fill in a form with the realization of their private signal, the choices they observed and the choice they made, and, once known, the option selected at random by the assistant.

The second part of the session was computerized. Electronic instructions detailed the course of the second part, and illustrations were provided concerning the random draw of the private signal, the implementation of choices and the feedback at the end of each repetition. A short summary of the instructions was read aloud. After that, the three repetitions of part 2 were ran.

The third and fourth parts of the session were conducted similarly to the second one except that short paper instructions replaced the electronic instructions. Participants were then asked to report their month and year of birth, their gender, and their academic major. Finally, participants privately retrieved their earnings.

Sessions in the Computer-Human treatment followed the same procedure except for the presence of preprogrammed computer algorithms. At the end of the session, participants had the possibility to earn 10 additional Euro by correctly identifying the strategy of computer algorithms. 4Table 1 summarizes our experimental treatments.

Human-Human

Computer-Human treatment treatment Notes: In each column, the number of choices per participant is reported for the four different parts separately. In the last three parts, averages are reported for observed.

Earnings are stated in Euro and they include a show-up fee of 5 Euro which corresponds to twice the usual amount due to lengthy sessions. Earnings in the Computer-Human treatment do not include the 10 Euro earned by correctly identifying the strategy of computer algorithms.

Table 1: Experimental Design

Theoretical Considerations

In this section, we provide a formal description of our social learning game and the predictions of a series of behavioral models. First, we consider the full rationality model. The standard predictions are mainly derived to describe the behavior of the computer players in the Computer-Human treatment as the latter follow the Bayesian rational strategy. Second, we extend the standard approach by allowing noisy optimizing behavior while maintaining the internal consistency of rational expectations. The quantal response equilibrium approach has been considered in past studies as a first good approximation to actual behavior in experimental social learning games (see especially [START_REF] Goeree | Self-Correcting Information Cascades[END_REF], and we agree that the introduction of a random component in decision-making is a reasonable starting point. Still, the existing experimental literature has also established that the main regularities observed in laboratory cascades are not captured in a fully satisfactory way by the quantal response equilibrium. We therefore extend noisy best reply with rational expectations in two directions. On the one hand we allow for disequilibrium beliefs as argued by [START_REF] Kübler | Limited Depth of Reasoning and Failure of Cascade Formation in the Laboratory[END_REF], and on the other hand we consider an equilibrium approach with non-Bayesian updating of beliefs. In the last part of this section, we illustrate the two final approaches to show that both have the potential to capture the full diversity of experimental regularities.

A Rich-Information Social Learning Game

There are two payoff-relevant states of Nature (henceforth states)-state B and state G, and two possible actions-"predict state B" simply denoted by B and "predict state Nature moves first and chooses a state which remains unknown to the players. Each player is then endowed with a private signal which corresponds to the realization of a random variable, denoted by sn , with support S = {b, g}, and whose distribution depends on the state. Conditional on the state, private signals are independently distributed across players. In state B (resp. state G), player n receives signal b (resp. signal g) with probability p < q n < 1 and signal g (resp. signal b) with probability 0 < 1 -q n < 1 -p. We refer to q n as player n's signal quality. There are two groups of players. Observed receive private signals of medium quality q n = 14/21 meaning that the signal indicates the true state of Nature in two thirds of the cases. Private signals of unobserved have quality q n ∈ {12/21, 14/21, 18/21}.

Time is discrete and, in each period t = 1, 2, . . . , T , k ≤ N players simultaneously choose an action. The action of exactly one observed is then publicly revealed at the beginning of the next decision-making period. The observed whose action is made public does not act in any subsequent period. Accordingly, players who act in period t ∈ {1, . . . , T } observe the history h t = (a 1 , . . . , a t-1 ) ∈ H t = {B, G} t-1 where a τ is the action which is public in period τ ∈ {1, . . . , t -1} and h 1 = ∅. Payoffs are realized at the end of period T such that observed receive the payoff from the action which is made public and unobserved receive the payoff from their action in exactly one randomly chosen period. Since payoff externalities are absent and we abstract from social preferences the game is similar to a social learning game where each player is randomly assigned to exactly one period and makes a once-in-a-lifetime decision.

For each player a belief for period t is given by the mapping µ t : {b, g}×H t → [0, 1] where µ t (s n , h t ) denotes the conditional probability assigned to state B given signal s n ∈ {b, g} and observed history h t ∈ H t . A behavioral strategy for period t is a mapping σ t (s n , h t ) : {b, g} × H t → [0, 1] where σ t (s n , h t ) denotes the probability that the player takes action B given signal s n and history h t . We say that player n follows private information at history h t when taking action B with s n = b or taking action G with s n = g. Alternatively, player n contradicts private information at history h t when taking action B with s n = g or taking action G with s n = b.

Predictions

Perfect Bayesian Rationality

We first assume that players are Bayes-rational, and that Bayesian rationality and the structure of the game are commonly known. Under these assumptions the game has a unique rationalizable outcome. In period 1, observed follow private information. Consequently, in period 2, observed follow private information only if the history is G and they choose action B independently of their private information if the history is B. Hence, after action B is revealed at the beginning of period 2, an information cascade starts in which all subsequent observed choose action B. 6 In a cascade, the beliefs of two players are identical when endowed with the same private signal. A similar reasoning implies that an information cascade in which all observed choose G starts at history GG. More generally, a B-cascade (resp. G-cascade) starts after one public B (resp. two G) not canceled out by previous public actions. The only history which does not lead to an information cascade is the history GBGB . . . GB, and the probability that no cascade has started by period 2k + 1 decreases exponentially.

Unobserved with signal quality q n = 12/21 follow private information only in the first period and in odd periods following history GB . . . GB, and otherwise they choose the action which has been publicly revealed most frequently. Obviously, unobserved with signal quality q n = 14/21 behave similarly as observed. Finally, unobserved with signal quality q n = 18/21 follow private information at all histories.

Notice that Bayes-rational players have a correct perception of the expected value of each action which implies that they follow (contradict) private information provided their belief is indicative of the same action as their signal with probability at least (at most) 1/2.

Almost Bayesian Rationality

We now derive predictions for the quantal response equilibrium model (McKelvey and[START_REF] Mckelvey | Quantal Response Equilibria for Extensive Form Games[END_REF]. We rely on the common logit specification given by

σ λ t (s n , h t ) = 1 + e λ (1 -2 µt(sn,ht)) -1
where λ ≥ 0 denotes players' sensitivity to payoff differences. Choices are random if λ = 0, they become more responsive to beliefs as λ increases and players best respond to beliefs as λ → ∞.7

Assuming that the sensitivity to payoff differences is commonly known, player n's belief at history h t with signal s n is given by

µ t (s n , h t ) =   1 + 1 -p p Pr (s n = s n | ω = G) Pr (s n = s n | ω = B) τ <t sτ ∈S Pr (s τ | G) σ λ τ (a τ | s τ , h τ ) sτ ∈S Pr (s τ | B) σ λ τ (a τ | s τ , h τ )    -1
where s τ and a τ are respectively the signal realization and the action of an observed whose action has been made public. The dynamics of beliefs and choices have the following properties: First, each public action conveys a noisy signal about the state of Nature as σ λ t (s n , h t ) strictly increases with µ t (s n , h t ). Actions reveal less information the noisier choice probabilities are (the smaller λ is). Second, since each action conveys some information each player is more likely to contradict than to follow private information after observing a large number of similar choices which point in the opposite direction of private information. Accordingly, "cascades" emerge and they do so no sooner than predicted by perfect Bayesian rationality. Even unobserved with high signal quality become more likely to contradict private information. Third, since no action is chosen with certainty (0 < σ λ t (s n , h t ) for each s n , q n , h t ) a cascade once started is broken with strictly positive probability in each subsequent period. Players who break a cascade are more likely to have a contradictory signal. Fourth, the longer a cascade the more likely players are to contradict private information. Fifth, cascades are self-correcting meaning that after the break of an incorrect cascade the new cascade which emerges is often a correct one.8 Finally, players have a correct perception of the available information meaning that players contradict private information with probability at least 1/2 if and only if the expected payoff from contradicting is larger than the expected payoff from following private information.

Limited Bayesian Rationality

So far we assumed that players' behavioral strategies are commonly known (or can be derived from commonly known traits). In general, a player's belief with signal s n at history h t is given by

µ t (s n , h t ) =   1 + 1 -p p Pr (s n | G) Pr (s n | B) τ <t sτ Pr (s τ | G) στ (a τ | s τ , h τ ) sτ Pr (s τ | B) στ (a τ | s τ , h τ )   -1
where Pr (

s n = b | B) = Pr (s n = g | G) = q n and
(B | s τ , h τ ) = σ τ (s τ , h τ ) and σ τ (G | s τ , h τ ) = 1 -σ τ (s τ , h τ ).
The formation of correct beliefs through iterative reasoning is highly demanding from a cognitive point of view. We now discuss the dynamics of beliefs and choices under limited strategic thinking. We consider the level-k model [START_REF] Stahl | On Players' Models of Other Players: Theory and Experimental Evidence[END_REF] according to which each player is one of a (potentially infinite) number of types (L 0 , )L 1 , L 2 , . . .. Although players are heterogeneous, each player's type is drawn from a common distribution. Type L k anchors its beliefs in a non-strategic type L 0 which captures instinctive responses to the game and adjusts them via thought-experiments with iterated best responses. Concretely, type L k noisy best responds to a belief formed under the assumption that other players are of type L k-1 . We assume that the structure of the game and quantal response functions are iteratively know up to level k for type L k . Finally, we assume that type L 0 noisy best responds to private beliefs only.9

Consider first the case where players best respond to beliefs. L 0 players ignore public information and best respond to private information. L 1 players therefore believe that each action perfectly reveals the underlying signal and they form beliefs according to a counting rule.10 It is easy to see that observed L 1 players and unobserved L 1 players with signal quality q n ∈ {12/21, 14/21} mimic the behavior of Bayes-rational players. However, the beliefs of L 1 players increase (decrease) with the number of public B (G) choices which eventually leads unobserved L 1 with high signal quality q n = 18/21 to contradict private information. The latter is not true for L k players such that k > 1. Indeed, beliefs and choices of L 2 players are the same for all signal qualities as those of Bayes-rational players.

Under noisy best reply, L 0 players follow private information with probability at least 1/2, but they occasionally make mistakes. Accordingly, L 1 players infer from each public action a signal which is noisier than the underlying private signal which leads them to form beliefs according to a counting rule with discounting. L 1 players now follow private information (with probability at least 1/2) at more histories. Contrary to almost Bayes-rational players, L 1 players' beliefs become extreme more quickly. For higher types differences are more subtle. For instance, L 2 players correctly infer that the second observed action conveys less (more) information if it matches (differs from) the first action. However, if the first three observed actions match they infer too little information from the third action assuming wrongly that the third player did not take into account the reduced information conveyed by the second action.

Notice that L k players have an incorrect perception of the expected value of actions unless an arbitrarily large fraction of the population is of type L k-1 .

Non-Bayesian Rationality

In the above behavioral models players update probabilities in a Bayesian way. However, individuals exhibit considerable heterogeneity in the way they revise their expectations in light of the same information (see for instance [START_REF] Delavande | Measuring Revisions to Subjective Expectations[END_REF]. [START_REF] March | Adaptive Social Learning[END_REF] shows that in social learning environments where knowledge about others and the information structure has to be acquired, alternative updating rules may be payoff-enhancing.

We here discuss the dynamics of beliefs and choices under an alternative model where players update beliefs in a non-Bayesian way. 11 Player n of type β > 0 forms beliefs according to

µ β t (s n , h t ) =   1 + 1 -p p Pr (s n | G) Pr (s n | B)   τ <t sτ Pr (s τ | G) στ (a τ | s τ , h τ ) sτ Pr (s τ | B) στ (a τ | s τ , h τ )   β    -1
.

Players are Bayesian with β = 1, they underweight public relative to private information if β < 1, and overweight public relative to private information if β > 1. Types are drawn from a common distribution W . We assume that the distribution is commonly known such that assessed choice probabilities στ (a | s τ , h τ ) = ∞ 0 σ β τ (a | s τ , h τ ) W (dβ) are correct averages across the distribution. In period 1, all types follow private information. In later periods, behavior is characterized by a cutoff β * (h t , q n ) such that players of type β < β * (h t , q n ) follow private information of quality q n at history h t whereas players of type β > β * (h t , q n ) contradict private information of quality q n at h t . In particular, players with sufficiently small β < 1 follow private information even with the low signal quality also at histories other than h 1 = ∅ or h t = GBGB . . . GB. On the other hand, players with sufficiently large β > 1 contradict private information even with the high signal quality at some histories. Since the cutoff is strictly increasing in the correctness of the signal, players respond to incentives but not perfectly so.

In general, whenever the (observed) population contains a sufficient mass of underweighters information cascades start later, but each action conveys more information and beliefs become more extreme. Hence, even underweighters may eventually contradict private information with the high signal quality. In fact, non-Bayesian players may contradict private information less often than Bayesrational players when endowed with a signal of low or medium quality and after short sequences of identical choices, and, at the same time, they may follow private information more often than Bayesrational players when endowed with a high signal quality and after long sequences of identical choices.

Illustrations

Figure 1 and 2 plots the expected payoff of contradicting private information, mean pay|contradict, against the probability to contradict private information, prop contradict, in the level-k and non- 11 The argument is derived from [START_REF] March | Behavioral Social Learning[END_REF].

Bayesian equilibrium model, respectively.12 The expected payoff of contradicting private information is calculated under the assumption that observed adopt the Bayesian rational strategy (as they do in the Computer-Human treatment). Each marker reflects a distinct situation (s n , q n , h t ) which occurs with strictly positive probability. Green (blue, red) markers illustrate predictions for the medium (low, high) signal quality. Situations where the number of observed actions not favored by the signal minus the number of observed actions favored by the signal is strictly greater than 2 are highlighted with darker markers.

For both figures, the left panel corresponds to λ = 4 and the right panel corresponds to λ = 15. Figure 1 illustrates the predictions in the level-k model. The first, second, third and fourth row shows the predicted choice probabilities for L ∞ , L 0 , L 1 and L 2 players. Figure 2 illustrates the predictions in the non-Bayesian equilibrium model. The first, second, third and fourth row shows the predicted choice probabilities for non-Bayesian players with β = 1/3, β = 2/3, β = 3/2, and β = 3.

Figure 1 shows that L ∞ as well as L 2 players with a sufficiently large λ respond appropriately to the underlying incentives. In contrast, L 0 players as well as L 1 and L 2 players with sufficiently small λ suboptimally follow private information when endowed with a private signal of low or medium quality. Finally, L 1 players naïvely herd when endowed with a high quality signal as they contradict private information if sufficiently many actions are observed which point in the opposite direction of their private signal.

Figure 2 shows that β < 1 players suboptimally follow private information with the low and medium signal quality. In contrast, β > 1 players suboptimally contradict private information with the high signal quality.

In summary, the level-k model and the non-Bayesian equilibrium model have the potential to predict a variety of regularities at the aggregate level. They may predict overweighting of private information with a low or a medium quality signal as well as naïve herding with the high signal quality. Notice that only the non-Bayesian equilibrium model predicts the occurrence of both phenomena at the individual level assuming that sufficiently many players underweight public information (which is not the case in our illustration). Of course, the two behavioral models are fully specified only once the distribution of types is known. With the help of our large experimental dataset, we aim at reliably uncovering the type distributions of those two models.

Results

We first examine some aggregate properties of our data, and then we measure how successful participants are in learning from others.

Descriptive Statistics

We start by presenting some summary statistics concerning the experimental choices in the first part of the experiment. Table 2 reports the proportion of herding choices when the size of the majority of previous choices against the private signal either equals one or is strictly greater than one. A majority of previous choices which point in the opposite direction of the private signal is called a contradicting herd. In line with the existing literature, observed give too much weight to their private information relative to the information conveyed by previous choices. Though the size of the contradicting herd Even with a low quality signal unobserved never fully herd, and herding behavior is again comparable in both treatments. Still, the most surprising result is the large proportion of herding choices when unobserved are endowed with a high signal quality and they observe a contradicting herd of size at least two. In the latter cases, unobserved usually do not understand the value of the available information in the Computer-Human treatment. Few experimental choices have been collected in the first part of the experiment since this part essentially served the purpose of letting participants gain experience with the social learning task (we collected 72 and 135 choices in the Computer-Human and Human-Human treatment, respectively). Moreover, a substantial proportion of choices seem to be the consequence of confusion. For example, observed contradict private information in about 13 percent of the cases where the majority of previous choices points either in no direction or in the same direction as the private signal. All remaining statistical analyzes rely exclusively on experimental choices made in the last three parts of the experiment. Figure 3 and 4 plots the size of the contradicting herd against the estimated probability of contradicting private information in the Computer-Human and Human-Human treatment, respectively. 13Probabilities have been obtained by estimating a logit regression model for each treatment and each role separately (Appendix 1 reports the regression results). The dependent variable is a dummy variable which takes value one if the choice contradicts private information and zero otherwise, and the explanatory variables are interaction effects between dummies for the sizes of the contradicting herd (from -7 to 7) and dummies for the signal qualities (we also included the dummy observed in the Human-Human treatment). The numbers of observations for each signal quality are shown on top of the regression lines. Figure 3 confirms that unobserved when endowed with a private signal of low or medium quality fall prey to the 'overweighting-of-private-information' bias in the Computer-Human treatment. The probability to contradict private information increases with the size of the contradicting herd until it reaches a plateau at about 0.8 for the low quality and 0.7 for the medium quality. Accordingly, unobserved do not sufficiently herd in situations where it is optimal to do so. Most remarkably, the tendency of unobserved to herd naïvely when endowed with a private signal of high quality is also confirmed. As with lower signal qualities, the probability to contradict private information increases with the size of the contradicting herd, and it reaches one half. Though a contradicting herd of size 5 is not stronger evidence against private information than a contradicting herd of size 2, unobserved seem to believe that every imitative choice is informative to some extent.

Size of the

Figure 4 shows that unobserved make similar choices in the Human-Human treatment than in the Computer-Human treatment, and that observed seem more willing to herd than unobserved who are endowed with the medium signal quality. Additionally, the comparison of the left half of the two figures suggests that unobserved doubt more the correctness of their private signal when the latter is confirmed by human choices than when it is confirmed by computer choices.

In summary, we observe the cascade phenomenon systematically reported in social learning experiments. Laboratory cascades occur among observed participants. Also in line with the existing literature, we find that observed are more willing to follow a contradicting herd the bigger the herd is. From the perspective of perfect Bayesian rationality, observed behave as if they discount the evidence conveyed by choices which are not part of an information cascade and they do not fully discount the evidence conveyed by cascade choices. The relative frequency with which they engage in cascade behavior equals the one usually reported in the literature, and this frequency is almost identical in the first part and the last three parts of the experiment (0.75 and 0.77 for contradicting herds of size at least 2, respectively). Thus, observed choices elicited under the direct-response and the (partial) strategy method are comparable.

Moreover, the qualitative behavior of observed and unobserved participants is similar when the latter are endowed with a weak or medium signal quality. However, unobserved with a medium signal quality exhibit an even stronger 'overweighting-of-private-information' bias than observed, and even more so in the Computer-Human treatment. This behavioral difference between the two groups of participants might result from the mistaken disposition of some unobserved to distinguish weak quality from medium quality choices when observing contradicting herds of size at least 2.

Finally, unobserved participants herd naïvely when endowed with a high signal quality. In the Human-Human treatment, the majority of high quality choices disregard the private signal and follow others as soon as the contradicting herd reaches a size of 4.

The Success of Social Learning

Choice frequencies are reliable indicators of the success of social learning only in the Computer-Human treatment. And even in the latter treatment, it remains unknown how large are the parts of earnings that participants forgo when they rely too much on private signals of weak or medium quality or when they herd naïvely with high signal quality. Building on [START_REF] Weizsäcker | Do We Follow Others when We Should? A Simple Test of Rational Expectations[END_REF], we now measure how successful participants are in learning from others. To do so, we first assess the value of the available actions for unobserved participants which incidentally provides additional information on the behavior of observed participants. Once the value of actions is available, we determine the extent to which unobserved participants respond to the underlying incentives in situations where they should contradict weak or medium quality signals and in situations where they should follow the high quality signal.

The Value of Contradicting Private Information

In the Computer-Human treatment, the expected value of contradicting private information in period t when endowed with private signal s n of quality q n ∈ {12/21, 14/21, 18/21} and observing history h t is given by

mean pay CH |contradict(s n , q n , h t ) =                  1 + p 1-p qn 1-qn τ <t q O σ * (aτ |b,hτ )+(1-q O ) σ * (aτ |g,hτ ) (1-q O ) σ * (aτ |b,hτ )+q O σ * (aτ |g,hτ )   -1 if s n = b   1 + 1-p p qn 1-qn τ <t (1-q O ) σ * (aτ |b,hτ )+q O σ * (aτ |g,hτ ) q O σ * (aτ |b,hτ )+(1-q O ) σ * (aτ |g,hτ )   -1 if s n = g,
where σ * is the Bayesian rational strategy and q O = 14/21 is the signal quality of the observed players (products in the above equation are assumed equal to one in the first period).

In the Human-Human treatment, we rely on a modified version of [START_REF] Weizsäcker | Do We Follow Others when We Should? A Simple Test of Rational Expectations[END_REF] counting technique to estimate the expected value of contradicting private information.14 Accordingly,

mean pay HH |contradict(s n , q n , h t ) =                  1 + p 1-p qn 1-qn τ <t q O P r(aτ |b,hτ ,B)+(1-q O ) P r(aτ |g,hτ ,B) (1-q O ) P r(aτ |b,hτ ,G)+q O P r(aτ |g,hτ ,G)   -1 if s n = b   1 + 1-p p qn 1-qn τ <t (1-q O ) P r(aτ |b,hτ ,G)+q O P r(aτ |g,hτ ,G) q O P r(aτ |b,hτ ,B)+(1-q O ) P r(aτ |g,hτ ,B)   -1 if s n = g,
where P r (a τ | s n , h τ , ω) is the relative frequency with which action a τ is chosen across all observed choices where the private signal is s n , the history is h τ and the state of Nature is ω ∈ {B, G} (products in the above equation are assumed equal to one in the first period). The empirical value of contradicting private information in the Human-Human treatment is a consistent estimate of the true expected value of contradicting private information whatever the behavioral model of observed players. Still, the precision with which mean pay HH |contradict(s n , q n , h t ) estimates the underlying expected value depends on the number of observations with identical (s n , q n , h t ). When few observations are available, the estimate might be far from its expected value. But relying only on situations (s n , q n , h t ) with many observations implies that relatively few values of mean pay HH |contradict(s n , q n , h t ) are available. Unless otherwise specified, statistical analyzes which use the expected value of contradicting private information in the Human-Human treatment exclude situations which appear in two or less distinct repetitions of the social learning game (for a total of 36 repetitions). 15We now compare the incentives to contradict private information for the unobserved participants in the two treatments. Table 3 summarizes the distribution of the empirical value of contradicting private information for each signal quality, simply denoted by mpc, in each treatment. The two distributions are largely comparable for each signal quality which confirms that the behavior of observed participants is reasonably close to the behavior of (almost) Bayes-rational players. As expected, the incentives to follow private information with a high signal quality are quite substantial and more so in the Computer-Human treatment. In contrast, when endowed with a low or medium signal quality unobserved should contradict private information in some situations.

To better appreciate the benefits associated with contradicting low or medium quality private information when observed choices point in the opposite direction, Table 4 reports the median values of mpc as a function of the size of contradicting herds. Contrary to the Computer-Human treatment, incentives to follow others increase with the size of the contradicting herd in the Human-Human treatment, until the herd size equals 4. This observation confirms that observed participants do Figure 5 and 6 plots the empirical value of contradicting private information against the proportion of choices which contradict private information in the Computer-Human and Human-Human treatment, respectively. For each situation (restricted to situations which appear in at least three distinct repetitions of the game in the Human-Human treatment), the figures contain a bubble with x-value the empirical value of contradicting private information and y-value the proportion of choices which contradict private information, and the bubble's size reflects the number of observations. Each figure also includes a regression line for each signal quality (details about the regressions are to be found in Appendix 2).16 When endowed with a private signal of low or medium quality, unobserved participants respond reasonably well to the underlying incentives in each treatment.

In the Human-Human treatment, the average participant follows private information only in situations where it is optimal to do so. Both the blue and green regression lines are basically Sshaped lines through (0.5, 0.5). We can never reject the hypothesis that participants exhibit a correct perception of the value of contradicting private information since the vertical distance between the regression line and (0.5, 0.5) is not significant (p-value equals 0.348 and 0.914 with low and medium quality). Moreover, participants use their information with identical success in situations where they should follow others than in situations where they follow their private signal. In situations where the empirical of contradicting private information is strictly larger than 0.5, the relative frequency of optimal choice is 0.794 and 0.703 with a low and medium signal quality, respectively. In situations with similar incentives but where the empirical value of contradicting private information is strictly smaller than 0.5, the relative frequency of optimal choice is 0.776 and 0.713 with a low and medium signal quality, respectively.17

In the Computer-Human treatment, participants follow others with more than probability 0.5 in situations where the empirical value of contradicting private information is larger than 0.560 and 0.532 with a low and medium signal quality, respectively. The vertical distance between the regression line and (0.5, 0.5) is significantly different from zero with a low signal quality (p-value equals 0.002) which confirms the existence of the 'overweighting-of-private-information' bias. However, we cannot reject the hypothesis that participants exhibit a correct perception of the value contradicting private information with medium quality (p-value equals 0.208). The slight tendency of participants to mistakenly discount the evidence conveyed by the predictions of computers is confirmed by their relative success of learning from others. In situations where the empirical value of contradicting private information is strictly larger than 0.5, the relative frequency of optimal choice is 0.755 and 0.607 with a low and medium signal quality, respectively. In situations with similar incentives but where the empirical value of contradicting private information is strictly smaller than 0.5, the relative frequency of optimal choice is 0.888 and 0.879 with a low and medium signal quality, respectively.18 To our surprise, participants are less successful in following computer than human choices in situations where it is empirically optimal to do so.

The fact that participants assess the expected value of actions only imperfectly is confirmed by a series of regression analyzes. These analyzes control for the value of contradicting private information to determine whether non payoff-relevant aspects of the situation influence participants' behavior. We find that, when endowed with a private signal of low or medium quality, unobserved contradict private information to a larger extent the bigger the size of the contradicting herd though the empirical value of contradicting private information is kept constant (details are available from the authors upon request). In conclusion, participants mistakenly adopt a sort of counting rule as they do not discount sufficiently the evidence conveyed by late cascade choices.

In contrast, when endowed with a private signal of high quality, unobserved participants respond rather badly to the underlying incentives in each treatment. Both figures show that once the size of the contradicting herd is big enough the average participant follows others though incentives clearly advise to follow private information. Our analysis of the social learning success establishes that participants forgo large parts of their earnings when herding naïvely. Indeed, in each treatment, unobserved herd naïvely on average in situations where the empirical value of contradicting private information approximately equals 1/3: In such situations, the evidence conveyed by the observable choices is so weak that the private information is correct more than twice as often as it is wrong. In situations with identical incentives but in the absence of big contradicting herds, participants largely follow private information whatever the quality of the private signal.

Conclusion

Since the seminal paper of [START_REF] Anderson | Information Cascades in the Laboratory[END_REF], the economic literature on experimental social learning games investigates whether participants are capable of making rational inferences in controlled settings. Notwithstanding the tendency to overweight private information relative to public information, the literature concluded that participants generally use their information efficiently and follow others only in warranted situations.

Our results severely undermine the robustness of this conclusion. We show that participants forgo large parts of earnings by following others in situations where they should contradict them. Our new evidence therefore suggests that participants are prone to a 'social-confirmation' bias and it gives support to the argument that they naïvely believe that each observable choice reveals a substantial amount of that person's private information. At the aggregate level, participants' behavior seems best describe by a counting rule which discounts the early cascade choices and does not fully discount the late cascade choices.

Thanks to the large amount of data collected at the individual level, we have classified the social learning behavior of each of our unobserved participants. Though a substantial fraction of participants (almost) always follow their private information, we also find a substantial fraction of socialconformists who follow contradicting herds of any size (details are available from the authors upon request). Clearly, some unobserved participants drew opposite conclusions from the same evidence. Further experimental work on social learning should dig deeper into this heterogeneity in informational inferences.

Appendix 2: Regression Results for the Success of Social Learning Table 5 shows the regression results for the Computer-Human treatment. The logit regression includes a constant, the empirical value of contradicting private information mean pay CH |contradict, dummy variables for the low and high signal qualities, and interaction effects between mean pay CH |contradict and the signal quality dummies. The regression uses all choices made in the fourth part of the experiment with robust standard errors clustered at the individual level. 6 shows the regression results for the Human-Human treatment. To correct for the measurement problem (mean pay HH |contradict is an imperfect indicator of the expected value of contradicting private information), we relied on an instrumental variable (IV) approach. The dataset of the Human-Human treatment has been partitioned into two subsets, the empirical value of contradicting private information has been computed for each subset to obtain mean pay HH |contradict 1 and mean pay HH |contradict 2 , and regression terms with the variable mean pay HH |contradict 1 have been instrumented by the corresponding terms with the variable mean pay HH |contradict 2 . The IV regression includes a constant, linear, squared and cubed terms of mean pay HH |contradict 1 , dummy variables for the low and high signal qualities, and interaction effects between the linear, squared and cubed terms of mean pay HH |contradict 1 and the signal quality dummies. 19 The regression uses robust standard errors clustered at the individual level, and choices made in the fourth part of the experiment are included whenever the underlying situation is observed in both subsets.

Estimate

Estimate

  G" simply denoted by G. Nature chooses state B with probability p = 11/20. The finite set of players is {1, . . . , N } with generic element n. For all players, action B has vN-M payoffs u (B, B) = 1 and u (B, G) = 0, and action G has vN-M payoffs u (G, B) = 0 and u (G, G) = 1. 5

Figure 2 :

 2 Figure 1: Predicted Probability to Contradict Private Information in the Level-k Model

Figure 4 :

 4 Figure 4: Estimated Probability to Contradict Private Information in the Human-Human Treatment

Figure 5 :Figure 6 :

 56 Figure 5: Proportion of Choices which Contradict Private Information in the Computer-Human Treatment

  στ (a τ | s τ , h τ ) are the choice probabilities for signal s τ and history h τ assessed by player n. If strategies are commonly known these probabilities coincide with probabilities σ τ (a τ | s τ , h τ ) where (with a slight abuse of notation) σ τ

Table 2 :

 2 Proportion of Herding Choices in the First Part of the Experiment (number of observations in parentheses)

Table 3 :

 3 Empirical Value of Contradicting Private Information in Each Treatment not systematically engage in cascade behavior after few contradicting choices but almost always do so after many contradicting choices. For contradicting herds larger than two, incentives advise to contradict private information. However, the expected gain from contradicting medium quality private information is small which indicates that low quality choices are more appropriate to identify the 'overweighting-of-private-information' bias.

	Quality of the					Percentile
	private signal	1%	5%	10%	25%	50%	75%	90%	95%	99%
	mpc CH 0.186 0.186 0.235 0.235 0.380 0.647 0.711 0.711 0.711
	Low
	mpc HH 0.183 0.183 0.190 0.209 0.424 0.681 0.706 0.715 0.715
	mpc CH 0.133 0.133 0.170 0.170 0.379 0.550 0.550 0.621 0.621
	Medium
	mpc HH 0.116 0.130 0.135 0.149 0.318 0.526 0.615 0.626 0.657
	mpc CH 0.048 0.048 0.064 0.064 0.120 0.289 0.289 0.353 0.353
	High
	mpc HH 0.042 0.047 0.050 0.070 0.145 0.322 0.347 0.358 0.389
	Quality of the		Size of the contradicting herd
	private signal		1	2	3		≥ 4
			mpc CH 0.647 0.647 0.647	0.647
	Low
			mpc HH 0.591 0.681 0.705	0.706
			mpc CH 0.550 0.550 0.550	0.550
	Medium
			mpc HH 0.490 0.587 0.614	0.615

Table 4 :

 4 Median Empirical Value of Contradicting Private Information and Size of the Contradicting Herd in Each Treatment 4.2.2 Overweighting of Private Information and Naïve Herding

Table 5 :

 5 Probability to Contradict Private Information in the Computer-Human treatment Table

  SE mean payHH |contradict 1 -3.299*** 0.803 mean pay HH |contradict 1 2 11.252*** 2.172 mean pay HH |contradict 1 3 -7.761*** 1.506 mean pay HH |contradict 1 × low -1.220 0.924 mean pay HH |contradict 1 2 × low 2.322 2.319 mean pay HH |contradict 1 3 × low -1.217 1.578 mean pay HH |contradict 1 × high 2.368*** 0.504 mean pay HH |contradict 1 2 × high

		-2.734	1.962
	mean pay HH |contradict 1 3 × high	0.381	1.627
	low	0.156	0.106
	high	-0.210*** 0.051
	constant	0.301*** 0.081
	Observations	6144
	R-squared	0.307

*** significant at the 1 percent level.

Table 6 :

 6 Probability to Contradict Private Information in the Human-Human treatment

We rely on a modified version of[START_REF] Weizsäcker | Do We Follow Others when We Should? A Simple Test of Rational Expectations[END_REF] counting technique to estimate the value of contradicting private information.

Even in the last part of the experiment, our setting relies only on the partial strategy method. Indeed, for relatively long decision-making sequences, the implementation of the full strategy method seems impractical in information cascade experiments. Participants would have to submit hundreds of predictions without ever becoming familiar with the environment. Moreover, long decision-making sequences are preferable since the main regularity observed in cascade experiments is the correlation of length and strength of laboratory herds[START_REF] Kübler | Limited Depth of Reasoning and Failure of Cascade Formation in the Laboratory[END_REF].

One pilot session was conducted in each treatment. We do not include these two pilot sessions since their structure is slightly different from the sessions reported here.

Assuming well-behaved non-expected utility maximizers does not significantly alter the predictions of our behavioral models.

Information cascades only develop in the observed sequence.

Similar results are obtained when relying on regular quantal response functions[START_REF] Goeree | Regular Quantal Response Equilibrium[END_REF].

See Goeree, Palfrey, Rogers, and McKelvey (2007) for a more precise characterization of the social learning outcome in quantal response equilibrium.

Assuming that L0 players choose randomly implies that L1 respond to private beliefs and the subsequent analysis carries over with the type hierarchy shifted upwards by one level.

See Eyster and Rabin (2010) for an extensive discussion of this type of behavior in social learning games.

For a given history ht, when endowed with the realization of the private signal g and b the expected payoff of contradicting private information is given by mean pay|contradict (sn = g, ht) = Pr (B | sn = g, ht) and mean pay|contradict (sn = b, ht) = Pr (G | sn = b, ht), respectively.

Note that contradicting herds of negative size are majorities of previous choices which point in the same direction as the private signal.

See Ziegelmeyer, March, and Kruegel (2011) for details.

The qualitative insights of our data analyzes do not change when the requirement on the precision of the estimate is strengthened. The results of those robustness checks are available from the authors upon request.

Only choices made in the fourth part of the experiment are included. Regression analyzes show an absence of significant behavioral change in the course of the experiment. Details are available from the authors upon request.

In situations where they should follow others, the range of the empirical value of contradicting private information is ]0.500, 0.715] and ]0.500, 0.657] with a low and medium signal quality, respectively (see Table3). Thus, the mirrored range below 0.5 which corresponds to situations with relatively weak incentives to follow private information is[0.285, 0.500[ and [0.343, 0.500[ with a low and medium signal quality, respectively. In all situations where the empirical value of contradicting private information is below 0.5, the relative frequency of optimal choice is 0.862 and 0.874 with a low and medium signal quality, respectively.

In situations where they should follow others, the range of the empirical value of contradicting private information is ]0.500, 0.711] and ]0.500, 0.621] with a low and medium signal quality, respectively (see Table3). Thus, the mirrored range below 0.5 which corresponds to situations with relatively weak incentives to follow private information is[0.289, 0.500[ and [0.379, 0.500[ with a low and medium signal quality, respectively. In all situations where the empirical value of contradicting private information is below 0.5, the relative frequency of optimal choice is 0.943 and 0.949 with a low and medium signal quality, respectively.

IV models for discrete outcomes do not deliver point identification of the values of parameters(Chesher,

2010).

The first author gratefully acknowledges financial support from the European Research Council.

Appendix 1: Regression Results for Figure 3 Notes: Robust standard errors clustered at the individual level. Two variables were omitted from the regression for the observed since the latter always followed private information. ***, ** significant at the 1 and 5 percent level, respectively.


E Testa

D Dauvergne
email: d.dauvergne@ipnl.in2p3.fr

A Bräuning-Demian

F Bosch

H Bräuning

M Cheval- Lier

C Cohen

A Gumberidze

S Hagmann

A L'hoir

R Kirsch

C Kozhuharov

D Liesen

P H Mokler

J.-C Poizat

C Ray

J.-P Rozet

Th Stöhlker

S Toleikis

M Toulemonde

P Verma

Electron gas polarization effect induced by heavy H-like ions of moderate velocities channeled in a silicon crystal

Keywords: 61.85.+p; 34.70.+e Channeling, highly charged ions, Radiative Electron Capture, REC, wake effect

Electron gas polarization effect induced by heavy H-like ions of moderate velocities channeled in a silicon crystal

Introduction

Highly charged ions of moderate velocities induce a strong perturbation of the electron gas in a solid. This perturbation consists in a dynamic screening of the projectile charge by the electrons of the medium. If the particle propagates much faster than the Fermi velocity of the electrons, a trailing polarization cone takes place, which is due to the collective excitation of the electron gas (wake effect). The wake potential manifests itself through the electric field, acting as the stopping force on the projectile, and is also responsible for a Stark splitting of bound states of an ion, which influences the transport of excited states in solids. For a review on the wake effect, see e.g. refs. [1,2,3] and references therein. In the vicinity of the projectile, the polarization of the electron gas produces a shift, relative to vacuum, of the continuum energy levels in the projectile frame. This reduces the absolute binding energies of electrons in an ion. Echenique et al. [2] estimated this shift, using the linear response theory, to be ( )

0 2 0 / 2 lim v v Z r e Z p p p r ω π ℏ - = - Φ →
, where Φ is the potential, Z p and v are the ion charge and velocity, respectively, Practically, transitions between bound states of an ionic projectile may be affected only by Stark splitting of these states, whereas transitions between continuum states and bound states are modified in energy. Among them, Radiative Electron Capture (REC) of target electrons provides a local probe for the electron gas polarization in a solid. First, since REC consists in the emission of a photon, the energy of which is the sum of the kinetic energy of the target electron in the ion frame, and the binding energy in the final bound state, it gives access to the energy shift (decrease of the emitted photon energy) [3,4].

Second, REC into deeply bound states of a heavy ion is a very localized process (at the scale of the final bound orbital size). So the measurement of absolute REC probabilities is a test for a possible enhancement of the local electron density at the projectile site.

The problem arising is that strong enough perturbations to be measurable are obtained for high Z p /v values, for which REC is hardly observable in solids, because it requires inner-shell vacancies.

Ion channeling leads to a non homogeneous flux inside a crystal, preventing ions from undergoing close impact parameter collisions with target atoms. Thus channeled ions sample mainly the quasi-free valence electron gas, and close interaction with core electrons is substantially attenuated. This allows REC to be the dominant electron capture process, even at low energy. Another interesting feature related to the impact parameter distribution is that detailed information can be obtained by analyzing the shape of REC lines, which depends on the longitudinal momentum distribution of the target electrons in their initial state (Compton profile). In particular this enables to identify the contributions of core electrons and valence or conduction electrons [6]. Finally, the knowledge of the ion flux in the crystal, combined with the precise measurements of the REC line intensities, allows one to evaluate the electron density at the ion site.

Experiment

We used 20 MeV/u U 91+ and 13 MeV/u Pb 81+ ions extracted from the GSI-ESR storage ring, for which Z p v 0 /v values are 3.26 and 3.57, respectively. The process for cooling, deceleration and slow extraction of H-like ions by radiative recombination inside the electron cooler has been described in ref. [7]. During the extraction cycle, a continuous beam of some 10 4 ions per second is sent onto the target with an angular divergence suitable for channeling experiments. The beam impact on the target is less than 3 mm in width, and 7 mm in height. A 9.6 µm thick (111) Si crystal (tilted at 35° for alignment along the <110> axis) was used as a target during the experiment with U 91+ ions. For the Pb 81+ ion experiment, a thin (0.8 µm) (100) silicon crystal was tilted at 45° to allow the same axial orientation. Transmitted ions were charge-and energy-analyzed by a magnetic spectrometer, and detected at the focal point by a 2D-position sensitive particle detector.

X-rays emitted at the target were detected by a 1 cm thick germanium detector at 90° from the beam direction. The detector was set at 135 mm from the beam impact in both experiments. The Doppler broadening of X-rays by the detector angular aperture was limited by vertical collimating slits of lead (6 mm) in the U 91+ experiment, and of tantalum (8 mm) for the Pb 81+ experiment.

4

The acquisition was done event by event, allowing for instance the selection of coincidences between X-rays and a given charge state at emergence.

Results

Part of the experimental details and results has been reported already in ref. [8] for the U 91+ experiment.

In particular the charge state distribution for <110> axial orientation of the 11 µm thick target showed a fraction of ~25% of frozen ions, and about the same for ions emerging as 90+. For the latter most of the capture was due to REC. An illustration is given in fig. 1, which shows X-rays recorded for both axial and random incidences during this experiment. The axial spectrum is recorded in coincidence with ions emerging from the crystal with the charge 90+, i.e. ions having captured only one electron. In the axial spectrum, the K-REC and L-REC peaks are observed with a very good statistics, whereas they are reduced by nearly two orders of magnitude for a random orientation. The non-radiative capture (or Mechanical Electron Capture, MEC) is the dominant capture process in random geometry. In this case, charge state equilibrium is reached and K-and L-shell vacancies are rapidly filled below the surface.

The signature of MEC is still present in the axial spectrum, with the presence of K and L lines, that come from a large part from decays after MEC into excited states (n>2). For well channeled ions, like ions emerging as 90+, most of the MEC events occur in the thin amorphous layers at the crystal surfaces. For a more detailed discussion on the X-ray spectra, see ref. [8].

The situation is similar for 13 MeV/u Pb 81+ ions channeled in a 1.1 µm crystal. In this case the frozen ion fraction was 60%, and the 80+ fraction was 20%. The target being much thinner, the statistics on REC are lower than in the former experiment. A spectrum recorded for the axial orientation, in coincidence with He-like transmitted ions, is shown in Fig. 2. The proportion of K lines is higher, due to the larger relative contribution of amorphous layers to the MEC single capture events.

Both spectra on Figs 1 and 2 are not corrected for detection efficiency.

The measurements of absolute energy shifts for REC peaks require precise energy calibration. This was made in various ways in the whole energy range described here. First radioactive γ-ray sources were used, without the beam on target. We also used the Pb and Ta Kα fluorescence lines (for instance the Pb Kα lines can be observed on the spectra of fig. 1). Additional information for the linear calibration offset is obtained by setting the differences between K-and L-REC peak energies equal to the Kα ener-gies. The precise knowledge of the K and L peak energies in the laboratory frame allows the correction of the Doppler shift of the photon energy emitted by fast projectiles: the derivative of the Doppler shift is maximum at 90°, and so a small deviation of the mean laboratory angle has to be taken into account.

The energy loss inside the target was taken into account to determine the mean energy of REC photons emitted during the path of the ions in the crystal. For Pb 81+ in the 1.1 µm thick target the energy loss was almost negligible.

As done in ref. [6], we have performed simulations based on the statistical flux equilibrium for channeled ions in the crystal, according to the experimental conditions (beam angular divergence, energy loss, Doppler shift and broadening, detector resolution). These simulations provide the full calculation of the REC line shape, conditioned by the local electron density sampled by channeled ions. The Compton profile is calculated using the local density approximation of a free valence electron gas. For core electron, an impact parameter dependent Fourier transform of spatial wave functions is performed.

Figure 3 shows the results of such a simulation for the K-REC line shape of U 91+ ions in the laboratory frame. This simulation was made assuming that all incident ions in channeling conditions could undergo REC in the crystal. Among them, ions with a high transverse energy can approach the target atoms sufficiently close, so that they can capture core electrons of silicon. The corresponding Compton profile is much broader than the Compton profile of valence electrons. As silicon atoms have 10 core electrons and 4 valence electrons, one can see that the calculated rate for the capture of core electrons is strongly reduced by channeling. In order to reproduce our experimental observations (angular scans), we used a beam angular divergence made of two components: a narrow one (85% of the beam), of rms σ x =σ y =0.2 mrad, and a broad one (15%), rms σ x =σ y =3.5 mrad (i.e. larger than the channeling critical angle). According to this, the mean sampled electron densities are calculated to be 0.16 e -Å -3 for valence electrons, and 0.053 e -Å -3 for core electrons. The experimental peak is superimposed on the calculations. One can see that the calculated core electron contribution is obviously overestimated, and that valence electrons represent almost all the contribution to the K-REC peak. Actually, as we already stated in ref. [8], ions with high enough transverse energy to approach the target atoms as close as 0.5 Å undergo many MEC capture events, which rapidly fill their inner-shell and prevent REC to occur.

So only ions with a restricted transverse energy are able to make REC in the target. The simulations show that their mean sampled electron density is 0.17 e -Å -3 . This is an important point since, as calculated for instance in ref. [4], the strong dependence of REC cross section on the relative velocity of tar-get electron would make REC peaks strongly asymmetric, which induces a negative average energy shift of the REC peak because of the Compton profile. This has to be taken into account to correctly estimate the wake shift. In the present case where mainly valence electrons are captured, the shift due to the Compton profile is minimum. The comparison between calculated and measured peaks in Fig. 3 exhibits a shift by about -100 eV for the experimental one, that we attribute to the wake effect.

The shift for the L-REC lines, which has to be the same as for K-REC, is also measured, and is found in agreement within the calibration uncertainties. Thus we present in table 1 the measured values in both experiments, which are averaged for K-and L-REC. These shifts were measured by selecting REC photons in coincidence with He-like emergent ions, i.e. ions with a small transverse energy. The advantages are: i) contributions of core electrons to REC are even smaller than in Fig. 3, ii) their energy loss is minimum, iii) no additional shift on the L-REC energy comes from the various lower charge states. The error bars come from the determination of the peak positions in the spectra, the X-ray energy calibration, the evaluation of the energy loss in the case of the uranium experiment, and the theoretical knowledge of the binding energies for the He-like ions (taken from ref. [9]). Calculations for the energy shift were made using the commonly admitted formula indicated in the introduction, and using an electron gas density equal to the average valence electron density of silicon (i.e. 6 . 16

= p ω ℏ eV).
This value assumes that the whole valence electron gas contributes collectively to the energy shift, i.e.

that the polarization is a long range effect. It also supposes that core electrons of silicon do not contribute significantly to the collective response, which may somehow underestimate this response. The agreement between experimental and theoretical values is quite good.

As for the absolute REC cross sections, they can be evaluated, although the absolute detector efficiency was not measured by another mean than by geometrical calculations. When looking at figs 1 and 2, one can get an absolute normalization of the X-ray yields, since all the ions having captured at least one electron in the crystal have emitted one -and only one -photon corresponding to the filling of the Kshell. These photons are K αβ... and K-REC (in this case the fluorescence yield is 100% for the initial Kvacancy). For a given charge state used to select X-rays, the yields of K-and L-REC photons in the spectra (corrected from the intrinsic detector efficiency), multiplied by the charge fraction, provide the K-and L-REC probabilities. Simulations, to be described in a forecoming paper, allow us to estimate the mean unperturbed electron density e ρ sampled by channeled ions, as a function of their emerging charge state. Accounting for the angular distribution of REC photons in the laboratory frame [10], one can then give the absolute value of the K-and L-REC probabilities. We use the Stobbe formulae [11] of REC cross sections σ REC , which are commonly used to estimate the absolute REC cross sections as a function of the adiabaticity parameter

( ) 2 0 / v Z v p = η
(assuming non relativistic velocities for both the projectile and the electron in a bound orbital, η is the ratio of the kinetic energy of a target electron viewed by the projectile to the binding energy in the final state) [10]. The K-and L-REC probabilities are found to lie between 20% and 60% above the values given by P(REC) = ( )

ℓ × × - - e REC ρ σ exp 1 ,
where ℓ is the crystal thickness. Moreover, the non-relativistic dipole approximation calculations of Stobbe formulae tend to be systematically above the experimental values by at least 25%, as reviewed in ref. [10]. Note that exact relativistic calculations have been performed by J. Eichler and A. Ichihara in the case of 20 MeV/u U 91+ ions [START_REF] Eichler | [END_REF]. Their K-REC cross section is 20% lower than Stobbe's one, and higher n-REC cross sections are in agreement within 10%. This justifies the use of Stobbe's values as a fairly good reference on one hand, and, on the other hand, makes our electron density enhancement even higher. Actually, some solid state measurements at high Z p /v values tend to lead to higher values of the REC cross sections. For instance, our former measurements of K-REC with 60 MeV/u Kr 36+ ions (Z p v 0 /v= 1.1) under channeling conditions agree perfectly with the Stobbe formula (with an absolute uncertainty of ± 10%) [6]. Tribedi et al. [5] used the linear response scaling [2] of the local electron density to explain the excess of REC cross sections in channeling relative to gas targets. We prefer to consider an increase of electron density instead of an increase of cross sections, which are defined for a single ion-electron (or ion-atom) collision. In the present case, this density enhancement would be by a factor 10, which is certainly not observed. However, our values are significantly above the calculations using a non-perturbed electron gas. We can already claim that the local density enhancement by the wake effect is not as localized as predicted by the linear response theory (which could be expected since we are not at all in a regime where Z p v 0 /v<<1). On the other hand, the induced potential at the projectile site results from an integration of the polarization over a very large scale (a typical scaling of the perturbation is given by v/ω p ~ 10Å), which makes the perturbation approximation more realistic for the REC energy shift than for the REC probability. However, in order to get a better understanding of the whole dynamic polarization effect on REC energies and probabilities, the comparison with REC Table 1 : measured and calculated energy shifts of the continuum due to the electron gas polarization effect. The calculated values are given for a plasmon energy of 16.6 eV.

,

  energy of the target electrons, and v 0 the Bohr velocity. For a free electron gas of density ρ e , where m e is the electron mass. Also related to the wake are the fluctuations of the electron density around the projectile. Still within the first order perturbation, the authors of ref.[2] also predicted that the relative enhancement of the local electron density ρ e should be

Figure 1 :

 1 Figure 1: X-ray spectra detected at 90° from the beam direction for U 91+ ions incident on a 11 µm thick silicon crystal. (a) <110> axial orientation, in coincidence with ions transmitted in the 90+ charge state. (b) random orientation. Both spectra are normalized to the same number of selected transmitted ions.

during collisions in gas is highly desirable, as well as a non perturbative theoretical description of the effect.

Summary and conclusion

We have measured K-and L-REC with highly-charged, decelerated heavy ions in channeling conditions, which allows to keep inner-shell vacancies, and to perform ion-electron interaction inside a solid target. We report on a very strong value of the energy shift due to the polarization of the dense electron gas sampled in the solid (~100 eV), in good agreement with theoretical expectations based on first order perturbation. Our data show an enhancement of the total REC probabilities with respect both to theoretical calculations (by 20-60% relative to the non-relativistic dipole approximation cross sections) and to measurements performed at Z p v 0 /v ~1 (by more than 50% compared to gas target experiments). This is likely due to the polarization effect around an ion, this effect being much smaller than predicted by the linear response theory.

We would like to thank J. M. Pitarke and M. Seliger for helpful discussions, and J. Eichler and A. Ichihara for providing us with REC cross sections. The support by GSI-IN2P3 collaboration agreement # 97-35 is acknowledged.