Centralizing information improves market efficiency more than increasing information: Results from experimental asset markets

1 Centralizing information improves market efficiency more than increasing information: Results from experimental asset markets* Iván Barreda-Tarrazonaa, 1 , Gianluca Grimaldaa, c, Andrea Morone a, b, Simone Nuzzob, Andrea Teglioa aLEE & Economics Department, Universitat Jaume I, Castellón, Spain b Dipartimento di Economia, Management e Diritto d’impresa, Università degli Studi di Bari, Aldo Moro, Italy c Institut für Weltwirtschaft, Kiel, Germany Abstract We study the relationship between market efficiency and the distribution of private information in experimental financial asset markets. Traders receive imperfect signals over the real value of an asset. Agents can share their information within a relatively small – compared to market size - group of agents. Both the number of signals and the way these are allocated among agents are manipulated in four experimental treatments. In two treatments signals are evenly distributed among agents. In two other treatments one group of ‘quasi-insider’ agents receives more signals than all other groups. In the baseline condition no signal is distributed. We show that centralizing information unambiguously achieves higher market efficiency than spreading information evenly. Furthermore, increasing the amount of information has no effect on efficiency either when information is symmetric or when it is asymmetric. We argue that two complementary mechanisms drive these results. First, having more private information ex ante induces traders to rely on their own signals, reducing the expected benefits of sharing information. Second, the presence of quasi-insider being common knowledge prompts agents to extract more information from market prices rather than their own private signals. This leads to swift information aggregation. Key words: Experimental Markets; Information Aggregation; Market Cooperation * Financial support by Universitat Jaume I (project P1.1B2015-48) and the Spanish Ministry of Economics and Competitiveness (projects ECO2013-44409-P and ECO2015-68469-R) is gratefully acknowledged. 1 Corresponding author: LEE & Economics Department, Universitat Jaume I, 12071 Castellón, Spain. E-mail address: [email protected]. 2 Introduction The capacity of markets to efficiently aggre gate privately dispersed information has been a central topic in economics since Adam Smith and von Hayek. In the context of financial markets, Fama (1965) defines a market as efficient whenever prices "fully reveal" the information dispersed in the market. Intuitively, efficiency can be achieved becau se traders owning private information on the real value of an asset – so-called “insiders” - will seek to profit whenever prices do not fully embody their own private information. If markets are efficient, insiders’ private i nformation is in the long run worthless and insiders cannot realize higher gains than other traders. In the short run, uninformed traders can try to infer the existence of insider information from the observation of trading activity, in order to also realize profits. This can either accelerat e the process of conv ergence toward the equi librium, or drive it astray if uninformed traders are mistaken i n their inferences, as the literature on informational cascades demonstrates (Bikhchandani et al., 1998). The experimental literature has extensively examine d the conditions at which financial markets achieve efficiency (see the next section for a review). However, this is normally done in settings where agents are either fully informed or not informed, and have no connections with others. In this paper we generalize on both of these conditions. We allow for all agents to have imperfect information and for some agents – whom we call “quasi-insiders” - to have close-to-perfect information. We vary the overall amount of information available in the market an d the way this is distributed among agents. In particular, in a two-by-two desi gn, i nformation is either equally distributed among traders or is unequally distributed. In the latter case, “quasi-insiders” receive a higher amount of information than other agents. Moreover, the overall quantity of information is also modified across our experimental conditions. Secondly, we introduce a simple network structure in our financial markets. Each agent is connected with two other agents and each has the option to share their own private information with the two other agents i n their group. The importance of networks in financial markets has been stressed both the oretically and empirically. According to Abou lafia, (1997), financial markets are embedded into a vast and dense network of customary codes of conducts, mutual expectation s of appropriate behavior, and trust among i ndividual agents. In fact, some accounts stress that the breakdown in the networks of trust among financial agents played a major role in precipitating a nd 3 aggravating the 2008 financial crisis (Kirman, 2010; Anand et al., 2013). According to this view, financial markets may share some characteristics with standard markets for goods, where social norms and trust networks can ca use the actual market price to depart from the Walra sian price (Greif, 1993). The effect of trust network in financial markets is a largely unexplored issue. We are the first, to the best of our knowledge, to examine it experimentally. Our main research question is whether and how the amount, distribution, and the spreading of informatio n through the networ k, i ncrease market effi ciency. Except for the baseline c ondition where no a gent receives any information, in all other experimental conditions all agents receive some noisy signals on the fundamental valu e of the asset. Such signals only reveal the real value of the asset with 70% prob ability. We consider two cases where each agent is either allocated the same number of signals as all oth ers or where some “q uasi-insiders” are endowed with a larger number of signals. The overall number of signals distributed in the m arket before tran sactions i s also manip ulated across experimental conditions. Our framework enables us to study two different mechanisms for the di ffusion of information in the market. The first is what we call a “leader ship mechanism”, where it is common knowledge that one group of quasi-insiders wi thin the market owns ex ante more informative signals than all other groups. In some cases quasi-insiders can expect to know the real value of the asset with probability greater than 95% 2 . We expect that this mechanism will have important consequences on “information di sclosure” (see next section) and thus on market efficiency. On the one ha nd, quasi-insiders will presumably try to gain from their informational advantage, and their trading activity should, ceteris paribus, drive the asset price in the direction of the fundamental value faster than i n other cases. On the other hand, les s informed traders may pay more attention to the price adjustments within th e market, because the presence of quasiinsiders should make them aware of the possi bility that prices move in the direction of the fundamental faster than in other cases. The second channel is what we call a “cooperation mechanism”, whereby agent s can sha re their private signal(s) within groups f ormed by three members, a nd can in 2 The term “insider” normally ch aracterizes traders who know with pro bability one the real value of the asset. S ince in our study traders h aving informational advantage can never be absolutely certain of the real value of the asset, we prefer to call th em “quasi-insiders”. See also section 2. 4 turn receive the information shared by the other two members of their group. This is done before trading starts, so agen ts can access the market with a larger number of per capita signals, if others in their group have decided to share. In the cooperation mechanism, information is therefore multiplied if agents decide to share. If coopera tion does occur, we expect that transactions will become more infor mative and, in aggregate, prices may incorporate the available i nformation. Manipulating the quantity of signals owned by quasi-insiders makes i t possible to speculate ov er the relative strength of these two factors. Our experimental design includes a baseline condition where no information is available to agents, and four treatment conditions that vary both the amount and the concentration of signals. Information is evenly distributed among agents in two of such treatments,, but the total number of signals is tripled in one treatment compared to the other. That is, in one treatment each agent receives one signal, while in the other treatment each agent receives three signals. In two other conditions information is unevenly distributed between one group of quasi-insiders and three groups of noninsiders. In these two conditions the total number of si gnals is kept constant. This enables us to evaluate the impac t on efficiency of modif ying the distribution of signals from even to uneven. In order to better appreciate the relevance of each of the two mechanisms illustrated above, we draw on two different price benchmarks. The first is the “Bayesian price”. This is the price that would result if all traders aggregated the information at their disposal rationally – namely, according to the Bayes rule – after traders have decided whether to sh are their signals within their group or not. More precisely, individual Bayesian prices are computed for each trader, and the market Bayesian price is calculated as the arithmetic mean of such individual prices. The second notion is what we call “Fama-market efficiency”, and draws on the idea set out at the beginning of the paper that prices should inc orporate all the information present in the market. For Fama-efficiency we do not consider signals being shared in the “ cooperation sta ge”, but we only consider the information available before such a stage. Here we compute what we call the “Fama-efficient” price as that resulting from the assumption that each agent knew the whole information present in the market. We find that the “leadership mechanism” unambiguously brings about more efficiency than the "cooperation mechanism”. Interestingly, we a lso find that increasing 5 the quantity of signals does not necessarily lead to appreciable gains in efficiency. In particular, efficiency is not higher in the symmetric treatment having thrice as many signals as the alternative symmetric treatment. Likewise, efficie ncy is not higher in the asymmetric treatment h aving overall twice as many signals as the alternative asymmetric treatment. Nonetheless, the two asymmetric treatments bring about appreciably more effici ency than the symmetric ones. We speculate that the main driver of thi s result is non-insiders extracting information from market prices more actively than in symmetric treatments. Our study is of interest for the theoretical issue of whether markets are capable of efficiently aggregating and disseminating private information. We innovate on previous literature (see n ext section) by genera lizing the standard fra mework in the two directions mentioned above. That is, we allow for all agents to receive imperfec t signals, comparing cases of equal distribution of the information and unequal distribution. We also introduce a network structure whereby agents can share their private information. We believe that our study is al so relevant for policy issues. Investigating how market efficiency is affected by increasing information or spreading it more evenly, and how information spreads within networks, are all important questions for the optimal management of financial markets. Both in normal times but, even more so, in times of “crisis”, financial authorities may decide to release additional pieces of information to stabilize markets. Our experimental evide nce may help understand how to do this optimally. The remain der of the paper is organized as follows. I n the next section we present a review of relevant literature and in the third section the experimental design. In section 4 we present the methodology of the analysi s, then in section 5 we report the results obtained. Sections 6 discu sses the results and 7 conclude. 2 Related Literature Experimental studie s dealing with informational efficiency are typically divided into three types. The first one is the dissemination of information from identically informed agents - normally referred to as “insiders” - to uninformed traders (Plott and Sunder, 1982). The second strand includes studies about infor mation aggregation among market participants with less than perfect i nformation (Plott and Sunder, 1988). The third on e focuses on th e simultaneous equilibrium in asset and informati on market s 6 (Sunder, 1992). Theoretically, the i nformation aggregation process can be expected to be more sluggish in achieving market efficiency than the dissemination one. In the dissemination case insiders’ transactions release unambi guous signals about the value of the asset, at least w hen th e presence of insiders is com mon knowledge. Conversely, in th e aggregation case the proce ss of retrieving information is by construction subject to errors. Consequently, making i nference on the t rue state of the world is more prob lemati c in the latter case. Comprehensive recent surveys on ex perimental financial markets can be found in No ussair and Tucker ( 2013), and Morone and Nuzzo (2016). Hayek (1945) and Muth (1961) argued that markets never fail in aggregati ng the available i nformation. In a pioneering work, Plott and Sunder (1988) studied information aggregation in three differently designed markets and showed that this is not generally the case. In particular, while the price mechanism ef ficiently aggregated the dispersed inf ormation both in markets where participants traded a complete set of Arro w-Debreu securities and in markets where traders had identical payoff structures, aggregation failed in single security markets where traders were paid different divi dends upon the realization of uncertainty. T he authors explain ed this result arguing that traders cannot infer the contingent s tate of the market from other agents’ trading behavior when their payoff structures differ. Forsythe and Lundholm (1990) found that, in spite of hete rogeneous dividend structures in incomplete markets, information was correctly aggregated whenever the dividend distribution was common knowledge among traders and the su bject s had pre viously experienced the trad ing institution. Other s tud ies found even more negative results on the c apacity of markets to aggregate information under more general conditions than the ones c onsidered in previous studies. O'Brien and Srivastava (1991) s howed t hat, even with unifor m an d common dividend distributions, markets did not manage to aggregate the dispersed information if some elements of complexity (multi-period assets, no common knowledge about information distribution) are introduced in the market design. Noeth et al. (1999) found tha t information aggregation might be hindered by the existence of “information traps”. In particular, misal igned patterns, in which actions are based on wrong beliefs a bout others’ informative set, can result in information not being correctly revealed into prices. Brandouy et al. (2000) provided further evidence a bout price formation, asymmetric information and traders’ behaviour, in the context of asymmetric and possibly misleadi ng information in a (double-auction) stock market. 7 They found that asymmetric information released its effect into the market only when it is common knowledge among market participants. Plott et al. (2003) found that information aggregation strictly de pends on the environment complexity. Wh ile the competitive equilibrium (rational expectations model) is very likely to hold in simpler contexts, private information based models are generally more accurate in more complicated environments. In a market where information about the intrinsic value of an asset is cumu latively distributed among traders, Huber et al. (2008) proved the exi stence of a wide range of levels of information for which acquiring additional information di d not produce hi gher gains. A positive relation sh ip between information and higher profits was detected only for very high information levels. Among the studies that analyzed the impact of insider information, Schotter and Yorulmazer (2009) found that releasing information to some insiders helped to decrease the rate of bank run s in an experiment over banking crisis. A s we shall see, we obtain a similar result in the context of financial markets, as th e presence of insiders raises efficiency (see section 4). Similarly to the studies dealing with informati on aggregation, in our experimental markets all agents are only imperfectly informed on the fundamental value of the asset. No trader is given enough in formation to know with certainty the future value of the asset. No netheless, we introduce two major novelties. First, in two of our experimental conditions “quasi-i nsiders” receive a larger number of signals than oth ers. Although we can not, strictly speaking, talk about a process of dissemination of information, we are nonetheless interested in studying the impact on market efficiency of centralizing information in the hands of few agents. Second, differently from all previous studies, in our design traders are given the chance to share their information set with the other members of their group before trading begins. We expect that trust and reciprocity may prove relevant motivations as found in the literature studying standard cooperation problems (Fehr and Fischbacher, 2002). In fact, grou p attachment and social identity may also play a role (Brewer, 2008). The presence of quasi-insider may eith er induce a stronger sense of identity in groups of non-quasi-in siders, or a heightened perception of the unfairness of the process (Fehr and Schmidt, 1999; Trautma nn, 2009; Krawczyk, 2011). In both cases we would expect groups of less-informed traders to increase cooperation in comparison with symmetric treatments. This would lead to smoothing the information distribution heterogeneity 8 and to increasing the flow of information among traders 3 . To the best of our k nowledge, our paper is the first studying information aggregation in a framework where cooperation, reciprocity and leadership all matter to agents’ choices and price dynamics. 3 Experimental Design 3.1 General Design We run 27 independent experimental markets where a total of 324 agents traded a generic financial asset. Each age nt was provided with 1000 tokens and ten units of asset. Each token was wor th 0,02 Euros. At the end of the trading period, the asset paid an uncertain dividend D, wh ich c ould be worth ten tokens or zero tokens, depending on two equally likely states of the world. At the beginning of the per iod, agents received partially informative signal(s) on the fundamental value of the security. Before trading started, in what we call the sharing stage, each trader independently decided whether or not to reveal her signal(s) to the other two members of her group or not. We designed four treatments in addition to a baseline condition w here no agent received any information. In treatment 1 (T 1) all agents received one signal; in treatment 2 (T2) basic-informed agents received one signal and quasi-i nsider agents received three signals; in treatment 3 (T3) all agents received three signals; in treatment 4 (T4) basic-informed agents received one signal and quasi-insider agents received nine signals. Three markets were run for the baseline condition, while six markets were run for each of the four treatments. This design allows us to consider several invariants for treatments comparison. T1 and T2 differ because of the presence of qua si-insiders but preserve the amount of information given to basicinformed agents. T3 and T4 differ because of the presence of quasi-insiders agents but preserve the total amount of information in the market. T1 and T3 do not include quasi-insiders agents b ut differ in the amount of information given to basic-informed agents. T2 and T4 both include quasi-insiders but differ in the amount of additional information given to them. 3 In a companion paper, we study in detail how sensitive the pre-trade cooperation mechanism is to the presence of quasi-in siders. 9 Each market included 23 trading periods, thre e of which were trial periods while all of the 20 ensuing periods were paid off. The experiment was programmed in z-Tree (Fischbacher, 2007) and was run at the Laboratory for Experimental Economics (LEE) of Universitat Jaume I (Castello n, Spain). Instructions are reported in the App endix G. 3.2 State of information In all cases except the baselin e, traders received partially informative signal(s) on the future value of the asset dividend before trading started. Signals were not 100% reliable. Assuming that the true dividend to be paid at the end of the period was ten (zero), the probabi lity of getting a signal indicating that the dividend would be ten (zero) was p. (1 – p) was therefore the probability of getting a private signal indicating that the dividend would b e ten ( zero) while the true value o f the dividend was instead zero (ten). In othe r words, p was the probability that the signal reveals the true value of the dividend, while 1-p is the c omplementary probability that the signal indicated a wrong value of the dividend. We set p e qual to 70%. The value of p was common knowledge among subjects. At the beginning of the experimental session, in ea ch market 12 traders were randomly assigned to four different grou ps, each composed by three traders. The group composition was fixed throughout the session. Before trading began, subjects went through a sharing stage, in which they simultaneously decided whether or not to share their signal(s) with others in their group. Information sharing could only occur with components of the same group. Moreover, if one trader decided to share her information, all his or her signal(s) would be shared within the group. No dec eption when sharing signals was allowed. Treatment T1 T2 T3 T4 Baseline Panel A Ex-Ante Number of signals distributed to “basic informed” traders 1 1 3 1 - Number of signals distributed to “quasi insider” traders - 3 - 9 - 16 Alternative Hypothesis 1(a): Keeping constant the total number of signals, prices exhibit a significant closer convergence to the efficient price when information is uniformly distributed. Alternative Hypothesis 1(b): Keeping constant the total number of signals, prices exhibit a significant closer convergence to the efficient price when quasi-insider agents are present in the market, i.e. when information is centralized. As a second step, considering those cases in which quasi-insider agents are present in the market (T2 and T4), we test whether an increase in the number of quasiinsiders’ per capita signals improves the convergence toward the efficient pri ce. Indeed, in T2 and T4, while basic informe d agents were provided wi th one signal each, quasiinsider agents (three subjects who belong to the same group) were given three an d nine signals each respectively. Therefore, we for mulate hypothesis 2 and its alternatives. Hypothesis 2: Other things being equal, when quasi-insider agents are provided with three signals each, prices exhibit the same deviation from the efficient price as when quasi-insider agents are provided with nine signals each. Alternative Hypothesis 2(a): Other things being equal, when quasi-insider agents are provided with three signals each, prices exhibit a significant closer convergence to the efficient price than when they are provided with nine signals each. Alternative Hypothesis 2(b): Other things being equal, when quasi-insider agents are provided with nine signals each, prices exhibit a significant closer convergence to the efficient price than when they are provided with three signals each. Finally, considering the cases where information is uniformly distributed among traders (T1 and T3), we test whether increasing the number of signals in the market impacts on market efficiency. This can be tested because each agent is provided with one and three signal(s) in T1 and T3, respectively. Our third hypothesis and it s alterna tives are stated below: Hypothesis 3: When information is uniformly distributed and traders are provided with one signal each, prices exhibit the same deviation from the efficient price as when traders are provided with three signals each. 17 Alternative Hypothesis 3(a): When information is uniformly distributed, prices exhibit a significant closer convergence to the efficient price when traders are provided with one signal each. Alternative Hypothesis 3(b): When information is uniformly distributed, prices exhibit a significant closer convergence to the efficient price when traders are provided with three signals each. 5 Results 5.1 Information sharing We first analyze how traders use the option to share their private information; secondly we present results on market outcomes. Figures 1 and 2 illustrate how subjects use the cooperation mechanism. They report the average number of signals s hared in each of the six markets comprising a given treatment for basic informed agents’ (Figure 1) and quasi-insider agent s (Figure 2). Figure 1: Information sharing by basic-informed agents Figure 2: Info sharing by quasi -insiders agents We first compare infor mation s haring patterns in T1 and T3, where no quasiinsider agent is present. Throughout our descriptive analysis, we consider each market as yielding one independent observation. Indeed, since the same group of people within a mar ket interact over several periods, within-group observations are serially interdependent. This property makes it suitable to consider each group (market) as an independent observation, e.g. 0.2 .4 .6 .8 1 1 2 3 4 Treatment Info Sharing (%) Median Line Only basic informed agents included Information sharing over treatments 0.2 .4 .6 .8 1 42 Treatment Info Sharing (%) Median Line Only quasi insiders included Information sharing over treatments 18 by computing the mean (m edian) of the variable of interest over the periods comprising a given market (see Fre chette, 2012). We note that traders cooperate significantly less in T 3 (twotailed k-sample median test: N = 6; Pearson chi square = 5.33; P = 0.021. See also Table A1, Appendix A). This is likely the conseq uence of traders’ initial information set being larger in T3 compared to T1. Agents can thus be more confident in T3 than T1 that their information s et is sufficient to i ndicate the true state of th e world. In other words, the expected benefits from cooperation is lower in T3 than T1, hence the incentives to share information are also lower. We a lso find that the level of information sharing among basic informed age nts does not significantly change when quasiinsi der agents are introduced in the m arket, as can be seen in the com parison between T1 - agents provided with one signal each - and T2 - basic informed and quasi-insider agents provided with one and three signals each, respectively - (two-tailed k-sample median test: N = 6; Pearson chi square = 0.00; P = 1.000. See also Table A2, Appendix A) and between T1 and T4 - basic informed and quasi-insider agents provided with one and nine signals each respectively – (two-tailed k-sample median test: N = 6; Pearson chi square = 0.00; P = 1.000. See also Table A3, Appendix A). Furthermore, moving from T2 to T4, the median pe rcentage of basic informed traders sharing their information set switches from 57.22% to 63.61%. Yet, this difference is not statistically significant (two-tailed k-sample median test: N = 6; Pearson chi square = 0.00; P = 1.000. See also Table A4, Appendix A). Finally, no significant difference (twotailed k-sample median test: N = 6; Pearson chi square = 1.33; P = 0.248. See also Table A5, Appendix A) emerges between quasi-insid ers’ information shari ng behavior in T2 and T4, although it is apparent from Figure 2 that sharing is lower in T4 tha n T2. This behavior presumably follows the same reasons as the drop in sharing for basic informed agents in T3 relative to T1. That is, a higher number of initial signals for each agent reduces their need to cooperate with others. Our conjecture (s ee section 2) that procedural unfairness in the asymmetric treatments may have led to stronger “group spirit” in basic-informed agents is thus disconfirmed by the data. As found in a companion paper, though, some other identity effects, which are not relevant for the present paper, seem nonetheless to emerge. 5.2 Market Efficiency 5.2.1 General Overview 19 Figures 5-8 report the box-plots of the RMSE distribution for e ach of the benchmark prices in each treatment. These graphs pool RMSE over periods and ma rkets. A more detai led overview can be found in Figures B1, B2, B3, B4, and B5 in Appendix B . There we show the actual evolution of the traded prices in relation to the benchmark prices, broken down by market and period. First, we note a clear difference between the baseline condition and all other treatments. The uninformed and dividend price RMSE distributions are shif ted downward and upward, respectively, in comparison to all other treatments. Through the use of a Tobit regression analysis (see Appendix C, Model C1), we find that the uninformed price RMSE in the baseline is significantly lower than that computed in each of the other treatments (P < 0.00 1 in all the four pairwise comparisons). On the contrary, we find that the dividend price RMSE distri bution in the baseline condition is significantly higher with respect to that computed in each of the other treatments (P < 0.05 in all the four pairwise comparisons). This preliminary analy sis shows that when no information is present in the market, trade prices remain significantly closer to the uninformed price and further away from the div idend price than when some information is present in the market. In particular, we note th at this difference is more pronounced for the uninformed price than the dividend price. It is relatively easier for markets with information to depart away from the uninformed price than to come closer to the fundamental in comparison to markets without information. In fact, trade prices in the baseline condition exhibit a random walk process around the expected valu e of the dividend dis tribution, and in n o case prices reach the dividend value (see Figure B 5, Appen dix B). We e mploy a Tobit regression model (see Appendix C, Model C2) to assess whether, when no information is in the market, the distance between trade prices and the uninformed price i s lower than the distance between trade prices and the fundamental value of the asset. This i s the standard assumption in markets with no inf ormation, and it is, not surprisingly, confirmed in our case (coeff. = -4.18; P < 0.001). This preliminary evidence e nsures that agents were able to exploit the available information and traded at prices that were further away from the uninformed price and closer to the fundamental asset value than in the baseline. 20 Figure 5: RMSE Distribution, Uninfor med Price Figure 6: RMSE Distribution, Bayes Price Figure 7: RMSE Distribution, Efficient Price Figure 8: RMSE Distribution, Dividend Among the treatments with information, we note some tendency for the uniformed price RMSE to increase as we move from T1 to T4, and correspondingly (though less markedly so) for the dividend price RMSE to decrease as we move from T1 to T4. This may signal that the combination of adding informati on and centralizing information helps agents to trade at pric es that are closer to the funda mental. Nevertheless, we note no clear pattern with respect to either the Bayes RMSE or the Fama-efficient price. We conjecture that this apparent lack of treatment differences in the Bayes and Fama-efficient RMSE may be due to learning effects. Learni ng may occur because agents update their decisi on-making rules as they accumulate trading experience. Agents c an improve their ability to infer infor mation from the other traders’ activity over the course of 20 periods. Agents may also update their cooperation strategies over time, thus also affec ting th e way markets spread i nformation. In fact, time series plots in Appen dix B typically exhibit proximity to uninformed or Bayesian price s in early periods, and proximi ty to the Fama-efficient price in the late periods of the session. For instance, in market 1 from ses sion 2 and T2 (see appendix B, Figure B2), uninformed trades dominate the first three periods, prices then converge to the Bayes price in 012345 RMSE Uninformed Price Baseline Treatment 1 Treatment 2 Treatment 3 Treatment 4 02468 RMSE Bayes Price Treatment 1 Treatment 2 Treatment 3 Treatment 4 02468 RMSE Efficient Price Treatment 1 Treatment 2 Treatment 3 Treatment 4 0246810 RMSE Dividend Price Baseline Treatment 1 Treatment 2 Treatment 3 Treatment 4 21 periods from four to nine, and prices track the efficient eq ui librium price for all later periods. This suggests a pattern whereby agents trade as if they were uninformed in the earliest periods, process their own private information in intermediate periods, and eventually man age to correctly pool the information dispersed in the market in the final periods. Learning may thus be relevant not only to account for individual behavior but also for its impact on market performance. For these reasons, we split our descriptive result s into the first and second block of ten periods in each market 5 and report on the benchmarks performan ce rates over the four treatments. We identify for each p eriod of each market which benchmark price is best able to approximate the actual tran saction prices. More precis ely, we select the benchmark price with the lowest RMSE value from actual prices 6 . Essentially, in each treatment we count how many times a given benchmark best approximates our data. Table 2 reports the percentages of each b enchmark being selected as the one with the lowest RMSE. Performance Rates Uninformed Price Bayes Price Efficient Price Dividend Treat. 1 First Half 45.00% 45.00% 10.00% 0.00% Treat. 1 Second Half 45.00% 38.33% 16.67% 0.00% Treat. 2 First Half 43.33% 46.67% 10.00% 0.00% Treat. 2 Second Half 21.67% 28.33% 50.00% 0.00% Treat. 3 First Half 50.00% 50.00% 0.00% 0.00% Treat. 3 Second Half 35.00% 46.67% 18.33% 0.00% Treat. 4 First Half 43.33% 48.33% 8.33% 0.00% Treat. 4 Second Half 15.00% 38.33% 46.67% 0.00% Baseline First Half 100.00% / / 0.00% Baseline Second Half 100.00% / / 0.00% 5 Box-Plots are reported in Appendix D. 6 In case of ties between RMSE for two or more benchmarks, we select the least efficient benchmark. This is on the o ne hand the most conservative criterion for our analysis and on the other hand permits performance rates to always sum up to 100%. 22 Table 2: Benchmark performance rates, group ed by treatment and first or second blo ck of ten periods. First, we note that the Dividend pri ce has never the lowest RMSE, denoting the difficulty of traders of achieving the fundament al price. With regards to the three other benchm ark s, in T1 the uninformed and the Bayes prices are the best performing benchmarks in both the first and the second 10 - block periods, with a performance rate of 45% and 45%, respectively, in the f irst 10-block peri od, and of 45% and 38.33% in the second 10-block period. The efficient pric e marginally i mproves from a performance rate of 10% in the early periods to a performance rate of 16.67% in late periods. Trading in T1 is thus still predominantly uninformed or based on private information. In T2 the uninformed and the Bayes prices are the b est performing ones in early periods (with a perfor mance rate of 43.33% and 46.67% respectively). Nevertheles s, when we move to late periods the efficient price becomes the b est tracked benchmark (with a performance rate of 50%). It is particularly interesting to note how uninformed trades decrease f rom 43.33% to 21.67% and efficient trades increase from 10% to 50% when moving from ear ly to late periods. This evidence shows that, over time, traders improve th eir ability to infer and aggregate the information dis persed in the mark et. In T3 results are remarkably similar to T1. Both the uninformed and the Bayes prices perform better in accounting for our data than the efficient price in both early and late periods. I n particular, in the second ten-round block, the Bayes price is the best trac ked bench mark with a performance rate of 46.67%. It is remarkable that, in spite of the number of initial signals being ex ante three times as high in T3, we observe market prices to have the same levels of proximity to the efficient price and the B ayes price as in T1. This is not due to the fact that the ex post number of signals is si milar in the two treatments. As Table 1, Panel B, shows, agents in T3 have a significantly (Mann Whitney U test: N 7 = 6; z = -2.882; P < 0.0039) larger amount of information (5 .90 signals per capita) than in T1 (2.18 signals per capita). The ex post ratio of number of signals is 2.70, which is less than the ex ante ratio of 3:1 bec ause agents shared on average less in T3 than T1. The bad performance of the efficient price suggests that traders are mainly concerned with processi ng their own priva te information than tryi ng to infer others’ infor mation through the observation of price signals. 7 Market averages of the ex-post signals distribution are used to account for within market correlation. 23 Finally, benchmarks in T4 perform similarly to T2. While the uninformed and the Bayes prices exhibit the highest performan ce rates in early periods (43.33% and 48.33% respectively), the efficient price performs better in late periods (with a performance rate of 46.67%). Even in this case it is remarkable how the efficient price performance rate switches from 8.33% to 46.67% moving from early to late periods. Here again we note that in spite of a larger number of signals being availab le in T4 compared to T 2 both ex ante - in a proportion of 2:1 - than ex post - in a proportion of 1.9:1 (Mann Whitney U test: N 8 = 6; z = -2.892; P < 0.0038) – the performance in terms of efficiency appear to be virtually the same. 5.2.3 Econometric Analysis In this section we perf orm a thorough ec onometric analysis of our hy potheses and of the conjectures that emerged from the descriptive analysis. For this purpose we use the following Tobit regression model: 𝑅𝑀𝑆𝐸𝑖,𝑡 =𝛼+∑𝛽𝑖 𝑛 𝑖=1 ∙𝑀𝑘𝑡𝑖+𝛾∙𝑃𝑒𝑟𝑖𝑜𝑑+∑𝜃𝑗 𝑘 𝑗=1 ∙𝑋𝑖,𝑗+𝜀𝑖,𝑡 The RMSE index of actual trader prices with respect to a given price benchmark is the dependent variable of the model. Our covariates include a dummy variable 𝑀𝑘𝑡𝑖 for each of n markets but one that is omit ted. In this way we control for both possi ble hydiosincracies across markets and for the clustering of our data at the market level. 𝑃𝑒𝑟𝑖𝑜𝑑 is a trend variable capturing the time effect; and 𝑋 𝑖,𝑗 is a vector of d emographics and attitudinal variables 9 that are averaged at the market level. We test f or treatment effec ts performing Wald tests over the differe nce between the sums of market dummy coefficients belonging to different treatments. That is, to test for the null hypothesis of absence of differences between two treatments, we consider the null hypothesis: 8 Market averages of the ex-post signals distribution are used to account for within market co rrelation. 9 A detailed d escription of the demographics and attitudinal variables is reported in the note below Table E1 (Appendix E). 24 𝐻0: 𝑍𝑟,𝑠 ≡∑βi,r 𝑛𝑟 i=1 −∑βi,s 𝑛𝑠 i=1 =0 where r and s identify the mark ets associated with two different treatments. 𝑛𝑟 and 𝑛𝑠 are the numbers of markets belonging to treatment r and s. Our design includes six markets for each treatment . 𝑛𝑟 and 𝑛𝑠 are therefore always equal to six, except for the treatment to which the omitted category of the model belongs (Treatment 1). Note that the possibility that 𝑛𝑟>𝑛𝑠 for comparisons involving Treatment 1 does not affect the estimation of our pairwise c omparisons, since the omitted cate gory coefficient is implicitly zero. Since our descriptive analysis highlighted the presence of different pri ce pa tterns between the first and second half market periods, we run the econometric model both in the first and the sec ond block of ten periods as well as over the entire set of market periods. All the regression outputs are reported in Appendix E; all the treatments pairwise comparisons are available in Appendi x F. Here we mainly foc us on the results derived from the s econd block of ten periods. Result 1: Hypothesis 1 is rejected. We find that, keeping information constant ex ante (36 signals in the market), when quasi-insider agents are active in the market (T4), actual prices exhib it a significantly closer convergence to the efficient price with respect to the case in which information is uniformly distributed (T3) (𝑍𝑇4,𝑇3 = -4.40; P = 0.014). Therefore , keeping constant the quantity of information in the market, a centralized information distribution in which some agents are provid ed with more information guarantees more efficiency than a uniform information distribution in which all subjects receive the same amount of information. Result 2: We cannot reject Hypothesis 2. We find that, other things being equal, when quasi-insider traders are given nine signals (T4), market efficiency is not significantly higher than when quasi-insider agents are provided with three signals each (T2) (𝑍𝑇4,𝑇2 = -0 .79; P = 0.710). I n other words, when i nformation is polarized, providing quasi-insider agents with a greater number of per-capita signals does not significantly increase market efficiency. Result 3: We cannot reject Hypothesis 3. When infor mation is unifor mly distributed, switchi ng from a market where subjects are provided with one signal each 25 (T1) to a market where three signals are released to each subject (T3) does not lead to a significant increase in the market efficiency level (𝑍𝑇3,𝑇1 = -1.58; P = 0.384). We conjecture that the common knowledge that information is uniformly distributed leads subjects not to recognize the presence of an informed market leader and, as a consequen ce, not to focus on the others’ trading activity. In fact, the sign of the coefficient indicates that the convergence to the efficient price is higher in T1 than in T3. This is remarkabl e, as the number of initial signals is three times higher in T3 than in T1 (keeping fixed the unifor m distribution in both treatments). We conjecture that the greater amount of information in T3 makes traders more confident in being able to correctly forecast the asse t dividend and makes them less prone to use the market trading activity as an inference tool. On the contrary, when provided with only one informative signal (as in T1), traders focus mo re on the market activity to improve their chance of properly inferring the asset fundamental value and, consequently, market efficiency is improved. With regard to the re maining be nchmarks, we find tha t additional information is not discarded but is somehow processe d by traders. This is evident from the increas ing pattern over treatments of the uninformed price RMSE in late peri ods. In other wor ds, moving from T1 to T4, as the quantity of information (total number of signals) increases, prices de part from the uninformed price. This trend is in place in all the pairwise comparisons (𝑍𝑇1,𝑇2 = -4.93; P = 0.00; 𝑍𝑇1,𝑇3 = -3.89; P = 0.000; 𝑍𝑇1,𝑇4 = - 8.29; P = 0.000; 𝑍𝑇2,𝑇4 = -3 .80; P = 0.0 01; see Table F8, Appen dix F) but on e (𝑍𝑇2,𝑇3 = 0.40; P = 0.717; see Table F8, Appendix F). Interestingly, even if not significant, the sign of the Wald test 𝑍𝑇2,𝑇3 appears to contradict the rule “more information less noise ”. Indeed, although in T3 traders receive twice as many signals as i n T2, prices come closer to the uninformed price when only basic-informed agents are active in the market (T3) 10 . Coherently with our results on the efficient price, we also find that even k eeping identical the amount of information within the market (i.e. in T3 and T4), asymmetric information distributions produce lower noise tha n th e case in which information is instead symmetrically spread out (𝑍𝑇3,𝑇4 = -4.21; P = 0.000; see Tab le F8, Appendix F). 10 Interestingly, when information is instead asymmetrically distributed, doubling the overall amount of signals leads prices away from the uninformed price (𝑍𝑇2,𝑇4 = -3.80; P = 0.001) 32 APPENDIX A T1 T3 Total Observations Lower than the median 1 5 6 Greater than the median 5 1 6 Total 6 6 12 Pearson chi-square (1) = 5.3333 P = 0.021 Table A1: Median test on information sharing (treaments 1 and 3) T1 T2 Total Observations Lower than the median 3 3 6 Greater than the median 3 3 6 Total 6 6 12 Pearson chi-square (1) = 0.0000 P = 1.000 Table A2: Median test on information sharing (treatments 1 and 2), only basic-informed included T1 T4 Total Observations Lower than the median 3 3 6 Greater than the median 3 3 6 Total 6 6 12 Pearson chi-square (1) = 0.0000 P = 1.000 Table A3: Median test on information sharing (treatments 1 and 4), only basic-informed included T2 T4 Total Observations Lower than the median 3 3 6 Greater than the median 3 3 6 Total 6 6 12 Pearson chi-square (1) = 0.0000 P = 1.000 Table A4: Median test on information sharing (treatments 2 and 4), only basic-informed agents included T2 T4 Total Observations Lower than the median 2 4 6 Greater than the median 4 2 6 Total 6 6 12 Pearson chi-square (1) = 1.3333 P = 0.248 Table A5: Median test on information sharing (treatments 2 and 4), only quasi-insiders agents included 33 APPENDIX B 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T1-S1-M1 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T1-S3-M1 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T1-S3-M2 34 Figure B1: Trade pri ces in Treatment 1 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T1-S3-M3 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T1-S10-M1 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T1-S10-M2 35 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T2-S2-M1 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T2-S4-M1 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T2-S4-M2 36 Figure B2: Trade pri ces in Treatment 2 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T2-S4-M3 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T2-S4-M4 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T2-S9-M1 37 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T3-S5-M1 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T3-S5-M2 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T3-S5-M3 38 Figure B3: Trade prices in Treat ment 3 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninfromed Price Bayes Price Efficient Price T3-S8-M1 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninfromed Price Bayes Price Efficient Price T3-S8-M2 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninfromed Price Bayes Price Efficient Price T3-S8-M3 39 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T4-S6-M1 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T4-S6-M2 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T4-S6-M3 40 Figure B4: Trade pri ces in Treatment 4 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 A Price Dividend Uninformed Price Bayes Price Efficient Price T4-S6-M4 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T4-S7-M1 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Bayes Price Efficient Price T4-S7-M2 41 Figure B5: Trade prices in the baseline condition 0246810 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Price Baseline-S11-M1 0.00 2.00 4.00 6.00 8.00 10.00 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Baseline-S12-M1 0.00 2.00 4.00 6.00 8.00 10.00 0120 240 360 480 600 720 840 960 1080 1200 1320 1440 1560 1680 1800 1920 2040 2160 2280 2400 Time Price Dividend Uninformed Baseline-S12-M2 48 Table E2 contd. T4S6M3 0.329 0.531 0.442* (0.364) (0.338) (0.248) T4S6M4 -0.681* -0.138 -0.334 (0.400) (0.337) (0.253) T4S7M1 -0.112 -0.543 -0.223 (0.613) (0.349) (0.361) T4S7M2 0.421 0.240 0.439 (0.527) (0.537) (0.365) age_market_mean 0.0938 0.0130 0.0596 (0.0706) (0.0615) (0.0480) deg_market_mean 0.370 -0.856 -0.170 (0.602) (0.627) (0.417) exp_market_mean -0.403** -0.263 -0.265** (0.200) (0.170) (0.126) fin_market_mean 0.213 0.119 0.132 (0.276) (0.286) (0.194) fut_market_mean -0.313 0.0487 -0.0907 (0.252) (0.283) (0.187) gender_market_mean -0.323 -0.212 -0.247 (0.318) (0.292) (0.227) Constant 1.143 3.057* 1.538 (1.897) (1.736) (1.301) Observations 240 240 480 Pseudo R2 0.0734 0.162 0.0826 Notes: The Bayes Price RMSE is the Dependent Variable. See Notes to Table E1 for variables’ definition. 49 Table E3: Regression analysis of Tobit model for price convergence toward the Uninformed Price VARIABLES Periods 1_10 Periods 11_20 Periods 1_20 period 0.0472*** 0.0526*** 0.0508*** (0.0159) (0.0193) (0.00665) T1S3M1 0.336 -0.832*** -0.276 (0.426) (0.235) (0.306) T1S3M2 0.0687 -0.116 -0.0985 (0.265) (0.265) (0.207) T1S3M3 0.836*** 0.730** 0.686*** (0.274) (0.354) (0.228) T1S10M1 1.173*** 0.267 0.672** (0.336) (0.416) (0.315) T1S10M2 1.010** 1.698*** 1.282*** (0.412) (0.511) (0.334) T2S2M1 0.621** 1.582*** 1.123*** (0.312) (0.342) (0.246) T2S4M1 0.180 1.566*** 0.846*** (0.322) (0.381) (0.279) T2S4M2 0.230 1.507*** 0.866*** (0.332) (0.314) (0.249) T2S4M3 0.282 1.108** 0.684** (0.307) (0.428) (0.279) T2S4M4 -0.205 0.980** 0.388 (0.317) (0.383) (0.275) T2S9M1 -0.0160 0.486 0.158 (0.284) (0.323) (0.211) T3S5M1 0.919*** 0.204 0.480* (0.279) (0.286) (0.245) T3S5M2 0.982*** 0.213 0.510* (0.352) (0.391) (0.298) T3S5M3 0.755** 1.852*** 1.214*** (0.298) (0.421) (0.277) T3S8M1 0.196 1.574*** 0.839*** (0.236) (0.388) (0.262) T3S8M2 0.471** 0.842*** 0.649*** (0.237) (0.249) (0.185) T3S8M3 0.917*** 2.138*** 1.507*** (0.269) (0.322) (0.248) T4S6M1 1.335*** 2.092*** 1.685*** (0.332) (0.394) (0.268) T4S6M2 1.001*** 1.882*** 1.372*** (0.229) (0.357) (0.226) 50 Table E3 contd. T4S6M3 1.571*** 2.541*** 2.058*** (0.263) (0.228) (0.194) T4S6M4 0.488 2.015*** 1.197*** (0.307) (0.306) (0.256) T4S7M1 1.110*** 1.000*** 1.047*** (0.386) (0.372) (0.284) T4S7M2 1.209*** 1.504*** 1.274*** (0.244) (0.371) (0.218) age_market_mean 0.00693 0.0325 0.0446 (0.0503) (0.0598) (0.0404) deg_market_mean 0.648** 0.950*** 0.881*** (0.300) (0.351) (0.262) exp_market_mean 0.228** 0.182 0.171* (0.110) (0.142) (0.0981) fin_market_mean 0.215 0.278 0.165 (0.172) (0.231) (0.148) fut_market_mean 0.283 0.158 0.307* (0.177) (0.262) (0.163) gender_market_mean -0.387* -0.261 -0.202 (0.214) (0.283) (0.193) Constant -0.397 -1.637 -1.605* (1.167) (1.483) (0.940) Observations 270 270 540 Pseudo R2 0.194 0.241 0.177 Notes: The Uninformed Price RMSE is the Dependent Variable. See Notes to Table E1 for variables’ definition. 51 Table E4: Regression analysis of Tobit model for price convergence toward the Dividend Price VARIABLES Periods 1_10 Periods 11_20 Periods 1_20 period -0.0691** -0.0655* -0.0728*** (0.0301) (0.0357) (0.0118) T1S3M1 -0.00707 -0.396 -0.117 (0.777) (0.515) (0.478) T1S3M2 0.434 -0.903* -0.129 (0.549) (0.460) (0.362) T1S3M3 -0.413 -1.469* -0.893 (0.794) (0.816) (0.579) T1S10M1 -0.789 -1.371* -1.116* (0.869) (0.721) (0.586) T1S10M2 -1.510** -2.065** -1.756*** (0.639) (1.023) (0.623) T2S2M1 -0.665 -2.528*** -1.652*** (0.533) (0.565) (0.402) T2S4M1 -0.157 -2.226*** -1.216** (0.614) (0.689) (0.488) T2S4M2 -1.231** -1.861** -1.605*** (0.528) (0.883) (0.532) T2S4M3 -0.538 -1.953** -1.277** (0.574) (0.779) (0.496) T2S4M4 -0.707 -1.937*** -1.311*** (0.451) (0.647) (0.409) T2S9M1 -0.374 -1.127* -0.763* (0.504) (0.605) (0.406) T3S5M1 -0.564 0.00251 -0.152 (0.496) (0.554) (0.406) T3S5M2 -0.625 -0.0954 -0.310 (0.814) (0.690) (0.540) T3S5M3 -1.084* -1.802* -1.354** (0.586) (0.927) (0.562) T3S8M1 -0.651 -2.906*** -1.693*** (0.432) (0.626) (0.395) T3S8M2 -0.589 -2.075*** -1.358*** (0.487) (0.502) (0.362) T3S8M3 -1.130** -3.409*** -2.218*** (0.518) (0.517) (0.383) T4S6M1 -1.191** -3.401*** -2.280*** (0.574) (0.557) (0.427) T4S6M2 -0.643 -2.192*** -1.422*** (0.580) (0.749) (0.484) 52 Table E4 contd. T4S6M3 -0.951** -3.004*** -1.991*** (0.396) (0.643) (0.398) T4S6M4 -1.032** -2.364*** -1.684*** (0.460) (0.587) (0.383) T4S7M1 -0.600 -2.494*** -1.557*** (0.814) (0.653) (0.550) T4S7M2 0.197 -1.264* -0.459 (0.581) (0.755) (0.481) age_market_mean 0.153 -0.132 -0.0195 (0.101) (0.103) (0.0709) deg_market_mean -0.113 0.460 0.0887 (0.621) (0.648) (0.472) exp_market_mean -0.273 -0.299 -0.217 (0.236) (0.248) (0.172) fin_market_mean 0.326 -0.179 0.104 (0.397) (0.437) (0.288) fut_market_mean -0.217 -0.527 -0.378 (0.374) (0.431) (0.287) gender_market_mean 0.288 0.201 0.0636 (0.421) (0.480) (0.325) Constant 2.011 10.08*** 6.809*** (2.341) (2.445) (1.642) Observations 270 270 540 Pseudo R2 0.0491 0.0806 0.0640 Notes: The Dividend Price RMSE is the Dependent Variable. See Notes to Table E1 for variables’ definition. Appendix F 53 Wald tests over the difference between the sums of market dummy coefficients belonging to different treatments. In order to test for the null hypothesis of absence of differences between two treatments, we c onsider: 𝐻0: 𝑍𝑟,𝑠 ≡∑βi,r 𝑛𝑟 i=1 −∑βi,s 𝑛𝑠 i=1 =0 where r and s identify the markets associated with two different treatments. 𝑛𝑟 and 𝑛𝑠 are the numbers of markets belongi ng to treatment r and s. Our de sign includes six markets for e ach treatment. 𝑛𝑟 and 𝑛𝑠 are therefore always equal to six, except for the treatment to which the omitted category of the model belongs (Treatment 1). Note that the possibility that 𝑛𝑟>𝑛𝑠 for comparisons involving Treatment 1 does not affect the estimation of our pairwise comparison s, since the omitted cate gory coefficient is implicitly zero. With reference to each pairwise comparison, tables from F1 to F12 report the statistics 𝑍𝑟,𝑠 as well as the related pvalu e in brackets (*** p<0.01, ** p<0.05, * p<0.1). In p articular, each pairwise comparison has to be read subtracting the colu mn variable from the row variable, e.g. T1 – T2, T1 – T3, T1 – T4, T2 – T3, and so on. Table F1: Pairwise comparisons of Rmse Efficient Price across treatments in periods from 1 to 10 Rmse Efficient Price in Periods 1_10 T1 T2 T3 T4 T1 -0.16 (0.938) -1.44 (0.444) -2.34 (0.238) T2 -1.28 (0.446) -2.18 (0.292) T3 -0.90 (0.617) T4 54 Table F2: Pairwise comparisons of Rmse Efficient Price across treatments in periods from 11 to 20 Rmse Efficient Price in Periods 11_20 T1 T2 T3 T4 T1 2.02 (0.300) -1.58 (0.384) 2.81 (0.136) T2 -3.60* (0.068) 0.79 (0.710) T3 4.40** (0.014) T4 Table F3: Pairwise comparisons of Rmse Efficient Price across treatments in periods from 1 to 20 Rmse Efficient Price in Periods 1_20 T1 T2 T3 T4 T1 1.39 (0.327) -1.37 (0.296) 0.63 (0.647) T2 -2.77** (0.031) -0.76 (0.613) T3 2.01 (0.120) T4 55 Table F4: Pairwise comparisons of Rmse Bayes Price across treatments in periods from 1 to 10 Rmse Bayes Price in Periods 1_10 T1 T2 T3 T4 T1 0.39 (0.771) -0.17 (0.871) -1.41 (0.199) T2 -0.57 (0.671) -1.80 (0.200) T3 -1.23 (0.292) T4 Table F5: Pairwise comparisons of Rmse Bayes Price across treatments in periods from 11 to 20 Rmse Bayes Price in Periods 11_20 T1 T2 T3 T4 T1 -5.90*** (0.000) -2.11** (0.023) -3.38*** (0.000) T2 3.79*** (0.002) 2.52* (0.057) T3 -1.27 (0.192) T4 56 Table F6: Pairwise comparisons of Rmse Bayes Price across treatments in periods from 1 to 20 Rmse Bayes Price in Periods 1_20 T1 T2 T3 T4 T1 -2.63*** (0.005) -1.32* (0.073) -2.47*** (0.001) T2 1.31 (0.162) 0.16 (0.867) T3 -1.14 (0.138) T4 Table F7: Pairwise comparisons of Rmse Uninformed Price across treatments in periods from 1 to 10 Rmse Uninformed Price in Periods 1_10 T1 T2 T3 T4 T1 3.07*** (0.001) 0.06 (0.939) -2.17** (0.013) T2 -3.14*** (0.000) -5.62*** (0.000) T3 2.47*** (0.002) T4 57 Table F8: Pairwise comparisons of Rmse Uninformed Price across treatments in periods from 11 to 20 Rmse Uninformed Price in Periods 11_20 T1 T2 T3 T4 T1 -4.93*** (0.000) -3.89*** (0.000) -8.29*** (0.000) T2 0.40 (0.717) -3.80*** (0.001) T3 -4.21*** (0.000) T4 Table F9: Pairwise comparisons of Rmse Uninformed Price across treatments in periods from 1 to 20 Rmse Uninformed Price in Periods 1_20 T1 T2 T3 T4 T1 -1.10 (0.168) -1.96** (0.011) -5.39*** (0.000) T2 -1.13 (0.138) -4.56*** (0.000) T3 -3.43*** (0.000) T4 64 At the end of e ach period you will be shown your individual gain s in the last period, and your accumulated gains, al so the average gains in your group, the average gains in your market and the average gains of the members of the type B group.