An Experiment on Network Density and Sequential Learning
AAn Experiment on Network Density andSequential Learning ∗ Krishna Dasaratha † Kevin He ‡ First version: September 4, 2019This version: October 5, 2019
Abstract
We conduct a sequential social-learning experiment where subjects take turns guess-ing a hidden state based on private signals and the guesses of a subset of their prede-cessors. A network determines the observable predecessors, and we compare subjects’accuracy on sparse and dense networks. Accuracy gains from social learning are twiceas large on sparse networks compared to dense networks. Models of naive inferencewhere agents ignore correlation between observations predict this comparative static innetwork density, while the finding is difficult to reconcile with rational-learning models. ∗ We thank J. Aislinn Bohren, Ben Enke, Drew Fudenberg, Ben Golub, Jonathan Libgober, MatthewRabin, Ran Spiegler, and Tomasz Strzalecki for useful comments. Financial support from the Eric M.Mindich Research Fund for the Foundations of Human Behavior is gratefully acknowledged. † Harvard University. Email: [email protected] ‡ California Institute of Technology and University of Pennsylvania. Email: [email protected] a r X i v : . [ ec on . T H ] O c t Introduction
In many economic situations, people form beliefs based on others’ actions. In these set-tings, agents typically do not observe all members of the society, but only a select subset —namely, their neighbors in an underlying social network. How the structure of this observa-tion network affects learning outcomes is a fundamental question for understanding sociallearning. While an extensive theoretical literature has explored this question for both naiveand rational agents (e.g., Golub and Jackson, 2010; Acemoglu, Dahleh, Lobel, and Ozdaglar,2011; Golub and Jackson, 2012), much less is known empirically.Density is one of the most basic properties of a network. How do learning patterns dif-fer between sparse networks, where agents usually observe very few neighbors, and densenetworks, where agents generally have abundant social information? On denser networks,agents observe more predecessors (both directly and indirectly), so their actions can incorpo-rate the private signals of more individuals. But whether this leads to more accurate learningultimately depends on how society aggregates these signals. In this work, we conduct anexperiment to compare social-learning outcomes on sparse and dense networks. We studya sequential social-learning environment where agents on an observation network take turnsguessing a state. We find that although later agents have fewer observations on sparser net-works, they nevertheless learn substantially better on sparse networks than dense networks.We place subjects into groups of 40 who act in order. Each group lives on a social network,with randomly-generated links that determine each subject’s observations. Each subject hasa 25% chance of observing each predecessor in the sparse treatment and a 75% chance in thedense treatment (and subjects know these probabilities). A hidden binary state is drawn foreach group. On her turn, each subject must guess the state using her private signal and thepast guesses of the predecessors she observes. Subjects were paid for accuracy.Prior to data collection, we pre-registered a measure of long-run learning accuracy: thefraction of the final 8 subjects in the group who correctly guess the state. Comparing thismeasure on 130 sparse networks versus 130 dense networks, we find that denser networks leadto worse learning accuracy. In dense networks, the average accuracy of the last 8 subjectsimproves on the autarky benchmark (i.e., the average accuracy if no one can observe others’actions) by 5.7%, but this improvement is 12.6% in sparse networks. Thus, the long-runaccuracy gains from social learning are twice as large in the sparse treatment as in the densetreatment ( p -value 0.0239).In addition to its direct implications about the role of network density in social learning,this finding provides indirect evidence supporting models of naive inference in which agentsneglect the correlations among their social observations (as in Eyster and Rabin, 2010).1otivated by a theoretical result from Dasaratha and He (2019), we compute predictionsof the naive model. Later agents exhibit higher accuracy on sparse networks than densenetworks in this model, as in our experimental evidence. The basic intuition is that anagent with correlation neglect ends up placing too much weight on the actions of the veryearly movers, as these actions are a common source of influence for many of the agent’spredecessors. When the network is denser, this over-weighting is more severe and so naiveagents’ guesses are less accurate in the long run.On the other hand, our experimental findings are inconsistent with the rational social-learning model. Acemoglu, Dahleh, Lobel, and Ozdaglar (2011)’s results imply that rationalagents learn asymptotically in environments matching our experimental setup. We adapttheir methods to provide lower bounds on the accuracy of rational agents 33 through 40 inthe sparse and dense treatments. These bounds imply that rational agents’ accuracy cannotimprove substantially from the dense-network treatment to the sparse-network treatment —in particular, the rational model does not predict a doubling of accuracy gain.Our data also show that network density has no statistically significant effect on the overall accuracy averaged across all 40 subjects in each group. This is because dense networksincrease the accuracy of subjects who move early in the group, even though they lower theaccuracy of subjects who move later. This reversal of the accuracy ranking between sparseand dense networks over the course of social learning is another prediction of naive inference. Our experimental results add to a growing body of evidence that humans do not properlyaccount for correlations in social-learning settings. Enke and Zimmermann (2017) show thatcorrelation neglect is prevalent even in simple environments where the observed informationsources are mechanically correlated. In a field experiment where agents interact repeatedlywith the same set of neighbors, Chandrasekhar, Larreguy, and Xandri (2019) find agents failto account for redundancies.Most closely related to the present work, the laboratory games in Eyster, Rabin, andWeizsacker (2018) and Mueller-Frank and Neri (2015) directly evaluate the naive inferencebehavioral assumption. Eyster, Rabin, and Weizsacker (2018) find that on the completeobservation network agents’ behavior is closer to the rational model than the naive model. Ona more complex network the naive model matches more observations than the rational model,and there is little anti-imitation (which would be required for correct Bayesian inference). Mueller-Frank and Neri (2015) find most observations are consistent with naive inference In the complex network, four agents move in each period after observing predecessors from previousperiods.
The state of the world ω ∈ { , } takes one of two possible values with equal probabilities.The set of agents is indexed by i ∈ N . Agents move in the order of their indices, each actingonce.On her turn, each agent i observes a private signal s i ∈ R , as well as the actions of someprevious agents. Then, i chooses an action a i ∈ { , } to maximize the probability that a i = ω given her belief about ω .Private signals ( s i ) are i.i.d . and Gaussian conditional on the state of the world. When ω = 1, s i ∼ N (1 , σ ). When ω = 0, s i ∼ N ( − , σ ). Here σ > i observes the action of each predecessor withprobability q . These observations are i.i.d . Agents observed by i are called the neighbors of i , and the sets of neighbors define a (random) directed network.We compare two kinds of agents: rational agents and naive agents. Rational agentsplay the unique perfect Bayesian equilibrium. Naive agents optimize given the followingmisspecified beliefs: 3 ssumption 1 (Naive Inference Assumption) . Each agent wrongly believes that each pre-decessor chooses an action to maximize her expected payoff based solely on her private signal,and not on her observation of other agents.
Equivalently, naive agents act as if each of their neighbors observes no other agents.Besides the error in Assumption 1, naive agents are otherwise correctly specified and optimizetheir expected utility given their mistaken beliefs.Assumption 1 was introduced in a sequential learning setting where agents observe allpredecessors by Eyster and Rabin (2010). Their work refers to this form of inference as“best-response trailing naive inference” (BRTNI).
Dasaratha and He (2019) suggest an empirical test for the naive inference assumption: inthe context of sequential learning on uniform random networks, does increasing the link-formation probability q cause more inaccurate long-run beliefs? In this paper, we experimen-tally test this comparative static in networks of 40 agents by comparing learning outcomesin sparse networks (where q = ) and dense networks (where q = ).The naive-learning model and the rational-learning model make competing predictionsabout this comparative static. The intuition for naive learning comes from Dasaratha andHe (2019), which suggests that overweighting due to correlation neglect is more severe ondense networks. We do not expect human subjects to behave exactly according to As-sumption 1 — for example, the meta-analysis of Weizsäcker (2010) reports that laboratorysubjects in sequential learning games suffer from autarky bias, underweighting their socialobservations relative to the payoff-maximizing strategy. However, the comparative staticprediction of the naive model remains robust even after introducing any fraction of autarkicagents. The prediction of the naive model is shown in Figure 1, which plots the probabilitiesthat each of the 40 naive agents will correctly guess the state in sparse and dense networkswith σ = 2 . Because naive agents’ actions only depend on the number of their predecessorschoosing each of the two actions and not the order of these actions, recursively calculatingthe distributions of actions is computationally feasible (see Appendix 6.2 for details). Asshown in Figure 1, early naive agents do worse under q = because there is very little social Dasaratha and He (2019) consider agents with a continuous action space, but we implemented a binaryaction space in the experiment for clarity. We felt it would be easier for subjects to make a binary choicethan to accurately report their exact belief. See the Appendix of a previous version of Dasaratha and He (2019), available at https://arxiv.org/pdf/1703.02105v5.pdf .
10 20 30 40 . . . . Learning on Erdos−Rényi Networks with Naive Agents
Agent P r obab ili t y o f c o rr e c t a c t i on q = 0.25q = 0.75 Figure 1: Learning accuracy on random networks with 40 naive agents, binary actions, and σ = 4. Blue and red curves show the expected accuracy of different agents on networkswith link probabilities q = and q = , respectively.information, but the comparison quickly switches as we examine later naive agents.On the other hand, the rational-learning model predicts that later agents will have eithersimilar or greater accuracy on the dense network compared to the sparse network. Acemoglu,Dahleh, Lobel, and Ozdaglar (2011)’s results imply that in an environment matching ourexperimental setup, rational agents will learn the true state in the long-run, regardless ofthe network density. We can confirm that 40 rational agents are enough to approach thisasymptotic learning limit when q = . To do this, we compute a lower bound for theprobability of correct learning for each agent i in the dense network of our experiment,assuming all agents are rational Bayesians (see Appendix 6.1 for details). This lower boundis based on (suboptimal) agent strategies that only depend on their own private signals andthe action of just one neighbor, as in the neighbor-choice functions in Lobel and Sadler(2015). This exercise shows that the 33 rd rational agent is correct at least 96.8% of thetime on dense networks, with the lower bound on accuracy continuing to increase up to the40 th agent, who is correct at least 97 .
5% of the time. In addition to suggesting that theasymptotic result of Acemoglu, Dahleh, Lobel, and Ozdaglar (2011) very likely holds by the40 th agent, the fact that this lower bound for accuracy on the dense network is so close toperfect learning proves the 40 th rational agent could not perform substantially better on thesparse network, contrary to the predicted improvement for the 40 th naive agent shown inFigure 1.Intuitively one might also expect more connections to also help rational agents in the We prove these bounds because we are not aware of a computationally feasible method of calculating orsimulating the probability that rational agents are correct. Rahimian, Molavi, and Jadbabaie (2014) showcomputing rational actions in another social learning environment is NP-hard. i does better onthe complete network than on any sparser network structure. We note, however, that exactcomparative statics of the rational model or variants are not known on random networks.We experimentally test the competing predictions of the naive and the rational modelsabout how long-run accuracy varies with network density. We thus provide indirect evidencefor the naive inference assumption, complementing the direct measurement of behavior inEyster, Rabin, and Weizsacker (2018) and Mueller-Frank and Neri (2015).Beyond providing another form of evidence, our experiment also contributes to under-standing social learning by using the welfare-relevant outcome, namely the long-run accuracyof actions, as the dependent variable. Even if individual behavior tends to match redundancyneglect models in simple or stylized settings, one might worry that the theoretical implica-tions of said models concerning aggregate learning need not hold in practice for complexenvironments. For a policymaker who can alter the observation network, for instance, ex-periments using welfare-relevant outcomes as their dependent variables give more explicitguidance as to the consequences of different policies. We conducted our experiment on the online labor platform Amazon Mechanical Turk (MTurk)using Qualtrics survey software.We pre-registered our experimental protocol and regression specification prior to thestart of the experiment in August 2017. Our pre-registration included the target sam-ple size (which was met exactly) and the dependent variable to measure the accuracy ofsocial learning. The pre-registration document can be found on the registry website athttps://aspredicted.org/yp6eq.pdf and is also included in the Appendix.We recruited 1040 subjects satisfying the selection criteria described in the Appendix.Each subject also needed to complete three comprehension questions (which were scenariosin the game with a dominant choice); MTurk users who incorrectly answered one or morecomprehension questions were excluded from the experiment. The experiment was carriedout in autumn 2017.In addition to comprehension questions, we restricted to subjects located in the UnitedStates who had completed at least 50 previous MTurk tasks with a lifetime approval rateof at least 90%. Subjects were not permitted to participate in more than one round of ourexperiment. There were at most 15 subjects who did not complete all trials, implying a6ompletion rate of at least 98 . trial consisted of 40 agents who were asked to each make a binary guess betweentwo a priori equally likely states of the world, L (for left) and R (for right). The stateswere color-coded to make instructions and observations more reader-friendly. Agents areassigned positions in the sequence and move in order. Each MTurk subject participated in10 trials, all in the same position (depending on when they participated in the experiment).The grouping of subjects into trials was independent across trials. Subjects received $0 . .
25 per correct guess, for a maximum possible paymentof $2 .
75. Subjects ordinarily took less than 10 minutes to complete their participation andearned on average $2.08, so the incentives were quite large for an MTurk task.In each trial, every agent received a private signal, which had the Gaussian distribution N ( − ,
4) in state L and the Gaussian distribution N (1 ,
4) in state R. These distributionswere presented visually in the instructions. Along with the value of their signal, subjectswere told the probability of each state conditional on only their private signal.Each trial was also associated with a density parameter, either q = or q = . Arandom network was generated for each trial by linking each agent with each predecessorwith probability q . Each MTurk subject was assigned into either the “sparse” or the “dense”treatment, then placed into 10 trials either all with q = or all with q = . So there were520 subjects and 130 trials for each treatment. Agents were told the actions of each linkedpredecessor and the link probability q (but not the full realized network, which could not bepresented succinctly).In each trial, agents viewed their private signal and any social observations and were askedto guess the state. States, signals, and networks were independently drawn across trials.Experimental instructions and an example of a choice screen are shown in the Appendix. Let y i,j be the indicator random variable with y i,j = 1 if agent i in trial j correctly guessesthe state, y i,j = 0 otherwise. Define ˜ y j := P i =33 y i,j as the fraction of the last eightagents in trial j who correctly guess the state. We test learning outcomes for the final eightagents because welfare depends on long-run learning outcomes in large societies and theseagents better approximate long-run outcomes. By using only her private signal, an agent cancorrectly guess the state 69.15% of the time. We call ˜ y j − . gain from social learning In fact, 94% of the subjects assigned to the first position (who have no social observations) correctly usetheir private signal.
71) (2)FractionCorrect FractionCorrectNetworkDensity -0.0923 -0.0923(0.0406) (0.0406)Constant 0.802 0.802(0.0227) (0.0218)Observations 260 260Adjusted R (1) without robust SEs; (2) with robust SEs Table 1: Regression results for the effect of network density on learning outcomes.in trial j , as this quantity represents improvement relative to the autarky benchmark.We find that the average gain from social learning is 8.73 percentage points for the q = treatment and 4.12 percentage points for the q = treatment. Social learning improvesaccuracy on the sparse networks by twice as much as on the dense networks. To test forstatistical significance, we consider the regression˜ y j = β + β q j + (cid:15) j where q j ∈ { , } is the network density parameter for trial j . Recall that each subject wasassigned into ten random trials with the same network density and in the same sequentialposition. This means for two different trials j = j , the error terms (cid:15) j and (cid:15) j are close toindependent since there are likely very few subjects who participated in both trials. Indeed,our estimates are identical whether we use robust standard errors or not.We estimate β = − .
092 with a p -value of 0.0239 (see Table 1). These findings areconsistent with naive updating but not with rational updating, as discussed in Section 2. This difference in the gains from social learning is not driven by different rates of autarkyamong the two treatments for the last eight agents. We say an agent goes against her signal if she guesses L when her signal is positive or guesses R when her signal is negative. Withinthe last eight rounds, there are 138 instances of agents going against their signals in the q = treatment, which is very close to the 136 instances of the same under the q = treatment.However, when agents go against their signals in the last eight rounds, they correctly guessthe state 81.88% of the time under the q = treatment, but only 71.32% of the time under We pre-registered average accuracy in the last 8 agents (i.e last 20% of agents) as the dependent variablefor the experiment, but the regression result is robust to other definitions of ˜ y j . When ˜ y j encodes averageaccuracy among the last m agents for any 4 ≤ m ≤
12 (i.e. between last 10% and last 30% of the agents),the estimate for β remains negative. q = treatment. This shows the observed difference in accuracy is due to social learningbeing differentially effective on the two network structures.However, the q = treatment yields better learning outcomes for early agents. Foragents 10 through 20, the average guess accuracy is 72.24% under the q = treatmentand 73.22% under the q = treatment. As such, if we replace the dependent variable inthe pre-registered regression with overall accuracy ¯ y j := P i =1 y i,j , then we do not find astatistically significant estimate for β ( p -value of 0.663). This result is consistent with thenaive-learning model: according to the predictions of the naive model shown in Figure 1,early agents are more accurate under q = , but later agents are more accurate under q = . The point of overtaking happens at a later round in practice than in theory, because ourexperimental subjects rely more on their private signal than predicted by the naive model, consistent with the meta-analysis of Weizsäcker (2010).Our experiment was designed to compare long-run learning accuracy on different networksinstead of measuring individual behavior. We do not directly test alternate behavioral modelsfor two reasons. First, given the complex signal and network structures, such tests will bevery noisy in our data. Second, because the spaces of possible networks and actions have veryhigh dimension, determining the action each agent would take assuming common knowledgeof rationality is computationally infeasible. However, in the next subsection we providesome evidence that our findings are driven by herding under naive inference rather thanother behavioral mechanisms. In this section, we present three pieces of evidence suggesting that naive herding is themechanism responsible for the difference in learning accuracy between the two treatments. (1) Distribution of overall accuracy . Figure 2 in Appendix 7 plots the distributionssubjects who correctly guess the state in the q = and q = treatments, across differenttrials. The distribution under q = has more extreme values than the one under q = ,and also a larger standard deviation (11.36 percentage points versus 9.12 percentage points).This is suggestive evidence for naive herding. With denser networks, we simultaneously findmore trials where agents do very badly overall (from herding on the wrong state) and moretrials where agents do very well overall (from herding on the correct state). (2) Effect of misleading early signals on the accuracy of later agents . Call aprivate signal misleading if it is positive while the state is L, or if it is negative while thestate is R. If naive herding is the mechanism, we would expect misleading signals received by The overall frequency of agents going against their signals was 36.8% of the predicted frequency underthe naive model. m j of the first fifth of agents who receive misleading signals in trial j , and its interaction effect with network density. That is, we estimate˜ y j = β + β q j + β m j + γ ( q j m j ) + (cid:15) j . The difference in the marginal effect of a misleading early signal for learning accuracy on thedense network ( q = ) versus on the sparse network ( q = ) is γ in the above specification.As reported in Table 4 in Appendix 7, we find γ = 0 .
05 with a p -value of 0.0923. Thismeans each misleading signal among the first fifth of agents harms the average accuracy ofthe last fifth of agents in the same trial by an extra 2.5 percentage points in dense networkscompared to sparse networks. (3) Average uncertainty . Based on simulation evidence, we expect naive agents toexhibit more agreement within denser networks. To test this prediction in the data, weconsider for each game a set of 30 moving windows centered around periods 6, 7, ... 35,with each window spanning 11 consecutive periods. For each game j and each window w , wecompute r j,w ∈ { , , ..., } as the fraction of 11 agents in the window who guessed R, andwe let u j,w := r j,w · (1 − r j,w ) be a measure of uncertainty within the window. In windowswhere agents exhibit a greater degree of agreement, we will see a lower u j,w . Under herding,we expect lower uncertainty on denser networks, as higher density accelerates convergenceto a (possibly mistaken) social consensus. We find in the data that the average u j,w is loweramong sparse networks than dense networks for all but 1 out of 30 windows. Numerically,the naive herding theory predicts lower average u j,w on denser networks in all 30 windows. Our study provides experimental evidence on how the density of the observation networkaffects people’s long-run accuracy in social-learning settings. We find that sparser networksdouble the accuracy gains from social learning relative to denser networks. This finding alsoprovides indirect evidence supporting naive inference.While the rational model predicts correct asymptotic social learning with minimal as-sumptions on the social network, we conjecture that in practice, many structural propertiesof the network can substantially alter long-run accuracy. Our empirical findings support this The value of u j,w would be unchanged if we instead defined r j,w as the fraction of the 11 agents inwindow w who correctly guessed the state. References
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Appendix6 Theoretical Predictions in the Experimental Envi-ronment
Consider 40 rational agents on a random network where each agent is linked to each of herpredecessors of the time, i.i.d. across link realizations. Agents know their own neighborsbut have no further knowledge about the realization of the random network. The signalstructure and payoff structure match the experimental design in Section 3.We provide a lower bound for the accuracy of agents 33 through 40 in the unique PBEof the social-learning game. We first show that when every player uses the equilibriumstrategy, all agents learn at least as well as when everyone uses any constrained strategy that chooses an action based on only own private signal and the action of the most recentneighbor. We then exhibit payoffs under one such strategy, which give a lower bound onrational performance.Fix an arbitrary sequence of constrained strategies ( σ i ) where σ i : S i × { , , ∅} → ∆( { , } ) is only a function of i ’s signal s i and the action of the most recent predecessor that i observes ( σ i ( s i , ∅ ) refers to i ’s play if i does not observe any predecessor). Let a i denote i ’s(random) action induced by this sequence of strategies. Let a i denote i ’s (random) actionwhen all agents use the PBE strategy. Claim . For all i , P [ a i = ω ] ≥ P [ a i = ω ] . Proof.
The proof is by induction on i and the base case of i = 1 is clear. Suppose the claimholds for i = 1 , ..., n . Conditional on agent n + 1 observing no predecessors, the claim againholds as in the base case, so we can check the claim conditional on n + 1 observing at leastone neighbor.Let j be the most recent neighbor that n + 1 observes. Then the rational agent observes s n +1 , a j for some j ≤ n , and perhaps some other actions while the constrained agent only uses s n +1 and a j in decision-making, where P [ a j = ω ] ≥ P [ a j = ω ] by the inductive hypothesis. Bygarbling the observed action a j , the rational agent could construct a random variable with the12ame joint distribution with ω as the less accurate action a j . Ignoring information other than s n +1 and the garbled a j , the rational agent n +1 could therefore follow a strategy that does aswell as agent n +1 under the strategy profile ( σ i ). So we must have P [ a n +1 = ω ] ≥ P [ a n +1 = ω ]when everyone uses the PBE strategy.We then numerically compute the values for P [ a i = ω ] under the optimal constrainedstrategy, which are displayed in Table 2.agent numberprobability correct 33 34 35 36 37 38 39 400.9685 0.9695 0.9705 0.9714 0.9723 0.9731 0.9739 0.9746Table 2: Lower bounds on the accuracy of rational agents on dense networks. Consider 40 naive agents on a random network where each agent is linked to each of herpredecessors with probability q , i.i.d. across link realizations. The signal structure and payoffstructure match the experimental design in Section 3.We will compute the accuracy of each agent by a recursive calculation. Because naiveagents’ actions do not depend on the order of predecessors, behavior depends only on thenumber of agents who have played L and the number of agents who have played R as wellas the network. We will compute the distribution over the number of agents from the first n who have played L and the number who have played R recursively.Assume the state is R . Let P ( k, k ) be the probability that k of the first n agents play L and k of the first n agents play R . We define P ( k, k ) = 0 if k < k < . The posteriorlog-likelihood of state R for a naive agent observing one action equal to R (and no signal) is ‘ = 2 σ · µ + σφ ( − µ/σ )1 − Φ( − µ/σ ) , where Φ and φ are the distribution function and probability density function of a standardGaussian random variable, respectively.Then we have the recursive relation P ( k, k ) = P ( k − , k ) P i ≤ k − ,i ≤ k B ( i, k − , q ) B ( i , k , q )Φ( σ ( i − i ) ‘ − µσ ) + P ( k, k − P i ≤ k,i ≤ k − B ( i, k, q ) B ( i , k − , q )[1 − Φ( σ ( i − i ) ‘ − µσ )] , where B ( i, k, q ) is the probability a binomial distribution with parameters k and q is equal to i . The first summand gives the probability of agent k + k choosing L after k − L and the remainder choose R , and the second summand gives the probability of agent k + k choosing R after k predecessors choose L and the remainder choose R . The binomialcoefficients correspond to the possible network realizations. Here we use naive inference,which implies that only the number of observed agents choosing each action matters forbehavior and not their order.From these distributions P ( · , · ) we can compute the probability that agent n chooses thecorrect action R : n X k =0 P ( k, n − k ) X i ≤ k,i ≤ n − k B ( i, k, q ) B ( i , n − k, q )[1 − Φ( σ ( i − i ) ‘ − µσ . These probabilities, which we compute numerically, are displayed in Table 3 for agents 33through 40.agent numberaccuracy with q = 1 / q = 3 / Relegated Figures and Tables
Distribution of overall accuracy for trials on sparse networks (p = 0.25) fraction of subjects correctly guessing the state F r equen cy Distribution of overall accuracy for trials on dense networks (p = 0.75) fraction of subjects correctly guessing the state F r equen cy Figure 2: Histograms of fractions of agents correctly guessing the state15 ependent variable:
FractionCorrectMisleadingEarlySignals 0.014(0.017)NetworkDensity 0.033(0.082)MisleadingEarlySignals × NetworkDensity − ∗ (0.030)Constant 0.768 ∗∗∗ (0.045)Observations 260R ∗∗ (df = 3; 256) Note: ∗ p < ∗∗ p < ∗∗∗ p < Instructions and an example choice follow. To avoid confusion, the instructions were modifiedfor player 1 in each round to exclude discussion of social observations.16 (cid:29) (cid:3) (cid:29)(cid:3) VRFLDO(cid:16)OHDUQLQJ(cid:3)JDPH (cid:17)(cid:3)$W(cid:3)WKH(cid:3)VWDUW(cid:3)RI(cid:3)HDFK(cid:3)URXQG(cid:15)(cid:3)WKH(cid:3)FRPSXWHUZLOO(cid:3)UDQGRPO\(cid:3)FKRRVH(cid:3)D(cid:3) GLUHFWLRQ(cid:3) IRU(cid:3)WKH(cid:3)URXQG(cid:15)(cid:3)ZKLFK(cid:3)LV(cid:3)HLWKHU(cid:3)/()7(cid:3)RU(cid:3)5,*+7(cid:17)(cid:3)(DFK(cid:3)GLUHFWLRQ(cid:3)LV(cid:3)HTXDOO\OLNHO\(cid:3)WR(cid:3)EH(cid:3)FKRVHQ(cid:17)(cid:3)$(cid:3)QHZ(cid:3)GLUHFWLRQ(cid:3)LV(cid:3)FKRVHQ(cid:3)IRU(cid:3)HDFK(cid:3)URXQG(cid:15)(cid:3)ZKLFK(cid:3)GRHVQ(cid:10)W(cid:3)GHSHQG(cid:3)RQ(cid:3)SUHYLRXVURXQGV(cid:17)(cid:3) (cid:3) ,Q(cid:3)HDFK(cid:3)URXQG(cid:15)(cid:3)D(cid:3)QXPEHU(cid:3)RI(cid:3)SDUWLFLSDQWV(cid:3)IURP(cid:3)$PD]RQ(cid:3)7XUN(cid:3)ZLOO(cid:3)WDNH(cid:3)WXUQV(cid:3)JXHVVLQJ(cid:3)WKH(cid:3)GLUHFWLRQ(cid:17)(cid:3) VLJQDO ·(cid:17)(cid:3)’LIIHUHQW(cid:3)SDUWLFLSDQWV(cid:3)ZLOO(cid:3)KDYHGLIIHUHQW(cid:3)VLJQDOV(cid:17)(cid:3):KHQ(cid:3)WKH(cid:3)GLUHFWLRQ(cid:3)LV(cid:3)/()7(cid:15)(cid:3)\RXU(cid:3)VLJQDO(cid:3)WHQGV(cid:3)WR(cid:3)EH(cid:3)D(cid:3)PRUH(cid:3)QHJDWLYH(cid:3)QXPEHU(cid:17)(cid:3):KHQ(cid:3)WKHGLUHFWLRQ(cid:3)LV(cid:3)5,*+7(cid:15)(cid:3)\RXU(cid:3)VLJQDO(cid:3)WHQGV(cid:3)WR(cid:3)EH(cid:3)D(cid:3)PRUH(cid:3)SRVLWLYH(cid:3)QXPEHU(cid:17) (cid:3) (cid:29)(cid:3)6LQFH(cid:3)WKLV(cid:3)LV(cid:3)D(cid:3)