Payo¤ consequences of social learning: A meta analysis Georg Weizsäcker (LSE) October 2006 I am grateful for funding by the ESRC, grant 1-SEC-T004
Motivation: Why a meta analysis on social learning? "Experiments on social learning are overstudied." Anonymous economist I, 2005 "So, do people learn from others when they should?" Anonymous economist II, 2006 We do not know the answer – when should they? This depends on other players’ unknown strategies. (To generate an answer, we relied on structural models or equi- librium predictions.) ! Use lots of data to estimate the expected payo¤ from learning from others, for a variety of situations.
Motivation II: We lack estimates the payo¤ impact of identi…ed biases The expected payo¤ estimates may also help to quantify the importance of behavioral di¤erences between people/situations: How much more or less would people earn if they behaved di¤erently? � Giving excessive weight to own information � Not realizing that others have learned � Trembling
� Higher/lower success in later positions of the game � Others? The problem there is, again, that when considering the payo¤ between di¤erent situations/behaviors, we need to control for the value of the available information: The likelihood ratios of the relevant states of the world are di¤erent in di¤erent situations. ! Control for the expected payo¤ from the available actions.
Games: All games follow the basic model by Bikhchandani, Hirshleifer and Welch (1992) and were conducted as controlled experiments. The …rst study by Anderson and Holt (1997, symmetric treatment) has the following parameters. � t = 0 : Nature draws one of two states of the world ("urns"), ! 2 f A; B g , with Pr( A ) = 1 = 2 . Of the balls in urn A , a fraction q A = 2 = 3 is labelled a and 1 � q A is labelled b . Urn B , analogously, has fractions of q B = 2 = 3 labelled b and 1 � q B labelled a . Nature’s draw is not revealed. � t = 1 : The …rst player makes predictions about the state of the world ("predicts the urn"): The player …rst receives a private signal, in the form of a ball drawn from the true urn. The player then chooses an action d 1 2 f A; B g .
� t = 2 : The next player observes d 1 , receives a signal from the same urn, and makes a prediction. � t = 3 :::T : (Etc. for later players.) All T = 6 players observe the predictions by all previous players, receive a signal, and make a prediction. E.g. for player 6 , the information could be AAAAAa , or ABAAAb . � If a player’s prediction coincides with the true urn, he or she gets a …xed amount U , here normalized to 1 . Otherwise, he or she gets u; normalized to 0 .
Data: I re-formatted the raw data from 12 di¤erent studies, all of which had the participants play this game in at least one treatment, with only slight modi…cations. � Anderson/Holt (1997, N = 810) : 54 participants playing 15 repetitions of the game. 18 participants played the symmetric version described above, and 36 participants played an asymmteric treatment with q A = 6 = 7 and q B = 2 = 7 . � Willinger/Ziegelmeyer (1998, N = 324 ): Replication of Anderson/Holt experi- ment, with q � q A = q B = 0 : 6 , 36 participants, and 9 repetitions. � Hung/Plott (2001, N = 890 ). Replication of Anderson/Holt with T = 10 and q = 2 = 3 , in three treatments with minor di¤erences in the experimental implementation. 40 participants and 22 : 25 repetitions on average.
� Nöth/Weber (2003, N = 9834 ). Variant of the game, where q is drawn sep- arately for each player from f 0 : 6 ; 0 : 8 g , and unknown to other players. 126 participants, and about 78 repetitions on average. � Kübler/Weizsäcker (2004, N = 482 ). Variant where decision makers also decide whether or not they receive a private signal. 36 participants and 15 repetitions, with T = 6 and q = 2 = 3 . Observations are dropped if the participant requested no signal. � Drehmann, Oechssler and Roider (2005, N = 2789 ) 1840 participants played in 8 di¤erent Internet-based treatments with di¤erent signal precisions and di¤erent values for Pr( A ) . 267 participants were management consultants who played among each other, with T = 7 . In addition, 1573 participants of di¤erent
background (mostly students or graduates of universities) played in games with T = 20 (mostly). Participants played up to three repetitions. � Ziegelmeyer et al (2002, N = 810 ). 54 participants and 15 repetitions with T = 9 , Pr( A ) = 0 : 55 and q = 2 = 3 . Subjects in one of the two treatments also reported beliefs about the state of the world. � Cipriani and Guarino (2003, N = 161 ). Variant where the players have an outside option, modelled after the decision not to trade in …nancial markets. 48 participants and 10 repetitions, with T = 12 and q = 0 : 7 . Observations are dropped of a player or any of his/her predecessor chose the (suboptimal) outside option.
� Oberhammer and Stiehler (2003, N = 876 ). Variant where the subjects also announce their willingness to pay for playing the game, with T = 6 , q = 0 : 6 , 36 participants, and about 24 repetitions on average. � Alevy, Haigh and List (2005, N = 1647 ). Replication of both Anderson/Holt treatments, with T = 5 or T = 6 , 15 repetitions, and with subjects of di¤erent backgrounds: 55 …nancial market professionals, and 54 undergraduate students. Treatments also di¤ered with respect to gains/losses framing. � Goeree et al (2006, N = 8760 ). 380 participants play four di¤erent long versions of the game, with T = 20 and T = 40 , q = 5 = 9 and q = 2 = 3 , and an average of 22 : 7 repetitions.
� Dominitz and Hung (2004, N = 2270 ). 90 participants, games with T = 10 and q = 2 = 3 . 30 participants played 20 repetitions, and the other 60 participants reported belief statements during the last 10 out of 20 repetitions. After dropping some cases from two datasets, the meta dataset contains 29653 individual decision situations f s i g i , made by 2795 participants in 34 treatments in 12 separate studies. All of them follow the observation of a private signal and a (possibly empty) string of previous choices made in the analogous situation. In all of them there are two actions and two possible payo¤s. But note the variety in environments, instructions, and paths of play. For each decision situation s i , the dataset contains the participant’s choice d i as well as the true state of the world ! i .
Additional variables for decision s i : treatment i : Two individual decisions ( s i ; s j ) made by two participants are in the same treatment if: (i) The participants received the same instructions for the current game. (ii) The participants (and their opponents) are drawn from the same pool. sitcount i : Let I i be the information available in decision s i (observable history within the current game, private signal, e.g. BBAb ). sitcount i denotes the number of times that a decision situation with the same information I i occured within the same treatment. In treatments with A=B symmetry, this includes situations with the symmetric information. ( AABa is viewed as identical to BBAb .) The variable sitcount describes how many "identical" situations appear. Let e S ( s i ) be the set of all situations s j that are identical to s i , i.e. where the ( treatment j , I j ) description is the same.
This ignores information about history of play before the current run of the game. Under the assumption that history does not enter, the information about others’ signals is identical across e S ( s i ) because all that the participants know about the behavior of other players is re‡ected in ( treatment, I i ) . Note that sitcount does not compare situations across treatments. Since we know the underlying true state of the world in each case, we can calculate the value of disregarding the own signal. disreg i : Indicates that the participant chose not to follow the own signal. est_E[payo¤ j disreg] i : Averaging across situations s j 2 e S ( s i ) , what the participant would have earned if he or she had chosen disreg i =1. Due to the payo¤ normalization, this is the frequency of the "other" state of the world being true, across s j 2 e S ( s i ) . prop_disreg i : The proportion of disreg i =1, across situations s j 2 e S ( s i ) .
1 .8 prop_disreg .6 .4 .2 0 0 .2 .4 .6 .8 1 est_E[payoff|disreg] prop_disreg Fitted values Figure 1: Proportion of disregarding own signal by estimated payo¤. Note: Only situations with sitcount i > 10 are included (339 distinct situations). Regression includes squared and cubed x-variable, weighted by observations.
1 .8 prop_disreg .6 .4 .2 0 0 .2 .4 .6 .8 1 est_E[payoff|disreg] prop_disreg Fitted values Figure 2: Proportion of disregarding own signal by estimated payo¤. Note: Only situations with sitcount i > 10 are included (15592 observations).
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