Abstract
We experimentally investigate social learning in a two-agent prediction game with both exogenous and endogenous ordering of decisions on a continuous action space. We are first in comparing exogenous and endogenous ordering within one framework, which enables a direct comparison of both structures in terms of informational efficiency, strategic delay and welfare. More efficient observational learning leads to more accurate predictions in the endogenous setting and increases informational efficiency compared to an exogenous setting. However, strategic delay induces waiting costs that offset these benefits and lead to a parity of exogenous and endogenous ordering in terms of welfare results. Our results hold relevance for the efficient design of decision regimes in contexts characterized by continuous action spaces.
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Notes
Other studies using Anderson and Holt’s (1997) urn experiment include Willinger and Ziegelmeyer (1998), Anderson (2001), Hung and Plott (2001), Oberhammer and Stiehler (2001), Nöth and Weber (2003), Kübler and Weizsäcker (2004), Cipriani et al. (2005), Drehmann et al. (2005), Alevy et al. (2007), Goeree et al. (2007), Ziegelmeyer et al. (2008), Dominitz and Hung (2009), and Fahr and Irlenbusch (2011) for group players. Çelen and Kariv (2004) use the basic frame, yet implement continuous rather than binary signals.
Many situations such as investment, market entry or forecasting are better characterized by endogenous ordering of choices. For instance, consider financial analysts forecasting future values of an economic variable. Analysts with little confidence in their private information may wait and observe other forecasts, as they are able to choose the point in time of their forecast. Since other analysts’ forecasts might reflect valuable information, analysts acting later tend to adjust their forecasts using the previous ones. Overall, information efficiency thus potentially improves. See Gul and Lundholm (1995) for an elaboration of these examples.
The finiteness of the action set is explained by Bikhchandani et al. (1998) who state that informational cascades are likely to be most important for decision situations with “an element of discreteness or finiteness” (p. 159) and that individuals tend to sort actions in discrete categories, even when they are actually continuous.
There are a number of further models for endogenous ordering of choices, with Chamley (2004) providing an overview. Closest to our investigation is Zhang (1997), who extends the basic model by informing agents about the precision of private information that is correlated with the true state. Agents with more precise information face higher waiting cost and thus act first. Zhang shows that for any given precision informational cascades will always occur in equilibrium. The equilibrium is inefficient due to excessive delay and imperfect revelation of private information. Frisell (2003) in turn introduces pay-off externalities. Strategic delay is reduced, as the advantage of being well informed decreases the stronger the pay-off externalities. For a sufficiently negative pay-off externality, the worst-informed agent acts first.
In an investigation under which conditions an informational cascade becomes fully revealing, Lee (1993) shows that for a continuous action space and binary signals, inefficient herding does not occur, because no information is left unused. Smith and Sorensen (2000) extend this analysis and allow for heterogeneous preferences and a more general structure of private signals. They show that inefficient herding occurs only when private signals are bounded in strength. On the other hand, if private signals are unbounded in strength, i.e. at least for some agents private signals are arbitrarily precise, social learning is successful due to the “overturning principle”: If it is possible for agents to receive a private signal which is overturning the sum of available social information (the history of previous actions), social learning continues and in the long run, beliefs and actions converge to the true state.
The time bonus is displayed to participants in an easily accessible form. For low cost and high cost, “’10...2 * project value” is displayed, for signal dependent, subjects are shown the exact time bonus.
Note that the matching of players across the seven repetitions of both experiments is held constant to enable reputation effects related to individual preferences. For instance, consider a risk-averse player who develops a reputation of always deciding first. Understood by the second player, this should have a profound impact on the overall results. We chose this setting since the opportunity to build up reputation effects is a central feature of actual social learning environments where the set of participants remains mostly constant.
Participants had the following fields of study: humanities 11 %, Law studies 12 %, social sciences 14 %, science 26 %, economics and business administration 35 %. The mean age was 23.7 years, and 55 % of participants were female.
The minimum average prediction error could be lower if timing reveals information, i.e. subjects conduct anticipation. However, decisions might be given simultaneously in Exp2 in which case the minimum error amounts to 25. Given the share of decisions in the same period the theoretical minimum error without anticipation is 18.48 in high cost, 16.67 in low cost and 16.9 in signal dependent. Obviously, actual absolute average deviations far outreach these hypothetical values.
We apply a Skillings–Mack (SM) test since we have repeated measures given by the predictions of the participants and also missing values when participants did not give a prediction in a project. For Exp1 we find (\(\hbox {SM} = 9.933, \hbox {p(no-ties)} = 0.1275\) and empirical p(ties) \(\sim \) 0.1280); for high cost (\(\hbox {SM} = 4.004, \hbox {p(no-ties)} = 0.6761\) and empirical p(ties) \(\sim \) 0.658); for low cost (\(\hbox {SM} = 6.048, \hbox {p(no-ties)} = 0.4178\) and empirical p(ties) \(\sim \) 0.421) and for signal dependent (\(\hbox {SM} = 4.504, \hbox {p(no-ties)} = 0.06088\) and empirical p(ties) \(\sim \) 0.585).
These results are partially driven by the outliers mentioned; therefore, we tested for the differences after taking out these values. However, we find the same significant effects (average expectation for Exp1 is 45.0 with \(\hbox {t} = -3.7452\) and \(\hbox {p} = .0002\); average expectation for high cost is 47.4 with \(\hbox {t} = -2.3252\) and \(\hbox {p} = .0208\); two-sided \(t\) test).
We run a pooled OLS regression applying robust Driscoll and Kraay standard errors. Hence, we control for unobservable heterogeneity, heteroskedasticity, serial correlation in the idiosyncratic errors of order (2) and cross-sectional dependence.
For high cost the marginal effect of an additional period is estimated to reduce signal strength by \(-\)2.9 with a constant of 56.6. The effects is not significant (\(\hbox {t} = -1.65; \hbox {p} = .151\)). For low cost the coefficient is 2.55 and significant at the 5 % level (\(\hbox {t} = 2.85; \hbox {p} = .021\)), while the constant is 50. Applying fixed effects procedure also yields a significant decrease of signal strength estimated to be \(-\)6.56 points per period in high cost. For the other treatments results change only slightly.
The regression procedure is implemented as before. For high cost, we get a marginal effect of .025 points (\(\hbox {t} = -0.02; \hbox {p} = .981\)) and for low cost the marginal effect is \(-.133\) points (\(\hbox {t} = -0.25; \hbox {p} = .807\)), thus both are not significant. Using fixed effect procedure does not change coefficients substantially and significances remain the same.
Note that, for the analysis of second movers, relying solely on decisions following potentially rational predictions might be problematic. Players in a decision pair might have observed earlier irrational decisions thus causing project-interdependent assumptions regarding the co-player’s behavior. However, as this keeps the analysis simple and our results in this section are very robust, we refrain from integrating project-interdependent effects.
Although we showed that expectations of first movers in the high cost are not related to the decision period, second movers adjust their expectation significantly by a rate of -8.8 points (\(\hbox {t} = -3.72; \hbox {p} = 0.01\)). Results for low cost show an insignificant marginal effect of 1.5 points (\(\hbox {t} = 1.01; \hbox {p} = 0.352\)) for decision period on the second mover’s expectation. The results do not change substantially when we only use decisions of second movers following potentially rational decisions of first movers or when we use a fixed effects procedure.
Therefore, one might consider a level-k approach to define an appropriate model of behavior in our experiment (Nagel 1995; Stahl and Wilson 1994, 1995; Crawford and Iriberri 2007), thus rationalizing predictions conditional on the first mover’s assumed depth of reasoning. However, even interdependent expectations regarding the depth of reasoning of the co-player might not explain the differences in informational efficiency between Exp1 and Exp2. However, an extensive analysis regarding the expectation on the depth of reasoning is beyond the scope of this paper.
The specific timing strategies used in the endogenous game might be interpreted within Ivanov et al.’s (2009) scheme of different types of subjects classified by rules of thumb. For instance, Ivanov et al.’s self-contained subjects fail to draw inferences from other subjects’ behavior; this rule of thumb also applies to our subjects predicting continuously in the very first period, thus systematically ignoring the other player’s decision. Subjects strictly deciding after the other player might reflect the subject type Ivanov et al. call foresight, i.e. acknowledging the advantage of observing the other player in advance. Finally, their myopic type can be interpreted in our design as subjects making use of the additional information available as second movers without actively seeking to become second mover as would a foresight subject. While our setting does not allow for a distinct assignment of player types according to rules of thumb, this approach provides a further line of interpretation for different patterns of individual timing behavior, potentially complementing a level-k approach. A deeper analysis of the factors affecting the choice of a rule of thumb and their predictable power for subjects’ behavior could thus provide valuable additional insight and should be addressed in further research.
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Appendix
Appendix
1.1 Instructions for experiment 1
1.1.1 The game
In this game you and a co-player will estimate the value of a project. The value of the project consists of two parts: your own information and your co-player’s information.
Your information and the information of your co-player are randomly determined numbers between 1 and 100. Therefore, project value that you have to estimate is always between 2 and 200. All of the possible information is equally likely.
There are seven projects in which you will estimate the project value. In every project, it will be randomly determined if you or your co-player will give the estimation first. The first estimation is always displayed to the other player. Once both players have made their estimation, the next project begins.
You will have the same co-player in all projects. You have a maximum of 1 min for each estimation. If you do not type in an estimation in time, you will not receive a payoff for this project!
1.1.2 The payoff
You will receive a precision bonus in every project, which depends on how precise your estimation was. The precision bonus depends on the deviation of your estimation from the correct project value. 1000 ECU equals a payoff of 1.70€. Additionally, you will receive an independent payoff of 2.50€. The following table clarifies the precision bonus:
Distance from correct project value | Precision bonus (in ECU) | Example: The project value is 100 | |
|---|---|---|---|
Your estimation was... | The precision bonus is... | ||
0–5 points | 2000 | ...96 | ...2000 ECU |
6–10 points | 1600 | ...109 | ...1600 ECU |
11–15 points | 1200 | ...87 | ...1200 ECU |
16–20 points | 800 | ...118 | ...800 ECU |
21–25 points | 400 | ...75 | ...400 ECU |
From 26 points | 0 | ...12 | ...0 ECU |
Example
At the beginning of a project your information is 45. Therefore, you know that the project value is at least 45 plus the information of your co-player. Your co-player decides before you and estimates a project value of 120. You decide after him and estimate a project value of 105. The correct project value is 95. Thus, you receive a precision bonus of 1600 ECU, as your estimation deviated from the correct project value by ten points.
1.2 Instructions for experiment 2
Note that the instructions refer to the high cost treatment. The differences from the other treatments are indicated as follows: information in square brackets corresponds to the signal dependent treatment, braces corresponds to the low cost treatment.
1.2.1 The game
In this game you and a co-player will estimate the value of a project. The value of the project consists of two parts: your own information and your co-player’s information.
Your information and the information of your co-player are randomly determined numbers between 1 and 100. Therefore, the project value that you have to estimate, is always between 2 and 200. All of the possible information is equally likely.
There are seven projects in which you will give an estimation of the project value. All projects have five rounds of 2 min each. You must decide in which round you want to give your estimation.
All projects end once both players have given their estimation. Subsequently, the next project starts. You will have the same co-player in all projects. The following table provides an example of the course of the game:
Project 1 | Project 2 | |||||
|---|---|---|---|---|---|---|
Round 1 | Round 2 | Round 3 | Round 4 | Round 5 | Round 1 | ... |
2 min | 2 min | 2 min | 2 min | 2 min | 2 min | ... |
At the beginning of each project, both players receive their information. Your co-player’s information is unknown to you. You will have to decide in every round if you want to give an estimation (YES/NO). If you allow 2 min per round to elapse, you will not get a payoff for this project! If you choose NO, please wait for the next round of the project. If you choose YES, you will be told if your co-player will give an estimation in the same round. Subsequently, you will enter your estimation. Meanwhile, you will see an overview of the last rounds and, if applicable, the estimation of your co-player. If you decide before your co-player, your estimation will also be shown to him. The following table exemplifies the course of the game and your possible actions:
Round 1 | Round 2 | Round 3 | Round 4 | Round 5 | |
|---|---|---|---|---|---|
Action by player 1 | NO | NO | NO | YES! Enters the estimation | Project completed! |
Action by player 2 | NO | YES! Enters the estimation | Wait for the co-player... | ||
1.2.2 The payoff
The total payoff consists of two parts: the accuracy bonus (I.) and the time bonus (II.). For every round you wait with your estimation, your time bonus will be reduced. The precision bonus is higher, the closer your estimation gets to the correct project value. 1000 coins equal a payoff of 0.80€ {1.20€}, [1.20€]. Additionally, you will receive an independent payoff of 2.50€.
I. Precision bonus
You receive a bonus in every project which depends on the precision of your estimation, based upon its distance to the correct project value. The following table clarifies the precision bonus:
Distance from the correct project value | Precision bonus (in ECU) | Example: The project value is 100. | |
|---|---|---|---|
Your estimation was... | The precision bonus is... | ||
0–5 points | 2000 | ...96 | ...2000 coins |
6–10 points | 1600 | ...109 | ...1600 coins |
11–15 points | 1200 | ...87 | ...1200 coins |
16–20 points | 800 | ...118 | ...800 coins |
21–25 points | 400 | ...75 | ...400 coins |
ab 26 points | 0 | ...12 | ...0 coins |
II. Time bonus
You receive a time bonus in every project, depending on the size of the project value {on the size of your information}. For every round you wait with your estimation, your time bonus will be reduced. The following table clarifies the time bonus:
Estimation in round | Time bonus | Example: The project value is 100. | |
|---|---|---|---|
Estimation in round... | Time bonus... | ||
1 | 10{5} \(\times \) project value [10 \(\times \) information] | ...1 | ...1000{500} ECU [1000 ECU] |
2 | 8{4} \(\times \) project value [8 \(\times \) information] | ...2 | ...800{400} ECU [800 ECU] |
3 | 6{3} \(\times \) project value [6 \(\times \) information] | ...3 | ...600{300} ECU [600 ECU] |
4 | 4{2} \(\times \) project value [4 \(\times \) information] | ...4 | ...400{200} ECU [400 ECU] |
5 | 2{1} \(\times \) project value [2 \(\times \) information] | ...5 | ...200{100} ECU [200 ECU] |
Example
At the beginning of a project, your information is 45. Therefore, you know that the project value is at least 45 plus the information of your co-player. Your co-player decides before you and estimates in round 3 that the project value is 120. You decide in round 4 and estimate that the project value is 105. The correct project value is 95. Therefore, you receive a time bonus of 380 (time bonus in round 4 \(=\) 4 \(\times \) project value) {190 (time bonus in round 4 \(=\) 2 \(\times \) project value)} [180 (time bonus in round 4 \(=\) 4 \(\times \) information)]. Additionally, you receive a precision bonus of 1600 coins, as your estimation deviates from the correct project value by ten points.
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Meub, L., Proeger, T. & Hüning, H. A comparison of endogenous and exogenous timing in a social learning experiment. J Econ Interact Coord 12, 143–166 (2017). https://doi.org/10.1007/s11403-015-0156-6
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DOI: https://doi.org/10.1007/s11403-015-0156-6