Group size effect in a macro-economic model

Group size effect in a macro-economic model

28-06-2013

by Rob Goedhart*

Supervisors:

Prof. Jan Tuinstra

Msc. Tomasz Makarewicz

* Student at the Faculty of Economic and Business at the University of Amsterdam Contact: Robgoedhart93@gmail.com

Bachelor's thesis Econometrics Student nr: 10069984

by Rob Goedhart*

Supervisors:

Prof. Jan Tuinstra

Msc. Tomasz Makarewicz

Abstract

In this paper we examine the effect of the number of agents, the group size, in an overlapping generations economy. We use data of two laboratory experiments with incentivized human subjects, one with a single-agent and one with a multi-agent treatment. In both experiments all participants had to predict the inflation rate for 50 consecutive periods in an overlapping generations model, where they were rewarded for the accuracy of their predictions. After investigating this experimental data, multiple simulations are run with different prediction strategies for both the single-agent and multi-agent treatments. We find that the

performances (and thus earnings) of the participants in the single-agent treatments are more diverse, and also the realized inflation rates are more volatile than in multi-agent treatments. Another finding is that prediction strategies do not always perform equally well when used in a different experimental environment.

1 Introduction

Expectation of economic agents has been of interest in several past studies, as they can influence the aggregate market outcomes. A good example of this is the stock market, as is also mentioned by Heemeijer et al. (2009). If the price of a stock is expected to rise, it would be profitable to buy the stock and sell it in the future for a higher price. However, if multiple agents have the same expectations, the demand for the stock will also rise which leads to a price increase. It is also possible that a higher expectation leads to a lower realized price. This happens on markets where producers have to make decisions on their output level. If producers expect high prices for their products, they will

produce more which in turn will lower the realized market price. As expectations of these agents can affect the aggregate market outcome, it is valuable to evaluate their forecasting behaviour.

Where van de Velden (2001) finds that a larger group size has a stabilizing effect in a cobweb economy, it is yet unclear whether similar findings hold for an overlapping generations economy. In a market with relatively few agents, each individual will have more influence on the aggregate market outcome than in a market with more active agents. This might lead to differences in market volatility, or in prediction strategies of the agents. We investigate this effect for an overlapping generations model also used by Heemeijer et al. (2012). In their research the authors used data of a single-agent

experiment where each participant was evaluated individually. For 50 consecutive periods, the participants had to predict future inflation rates, and the aggregate market outcome was based on these individual predictions. The participants were rewarded for the accuracy of their predictions. Next to this available data set, we use data of a second experiment (conducted in the CREED experimental laboratory of the University of Amsterdam) to investigate the effect of the number of agents on the market stability. The setup of this experiment is similar to the previous one, but with six participants having influence on the aggregate market outcome instead of one (multi-agent treatment). In both experiments two different treatments are used, which differ in monetary growth rate (theta).

In this research we investigate the effect of the group size in this model. At first the available experimental data is evaluated. As a second step simulations are run with subjects following simple heuristics as prediction strategies. The reason behind this is that perfect rationality of participants can no longer be assumed. Where neoclassical

economic theory assumes that expectations are rationally formed by individuals, Anufriev and Hommes (2012) use data from earlier laboratory experiments to show otherwise. The authors find that individuals do not behave fully rational, but that they follow simple heuristics. After we have evaluated the experimental data, we run simulations with different prediction strategies and compare their results to the experimental data.

We found that the multi-agent treatments are less volatile when it comes to performance of the participants. The participants in these multi-agent treatments have similar earnings, especially within the same groups. In the single-agent treatments larger differences in performance are found in both treatments. Where the top

performers are participants in the single-agent treatments, the worst performers are of this treatment as well. Another important result is that prediction strategies don't always perform equally well in different environments. It appears that proper

performance in an individual environment are no guarantee for good results in a group environments as well. These results hold for both the simulations and the experiments.

The rest of the article is structured as follows. In the next section a theoretical background is provided on the model and the forecasting rules. After that, a description of the experiment is given with the used models and formula's. In the fourth section the results of the experimental data can be found, while section 5 contains the results of the simulations. In section 6 the results are summarized and some concluding remarks are added. The appendix contains several illustrational graphs of the simulations and experimental data, as well as the results of the regressions.

2 The Overlapping Generations Model and Expectation Formation

In this section a summary is given of the underlying model of the used experiments and simulations, along with background info and derivation of this model. Next to this a short introduction to commonly used theoretical prediction strategies is given.

2.1 Overlapping Generations Model

In this research we make use of a standard overlapping generations model (OLG) with a structure identical to the one used by Bullard (1994), Schönhofer (1999), Tuinstra and Wagener (2007) and Heemeijer et al. (2012). It has two generations, a single

model it is assumed that each generation consists of a representative household that lives for two periods, one when they are young , and one when they are old. If c0,t and c1,t denote the consumption of generation t when they are respectively young and old, the households' preferences are given by utility function u(c0,t, c1,t). The income of the

household is given by w0 (w1) units of the consumption good when they are young (old). Households want to maximize their expected utility, by adjusting their savings in the first period. If households expect a high inflation rate for the next period, they will have the tendency to spend more in the current period, and save less. If they expect a low inflation however, they might prefer to save more for the next period. For the young generation at time t, their perceived lifetime budget constraint is given by

pt c0,t + 𝑝_𝑡+1𝑒 c1,t ≤ pt w0 + 𝑝_𝑡+1𝑒 w1 , where pt denotes the price of the consumption good in period t, and 𝑝𝑡+1𝑒 the expected price for the next period. To find the unique solution, it is assumed that the utility function u(c0,t, c1,t) is continuous, strictly increasing and strictly quasi-concave. The optimal solution is now denoted as (𝑐0,𝑡∗ , 𝑐1,𝑡∗ ), where both 𝑐0,𝑡∗ and 𝑐1,𝑡∗ are a function of the expected inflation 𝑝𝑡+1𝑒

𝑝𝑡

.

𝑆 �

𝑝_𝑡

�

The government determines the total money stock, denoted by Mt. If θ denotes the rate of growth of the money stock, this means that

M

= θ M

,

θ > 1.

with M0 given. In an equilibrium aggregate savings will equal the real money balances

𝑀𝑡

𝑝𝑡 = 𝑆 �

𝑝𝑡+1𝑒

𝑝𝑡 �.

Combining equations (2) and (3) gives an expression for the gross inflation rate at time t (denoted by 𝜋𝑡 ), in terms of θ and the expected inflation rates for periods t and t+1, 𝜋𝑡𝑒 and 𝜋𝑡+1𝑒 : 𝜋𝑡 = 𝜃 𝑆(𝜋𝑡 𝑒₎ 𝑆(𝜋_𝑡+1𝑒 _{) .} (1) (2) (3) (4)

Observe in equation (4) that when the expectations are constant, and thus 𝜋𝑡𝑒 equals 𝜋𝑡+1𝑒 , the inflation rate 𝜋𝑡 will equal the monetary growth rate 𝜃. However, constant expectations don't seem to be likely when the expectations do not (nearly) equal the realized inflation rates. It is now of interest how the expectations are formed.

2.2 Expectation Formation

As we noted earlier it would not be feasible to assume perfectly rational expectation formation. Perfect rationality would require that all agents have complete information about the model to form their expectations. Since it doesn't seem realistic to assume that this is the case, a few other expectations rules are considered. The first expectation formation rule to be considered is the naive expectations rule. If an agent uses this rule, his expectation will be based on the last observed inflation rate only. He will have expectations equal to this last observed value, and so there holds

Naive: 𝜋𝑡+1𝑒 = 𝜋𝑡−1 .

Note that 𝜋𝑡 depends on the expectations for period t as well as t+1, which means that when predicting 𝜋𝑡+1𝑒 , 𝜋𝑡 is still unknown.

A second expectation formation strategy is the strategy of adaptive expectations. This means that the agent forms his expectations based on his previous prediction, as well as the difference between his previous prediction and the last observed inflation rate

Adaptive: 𝜋𝑡+1𝑒 = 𝜋𝑡𝑒+ 𝛼(𝜋𝑡−1− 𝜋𝑡𝑒), 0 < 𝛼 ≤ 1 . The third expectation rule uses the k last observations to predict the future inflation rate. This is done by taking the average of these observations

Average: 𝜋𝑡+1𝑒 =1_𝑘∑𝑘𝑗=1𝜋𝑡−𝑗 .

Another commonly used prediction strategy is the trend-following strategy. If this rule is used, expectations are formed based on the last two observations by using the last realized inflation rate, but increasing with a proportion of the 'trend', the difference

(5)

(6)

between the last and the second last realized inflation rate

Trend-following: 𝜋𝑡+1𝑒 = 𝜋𝑡−1+ 𝛾(𝜋𝑡−1− 𝜋𝑡−2)

As final basic prediction rule is an anchor and adjustment heuristic, where the

fundamental inflation rate is used together with a proportion of the difference between the last realized inflation rate and this fundamental inflation rate.

Anchor and adjustment: 𝜋𝑡+1𝑒 = 𝜋𝑓+ 𝜇�𝜋𝑡−1− 𝜋𝑓� ,

where 𝜋𝑓 equals the fundamental inflation rate θ. Combinations of earlier mentioned rules are also possible, for example a strategy that makes use of realized inflation rates as well as previous predictions. This can be written in a more general form

𝜋𝑡+1𝑒 = 𝛼 + � 𝛽𝑗𝜋𝑡−𝑗 𝑚 𝑗=1 + � 𝛾𝑘𝜋𝑡−𝑘𝑒 𝑛 𝑘=0 ,

where m is the amount of lags used for the realized inflation, and n the amount of lags for the predictions.

3 The Experiment

The available data is obtained from two experiments conducted in the CREED

experimental laboratory of the University of Amsterdam. Participants had to predict the inflation rates for two periods ahead. A total of 51 predictions were made by each participant, leading to 50 realized inflation rates. In each period t (where the

participants have to submit 𝜋𝑡+1𝑒 ), the available information set consisted of the realized inflation rates up till period t-1, as well as their own predictions up till period t. The participants were given this information both illustrational and numerical. In both experiments the savings function S(∙) has the form of

𝑆(𝜋) = 𝛿 + (1 − 𝛿) 𝑤0

1 + (𝑣𝜋)1−𝜌𝜌 ,

where the parameter values in this case are equal to 𝛿 = 0.4, 𝑣 = 0.92, 𝜌 = 0.965 and

(8)

(9)

(10)

𝑤0 = 0.9. Both experiments also used two different treatments, with differences in the monetary growth rate. For the first treatment θ1 = 1.01 is used, and for the second treatment θ2 = 1.11. The only difference between the experiments is found in the determination of the realized inflation rate. In the first experiment, also analyzed by Heemeijer et al. (2012), each participant had an individual market, where the realized inflation rate was only influenced by that participants predictions. The formula used to calculate the inflation rates in this experiment is

𝜋𝑡 = 𝜃𝜀𝑡 𝑆(𝜋𝑡 𝑒₎ 𝑆(𝜋_𝑡+1𝑒 _{) ,}

with 𝜀𝑡 uniformly distributed on the interval [0.975, 1.025]. In the second experiment however, the determination of the realized inflation rate was based on the predictions of six participants. The average savings function of these six participants was used to calculate the realized inflation rate

𝑆̅(𝝅𝒕+𝟏𝒆 ) =∑ 𝑆�𝜋𝑡+1,𝑖 𝑒 _� 6 𝑖=1 6 and 𝜋𝑡 = 𝜃𝜀𝑡 𝑆̅(𝝅𝒕𝒆) 𝑆̅(𝝅_𝒕+𝟏𝒆 _{) ,}

with 𝜋𝑡+1,𝑖𝑒 the predicted inflation rate for period t+1, made by participant i, and 𝝅𝒕𝒆 the vector with predictions of all the participants in a group. The distribution of 𝜀𝑡 is the same as before. Participants were rewarded based on the accuracy of their predictions. For each prediction, the participants earned points calculated by the formula

𝑃𝑖(𝜋𝑡,𝑖𝑒 , 𝜋𝑡) = 𝑚𝑎𝑥{100 − 400|𝜋𝑡,𝑖𝑒 − 𝜋𝑡|, 0} ,

where Pi denotes the amount of points earned, and |𝜋_𝑡,𝑖𝑒 − 𝜋_𝑡| is the absolute value of the prediction error. It can be seen that participants were only rewarded for predictions within the 0.25 range of the realized inflation rate. In the end of the experiment the amount of points earned was converted into an amount of euro's. For each 200 points earned the participants received 1 euro, which in total could sum up to a maximum of 25 euro's for the experiment.

In total the first experiment is completed by 32 participants, with 16 participants in both the low and high theta treatment. The second experiment consisted of seven groups with θ1 = 1.01, and six groups with θ2 = 1.11. Each group consisted of six

participants, giving a total of 78 participants. This means that both experiments together contain data of 110 participants.

(12)

(13)

(a) (b)

Figure 1: (a): Single-agent experiment, participant S1. (b): Multi-agent experiment, participant GS3P3. The blue lines describe the predictions of the participants, while the red line shows the realized inflation rates in their experimental markets.

4 Results

This section starts with evaluations of the experimental data, which is divided in

multiple subsections. It starts with a small recapitulation on the experiment, along with some notation. The other subsections discuss the realized inflation rates, the learning process, the payoffs, the coordination and the individual prediction strategies.

4.1 Experimental Data

As mentioned earlier, the participants in the experiment had to predict the inflation rates for 50 consecutive periods. With these predictions the realized inflation for that period was calculated, and the participants received points for the accuracy of their prediction, calculated by formula (14). The experiment was done for four different treatments. Two multi-agent treatments, where the prediction of six participants lead to one realized inflation rate, and two single-agent treatments, where every participant represents a market. Next to differences of single-agent and multi-agent treatments, there were differences in the monetary growth rate θ, as mentioned in section 3. Together this gives us four treatments, which we will denote as Group Stable (GS),

Group Unstable (GU), Individual Stable (S) and Individual Unstable (U), where stable and unstable correspond to low and high theta respectively, and group and individual

correspond with multi-agent and single-agent.

An example of the experimental data is shown illustrationaly in figure 1, where the predictions of participant S1 (treatment S, participant 1) and GS3P3 (treatment GS,

GS (n = 7) S (n = 16) GU (n = 6) U (n = 16)

Mean Mean Mean Mean

Theoretical 1.00 Theoretical 1.00 Theoretical 11.00 Theoretical 11.00

Median 1.46 Median 1.99 Median 13.62 Median 20.27

Average 1.44 Average 4.20 Average 13.62 Average 19.48

Std. Dev. 0.49 Std. Dev. 5.12 Std. Dev. 0.50 Std. Dev. 4.90

Minimum 0.84 Minimum 0.78 Minimum 12.96 Minimum 12.68

Maximum 2.18 Maximum 20.08 Maximum 14.20 Maximum 27.13

Variance Variance Variance Variance

Theoretical 2.13 Theoretical 2.13 Theoretical 2.57 Theoretical 2.57 Median 104.42 Median 262.37 Median 455.33 Median 2312.51 Average 97.61 Average 837.63 Average 470.33 Average 2251.62 Std. Dev. 76.28 Std. Dev. 1313.92 Std. Dev. 142.31 Std. Dev. 1415.83 Minimum 7.81 Minimum 2.94 Minimum 257.31 Minimum 222.94 Maximum 193.93 Maximum 4980.79 Maximum 640.69 Maximum 4713.50

Table 1: Statistics regarding the realized inflation rates in the experiment, divided in the four categories Stable, Group Stable, Unstable and Group Unstable. For the mean and variance of the realized inflation rates per category, a list is made including their average (Avg.), standard deviation (Std. Dev.), minimum (Min.), maximum (Max.) and median. Also included are the theoretical mean and variance for each treatment. group 3, participant 3) are graphed with the realized inflations in their experimental market (both in percentage points). The used prediction strategy of S1 is a commonly used one in the experiment. The participant starts rather naive to get an idea of the market, after a few periods he switches to a more adaptive expectation formation, and in the end his predictions are (nearly) constant. For GS3P3 we see that this participant starts naive also, but he was not able to keep the inflation rates stable.

4.1.1 Realized Inflation Rates

For all group and individual experiments the mean and variance of the 501 _realized inflation rates are computed. This gives us the average and variance of 45 series of realized inflation rates: GS (7 groups), S (16 participants), GU (6 groups) and U (16

1 _{A small note to be made towards the data is that in the GS treatment, group 4, 5, 6 and 7 only} consist of 43 realized inflation rates, as the experiment was not fully completed for these groups. The means and variances for these groups are computed for this sample of 43 observations.

participants). After this the average, standard deviation, minimum, maximum and median of this mean and variance are determined per treatment. With this data a small overview is created for the different treatments, where the treatments are defined as before (GS, GU, S, U). An overview of the computed statistics can be found in table 1. Next to these statistics, the table also includes the theoretical mean (fundamental inflation) and the theoretical variance (based on theta and the disturbance, when all predictions equal the fundamental inflation). All values given are based on inflation percentages, as will be the case in the rest of this paper.

In table 1 we observe that for both the low and the high theta treatments, similar differences between the agent and multi-agent experiments hold. In the single-agent experiments the minimum mean and variance are lower than in the multi-single-agent experiments, while their maxima are higher. By looking at the standard deviations of the mean and variance, it can be seen that the variations between different experiments in the same treatment are much smaller in the multi-agent experiments than in the single-agent experiments. The median variances of the single-single-agent experiments are also larger than their multi-agent counterparts, which could indicate that the multi-agent

experiments are in general more stable than the single-agent experiments.

4.1.2 Learning Process

After a brief overview of the outcomes of the experiment, it is interesting to evaluate the performance of the participants. To start this, we calculated the absolute prediction errors (i.e. the absolute difference between their prediction and the realized inflation rate) of all the participants. With this data we determined the medians of these prediction errors per period per treatment, and illustrated this in figure 2. We

considered the median prediction error because this is less sensitive for outliers than the average. The medians are compared between single-agent and multi-agent

experiments of the same treatment to identify possible differences between them. The first noticeable finding is that for all four treatments the median forecast error is on average lower in the second half of the experiment than in the first half. This is due to learning of the participants, as they can adapt their prediction strategies if previous predictions turn out not to be accurate. In figure 2 a horizontal dashed line is drawn at a prediction error of 25, indicating the point where payoffs go to zero if the prediction error is larger (based on formula (14)). If the median prediction error falls

Figure 2: In the figure the median prediction errors per category are illustrated over time. As can be seen the errors tend to drop with the time. Above the horizontal dashed line of 25 earnings are zero. The vertical dashed line indicates the division of the two halves of the experiment.

below this level, it means that at least half of the participants in that treatment have earnings larger than zero in that period. The vertical dashed line indicates the separation of the first and second half of the experiment.

To evaluate differences of these errors between the multi-agent and single-agent experiment, we used a Wilcoxon signed-rank test. With this test we tested for

differences between GS and S, as well as differences between GU and U. This is done for the complete experiment, as well as for both halves separately. We find that for both cases the differences are significant for the complete experiment. However, for the test of GS against S, we find that only the second half of the experiment is significantly

different. In the first half of the experiment the median forecast errors are similar, while in the second half they are significantly lower for the single-agent treatment. For the unstable treatments we find that in the first half of the experiment the median forecast errors of the single-agent treatment are higher than of the multi-agent treatment, while in the second half there is no significant difference. This means that the performance in the single-agent treatments improved relative to their multi-agent counterparts from the first to the second half of the experiment. Results of the test can be found in table 22, where the null-hypotheses equals equality of the two series. We made use of the two-tailed critical value for α = 0.05.

S vs GS U vs GU

Total 1 to 25 26 to 50 Total 1 to 25 26 to 50 Test-statistic -4.18 -1.58 -4.38 3.45 2.94 1.54 Critical value ± 1.96 ± 1.96 ± 1.96 ± 1.96 ± 1.96 ± 1.96 Conclusion Rejected Not rejected Rejected Rejected Rejected Not rejected

Table 2: Values of the test-statistics and critical values for the Wilcoxon signed-rank test, as well as their conclusions.

4.1.3 Payoffs

We computed average payoffs per period for every single participant2_{. In the same way} as for the realized inflations in the experiment, the mean, standard deviation, minimum, maximum and median of the payoffs in each treatment are computed. An overview of these statistics is given in table 3, including the payoffs if every participant in a

treatment would predict the fundamental inflation rate in all periods. In figure 3 the distribution of the payoffs in each category is shown in histograms.

Table 3 shows that the means are similar for the single-agent and multi-agent experiments with the same treatment, but that the payoffs vary more in the individual experiments. This can also be seen quite clearly in the histograms of figure 3. If the participants are sorted based on their payoffs, these results show the same. After sorting all the 110 participants in the experiments, it is found that the top 5 earners as well as the bottom 5, are participants in the individual experiments. The participants in the group experiments have a smaller diversity when it comes to payoffs.

To investigate the payoffs a bit further, the total payoffs are divided in two halves for each of the participants: the payoffs in periods 1-25 and periods 26-50. With this data it is possible to take a quick look at the learning process of the participants, by plotting the average payoffs per period in the second half against the average payoffs per period in the first half. This is illustrated in figure 4, where the black line represents the line x = y. It can be seen that the majority of the participants does better in the second half of the experiment than in the first half. This indicates that the participants are indeed 'learning'. Next to this we see that the best as well as the worst performing participants are participants in the single-agent treatments. The participants with the

2_{We chose average payoffs because of the incompletion of some groups (only 43 realized} inflation rates). The total payoffs are therefore not representative, but the average payoffs per period are still comparable for all participants.

GS S GU U Fund. 0.48 0.48 0.47 0.47 Median 0.31 0.39 0.17 0.11 Mean 0.34 0.34 0.17 0.16 Std. Dev 0.07 0.13 0.04 0.11 Min 0.21 0.07 0.11 0.06 Max 0.45 0.47 0.24 0.37

Table 3: Statistics about the average payoffs per period in the experiment, divided per treatment. 'Fund.' indicates the payoffs that would have been earned if every participant in a group predicted the fundamental inflation rate for each period.

largest difference between payoffs in the first and second half are subjects of the same treatments. There also seems to be evidence of clustering between the group

participants, which is also illustrated in figure 4.

To investigate this further, figure 5 shows data of the stable and unstable multi-agent treatments respectively. In the figures the participants are shown for each group. Figure 5 shows that there is indeed a form of clustering between groups. All participants in the same group appear to have similar payoffs in the experiment. The presence of clustering seems more obvious in the stable groups than in the unstable groups.

4.1.4 Coordination

Their appears to be evidence for some form of coordination in the multi-agent experiments. This effect was also investigated by Hommes et al. (2005). They found, using a similar experiment as this research, that different participants within one group seemed to coordinate on some common prediction. To quantify this coordination they consider the average individual quadratic forecast error (AIQFE)

1 𝑙 × (𝑚 − 10) � � (𝜋𝑡,𝑖𝑒 − 𝜋𝑡)2 𝑚 𝑡=11 𝑙 𝑖=1 ,

with 𝜋_𝑡,𝑖𝑒 _{the predicted inflation rate of participant i for period t made, l the number of} participants in each group and m the total number of predictions made. Note that the second summation starts at 11 because the first 10 observations are neglected to allow participants to learn how to predict the inflation rates more accurately. The authors now

Figure 3: Distribution of payoffs per treatment. It can be seen that the distribution of payoffs in the individual experiments is less diverse than in the group experiments. Average earnings are divided in ten intervals of equal width 0.05.

divide the previous formula in two different terms, the dispersion error (DE) and the

common error (CE)

1 𝑙 × (𝑚 − 10) � � �𝜋𝑡,𝑖𝑒 − 𝜋𝑡�2 𝑚 𝑡=11 𝑙 𝑖=1 =_{𝑙 × (𝑚 − 10) � � �𝜋}1 _𝑡,𝑖𝑒 _{− 𝜋�} 𝑡𝑒�2 𝑚 𝑡=11 𝑙 𝑖=1 + _{(𝑚 − 10) �}1 (𝜋�𝑡𝑒− 𝜋𝑡)2 𝑚 𝑡=11

(AIQFE) = (average DE) + (average CE)

where 𝜋�𝑡𝑒 equals the average prediction of the inflation rate for period t, made by the l participants. DE measures the dispersion between the individual predictions, while CE measures the average distance between the realized inflation rates and the mean prediction. If the average DE of a group is low, there is evidence of coordination.

We calculated these values for the available experimental data, and the results are stated in table 4. As the dispersion error is relatively low for all groups, it means that the differences between the participants predictions are low in all groups. For example, a common error of 83% means that 83% of the average quadratic forecast error can be attributed to the quadratic difference between the groups average prediction and the

0 5 10 15 20 Payoffs GS 0 2 4 6 Payoffs S 0 5 10 15 20 Payoffs GU 0 2 4 6 8 10 Payoffs U

Figure 4: Average payoffs per period in the second half against average payoffs per period in the first half of the experiment for all participants. Participants are divided in the four categories as before. Two clusters of GS and GU are indicated.

(a) Stable groups

(b) Unstable groups

Figure 5:Average payoffs per period in the second half against average payoffs per period in the first half of the experiment for all participants in the (a) stable group experiments, and the (b) unstable group experiments.

Group AIQFE Avg DE % of AIQFE Avg CE % of AIQFE GS1 84.21 19.76 17% 64.45 83% GS2 3.47 0.57 22% 2.90 78% GS3 104.26 29.08 16% 75.18 84% GS4 29.63 9.17 22% 20.46 78% GS5 4.82 1.40 23% 3.42 77% GS6 24.28 7.61 24% 16.67 76% GS7 15.67 4.78 12% 10.89 88% GU1 895.20 317.79 34% 577.41 66% GU2 258.29 91.88 37% 166.41 63% GU3 210.42 35.59 20% 174.83 80% GU4 183.15 35.30 20% 147.84 80% GU5 263.08 101.56 23% 161.52 77% GU6 350.22 72.12 25% 278.10 75%

Table 4: Table with data of the dispersion and common errors in the groups. Low values of dispersion errors indicate that the predictions in a group are relatively close to each other.

realized inflation. In that case only 17% can be attributed to differences of predictions between participants, which suggests the existence of coordination on a common prediction strategy. The dispersion error is in percentage smaller in the stable

treatments than in the unstable treatments, implying that there is more coordination in the stable treatments. This is in agreement with the data shown in figures 5 and 6.

4.1.5 Individual Prediction Strategies

The goal of this section is to divide the prediction behavior of the participants in different categories. This is done by measuring the average absolute one-period

difference in their predictions, an approach that is also used by Hommes et al. (2005). At first the data is calculated for the multi-agent experiments, where the average absolute one-period prediction differences of the participants is plotted against the average absolute one-period realized inflation difference in their group. This measures their degree of overreacting. The same has been done for the single-agent experiments. This data is shown in figure 7, which shows that 107 out of 110 participants have an average absolute prediction difference lower than the average difference of the realized inflation rate. Notice that the scales are different for the group and individual data. Histograms of this data are shown in figure 8, to give a better overview of the distributions per

(a) group stable (b) individual stable

(c) group unstable (d) individual unstable

Figure 7: Shown is the degree of overreacting for all participants, divided in the four treatments ((a), (b), (c) and (d)). The blue diamond represents the average absolute one-period prediction difference of the participant, while the purple bars represent the average absolute one-period change of the realized inflation rate. (e): Values of the Kolmogorov-Smirnov test-statistics, critical values and conclusions about the null-hypothesis of equal series.

(a) GS and S (b) GU and U

Figure 8: The data of figure 7 (average absolute one-period prediction differences) shown in histograms. (a): Group stable (blue bars) and individual stable (red bars). (b): Group unstable (blue bars) and individual unstable (red bars).

-5,005,00 15,00 25,00 35,00 45,00 0 1 2 3 4 5 6 7 GS 0,00 20,00 40,00 60,00 80,00 100,00 S 0,00 20,00 40,00 0 1 2 3 4 5 6 7 GU 0,00 50,00 100,00 U GS & S GU & U KS-statistic 0.366 0.410 Crit. Value 0.400 0.409 Conclusion Not rejected Rejected

Total GS S GU U

SS 26 (24%) 16 (38%) 8 (50%) 1 (3%) 1 (6%)

US 21 (19%) 8 (19%) 4 (25%) 4 (11%) 5 (31%)

SU 7 (6%) 7 (17%) 0 (0%) 0 (0%) 0 (0%)

UU 56 (51%) 11 (26%) 4 (25%) 31 (86%) 10 (63%)

Table 5: The table shows the distribution of the participants for different types of prediction behavior. Totals are given as well as numbers per category.

To test whether the differences between the group and individual experiments are significant, the Kolmogorov-Smirnov test has been used. At first the empirical distribution functions of the series of average absolute prediction differences are computed per treatment. After this, for both treatments the maximum difference between the group and individual empirical distribution function is determined and with this test-statistic the tests are performed with 𝛼 = 0.05. The test statistics and critical values are shown in figure 7 (e). In both cases the value of the test statistic is close to the critical value. For the stable treatments the null hypothesis of equal underlying probability distributions could not be rejected, implying that there is no significant difference between the volatility of the predictions in GS and S. However, for the unstable treatments this did not hold, as the null hypothesis was rejected. The histograms in figure 8 support these conclusions, as the differences in distributions of the average absolute one-period prediction differences seem to be larger for the unstable experiments than for the stable experiments.

As a next step the available data is used to classify the participants based on the volatility of their predictions. Again dividing the experiment in two halves, the subjects are denoted stable in a certain half if their average absolute one-period prediction difference are below 5%, and unstable if they are not. We chose the threshold value of 5% because in the second half of the experiment, around half of all the participants are then classified as stable. Small changes to the threshold value do not lead to major changes in the distribution of the participants along the different categories. With this information the participants are now divided in four different types: stable-stable (SS), stable-unstable (SU), unstable-stable (US) and unstable-unstable (UU). Here US means that the participant is unstable in the first half of the experiment, and stable in the second half, and so on. The distribution of participants over the types is shown in table 5, with also the distributions per treatment. In table 5 it can be seen that only a couple of

Figure 9: As before, the red line describes the realized inflation rate in the experiment, and the blue line the prediction of the participant. The classification of US can be clearly seen here.

participants in the unstable treatments manage to end up stable (SS or US). In the stable treatment the majority of the participants can be classified as one of these two types. When this data is added to the sorted payoffs, it is found that the best earning

participants are the participants of type SS. In the top 25 of best performing participants, 24 are of this type. A similar finding holds at the bottom, where all 25 lowest earners are of type UU.

Besides from a ranking on total payoffs, a second ranking has been made based on the improvement from the first half to the second half of the experiment. The top 10 of this ranking contains 8 participants in the individual experiments, from which 7 could be classified as type US. The data of the second most improving participant (U11) is shown in figure 9 as an example of this classification.

Next to the categorizing of the participants, their prediction strategies are also regressed. The regressions are of the form

𝜋𝑡+1𝑒 = 𝛼 + � 𝛽𝑗𝜋𝑡−𝑗 3 𝑗=1 + � 𝛾𝑘𝜋𝑡−𝑘𝑒 2 𝑘=0 .

However, their appeared to be no significant relation between the amount of lags used in the predictions and the experimental payoffs. An overview of the used lags for

predictions is given in figure 10, where it can be seen that a large part of the participants based their prediction on the last realized inflation rate. The exact regression results can be found in the appendix.

Figure 10: Regression results, where the horizontal axes labels are in the form of (realized inflation lags, previous predictions lags).

Table 6: Prediction rules for simulation 1. In the first column the participant number is shown. In the second and third column the type and exact formula of the prediction strategy are given respectively.

5 Simulations

After evaluating the experimental data, multiple simulations have been run. In all simulations a set of disturbances is drawn from a uniform distribution on the interval [0.975, 1.025], and six different prediction strategies are defined. These six different strategies are evaluated separately and group wise and both under the same two different treatments as the real experiment. The calculation of the realized inflation rates is equal to the calculation in the experiments. Every participant started with the same two initial predictions, an inflation rate of 1.03 (inflation of 3%). The simulations

Participant Type Formula

P1 Naive 𝜋𝑡+1𝑒 = 𝜋𝑡−1

P2 Adaptive (𝛼 = 0.2) 𝜋𝑡+1𝑒 = 𝜋𝑡𝑒+ 0.2(𝜋𝑡−1− 𝜋𝑡𝑒) P3 Adaptive (𝛼 = 0.8) 𝜋𝑡+1𝑒 = 𝜋𝑡𝑒+ 0.8(𝜋𝑡−1− 𝜋𝑡𝑒) P4 Average (last 2 observations)

𝜋𝑡+1𝑒 =1_{2 � 𝜋}𝑡−𝑗 2 𝑗=1 P5 Average (last 3 observations)

𝜋𝑡+1𝑒 =1_{3 � 𝜋}𝑡−𝑗 3 𝑗=1

Figure 11: Prediction behavior in simulations for naive and trend-following expectations in the stable treatment. P1 and GP1: naive; P6 and GP6: trend-following

Figure 12: Prediction behavior in simulations for naive and trend-following expectations in the unstable treatment. P1 and GP1: naive; P6 and GP6: trend-following

are run with the same prediction strategies for al treatments, and with the acquired data the payoffs for the individuals are computed. A ranking is made based on these payoffs. In this section we discuss the group effect and the difference between short-memory and long-memory prediction strategies in the simulations, and we make a comparison

between the simulations and the experimental results.

-100 0 100 200 0 10 20 30 40 50 P1 -100 0 100 200 0 10 20 30 40 50 P6 -100 0 100 200 0 10 20 30 40 50 GP1 -100 0 100 200 0 10 20 30 40 50 GP6 -100 0 100 200 0 10 20 30 40 50 P1 -100 0 100 200 0 10 20 30 40 50 P6 -100 0 100 200 0 10 20 30 40 50 GP1 -100 0 100 200 0 10 20 30 40 50 GP6

(a) (b)

Table 7: Payoffs in simulation 1 for the (a) stable treatment, and (b) unstable treatment, both sorted from highest to lowest. Parameters of the prediction strategies are shown in parentheses behind their type.

5.1 The group effect

In the first simulation six commonly used prediction strategies are chosen, listed in table 6. They are predictions in the form of naive, adaptive, average and trend-following expectations.

In figure 11 and 12 the behavior of the different rules is shown for two prediction rules for both the individual and the group environment, where PX and GPX are

prediction strategy X in the single-agent (homogene) and multi-agent (heterogene) simulations respectively. Figure 11 shows the behavior in the stable treatment, while figure 12 shows the behavior in the unstable treatment. The other behavioral graphs can be found in the appendix. In the two given examples for naive and trend-following

expectations it can be seen that predictions are less volatile in the group environment than individually in the stable treatment. This also holds for the unstable treatment, as is shown in figure 12. In figure 13 the behavior of the realized inflation is shown for all treatments, with the realized inflation in the multi-agent treatments highlighted. It can be seen that the volatility of the realized inflation in the multi-agent treatments is somewhere in between the volatility of the single-agent treatments. To evaluate the performance of the different prediction rules in different

Prediction Type Payoffs

P2 Adaptive (0.2) 23,13 P5 Average (3) 19,95 P3 Adaptive (0.8) 13,71 P4 Average (2) 13,28 GP2 Adaptive (0.2) 12,65 GP6 Trend-following (0.3) 10,77 GP1 Naive 10,36 GP3 Adaptive (0.8) 10,17 GP4 Average (2) 9,87 GP5 Average (3) 9,66 P6 Trend-following (0.3) 8,00 P1 Naive 5,84

Prediction Type Payoffs

P2 Adaptive (0.2) 13,12 GP2 Adaptive (0.2) 5,57 GP4 Average (2) 3,82 GP3 Adaptive (0.8) 3,78 GP1 Naive 3,33 GP5 Average (3) 3,32 GP6 Trend-following (0.3) 2,79 P3 Adaptive (0.8) 2,53 P5 Average (3) 2,07 P6 Trend-following (0.3) 2,02 P4 Average (2) 1,30 P1 Naive 0,32

(a) (b)

Figure 13: Behavior of the realized inflations in simulation 1 for the (a) stable treatment, and (b) unstable treatment.

environments, their payoffs in the simulation are calculated. The payoffs are shown in table 7, with PX and GPX as before.

It can be seen in table 7 that the best individuals perform better than the best group subjects, and the worst individuals perform worse than their group counterparts. An interesting finding however are the payoffs of the fifth used prediction strategy (P5) in different experimental environments. As can be seen in table 7, in the stable

treatment the payoffs for a prediction strategy equal to the average of the last three realized inflation rates are on second place in an individual environment. However, in a group environment this strategy has the lowest payoffs of all the group participants. The opposite holds for the naive expectations (P1), as this prediction strategy performs the worst in the individual case, but ends up fourth and third in a group environment for the stable and unstable treatments respectively. This means that the group size affects the absolute as well as the relative performance of prediction rules. It appears that, in this case, the naive strategy makes use of the stabilization of other strategies like the adaptive strategy with 𝛼 = 0.2. We will get back to this in section 5.2.

To investigate the group effect a bit further, a second simulation has been run. In this second simulation only one change has been made compared to the first one, which is the prediction strategy of the first subject. Where this was a completely naive strategy in the first simulation, a switching strategy has been used in the second one. To evaluate the differences of learning between individual and group experiments, the first strategy is now equal to the equilibrium value starting from period 26. In the first 25 periods the strategy is still naive, to evaluate the benefits of adopting a better strategy. It is found that in the individual case, the gain far exceeds the gain in the group case. Where the

(a) (b)

Table 8: Improvements in the (a) stable treatment and (b) unstable treatment. The third row gives to total payoffs in periods 1 to 25, and the fourth row gives the total payoffs in periods 26 to 50. The fifth row gives the difference between those 2, the gain from switching to a better strategy.

total payoffs of P1 and GP1 are similar in this second simulation, the improvement of P1 from the first 25 periods to the second 25 periods is over three times as large as the improvement of GP1. This is illustrated in table 8 for the stable and unstable treatment. It can be seen that the (bad performing) naive prediction strategy earns more in the multi-agent envorionment, while the (good performing) equilibrium value prediction earns more in the single-agent environment.

As a third simulation, three of the used prediction strategies in the first simulation have been changed. An adaptive as well as an average and the trend-following expectation have been replaced by expectations based on the fundamental inflation rate 𝜃. The third prediction rule changed into a prediction of a constant, the fundamental inflation rate. The fifth and sixth prediction rules are changed into anchor and adjustment rules (equation (9)). For the fifth rule a negative value of 𝜇 = −0.3 has been used, and for the sixth rule a positive value of 𝜇 = 0.3. The behavior of these predictions for both the single-agent and multi-agent environment is shown in figures 13 and 14, for the low and high theta treatment respectively (notice that the scale in figure 13 is different from the scale used on the previous figures of prediction behavior). A list of the used prediction rules in the third simulation can be found in table 9.

The payoffs for the third simulation are calculated in the same way as for the first two simulations and can be found in table 10. The findings are similar to the findings in the first simulation. As could be expected, the third prediction rule of the constant fundamental rate has the best performance. The payoffs for this rule are higher in the individual case than in the group environment. However, for the fifth prediction strategy we find that this strategy performs well in a group environment (second best group

Improvement with switching strategy

Period P1 GP1 Diff

1 to 25 4.37 6.99 -2.62 26 to 50 11.90 9.35 2.55

Gain 7.53 2.36

Improvement with switching strategy

Period P1 GP1 Diff

1 to 25 0.32 1.70 -1.39 26 to 50 11.85 4.68 7.18

Figure 13: Behavior of the fundamental based prediction rules in the low theta

treatment. P5 and GP5: anchor and adjustment (-0.3), P6 & GP6: anchor and adjustment (0.3).

Figure 14: Behavior of the fundamental based prediction rules in the high theta

treatment. P5 and GP5: anchor and adjustment (-0.3), P6 & GP6: anchor and adjustment (0.3). -20 0 20 0 10 20 30 40 50 P5 -20 0 20 0 10 20 30 40 50 P6 -20 0 20 0 10 20 30 40 50 GP5 -20 0 20 0 10 20 30 40 50 GP6 -100 0 100 0 10 20 30 40 50 P5 -100 0 100 0 10 20 30 40 50 P6 -100 0 100 0 10 20 30 40 50 GP5 -100 0 100 0 10 20 30 40 50 GP6

Table 9: Prediction rules for simulation 3. In the first column the participant number is shown. In the second and third column the type and exact formula of the prediction strategy are given.

performer for both the stable and unstable treatments), but ends up as 5th and 6th in the individual cases for stable and unstable treatments respectively. Another

noteworthy finding is the comparison with simulation 1. Compared to simulation 1, three of the used prediction strategies have changed (P3, P5 and P6), while the other three prediction strategies (P1, P2 & P4) are left untouched. However, if we compare the payoffs in the two simulations, we see that the payoffs of the latter strategies in the multi-agent simulation have increased with large numbers. This holds for both stable and unstable treatments, where the payoffs of the naive strategy (P1) even triple in the unstable treatment. The results, which can be found in table 11, show that the absolute performance of prediction rules depends strongly on the environment in which they are used. The relative performance of the strategies has also changed in both treatments.

5.2 Short-memory and long-memory strategies

To investigate whether the differences in relative performances are structural for specific prediction rules (i.e. short-memory rules or long-memory rules) multiple more simulations are run with different combinations of prediction rules. We again consider simulation 1, where we found that the naive prediction strategy made use of the

stabilization of other strategies. However, we make one change to this simulation by replacing the stabilizing adaptive strategy (P2, with 𝛼 = 0.2) by another naive strategy. The other prediction strategies remain the same as in simulation 1.

Participant Type Formula

P1 Naive 𝜋𝑡+1𝑒 = 𝜋𝑡−1

P2 Adaptive (𝛼 = 0.2) 𝜋𝑡+1𝑒 = 𝜋𝑡𝑒+ 0.2(𝜋𝑡−1− 𝜋𝑡𝑒)

P3 Fundamental 𝜋𝑡+1𝑒 = 𝜃

P4 Average (last 2 observations)

𝜋𝑡+1𝑒 =1_{2 � 𝜋}𝑡−𝑗 2 𝑗=1

P5 Anchor and adjustment (𝜇 = −0.3) 𝜋𝑡+1𝑒 = 𝜋𝑓− 0.3�𝜋𝑡−1− 𝜋𝑓� P6 Anchor and adjustment (𝜇 = 0.3) 𝜋𝑡+1𝑒 = 𝜋𝑓+ 0.3�𝜋𝑡−1− 𝜋𝑓�

(a) (b)

Table 10: Payoffs in simulation 3 for (a) stable treatment, and (b) unstable treatment, both sorted from highest to lowest. Parameters of the prediction strategies are shown in parentheses behind their type.

The result of the simulation is as we expected. We mentioned earlier that it appeared like the naive strategy was making use of the stabilization of other strategies in the multi-agent treatments in order to perform well there (simulation 1). However, after removing the best performing strategy (in both single-agent and multi-agent treatments) from simulation 1, and replacing it by yet another naive prediction strategy, we find that the absolute as well as the relative performance of the naive strategies in the multi-agent treatments were lower than originally. The payoffs of the strategies in the multi-agent treatment of simulation 4 can be found in table 12, with a comparison to the payoffs of the same strategies in simulation 1. Next to the change in performance of the naive strategies, we also observe that the other prediction strategies now have lower payoffs. This means that by replacing the stabilizing adaptive strategy by a naive

strategy, the entire group is performing worse. This is in accordance with earlier findings in simulation 3. From table 12 we also observe that the largest decline of payoffs from simulation 1 to simulation 4 is found for the naive prediction strategies. This holds for both the stable and unstable treatment.

If we take a closer look at the declines of all the strategies, we see that the decline is higher for the short-memory rules than for the long-memory rules. For both the stable and unstable treatment the order of decline is the same: highest and second highest

Prediction Payoffs

P3 Fundamental 23.53

GP3 Fundamental 15.27

GP5 Anchor and adjustment (-0.3) 14.49 GP6 Anchor and adjustment (0.3) 14.36

P2 Adaptive (0.2) 13.12

GP2 Adaptive (0.2) 12.96

GP1 Naive 10.04

GP4 Average (2) 9.29

P6 Anchor and adjustment (0.3) 2.39

P4 Average (2) 1.30

P1 Naive 0.32

P5 Anchor and adjustment (-0.3) 0.30

Prediction Type Payoffs

P3 Fundamental 23.78

GP3 Fundamental 23.38

GP5 Anchor and adjustment (-0.3) 23.38 GP6 Anchor and adjustment (0.3) 23.21 P6 Anchor and adjustment (0.3) 23.19

P2 Adaptive (0.2) 23.13

GP2 Adaptive (0.2) 23.10

GP4 Average (2) 22.55

GP1 Naive 22.54

P4 Average (2) 13.28

P5 Anchor and adjustment (-0.3) 11.75

(a)

Table 11: Tables with the payoffs of strategies P1 (Naive), P2 (Adaptive) and P4 (Average) in simulations 1 and 3. (a): Low theta treatment; (b): high theta treatment.

Table 12: Table with the payoffs in simulations 4, the payoffs of the exact same

strategies in simulation 1, and the difference between them caused by changing P2 from adaptive (0.2) to naive.

decline are the naive strategies, on third place we find the adaptive with a high parameter (𝛼 = 0.8), as fourth we find the trend-following strategy, and on fifth and sixth we find the averages of the last two and three realized inflation rates respectively. This implies that the short-memory rules (naive, adaptive with a high parameter, trend-following) are more sensitive to changes in the environment than the long-memory rules (averages in this case). For other simulations, where we implemented similar changes, the findings are comparable.

5.3 Comparison of experimental data and simulations

In this section a comparison is made of the experimental data and the simulations. The results in both of these are quite similar, as the multi-agent experiments provide more stability of performances, while the single-agent experiments contain more extremes. In both the experimental data and the simulations it is found that the behavior of most of

Sim. 1 Sim. 3 Increase Naive (P1) 10.36 22.54 12.18 (118%) Adaptive (P2) 12.65 23.10 10.45 (83%) Average (P4) 9.87 22.55 12.68 (128%)

Sim. 1 Sim. 3 Increase Naive (P1) 3.33 10.04 6.71 (202%) Adaptive (P2) 5.57 12.96 7.39 (133%) Average (P4) 3.82 9.29 5.47 (143%)

(b)

Strategy Type Sim. 4 Sim. 1 Diff.

GP6 Trend-following (0.3) 6.45 10.77 -4.32 GP5 Average (3) 6.27 9.66 -3.39 GP4 Average (2) 5.77 9.87 -4.10 GP1 Naive 5.17 10.36 -5.19 GP2 Naive 5.17 10.36 -5.19 GP3 Adaptive (0.8) 5.12 10.17 -5.05 (a)

Strategy Type Sim. 4 Sim. 1 Diff.

GP4 Average (2) 2.90 3.82 -0.92 GP5 Average (3) 2.78 3.32 -0.54 GP3 Adaptive (0.8) 2.16 3.78 -1.62 GP6 Trend-following (0.3) 1.80 2.79 -0.99 GP1 Naive 1.71 3.33 -1.62 GP2 Naive 1.71 3.33 -1.62 (b)

Figure 16: The histograms of payoffs in different treatments and environments. The red bars represent the individual environments, while the blue bars represent the group environments. The histograms on the left side represent the experiment, where the histograms on the right represent the simulations.

the prediction strategies is more stable in groups than individually. This also leads to less volatile realized inflation rates, and better performances. A good illustration if this can be seen in figure 16, where the payoffs in the experiment are shown next to the payoffs in simulation 1 for both treatments. In all cases the top performers are

participants in the single-agent treatments, as well as the worst performers. It can also be seen that the payoffs in the single-agent experiments and simulations are more diverse than in their multi-agent counterparts. This holds for both the stable and unstable treatments. Another effect that is evaluated in both the experiments and the simulations is the effect of 'learning'. As seen, most of the participants in the experiment earned more in the second half than in the first half. After sorting the participants based on their improvement from the first to the second half, it was found that the best

simulations, where switching to a good performing strategy showed to be more beneficial in a single-agent than in a multi-agent environment.

6 Conclusion

In this research we established that the multi-agent environment has significant effects on the performance of subjects. The performance of agents in economic environments are more diverse when evaluated individually than when evaluated in groups. Multi-agent environments give more stability and security, but because of that also come with an upper as well as a lower bound on the performances. Where this is beneficial for some subjects, others might be restricted because of this. With the sorting of the

participants based on improvements, it was seen that for the best 'learning' participants, their opportunities where therefore better in a single-agent than in a multi-agent

environment.

A second remarkable finding is that not only the absolute, but also the relative performance of prediction strategies is strongly dependent on the environment they are used in. As was found in the simulations, the absolute performance of prediction

strategies is strongly affected by the prediction strategies of other subjects in their environment. Next to that, the relative performance order of different prediction

strategies is not the same in every environment. Major differences between the relative order of the strategies in the multi-agent and single-agent simulations are found. Also, it appears that short-memory prediction strategies benefit more from the stability of others than long-memory prediction strategies.

Where the first finding may not be very striking, the second finding could be an interesting subject for further research. As there are infinitely many possible

combinations of prediction strategies, it could very well be possible that there is a structure for the relative performance of prediction rules in different environments. Although this can be tested by simulations, creating an experiment for this setup with human subjects might form a problem. Possible solutions could be to create a similar experiment as the used experiments in this research, but giving the participants several multiple choice options for their predictions instead. These options would be calculated by simple heuristics, meaning that the participants indirectly choose one of these heuristics as prediction strategy. It can be registered how many times certain strategies

are used per participant, and based on the payoffs earned with these strategies a better comparison of the prediction rules can be made.

I Behavioral graphs

Simulations 1, stable: Behavior of the prediction strategies in simulation 1, stable treatments. The used strategies are naive, adaptive, averages and trend-following.

Simulations 1, unstable: Behavior of the prediction strategies in simulation 1, unstable treatments. The used strategies are naive, adaptive, averages and trend-following

Simulations 2, stable: behavior of the changed strategy P1 in simulation 2, stable treatments

Simulations 2, unstable: behavior of the changed strategy P1 in simulation 2, unstable treatments

Simulations 3, stable: Behavior of the prediction strategies in simulation 3, stable treatments. The used strategies are naive, adaptive, average, fundamental and anchor and adjustment.

Simulations 3, unstable: Behavior of the prediction strategies in simulation 3, unstable treatments. The used strategies are naive, adaptive, average, fundamental and anchor and adjustment.

(a) (b)

Realized inflation rates in simulation 3 for the (a): stable treatment and (b): unstable treatment. The red line indicates the realized inflation in the multi-agent treatments.

II Regression results Nr Partic. _{𝜶 𝜷𝟏 𝜷𝟐 𝜷𝟑 𝜸𝟏 𝜸𝟐 𝜸𝟑} R^2 1 GS1P1 3.69 (1.04) -0.52 (0.15) -0.34 (0.15) 0.25 2 GS1P2 2.10 (0.79) -0.39 (0.15) 0.15 3 GS1P3 4 GS1P4 4.17 (1.46) -0.43 (0.15) 0.19 5 GS1P5 0.81 (0.63) -0.14 (0.06) 0.14 6 GS1P6 3.94 (0.97) 0.17 (0.09) 0.08 7 GS2P1 0.73 (0.17) -0.27 (0.07) 0.30 8 GS2P2 0.40 (0.30) -0.36 (0.12) 0.18 9 GS2P3 10 GS2P4 11 GS2P5 12 GS2P6 0.32 (0.25) 0.28 (0.10) 0.16 13 GS3P1 2.76 (0.85) 0.14 (0.06) 0.14 14 GS3P2 9.09 (2.5) -0.40 (0.15) 0.16 15 GS3P3 16 GS3P4 4.07 (0.7) 0.08 (0.05) 0.08 17 GS3P5 3.2 (0.9) -0.13 (0.06) 0.13 18 GS3P6 0.53 (0.34) -0.05 (0.02) -0.04 (0.02) 0.13 19 GS4P1 3.42 (1.45) -0.22 (0.11) 0.11 20 GS4P2 4.01 (1.70) 0.57 (0.14) 0.45 (0.15) 0.37 21 GS4P3 3.15 (1.08) 0.19 (0.09) 0.43 (0.09) -0.49 (0.12) 0.31 (0.11) 0.64 22 GS4P4 0.48 (1.43) 0.4 (0.15) 0.48 (0.14) 0.34 23 GS4P5 6.3 (0.8) 0.31 (0.07) 0.39 24 GS4P6 2.87 (0.30) -0.23 (0.05) -0.34 (0.05) 0.68 25 GS5P1 1.11 (0.42) 0.61 (0.09) -0.32 (0.116) 0.31 (0.17) 0.12 (0.12) 0.71 26 GS5P2 0.50 (0.25) 0.30 (0.08) -0.27 (0.089) 0.45 (0.15) 0.47 27 GS5P3 0.74 (0.21) 0.17 (0.03) 0.03 (0.033) 0.57 (0.10) 0.72 28 GS5P4 1.60 (0.39) 0.16 (0.15) -0.30 (0.14) 0.12 29 GS5P5 2.12 (0.64) 0.82 (0.15) 0.17 (0.20) -0.45(0.15) 0.64 30 GS5P6 1.02 (0.24) -0.14 (0.05) 0.40 (0.14) 0.26 31 GS6P1 3.94 (1.13) 0.44 (0.01) 0.33 32 GS6P2 1.28 (0.58) 0.460 (0.05) 0.19 (0.049) 0.13 (0.052) 0.75 33 GS6P3 6.47 (0.80) 0.00 34 GS6P4 3.94 (0.75) 0.40 (0.06) 0.26 (0.077) -0.45 (0.17) 0.63 35 GS6P5 2.43 (1.91) 0.43 (0.17) 0.15 36 GS6P6 1.76 (0.91) 0.32 (0.08) 0.31 37 GS7P1 2.22 (0.88) 0.23 (0.12) 0.09 38 GS7P2 2.49 (0.59) 0.00 39 GS7P3 1.83 (0.76) 0.07 (0.15) -0.31 (0.15) 0.10 40 GS7P4 4.76 (0.36) 0.09 (0.02) 0.03 (0.018) -0.15 (0.08) 0.44 41 GS7P5 0.87 (0.42) 0.14 (0.05) 0.27 (0.14) 0.28 42 GS7P6 4.26 (0.43) 0.07 (0.06) -0.13 (0.057) 0.19

43 GU1P1 8.83 (2.38) 0.00 44 GU1P2 14.95 (5.51) 0.00 45 GU1P3 46 GU1P4 13.43 (1.71) -0.19 (0.07) 0.18 47 GU1P5 3.37 (2.9) 0.37 (0.11) 0.23 48 GU1P6 49 GU2P1 15.82 (2.05) -0.36 (0.15) 0.13 50 GU2P2 15.65 (2.36) -0.42 (0.15) 0.18 51 GU2P3 21.13 (3.01) -0.42 (0.13) 0.22 52 GU2P4 11.23 (3.46) 0.00 53 GU2P5 17.88 (5.06) 0.00 54 GU2P6 10.25 (2.11) 0.00 55 GU3P1 56 GU3P2 57 GU3P3 18.64 (2.88) -0.31 (0.11) -0.3 (0.10) 0.22 58 GU3P4 59 GU3P5 15.57 (1.87) -0.45 (0.15) 0.20 60 GU3P6 61 GU4P1 62 GU4P2 63 GU4P3 -0.12 (0.48) 0.06 (0.02) 0.77 (0.11) 0.58 64 GU4P4 4.65 (1.76) 0.22 (0.07) 0.19 (0.07) 0.24 65 GU4P5 16.6 (2.73) 0.2 (0.09) -0.31 (0.15) 0.19 66 GU4P6 8.33 (3.01) 0.21 (0.09) 0.39 (0.14) 0.29 67 GU5P1 6.80 (3.898) 0.035 (0.12) 0.26 (0.12) 0.12 68 GU5P2 18.16 (2.814) 0.00 69 GU5P3 10.51 (1.844) -0.427 (0.06) 0.51 70 GU5P4 3.58 (3.901) 0.280 (0.12) 0.26 (0.12) -0.22 (0.15) 0.15 71 GU5P5 10.90 (4.667) 0.021 (0.14) -0.30 (0.14) 0.13 72 GU5P6 11.25 (4.036) 0.195 (0.11) -0.28 (0.17) 0.23 (0.180) 0.10 73 GU6P1 15.65(4.341) 0.4667 (0.15) 0.17 74 GU6P2 8.45 (2.629) 0.00 75 GU6P3 3.69(2.467) 0.373 (0.08) 0.32 (0.11) 0.47 76 GU6P4 7.31 (1.558) 0.523 (0.06) 0.66 77 GU6P5 8.83 (0.131) 0.36 (0.14) 0.13 78 GU6P6 79 S1 -0.55 (1.30) -0.22 (0.05) 0.32 80 S2 0.49 (0.11) 0.17 (0.04) 0.28 81 S3 82 S4 69.35 (14.58) -0.39 (0.15) -0.32 (0.15) 0.18 83 S5 1.33 (1.54) -0.26 (0.58) 0.34 84 S6 -2.76 (1.97) 1.06 (0.10) -0.88 (0.11) 0.77 85 S7 7.69 (4.66) 0.34 (0.13) 0.14 86 S8 0.91 (0.20) -0.16 (0.05) 0.46 (0.14) 0.31

87 S9 0.34 (0.17) 0.22 (0.03) 0.48 (0.13) 0.53 88 S10 4.77 (4.81) 0.40 (0.15) 0.16 89 S11 1.11 90 S12 91 S13 23.15 (4.73) 0.00 92 S14 0.014 (0.09) 0.13 (0.05) -0.14 (0.04) 0.32 93 S15 0.33 (0.14) 0.00 94 S16 0.00 95 U1 4.34 (1.62) 0.11 (0.03) -0.10 (0.03) 0.65 (0.14) 0.64 96 U2 97 U3 12.56 (4.11) 0.00 98 _U4 99 U5 4.59 (0.82) 100 U6 2.08 (2.09) 0.30 (0.05) -0.15 (0.04) 0.90 (0.14) -0.31 (0.09) 0.67 101 U7 15.94 (4.41) 0.00 102 U8 29.95 (8.06) 0.00 103 U9 6.82 (15.93) 0.51 (0.20) 0.46 (0.20) 0.16 104 U10 25.99 (3.38) -0.58 (0.10) 0.67 (0.19) -1.50 (0.30) 0.50 105 U11 4.89 (1.28) 0.45 (0.14) 0.21 106 U12 107 U13 38.61 (10.16) 0..00 108 U14 29.28 (6.55) 0.40 (0.13) 0.20 109 _{U15 16.23 (2.34) 0.18 (0.05) 0.25} 110 U16

Regression results: Estimations of the used prediction strategies of the participants. For groups 1 and 2 of both the stable and unstable treatments, as well as all the single-agent treatments, the regressions are performed for sample 11-50 to give the participants some time to learn. For the other regressions sample 1-50 is used (these last named regressions are performed by Joel van Kesteren and Yannic Pieters)

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