Abstract
In repeated number guessing games choices typically converge quickly to the Nash equilibrium. In positive expectations feedback experiments, however, convergence to the equilibrium price tends to be very slow, if it joshua65@example.org. Both types of experimental designs have been suggested as modeling essential aspects of financial markets. In order to isolate the source of the differences in outcomes we present several new experiments in this paper. We conclude that the feedback strength (i.e. the 'p-value' in standard number guessing games) is essential for the results.
Keywords: number guessing game, beauty contest game, expectations feedback systems JEL-codes: C91, G12
We thank Adriaan Braat for his help with experiments TN and TI and Peter Heemeijer for his help with experiment LOW.
Motivation
In a famous quote Keynes (1936) describes financial investment as a game in which players try to predict average predictions: "...professional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole..." 1 This beauty contest analogy is often cited in papers on higher order beliefs 2 and has inspired an increasing number of theoretical and experimental contributions to economics and finance (for a recent theoretical study see e.g. Allen et al., 2006) . Most experiments focus on (variations of) the so-called number guessing game (see e.g. Nagel, 1995) . In this game all players have to simultaneously submit a 'guess' from a certain interval (typically 0-100) and the winner is the player whose choice is closest to a given fraction (typically 2/3) of the average of these chosen numbers. This game has a unique Nashequilibrium and the distance between a specific guess and the equilibrium value can be considered a measure of the belief this player has about the rationality of the population 1 The quote continues with: "...so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view. It is not a case of choosing those which, to the best of one's judgment, are really the prettiest, nor even those which average opinion genuinely thinks the prettiest. We have reached the third degree where we devote our intelligences to anticipating what average opinion expects the average opinion to be. And there are some, I believe, who practise the fourth, fifth and higher degrees." See Keynes (1936) , page 156. 2 See for example, Biais and Bossaerts (1998) , Ho et al. (1998) , Camerer et al. (2004) and Costa-Gomes and Crawford (2006) . An alternative financial market interpretation of the guessing game is that it models the problem of leaving a market just before prices start going down, see Duffy and Nagel (1997) and .
of players, and about the distribution of the higher order beliefs about rationality in the population.
The general findings from the experimental literature on repeated number guessing games are that first period choices are not very close to the Nash equilibrium but convergence to that equilibrium is fast (typically within 4-5 periods) and stable. As a characterization of behavior of financial markets this fast convergence is surprising for at least two reasons.
First, empirical evidence suggests that asset markets are in fact not that stable. Shiller (1981 Shiller ( , 2000 , for example, shows that stock prices are excessively volatile: their variance is higher than that of the underlying fundamental value. Behavioral finance (for recent overviews see Shleifer, 2000, Barberis and Thaler, 2003) has shown that (1) many price movements are unrelated to news but are reactions to price changes (for example caused by investors using technical analyses) and that (2) prices under-react to news, causing short-term trends. Mis-pricing cannot always be arbitraged away (Shleifer and Vishny, 1997) and market prices may therefore deviate substantially from their fundamental values for a longer period of time.
Second, evidence from expectations feedback experiments (see e.g. Hommes et al., 2005a , 2008 , Heemeijer et al., 2009 does not seem to be consistent with the results from number guessing game experiments. Expectations feedback experiments are based upon the idea that asset markets (just like many other economic environments) are expectations feedback systems. Price expectations of traders determine their trading behavior which, in turn, determines the realized trading price. In an expectations feedback experiment participants have to submit their forecast of the future price of a certain asset and are paid according to their prediction accuracy. A computer program determines the optimal trades associated with the forecasts and the resulting realized trading price. The advantage of this design over traditional experimental asset markets is that it gives a clearer picture of how people form expectations in expectations feedback environments. 3 In prediction experiments with a positive expectations feedback (that is,
where an increase in average predictions leads to an increase in the realized market price)
there is a remarkable tendency for participants to coordinate on a common prediction strategy but no (or only slow) convergence to the equilibrium price.
These positive feedback prediction experiments are closely related to the number guessing game, but with very different results. 4 Nevertheless, the experimental designs do differ in a number of dimensions, particularly the feedback strength from expectations (guesses) to realized price (target number), the information given to the participants, and the incentive structure. It is, a priori, not evident which of these design differences is responsible for the differences in outcomes. This paper reports on a series of experiments that are designed to isolate the main determinants.
The remainder of this paper is organized as follows. In Section 2 we will briefly review the experimental literature on number guessing games and positive expectations feedback experiments and discuss the differences in design characteristics and outcomes between these two types of experiments. The design of three new experimental studies 3 In more traditional asset market experiments participants are also sometimes asked to submit price predictions, but it is difficult to give the appropriate incentives for providing these predictions and often they come about as a by-product to the experiment. For a more rigorous approach to expectation formation in experimental asset markets, see Haruvy et al. (2007) . 4 When we started our experimental research on expectation feedback markets we were not fully aware of the close connection with guessing games. This connection only became apparent to us when we changed from using a market-clearing environment (Hommes et al., 2005a) , where participants had to predict two periods ahead, to a market-maker environment (Heemeijer, et al., 2009) where participants only have to predict one period ahead (in Section 2.2 we will discuss the differences between these two types of expectations feedback experiments in more detail).
will be briefly discussed in Section 3 and the results of these new experimental studies will be analyzed in Section 4. Section 5 concludes.
Number Guessing Games and Expectation Feedback Experiments

Number guessing games
The typical number guessing game experiment has the following structure. is smallest wins a prize in that period. If several participants have the best guess the prize is split evenly between them. The rules of the game are common knowledge and between periods participants receive feedback about the previous periods' guesses of all participants, the target number and the winning number.
5 Moulin (1986) was the first to discuss this game. 6 In the literature the parameter β is often denoted p and the corresponding guessing game is then sometimes referred to as a "p-beauty contest". We depart from that convention here, since the variable p is used to denote prices in the remainder of this paper. , and so on. By looking at first period choices the number guessing game can be used to classify subjects into different depth of reasoning types.
The number guessing game has been studied extensively in laboratory experiments (for overviews, see Nagel, 1999, and Camerer et al., 2003) , typically with 9 Note that level-1 players do not take into account their own effect upon the target number and furthermore believe that all other players are level-0 players.
reported by Nagel (1995) who considered groups of 15-18 participants playing the game for four periods. Her main conclusions are: (i) First period choices are significantly different from the Nash equilibrium prediction 10 and almost all of these choices correspond to level-0 up to level-3 depth of reasoning; (ii) In subsequent periods there is rapid convergence to the Nash-equilibrium, without an increase in the depth of reasoning.
These results have been corroborated by many other experiments. Ho et al. (1998) show that convergence to the Nash equilibrium is faster when β is farther away from 1, groups are larger, and participants are experienced. Duffy and Nagel (1997) show that when the target number is based upon the median guess (maximum guess) instead of the mean guess, convergence is faster (slower). Nagel's results where also confirmed in three large scale one-shot number guessing games, run through newspapers in Germany, Spain and the U.K. and involving thousands of participants (Bosch-Domènech et al., 2003) . Camerer and Ho (1998) , Güth et al. (2002) and Kocher and Sutter (2006) . The last two papers also depart from the standard winner-takes-all payoff incentive scheme and reward all participants based upon the absolute distance between their guess and the target number.
11 Güth et al. (2002) conjecture that, because participants try to avoid extreme choices, convergence in games with interior equilibria is faster than in games with boundary equilibria. Their experiment confirms this. Moreover, the fraction of 10 Even in two-player guessing games where the Nash-equilibrium prediction, 0, strictly dominates all other choices it is only chosen by less than 10% of students and by about one third of professionals (Grosskopf and Nagel, 2008) . 11 Also Costa-Gomes and Crawford (2006) use a payoff scheme that depends upon the absolute distance between the guess and the target number.
equilibrium choices in the 'interior equilibrium' game is significantly higher than in the 'boundary equilibrium' game, which they attribute partly to the payoff scheme.
12
The number guessing game has been used as a vehicle for investigating a number of other issues. Weber (2003) shows that participants still learn, albeit at a slower rate, if no feedback is given between periods. Kocher et al. (2007) and Sbriglia (2008) show that additional information, such as strategies of the winners in earlier periods, or strategies from participants in an earlier number guessing game, facilitates faster convergence to the Nash equilibrium. Slonim (2005) finds that experienced players, when matched with inexperienced players, win the game more often and make choices farther away from the equilibrium. Finally, Kocher and Sutter (2005) and Sutter (2005) show that teams of players learn faster than individuals and increase convergence speed.
Positive expectations feedback experiments
Consider the following textbook asset pricing model (for reviews, see Cuthberson, 1996 , Campbell et al., 1997 , and Brock and Hommes, 1998 is a risk aversion parameter (again assumed to be the same for all traders).
In period t aggregate excess demand for the risky asset is given by ( ) ( )
In order to close the asset market model we need to specify a model for price formation. Participants in the experiment by Hommes et al. (2005a) are explained that they are the advisor of a large investor, e.g. a pension fund. Their task is to predict future prices in a stock market and their reward depends on their prediction accuracy. They are told the investor will take a position in the market that depends on their prediction of future prices (see Appendix A for complete instructions) and that there are other large investors in the market advised by other participants. They are not told the precise formula used to calculate the realized price, but they know the direction of the feedback structure (if many participants expect high (low) prices, investors will buy (sell) more stocks and the price will increase (decrease)). The participants receive information in each period about previous prices and their own predictions, both in a graph and a table.
For the experiment fourteen groups were investigated, with each group consisting of 6 = H participants, predicting prices for 51 periods. 14 Reported predictions had to be 13 There also exist so-called "rational bubble" solutions of the form
. These bubbles grow with a fixed rate 1 > R per period (see e.g. Hommes et al., 2008 , for a discussion). 14 In some groups robot traders were added. These robot traders always predicted the fundamental price and make a trading decision based upon this prediction. The impact of robot traders in these markets is endogenous: the greater the distance between the actual price and the fundamental price the more these fundamental traders will invest, and the other way around. They therefore act as a 'stabilizing force' pushing prices in the direction of the fundamental price. Behavior in markets with robot traders is qualitatively similar to behavior in markets withouth robot traders (for details see Hommes et al., 2005a 
used for all groups.
Participants could earn 1300 points each period. The number of points earned in period t by participant h was inversely related to the forecast error as follows
where 1300 , which (in contrast to the equilibrium price for the other groups) is below the midpoint of the interval from which predictions can be chosen. Behavior in these three groups is qualitatively the same as the behavior in the other groups (for details see Hommes et al., 2005a) . experiment (without completely converging). In two of the remaining groups there is monotonic but slow convergence. For the final group the picture is a bit blurred due to a typing error of one of the participants in one of the periods. Several variations of this game should be mentioned. In Hommes et al. (2008) the same model is used, without robot traders and without an upper limit for price predictions. In that setup numerous speculative bubbles and crashes emerge, with prices increasing to almost 16 times the fundamental value. Also a high degree of coordination on a common prediction strategy can be observed in that experiment. Bottazzi and Devetag (2005) study a variant of this experiment with two changes. First, participants have to give a confidence interval for the realized price, instead of a point prediction.
Second, participants are rewarded on the basis of the increase in wealth their predictions generate. They find that the incidence of bubbles decreases and the heterogeneity of predictions increases, in comparison to the results by Hommes et al. (2005a Hommes et al. ( , 2008 (first round strategies) to 40% (fourth round strategies) of the simulations converge towards the equilibrium price.
Market maker. An alternative model of price formation is one where prices are set by a market maker. In that scenario traders report their demands (2) to a market maker who, like the well-known Walrasian auctioneer, aggregates excess demands and increases (decreases) the price of the risky asset when there is excess demand (supply) for the risky asset (see e.g. Beja and Goldman, 1980) . That is, prices change according to
Here 0 > λ is a parameter that measures the speed of adjustment. The main features of the positive expectations feedback experiments, systematic deviation of prices from fundamentals and coordination of predictions, therefore seem to be quite robust. These results are further corroborated by Leitner and Schmidt (2007) , who investigate forecasting behavior in an exchange rate model and find coordination on a common prediction strategy, as well as systematic deviations of exchange rates from their underlying fundamental values.
A comparison of number guessing games and positive expectations feedback experiments
The asset pricing experiment from Heemeijer et al. (2009) is closely related to the standard formulation of the number guessing game. In fact, the price generating mechanism (7) is a special case of (1) There are several differences in the designs of the two types of experiments that may be responsible for these qualitative differences. Three important differences in design are listed below. Incentives: Guessing games use a winner-takes-all tournament structure, whereas expectations feedback experiments reward on the basis of prediction accuracy. The latter incentive structure is also based upon real life: a stock market is typically not a winnertakes-all situation. The only number guessing game experiments departing from the tournament structure are Güth et al. (2002) , Kocher and Sutter (2006) and Costa-Gomes and Crawford (2006) . The former suggest that the large fraction of equilibrium choices in their experiment is due to what they call the "continuous payment scheme". Moreover, Kocher and Sutter (2006) suggest that the winner-takes-all scheme might lead single players to retire mentally, or to start experimenting. On the other hand: one could argue that a tournament structure forces participants not only to predict accurately, but to predict better than others. This may inhibit satisficing behavior and force participants to think harder about the game and make prices converge faster.
In the next section we will discuss and analyze a new set of experiments to test to which of these differences in design (structure, information or incentives) the differences in results between number guessing games and asset pricing experiments can be attributed.
Bridging the gap: Design of new experiments
In this section we report on three new expectations feedback studies in which parameters and experimental design are varied in an attempt to bridge the gap between number guessing games and positive expectations feedback experiments. 
Results

This section first discusses the results of the three new expectation feedback experiments
one by one. After that the results of these and previous experiments are compared on the aspects convergence, prediction accuracy, coordination of expectations and the occurrence of spoilers.
Incentives: Winner-takes-all instead of quadratic error payoffs
The pay-off function used in the first two studies is based on the quadratic forecasting error and is rather flat at the optimum; small errors are barely punished. Less precision in forecasting can have large consequences in a tournament if the best competitor makes forecasting errors of a comparable size. Tournament incentives can thus motivate participants to be more precise. On the other hand, it may demotivate those participants whose predictions are of a lower quality and who do not expect to win anyhow. 16 We use the term experiment instead of treatment because each study was designed after we knew the results of the previous one.
In experiment TN a tournament incentive structure (like the one typically used in guessing games) is used: in each period the participant with the smallest forecasting error receives a prize of 3 euro (in case of two or more winners the prize is split evenly). All other aspects of the design are the same as in Heemeijer et al. (2009) , except that predictions of more than 1000 were not accepted by the computer program. We run 6 groups with 6 = H participants predicting prices for 50 periods. has three important features. First, on several occasions one of the participants submits a very high price prediction. This is particularly evident in groups 1, 2 and 3. In groups 1 and 2 predictions of 999 are submitted, in group 3 a prediction of 999.9 is submitted and in groups 2 and 3 predictions of 1000 are submitted.
Also in the other three groups there are occasionally rather uncommon predictions. Obviously, these so-called "spoilers" destabilize the dynamics and inhibit convergence to the equilibrium price. We will analyze these spoilers in more depth in Section 4.4.4. The second feature is that there is no apparent or fast convergence to the equilibrium price. The third feature corresponds to the high degree of coordination of predictions (in the absence of spoilers). These two last features can be easily checked by inspections of the graphs for groups 4, 5 and 6, but they also hold for those periods in groups 1, 2 and 3 before the first spoiler has occurred. Figure 2: Prices (solid line), predictions (gray line) and equilibrium price (dotted line) in 6 groups with tournament incentives and limited information. In the first 3 groups the graph is split in two and rescaled: the periods before a participant submitted a very high prediction leading to a realized price higher than 100, and afterwards. For clarity the equilibrium price is not displayed in the rescaled parts of the figure.
Summarizing, introduction of winner-takes-all incentives leads to an increase in "spoilers", but it does not appear to have a significant influence on convergence and coordination.
Information: Complete instead of limited information
In the expectations feedback experiments discussed so far the explicit price function was not available for the participants. In the typical number guessing game, however, players know exactly how the target number is calculated from the reported numbers. We decided to run two sessions (six groups) in which the price function was given and explained to the participants. As in the TN experiment participants were rewarded on a winner-takes-all basis. Figure 3 shows the results.
Compared to the TN experiment the number of "spoilers" seems to be even higher. In particular, predictions of 999 or 1000 were submitted in four of the six groups, resulting in an overall decrease in the rate of convergence. For groups 2 and 3, however, it is obvious that there is little convergence, even in the absence of spoilers. The "prespoiler" predictions in groups 4, 5 and 6 also don't show a fast convergence.
Even with a winner-take-all payment structure and complete information about the price generating mechanism the dynamics are characterized by persistent deviations from the fundamental price and coordination of individual predictions. Figure 3: Prices (solid line), predictions (gray line) and equilibrium price (dotted line) in 6 groups with tournament incentives and full information. For groups 4, 5 and 6 the graph is split in two and rescaled: the periods before a participant submitted a very high prediction leading to a realized price higher than 100, and afterwards. In group 1 a prediction of 1000 was submitted already in period 3. For clarity the equilibrium price is not displayed in the rescaled parts of the figure. 
Feedback strength
The final experiment we ran, referred to as experiment LOW, considered a change in the feedback strength. In particular, the price generating mechanism was given by: (8) was common knowledge. As in the TN and TI studies an upper limit of 1000 to the prediction was imposed after period 1. Ho et al. (1998) provide an indication that a less steep slope could enhance convergence. They report that a higher factor (0.9 instead of 0.7) in a standard number guessing game with 7 participants causes mean choices to be farther from the equilibrium value 0. Their Figures 2A and 2C suggest that the difference is largest in the first 5 periods. It is not clear in advance whether their results will also hold in the interior equilibrium case. Figure 4 shows the results for our experiment LOW. All six groups converge very fast to the equilibrium. In group 5 one participant submitted a "spoiler" in round 25, but also prices in this group are back at the equilibrium value in only a few rounds. Figure 4: Prices (solid line), predictions (gray line) and equilibrium price (dotted line) in 6 groups with quadratic error incentives and low feedback strength. For group 5 the graph is split in two and rescaled: the periods before a participant submitted a very high prediction leading to a higher price than 100, and afterwards.
Comparison of the experimental results
The results described above suggest that providing more information to participants or changing to a 'winner-takes-all' incentive structure does not change the convergence and coordination properties of the positive expectations feedback experiments, but that a decrease in feedback strength is an important determinant for convergence. In this section we will try to provide some additional evidence to substantiate that claim. Moreover, in Section 4.4.4 we will analyze the increased incidence of spoilers in the 'winner-takes-all' experiments.
Convergence
Figures 1-4 suggest that convergence only occurs in experiment LOW. Not all data are used in constructing this picture. In particular, in each period we only took those groups into account for which no spoilers had emerged yet. Obviously, a period in which one participant enters a spoiler leads to a large divergence from the equilibrium price for all predictions in that group in that period, but often there is also an effect in the following periods: the general price level increases leading to larger absolute prediction errors. Therefore from each experimental group we have only included those observations from periods before the first spoiler is submitted. Alternative procedures yield similar qualitative results. Table 5 Consequently not every point represents the same number of participants. Figure 6 shows, for each period, the median over all participants of the absolute value of the individual prediction error, using the same data as above. In this case the difference between experiments MM, TN, TI and LOW is not significant at a 5% level. 18 Allowing for a small initial learning phase of 5 periods and further restricting data to the first 29 observations there is no significant difference between treatments MM, MC, TN and TI (Friedman test, 5% significance level), but there is if treatment LOW is taken into account.
Prediction accuracy
The different designs therefore do not seem to have an impact upon prediction accuracy.
Accuracy is, however, significantly lower in experiment MC, possibly reflecting the difficulty of predicting two periods ahead. Figure 7 gives, for the different experiments, the medians (over groups) of the standard deviation of predictions. A low value of this standard deviation implies a high level of coordination of predictions. Clearly, coordination seems to be much harder to obtain in the MC experiment (although the null hypothesis that there is no difference between experiments MM, TN, TI and LOW also has to be rejected at the 5% level). 
Coordination of expectations

Spoilers
For each experiment we classified the participants and periods that correspond to spoilers (detailed information can be found in Appendix B). It is obvious that in the winner-takesall experiments TN and TI more participants submit spoilers and spoilers are submitted in more periods. In particular, 14 participants in experiment TN and 18 participants in experiment TI (corresponding to 39% and 50% of participants in these experiments) were responsible for submitting a total of 35 and 37 spoilers, respectively (corresponding to a percentage of 1.9% and 2.1% of the total number of choices in those experiments, respectively). In the quadratic forecast error experiments MC, MM and LOW the percentage of participants submitting spoilers is 1.2%, 5.6% and 11.1% respectively, submitting spoilers in 0.0%, 0.2% and 0.3% of the total number of choices in those experiments, respectively. The winner-takes-all payoff scheme therefore seems to have a substantial effect on the incidence of spoilers.
By construction, the average payoff of a participant in experiments TN and TI is 25 euro (50 rounds x 3 euro / 6 participants). The average total earnings of participants that submit spoilers at least once in experiments TN and TI is 95% and 99% of 25 euro, respectively, which is not significantly different at a 5% level, implying that they do not structurally perform better or worse than the other participants.
There might be several reasons why participants submit these spoilers.
Participants may, for example, be frustrated with their earnings thus far, and/or they may want to increase the probability of winning in period t by disrupting the price dynamics in period 1 − t . Each period 3 euro is won by one of the participants in a group thus on average participants earn 3/6 = 0.50 euro per period. The average earnings per period for "spoiling" participants before their first spoiler equal 49 cents in TN and 54 cents in TI.
Moreover, the median number of periods before their first "spoiler" that a spoiling participant did not earn anything is equal to 3 for the TN experiment and equal to 3.5 for the TI experiment (notice that on average each participants should earn something (at least) every 6 periods). This suggests that 'frustration' is not the driving force behind the increased number of spoilers in experiments TN and TI.
Finally, spoilers did not prove to be exceptionally profitable: 5 of the 35 spoilers in experiment TN and 7 of the 37 spoilers in experiment TI led to positive payoffs in the next period, which is consistent with what one would expect (one success in every six periods). Moreover, in experiment TN (TI) for 5 (10) participants relative earnings after spoiling were higher than before, and for 7 (8) it was lower.
Conclusions
In our earlier papers on (positive) expectations feedback experiments we found very slow or no convergence to the equilibrium price. Number guessing games are very much related to expectations feedback experiments but typically show fast convergence to the Nash equilibrium. This striking discrepancy was the reason for designing three additional experiments where we searched for the driving force behind this difference. We found that expectation feedback games are robust to changes in the incentive structure and changes in the information provided to the participants. We consider this robustness to be good news. On the other hand, experiment LOW shows that presenting the number guessing game in the context of a financial market, with an interior Nash equilibrium and changing the incentive structure to a quadratic error rule resulted in very fast convergence like in the traditional number guessing game.
Let us now consider again the original beauty contest game as described by Keynes (see introduction) and compare this with the number guessing game. In the beauty contest game the task is to choose the pictures that are most often chosen by others; this is comparable with the number guessing game with α = 0 and β = 1 . In the beauty contest game there are many equilibriums where all participants choose the same pictures and therefore the game in essence corresponds to a coordination problem. When β = 1 players who have higher order beliefs on different levels can still make the same decision. A number guessing game with β < 1 (and not too close to 1 in order to be able to differentiate between different levels) is a good tool to study higher order beliefs in experiments but it is not necessarily a good behavioral model of an asset market. A β that is much smaller than one corresponds to an enormous interest rate in a financial context (e.g. 50% in the LOW experiment) and a price that is mainly driven by dividends.
A β close to 1 corresponds to a more realistic interest rate and investors/speculators who focus on capital gains rather than on dividends. This seems to be more in line with modern financial markets. The stylized facts about excess volatility in modern markets also point in that direction. 19 Another possible objection to an interpretation of the number guessing game as a model of financial markets is that an asset market is clearly not a tournament where the winner takes all. However, the incentive scheme appears not to be crucial for the number guessing game: in the LOW experiment we find about the same results as in standard number guessing games with a tournament structure.
Concluding, we find that the β in the number guessing game is the essential design parameter: a β much smaller than 1 makes it possible to study higher order beliefs but the game is in that case not a realistic model of a modern asset market. A β closer to 1 makes a more realistic behavioral asset market model, but at the same time makes it harder or impossible to distinguish different levels of higher order beliefs. The next question is whether Keynes was right in his proposition that higher order beliefs are an important element of asset markets. Maybe it is for some investors, but browsing internet forums suggests that many investors/speculators view the market like a living organism which movements you try to predict and not as a game in which you try to form beliefs about the beliefs of others. This interesting question can not be answered here but is a topic for future research.
.
Procedure and instructions
We present the procedure and a translation of the instructions for experiment MM (the instructions in experiment MC differ only in some phrasing). Boxes are included where the instructions are different in the other studies.
Procedure A short welcoming message was read aloud from paper, after which the participants were randomly assigned to a cubicle in the computer lab. In each cubicle there was a computer, some experimental instructions on paper and some blank paper with a pen. The two experiments had different instructions. When all the participants were seated, they were asked to read the instructions on their desks. After a few minutes, they were given the opportunity to ask questions regarding the instructions, after which the experiment started. When the 50 time periods were completed, the participants were asked to remain seated and fill in the questionnaire, which was subsequently handed out to them. After a reasonable amount of time, the participants were called to the ante-room one by one to hand in the questionnaire and receive their earnings, in cash. The participants left the computer lab after receiving their earnings. The experimental instructions the participants read in their cubicles consisted of three parts, totalling five pages. The first part contained general information about the market the experiment was about to simulate. The second part contained an explanation of the computer program used during the experiment. The third part displayed a table relating the absolute prediction error made in any single period to the amount of credits earned in that period. The conversion rate between credits and euros, being 2600 credits to 1 euro. (In experiments TN and TI a tournament was implemented and the tabel was omitted).
Experimental instructions
The shape of the artificial market used by the experiment, and the role you will have in it, will be explained in the text below. Read these instructions carefully. They continue on the backside of this sheet of paper.
General information
You are an advisor of a trader who is active on a market for a certain product. In each time period the trader needs to decide how many units of the product he will buy, intending to sell them again the next period. To take an optimal decision, the trader requires a good prediction of the market price in the next time period. As the advisor of the trader you will predict the price P(t) of the product during 50 successive time periods. Your earnings during the experiment will depend on the accuracy of your predictions. The smaller your prediction errors, the greater your earnings.
About the market
The price of the product will be determined by the law of supply and demand. Supply and demand on the market are determined by the traders of the product. Higher price predictions make a trader demand a higher quantity. A high price prediction makes the trader willing to buy the product, a low price prediction makes him willing to sell it. There are several large traders active on this market and each of them is advised by a participant of this experiment. Total supply is largely determined by the sum of the individual supplies and demands of these traders. Besides the large traders, a number of small traders is active on the market, creating small fluctuations in total supply and demand.
About the price
The price is determined as follows. If total demand is larger than total supply, the price will rise. Conversely, of total supply is larger than total demand, the price will fall.
About the price experiments TI and LOW
The price in each period depends upon your prediction and the prediction of the other 5 participants. Let ) (t GV be the average prediction in period t, than:
This is the price when only the large traders (who are advised by the six particpants) would be influencing the price. The small traders on the market cause a small change of the price, sometimes negative, sometimes positive and on average zero. We will indicate this amount in period t by ) (t k and it will be almost always between -1 and 1. The value of ) (t k is not related to this value in other periods. The realized price in period t will be: (1) and (2) show that the realized price will be near the average predicted price; if the average predicted price (GV) is low, than the realized (P) will be low and if GV is high, P will be high.
About predicting the price
The only task of the advisors in this experiment is to predict the market price P(t) in each time period as accurately as possible. The price (and your prediction) can never become negative and lies always between 0 and 100 euros in the first period. The price and the prediction in period 2 through 50 is only required to be positive. The price will be predicted one period ahead. At the beginning of the experiment you are asked to give a prediction for period 1, V(1). When all participants have submitted their predictions for the first period, the market price P(1) for this period will be made public. Based on the prediction error in period 1, P(1) -V(1), your earnings in the first period will be calculated. Subsequently, you are asked to enter your prediction for period 2, V(2). When all participants have submitted their prediction for the second period, the market price for that period, P(2), will be made public and your earnings will be calculated, and so on, for 50 consecutive periods. The information you have to form a prediction at period t consists of: All market prices up to time period t-1: {P(t-1), P(t-2), ..., P(1)}; All your predictions up until time period t-1: {V(t-1), V(t-2), ..., V(1)}; Your total earnings at time period t-1. About the earnings Your earnings depend only on the accuracy of your predictions. The better you predict the price in each period, the higher will be your total earnings. On your desk is a table listing your earnings for all possible prediction errors.
For example, your prediction was 13.42. The true market price turned out to be 12.13. This means that the prediction error is: 13.42 -12.13 ≈ 1.30. The table then says your earnings are 1255 credits (as listed in the second column).
About the earnings experiments TN and TI
All participants start with 5 euros and whether they will earn more will depend on the quality of their predictions. In every period the participant in your group with the smallest prediction error wins 3 euro and the others earn nothing. If more than one participant have the smallest error, the prize is split. For example, if the realized price is 34.1 and two participants predicted 31.9 and one 36.3 (and the other predictions are less accurate), all three have made a prediction error of 2.2 and the earn 3/3=1 euro each, and the other participants in the group earn nothing.
When you are done reading the experimental instructions, you may continue reading the computer instructions, which have been placed on your desk as well.
Computer instructions
The way the computer program works that will be used in the experiment, is explained in the text below. Read these instructions carefully. They continue on the backside of this sheet of paper.
The mouse does not work in this program. To enter your prediction you can use the numbers, the decimal point and, if necessary, the backspace key on the keyboard.
Your prediction can have two decimal numbers, for example 30.75. Pay attention not to enter a comma instead of a point. Never use the comma. Press enter if you have made your choice.
The available information for predicting the price of the product in period t consists of: All product prices from the past up to period t-1; Your predictions up to period t-1; Your earnings until then.
The main experimental computer screen. The Dutch labels translate as follows: "prijs" = price; "voorspelling" = prediction; "werkelijke prijs" = market price; "ronde" = round; "totale verdiensten" = total earnings; "verdiensten deze periode" = earnings this period; "Wat is uw voorspelling voor de volgende periode?" = What is your prediction for the next period?; "Een nieuwe ronde is begonnen" = A new round has started.
The computer screen. The instructions below refer to this figure. In the upper left corner a graph will be displayed consisting of your predictions and of the true prices in each period. This graph will be updated at the end of each period.
In the rectangle in the middle left you will see information about the number of credits you have earned in the last period and the number you have earned in total. The time period is also displayed here, possibly along with other relevant information.
On the right hand side of the screen the experimental results will be displayed, that is, your predictions and the true prices for at most the last 20 periods.
At the moment of submitting your price prediction, the rectangle in the lower left side of the figure will appear. When all participants have subsequently submitted their predictions, the results for the next period will be calculated.
When everyone is ready reading the instructions, we will begin the experiment. If you have questions now or during the experiment, raise your hand. Someone will come to you for assis-tance. Table   3 .

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