Know thy self, know thy enemy : improving best response performance through learning about opponent rationality in an experimental Beauty Contest Game

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Know thy self, know thy enemy:

improving best response performance through

learning about opponent rationality in an

experimental Beauty Contest Game

Master’s Thesis in Economics (15 ECTS)

Behavioural Economics & Game Theory

by

Tomek Dabrowski

University of Amsterdam

under the supervision of

Prof. Dr. A.J.H.C. Schram

University of Amsterdam

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Abstract: This paper aims to find out if deviations from equilibrium behaviour are caused by doubts about others’ rationality and difficulties in predicting their behaviour, or simply by bounded rationality and limited computational ability. In order to test this, a Beauty Contest game is designed in which some subjects are allowed to learn about the rationality of their opponents by providing the former with information about their opponents’ guesses in previous rounds. The design prevents that learning about optimal choices occurs through

extrapolation based on previous rounds by changing the environment from one round to the next, leaving only learning about the opponents’ rationality. The results indicate that over time, the informed subjects outperform uninformed subjects by 24% – 41%. Additionally, uninformed subjects do not seem to differ significantly in behaviour from a control group of subjects without information exchange, implicating that the improvement is not at the expense of uninformed subjects. These results suggest that subjects are rational but that doubts about the rationality of others move them away from equilibrium guesses.

(JEL C72; C91)

Tomek Dabrowski**

Amsterdam, July 11, 2015

Keywords: beauty contest game; guessing game; experimental economics; behavioural economics; rationality; decision-making; best response; learning.

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*_{!Special thanks to: Arthur Schram for his guidance and supervision; fellow students Nandi Oud}

and Dovile Venskutonyte for the collaboration on the preliminary experiment and paper on which this thesis is based, and for their approval of continued research into the jointly developed topic; Joep Sonnemans for helpful talks regarding the research topic; former middle school teachers Teunis Bloothoofd, Ilse Visser, Paco Prins, Ton van Drielen, Cor Steffens, Sandra Grootenboer, Coosje Martens and Otto Kelderman for supplying students participating in the experiment, and Ton van Drielen in particular for sharing his views on this subject; all the participating students from the SGL schooling community in Lelystad for their cooperation and enthusiastic

participation; best friend and fellow student Artur Olszewski for motivating and enlightening conversations concerning writing this thesis and for pointing out articles related to the subject. Contact: tomekdabro@gmail.com!

Introduction 1. Introduction

The ability to correctly respond in a situation where multiple agents through their cumulative actions simultaneously determine a common outcome is important in everyday social life and economic decision-making. From something as complex as investing in a firm on the stock exchange (which requires predicting whether and how much others will invest, as all decision taken together will directly influence the price), to something as simple as deciding where to stand on a railway platform to wait for an incoming train (where your position and that of all others will determine whether you will be able to sit or will be standing in a full compartment): these situations all require rational decision-making in order to best respond to how others will behave and maximise your own expected payoff. Such decisions are based on making assumptions on how other people will act in certain strategic circumstances.

While game theoretic solution concepts such as the Nash equilibrium offer clear ways of mutual best responding in a variety of strategic situations, experimental results show that people do not necessarily behave according to these concepts and therefore do not best respond (when solely taking into account the payoffs they can obtain). This could potentially be caused by the fact that people are simply boundedly rational and have trouble in calculating what their optimal choice should be. Another reason could be that people are rational (to some humanly possible limit), but are having doubts that others are going to choose equilibrium strategies, either because of doubt about the other players’ rationality, or by the belief that other players doubt the rationality of their opponents (Grosskopf & Nagel, 2007). Either sources of deviation from

equilibrium strategies can result in players not best responding.

There is some discussion in the literature as to which explanation to support. Grosskopf and Nagel (2007) argue based on the observation that subjects seem to converge towards equilibrium by learning through information about the other player’s choices and implementing this knowledge, as opposed to self-initiated rational reasoning, that subjects are most likely to simply be boundedly rational. Ohtsubo and Rapoport (2006) review the evidence of various experimental games, and by suggesting that subjects might be

under-Introduction

or overestimating their opponents in their mental model of them, they support the theory that subjects are rational but have trouble predicting how others will behave. Sbriglia (2008) evaluates the impact of information on reasoning in Beauty Contest games, and shows that non-winners often imitate the reasoning of winners, while more sophisticated players assume others will imitate winning strategies and adapt to such behaviour. This suggests that some people are boundedly rational, while other people are more rational but might make mistakes because they cannot correctly predict how rational others are, thus resulting in Sbriglia (2008) supporting both explanations.

A simple game-theoretic tool that resembles the real life situations described in the first paragraph is the experimental Beauty Contest game (hereafter BCG), also called the guessing game: it is often used to test the bounds of rationality and reasoning (Stahl, 1996; Ho, Camerer & Weigelt, 1998; Güth, Kocher & Sutter, 2002; Bosch, Montalvo, Nagel & Satorra, 2002). This game is a very suitable tool for looking at human reasoning processes, because of (1) its simplicity, yet ability to create an interactive game which requires

anticipation of what others will do; (2) the ease with which learning and

reasoning can be observed; and (3) the clear indicator for performance, as ‘the winner takes it all’ (Kocher, Sutter & Wakolbinger, 2014). In its simplest form, the BCG entails two or more players choosing an integer number on a set interval, for instance [0, 100], their target being the average of the guesses of all

participants times a multiplier p. The person closest to the target wins, or the reward is allotted between tied players (Ohtsubo & Rapoport, 2006).

This study aims to investigate if people are able to improve their

response when given the opportunity to learn about the rationality of opponents in a competitive environment. In a paper in collaboration with Dovile

Venskutonyte and Nandi Oud (see Appendix I), on which this study is based, some subjects participating in a BCG received information about guesses of others and were allowed to change their guess for the same round after receiving this information. While an improvement in guess accuracy was observed, the sentiment prevailed that such information might have simply improved the learning opportunity instead of allowing inference of the rationality

Introduction

of opponents, thus hurting internal validity. The challenge therefore was to devise a BCG experiment that would provide subjects with information about opponents, but without making the target of the game obvious or making opponents completely predictable. To avoid these problems, the experimental setup described in this paper allows supplying some subjects with information about what guesses opponents made in previous rounds, however with a slightly different strategic design each round (realised by changing the multiplier

p from round to round). Such a design improves the ability of these informed

subjects to learn about others’ rationality in the BCG.

Unlike previous studies, this design also limits the increase in learning opportunities that arise when players are given information about the opponents’ choices. In a setup where learning from previous rounds of exactly the same strategic game, in other words with the same multiplier p, is allowed for, subjects can use the information from previous rounds to simply infer the

direction everyone is moving in over time and in this way learn what the target is, without think about the strategies others use. However, the challenge should lie in predicting how others will act only by learning about their choices. Changing p in each round of the game prevents subjects from extrapolating results round after round, by making each of them effectively a first round. This design also mitigates the problems found in the collaborative paper preceding this research, and compared to previous studies it offers a different and perhaps cleaner way of studying if subjects are able to improve performance when given the

opportunity to learn about the opponents’ rationality.

The results show that in the last few rounds, subjects who have received information about the rationality of their opponents outperform uninformed subjects by 24% to 41%. Uninformed subjects do not seem to display different behaviour from a control group of subjects that were not involved in any transfer of information. This indicates that the increase in performance does not take place at the expense of uninformed subjects. A comparison with research by Camerer (2003) suggests that the design is successful in making each of the rounds played as if it were a first round. The findings suggest that subjects are

Introduction & Literature

rational but have doubts that others are as well, and have troubles in predicting the opponents’ behaviour.

The setup of this research paper is as follows. The second section provides an overview of the relevant literature, in which the existing studies are evaluated and the contribution of this study compared to previous research is outlined. The third section describes the methodology, including the

experimental design and procedures and a formulation of hypotheses. The fourth section presents the results of this study. In the fifth section, the main findings including their limitations are evaluated. The sixth section summarizes the main conclusions that can be derived from this research and offers some suggestions for possible future research.

2. Literature

Herbert A. Simon writes in his 1955 paper “A Behavioural Model of Rational

Choice” about the perfect rationality of the ‘economic man’ of traditional

economics, and how this rationality should be made compatible with actual behaviour displayed by people through a realistic view of access to information and computational capabilities. He argues that, when substituting for the ‘economic man’ a choosing organism of limited knowledge and ability, choices made by such an organism create discrepancies between a simplified model and the reality. These discrepancies can explain many phenomena in human behaviour (Simon, 1955). Almost half a decade later, Simon refines his view of bounded rationality, in which “choices people make are determined not only by

some consistent overall goal and the properties of the external world, but also by the knowledge that decision makers do and don't have of the world, their ability or inability to evoke that knowledge when it is relevant, to work out the consequences of their actions, to conjure up possible courses of action, to cope with uncertainty (including uncertainty deriving from the possible responses of other actors), and to adjudicate among their many competing wants. Rationality is bounded because these abilities are severely limited” (Simon, 2000). Building

on this view, Kahneman attempts to provide a map of bounded rationality, with at its base (1) heuristics used by people in decision-making, and biases they are

Literature

affected by in tasks of judgment under uncertainty, including predictions and evaluations of evidence; (2) prospect theory, modelling of choice under risk; and (3) framing effects and their implications for rational-agent models (Kahneman, 2003). What Kahneman and Simon have in common is their view of human rationality as bounded, with these bounds being the reason why their choices do not always necessarily obey to optimal responses predicted by economic theory.

Of the many factors limiting human rationality, one suggested by

Grosskopf and Nagel is doubt about the rationality of others, and difficulties with predicting how these others behave instead. While limited computational abilities could simply cause subjects to not realise what equilibrium behaviour should be and therefore to not best respond to others’ choices, doubting the rationality of co-players could be causing similar deviations from optimal best response behaviour (Grosskopf & Nagel, 2007). Supporting this hypothesis is research done by Weizsäcker, in which he finds that on average, subjects consistently seem to play as if they significantly underestimate their opponent’s rationality (Weizsäcker, 2003).

To be able to distinguish between and compare different levels of rationality, Nagel describes a model of iterated dominance defining players as strategic of a certain degree. When basing their choice on the [0, 100] interval, a subject is rational of degree 0 if they choose the number 50, the expected choice of a player choosing randomly from a symmetric distribution. A player is rational of the degree n if they choose a number 50pn_{, and subsequently their}

iterated best response is of degree 1, 2, ..., n. Choosing a number according to degree n = 1 corresponds with best responding to a player of degree 0,

however believing that others will also behave in such a way should result in choices of higher degrees (Nagel, 1995). As n increases, choices converge to 0, the Nash equilibrium through iterated domination if 0 ≤ p < 1. This strategy emerges through iterated elimination of dominated strategies, where choosing each lower number is better than choosing a higher number. For p > 1, the highest number chosen is the Nash equilibrium (Camerer, 2003). This model has

Literature

been widely used to model rationality in games (Ho, Camerer & Weigelt, 1998; Kocher & Sutter, 2005; Ohtsubo & Rapoport, 2006; Sbriglia, 2008).

When comparing the performance among subjects in a BCG, another approach used by Burnham et al. (2009) is calculating the absolute deviation from each round’s target for each subject, in order to obtain a consistent indication of performance. This paper will apply the approach of iterated dominance to compare results with previous research, and the absolute deviation-approach to compare best response performance of subjects with each other.

A different set of papers discusses the question whether people have trouble predicting the rationality of others or are boundedly rational themselves. Grosskopf and Nagel (2007) conduct a round of experiments in order to study behaviour in repeated fixed pair two-person BCG’s and repeated many-person BCG’s, strategically different from one another. Utilising four different treatments (full information or only own payoff as feedback, and first two-player and then many-player rounds or other way around), they find that the majority of players learns to play the equilibrium strategy with choices converging to 0 with full information, as opposed to noisier learning behaviour with restricted information. While Grosskopf and Nagel do find that players increase their choices when moving from the two-person BCG to the many-person BCG, this also happens when moving the other way. While the first result would confirm the hypothesis that people doubt the rationality of others, the second result violates it as such behaviour does not adhere to choice dominance. Therefore, the authors conclude that subjects do not reason rationally but for the most part are boundedly rational, as they seem to converge towards equilibrium by learning through information about the other player’s choice and implementing this knowledge, as opposed to self-initiated rational reasoning (Grosskopf & Nagel, 2007).

Ohtsubo and Rapoport (2006) investigate depths of reasoning on the basis of other studies using the BCG and the Investment game. They find that taken together, the results show that (1) people apply relatively low levels of strategic reasoning, and (2) there are individual differences in the depth of such

Literature

reasoning. On average, people seem to display only two steps of strategic reasoning in the BCG, and three to four in the Investment game. However, anecdotal evidence suggests the presence of deeper levels of reasoning that seem to be related to the ability of some subjects to process information more efficiently. Ohtsubo and Rapoport suggest that these “subjects, who go through several levels of reasoning and figure out the equilibrium solution to the game, will in general not invoke the maximum depth of reasoning precisely because they do not assume—and perhaps should not assume—that the other n − 1 players are as smart as they are.” By thinking like this, they might under- or overestimate their opponents in their mental model, and therefore doubt others are rational and have trouble predicting their behaviour (Ohtsubo & Rapoport, 2006).

Sbriglia (2008) evaluates the impact of information on levels of reasoning in a series of many-person BCG’s. As opposed to only seeing the average guess and target value the treatment involves a short message from the round’s winner explaining the reasoning they used when choosing their number. After this message, the winner stopped playing. The results show that non-winners often imitate the depth of reasoning of winners in previous rounds of which they received information, which seems to be a more effective strategy than only using the information on previous-round average guesses and target numbers. More sophisticated players, on the other hand, assume others will imitate the strategies from previous rounds and respond with a strategy exhibiting a higher level of rational reasoning. In this way, some players use the additional

information they receive to improve their decision-making, and seem to be able to better predict other people’s behaviour (Sbriglia, 2008).

Where this research differs from previous studies is that those allow for learning from previous rounds, with the multiplier not changing across rounds. However, in such a setup subjects can use the information from previous rounds to simply infer the direction everyone is moving over time, instead of predicting others’ rationality. In line with subjects doubting the rationality of others, Bosch et al. (2002) perform an analysis of comments by newspaper readers

Literature & Experimental Design

subjects discover that 0 is the equilibrium, but a large proportion of them (81%) choose a number larger than this equilibrium, which could be the result of an expectation that others will behave irrationally (Bosch et al., 2002). The keyword here is ‘one-shot’, as such a setup does not allow for learning from previous rounds and requires the cleanest form of predicting other players’ behaviour. Therefore, the experiment described in this paper is setup in such a way as to make each round effectively a first round by varying the multiplier, in the range 0 ≤ p < 2, thus incorporating different equilibria processes. This setup builds on experience from a small-scale study in collaboration with Nandi Oud and Dovile Venskutonyte mentioned in the introduction. The results showed an

improvement in decision-making by subjects receiving additional information, however, we found that the information some subjects had been receiving could simply have helped them because it was additional information. This was

contrary to the original purpose of allowing some subjects to infer from the information provided the rationality of opponents: it could have been that the learning opportunity of informed subjects was improved instead of the

opportunity to better predict others. A solution for this problem chosen here is disentangling information about the game in general, and information about the rationality of co-players. From this notion comes the idea of an experimental setup in which each round a different game is played, resulting in much less opportunity to learn and increasing the importance of predicting others’ behaviour.

3. Experimental Design

The experiment was conducted at a high school, the Scholengemeenschap Lelystad (SGL) in Lelystad, in the Netherlands during two weeks in May 2015. With permission of several former teachers, students from multiple classes were recruited for the experiment. The participants were aged 15-18 years old

(average age 16,3), and came from classes from the highest two educational levels in the Dutch education system: VWO (pre-university secondary education; 36 students from the fifth grade and 44 students from the fourth grade) and HAVO (higher general secondary education, pre-university of applied sciences

Experimental Design

education, one level below VWO; 8 students from fourth grade). The students came from classes that were diverse in educational profiles (focused on economics, social studies and languages or physics, chemistry and

mathematics). Taken together, this yielded a subject pool of 88 students (of which 47,7% was female), playing in groups of four, resulting in 22 groups. This study makes use of a between-subjects design, comparing participants from the treatment groups who receive additional information (informed subjects) to those from the treatment groups who do not (uninformed subjects). Furthermore, control groups are used to compare these subjects (control subjects) to the uninformed subjects. Of the 22 groups, 5 were control groups, and as each treatment group had one informed subject this gives 17 informed subjects in total. A more detailed view of age and gender across groups can be found in Table 1. Based on a questionnaire given at the end of the experiment (see Appendix III), none of the students had seen or played the BCG before.

Participants were selected to participate in a pen-and-paper experiment from a classroom using a random-number generator to ensure a random group of four students each time. As an incentive, the students were informed they had the possibility to win a small amount of money. Five out of the twenty-two groups were selected to be control groups. In two cases, the control groups came right after a treatment group because time allowed to do both groups in the same class hour; in the other three cases, at the beginning of the hour it was randomly determined that a control group would go first. Participants were unaware of the fact in which treatment they were. The chosen four students were asked to bring a pen, and were seated apart from and facing away from each other. No communication was allowed. The instructions (see Appendix II) were distributed, and the participants were asked to read these carefully. During reading, participants were one by one asked to draw a card to assign an identity (spades, hearts, diamonds or clubs) to ensure their anonymity, and were asked to circle this identity on the answer sheet (see Appendix III) also being

distributed. After this, the participants were given time to read the instructions and ask questions, which were answered privately. After everyone indicated they were done reading and all questions had been answered, a short recap

Experimental Design

was given highlighting the procedures, the order of experimental parts, the payoff structure and the importance of the changed multiplier in each round. Up to this point, the treatment groups and the control groups did not differ from one another.

For the control sessions, the experiment began with two practice rounds (not counting towards payment) followed by eight rounds of a variation of the Beauty Contest game, discussed previously. In this variation, participants were asked to guess an integer number from the interval [0, 100] closest to the target number, calculated using the following formula:

target = (average guess of group of four) x (multiplier)

The multiplier p was different each round, and chosen based on the

experimental design by Coricelli and Nagel (2009): p took the values (1/2; 3/4; 1/5; 2/3; 1/3) for 0 ≤ p < 1, and (3/2; 5/3; 7/4; 4/3; 6/5) for p > 1. The first two practice rounds were played with multipliers 1/2 and 3/2 respectively, while the remaining eight multipliers were assigned randomly to the eight remaining rounds; however the order was the same for each group to avoid order effects.

During the practice rounds and eight subsequent rounds, the procedure was as follows: the round number and multiplier were announced (both also indicated on the answer sheet), after which the participants were given time to fill in a number on the answer sheet. After this, the guessed numbers were

collected by walking around, the target was calculated using and Excel sheet prepared beforehand and announced, and the round number and multiplier of the next round were announced. After the practice rounds the participants were once again asked if everything was clear, before proceeding with the remaining eight rounds of the experiment in a similar fashion.

After all rounds were finished, the subjects were asked to fill out the questionnaire at the bottom of the answer sheet. This questionnaire focused on demographic characteristics (gender, age, education level and class), strategy and information use and whether they had seen the game before. In the meanwhile, one student was asked to randomly determine which round would be paid out to the winner of that round, by drawing a card from a stack of cards

Experimental Design

numbered ace, 2, 3, … 8. The chosen round was then announced. After everyone had finished with the questionnaire, the subjects were asked not to share details of the experiment with classmates who had not yet participated, and then were called out one by one by their identifying card suit. They were thanked for their participation and given either a hand or the reward of three euros, after which they went back to the classroom. This was done in such a way that other participants could not see whether someone was receiving a payment or not.

For the treatment sessions, similar to the control groups each group started out with two practice rounds followed by eight rounds that counted towards payment. After the practice rounds, the participants of a treatment group received additional instructions: after the announcement of the round number and multiplier, everyone writing down their guess, collecting the guesses and announcing the target, one subject would receive additional information about the round just played. This subject, determined randomly but constant throughout the remainder of the experiment, would after each round receive the guesses of their three opponents. To ensure no one would know who was receiving information, everyone was visited and wrote down on their answer sheet either the information, or three X’s to indicate ‘no information’. After this, the next round commenced with announcing the round number and the multiplier. The remaining part of the session was the same as with the control group. Overall the treatment sessions took longer than the control sessions because of the additional time it took to write down information for every subject.

In order to do evaluate the results and determine whether subjects are boundedly rational or only have doubts about the rationality of others and cannot predict their reasoning, the following hypotheses are formulated: (1) In the treatment groups, the informed subjects will outperform the uninformed subjects in guessing the target number in several Beauty Contest games with varying multipliers. This result would indicate people are rational to some extent, but cannot predict others’ rationality and therefore make mistakes in best responding in strategic situations.

Experimental Design & Results

(2) Uninformed subjects from the treatment groups will not behave

significantly differently from subjects in the control group, meaning that informed subjects do not improve at the expense of uninformed subjects. This is

necessary to show that informed players actually improve their decision-making as opposed to simply benefiting from a poorer performance of the uninformed.

4. Results

Table 1 gives a summary of the descriptive data, and shows that the average age of subjects is quite similar across groups. While this is also to some extent the case for gender, the fact that the percentage of females is higher among informed subjects than among uninformed subjects is of some importance.

Table 1 Descriptive data

Informed subjects Uninformed subjects Control subjects

Number of subjects 17 51 20

Average age 16,18 16,39 16,10

Percentage of females 52,9% 43,1% 55%

Table 2 shows the average absolute deviations from the target for all rounds and for specific round averages of interest. The top half of the table shows the average absolute deviation across rounds for the informed,

uninformed and control subjects. The bottom half is useful for comparing results of low- versus high-multiplier rounds, as well as the average deviations across groups for the last four, three and two rounds. When looking at the average deviations for the low- and high-multiplier rounds, it seems uninformed and control subjects do better with low multipliers as opposed to high multipliers, while informed subjects’ performance does not appear to be affected by the multiplier. The last three averages show the differences from learning about opponent rationality, which ought to show up in the last few rounds.

Results

Table 2 Average absolute deviations from target across rounds, with standard errors

Informed subjects Uninformed subjects Control subjects

Round 1 (p = 5/3) 22,26 (20,51) 26,42 (18,67) 25,60 (22,65) Round 2 (p = 3/4) 20,07 (16,01) 15,74 (14,21) 14,59 (15,92) Round 3 (p = 1/5) 21,03 (23,64) 15,65 (19,66) 14,15 (18,70) Round 4 (p = 7/4) 14,71 (22,88) 14,31 (20,57) 14,20 (22,62) Round 5 (p = 2/3) 19,47 (13,43) 15,68 (14,84) 15,69 (13,02) Round 6 (p = 4/3) 17,25 (13,46) 21,05 (15,34) 20,2 (15,50) Round 7 (p = 6/5) 13,11 (11,79) 12,60 (15,13) 13,86 (13,95) Round 8 (p = 1/3) 6,29 (5,90) 13,03 (13,30) 12,61 (15,03) Total 16,78 (10,86) 16,81 (10,32) 16,36 (11,13) Rounds with 0 ≤ p < 1 16,72 (11,74) 15,02 (10,42) 14,26 (11,34) Rounds with p > 1 16,83 (12,35) 18,60 (13,32) 18,47 (13,65)

Final four rounds 14,03 (6,40) 15,59 (10,16) 15,59 (9,81)

Final three rounds 12,22 (6,38) 15,56 (11,28) 15,56 (11,55)

Final two rounds 9,70 (6,65) 12,82 (12,63) 13,23 (12,21)

Notes: Average (standard error); multiplier given for each round. Table 3 reports the total

average absolute deviations divided according to age and gender. It is worth noticing here that while overall performance seems to improve with age, the subjects aged 18 seem to behave out of line in doing a lot worse than all the other age groups. When looking at gender differences, it seems that females perform

somewhat worse compared to males.

Table 3 Total average absolute deviations from target across demographics Age 15 17,43 (10,68) Age 16 16,40 (10,25) Age 17 14,22 (7,92) Age 18 24,33 (15,60) Male 16,08 (10,03) Female 17,38 (11,05)

Results

Notes: Light grey bars represent informed players; dark grey bars represent uninformed players; dashed line indicates outlier cut-off. 36 00 10 20 30 40 50 60 32 20 41 9 28 66 67 11 36 55 45 26 46 22 4 8 39 47 25 64 3 50 2 13 5 53 6 44 60 7 49 1 33 59 30 57 40 52 43 56 51 37 42 48 62 38 63 16 58 18 61 12 19 21 27 10 68 54 14 24 23 35 65 17 31 34 29 15

Figure 1 Eight-round average absolute deviation from target for each subject (sorted descending)

0 5 10 15 20 25 30 1 2 3 4 5 6 7 8

Figure 2 Average absolute deviation across rounds for each subject group

Informed Uninformed Control

Results

Figure 1 depicts the individual eight-round average absolute deviation for each subject, in decreasing order. It also visualises the outlier cut-off value of 36, as the four subjects with an average absolute deviation above this value seem to stand out from the rest. This value is later used in the regression analysis. The figure also shows the division of informed subject averages across the treatment group.

Figure 2 provides a graph of the first half of Table 2, showing the development of the average absolute deviation for each group across rounds. While the uninformed subjects and the control subjects seem to move in a similar fashion, the informed subjects diverge from this trend. Also, while

uninformed and control subjects seem to improve quickly and then stabilise, the informed subjects improve more slowly but performance improves more rapidly towards the end.

To compare the results with previous research, Table 4a shows the relative frequency of the choices for each round for the uninformed subjects and control subjects, as they are not affected by receiving additional information and therefore most comparable. While the design of this study is unique and

comparison with similar multiplier dynamics is impossible, it is valuable to look at choice evolution over time. Compared to subjects with basic information from a BCG by Sbriglia (2008) shown in Table 4b, similar behaviour is observed where over time, participants move towards the equilibrium value 0 with a low multiplier (in rounds 2, 3, 5 and 8), but do not quite reach it. The behaviour is however less pronounced, because of the smaller number of rounds with a low multiplier and high-multiplier rounds in between. Furthermore, when looking at round 5 with multiplier 2/3, the guesses are in the 20 – 50 range shown by Camerer (2003) to be appropriate for first-round results, though the spread is somewhat larger. Overall, similar to what is found by Ho, Camerer and Weigelt (1998), on average choices are closer to the equilibrium for games with p farther from 0 (rounds 1, 4, 6 and 7 with p >1 as opposed to rounds 2, 3, 5 and 8 with p < 1), as can be seen in Table 4a.

Results

Table 4a Relative frequency of the choices for each round for the uninformed subjects and control subjects

1 2 3 4 5 6 7 8 0 – 3 – – 4,2 – – – – 5,6 4 – 10 – – 46,5 1,4 – – – 36,6 11 – 20 1,4 2,8 23,9 1,4 5,6 – – 9,9 21 – 30 1,4 14,1 8,5 1,4 21,1 1,4 5,6 21,1 31 – 40 7,0 21,1 5,6 1,4 31,0 – 1,4 7,0 41 – 50 8,5 19,7 1,4 4,2 15,5 8,5 – 1,4 51 – 60 11,3 9,9 4,2 2,8 15,5 8,5 4,2 1,4 61 – 70 11,3 9,9 – 2,8 5,6 21,1 11,3 – 71 – 80 19,7 15,5 2,8 8,5 1,4 16,9 12,7 2,8 81 – 90 19,7 4,2 2,8 21,1 1,4 22,5 23,9 – 91 – 96 5,6 – – 11,3 – 4,2 11,3 – 97 – 100 14,1 2,8 – 43,7 2,8 16,9 29,6 – Total 100 100 100 100 100 100 100 100

Notes: 51 uninformed subjects, 20 control subjects, 71 subjects in total.

Table 4b Relative frequency of the choices in sessions 1A/1B (Sbriglia, 2008)

1 2 3 4 5 6 0 – 3 20/5,6 –/11,1 –/11,1 –/5,6 –/5,6 5/16,7 4 – 10 5/5,6 25/11,1 5/22,2 5/66,7 45/50,0 90/77,8 11 – 20 15/5,6 35/27,8 75/44,4 85/11,1 50/44,4 5/5,5 21 – 30 15/22,2 10/16,7 10/16,7 –/5,6 –/– –/– 31 – 40 25/22,2 10/22,2 –/5,6 –/– –/– –/– 41 – 50 10/27,8 10/– –/– –/– –/– –/– 51 – 60 10/– 5/5,6 –/– –/– –/– –/– 61 – 70 –/5,5 5/– 5/– –/– –/– –/– 71 – 80 –/– –/5,5 –/– –/– –/– –/– 81 – 90 –/5,5 –/– 5/– –/– –/– –/– 91 – 100 –/– –/– –/– 5/11,1 5/– –/–

Results

Table 5 provides the results of six Wilcoxon rank-sum tests conducted to test for a difference in absolute average deviations between the informed and the uninformed subjects in the second half of the rounds, as well as for a difference between the uninformed subjects and the control subjects across all rounds. None of the tests rejects the null hypothesis of no difference in choices.

Table 5 The difference in absolute average deviations

between informed and uninformed, and uninformed and control subjects

p-value Difference between informed and

uninformed subjects

Rounds 5, 6, 7 and 8-average 0.899

Rounds 6, 7 and 8-average 0.432

Rounds 7 and 8-average 0.533

Difference between uninformed and control subjects

Rounds 1 and 2-average 0.482

Rounds 3, 4, 5 and 6-average 0.898

Rounds 7 and 8-average 0.808

Notes: Wilcoxon rank-sum test. An asterisk (*) means significance at 10%; a double asterisk (**) means significance at 5%. Table 6 provides the results of three Tobit regressions with a lower limit of 0 to account for all deviations made non-negative, all clustered on the group level of four subjects. The first simply tests the effect of the additional information given to informed subjects (independent variable) on average absolute

deviations (dependent variable), averaged over rounds 6, 7 and 8. It does however not yield a significant result. The second regression adds to the information dummy information about demographics, gender and age. This results in a negative information dummy coefficient at 10% significance: coefficient = –3.350, p = 0.061. The third regression also adds the outlier dummy, where subjects with an average absolute deviation over 36 are

Results

regression yields a significant negative information coefficient at 5% significance: coefficient = –5.145, p = 0.013. When looking at the two significant coefficients of the target variable, the information dummy, the results show that informed subjects experience a decrease in average absolute deviation of 3.4 to 5.1. To put this into perspective, the average absolute deviation across all subjects for the last three rounds was 14.45, yielding a 24% to 35% increase in performance among informed subjects. Only the Model 3 regression has a significant

F-statistic (p = 0.016). None of the models has either left- or right-censored observations.

Table 6 The effect of information on best response accuracy in the final three rounds

Model 1: information dummy

Model 2:

additional gender and age dummy

Model 3: additional outlier dummy (average absolute deviation > 36) Information dummy –3.346 [2.064] (0.110) –3.350* [1.756] (0.061) –5.145** [2.005] (0.013) Gender dummy [2.737] 3.934 (0.155) [2.053] 2.394 (0.248) Age 1.767 [1.462] (0.231) 0.495 [1.039] (0.635) Outlier dummy 21.303** [8.493] (0.015) Pseudo R2 _{0.003 0.010 0.046} Prob > F 0.110 0.216 0.016 Left-censored 0 0 0 Uncensored 68 68 68 Right-censored 0 0 0

Notes: Tobit regressions, lower limit of 0, standard errors clustered on the four-person group level. Model 1: the effect of information about previous round guesses on best response accuracy in an average of absolute deviations from target of rounds 6, 7 and 8. Model 2 adds age and a dummy for gender. Model 3 further adds a dummy for outliers. Coefficient [clustered robust standard errors] (p-value). An asterisk (*) means significance at 10%; a double asterisk (**) means significance at 5%.

Results

In Table 7, the results are provided of two additional Tobit regressions based on Model 3 from Table 6 including information, gender, and age and outlier dummies. Model A and Model B are based on the average absolute deviations of the rounds 5, 6, 7 and 8-average and the rounds 7 and 8-average, respectively. In Model A, this yields a negative coefficient for the information dummy at 5% significance: coefficient = –3.490, p = 0.041. In model B, this results in an also negative coefficient at 5% significance: coefficient = –4.863, p = 0.026. Across models, these results yield an increase in performance ranging from 23% in Model A (average absolute deviation across all subjects for the last four rounds was 15.07), to 35% in Model 3, to 41% in Model C (average

absolute deviation across all subjects for the last two rounds was 11.92). Finally, both models display a significant F-statistic at the 5% level (Model A: p = 0.030; Model B: p = 0.034). Again, all observations of both models are uncensored. Table 7 The effect of information, gender and age (taking into account

outliers) on best response accuracy in the final four and final two rounds

Model A: rounds 5, 6, 7 and 8-average

Model B: rounds 7 and 8-average Information dummy –3.490** [1.674] (0.041) –4.863** [2.131] (0.026) Gender dummy 1.794 [1.937] (0.358) 0.197 [2.021] (0.922) Age 0.312 [0.892] (0.728) 1.513 [1.243] (0.228) Outlier dummy 23.194** [7.515] (0.003) 26.144** [9.601] (0.008) Pseudo R2 _{0.065 0.054} Prob > F 0.030 0.034 Left-censored 0 0 Uncensored 68 68 Right-censored 0 0

Notes: Tobit regressions, lower limit of 0, standard errors clustered on the four-person group level. Model A and B: the effect of information about previous round guesses, a gender dummy, age and an outlier dummy on best response accuracy in an average of absolute deviations from target of rounds 5, 6, 7 and 8 and rounds 7 and 8, respectively. Coefficient [clustered robust standard errors]

(p-value). An asterisk (*) means significance at 10%; a double asterisk (**) means significance at 5%. !

! !

Results

Table 8, finally, shows the results of Tobit regression analyses comparing the uninformed subjects and the control subjects. As seen in Figure 2, these two groups do not diverge from each other much across rounds, and this is

reflected in the regression results. To provide an overview of all stages of the experiment, differences in average absolute deviations of the uninformed and control subjects have been analysed for the first two round average, the middle four round average and the final two round average to reflect the beginning, middle and final stages of the experiment, respectively. As can be seen from the regression results, there is a statistically significant difference between the uninformed subjects and the control subjects only in the average from rounds 3, 4, 5 and 6., at 10% significance: coefficient = 3.190, p = 0.082.

!

Table 8 The difference between informed subjects and control subjects across all eight rounds played

Model I: rounds 1 and 2-average

Model II: rounds 3, 4, 5 and 6-average

Model III: rounds 7 and 8-average Treatment dummy 2.885 [3.175] (0.367) 3.190* [1.806] (0.082) 0.890 [1.922] (0.645) Gender dummy 2.524 [3.259] (0.441) 1.040 [2.627] (0.694) –1.037 [1.998] (0.606) Age –0.047 [1.674] (0.978) –1.629 [1.065] (0.131) 2.916** [1.104] (0.010) Outlier dummy 26.188** [5.907] (0.000) 32.502** [2.286] (0.000) 37.530** [6.749] (0.000) Pseudo R2 _{0.031 0.075 0.095} Prob > F 0.001 0.000 0.000 Left-censored 0 0 0 Uncensored 71 71 71 Right-censored 0 0 0

Notes: Tobit regressions, lower limit of 0, standard errors clustered on the four-person group level. Model I, II and III: the difference between informed and control subjects with age and dummies for treatment group, gender and outliers in the rounds 1 and 2-average, rounds 3, 4, 5 and 6-average and rounds 7 and 8-average, respectively. Coefficient [clustered robust standard errors] (p-value). An asterisk (*) means significance at 10%; a double asterisk (**) means significance at 5%.

Discussion 5. Discussion

When looking at the descriptive data in Table 1, one thing that stands out is the fact that a somewhat higher fraction of the informed subjects was female when compared to the uninformed subjects. At the same time, Table 3 shows that females had a somewhat higher average absolute deviation from target. This is in line with findings by Burnham et al. (2009). Another point to consider is the other remarkable result from Table 3, showing that subjects aged 18 perform on average significantly worse than 15, 16 and 17-year olds in term of average absolute deviation. In the Netherlands, if a student fails to perform well enough in a given year, they have to redo this year in order to finish it with passing grades. An intuition for the 18-year olds performing worse than their younger classmates is therefore the fact that these subjects most likely failed a year and had to redo it, meaning that learning could be more difficult for them relative to their classmates. Out of the 17 informed subjects, 2 were 18 (11.8%), whereas out of the 51 uninformed subjects, 5 were 18 (9.8%). These two factors together could have biased the average deviations of the informed subjects upward, and therefore it makes sense to include gender and age in the regression analysis.

The data also show that a few subjects displayed an average absolute deviation from target across all rounds quite a bit higher compared to others. As shown in Figure 1, the average absolute deviations of subjects 67, 66 and 28 were 32.5, 33.7 and 34.8 respectively, whereas the average absolute deviations of the four outlier subjects 9, 41, 20 and 32 jump to 40.4, 44.2, 47.4 and 52.4, respectively. While the former deviations are large, they are still to be reasonably expected when comparing with the other subjects. However, the four outlier subjects appear to fall out of line.

This observation is supported when plotting leverage (a measure of how far an independent variable deviates from its mean) against the normalised squared residuals from a simple regression of the information and gender dummies and age on the average absolute deviations to identify outlier

observations, as is shown in Figure 3: players 41, 32 and 9 are outliers in both leverage and squared normalised residual, while player 20 is in squared

Discussion

normalised residual. Based on these two forms of evidence, it was decided to classify these four players as outliers through a dummy variable.

Figure 3 Leverage-versus-residual-squared plot

Notes: numbers represent subject numbers. When looking at the results of the Wilcoxon rank-sum tests in Table 5, the informed subjects do not seem to make significantly different choices when compared to uninformed subjects. This is also the case when comparing uninformed subjects and control subjects. However, this analysis does not include a correction for group effects (use of clusters), and also does not account for differences caused by gender and age, and is not corrected for outliers.

These factors are taken into account in the regression analyses in Tables 6 and 7. From those results, it is clear that in the second half of the rounds played, informed subjects outperform the uninformed subjects. In Model 2 without the outlier dummy, informed subjects outperform the uninformed subjects by 24% at 10% significance, and in Models 3, A and B including the outlier dummy, informed subjects outperform the uninformed subjects by up to 41% at 5% significance. This result is also visible in Figure 2, where informed subjects appear to decrease average absolute deviations more when compared to uninformed subjects. This also shows from the final round-averages in the

Discussion

bottom half of Table 2. Furthermore, average deviation dynamics shown in Figure 2 imply that informed subjects perform somewhat worse in the first few rounds, but near the end start outperforming the informed as well as the control subjects. Underlying this result could be a mechanism in which informed

subjects are initially overwhelmed with the additional information and perform worse in the first few rounds, but later on learn to use this information and improve their best response relative to the other subjects. At the same time, Table 8 shows that the uninformed and the control subjects seem not to differ in performance. While the coefficient of the difference between uninformed and control subjects in the rounds 3, 4, 5 and 6-average is significant at 10%, this seems to be by chance as the other two coefficients are not statistically significant.

Overall, the results imply that in the final rounds the informed subjects have a lower average absolute deviation from target compared to uninformed subjects, confirming hypothesis 1. Also, the performance of uninformed subjects does not differ significantly from control subjects, confirming hypothesis 2 and thereby the condition that informed subjects must not increase performance at the cost of uninformed subjects.

Average choices ranked by degree of iterated dominance in Table 4a are in line with previous findings when looking at (1) differences between higher and lower p’s (Ho, Camerer & Weigelt, 1998); (2) the range of guesses in first rounds with p = 2/3 (Camerer, 2003); and (3) the shift of relative frequency of choices towards equilibrium in Table 4b (Sbriglia, 2008). One important similarity is with research by Camerer (2003), showing that most choices in round 5 with

multiplier p = 2/3 match first-round results found by Camerer. This indicates that indeed, subjects play subsequent rounds as if they were first rounds. As a consequence, it can be said with more certainty that the differences in performance between informed and uninformed subjects stem from learning about opponent rationality instead of extrapolation on the basis of previous rounds. The latter was found to be a flaw in the paper at the base of this research in collaboration with Dovile Venskutonyte and Nandi Oud.

Discussion

An interesting result appearing in the bottom half of Table 2 is the fact that both uninformed and control subjects seem to perform somewhat better in low-multiplier rounds than in high-multiplier rounds, whereas informed subjects perform almost exactly the same in terms of absolute average deviation. While this is only an intuition, it seems that additional information in the form of seeing last round’s guesses by others given to informed subjects must somehow be changing their behaviour. This could potentially be explained by the order of the multipliers across rounds: the rounds with a low multiplier 2, 3 and 5 are in the first part of the game, where informed subjects appear to be performing worse when compared to uninformed subjects. However, of the last three rounds, rounds 6 and 7 are with a high multiplier, and by this time informed subjects seem to begin outperforming uninformed subjects. This order effect could be causing the shift in average deviations across low and high multiplier groups.

Another remarkable result is that informed subjects seem to outperform uninformed subject in round 1, while before that round no information was distributed yet, and also nobody knew who would be receiving information and who would not at that time. When performing a regression similar to models 3, A and B on first round results, this yields a negative information coefficient at 10% significance: –6.590, p = 0.072. It is however unclear how to explain such a result, as up to that point there were no differences in treatment between the informed and uninformed subjects.

A strength of this study, when compared to other studies using students, stems from the fact that a study by Belot, Duch and Miller shows that student and non-student subject pools behave differently in the choices they make, because students are more likely to behave selfishly and in line with the notion of a rational agent than non-students (Belot, Duch & Miller, 2015). While the

subjects from this study are quite young and could have had some difficulties with the experiment regarding mathematical skills, they may represent the

broader populace more accurately than student subject pools, especially as they had diverse educational profile backgrounds.

There are a few shortcomings to be addressed regarding this research. Because subjects playing against each other came from the same class, it may

Discussion & Conclusion

have been that some subjects placed together in a group knew each other, influencing their ability to predict their opponents. This was not controlled for, while it could have influenced results. Another variable that could have been controlled for was ability, for instance in terms of average grade. While age captured some of the subjects’ abilities from education, more accurate information could have been collected that at the same time could have been used to see if the worse performance of subjects aged 18 was indeed caused by them having more trouble learning, and may have potentially explained the performance difference in informed and uninformed subjects in the first round. Also, while subjects were asked not to share details about the experiment with classmates who had not yet participated, there was no way to control such behaviour, and some information transferring may have taken place.

6. Conclusion

While it is clear that people do not necessarily always best respond in situations that require strategic decision-making, it is debated whether this is caused by the bounded rationality of people, or because they have trouble predicting rationality of others and therefore cannot best respond when making a decision.

This research set out to investigate if allowing subjects to learn about the rationality of others in previous rounds of strategically different Beauty Contest games would induce those subjects to improve their best response. Based on results showing that in the final few rounds of the game, informed subjects outperform uninformed subjects, and at the same time uninformed subjects do not seem to display different behaviour from that of control subjects, it seems that people can indeed improve best response when learning about the

rationality of others. In the final three rounds of an eight-round game, informed subjects on average decrease the distance to the target value by 24% to 41% compared to uninformed subjects. Comparison of the results with research by Camerer (2003) shows that the experimental design is successful in making each of the rounds played effectively a first round and thus preventing subjects from predicting next round behaviour by extrapolation, a flaw found in previous research at the basis of this study.

Conclusion

Future research should focus on refinement of the design in which subjects can learn about rationality of their opponents in order to improve their prediction ability, without giving them information that would make an

experimental game easier. A different way in which this also could be achieved could be to acquaint players with one another through one game, and then test their abilities to predict opponents in another, different from the first. Using different kinds of games to test the findings in this paper could further solidify the result that people have trouble predicting the rationality of others.

References

1. Belot, M., Duch, R., & Miller, L. (2015). A comprehensive comparison of students and non-students in classic experimental games. Journal

of Economic Behavior & Organization, 113, 26-33.

2. Bosch-Domenech, A., Montalvo, J.G., Nagel, R., & Satorra, A. (2002). One, Two, (Three), Infinity, ...: Newspaper and Lab Beauty-Contest Games. American Economic Review, 92, 1687-1701.

3. Burnham, T.C., Cesarini, D., Johannesson, M., Lichtenstein, P., & Wallace, B. (2009). Higher cognitive ability is associated with lower entries in a p-beauty contest. Journal of Economic Behaviour &

Organization, 72, 171-175.

4. Camerer, C. (2003). Behavioural game theory. Princeton, New Jersey: Princeton University Press.

5. Coricelli, G., & Nagel, R. (2009). Neural correlates of depth of strategic reasoning in medial prefrontal cortex. Proceedings of the National

Academy of Sciences, 106(23), 9163-9168.

6. Grosskopf, B., & Nagel, R. (2007). Rational Reasoning or Adaptive Behavior? Evidence from two-person beauty contest games. Evidence

from Two-Person Beauty Contest Games (June 2007). Harvard NOM

research paper, (01-09).

7. Güth, W., Kocher, M.G., & Sutter, M. (2002): Experimental Beauty Contests with homogeneous and heterogeneous players and with interior and boundary equilibria. Economics Letters, 74, 219-228.

8. Ho, T., Camerer, C., & K. Weigelt (1998): Iterated Dominance and Iterated Best Response in Experimental “p-Beauty Contests”.

American Economic Review, 88, 947-969.

9. Kahneman, D. (2003). Maps of Bounded Rationality: Psychology for Behavioral Economics. The American Economic Review, 93(5), 1449-1475.

10. Kocher, M.G., & Sutter, M. (2005). The Decision Maker Matters: Individual Versus Group Behaviour in Experimental Beauty Contest Games. The Economic Journal, 115(500), 200-223.

11. Kocher, M., Sutter, M., & Wakolbinger, F. (2014). Social Learning in Beauty-Contest Games. Southern Economic Journal, 80(3), 586-613. 12. Nagel, R. (1995). Unraveling in Guessing Games: An Experimental

Study. The American Economic Review, 85(5), 1313-1326.

13. Ohtsubo, Y., & Rapoport, A. (2006). Depth of reasoning in strategic form games. The Journal of Socio-Economics, 35, 31-47.

14. Sbriglia, P. (2008). Revealing the depth of reasoning in p-beauty contest games. Experimental Economics, 11, 107-121.

15. Simon, H.A. (1955). A Behavioural Model of Rational Choice. The

Quarterly Journal of Economics, 69(1), 99-118.

16. Simon, H.A. (2000). Bounded Rationality in Social Science: Today and Tomorrow. Mind & Society, 1(1), 25-39.

17. Stahl, D.O. (1996). Boundedly Rational Rule Learning in a Guessing Game. Games and Economic Behaviour, 16, 303-330.

18. Weizsäcker, G. (2003). Ignoring the rationality of others: evidence from experimental normal-form games. Games and Economic Behaviour,

Appendix

Appendix I: Paper by Nandi Oud, Dovile Venskutonyte and Tomek Dabrowski

Effects of having current information about common

rationality

30 January 2015 Tomek Dabrowski 10178902

Experimental Economics Dovile Venskutonyte 10086102

Team 1G Nandi Oud 10002425

1. Introduction

In everyday social and economic decision-making situations, it is useful to predict what others might do when an outcome depends on the sum of decisions made by multiple agents acting simultaneously. For instance, when investing in a new firm on the stock exchange during an initial public offering, it is certainly helpful to know or be able to predict whether others will invest, since their decisions will directly influence the price.

General economic theory and game theory assume that people are perfectly rational in their decision-making. This would mean that one could simply assume others are perfectly rational and act accordingly. However, it can be derived from experimental results that this is often not the case and people deviate from perfect rationality. On the one hand this could be attributed to people being boundedly rational and not able to find the optimal solution to a problem, on the other hand people might be perfectly rational but have doubts that others are as well (Grosskopf & Nagel, 2007).

In previous research, playing multiple rounds and revealing results from previous rounds or even revealing strategies used by winners is used to allow for learning. However, one could argue that in everyday situations, it is possible to infer and predict what others might do before they make their decision. This information could then be used to make a better assessment of what the common rationality level is, what other participants will decide and then act accordingly. Translated into our research this means a new element is added that allows subjects to adapt immediately in each round instead of only learning from previous rounds. The research question we will therefore try to answer is the following: do current cues about common rationality improve decision-making ability when outcomes depend on aggregate choices?

A strategic game often used to study the bounds of human rationality is the Beauty Contest game, or BCG (Grosskopf & Nagel, 2007). Imagine the following situation: multiple candidates apply for a job position at a firm, and are all invited to a job interview. It is in their best interest to be hired, and additionally negotiate a

satisfying wage. When at the interview, each candidate can indicate how much she or he would like to get paid. This estimate should not be too low: the chance of getting hired will increase but the wage might be too low. However, too high of an estimate

Appendix I: Paper by Nandi Oud, Dovile Venskutonyte and Tomek Dabrowski might deter the firm from hiring the candidate. The optimal choice would be to estimate a wage around the average estimate of all candidates.

The description of this potential situation can be matched closely to the BCG: subjects within a group have the goal to guess a number on a certain interval that is closest to the target number. The target number is a proportion of the average guess of the group, with or without adding a constant. The BCG has a unique sub-game perfect equilibrium, which should be reached when everyone is perfectly rational. However, this is not always the case, because rationality is not common knowledge and not everybody expects everybody else to be perfectly rational. Therefore it is not a best response to play according to the optimal strategy (Sbriglia, 2008). Using the BCG in a laboratory experiment and having full control over the setup and the participants allows us to conduct an analysis of the rationality level, the decision-making process and its results, data that would otherwise be very hard to collect in a setting outside of the laboratory.

Our main result suggests that real-time information about the common rationality levels of a group positively affect the ability of subjects to successfully predict choices of others. This in turn is an indication that people are to some degree perfectly rational, but have doubts that others are as well and cannot predict what level of rationality the group will exhibit. However, it is important to note that due the small sample size and other shortcomings it is hard to tell how reliable this result is for conclusions about the general population. Careful design of a larger-scale experiment with a specific approach to disentangling different information effects might help increase knowledge about this topic.

2. Experimental Design and Procedures

Economics students of the University of Amsterdam’s Masters program were asked to participate in a pen and paper experiment. To test our research question we conducted a variation of the Beauty Contest game in a group of seventeen

participants, who were split into a control group of eight and a treatment group of nine. We used a between-subjects design since we compare participants in the

treatment group who received information to those from the same group who did not. Furthermore we also make use of a control group to see whether the subjects from the treatment group without information behave differently from those in the control group (explanation follows).

At the beginning, the participants were randomly assigned to the control or treatment group by a blind draw of playing cards. Black cardholders were assigned to the control group and red cardholders were in the treatment group. Participants were unaware of to which group they were assigned. The number of the card provided the participants with an ID to ensure anonymity. First, the red cardholders (treatment group) were asked to take a seat and from this point onwards communication was not permitted and actions of others were not visible. Instruction sheets were handed out individually, however they were also read out loud to ensure all participants knew that the information was common. As an incentive, the participants were informed that in a randomly determined round of the game, the player with the smallest deviation from the target number would be given a payoff of three euros. These conditions applied to both groups.

Appendix I: Paper by Nandi Oud, Dovile Venskutonyte and Tomek Dabrowski

Control Group

The control group played a variation of the Beauty Contest Game, discussed previously. The usual form of the game is for participants in a group to guess an integer number from the interval [0; 100] that they think will be closest to the target number, which is a fraction (p) of the group average. In our game a constant of 30 was added resulting in the target formula:

!"#$%! =!_!∗ (!"#$!%#!!"#$$ + 30)

Answer sheets were distributed to the participants with space for eight rounds. Before the first round participants were told that a maximum of eight rounds would be played, but that the experiment may end after fewer rounds. In round one each

participant filled in his or her guess for the first round on the answer sheet. Participants were given approximately one minute to make their guess. The experimenter came to each participant and wrote down his or her answer. The experimenters calculated the average and the target. Then the average and the target numbers were announced to all participants twice. Participants were informed that the first round ended, that the second round would begin and that they could fill in their guesses for the second round. Rounds two to five continued in the same fashion.

After the average and the target for the fifth round were announced the

experiment ended. The experimenters calculated the results and asked one participant to roll a die, which would determine the payoff round. The participant with the smallest deviation from the target number in that round was given a payoff of three euros. Participants were called one by one according to their ID and were given a questionnaire. One participant was given his or her payoff in addition to the questionnaire.

Treatment group

The experiment in the treatment group proceeded in the same fashion as in the control group, with the exception of the treatment being applied. Three participants received the guesses of three other participants in their group. The ones who received information and the ones who provided it were randomly determined. The providers of information were not entirely the same for all three receivers, but randomly drawn from the group of six, who did not receive information. All participants were

informed that three of them would receive information and that the rest may or may not provide it, but they were not informed who specifically would receive or provide information.

After the guesses in the first round were made, the experimenter collected all the answer sheets. Information was written on the answer sheets of the three receivers and all answer sheets were redistributed. After approximately one minute the

experimenter came to each participant (in order not to reveal who received information) and collected the revised guesses. The experimenters calculated the average and target numbers and announced them to the group. Rounds two to five continued in the same fashion, with the same three participants receiving the choices of that round from the same three providers. The entire experiment lasted for

approximately ninety minutes. The control group took about half an hour and the treatment group finished in about forty-five minutes.

Appendix I: Paper by Nandi Oud, Dovile Venskutonyte and Tomek Dabrowski

Design improvements

The experiment went smoothly, no miscalculations or corrections were made during the experiment and the time to calculate the average and target numbers was short enough. However, there were a few problems that future researchers should take note of.

First, instructions sheets were given to each participant as well as read out loud, but there was a small difference between the two. Participants were informed about the number of rounds only by the experimenter and not in the written

instructions. This difference may have led to confusion and should have been avoided.

Second, participants were asked to take seats in specific rows, but individual seating was not specified. This could lead to communication, especially considering that participants knew each other. However we have no reason to believe that there was any communication, since the room was silent and there were barriers between participants.

Third, one participant in the questionnaire indicated that a control question would have been desirable, since there was confusion about the constant term of 30. Since many participants knew the original BCG design they may not have paid careful attention to the instructions.

3. Theoretical Analysis and Predictions

As was indicated in the introduction, the literature is not conclusive on the topic of people being either boundedly rational or of them being perfectly rational but doubting others are as well. Grosskopf and Nagel (2007) argue that the first is more probable and that people improve upon playing strategies by learning and adapting. Ohtsubo and Rapoport (2006) however state that subjects apply a mental model about other players’ reasoning ability in order to perform well and thus the second statement is closer to the truth. Sbriglia (2008) concludes from her research that both imitative strategies and best responses to imitative strategies are used by subjects, thus both being boundedly rational and doubting others’ rationality influences subjects’ strategies.

Reasons for non-equilibrium behaviour

There are various explanations for why players may not choose the sub-game perfect Nash equilibrium (NE) strategy. First, it is possible that some players are simply unable to calculate the NE or are confused about what equilibrium behaviour is. Considering that our sample consisted of economics students, who have seen the p = 2/3 Beauty Contest game before, some may not take into account that in our model a constant of 30 was added, resulting in a sub-game perfect equilibrium of 60. Instead they might play according to the standard, most often used variant of the BCG

without a constant and subsequently choose to target 0 (zero), the sub-game perfect NE of the simplest form of this game. From feedback we collected with the

questionnaire we learnt that this was indeed the case for at least 1 player, and by looking at first-round choices possibly more participants did the same.

Grosskopf and Nagel (2007) study a two-player BCG, where it is much clearer that 0 is the optimal choice, because in order to win you always have to choose a number lower than your competitor. They find that even though the choices

eventually converge to 0, there is a persistence in choices above 0, and convergence is often due to one rational player solving the game and then being imitated by his or her