Documento de trabajo E2003/47 Teaching Nash Equilibrium and Strategy Dominance: A Classroom Experiment on the Beauty Contest Virtudes Alba Fernández Pablo Brañas Garza Francisca Jiménez Jiménez Javier Rodero Cosano Consejería de Relaciones Institucionales centrA: Fundación Centro de Estudios Andaluces Documento de Trabajo Serie Economía E2003/47 Teaching Nash Equilibrium and Strategy Dominance: A Classroom Experiment on the Beauty Contest* Virtudes Alba Fernández Pablo Brañas Garza U. Jaén & centrA U. Jaén, centrA & LINEEX Francisca Jiménez Jiménez Javier Rodero Cosano U. Jaén, centrA & LINEEX centrA & LINEEX RESUMEN El objetivo de esta investigación es mostrar cómo el uso de experimentos de clase puede ser un buen instrumento pedagógico para la enseñanza del concepto de Equilibrio de Nash. El juego utilizado para nuestros propósitos es una versión repetida del juego del concurso de belleza (BCG), un simple juego de adivinanzas cuya repetición permite que los estudiantes reaccionen a las opciones que elijan sus contrincantes de forma que se converja hacia la solución de equilibrio. Llevamos a cabo el experimento con estudiantes de carrera sin conocimiento previo de teoría de juegos. Tras cuatro rondas, se observó en todos los grupos una clara tendencia decreciente en el número medio escogido, por lo que podemos afirmar que a través de la repetición del BCG, los estudiantes aprenden rápidamente cómo alcanzar la solución de equilibrio. Palabras clave: Experimentos de clase, Juego de Concurso de Belleza, Enseñanza, Equilibrio de Nash. ABSTRACT The aim of this investigation is to display how the use of classroom experiments may be a good pedagogical tool to teach the Nash equilibrium (NE) concept. The basic game for our purposes is a repeated version of the Beauty Contest Game (BCG), a simple guessing game whose repetition lets students react to other players’ choices and to converge iteratively to the equilibrium solution. We performed this experiment with undergraduate students without any previous knowledge about game theory. After four rounds, we observed in all groups a clear decreasing tendency in the average chosen number. So, our findings prove that, by playing a repeated BCG, students quickly learn how to reach the NE solution. Keywords: Classroom Experiments, Beauty Contest Game, Teaching, Nash Equilibrium. JEL classification: A22, C99, D83 * Authors wish to thank Rosemarie Nagel, Curro Martínez Mora and Quique Fatás for their helpful suggestions; also, Ana & Pili López López for their contribution in the data gathering and processing. We gratefully acknowledge financial support from University of Jaén R+D program (# 20210/148). Fundación Centro de Estudios Andaluces Reasons for this study Over the last years, Game Theory (GT) has become an essential part of intermediate microeconomics theory. Obviously the key concept in Game Theory is the Nash Equilibrium (NE). Therefore, this powerful notion is an inevitable part of our teaching. However, educators know that it is not easy to explain NE in class. Students usually ﬁnd diﬃculties in understanding this idea and, consequently, the implicit features of this concept. Generally, the teaching of GT begins with the study of strategic interactions among players. Once they understand that each player’s payoﬀs depend not only on their own actions but also on those by other players, the next step is to explain the process of elimination of dominated strategies. For this, we only require two basic assumptions: rationality (maximization) and common knowledge (of rationality). Both concepts let us solve (dominance solvable) games and also to predict some particular behavior. However, there exist games that are not dominance solvable even if they have a NE. In this case, NE requires an additional hypothesis: common knowledge of players’ beliefs about rivals actions. So, the best response become the suitable mechanism to reach the solution. Best response is the best strategy that a player may choose given the strategies chosen by his rivals. The NE is reached when all agents play (some of) their best response. So, NE is self-enforcing because no player has incentives to deviate from it. Clearly, the use of NE eliminates the circular reasoning (player 1 thinks that player 2 thinks that player 1 thinks . . . ). This paper proposes a pedagogical tool to simplify the teaching of NE: successive repetitions of a dominance solvable game, the Beauty Contest Game (BCG). This particular game is very useful to show the two prior procedures intuitively. Concretely, we perform four repetitions of it as a 1 E2003/47 classroom experiment, previous to a game theory teaching class. Our ﬁndings clearly illustrate that students unexperienced (in GT) apply at each round both iterative elimination of dominated strategies and bestreply behavior to the previous choices. On average, we observe a recursive approximation to the Nash prediction, which we name learning eﬀect. Therefore, using the repetition of this static game guarantees us that students learn how to solve it in some way or the other. Theoretical Background Since the introduction of the concept in Nash (1951), the Nash equilibrium is not only the standard tool for the economic scientiﬁc community but also the basis for a systematic teaching of the discipline in the modern era. In game theory, it is a common practice to assume that individuals “magically” choose a set of actions such that all the (inﬁnitively recursive) predictions become true. So as to reach this hard-to-believe solution, theorists can follow one of the two original Nash interpretations: agents are either perfectly rational, or there is an evolutionary equilibrium1 . As it is well known, the ﬁrst one relies on the assumption that all agents are able to compute the game equilibria and they reach one of them. Instead, the second one presupposes the existence of a big population of simple people that play the game in an evolutionary framework: they pick up one strategy randomly, if the outcome is good they will repeat it again (or they disappear if the strategy is bad). After some time, the game will converge to one of the Nash equilibriums. The BCG is a good example of the underlying convergence to both interpretations. From a rational point of view, it is not credible that the subjects will solve this problem (few people can at the ﬁrst try!) but if 1 Although Nash only uses the rationality explanation in the published version due to space limitations. 2 Fundación Centro de Estudios Andaluces it is repeated, most of the times the solution converges to equilibrium. Beauty Contest Game is a simple guessing game that facilitates the evaluation of the individuals’ level of reasoning. The basic BCG is as follows: a certain number of subjects are invited to play a game and simultaneously all of them have to choose a number from an interval (generally [0, 100]). The winner is the player whose number is closest to p-times the mean of all the chosen numbers, being p > 0. The winner receives a ﬁxed prize, the losers get nothing. Under these rules the unique Nash Equilibrium is zero for all players2 . Figure 1, taken from Ho et al. (1998, pg. 951), shows the convergence to the zero theoretical solution from a dominance iterative point of view3 . Any number chosen between 66.6 and 100 is dominated by 66.6 (100 ∗ 2/3, if p = 2/3), so they say that the interval [66.6, 100] corresponds to an irrational behavior (R(0) for us). Rational individuals will always choose a number in the [0, 66.6] interval. Applying the same reasoning, R(1) players will choose a number below 66.6 (but above 44.4). Since 44.4 will dominate again any number between 44.4 and 66.6, we say that any number below 44.4 (and above 44.4 · 2/3 = 29.6) corresponds to a R(2) individual. Following this iterative reasoning level process ad inﬁnitum, we get the theoretical Nash equilibrium (0, with R(∞)). Then, this game is dominance solvable4 . 2 This is not as obvious as it looks, Bosch-Domènech et al. (2000) tries an one-shot BGC with game theoretics. The answers were very diﬀerent to zero but better than the usual ones! 3 See Rapoport and Amaldoss (2000) for an experimental study of iterative elimination of (strongly) dominated strategies. 4 In the real world, there is even a higher rationality level. An individual who knows the zero-solution by any of these methods, can guess that most people would not achieve the zero solution, so the “intelligent” individuals will link their answer to their estimation of the average rationality level. We can call this rationality level ∞-plus. Grosskopf and Nagel (2001) argue that most individuals think that other people are not fully rational and this is the reason why the equilibrium is not reached immediately. 3 3 E2003/47 R(4) R(3) ··· 0 13.2 19.7 29.6 R(2) R(1) 44.4 R(0) 66.6 100 Figure 1: Iterated reasoning of individuals by eliminating dominated strategies The BCG original idea was ﬁrst mentioned by Keynes (1936) when he wanted to express that a clever investor has to “anticipate the basis of conventional valuation a few months hence, rather than . . . over a long term of years” (page 155), so she could act in the stock market before the rest of investors do. The formal game model was introduced by Moulin (1986). As we noted above, the unique equilibrium of the game is 0 for p < 1 and it is obtained by iterated elimination of the weakly dominated strategies. After this basic framework, some experimental researchers started an investigation area on BCG or “p-beauty” (see Ho et al., 1998). The ﬁrst experimental study is found in Nagel (1994, 1995)5 . Other works have been carried out by Bosch and Nagel (1997a,b); Bosch-Domènech et al. (2000); Duﬀy and Nagel (1997) and Ho et al. (1998). See also Nagel (1998) for a survey of the literature. There are diﬀerent BCG experiments. Sometimes subjects are students, other times professors or newspaper readers (which presumably have diﬀerent education levels). Some studies are one-shot, others are repeated, communication versus non-communication, laboratory versus ﬁeld experiments. Generally, BCG has been run with individual subjects. However Kocher and Sutter (2001) compares the individual versus group behavior in this 5 The main purpose of Nagel (1994, 1995) was to contrast an iterated best-reply domi- nance model (IBRB) 4 Fundación Centro de Estudios Andaluces type of game. Concretely they ﬁnd that groups do not apply deeper levels of reasoning but that they learn faster instead. Repetition permits the individuals to learn dynamically from other people’ expected reasoning behavior. Our experiment is very similar to Ho et al. (1998). In their design the information is based on the previous period choices. Therefore, the learning process in Ho et al. (1998) is based on an evolutive game. Instead, Weber (2003) argue that learning in the BCG could happen even without feedback. Even when subjects do not receive any information between periods, authors found that there is convergence towards the NE of the game. Over the last years, the research on the BCG has returned to the original idea of Keynes. Hirota and Sunder (2002) explore experimentally whether price bubbles in security markets are generated by a beauty-contest mechanism: investors have to create beliefs about the others’ beliefs (second-order beliefs) and these beliefs depend on the third-order beliefs which in turn depend on the fourth-order beliefs, and so on, if the dividends are paid beyond their personal investment horizons. They conclude that when the realization dividend is distant and well beyond the investors investment horizon, investors ﬁnd diﬃcult to induct the fundamental value of securities from the future to the present (backward induct). This diﬃculty gives rise to price bubbles because, in this case, investors adjust their expectations on the basis of observed prices (forward induction). Up to now, the BCG has been used to study the depth of reasoning level of individuals. However in this paper we propose a new application of this framework in this paper: classroom experiments. 5 E2003/47 The experiment The experiment involves a repeated version of the Beauty Contest Game in several groups of students. The performance of this classroom experiment is useful for the teaching of Nash Equilibrium; speciﬁcally to the iterated elimination of dominated strategies. Given that Nash equilibrium is a general topic in both intermediate microeconomics and industrial organization (and, obviously, in Game Theory too), we expect that this pedagogical tool helps educators. Three aspects are crucial to run this experiment: 1. The size of the group should be large enough to reduce the eﬀect of any individual guess on the average (of guessing numbers); on the other hand, very large group causes increasing monitoring cost to the instructor6 . Nagel (1995) uses 12 and 17 subjects in each group; Ho et al. (1998) reduces the group size to 7 subjects and 3 subjects (like Kocher and Sutter, 2001). Following these papers we use small (5-6 subjects) groups and large groups (10-11 subjects)7 . Each subject will play a repeated BCG against the rest of the member of his group. There is not any relation among groups; that is, each group has its own game. 2. In order to motivate students you should design a mechanism of rewards. In this classroom experiment we gave the winner of each round 0.25 extra-credit points as an additional mark in the midterm exam of the ﬁnal grade. In the case of several winners at the same round (within the same group) the prize was split among them. 6 Furthermore, Ho et al. (1998) illustrates that reduced groups (3 subjects in its re- search) need longer time to converge. 7 It is diﬃcult to reproduce identical size groups in class. Usually, the instructor does not have the exact number of students required. 6 Fundación Centro de Estudios Andaluces Our experiment was performed in Intermediate Microeconomic Theory during the 2003 course (spring semester) in three diﬀerent classes: Business (morning and afternoon groups, B-1 and B-2) and Business + Law (one single group, B+L). All groups shared identical subject handout, lecturer and marks system (ranging from 0 to 10). Note that this is an important subject in both degrees. 3. Last, another important issue is the number of rounds that experimental subjects have to play the BCG. As our argument is for students learn Nash equilibrium through the iterated elimination of dominated strategies, we need some rounds to observe this learning eﬀect. Kocher and Sutter (2001) proves that 4 rounds are enough to approximate the theoretical prediction. Finally, our reduced group size guarantees that four rounds are enough to observe learning. The performance of this experiment lasts about an hour and the postexperimental session is during following half an hour8 . Therefore, the whole class takes over 1.5 hours. Procedures 1. The instructor must deﬁne the size of each group. It is irrelevant if one group diﬀers slightly in its size with respect to the others; Nevertheless, it would be desirable to get similar sizes. Our results will show that 10 is an adequate number. 2. Given any deﬁned size, the instructor selects randomly one monitor per group from the students pool. They will be awarded 0.25 points. The monitor will help the instructor recording and monitoring the experiment in his group. Each monitor will be given a calculator and 8 Note that it is worth having both sessions consecutively. 7 E2003/47 a “monitoring sheet” (see the appendix) on which he will track the guessed number given by each experimental subject (of his group). Note that this monitoring sheet is crucial to explaining results after the experimental session. Since the average and winning numbers are recorded on this sheet, the instructor can summarize the results immediately. These easily illustrate basic concepts of Game Theory as iterated elimination of dominated strategies. 3. The third task is to create groups, that is, to randomly allocate each experimental subject within the group so as to avoid that friends sit together. 4. In order to facilitate monitor’s tasks, the instructor should locate each group in one row (or column). 5. The lecturer must discuss instructions and procedures with the monitors. Afterwards, monitors give the instruction sheet and the individual sheet (see appendix) to the experimental subjects. 6. The instructor must explain instructions orally. It is important to avoid numerical examples. In this kind of games the generation of focal points is immediate. Any doubt is answered publicly. Experimental subjects are informed that any communication is absolutely forbidden9 . 7. When everybody is fully aware of the rules, the experiment can begin. Round 1: Experimental subjects must ﬁll their guess number in their answer sheet. Then, the monitor collects all guessing numbers and calculates mean and 2/3 ∗ mean. By comparing this value to those reported by the experimental subjects, he determines the winner. The 9 Subjects are required to keep the maximum level of conﬁdentiality for their own sake (note that if any subject knows his rivals guess he can use best reply rule). 8 Fundación Centro de Estudios Andaluces whole group is informed about the mean, 2/3 ∗ mean and the winner guess number (not the winning subject). Recall that, in order to analyze these results with the students during the post-experimental session, the monitors should record all this information in the decision sheet. 8. Round 2: Experimental subjects are then informed that they will play another round. The procedure is identical to round 1. The subjects are informed that this second round is independent of round one. 9. Round 3 and 4: Identical procedures. Post Experimental Discussion Immediately after the experimental session, we begin the post-experimental discussion. This second part introduces the theoretical background which involves three topics: a. First of all, we present students graphically the iterative elimination of dominated strategies underlying the BCG by means of graphics (1). By explaining this process, students can understand that if rationality is common knowledge, nobody will choose a number within the interval [66.6 − 100] because this subset of numbers is dominated by the [0 − 66.6] set. Following this reasoning ad inﬁnitum —as it was explained previously— students fully capture this idea. b. Once common knowledge rationality is clearly deﬁned, it is time to explain best response. Students observe that their own performance depends on their beliefs about other players’ actions. Furthermore, they realize that the other players’ behavior also relies on beliefs about their own actions and so on. 9 E2003/47 As an example, we present some preliminary results of their own responses. We can use some monitoring sheets (see appendix) to illustrate the average behavior of these groups. Since the average chosen number is decreasing in the course of the successive rounds, students realize that GT approximately predicts their performance. c. Now we have all the required ingredients to cook the Nash Equilibrium concept. Since students have experienced the recursive “dynamics” of the NE, we explain them what the abstract concept implies and how this can be reconciled with the real world. In the theoretical model, we suppose that everything is instantaneously adjusted; the BCG must be explained as a slow motion picture: the repetition lets individuals reach the theoretical solution step by step. Perfectly rational agents do not need this, but nobody is fully rational. This is the second remarkable point: repetition could be a good substitute for rationality. Finally, as an anecdote, we told about the Nash movie, A Beautiful Mind. The ﬁlm tries to explain the NE to the regular audience with an allegorical sequence. We see John with friends in the pub when some girls appear; if all the fellows tried to get oﬀ with the most beautiful girl, all the females would ﬂy away and none of the males would get her. The BCG can be seen as a similar symbolic game: individuals have to compete to outguess the rest of the participants, but without cooperation, costly errors take place. The knowledge of game theory concepts will reduce these costs10 . 10 It is important to remind students about the importance of the intellectual achieve- ment they have reached: John Nash won the Nobel Prize thanks to this simple discovery. Myerson (1999) gives an interesting historical perspective of this portentous innovation. 10 Fundación Centro de Estudios Andaluces Results The experimental BCG was performed in three diﬀerent sessions: two samples of undergraduate ﬁrst-course students in Business (B-1 and B-2) and a sample of undergraduate ﬁrst-course students in a mixed Business+Law degree (B+L). Table 1 shows the distribution of subjects within groups participating in the experimental sessions. Number of Subjects per group & session Session B-1 Session B-2 Session B+L Group 1 10 6 10 Group 2 11 6 8 Group 3 9 6 9 Group 4 9 5 8 Group 5 10 6 10 Group 6 10 6 — Session Pop. 59 35 45 Table 1: Distribution of groups size Our whole sample consists of 139 subjects. Each session consists of 6 (5 in B+L) groups of students. All groups performed four rounds of the BCG. As our main objective is the analysis of a classroom procedure and not the analysis of experimental results, we will only consider the data at an aggregate level. Table 2 shows the average guessed number (and standard deviation) in terms of sessions and rounds. As one might expect the average of chosen numbers —in each session— is decreasing throughout the successive rounds11 . This is true for the three cases. Although we should observe that the path to the convergence (to zero) 11 Although the initial average observed in our experiment is similar to that of Kocher and Sutter (2001), the ﬁnal one is very diﬀerent! Their averages are approximately around 7 while ours are clearly higher (from 9.7 to 19.1). 11 E2003/47 Session B-1 Session B-2 Session B+L Round Average St. Dev. Average St. Dev. Average St. Dev. 1 38.885 19.348 20.782 17.611 37.067 18.891 2 32.087 14.235 17.060 12.362 29.008 11.554 3 22.660 16.750 13.042 7.318 22.712 11.577 4 14.419 13.360 9.748 7.503 19.142 13.944 Table 2: Average number and standard deviation per round is not identical. Figure 2 illustrates each experimental session trend (data from table 2). At ﬁrst sight, sessions B-1 and B+L behave similarly but B-2 trend shows a ﬂatter slope. As our main interest is to study the learning eﬀect we should analyze the average variation round after round. We deﬁne speed of convergence as: (µi − µi−1 )/µi−1 being µi the average guessed number in round i (i = 1, . . . , 4). This speed is shown in absolute value in table 3. B-1 B-2 B+L (µ2 − µ1 )/µ1 0, 17 0, 18 0, 22 (µ3 − µ2 )/µ2 0, 29 0, 24 0, 22 (µ4 − µ3 )/µ3 0, 36 0, 25 0, 16 Table 3: Speed per Rounds and Sessions Interestingly, all sessions share a similar initial speed of convergence close to 20%. From round 2 to 3 (and from 3 to 4) this speed varies dramatically in each group. So, we observe learning eﬀects in all rounds and sessions. After this descriptive approximation, we analyze statistically the results obtained. Remember that our main objective is to contrast if playing a BCG repeatedly a certain number of times induces students to modify their behavior towards the Nash equilibrium. Therefore, we expect the average number in a round to be diﬀerent to the following one and, moreover, the 12 Fundación Centro de Estudios Andaluces 50 40 30 20 10 0 round 1 round 2 round 3 B-1 B-2 round 4 B+L Figure 2: Evolution of the session averages diﬀerence between one round and the following one to be greater than zero, that is, to be decreasing. In order to contrast that, we formulate as null hypothesis: H0 : µij − µkj = 0, i < k and i, k = 1, ..., 4 and as alternative hypothesis: H1 : µij − µkj > 0, i < k and i, k = 1, ..., 4 In the appendix we develop the statistical analysis in depth. Summarizing, we check if each population follows the Normal distribution; given that some of them do not follow it, we use two alternative statistical tests (parametric and non parametric ones) to test our null hypothesis. Our main results show that: 13 E2003/47 • In session B-1 we do observe learning eﬀect in the whole experiment. • In session B-2 we do observe learning eﬀect in the last two rounds. • In session B+L we also observe learning during all the whole rounds. Consequently, these results indicate that this tool is powerful to induce students to reach the Nash Equilibrium. Final Remarks In this paper we have formulated an interesting classroom experiment: a repeated version of the Beauty Contest Game. Our experience is that this kind of games are a good way to introduce the intricateness of the equilibrium concept. Not only does it permits students to see the mechanism in action but also to appreciate the diﬀerence between a theoretical abstraction and the real world dynamics towards equilibrium. The BCG is one of those tricky games worth playing: although it is plain simple in its formulation and the solution is always obvious ex-post, by amusing oneself in the game one realizes the diﬃculties of outsmarting other people. The classroom experiment needs some careful preparation to make conveniently mixed groups. However, once everything is ready to start, if the monitors are well trained, rounds should run quite smoothly and increasingly fast. Although for our purposes four rounds are enough, it would be easy to reach ten rounds if necessary. Continuous repetitions could be very rewarding for students, as this will let most of them to reach for themselves the theoretical solution before the proper explanation is given. Once the BCG is done and the post-experimental session is concluded we can run other experiments such as the Traveler’s dilemma (that we ac- 14 Fundación Centro de Estudios Andaluces tually did). Students realize how game theory training helps them to solve interesting (and sometimes lucrative) puzzles. Appendix Experimental Instruments All these instructions were originally written in Spanish. Instructions You are going to participate in a microeconomic experiment. You have been randomly assigned to a (6-10) person group. In this game you will have to make decisions repeatedly in four rounds. Your partners will be the same during the four stages. Moreover, your group has been assigned to a monitor that will control the procedure. The rules of the game in each period are the following: you should choose a (integer or decimal) number in the interval [0, 100]. Zero and one hundred are allowed choices. Once the monitor has collected the choices of your group, the winner will be determined. The winning number is the closest to 2/3 of the average of all group numbers: 2 x= · 3 n i=1 n xi . The winning prize will be 0.25 extra points in each round for your ﬁnal Micro II exam. If two or more people are equally close to x, the prize will be split equally among them. A person so lucky as to guess the right number the four times will get a whole point! First you must write down the group and code you were given at the top line of the attached strip. Then at each round, you must select your number. When all of you have ﬁnished, the monitor will pick up the responses. Afterwards, you will receive back your strip with the additional information: the 15 E2003/47 average of your group, 2/3 of this and the winning number. This procedure will be repeated four times. If you have any question, please raise your hand and the instructor will come to you. You are not allowed to speak during the experiment. Monitoring Sheet Monitor: Group: Full Name Session: Round 1 Subject 1 Subject 2 Subject 3 Subject 4 Subject 5 Subject 6 Subject 7 Subject 8 Subject 9 Subject 10 Average 2/3 of average 16 Round 2 Round 3 Round 4 Fundación Centro de Estudios Andaluces Individual Sheet Code: Group: First Round Group Average: Your Choice: 2/3 of the average Winning Number: Second Round Group Average: Your Choice: 2/3 of the average Winning Number: Third Round Group Average: Your Choice: 2/3 of the average Winning Number: Fourth Round Group Average: Your Choice: 2/3 of the average Winning Number: 17 E2003/47 Statistical Analysis Let Xij be the number chosen by a student at round i in session j, where 1 = 1, ..., 4 and j = 1, 2, 3. Let Fij (X) be the distribution function associated to each variable Xij . First, we contrast if each Fij (X) follows a normal distribution with average µij and standard deviation σij . For that, we use the Kolmogorov-Smirnof goodness-of-ﬁt test. The following table shows the results: B-1 B-2 B+L Round Z-statistic p-value Z-statistic p-value Z-statistic p-value 1 1.116 0.165 0.937 0.344 1.135 0.152 2 1.063 0.208 0.713 0.690 0.681 0.742 3 1.579 0.014(*) 0.384 0.998 0.792 0.556 4 1.285 0.074 0.709 0.696 0.867 0.440 (*) signiﬁcant at 5% Table 4: Kolmogorov-Smirnof goodness-of-ﬁt results As we can see in table 4, individual choices adjust to a normal distribution except for the ﬁrst session (B-1) at the third round of the game. Using a parametric test for the equality of means between paired rounds, the results are the following: B-1 t-stat. B-2 p-value t-stat. B+L p-value t-stat. p-value Round 1 vs. 2 2.315 0.012 1.252 0.109 2.951 0.002 Round 2 vs. 3 - - 2.107 0.021 3.148 0.001 Round 3 vs. 4 - - 2.859 0.003 1.758 0.042 Round 2 vs. 4 7.586 0 - - - - Table 5: Parametric test for the mean equality 18 Fundación Centro de Estudios Andaluces Observe that for the ﬁrst session (B-1) at the third round it is not possible to apply this test because it does not follow a normal distribution. Table 5 shows that the diﬀerence between successive average choices is statistically positive in all cells except for the ﬁrst two rounds in the second session (B-2). In this last case, the null hypothesis of mean equality is accepted. So as to involve the third round of the ﬁrst session in the statistical analysis, we apply the Wilcoxon non-parametric test using the same former hypothesis. Results are summarized in the following table: B-1 Z-stat. B-2 p-value Z-stat. B+L p-value Z-stat. p-value Round 1 vs. 2 -2.378 0.008 -1.077 0.1414 -2.467 0.006 Round 2 vs. 3 -4.612 0 -2.001 0.022 -2.871 0.002 Round 3 vs. 4 -5.225 0 -3.459 0 -3.45 0 Table 6: Wilcoxon non-parametric test for the mean equality References A. Bosch and R. M. Nagel. El juego de adivinar el número x: Una explicación y la proclamación del vencedor. Expansión, pages 42–43, 16th June 1997a. A. Bosch and R. M. Nagel. Guess the number: Comparing the FT’s and Expansión results. Financial Times, page 14, 30th June 1997b. A. Bosch-Domènech, R. M. Nagel, A. Satorra, and J. Garcı́a-Montalvo. One, two, (three), inﬁnity: Newspaper and lab beauty-contest experiments. Working paper, UPF, Barcelona, 2000. J. Duﬀy and R. M. Nagel. On the robustness of behavior in experimental beauty contest games. Economic Journal, 107:1684–1700, 1997. 19 E2003/47 Brit Grosskopf and Rosemarie Nagel. Rational reasoning or adaptative behavior? evidence from two-persons beauty contest games. 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Games and Economic Behavior, 44:134–144, 2003. 21 centrA: Fundación Centro de Estudios Andaluces Documentos de Trabajo Serie Economía E2001/01 “The nineties in Spain: so much flexibility in the labor market?’’, J. Ignacio García Pérez y Fernando Muñoz Bullón. E2001/02 “A Log-linear Homotopy Approach to Initialize the Parameterized Expectations Algorithm’’, Javier J. Pérez. E2001/03 “Computing Robust Stylized Facts on Comovement’’, Francisco J. André, Javier J. Pérez, y Ricardo Martín. E2001/04 “Linking public investment to private investment. The case of the Spanish regions”, Diego Martínez López. E2001/05 “Price Wars and Collusion in the Spanish Electricity Market”, Juan Toro y Natalia Fabra. E2001/06 “Expedient and Monotone Learning Rules”, Tilman Börgers, Antonio J. Morales y Rajiv Sarin. E2001/07 “A Generalized Production Set. The Production and Recycling Function”, Francisco J. André y Emilio Cerdá. E2002/01 “Flujos Migratorios entre provincias andaluzas y entre éstas y el resto de España’’, J. Ignacio García Pérez y Consuelo Gámez Amián. E2002/02 “Flujos de trabajadores en el mercado de trabajo andaluz’’, J. Ignacio García Pérez y Consuelo Gámez Amián. E2002/03 “Absolute Expediency and Imitative Behaviour”, Antonio J. Morales Siles. E2002/04 “Implementing the 35 Hour Workweek by means of Overtime Taxation”, Victoria Osuna Padilla y José-Víctor Ríos-Rull. E2002/05 “Landfilling, Set-Up costs and Optimal Capacity”, Francisco J. André y Emilio Cerdá. E2002/06 “Identifying endogenous fiscal policy rules for macroeconomic models”, Javier J. Pérez y Paul Hiebert. E2002/07 “Análisis dinámico de la relación entre ciclo económico y ciclo del desempleo en Andalucía en comparación con el resto de España”, Javier J. Pérez, Jesús Rodríguez López y Carlos Usabiaga. 22 E2002/08 “Provisión eficiente de inversión pública financiada con impuestos distorsionantes”, José Manuel González-Páramo y Diego Martínez López. E2002/09 “Complete or Partial Inflation convergence in the EU?”, Consuelo Gámez y Amalia Morales-Zumaquero. E2002/10 “On the Choice of an Exchange Regime: Target Zones Revisited”, Jesús Rodríguez López y Hugo Rodríguez Mendizábal. E2002/11 “Should Fiscal Policy Be Different in a Non-Competitive Framework?”, Arantza Gorostiaga. E2002/12 “Debt Reduction and Automatic Stabilisation”, Paul Hiebert, Javier J. Pérez y Massimo Rostagno. E2002/13 “An Applied General Equilibrium Model to Assess the Impact of National Tax Changes on a Regional Economy”, M. Alejandro Cardenete y Ferran Sancho. E2002/14 “Optimal Endowments of Public Investment: An Empirical Analysis for the Spanish Regions”, Óscar Bajo Rubio, Carmen Díaz Roldán y M. Dolores Montávez Garcés. E2002/15 “Is it Worth Refining Linear Approximations to Non-Linear Rational Expectations Models?” , Alfonso Novales y Javier J. Pérez. E2002/16 “Factors affecting quits and layoffs in Spain”, Antonio Caparrós Ruiz y M.ª Lucía Navarro Gómez. E2002/17 “El problema de desempleo en la economía andaluza (19902001): análisis de la transición desde la educación al mercado laboral”, Emilio Congregado y J. Ignacio García Pérez. E2002/18 “Pautas cíclicas de la economía andaluza en el período 19842001: un análisis comparado”, Teresa Leal, Javier J. Pérez y Jesús Rodríguez. E2002/19 “The European Business Cycle”, Mike Artis, Hans-Martin Krolzig y Juan Toro. E2002/20 “Classical and Modern Business Cycle Measurement: The European Case”, Hans-Martin Krolzig y Juan Toro. E2002/21 “On the Desirability of Supply-Side Intervention in a Monetary Union”, Mª Carmen Díaz Roldán. E2003/01 “Modelo Input-Output de agua. Análisis de las relaciones intersectoriales de agua en Andalucía”, Esther Velázquez Alonso. E2003/02 “Robust Stylized Facts on Comovement for the Spanish Economy”, Francisco J. André y Javier Pérez. 23 E2003/03 “Income Distribution in a Regional Economy: A SAM Model”, Maria Llop y Antonio Manresa. E2003/04 “Quantitative Restrictions on Clothing Imports: Impact and Determinants of the Common Trade Policy Towards Developing Countries”, Juliette Milgram. E2003/05 “Convergencia entre Andalucía y España: una aproximación a sus causas (1965-1995). ¿Afecta la inversión pública al crecimiento?”, Javier Rodero Cosano, Diego Martínez López y Rafaela Pérez Sánchez. E2003/06 “Human Capital Externalities: A Sectoral-Regional Application for Spain”, Lorenzo Serrano. E2003/07 “Dominant Strategies Implementation of the Critical Path Allocation in the Project Planning Problem”, Juan Perote Peña. E2003/08 “The Impossibility of Strategy-Proof Clustering”, Javier Perote Peña y Juan Perote Peña. E2003/09 “Plurality Rule Works in Three-Candidate Elections”, Bernardo Moreno y M. Socorro Puy. E2003/10 “A Social Choice Trade-off Between Alternative Fairness Concepts: Solidarity versus Flexibility”, Juan Perote Peña. E2003/11 “Computational Errors in Guessing Games”, Pablo Brañas Garza y Antonio Morales. E2003/12 “Dominant Strategies Implementation when Compensations are Allowed: a Characterization”, Juan Perote Peña. E2003/13 “Filter-Design and Model-Based Analysis of Economic Cycles”, Diego J. Pedregal. E2003/14 “Strategy-Proof Estimators for Simple Regression”, Javier Perote Peña y Juan Perote Peña. E2003/15 “La Teoría de Grafos aplicada al estudio del consumo sectorial de agua en Andalucía", Esther Velázquez Alonso. E2003/16 “Solidarity in Terms of Reciprocity", Juan Perote Peña. E2003/17 “The Effects of Common Advice on One-shot Traveler’s Dilemma Games: Explaining Behavior through an Introspective Model with Errors", C. Monica Capra, Susana Cabrera y Rosario Gómez. E2003/18 “Multi-Criteria Analysis of Factors Use Level: The Case of Water for Irrigation", José A. Gómez-Limón, Laura Riesgo y Manuel Arriaza. E2003/19 “Gender Differences in Prisoners’ Dilemma", Pablo Brañas-Garza y Antonio J. Morales-Siles. E2003/20 “Un análisis estructural de la economía andaluza a través de matrices de contabilidad social: 1990-1999", M. Carmen Lima, M. Alejandro Cardenete y José Vallés. 24 E2003/21 “Análisis de multiplicadores lineales en una economía regional abierta", Maria Llop y Antonio Manresa. E2003/22 “Testing the Fisher Effect in the Presence of Structural Change: A Case Study of the UK", Óscar Bajo-Rubio, Carmen Díaz-Roldán y Vicente Esteve. E2003/23 "On Tests for Double Differencing: Some Extensions and the Role of Initial Values", Paulo M. M. Rodrigues y A. M. Robert Taylor. E2003/24 "How Tight Should Central Bank’s Hands be Tied? Credibility, Volatility and the Optimal Band Width of a Target Zone", Jesús Rodríguez López y Hugo Rodríguez Mendizábal. E2003/25 "Ethical implementation and the Creation of Moral Values", Juan Perote Peña. E2003/26 "The Scoring Rules in an Endogenous Election", Bernardo Moreno y M. Socorro Puy. E2003/27 "Nash Implementation and Uncertain Renegotiation", Pablo Amorós. E2003/28 "Does Familiar Environment Affect Individual Risk Attitudes? Olive-oil Producer vs. no-producer Households", Francisca Jiménez Jiménez. E2003/29 "Searching for Threshold Effects in the Evolution of Budget Deficits: An Application to the Spanish Case", Óscar Bajo-Rubio, Carmen Díaz-Roldán y Vicente Esteve. E2003/30 "The Construction of input-output Coefficients Matrices in an Axiomatic Context: Some Further Considerations", Thijs ten Raa y José Manuel Rueda Cantuche. E2003/31 "Tax Reforms in an Endogenous Growth Model with Pollution", Esther Fernández, Rafaela Pérez y Jesús Ruiz. E2003/32 "Is the Budget Deficit Sustainable when Fiscal Policy is nonlinear? The Case of Spain, 1961-2001", Óscar Bajo-Rubio, Carmen Díaz-Roldán y Vicente Esteve. E2003/33 "On the Credibility of a Target Zone: Evidence from the EMS", Francisco Ledesma-Rodríguez, Manuel Navarro-Ibáñez, Jorge Pérez-Rodríguez y Simón Sosvilla-Rivero. E2003/34 "Efectos a largo plazo sobre la economía andaluza de las ayudas procedentes de los fondos estructurales: el Marco de Apoyo Comunitario 1994-1999", Encarnación Murillo García y Simón Sosvilla-Rivero. E2003/35 “Researching with Whom? Stability and Manipulation”, José Alcalde y Pablo Revilla. E2003/36 “Cómo deciden los matrimonios el número óptimo de hijos”, Francisca Jiménez Jiménez. 25 E2003/37 “Applications of Distributed Optimal Control in Economics. The Case of Forest Management”, Renan Goetz y Angels Xabadia. E2003/38 “An Extra Time Duration Model with Application to Unemployment Duration under Benefits in Spain”, José María Arranz y Juan Muro Romero. E2003/39 “Regulation and Evolution of Harvesting Rules and Compliance in Common Pool Resources”, Anastasios Xepapadeas. E2003/40 “On the Coincidence of the Feedback Nash and Stackelberg Equilibria in Economic Applications of Differential Games”, Santiago J. Rubio. E3003/41 “Collusion with Capacity Constraints over the Business Cycle”, Natalia Fabra. E3003/42 “Profitable Unproductive Innovations”, María J. Álvarez-Peláez, Christian Groth. E3003/43 “Sustainability and Substitution of Exhaustible Natural Resources. How Resource Prices Affect Long-Term R&DInvestments”, Lucas Bretschger, Sjak Smulders. E3003/44 “Análisis de la estructura de la inflación de las regiones españolas: La metodología de Ball y Mankiw”, María Ángeles Caraballo, Carlos Usabiaga. E3003/45 “An Empirical Analysis of the Demand for Physician Services Across the European Union”, Sergi Jiménez-Martín, José M. Labeaga, Maite Martínez-Granado. E3003/46 “An Exploration into the Effects of Fiscal Variables on Regional Growth”, Diego Martínez López. E3003/47 “Teaching Nash Equilibrium and Strategy Dominance: A Classroom Experiment on the Beauty Contest. Virtudes Alba Fernández, Francisca Jiménez Jiménez, Pablo Brañas Garza, Javier Rodero Cosano 26 centrA: Fundación Centro de Estudios Andaluces Normas de publicación de Documentos de Trabajo centrA Economía La Fundación Centro de Estudios Andaluces (centrA) tiene como uno de sus objetivos prioritarios proporcionar un marco idóneo para la discusión y difusión de resultados científicos en el ámbito de la Economía. Con esta intención pone a disposición de los investigadores interesados una colección de Documentos de Trabajo que facilita la transmisión de conocimientos. La Fundación Centro de Estudios Andaluces invita a la comunidad científica al envío de trabajos que, basados en los principios del análisis económico y/o utilizando técnicas cuantitativas rigurosas, ofrezcan resultados de investigaciones en curso. Las normas de presentación y selección de originales son las siguientes: 1. El autor(es) interesado(s) en publicar un Documento de Trabajo en la serie de Economía de centrA debe enviar su artículo en formato PDF a la dirección de email: wpecono@fundacion-centra.org 2. Todos los trabajos que se envíen a la colección han de ser originales y no estar publicados en ningún medio de difusión. Los trabajos remitidos podrán estar redactados en castellano o en inglés. 3. Los originales recibidos serán sometidos a un breve proceso de evaluación en el que serán directamente aceptados para su publicación, aceptados sujetos a revisión o rechazados. Se valorará, asimismo, la presentación de¡ trabajo en seminarios de centrA. 4. En la primera página deberá aparecer el título del trabajo, nombre y filiación del autor(es), dirección postal y electrónica de referencia y agradecimientos. En esta misma página se incluirá también un resumen en castellano e inglés de no más de 100 palabras, los códigos JEL y las palabras clave de trabajo. 5. Las notas al texto deberán numerarse correlativamente al pie de página. Las ecuaciones se numerarán, cuando el autor lo considere necesario, con números arábigos entre corchetes a la derecha de las mismas. 6. La Fundación Centro de Estudios Andaluces facilitará la difusión electrónica de los documentos de trabajo. Del mismo modo, se incentivará económicamente su posterior publicación en revistas científicas de reconocido prestigio. 28

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