GO figure: Analytic and strategic skills are separable

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GO figure: Analytic and strategic skills are separableR Sascha Baghestanian a,∗, Seth Frey b,c a

Department of Economics, Goethe University, House of Finance, Grüneburgplatz 1, Frankfurt, Germany Cognitive Science Program, Indiana University, Indiana c Stampfenbachstrasse 48, 8006 Zürich, Switzerland b

a r t i c l e

i n f o

Article history: Received 30 November 2013 Revised 11 June 2015 Accepted 12 June 2015 Available online xxx JEL classification: C81 C83 C90 D01 D03 D87

a b s t r a c t We measure the game behavior and analytic reasoning skills of expert strategic reasoners: professional GO players. We argue for a distinction between what we call “strategic” and “analytic” reasoning skills and present separate measures to elicit strategic and analytic abilities. The paper investigates the behavior of our subject pool in many different types of one-shot games, including the Traveler’s Dilemma, Centipede, Kreps, and Matching Pennies games. We observe that increased strategic skill predicts a greater probability of Nash behavior, while greater analytic skill predicts more cooperative play, even when such behavior is inconsistent with individual rationality. © 2015 Elsevier Inc. All rights reserved.

Keywords: Experimental economics Cognitive ability Economic psychology Social preferences Centipede Traveler’s Dilemma Matching Pennies Kreps

1. Introduction In strategic decision-making environments, the reasoning processes that rationalize Nash equilibrium play are cognitively demanding. A growing body of literature in behavioral economics is investigating the link between strategic behavior and cognitive abilities via controlled experiments. This literature provides evidence for a positive correlation between cognitive abilities and Nash equilibrium play in certain economic games (Agranov et al., 2013; Burnham et al., 2009; Gill and Prowse, 2013; Rydval et al., 2009). Still, cognitive ability is not a monolith. For instance, in p-beauty contests subjects with higher scores on various cognitive ability tests

R We would like to thank Valerio Capraro, Martin Dufwenberg, Pablo Brañas-Garza, Ignacios Palacios-Huerta, participants of the cognitive lunch at Indiana University Bloomington, participants of the level-k workshop at the WZB Berlin and two anonymous referees and the associate editor for very helpful suggestions. ∗ Corresponding author. Tel.: +49 69 798 34505. E-mail addresses: [email protected], [email protected] (S. Baghestanian), [email protected] (S. Frey).

are more likely to adopt Nash equilibrium strategies (Brañas-Garza et al., 2012; Burnham et al., 2009; Gill and Prowse, 2013; Rydval et al., 2009). In experimental Prisoner’s Dilemmas, on the other hand, subjects with higher cognitive ability scores tend to behave more cooperatively (Burks et al., 2009; Jones, 2008). In experimental dictator games Chen et al. (2013) report that subjects with higher grade point averages are more selfish, whereas subjects with higher scores in the math portion of the SAT are more generous. These results suggest that cooperative and competitive behavior may correlate with different elements of a subject’s cognitive skillset. Consistent with these findings, Decety et al. (2004) show that cooperative and competitive behavior engage distinguishable neural processes. In this study, we further explore the relationship between reasoning skills and competitive as well as cooperative behavior. We posit two types of reasoning—“strategic” and “analytic”—and investigate their potentially heterogeneous influences on choices in games which evoke a conflict between selfish and cooperative behavior. To explore these two types of reasoning, we measure the analytic reasoning skills of professional players of the East Asian board game GO and test their behavior in various one-shot games.

http://dx.doi.org/10.1016/j.socec.2015.06.004 2214-8043/© 2015 Elsevier Inc. All rights reserved.

Please cite this article as: S. Baghestanian, S. Frey, GO figure: Analytic and strategic skills are separable, Journal of Behavioral and Experimental Economics (2015), http://dx.doi.org/10.1016/j.socec.2015.06.004

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We focus on GO players for two reasons. First, international GO players are rated under a variation of chess’ Elo system, which allows us to quantify their strategic abilities. Elo is a standardized ordinal scheme defined in terms of relative skill, in which higher ranks go to better players. Systems like the Elo ratings have provided experimentalists with a convenient way to quantify (ex-ante) strategic abilities (Levitt et al., 2011; Palacios-Huerta and Volij, 2009). We follow this previous work by interpreting Elo as a measure of strategic skills. Second, even if strategic and analytic reasoning skills are separable, we would still expect GO players to be above average at both. Indeed, GO professionals must excel at the strategic ability to model an opponent. Other abilities, like suppressing heuristic reasoning, applying deduction and induction, accessing memory, and focusing attention all involve in-game reasoning that doesn’t necessarily incorporate the goals or abilities of an opponent. So if analytic and strategic skills are separable within GO professionals, any non-Nash behavior they exhibit is unlikely to be due to errors or reasoning failures. To elicit the analytic skills of our subjects, we use an incentivized version of the cognitive reflection test (CRT) (Frederick, 2005). We use these two measures to show that the exceptional analytic and strategic skills of GO players (CRT scores and Elo ratings) predict different behaviors. We suggest that these two measures reveal at least two kinds of reasoning. We examine the behavior of our subject pool in Traveler’s Dilemmas and Centipede games. We also elicit our subjects’ choices in Kreps and Matching Pennies games. The first two games allow us to explore the relationship between choices and different cognitive skills in games which evoke the Prisoner’s Dilemma’s conflict between cooperative and individually rational behavior. We propose that strategic skill predicts Nash behavior and explore whether analytic skill correlates with cooperative behavior in these decisionmaking environments. The last two games, in particular the Kreps game, serve as robustness checks. If our subjects tend to play Nash equilibrium strategies in these games, in which standard subject pools regularly fail to behave rationally (Goeree and Holt, 2001), then limited reasoning abilities would be less likely to explain observed deviations from Nash in the first two. Our main results can be summarized as follows. In games which evoke a conflict between cooperative and individually rational choices (Traveler’s Dilemma and Centipede game), strategic skill correlates positively with a greater likelihood of Nash behavior. Greater analytic skill, on the other hand, correlates positively with more efficient and cooperative play. Our results from the Kreps and Matching Pennies games suggest that deviations from Nash equilibrium play in Traveler’s Dilemma and Centipede game are not generated by limited reasoning skills. Where Nash behavior is efficient, as in the Kreps game, our subjects outperform standard subject pools in coordinating on efficient equilibrium outcomes. The remainder of the paper is organized as follows. Section 2 briefly introduces the game of GO. Section 3 describes the one-shot games we used in our experimental design. Section 4 describes the design and Section 5 presents our results. Section 6 provides a discussion and Section 7 concludes. 2. The game of GO GO is an East Asian board game noted for its strategic complexity. Two players place black and white “stones” on intersections of a 19 by 19 grid-board. The goal of the game is to encircle a larger total area of the board than the opponent. When a game ends, captured stones are subtracted from the number of controlled intersections to determine who has more points (Bozulich, 2001). Although GO, in contrast to chess, has only two kinds of game pieces, it is more complex than chess from a strategic point of view. More specifically, the number of legal positions in chess is estimated

to be between 1043 –1047 , with a game-tree complexity of approximately 10123 . By comparison, the upper bound on the number of legal board positions in GO is approximately 10170 , with a game48 171 tree complexity between 1010 –1010 . While in chess the average number of legal moves per turn is 37 (Keene and Levy, 1991), the average number of moves per turn throughout most of a GO game exceeds 150. If game skills, like backward-induction and iterated elimination of dominated strategies, should emerge from expertise in searching game trees (Palacios-Huerta and Volij, 2009), then GO will provide at least as compelling a training ground as chess. GO players can be ranked based on their Elo ratings (see Table A.7 for details). The larger the Elo-score of a player, the higher his ranking. Since the ability to model other strategic reasoners is central to GO proficiency, we treat a player’s Elo rating as his “strategic skill” in the remainder of the paper. 3. Reflection test, games, and model predictions In our experimental design we administered one logic test and four different game classes that vary in terms of the properties of their underlying Nash equilibria.1 3.1. Cognitive reflection test (CRT) The Frederick CRT is a simple, powerful, and well-validated measure consisting of three questions (Frederick, 2005). Each question has an incorrect answer that immediately suggests itself and a less obvious correct answer. It was originally designed to measure the dominance of a subject’s logical, analytic reasoning processes relative to automatic, heuristic processes. The questions are as follows •

•

•

A bat and a ball cost $1.10 in total. The bat costs $1 more than the ball. How much does the ball cost? If it takes five machines five minutes to make five widgets, how long would it take 100 machines to make 100 widgets? In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?2

Frederick (2005) ran a series of experiments and tested the performance of 3428 students on these questions. He showed that the number of correct answers, 0–3, is strongly correlated with the results from other measures of analytic reasoning, like the Wonderlic Personnel Test (WPT), the “need for cognition scale” (NFC; Cacioppo et al., 1984; 1982), the SAT, and the American College Test (ACT).3 We introduce the CRT as a measure for “analytic skills”. 3.2. Centipede and Traveler’s Dilemma The first class of games we consider is a normal-form representation of a Centipede game, following Nagel and Tang (1998) (see Table 1). In this version of the game, agent A selects an odd number between 1 and 13 and agent B selects an even number between 2 and 14. The unique Nash equilibrium outcome of this game is (1, 2) for the choices of players A and B, respectively. The Nash equilibrium is Pareto dominated by various other outcomes, in particular (13, 14), which corresponds to the cooperative and efficient (sum-of-utilitymaximizing) outcome. Standard subject pools playing the Centipede 1 For the games we used the designs and materials of Goeree and Holt (2001) and Nagel and Tang (1998) with random matching and no feedback. 2 The correct answers are 5, 5, and 47. Very frequently subjects jump into the conclusion that the correct answer to question one (for example) is 10 cents. 3 Oechssler et al. (2009) show that individuals with low CRT scores are more likely to be subject to the conjunction fallacy (Kahneman and Tversky, 1983) and to conservatism with respect to probability updating. Bergman et al. (2010) demonstrate that anchoring effects Kahneman and Tversky (1974) decrease with higher CRT scores.

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S. Baghestanian, S. Frey / Journal of Behavioral and Experimental Economics 000 (2015) 1–10 Table 1 Tabular form Centipede game, Nagel and Tang (1998). Smaller choice

Possible higher choices of opponent

Payoff of A

Payoff of B

Sum

1 2 3 4 5 6 7 8 9 10 11 12 13

{2, 4, 6, 8, 10, 12, 14} {3, 5, 7, 9, 11, 13} {4, 6, 8, 10, 12, 14} {5, 7, 9, 11, 13} {6, 8, 10, 12, 14} {7, 9, 11, 13} {8, 10, 12, 14} {9, 11, 13} {10, 12, 14} {11, 13} {12, 14} {13} {14}

40 20 80 30 160 60 320 110 640 220 1280 440 2560

10 50 20 110 40 220 80 450 160 900 320 1800 640

50 70 100 140 200 280 400 560 800 1120 1600 2240 3200

game neither follow the Nash prediction nor behave cooperatively (Nagel and Tang, 1998). The second class of games we consider are Traveler’s Dilemmas (Basu, 1994). In Traveler’s Dilemmas, two players simultaneously pick an integer between an upper and a lower bound. The player with the lower of the two numbers receives his bid plus a reward R > 1. The agent with the higher number loses, receiving the winning bid minus R. If both agents happen to select the same number their payoff is identical and simply equal to their bids. This game has a unique Nash equilibrium, the lower bound, for every R > 1. Under common knowledge of rationality players should “coordinate” to the Nash equilibrium, via iterated removal of nonrationalizable strategies. As in the Centipede game, the unique Nash equilibrium for Traveler’s Dilemmas is Pareto dominated by other outcomes, in particular by the cooperative and most efficient outcome in which both players choose the corresponding upper bound. Experimental evidence suggests that subjects are sensitive to the value of R. While low values of R result in choices consistent with cooperative behavior, large values of R generate choices consistent with Nash equilibrium play.4 Our participants played Traveler’s Dilemmas with two different values of reward-parameter R but the same upper- and lower- bounds (1000 and 500, respectively). We use R = 100 (“low-R treatment”) and R = 300 (“high-R treatment”) for the reward parameters. Agents should not be sensitive to changes in R: for both parameterizations the Nash prediction is the minimum bid, 500, and the prediction for agents who maximize the sum of utilities and behave cooperatively is the maximum bid of 1000. 3.3. Kreps and Matching Pennies: Robustness The third family of games we consider, the Kreps game, admits pure and mixed Nash equilibria (see Fig. 1 for the version we use). Its pure strategy Nash equilibria are (T, L) and (B, R). While the most efficient game outcome is equilibrium (T, L), the most frequent outcome within standard subject pools is the Pareto-dominated non-Nash outcome (T, NN) (Goeree and Holt, 2001).

4 The Traveler’s Dilemma has been studied extensively in the experimental literature. Goeree and Holt (2001) test the Nash prediction for this game with a lower bound of 180 and an upper bound of 300. They ran two different treatments; one with R = 5 and the other with R = 180. More than 80% of subjects in the low-R treatment picked the upper bound 300 (average: 280). Conversely, more than 80% of all subjects in the high-R treatment picked the lower bound, 180 (average: 201). Capra et al. (1999) present similar results on the Traveler’s Dilemma considering a broad range of reward parameters. Their results indicate a negative relationship between bids and the reward parameter. Brañas-Garza et al. (2011) investigate self reported motivations behind choices in the Traveler’s Dilemma to explain the observed deviations from Nash equilibrium play.

T B

L 1000, 700 600, 100

M 600, 690 620, 400

3

NN 620, 660 660, 660

R 640, 100 700, 680

Fig. 1. Kreps game parametrization.

The last class of games, Matching Pennies, admits a unique Nash equilibrium in mixed strategies. We depict the symmetric and asymmetric versions of the Matching Pennies game we use in our experiments in Fig. 2. The mixed-strategy Nash equilibrium for the game in panel (a) involves randomizing over both alternatives with equal probabilities. In the asymmetric Matching Pennies game (panel (b) of Fig. 2), the Nash equilibrium strategy for the row player remains (0.5, 0.5), while that for the column player changes to (0.16, 0.84). Pooling over play frequencies, standard subject pools tend to play the mixed strategy Nash equilibrium in the symmetric game and to coordinate on the outcome (U, R) in the asymmetric one (Goeree and Holt, 2001). 4. Experimental design We recruited subjects from among the participants at the 28th annual US GO Congress in Black Mountain, NC (August 2012). The congress’s 442 attendees consisted of 333 men and 109 women, mainly representing the US, Canada, China, Japan, Mexico, and Korea. Other attendees came from New Zealand, the Netherlands, and Germany. The conference organizers announced our experiment each morning of the competition, and 46 participants enrolled. We ran two sessions on the second day of the Congress. Subjects were randomly seated on chairs equally spaced throughout the experiment room. We spaced the chairs at 1.5 m from each other and provided folded hardpaper to serve as barrier and writing pad. After collecting consent forms we distributed the general instructions and a questionnaire asking subjects to provide information about their age, handedness,5 GO ranking (Elo), years of experience playing GO, whether they also play chess professionally, and whether they had ever participated in a social science experiment. Subjects who reported chess expertise were asked to report their Elo ranks for both games. Each subject received a separate instruction form and a unique ID number. We chose to control for experimenter effects, ensuring common knowledge of anonymity by assigning ID’s randomly and blindly.6 The general instructions, which were read aloud, explained that everyone would participate in a preliminary test followed by six consecutive decision-making environments, and that their payoff within every environment depended on their own decision and on the decision of a partner in the room. Specifically, we informed subjects that they were randomly and anonymously matched, potentially with a different partner, for every game. Before experimental sessions started, subjects were matched randomly to each other. Matches were assigned anew for each game. In the Kreps game, the asymmetric Matching Pennies, and the Centipede game, subjects were randomly assigned to be either row or column players and type-A or type-B players, respectively. The random assignment was determined before the experiment started and participants were informed about their types on their instruction sheets. Participants were given their own sequence of anonymous

5 We phrased the question on a subjects’ handedness in the following way: ‘would you consider yourself predominantly (a) left-handed or (b) right-handed in daily activities such as writing, carrying a bag etc”. We elicit handedness to control for the correlation between left-handedness and analytic skills, as suggested by various neuroscientific studies (see Annett (2002) for an overview). 6 Participants entered their names on a list to help us prevent non-participants from later collecting participants’ winnings, but they saw that their names could not have been connected to their IDs.

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U D

L 400, 200 200, 400

R 200, 400 400, 200

Symmetric Treatment

U D

L 1200, 200 200, 400

R 200, 400 400, 200

Asymmetric Treatment

Fig. 2. Matching Pennies game parametrization.

Fig. 3. GO, Elo, and CRT. Y-axes are frequencies.

partner IDs to note on their instruction sheets for every game. We chose this matching procedure because it was important to us to control for confounds like pairwise collusion and reputation considerations. At no point during the experiment did subjects receive any feedback on their earnings or on the behavior of their partners. In permitting the full anonymity of our subjects, we were forced to rely on self-reports for all Elo ranks and other questionnaire data. However, because their anonymity was common knowledge, subjects had less incentive to misrepresent themselves. Palacios-Huerta and Volij (2009) and Levitt et al. (2011) both used non-blind procedures that enabled objective Elo scores at the cost of permitting reputation to influence subjects’ game behavior. Section 5.2.2 looks more closely at the relations of these studies to ours. Participants received sheets for all six games in the same packet. To control for learning and order effects, we instructed subjects that they could play the games in any order and revise any of their choices before submitting their game packets. This design choice allowed for gradual learning as well as discontinuous epiphanies (Dufwenberg et al., 2010); subjects could at some point “figure out” or learn how to play a certain game, based on their experience in another game. Participants were also given time to review their print copy of the instructions and to ask questions. Following the evidence provided by Laury (2005), participants were informed that they would be paid for only two of the six games, and that after the experimental session volunteers would determine which two games by pulling two of six numbered paper scraps from an opaque bag. To support the credibility of the procedure, volunteers verified that the other sheets of paper in the bag held the remaining numbers. After reading the instructions and collecting the questionnaires, we distributed a handout with the three questions of Frederick’s CRT. Participants were informed that each correct answer to the quiz would be rewarded with 25 cents. They were also told that if they answered all questions correctly they would receive an additional 25 cents reward, which would be added to their total earnings. In total, payment consisted of a $5 participation fee, the sum of the earnings of the two selected games, and the earnings from the CRT. Subjects earned an average of $20.90. The maximum payoff amounted to

$33.60 and the minimum payoff was $8.10. All participants received their payments in sealed envelopes on the next day of the Congress. We did not use an experimental currency in our design. All payoffs in the games we used were in cents of a dollar. Each session lasted about 60–75 min, including instructions and other procedural overhead. 4.1. Subject-pool description Within the 46 GO players in our sample, the median Elo rating was 2000, with minimum 300 and maximum 2700. For reference, Palacios-Huerta and Volij report that their subject pool had an average Elo of about 2200. Both chess and GO players are typically considered strong club players if their Elo scores are higher than 2000. Fig. 3 shows the distribution of our subjects’ Elo ratings. The average age of participants was 34 years, with an average of 15 years of experience playing GO. 28 subjects were male and 18 were female. Eight subjects reported that they play chess professionally and participate in tournaments regularly, with a median (selfreported) chess Elo rating of 1925. Only one player reported participating in a social science experiment before. At first glance, the size of our subject pool might appear to be small. But the related work of Nagel and Tang (1998), Goeree and Holt (2001), and Capra et al. (1999) used similarly sized subject pools (60, 50, and 61, respectively). The effects we observe are strong enough that many of our results are significant despite the small sample size. Moreover, in Section 5.2.2 we compare our results to those of Palacios-Huerta and Volij (2009), who recruited 422 chess professionals. 5. Results 5.1. Elo and CRT in GO players Fig. 3 depicts the distributions of correct answers on the CRT in our subject pool, in the original subject pool of Frederick (2005) (3428 students), and in a pool consisting of 44 students of Indiana University (IU), that were incentivized with $2 for every correct answer (Baghestanian et al. 2012). Note that the incentivized sample of

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IU students is comparable to the large sample analyzed by Frederick. A standard t-test and a non-parametric Mann–Whitney test indicate that professional GO players give significantly more correct answers on the CRT than the standard IU student subject pool red (p-value < 0.01 for both tests). Our subject pool exhibits exceptional CRT scores. Given the correspondence of the CRT to other measures of reasoning skill, we argue that professional GO players are particularly skilled at abstract analytic reasoning. Next, we investigate the relationships between individual Elo rankings, CRT scores, and other covariates. For that purpose we estimate several nested linear models, using individual Elo ratings as the dependent variable in our regressions. The results are shown in Table A.1. In all our nested model specifications in columns (1)–(5), we observe a positive correlation between Elo and years of GO playing experience (p-value < 0.05), and a negative correlation between Elo and CRT scores (p-value < 0.1 or < 0.05). The negative relationship between Elo and CRT is consistent with a claim that the skills necessary to perform well on the CRT are different from the skills necessary to achieve a high Elo rating.7 For column (6) we construct a new dummy H variable, CRTH L , and replace the CRT variable with it. The dummy CRTL takes the value 0 for subject i if CRTi ∈ {0, 1} and it takes the value 1 for subject i if CRTi ∈ {2, 3}. The magnitude of the negative correlation between Elo and CRT almost triples if we consider this new dummy variable and it is significant at a 1% level. Lastly, we explore the relationship between Elo ranks and correct answers to each individual question on the CRT. We want to rule out the possibility that just a subset of the CRT questions are driving our results. For that purpose, we replace the CRT variable in our final regression specification with a dummy variable for each of the three questions separately. Each dummy variable takes a value of one if the respective question was answered correctly and is zero otherwise. The results are shown in columns (7)–(9). Independently of the question, we observe that the coefficients on all three dummy variables are negative. However, only the coefficients for questions one (bat and ball) and three (lily pad) are significant at a 10% level. Even if we depart from the multivariate regression framework, we obtain similar results. The average Elo rank, conditional on a CRT score of zero or one, is 2095.5 and the average Elo rank, conditional on a CRT score of either two or three, is 1714.3. A t-test suggests that these averages are significantly different at a 10% level (p-value = 0.056). A qualitatively similar result can be established if we consider medians instead. The median Elo score of subjects with a CRT score of zero or one is 2100, whereas the median Elo of participants with a CRT score of two or three is 1900. A Mann-Whitney suggests that this difference is insignificant (p-value = 0.10). Considering simple correlations between Elo and CRT we observe a Spearman rank correlation coefficient of −0.29, which is significant at a 10% level (pvalue = 0.056) and a Kendall-τ of −0.152 which is not significant at a 10% level (p-value = 0.21). All these tests either suggest a negative and significant or a negative and insignificant relationship between Elo and CRT. None of the tests suggest a positive relationship between the two variables. In summary, our results suggest a negative or a non-positive relationship between Elo ranks and CRT scores. We discuss this finding later in our discussion section. 5.2. Centipede and Traveler’s Dilemma: Results 5.2.1. Centipede game Table A.2 summarizes our findings on the Centipede game. We illustrate the fractions of type-A and B players who chose the cor7 Gender, age, and left-handedness have no significant effects on Elo ranks in any specification.

responding nodes. Notice that a type-B player who chooses 1 corresponds to a player who terminates the Centipede game in extensive form at the first node. Table A.2 also compares our results with those of Nagel and Tang (1998), who used a standard student subject pool in their experiments. Our GO players do not follow the Nash prediction. On the contrary, more than 80% of our B-type subjects make choices which are greater than or equal to ten and more than 70% of our A-type subjects make choices which are greater than or equal to nine. Next, we explore the relationship between individual choices in the Centipede game and various other covariates, again using several nested linear model specifications. The results are shown in Table A.3 in the Appendix. The results from our nested model specifications in columns (1)– (6), show that an increase in CRT scores correlates with an increase in the chosen node numbers (p-value < 0.1 or < 0.05), whereas an increase in Elo correlates negatively with the chosen nodes (p-value < 0.1). In other words, an increase in an individuals’ CRT score correlates with increasingly cooperative choices, while an increase in Elo correlates with choices that are closer to the Nash prediction. We also observe that male and left-handed subjects tend to choose higher nodes. However, the latter two correlations become insignificant if we drop the two subjects with the highest and most extreme Elo scores (= 2700). Dropping these two observations also increases the significance of the negative relationship between Elo and choices (column (7); p-value < 0.05).8 In summary, an increase in strategic skills (Elo) correlates positively with an increase in the likelihood that our subjects’ choices are closer to the unique Nash equilibrium strategy.9 An increase in analytic skills (CRT) correlates positively with an increase in the likelihood that the choices of subjects are closer to the cooperative outcome.10 5.2.2. Comparison with chess players In this subsection we compare our findings on the Centipede game with the closely related findings of Palacios-Huerta and Volij (2009) (accessed from the AEA database). Palacios-Huerta and Volij (2009) used the Centipede game to test the reasoning abilities of chess players. They found that chess players terminated the game at the initial node in 70% of all cases whenever they knew they were playing against other chess players. The study’s main finding was that subjects consistently deviate from the Nash prediction when they lack common knowledge of rationality; against non-chess players, only around 30% of chess players followed the Nash prediction.11 We focus our comparison on the sessions that consisted only of chess players (422 subjects; 211 pairs). While we used the 14-node normal-form Centipede game introduced in Nagel and Tang (1998) in our experiments, subjects in Palacios-Huerta and Volij played a 6-node extensive-form

8 We also include a variable which interacts Elo and CRT scores in one of our specifications to control for the joint distribution of the two variables. The inclusion of the interaction term renders both the CRT term and the interaction term insignificant but keeps the coefficient associated with Elo negative. Since the CRT score and interaction variable correlate quite strongly (Spearman-ρ : 0.749, p-value < 0.01), the insignificance of those terms together could be driven by the combination of colinearity and relatively small sample size. We thus dropped this particular specification from our presentation of results for the sake of brevity and did not include the interaction term in our following regressions for the same reasons. 9 However, we observe significant deviations from Nash equilibrium play overall. 10 We performed several robustness checks but since none of them changed our results qualitatively, we do not present them for the sake of brevity. For instance, we used ordered and multinomial probit models to control for non-linearities. For the same 2 purpose we also included Elo in our specifications. We also replaced CRT with CRTH , where CRTH is a dummy variable, that takes a value of 1 if a subject had a full score on the CRT. 11 Levitt et al. (2011) offer some conflicting results, but their design did not control for anticipation, supergame behavior, or reputation effects.

Please cite this article as: S. Baghestanian, S. Frey, GO figure: Analytic and strategic skills are separable, Journal of Behavioral and Experimental Economics (2015), http://dx.doi.org/10.1016/j.socec.2015.06.004

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T (96) B(4)

L(64) 1000, 700 600, 100

M (27) 600, 690 620, 400

N N (4.5) 620, 660 660, 660

R(4.5) 640, 100 700, 680

Fig. 5. Behavior in the Kreps game (with choice frequencies in parentheses).

Fig. 4. Behavior in the Traveler’s Dilemma.

representation of the game. However, despite the differences in these two designs, we find a remarkably similar relationship between Elo and choices. We estimate the parameters of the following model, using the dataset in Palacios-Huerta and Volij (2009):

Ni = α0 + α1 Eloi + ui .

(1)

For pair i, Ni corresponds to the node at which they ended the Centipede game and Eloi is their average Elo rating.12 Palacios-Huerta and Volij (2009) did not elicit the analytic skills of their subjects and so we omit this variable in (1). We estimate the model in (1) linearly and using an ordered probit specification. The results of those regressions are in columns (3) and (4) of Table A.4. Looking at the correlations between Elo and N for both subject pools we observe that the coefficients from a linear model specification, shown in columns (1) and (3), are essentially indistinguishable. Comparing the ordered probit regressions in columns (2) and (4), we observe that the estimated coefficient from our data is less than half the estimate derived for the data of Palacios-Huerta and Volij (2009). This effect arises naturally in this context since our games had more than twice as many “nodes” than the games in Palacios-Huerta and Volij (2009). In summary, higher Elo predicts lower Centipede game choices among both chess and GO professionals. The behavior of subjects with high strategic skills does not seem to depend on an extensiveor normal-form representation of the game. 5.2.3. Traveler’s Dilemma Fig. 4 shows the bids for the different treatments. The solidcolored bars depict the bid distribution for the low-R treatment in which the median bid was 912 cents. For the high-R treatment, the median bid was 730 cents. We replicate common empirical regularities for this game: even within a pool of GO professionals subjects do not follow the Nash prediction and the reward parameter affects the bids significantly, with the greater reward predicting behavior closer to the Nash prediction.13 Our main objective is to test whether the separable effects of Elo and CRT scores on choices, observed in the Centipede game, are also present in the Traveler’s Dilemma. Hence, we again estimate various linear model specifications with individual choices in our Traveler’s Dilemmas as our dependent variable. Tables A.5 and A.6 in the Appendix show the estimation results for both treatments separately. Our results from the nested model specifications, shown in columns (1)–(7) in Table A.5, indicate that an increase in CRT scores 12

Note that ui denotes an iid error term. Both a paired-sample t-test and a Wilcoxon ranksum-test indicate that the difference between the bids across treatments is significant on a 1% level. The test statistic for the Wilcoxon test is 1526.7 with a p-value < 0.001. The t-test statistic is 7.0237 with a p-value < 0.001. 13

correlates positively with higher bids in the Traveler’s Dilemma with R = 100 (p-value < 0.05). For the same treatment, Table A.5 also shows that choices are negatively correlated with strategic skills (Elo), although the relationship is non-linear (column (7); p-value < 0.05). In Table A.7, column (7), a similar negative relationship is also observed for our R = 300 treatment (p-value < 0.05). A deeper analysis of the data reveals that the positive correlation between choices and CRT scores is driven by subjects who had a full score on the CRT. In column (8) in Table A.6 we replace our CRT variable with a dummy variable that takes a value of one if a subject had a full score on the reflection test. The coefficient of this variable is more than twice as high as the coefficient on the overall CRT score, suggesting that top performers contribute substantially to the positive relationship between CRT and choices. Column (9) uses a Tobit specification to adjust for the fact that choices in Traveler’s Dilemmas are bounded from above and below. The change in estimation method does not affect our results qualitatively. Looking at the various specifications for both treatments separately indicates that CRT scores correlate positively with cooperative choices if R = 100 and Elo correlates negatively with choices if R = 300 and if R = 100. In summary, we observe patterns in Traveler’s Dilemmas and Centipede games consistent with a claim that cooperative behavior increases in CRT scores and equilibrium behavior increases in Elo ranks.

5.3. Kreps game and Matching Pennies: Results 5.3.1. Kreps game The Kreps game has multiple equilibria, two in pure strategies. If, compared to standard subject pools, GO professionals exhibit more Nash-consistent behavior, then errors and cognitive limits are less satisfactory explanations for their behavior in the Traveler’s Dilemma and Centipede games above. This is indeed what we observe (Fig. 5; the empirical frequencies at which each strategy was played in our experiments are given in the brackets next to the corresponding strategy). Subjects coordinated on the Pareto efficient Nash equilibrium most of the time—96% of the row players chose T and 64% of the column players picked L. The NN strategy was played in less than 5% of choices. In contrast, the more typical subjects in Goeree and Holt (2001) selected the T, L, and NN strategies in 84%, 24%, and 64% of choices, respectively, and a majority of their outcomes were Pareto dominated and non-Nash.14 We compare the distribution of choices from our GO players to the results in Goeree and Holt (2001), who have a similarly sized subject pool (50). A Kruskal–Wallis test suggests that the distributions of column player choices differ significantly at a 1% level (p-value < 0.01). The distributions of row player choices do not differ significantly (pvalue = 0.18).15

14 However, in contrast to Goeree and Holt (2001) we doubled the payoffs associated with each choice. 15 For the sake of completeness we also checked whether there are significant differences in Elo ranks or CRT scores among our participants, conditional on their choices in the Kreps game. A Kruskal–Wallis test on the equality of distributions of Elo ranks and CRT scores, conditional on choices, suggests that there are no significant differences (Elo: p-value = 0.61; CRT: p-value = 0.37). We also tested these differences for row and column players separately, yielding similar insignificant results.

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U (50) D(50)

L(60.8) 400, 200 200, 400

R(39.2) 200, 400 400, 200

Symmetric Treatment

U (80) D(20)

L(65) 1200, 200 200, 400

7

R(35) 200, 400 400, 200

Asymmetric Treatment

Fig. 6. Behavior in the Matching Pennies Game (with choice frequencies in parentheses).

5.3.2. Matching Pennies Our results for the two parameterizations of Matching Pennies are depicted in Fig. 6. Under the symmetric treatment, using a standard proportion z-test, we cannot reject the null hypothesis that each strategy is played with equal probability (0.5), as has been observed elsewhere (Goeree and Holt, 2001). Also as in standard subject pools, we observe that row players in the asymmetric treatment were significantly more likely to choose U instead of D (even though a rational row player would play uniformly random). However, 65% of our column players choose L in the asymmetric game (compared to 16% in Goeree and Holt (2001)). We observe that our subjects coordinate on the sum of utility maximizing outcome (U,L) more frequently than standard subject pools. We again compare the distribution of choices in our asymmetric Matching Pennies games to the results in Goeree and Holt (2001). A Kruskal–Wallis test suggests that the distributions of column player choices differ significantly at a 1% level (p-value < 0.01). The distributions of row player choices differ only marginally (p-value = 0.10).16 In summary, our sample of GO players behaves differently from standard subjects in asymmetric Matching Pennies games. Overall, we observe more coordination towards the efficient, sum-of-utilitymaximizing-outcome. In symmetric Matching Pennies games and Kreps games, we observe that our subjects tend to play Nash equilibrium strategies. Thus, the observed deviations from Nash behavior in Traveler’s Dilemmas and Centipede games are seemingly not driven by our subjects’ general inability to identify Nash equilibrium strategies. 6. Discussion By examining and contrasting behavior across all of the games we tested, some regularities suggest themselves. The first observation addresses robustness issues associated with common behavioral patterns in certain game classes. The other observations summarize our findings regarding the influences of strategic and analytic skills. Observation 1. Despite their impressive analytic and strategic skills, our participants deviate from Nash equilibrium play in Traveler’s Dilemmas, Centipede, and asymmetric Matching Pennies games, but not in the Kreps or symmetric Matching Pennies games. Confusion, errors, and limited computational ability are common explanations for non-Nash behavior in standard subject pools, but they are harder to accept in populations of expert reasoners, particularly given how our subjects distinguished themselves from standard subjects pools in the Kreps game. Where standard pools mostly play Pareto-dominated, non-Nash strategies in the game, ours preferred the Pareto-dominant Nash equilibrium. Consequently, our analysis suggests that GO players’ deviations from equilibrium behavior in

16 We again checked whether there are significant differences in Elo ranks or CRT scores among our participants, conditional on their choices in the asymmetric Matching Pennies game. A Kruskal–Wallis test on the equality of distributions of Elo ranks and CRT scores, conditional on choices, suggests that there are no significant differences (Elo: p-value = 0.96; CRT: p-value = 0.89). We also tested these differences for row and column players separately, generating similar insignificant results. We performed a similar analysis for the symmetric version of the Matching Pennies game, which also yielded no significant results.

Traveler’s Dilemmas and Centipede games are neither accidental nor due to limited cognition. Observation 2. There is a significant relationship between Elo (strategic skills) and the rationality of chosen actions in Centipede and Traveler’s Dilemma games. An increase in Elo correlates with an increased accuracy of the corresponding Nash prediction. Observation 2 is based on the consonance between our results from the Centipede and the Traveler’s Dilemma games. Our results on the Centipede game are consistent with the findings of PalaciosHuerta and Volij (2009) in that a linear fit suggests a positive relationship between combinatorial game expertise and Nash-consistent behavior, down to the value of the magnitude of the effect (see Table A.5). Observation 3. There is a significant relationship between CRT (analytic skills) and the efficiency of chosen actions in Centipede and Traveler’s Dilemma games. An increase in CRT correlates with a decreased accuracy of the corresponding Nash prediction and an increased accuracy of the cooperative choice prediction. In our subject pool, strategic and analytic skills correlate differently with behavior in one-shot games that evoke the Prisoner’s Dilemma’s conflict between the group’s efficiency and the individual’s rationality. Overall, individual willingness to cooperate correlates positively with analytic skills in the games we consider. On the other hand, individual willingness to compete, as reflected in an increased accuracy of the Nash prediction, correlates positively with strategic skills. These results indicate that subjects with higher analytic reasoning scores are more driven by efficiency considerations in Centipede games and Traveler’s Dilemmas. Both Elo ratings and CRT scores measure abstract reasoning abilities, but some part of CRT proficiency is orthogonal to Elo rank. Indeed, many of our results seem to be driven by the non-positive relationship between strategic and analytic skills in our subject pool. One possible explanation for this finding could be related to the observation that strategic skills necessary to achieve a high Elo rating are learned over time. The positive coefficient associated with years of GO-playing experience, when regressed against Elo, supports this claim. Analytic skill, on the other hand, could be related to biological traits which could affect the ability to suppress impulsive responses. More specifically, Bosch-Domènech et al. (2014) show that there is a significant negative relationship between CRT scores and second-tofourth digit ratios (2D:4D), which serve as putative markers for exposure to prenatal sex hormones. This difference between a learned skill and a skill related to an innate biological trait, may contribute to the observed separability of strategic and analytic skills in our data. 7. Conclusion We examined the behavior of professional GO players in four families of one-shot games. Overall—in the Centipede, Traveler’s Dilemma, and asymmetric Matching Pennies games—our subjects did not adopt Nash equilibrium play. In Centipede games and Traveler’s Dilemmas, GO players display Nash behavior with increased strategic skills (Elo), and efficient behavior with increased analytic skills (CRT). We argue that expertise in strategic reasoning correlates with Nash

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behavior, and that expertise in analytic reasoning, keeping strategic skills constant, correlates with cooperative behavior in games that evoke the Prisoner’s Dilemma’s conflict between cooperative and individually rational behavior. Our subject pool is non-representative in that their Elo ranks and CRT scores belong to the higher ends of their

distributions. Future work needs to address whether standard subject pools exhibit the same separability of strategic and analytic skills.

Appendix A

Table A1 Predictors of Elo: the dependent variable in the regression results shown above are individual Elo scores. The explanatory variables are CRT (a subject’s CRT score), Gender (a dummy variable that takes a value of 1 if a subject was male and is 0 otherwise), Age (a subject’s age), YearsGo (a subject’s years of GO-playing experience) and LeftHanded (a dummy variable for handedness, which takes the value 1 if a subject is left-handed and is 0 otherwise). The dummy CRTH L takes the value 0 for subject i if CRTi ∈ {0, 1} and it takes the value 1 for subject i if CRTi ∈ {2, 3}. The dummy variable BatBall takes a value of 1 if a subject answered the first question of the CRT correctly and is 0 otherwise. The dummy variable Machine takes a value of 1 if a subject answered the second question of the CRT correctly and is 0 otherwise. The dummy variable Liliypad takes a value of 1 if a subject answered the third question of the CRT correctly and is 0 otherwise. The Table presents OLS estimates with heteroscedasticity-corrected standard errors in parentheses (white standard errors). Columns (1)–(5) show nested models, column (6) uses the final specification from column (5) and replaces CRT with CRTH . (1) CRT

(2) ∗

(3) ∗

−139.2 (72.73)

(4) ∗∗

−143.3 (73.60)

(5) ∗∗

−150.8 (72.82)

−134.1 (62.85)

(6)

(7)

(8)

−131.9 (63.16)

CRTH L

−374.7∗∗∗ (142.2) −255.9∗ (154.4)

BatBall

−156.1 (177.7)

Machine Lilypad Gender

120.1 (169.5)

Age

106.3 (163.7) 5.382 (3.99)

121.3 (156.2) −16.38 (10.4) 26.87∗∗ (11.73)

46 0.089

46 0.206

YrsGo

134.9 (157.2) −17.02 (10.43) 27.05∗∗ (11.66) 60.19 (137.3) 46 0.208

LeftHanded N R2

(9)

∗∗

46 0.058

46 0.069

138.3 (152.5) −18.67∗ (10.71) 28.08∗∗ (11.80) 36.04 (129.82) 46 0.233

154.8 (159.1) −17.1 (10.98) 27.04∗∗ (12.2) 116.5 (157.07) 46 0.198

133.9 (155.7) −18.64 (11.5) 28.19∗∗ (12.75) 68.12 (153.6) 46 0.172

−284.3∗ (153.9) 97.59 (160.4) −17.95 (11.02) 27.73∗∗ (12.21) 26.61 (144.51) 46 01.93

Standard errors in parentheses. ∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01. Table A2 Behavior in the Centipede game. Go players

Nagel and Tang

B-choices

Fractions

A-choices

Fractions

B-choices

Fractions

A-choices

Fractions

2 4 6 8 10 12 14

0 0.045 0.045 0.091 0.318 0.182 0.318

1 3 5 7 9 11 13

0 0 0.083 0.208 0.167 0.208 0.333

2 4 6 8 10 12 14

0.022 0.027 0.14 0.313 0.312 0.143 0.043

1 3 5 7 9 11 13

0.005 0.005 0.053 0.325 0.258 0.192 0.162

Table A3 Centipede game choices and cognitive skills: the dependent variable in the regression results shown above are individual choices in the Centipede game. The explanatory variables are CRT (a subject’s CRT score), Elo (a subject’s Elo rating), LeftHanded (a dummy variable for handedness, which takes the value 1 if a subject is left-handed and is 0 otherwise), Age (a subject’s age), Gender (a dummy variable that takes a value of 1 if a subject was male and is 0 otherwise) and YearsGo (a subject’s years of GO-playing experience). The Table presents OLS estimates with heteroscedasticity-corrected standard errors in parentheses (White standard errors). Columns (1)–(6) present estimated parameters from nested OLS models. Column (7) drops two extreme observation with the highest Elo ranks (= 2700). (1)

(2)

(3)

(4)

(5)

(6)

(7)

CRT

0.789∗∗ (0.337)

0.656∗ (0.340) −0.001∗ (0.0006)

0.688∗∗ (0.320) −0.001∗ (0.0006) 1.239∗∗ (0.591)

0.722∗∗ (0.325) −0.001 (0.00066) 1.359∗∗ (0.669) −0.0154 (0.029)

0.676∗∗ (0.333) −0.0011∗ (0.0006) 1.750∗∗∗ (0.593) −0.0216 (0.029) 1.493∗ (0.779)

46 0.080

46 0.116

46 0.159

46 0.166

46 0.229

0.675∗∗ (0.332) −0.0011∗ (0.00068) 1.761∗∗∗ (0.593) −0.0304 (0.0401) 1.507∗ (0.795) 0.0112 (0.036) 46 0.230

Elo LeftHanded Age Gender YrsGo N R2

0.877∗∗ (0.366) −0.0014∗∗ (0.00065) 0.785 (0.970) −0.0140 (0.0410) 1.373 (0.867) −0.00423 (0.037) 44 0.247

Standard errors in parentheses. ∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01.

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9

Table A4 Comparison (Palacios-Huerta and Volij, 2009) (Chess vs. GO): the dependent variable in the regression results shown above are either individual choices in the Centipede game (for GO players) or end-node outcomes (for chess players). The explanatory variable is Elo (a subject’s Elo rating for GO players and average Elo rating for chess players). The Table presents OLS and ordered probit estimates with heteroscedasticity-corrected standard errors in parentheses.

Elo Observations R2 F statistic Pseudo R2 Wald χ 2

Dependent variables: Centipede choice Go players Linear Ordered probit (1) (2)

Dependent variables: Centipede end node Chess players (Palacios-Huerta and Volij, 2009) Linear Ordered probit (3) (4)

−0.001227∗∗ (0.0006) 46 0.06 4.21∗∗

−0.001242∗∗ (0.0005) 211 0.03 5.56∗∗

−0.005∗∗ (0.00025) 46

−0.016∗∗ (0.0008) 211

0.02 4.06∗∗

0.011 3.83∗∗

Note: ∗ p < 0.1; ∗∗ p < 0.05; ∗∗∗ p < 0.01. Table A5 Traveler’s Dilemma choices and cognitive skills, R = 100: the dependent variable in the regression results shown above are individual choices in the Traveler’s Dilemma for R = 100. The explanatory variables are CRT (a subject’s CRT score), Elo (a subject’s Elo rating), LeftHanded (a dummy variable for handedness, which takes the value 1 if a subject is left-handed and is 0 otherwise), Age (a subject’s age), Gender (a dummy variable that takes a value of 1 if a subject was male and is 0 otherwise) and YearsGo (a subject’s years of GO-playing experience). The dummy CRTH takes the value 1 for subject i if CRTi = 3. Columns (1)–(7) present estimated parameters from nested OLS models. Column (8) replaces CRT with CRTH . Column (9) uses a Tobit specification. For the Tobit specifications pseudo R2 are reported. Heteroscedasticity-corrected standard errors are in parentheses (White standard errors for OLS regressions).

CRT

(1)

(2)

(3)

(4)

(5)

(6)

(7)

34.39∗ (18.57)

34.77∗ (18.74) 0.00273 (0.0268)

34.85∗ (18.76) 0.00262 (0.0267) 3.172 (47.19)

35.59∗ (18.40) 0.00388 (0.0286) 5.720 (46.17) −0.325 (1.187)

33.86∗ (17.75) −0.000765 (0.0294) 20.15 (44.04) −0.554 (1.294) 55.15 (44.04)

33.73∗ (18.14) −0.0104 (0.0339) 22.19 (43.02) −2.143 (2.676) 57.74 (44.38) 2.004 (2.524)

36.31∗∗ (15.79) −0.353∗∗∗ (0.097) −14.81 (26.75) −2.647 (2.728) 50.08 (39.33) 3.198 (2.600) 0.000113∗∗∗ (0.0000357)

46 0.067

46 0.068

46 0.069

46 0.107

46 0.118

46 0.235

Elo LeftHanded Age Gender YrsGo 2

Elo

CRTH N R2

46 0.067

(8)

(9)

−0.376∗∗∗ (0.095) −22.02 (25.36) −3.624 (2.624) 52.48 (39.27) 4.352∗ (2.569) 0.000118∗∗∗ (0.0000372) 79.27∗∗∗ (31.13) 46 0.248

−0.525∗∗∗ (0.187) −3.733 (58.12) −6.435∗ (3.229) 67.20 (48.80) 8.405∗∗ (3.668) 0.000157∗∗∗ (0.0000338) 80.84∗ (47.56) 46 0.04

Standard errors in parentheses ∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01. Table A6 Traveler’s Dilemma choices and cognitive skills, R = 300: the dependent variable in the regression results shown above are individual choices in the Traveler’s Dilemma for R = 300. The explanatory variables are CRT (a subject’s CRT score), YearsGo (a subject’s years of GOplaying experience), Age (a subject’s age), Gender (a dummy variable that takes a value of 1 if a subject was male and is 0 otherwise) LeftHanded (a dummy variable for handedness, which takes the value 1 if a subject is left-handed and is 0 otherwise) and Elo (a subject’s Elo rating). The dummy CRTH takes the value 1 for subject i if CRTi = 3. Columns (1)–(7) present estimated parameters from nested OLS models. Column (8) replaces CRT with CRTH . Column (9) uses a Tobit specification. For the Tobit specifications pseudo R2 are reported. Heteroscedasticity-corrected standard errors are in parentheses (White standard errors for OLS regressions).

CRT

(1)

(2)

(3)

(4)

(5)

(6)

(7)

17.91 (29.78)

8.437 (30.94) −0.0680 (0.0477)

10.80 (28.78) −0.0715 (0.0442) 93.77 (62.67)

13.63 (29.02) −0.0666 (0.0454) 103.6∗ (61.91) −1.249 (1.882)

11.31 (29.36) −0.0729 (0.048) 123.0∗∗ (57.39) −1.558 (1.881) 74.23 (64.13)

11.19 (29.47) −0.0821 (0.0547) 124.9∗∗ (56.31) −3.076 (3.600) 76.71 (64.69) 1.916 (3.241)

14.29 (27.62) −0.492∗∗∗ (0.129) 80.55∗∗ (39.04) −3.681 (3.527) 67.53 (62.28) 3.348 (3.119) 0.000136∗∗∗ (0.0000540)

46 0.041

46 0.086

46 0.094

46 0.122

46 0.126

46 0.195

Elo LeftHanded Age Gender YrsGo 2

Elo

CRTH N R2

46 0.007

(8)

(9)

−0.507∗∗∗ (0.134) 76.86∗ (39.71) −4.389 (3.664) 67.81 (60.76) 4.092 (3.176) 0.000140∗∗∗ (0.0000498) 51.01 (59.90) 46 0.204

−0.854∗∗ (0.445) 236.8 (145.4) −7.220 (7.456) 205.0 (128.7) 6.559 (8.098) 0.000224 (0.000142) 47.29 (31.27) 46 0.03

Standard errors in parentheses ∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01.

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S. Baghestanian, S. Frey / Journal of Behavioral and Experimental Economics 000 (2015) 1–10 Table A7 Professional GO ratings and titles: The table shows the traditional GO ratings and how they can be converted into Elo scores using the EGF (European Go Federation) conversion table. Traditionally, GO ranks players on a kyu-dan scale, not unlike martial arts. Kyu ratings are beginner’s ranks and Dan ratings are advanced ranks. The upper part of the panel describes kyu and dan ratings. The lower part of the panel maps kyu and dan ratings into Elo scores. Rank type

Range

Stage

Professional-dan Amateur-dan Single-digit-kyu Double-digit-kyu Double-digit-kyu

1-9p (where 10p is special title) 1-7d (where 8d is special title) 10-1k 20-11k 30-21k EGF Elo ratings and traditional GO ranks EGF rating 2940 2820 2700 2600 2500 2400 2300 2200 2100 2000 1900 1800 1500 1000 500 100

Professionals Advanced player Intermediate club player Casual player Beginner

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GO rank 9 dan professional 5 dan professional 7 dan amateur or 1 dan professional. 6 dan (amateur) 5 dan 4 dan 3 dan 2 dan 1 dan 1 kyu 2 kyu 3 kyu 6 kyu 11 kyu 16 kyu 20 kyu

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Please cite this article as: S. Baghestanian, S. Frey, GO figure: Analytic and strategic skills are separable, Journal of Behavioral and Experimental Economics (2015), http://dx.doi.org/10.1016/j.socec.2015.06.004