Abstract
We examine strategic awareness in experimental games, that is, the question of whether subjects realize they are playing a game and thus have to form beliefs about others’ actions. We conduct a beauty contest game and elicit measures of cognitive ability and beliefs about others’ cognitive ability. We show that the effect of cognitive ability is highly non-linear. Subjects below a certain threshold choose numbers in the whole interval and their behavior does not correlate with beliefs about others’ ability. In contrast, subjects who exceed the threshold avoid choices above 50 and react very sensitively to beliefs about the cognitive ability of others.
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Notes
There is some evidence that individuals do not consider other individuals behavior when making decisions. For example, individuals compete too much on easy tasks and too little on hard tasks (Moore and Cain 2007), ignore competitors’ characteristics that are payoff-relevant (Camerer and Lovallo 1999) or take competitors behavior as given (Goldfarb and Xiao 2011).
This finding is in line with other recent evidence on the presence of non-strategic players in the BCG. For example, a study by Agranov, Caplin and Tergiman (2015) identifies a large proportion of k = 0 players (43 percent) by observing players’ provisional BCG choices within a 3 min time frame. Related, Burchardi and Penczynski (2014) use communication protocols among team members playing a BCG to classify strategic and non-strategic reasoning (about 33 percent).
Branas-Garza, Garcia-Munoz and Gonzalez (2012) use the Raven test and the cognitive reflection test (CRT) to measure subjects’ cognitive ability and relate the test scores to behavior in six BCGs with varying p. They find no effect for cognitive ability in the first two beauty contests (with p = 2/3 and p = 1/8), but in the remaining BCGs (p = 1/5, p = 1/3, p = 1/2, p = 3/4) they observe a negative correlation of CRT and chosen numbers. Similarly, Gill and Prowse (2015) find no relationship between Raven’s test scores and choices in the first round of a repeated BCG (p = 7/10), but they observe that more cognitively able subjects converge more frequently to equilibrium over time. Georganas, Healy and Weber (2015) find that CRT scores are related to level-2 play in a variety of 2-person BCGs.
Alaoui and Penta (2015) show theoretically that their experimental results are consistent with individuals engaging in a cost-benefit analysis of applying additional rounds of reasoning. See also Strzalecki (2014) for an alternative theoretical approach in which behavior also depends on own bounded rationality and beliefs about opponents’ bounds.
In four sessions subjects were unintentional matched in groups of twelve and thus in total there are 36 groups in the BCG. While the average choice turned out to be similar in groups of twelve (42.3) and six (45.8), the regression analysis controls for group-size effects.
The three questions are the following: A bat and a ball cost 1.10 Euro in total. The bat costs 1.00 Euro more than the ball. How much does the ball cost? If it takes 5 machines 5 min to make 5 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?
The used questions can be found in Appendix D. For general information about the test see www.wonderlic.com.
In one session there was a false fire alarm during the WPT and hence the test scores are not used in the analysis. The other tasks were not affected because they were completed before the false alarm.
Branas-Garza et al. (2012) report an average choice of 43.2 \( (p = 2/3\;{\text{BCG}}) \) and (Gill and Prowse 2015) find an average choice of 44.2 \( (p = 7/10\;{\text{BCG}}) \). Using two-player teams in a \( p = 2/3\;{\text{BCG,}} \) Burchardi and Penczynski (2014) find that the suggested average choices of subjects in a team are 43.9 (the final average choice was 39.7 after communication).
Other studies find similar shares. For example, Branas-Garza et al. (2012) find that 16 percent of subjects choose a number in the interval [2/3*100, 100], Burnham et al. (2009) find that about 20 percent of subjects choose a number in the interval [1/2*100, 100] and Allred et al. (2014) find that 33 percent of their subjects choose a number in [2/3*10, 10]. Relatedly, Agranov et al. (2015) show that when subjects can modify their choice as often as they want within a three-minute time frame, more than 50 percent of subjects choose at least once an action in the interval [2/3*100, 100]. In their setup, non-strategic subjects switch their choices more than five times as often as strategic subjects and the average choices of non-strategic subjects are remarkably close to 50. Similarly, we find that, in particular, the choices of subjects with a CRT score of 0 or 1 are on average close to 50.
The average score in the 3-item CRT is 1.49. About 48 % of the subjects have one or less answer correct and 52 % of the subjects have 2 or 3 answers correct. Other studies find similar average scores. For instance, the mean score of the whole sample in Frederick (2005) was 1.24 (with a maximum of 2.18 and a minimum of 0.57), whereas Hoppe and Kusterer (2011) and Oechssler et al. (2009) report slightly higher scores of 1.84 and 2.05 among German students, respectively.
The “Bat and Ball” question (BBQ) refers to the following question: A bat and a ball cost 1.10 Euro in total. The bat costs 1.00 Euro more than the ball. How much does the ball cost?
It is also the case that subjects who answered the BBQ correctly were more likely to win the prize (p = 0.04, χ²-test).
That the BBQ is an important indicator for choices in the BCG is also confirmed by testing the predictive power that each of the three CRT questions provides, conditional on the other questions. That is, including a dummy variable for each correctly answered question reveals that the BBQ has by far the largest the impact on choices in the BCG and we can reject the hypothesis that all three questions are equally predictive (Wald test, p = 0.058, two sided).
A similar problem may arise by using post-experimental questions. Even though such questions cannot influence behavior in the game, they can induce biased responses because subjects may realize ex-post that the decision environment was strategic. Note that in the BCG it is not possible to distinguish cleanly between non-strategic and strategic behavior, since low numbers might be evidence for both.
It also seems possible that administering the CRT before the BCG makes subjects who gave the correct answers more suspicious and raises their awareness in the subsequent BCG, because giving the correct answer often requires suppressing the intuitive, but wrong answer. Note that, in principle, changing the order of the tasks may lead to the related issue of BCG-play influencing awareness in the CRT. However, as pointed out by a referee, it seems unlikely that a potentially higher awareness due to the task order would only affect subjects with the maximum CRT score of three.
Relatedly, the BBQ (and more generally the CRT) is a good predictor for rational thinking, i.e., individuals’ immunity to cognitive biases (see e.g., Toplak et al. 2011).
Gill and Prowse (2015) find a similar result in an environment that facilitates learning. Using data from a repeated BCG they find that subjects with higher cognitive ability vary their level of reasoning depending on the known cognitive ability of their opponents, whereas low ability subjects do not react at all to the cognitive ability of their opponents. However, they do not find differences between high and low ability subject for initial responses in the BCG.
For example, different framing of instructions may induce strategic awareness. Suggestive evidence comes from 2-person guessing games where the choice of zero is weakly dominant (see e.g., Grosskopf and Nagel 2008). Chou et al. (2009) show that, while standard descriptions of the payoff function leads to frequent violations of weak dominance, framing the game as a “battle”—you win the battle if your chosen location is higher than your opponent’s—improves compliance dramatically. However, their setup leaves open whether subjects just followed the advice in the instructions, or whether the variation indeed induced subjects to reason strategically.
Although we can rule out that our finding is affected by reverse causality because subjects were not aware of the BCG when stating their beliefs about others’ cognitive ability, exogenous variation in beliefs would be useful to account for unobserved factors that may influence subjects’ beliefs about the cognitive skills of others.
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Acknowledgments
We thank the Editor, Jacob Goeree, and two anonymous reviewers for thoughtful comments. We are also grateful to Terry Burnham, Brit Grosskopf, Rosemarie Nagel, Joerg Oechssler and Andrew Schotter for helpful conversations as well as seminar participants for comments. We thank David Cesarini, Pablo Branas-Garza and Teresa Garcia-Munoz for sharing their data. Financial support from the Deutsche Forschungsgemeinschaft (DFG) through the SFB 649 “Economic Risk” is gratefully acknowledged.
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Fehr, D., Huck, S. Who knows it is a game? On strategic awareness and cognitive ability. Exp Econ 19, 713–726 (2016). https://doi.org/10.1007/s10683-015-9461-0
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DOI: https://doi.org/10.1007/s10683-015-9461-0