2.2. BCG AND COGNITIVE CAPACITY
This game implicitly requires each player to form his/her expectations of other players’ expectations. If other players are doing the same thing, the game then suffers from the familiar infinite regress problem. Under the homogeneous rational expectations hypothesis, a Nash equilibrium will be reached where everyone chooses an equilibrium of zero, which is the result of 50 (the middle point between 0 and 100) post-multiplied by pk when k goes to infinity. However, the resultant beauty contest experiments have demonstrated great deviations from this game-theoretic prediction (Nagel, 1998, 2008). For example, the experiment run by the Financial Times in 1997 in collaboration with Richard Tayler showed that the most popular guesses are the ones close to the above multiplication with k being two or three, far away from being infinity (Thaler, 2000).
This has motivated some recent progress in cognitive economics, such as Crawford’s level-k reasoning and Camerer’s cognitive hierarchies (Camerer, Ho and Chong, 2004). These two models exhibit similar features in that they reveal the nature of the step-by-step elimination process, but they are different based on the assumptions of human cognitive ability. Crawford’s level-k model assumes that different “k” arise from non-standard beliefs, rather than irrationality (Costa-Gomes, Crawford and Broseta,2001; Costa-Gomes and Crawford,2006). On the contrary, Camerer’s cognitive hierarchy (CH) model assumes that the hierarchy probably arises from the players’
inability to realize the existence of higher level players and probably attribute this to the brain’s limits, such as working memory constraint (Camerer, Ho and Chong,2004).2 The CH model also presumes that the difference between the perceived and actual level distribution shrinks as the “k” increases. This implies that smarter people are endowed by nature with a better model of others’ thinking and are more capable of making guesses near the target.
2.2 Beauty Contest Game and Cognitive Ca-pacity
A number of studies have investigated the relevance of cognitive capacity or intelligence in BCG, but the results are mixed (see the summary in Table
2Devetag and Warglien (2003) also notice these two different interpretations of the observed cognitive hierarchies or level-k reasoning. They cite Costa-Gomes, Crawford and Broseta(2001) as an example of the interpretation which attributes observed behav-ioral heterogeneity to the differences in preferences, decision rules and beliefs, while they themselves are inclined to consider the alternative, which attributes the observed behav-ioral heterogeneity to computational limits. Other work that also addresses this difference includesGrosskopf and Nagel(2008).
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Table 2.1: Literature on the Effect of Cognitive Capacity on Strategic Thinking
Authors Games Measures Relevance
Camerer(1997) Beauty contest SAT math Numbers: null
Ohtsubo(2002) Beauty contest Imposing memory task (a
Theory-of-Mind test)
Numbers:↓; Performances:↑
Devetag and Warglien(2003) Normal-form game solvable by iterated dominance; Dirty faces;
Extensive-form game solvable by backward induction
Wechsler digit span task Performances:↑
Burnham et al.(2009) Beauty contest Standard psychometric test of general intelligence
Numbers:↓; Performances:↑
Coricelli and Nagel(2009) Beauty contest Mathematical calculation task Performances: null Rydval et al.(2009) Simplified beauty contest Operation span task Dominance:↑
Georganas et al.(2010) 2-person guessing game;
Undercutting game
General intelligence test; Eye gaze test (a Theory-of-Mind test); Wechsler digit span task;
Cognitive reflection test (CRT);
One-player takeover game
Level-k: null; Earnings:↑ (only Eye gaze test and CRT)
Schnusenberg and Gallo(2011) Beauty contest CRT Numbers:↓; Clustering:↑ (only
matters for initial responses in both measures)
Gill and Prowse(2012) Beauty contest Raven test Numbers:↓ (Own-matched
groups); Level-k:↑; Earnings:↑
Branas-Garza et al.(2012a) Beauty contest Raven test; CRT Numbers:↓(CRT);
Dominance:↑(CRT);
Level-k:↑(CRT)
Due the space limitations, the last column is written in a very compact manner. What is written before the colon includes the key behavioral variables examined and tested by the respective paper. After the colon is the relation found between the cognitive ability and the variable under examination. The positive relation is represented by an upward arrow, whereas the negative relation is represented
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2.2. BCG AND COGNITIVE CAPACITY
2.1). Camerer (1997), the first study in this direction, did not find its rele-vance. He demonstrated that Caltech undergraduates, of which half scored the maximum in SAT math, did not choose numbers much closer to the Nash equilibrium than average people, although they did exhibit the lowest median and mean guesses among all subjects in the pool. Coricelli and Nagel(2009) found similar results in an fMRI (functional magnetic resonance imaging) study of BCG demonstrating that mathematical proficiency, defined as the accuracy in calculation task, was unrelated to the ability to match the right guess.3
On the other hand, Burnham et al. (2009) demonstrated the significance of cognitive capacity. In his study subjects were given a standard psycho-metric test of cognitive ability. It was then found that subjects with higher cognitive ability exhibited significantly lower beauty contest entries, and the average guesses of the smartest group turned out to be the closest to the tar-get number. Schnusenberg and Gallo(2011) also showed that cognitive abil-ity, measured by the cognitive reflection test (Frederick,2005), contributes to lower and more clustering beauty contest entries in the first round, although the effect will not be sustained in the later periods.
These mixed results may not necessarily be conflicting because different measures of cognitive capacity are employed by these studies. In Camerer (1997) andCoricelli and Nagel(2009), the measure is mainly limited to math-ematical capability, whereas inBurnham et al. (2009) andSchnusenberg and Gallo (2011), it is based on various kinds of psychological IQ tests. Hence, one possible reconciliation is that, while the BCG is formed as a problem involving numerical calculations, it may rest little on the specific mathemat-ical ability; instead, some general components of intelligence, for example, the ability to suppress an intuitive and spontaneous incorrect answer so as to leave room for a reflective and deliberative one, as Schnusenberg and Gallo (2011) suggest, are what count. Motivated by this initial evidence, this study will take a further examination along this research line to corroborate on the earlier findings. What, however, distinguishes this paper from the earlier ones is the measure of cognitive capacity. We consider a measure that is not exactly the same but closely related to IQ or general intelligence, namely, working memory capacity(WMC).4
3Coricelli and Nagel (2009) actually introduce a measure for subjects’ capability to guess a number that could potentially win against a large population of opponents. They even invent a new term for this measure, called strategic IQ.
4For the studies showing the strong connection between working memory capacity and general intelligence, the interested reader is referred toKyllonen and Christal(1990),Engle et al.(1999), andConway et al.(2002).