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Journal of Economic Interaction and Coordination

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01-04-2014 | Regular Article

Cognitive capacity and cognitive hierarchy: a study based on beauty contest experiments

Authors: Shu-Heng Chen, Ye-Rong Du, Lee-Xieng Yang

Published in: Journal of Economic Interaction and Coordination | Issue 1/2014

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Abstract

Recent developments in behavioral experiments, in particular game experiments, have placed human cognition in a pivotal place. Two related ideas are proposed and are popularly used in the literature, namely, cognitive hierarchy and cognitive capacity. While these two often meet in the same set of experiments and observations, few studies have formally addressed their relationship. In this study, based on six series of 15- to 20-person beauty contest experiments and the associated working memory tests, we examine the effect of cognitive capacity on the observed cognitive hierarchy. It is found that cognitive capacity has a positive effect on the observed cognitive hierarchy. This effect is strong in the initial rounds, and may become weaker, but without disappearing, in subsequent rounds, which suggests the possibility that cognitive capacity may further impact learning. We examine this possibility using the Markov transition dynamics of cognitive hierarchy. There is evidence to show that subjects with different cognitive capacities may learn differently, which may cause strong convergence to be difficult to observe.

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For example, McCain (2010) used the Brock-Hommes adaptive belief system (Brock and Hommes 1998) to build an agent-based model to simulate the evolution of the distribution over different depths of reasoning. In this model, agents can in principle choose whatever depth of reasoning they prefer; however, based on the experienced rewards and costs they may choose different depths of reasoning at different points in time. The point is that all depths are available for all agents, regardless of their personal attributes, not to mention cognitive capacity.

(Moulin (1986), p. 72), introduces the game called “guess the average” as an example to illustrate the idea of the successive elimination of dominated strategies. The beauty contest game has an interesting history. Its origin can be more complex than what one generally thought. The interested reader is referred to Buhren et al. (2012).

Recently, a semi-parametric cognitive hierarchy model which can encompass both the level-\(k\) model and the cognitive hierarchy model has been proposed by Hahn et al. (2010).

Devetag and Warglien (2003) also notice these two different interpretations of the observed cognitive hierarchies or level-\(k\) reasoning. They cite Costa-Gomes et al. (2001) as an example of the interpretation which attributes observed behavioral heterogeneity to the differences in preferences, decision rules and beliefs, while they themselves are inclined to consider the alternative, which attributes the observed behavioral heterogeneity to computational limits. Other work that also addresses this difference includes Grosskopf and Nagel (2008).

Coricelli 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.

For the studies showing the strong connection between WMC and general intelligence, the interested reader is referred to Kyllonen and Christal (1990), Engle et al. (1999), and Conway et al. (2002).

Although there has been great confusion about short-term memory and working memory in the literature of psychology, the distinction between them may be only a matter of semantics (Cowan 2008). Cowan argues that working memory has been conceived and defined in three slightly discrepant ways: as short-term memory applied to cognitive tasks, as a multi-component system that holds and manipulates information in short-term memory, and as the use of attention to manage short-term memory. In short, working memory contains short-term memory and other processing mechanisms that help to make use of short-term memory.

Among a total of 20 papers included in the survey by Nagel (2008) or in this paper, the number of subjects in the beauty contest experiment runs from a minimum of 2 to a maximum of 3,696. Barely half involve less than 10 subjects; for the others the number of subjects is either greater or much greater than 10.

The number of agents and their heterogeneities can contribute to the complexity of games and may have a further impact on the outcomes of the game (Ho et al. 1998; Guth et al. 2002; Kovac et al. 2007).

Branas-Garza et al. (2012) have only one round for each of their six different versions of the beauty contest game, but the use of this multiple-version design may still allow subjects to learn from their experiences such as the learning observed in the multiple-stage game. Nevertheless, the learning dynamics, i.e., how learning can happen by carrying over the experience from one version of the game to other subsequent versions, is not the focus of their analysis.

Gill and Prowse (2012) found that, while in the first five rounds there is no significant difference in earnings between high and low cognitive ability subjects, their earnings difference becomes bigger in the last five rounds, and the difference is statistically significant.

See http://eel.nccu.edu.tw/.

See http://eel.nccu.edu.tw/Registration/.

In Eq. (2), the upper limit of the level \(d=0\) is bounded from the right side by \(m(t-1)\), instead of \(m(t-1)p^{-0.5}\). About this asymmetry, Nagel (1995) indicates that the results, for the first period, would not change if a symmetric bound were to be taken instead (ibid, p. 1317). For the later periods, it is pointed out that the chosen numbers tend to be below the mean of the previous period (ibid, p. 1320). To make our results comparable with those of Nagel (1995) (Sect. 3.2), the same asymmetric bound is taken in our analysis.

The only exception is period 7 (\(\tau _{b}\)), but it is also insignificant.

All the thresholds or percentiles are computed based on our ESD which includes the WMC scores of 740 subjects.

This is shown in the last column of the upper panel, period 10, and the last column of the lower panel, window “9–10”.

This finding is different from Gill and Prowse (2012), who find that the effect of cognitive capacity on the earning performance becomes even more significant in the later rounds. As we mention in Sect. 1.4, the complexity of the game or the group size of the game may have effects on the result.

These results are also comparable to what was found in Gill and Prowse (2012). In their case, the mode of the high group lies in the rule learners (Stahl 1996) who switch from level-1 to level-2 in their cross-matched group, i.e., the group mixed with both high and low cognitively able subjects. For the low ability group, Gill and Prowse (2012) found that the mode appears at level 1 accounting for 40.8 % of that group, while level 2 only accounts for 28 % of the group. The only sharp difference between us is that we have a relatively large proportion of low WMC subjects belonging to the classes “\(d \le 0\)” (27.5 % vs. only 2 %). This difference may be accounted for by two possible reasons. One is the evolutionary pressure from the existence of the matched high cognitively capable opponents. The other comes from the information disclosure; Gill and Prowse (2012) revealed the choices of all subjects, that may help them to locate a better guess.

For a literature survey, the interested reader is referred to Eckel and Grossman (2008a).

The MU aims at measuring simultaneous storage and transformation in working memory (Oberauer et al. 2000). To take one more step in reasoning, players are required to retain the results of former reasoning (storage) while performing the next step (transformation). In this sense, there seems to be a natural link between MU and level-\(k\) reasoning. The significance of MU in the quality of economic decision making is also found in the double-auction experiments (Chen et al. 2012).

In Kocher and Sutter (2006) and Agranov et al. (2011), the RT is treated as a control variable in the beauty contest experiment, i.e., the time allowed for making a decision (time constraints or time pressure) is given. Under this design, Kocher and Sutter (2006) find that increasing time pressure reduces the quality of decisions, in terms of the distance to equilibrium, the standard deviation around the winning number, the guessing errors, and the payoffs. Agranov et al. (2011) also find the strategic levels are advanced as the subjects have more time to make beauty contest choices. Our case is more similar to Rubinstein (2007), who allows subjects to decide how much time for making a decision (guess), and hence the RT is not given at the outset. Rubinstein (2007) finds that both level-1 reasoning (guesses of 33 and 34) and level-2 reasoning (a guess of 22) take longer RTs, while the level-0 reasoning (a guess of 50) and level “\(<\)0” (guesses of more than 50) correspond to shorter RTs.

Psychologists and neuroscientists have long studied what is known as the dual system model, in which System 1 is more intuitive, automatic, quick and effortless, and System 2 is more effortful and demands attention. This model has also drawn the attention of behavioral economists in recent years (Kahneman 2011). In our experiments, subjects spending more time producing a higher level of reasoning may be an indication of the workings of System 2. Schnusenberg and Gallo (2011) also find that System 2 users, characterized by the subjects answering more questions correctly in CRT, will pick numbers close to zero in the initial stage.

For example, it is found that women give more than men in the dictator game (Eckel and Grossman 2008a), and women are more risk averse than men, although the latter is only evidenced in field experiments and is less conclusive in laboratory experiments (Eckel and Grossman 2008b).

While the subsequent level distributions between the two groups are less significant, for the two-tailed groups, the bottom and top one-fourth (\(P_{25}\) and \(P_{75}\)), the difference in their level distribution is still quite noticeable in some periods, say, periods 6 and 10 (Figs. 13 and 17).

From an aggregate viewpoint, the uniform-like distribution, the distribution with almost no modes, is interpreted as random behavior. In other words, we view the subjects with WMCs in the bottom one-fourth guessing as if they randomly picked a level in the possible domain. This interpretation is only for the convenience of understanding aggregate behavior, and certainly does not exclude the possibility of the sophisticated thinking of some individuals in this group. What is also worth mentioning is that randomly-behaving agents have been coupled with zero-intelligence agents in the agent-based economic literature (Chen 2012).

The only exception occurs when \(i=1\). In this case, \({\hat{p}}_{1,2}= 0.3394\), but \({\hat{p}}_{1,1}=0.3578\).

The only exception is period 10, where the test for the guessing error is significant, but the test for the level distribution in insignificant. This shows that the discrepancy between the two is still possible. One possible reason for this inconsistency is that in the final period the length of each level, depending on the reference point in period 9 (\(m(9)\)), can become rather small. In our experiments, \(m(9)\) ranges from a maximum of 14.31 to a minimum of 4.31 only. Needless to say, the latter case may be too tight to sensitively distinguish the six levels well.

Georganas et al. (2010) also report this Markov transition matrix for undercutting games and two-person guessing games. In the former game, they find a strong tendency to switch to level-1, instead of level-2, regardless of the current type of subjects, even from the Nash type. However, they have a totally different focus of these Markov transition matrices. They wonder if the level is fixed just as the human’s psychological traits. Therefore, the elements in the diagonal of the matrix, which give the frequency of no-switch behavior, turn out to be their focus.

This may be also caused by the insensitivity issue due to narrow and narrower intervals. See footnote 28.

If we consider level reasoning as a choice variable, then we can certainly go further to examine the intriguing dynamics between this choice made by the subjects and the aggregate result after pooling their choices. This would allow us to formally build a learning model, such as reinforcement learning or generalized reinforcement learning (Camerer and Ho 1999), to account for the observed correspondence between the empirical level distribution and the target-\(d\) distribution. Nonetheless, including this analysis will make this paper overloaded, and hence it is better to leave it to a separate paper.

It is hard to see where Branas-Garza et al. (2012) stand at this point. They did run a number of regressions to examine the effect of cognitive capacity on the reasoning level. By pooling all data together from their six \(p\)-beauty contest games and assuming that subjects were able to learn in the Weber sense, i.e., learning without feedbacks (Weber 2003), then the whole of the data will be dominated by the data generated by experienced subjects. If so, the effect of cognitive capacity should be minimal and the corresponding coefficient should be insignificant. Hence, their regression results of having a significant coefficient of cognitive capability using CRT may be regarded as evidence that gap in cognitive capacity remains even though agents are able to learn. However, since they did not provide the analysis by separating data into initial periods and later periods, something equivalent to our Table 2, further confirmation of this is infeasible.

In addition, they also present subjects with the guess distribution of the previous round which is not available in our design.

In a separate study, we also apply experience-weighted attraction learning to our data and estimate the key parameters by separating the high group from the low group. However, given the size of this paper, the result will be presented in a separate paper.

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Title: Cognitive capacity and cognitive hierarchy: a study based on beauty contest experiments
Authors: Shu-Heng Chen
Ye-Rong Du
Lee-Xieng Yang
Publication date: 01-04-2014
Publisher: Springer Berlin Heidelberg
Published in: Journal of Economic Interaction and Coordination / Issue 1/2014
Print ISSN: 1860-711X
Electronic ISSN: 1860-7128
DOI: https://doi.org/10.1007/s11403-013-0113-1

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