4.2 WMC and Cognitive Hierarchy
4.2.1 Static Analysis: Conditional Distribution and Regres-
4.2. WMC AND COGNITIVE HIERARCHY
doing so, we shall first summarize how well the level-k reasoning classification applied in this study has captured the behavior observed in our laboratory, particularly in a comparison with the one found by Nagel (1995).
As mentioned in Section 3.3, by following Nagel (1995), we classify our subjects into six levels of reasoning, d < 0, d = 0, d = 1, d = 2, d = 3 and d > 3, based on their guessing numbers. We find that, on average, 78.7% of gi(t) can be classified as levels from d = 0 to d = 3, which is similar toNagel (1995)’s findings. For period to period, this covering ranges from 72.22% to 87.96%. Note that the interval covering levels from d = 0 to d = 3 shrinks;
it is 37.90 in width when t = 1, and is 4.51 in width when t = 10. Yet the covering capacity of these intervals remains unchanged; it covers 77.78% of the observations when t = 1 and 76.85% of them when t = 10. The essence of our level-k classification, based onNagel(1995), is to assume that subjects will anchor their guesses for the next period to the pervious average guess, m(t − 1). The summary statistics above show that this assumption is very much in line with the practical reasoning processes applied by most subjects.
4.2.1 Static Analysis: Conditional Distribution and Regression
To obtain a general idea of the possible relation between cognitive capacity and cognitive hierarchy, we continue using the previous three thresholds to divide subjects into the high WMC groups and the low WMC groups. We then examine how their behavior in terms of cognitive hierarchy may differ between the two groups. Figure4.2 provides the empirical distribution (his-togram) of the level of reasoning based on our six-level classification. Panel (a) of the figure is the empirical distribution drawn by pooling together all subjects’ classification, di(t), over the 10 periods; in other words, it is drawn based on a total of 1,080 observations (108 subjects over 10 periods). Panel (a) shows that the mode of this distribution is level two (d = 2), which accounts for 33.52% of the population, followed by level one (d = 1) and level three (d = 3), which account for another 22.13% and 13.24% of the population, respectively.
In addition to the general distribution, conditional distributions, con-ditioned on different WMCs, are given from panels (b) to (g). They are separately presented in two blocks: the ones under high WMCs are placed in the middle block (panels (b)-(d)) and are blue-colored, whereas the ones under low WMCs are placed in the bottom block and are gray-colored (pan-els (e)-(g)). To make the comparisons easier, the distributions of the high and the low groups divided by the same thresholds are put in an equivalent
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4.2. WMC AND COGNITIVE HIERARCHY
d<0 d=0 d=1 d=2 d=3 d>3
Figure 4.2: Level-k distribution conditional on WMC percentiles
Panel (a) presents the unconditional distribution of subjects’ levels of reasoning. Panels (b) to (g) presents the conditional distributions. Panels (b)-(d) show the level distributions of the high WMC subjects characterized by the first-half (higher than the mean), top one-third (higher than the 67th percentile (P67)), and top one-fourth (higher than the P75), whereas panels (d)-(f) show the level distributions of low WMC subjects characterized by the second-half (lower than the mean), bottom one-third (lower than the P33), and bottom one-fourth (lower than the P25).
position of the two blocks. In this way, the pairwise difference between the two groups in their reasoning level distributions is more easily focused upon.
For all the high WMC groups, level two is a clear peak in their reasoning level distribution, whereas, for the low WMC groups, level two is just barely the mode and its probability value is only marginally different from that of level one. On the other hand, the zero level has a relatively high frequency in the low WMC groups as opposed to that of the high WMC groups. Pearson’s chi-squared test shows all of these distributions to be pairwisely significantly different with a p value of less than 0.01. This comparison already provides some initial evidence showing that cognitive hierarchy is not independent of
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4.2. WMC AND COGNITIVE HIERARCHY
cognitive capacity.5
The analysis based on the above conditional distribution can be further advanced by including more possible explanatory variables. In this case, we use regression analysis to study the effect of cognitive capacity on cognitive hierarchy when other variables are controlled. In the regression model, the reasoning level, as the dependent variable, is run against the working mem-ory capacity and other explanatmem-ory variables, including response time (RT), gender, and period. Response time has been used as a measure to study mental structure by psychologists since the mid 19th-century (Luce, 1991), but it was only very recently tha it began to play an important role in exper-imental economics, in particular, with regard to behavior involving either a moral dilemma or deliberate thinking (Chong, Camerer and Ho,2005;Kocher and Sutter,2006;Rubinstein, 2007;Coricelli and Nagel, 2009;Piovesan and Wengstrom,2009;Arad and Rubinstein, 2010;Agranovy, Caplin, and Tergi-man, 2011; Branas-Garza et al., 2012b). Our incorporation of this variable is motivated by the latter involvement.
As to gender, it is a common demographic variable that has been studied in many game experiments such as public goods, ultimatum games, and dictator games.6 In the beauty contest, it has also been considered as a control variable (Burnham et al., 2009). Finally, because the dependent variable di(t) itself is a function of time, to filter out the possible effect of the subjects’ experience on reasoning level, period t is also included in the regression.
The working memory capacity used in our paper is taken fromLewandowsky et al. (2010), which is an aggregate of the five individual components men-tioned in Section 3.1. Therefore, instead of taking the WMC as a single aggregate variable, we can also run the regression against its five individual components to see what are the precise cognitive functions which affect the cognitive hierarchy. This consideration gives us two sets of explanatory vari-ables, one using the aggregate variable, WMC, and the other using its five
5These 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 ≤ 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.
6For a literature survey, the interested reader is referred toEckel and Grossman(2008a).
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4.2. WMC AND COGNITIVE HIERARCHY Table 4.2: Level-k thinking and working memory capacity
Dep. variable: di(t)
Model I Model II
Regressor Coefficient Std. Error Coefficient Std. Error
WMC 0.321∗∗∗ 0.068
DSpan 0.099 0.076
SSTM 0.049 0.051
MU 0.155∗∗ 0.067
SentSpan 0.051 0.066
OpsSpan −0.025 0.073
RT 0.149∗∗∗ 0.054 0.156∗∗∗ 0.055
Gender −0.144 0.109 −0.146 0.115
Period 0.036∗ 0.019 0.036∗ 0.019
LR statistic 38.905∗∗∗ 42.225∗∗∗
Sample 1080 1080
∗,∗∗and∗∗∗denote the significance of estimates at the 10 percent, 5 percent and 1 percent levels, respectively.
components. To differentiate between these two regression models, we shall define Model I as the one using WMC as a whole, and Model II as the one using the five component variables. Since the level classification di(t) is an ordinal variable, the ordered logit regression is applied to Models I and II.
Would cognitive capacity matter for cognitive hierarchy? The answer from Table 4.2 seems to be positive. Reading through the two regression models, we can find that cognitive capacity either in terms of WMC as a whole (Model I) or some of its components (Model II) significantly positively affects the cognitive hierarchy (reasoning level). In addition to the signif-icance of WMC, we further learn from Model II that it is the component memory updating (MU) that plays a crucial role, since of the five variables MU is the only one which is found to be statistically significant.7 Most of the
7The memory updating task aims at measuring simultaneous storage and transforma-tion 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
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4.2. WMC AND COGNITIVE HIERARCHY
others also have a positive influence, but they do not comp up to a significant degree.
Notice that the above result is obtained after period t has been taken into account. As we can see from Table 4.2, period t also has a positive effect on the reasoning level, showing the significance of learning. Nevertheless, even though learning has been taken into account, the net effect of WMC on the reasoning level is still positive. This finding may corroborate our early finding (Section 4.1) that learning can make the gap among subjects with different WMCs smaller, but cannot completely eliminate it. The significance of WMC remains even with learning.
While cognitive capacity is found to be significant in the determination of cognitive hierarchy, it is not the only variable that matters. Maybe the most significant finding from the multiple regression is to identify other important variables, in our case, the response time (RT). Our finding on the positive effect of the response time is consistent with the findings in the existing literature (Kocher and Sutter, 2006; Rubinstein, 2007; Agranovy, Caplin, and Tergiman, 2011).8
This finding helps us understand why the low WMC groups also exhibit a positive distribution on higher levels of thinking as shown in Figure 4.2 (panels (e), (f) and (g)). Subjects who do not impulsively react, but are more prudent and spend some time to reflect upon what happened and what will happen may be compensated for their initial deficit in WMC.9 Finally,
memory updating 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).
8In Kocher and Sutter (2006) and Agranovy, Caplin, and Tergiman (2011), the re-sponse time 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. Agranovy, Caplin, and Tergiman (2011) also find the strategic levels are advanced as the subjects have more time to make beauty contest choices. Our case is more similar toRubinstein(2007), who allows subjects to decide how much time for making a decision (guess), and hence the response time 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 response times, while the level-0 reasoning (a guess of 50) and level “< 0” (guesses of more than 50) correspond to shorter response times.
9Psychologists 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
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4.2. WMC AND COGNITIVE HIERARCHY Table 4.3: Do High and Low WMC subjects exhibit similar level/guess dis-tributions?
Higher vs. Lower Mean P67 vs. P33 P75 vs. P25
Period Levela Guessb Levela Guessb Levela Guessb
1 0.0368∗ 0.0688 0.0417∗ 0.1051 0.0004∗∗ 0.0087∗∗
2 0.0038∗∗ 0.0145∗ 0.0076∗∗ 0.0720 0.0041∗∗ 0.0243∗
3 0.7212 0.4045 0.2848 0.1701 0.0502 0.083
4 0.2452 0.0452∗ 0.1379 0.0679 0.1172 0.0894
5 0.0098∗ 0.3119 0.0010∗∗ 0.1251 0.0016∗∗ 0.2249
6 0.3044 0.2565 0.2598 0.3703 0.0630 0.0799
7 0.4686 0.1789 0.5990 0.1462 0.4283 0.2249
8 0.6579 0.9999 0.9394 0.9999 0.9767 0.6459
9 0.3492 0.6050 0.6362 0.6912 0.7346 0.5464
10 0.1156 0.6890 0.2270 0.8493 0.1406 0.6207
∗ and ∗∗ denote the significance of the estimates at the 5 percent and 1 percent levels, respectively.
a p-value from Pearson’s chi-squared test with simulated p-value. The p-values are com-puted by running Monte Carlo simulations ten times, each with 107 replicates, and then taking the average of the ten simulated p.
b p-value from the Kolmogorov-Smirnov test.
while gender is found to be significant in many experimental studies10, it is not influential in the behavior of sophisticated reasoning. This result is also consistent withBurnham et al. (2009).