Running head:  REGULATION AND IRRATIONALITY

College Students’ Awareness of Irrationality on Gambling Tasks
Eric Amsel, Jason Close, Eric Sadler
Weber State University
Paul Klaczynski
Pennsylvania State University 

Contact information:
	Eric Amsel
	Department of Psychology
Weber State University
1202 University Circle
Ogden, UT, 84408-1202 

Abstract
Two studies examined the role of awareness of irrationally on gambling judgments and behavior. In Study 1, 197 college students completed a questionnaire in which they were asked to select between two equal ratios (1/10 vs. 10/100) on a lottery task and to judge their certainty of the rationality of each response option. In Study 2, 199 college students completed the same lottery task or a parallel non-gambling version of it. Most participants (89% and 85% in Study 1 and 2 respectively) could be categorized into those whose regulation of dual rational and experiential processing of information was Competent (representing only the rationally-based, correct response option as rational), Conflicted (representing both rationally-based correct and experientially-based incorrect response options as rational), or Flawed (misrepresenting an incorrect experientially-based response option as the only rational one).  The regulatory styles were consistent across task content and predicted judgments on each task.  Compared to others, those who were Flawed in regulating the dual processing of ratio information made more subjective and biased gambling judgments and more often engaged in formal and informal gambling activities.  The results were interpreted as pointing to the importance of self-regulatory processes in gambling-related decision-making and behavior.

College Students’ Awareness of Irrationality on Gambling Tasks
One of the more compelling and perplexing questions about gambling is why otherwise competent and intelligent young adults are vulnerable to forms of irrational decision-making and behavior exposed by their predilections for games of chance. Many college students claim to gamble in order to earn money (Neighbors, Lostustter, Cronce, & Larimer, 2002), despite the obvious futility of such a goal.  A recent assessment of over 20,000 student-athletes indicates that over 60% of the men and 40% of the women have participated in a wide variety of gambling activities (NCAA, 2004). The participation rate remains high (over 30% for men and 19% for women) for sports betting, an activity that is specifically outlawed by NCAA bylaws, the violation of which can result in a student-athletes losing NCAA eligibility. More generally, young adults are vulnerable to gambling problems and have a rate of gambling problems that is reported to be three times higher than the general population (Shaffer & Hall, 2000). 
College students’ vulnerability to gambling-related irrational decision-making and behavior may reflect a specific problem among particular students who reason illogically about probability or possess irrational gambling beliefs or misconceptions. On the basis of such cognitive propensities, researchers have distinguished between gamblers and non-gamblers or between problem and social gamblers in terms of their demonstration of such cognitive limitations (Calderwood, Dragonetti, & Tsanos, 1997; Joukhador, Maccallum, & Blaszczynski, 2003; Sharpe & Tarrier, 1993; Sylvain, Ladouceur, & Boisvert, 1997; Toneatto, Blitz-Miller, Walker, 1992).  Although valuable in pointing to the general differences between groups, the research is based on the questionable assumption that vulnerability to irrational gambling decision-making and behavior is limited only to those with these cognitive propensities.
A very different approach to understanding irrational decision-making and behavior has been emerging over the past several years.  Falling under the broad rubric of  “Dual Process Theories” (c.f., Chaiklen & Trope, 1999), the central notion is that two cognitive systems are simultaneously involved in the processing of information (c.f., Evans & Over, 1996; Klaczynski, in press; Kirkpatrick & Epstein, 1992; Stanovich & West, 2000). One of process is generally considered experiential. Experiential processing is characterized as the automatic, spontaneous, and largely unconscious processing of information holistically, minimally, and/or emotionally, and is believed to involve the activation of semantic and procedural memories (derived from experience) and informal heuristics. The experiential process can be thought of as an intuition, premonition, or “gut feeling” about a situation that is often experienced immediately. Rational processing, by contrast, involves the effortful, systematic, and largely conscious processing of information analytically, methodically, and dispassionately according to the an individual’s understanding of the rules of logic and mathematics. These processes are believed to act in parallel and interact in all manners of cognitive activities from impression formation to decision-making.  Most theorists consider experiential processing the default system—the system on which people typically rely in everyday judgment and decision situations (e.g., Evans & Over, 1996; Stanovich, 1999).  However, situational cues and motivational dispositions conspire to determine which form of processing is predominant in particular circumstances.  
The Dual Process approach to irrationality suggests that, rather than being unique to a particular subset of people, the entire population is vulnerable to irrational thinking. In three experiments, Kirkpatrick and Epstein (1992) tested a dual-process model of gambling behavior by offering college students a choice between two equal gambles.  Students were given a choice to play a lottery with a 10/100 chance of winning (10 winning and 90 losing tickets in a large tray from which they had one blindfolded draw) or with a 1/10 chance (1 winning and 9 losing tickets in a small tray). Despite being told of their equal probabilities, college students most often  chose the gamble with more winning targets (10/100).  In justifying their choices, students argued their that chances were better with the tray that had the larger absolute number of winners  (i.e., the 10/100 tray). Indeed, some participants were so compelled by the perceived difference between the equal gambles that they reported being willing to pay to guarantee their choice of a preferred tray.  Kirkpatrick and Epstein (1992) note that the “ratio-bias” effect (i.e., the tendency to prefer one of two equal gambles) was strongest in conditions that minimize the rational processing of gambling information and weakest in conditions that maximize it, a result that has subsequently been replicated (Alonso & Fernandez-Berrocal, 2003; Klaczynski, 2001b; Epstein & Panici, 2000; Panici & Epstein, 1999a, 1999b; Reduski & Volksdorf, 2003).
 It thus appears that illogical reasoning and irrational beliefs or misconceptions about gambling are not unique to problem or regular gamblers and that such reasoning is not entirely absent in non-gamblers or in social-gamblers. Instead, the results point to situations in which college students as a whole bypass rationally processing gambling information in favor of experiential processing such information. However, in our view those who gamble regularly or problematically may have particular difficulties regulating the dual processes involved in gambling-related reasoning and decision-making.  For example, Denes-Raj and Epstein (1994) demonstrated that self-reported frequency of engaging in informal and formal gambling was predicted by students’ preference for gambles with higher absolute frequency of winners but lower likelihood of winning (e.g., 9/100, 8/100 or 7/100 vs. 1/10 gambles).  Perhaps students who are vulnerable to irrational gambling-related decisions and behavior are those with limited ability to rein in their tendencies to experientially process of ratio information and, as a consequence, have difficulty engaging in rational processing.
The present study examined irrational gambling decision-making and behavior by exploring the role of college students’ awareness of experiential and rational processing of gambling information.  Given the assumption that everyone is vulnerable to making irrational decisions and behavior, we contend that such awareness is central for the competent regulation of the dual processes.  Competent regulation, in turn is necessary to limit students’ potential vulnerability to gambling-related problems.  
Consider, as an example, students who have little awareness of the dual processing of gambling information.  Such student may unknowingly bypass rational processing and instead rely heavily on experiential processing because experiential processing is cognitively automatic, undemanding, fast, and intuitively appealing. In doing so, they may consider their “gut feelings” evoked by the spontaneous activation of judgment heuristics (e.g., “more winners = better chances) to be products of rationally analyzing gambling information.  Consequently, they may believe that they have made rational decisions without being aware that their intuitions are, in fact, not products of true rational analyses. We characterize these people has having flawed regulatory styles because they unknowingly make irrational decisions due to misrepresentations of their own cognitive activities.  Students with flawed regulatory styles may be more vulnerable to continued gambling-related irrational decision-making and behavior than others, who knowingly bypass rational processing in favor of experiential processing of gambling information.  The latter type of gambler may also make irrational decisions by relying on “gut feelings,” but may do so knowing that their “gut feelings” conflict with rational analysis. Such individuals would represent themselves as having made irrational decisions because, despite their awareness that gambling information could have been processed rationally, they believe that their intuitions are superior guides to behavior.   The availability of such rational knowledge may serve to limit continued irrational decision-making and behavior.  More specifically, the awareness of the potential for rational processing, in combination with misbegotten gambles may produce feedback of which these individuals are aware and on which they can subsequently make more effective gambling decisions.  We characterize these people as having competent regulatory styles because, even though they make irrational decisions, they do so with full awareness of the irrationality of their decisions.  
Finally, consider other students who bypass rational analysis in favor of an experiential processing because they experience conflict between their “gut feelings” and rational analysis. In these cases, individuals would represent as rational both their experiential and rational processing of gambling information.  Because of the cognitive and emotional dissonance this recognition creates, these individuals might reconcile experiential/rational processing conflict by portraying both experiential and rational processing as rational.  We characterize these people as having a conflicted regulatory style because the decisions they make reflect an arbitrary choice between response options generated from rational and experiential processes.  Such a conflict has been documented among adolescents who offered a rational analysis of the limits of a statistical generalization but intuitively believed that that the generalization was intelligent (Klaczynski, 2001a).
Two studies examine college students’ regulation of gambling information.  Students completed Kirkpatrick and Epstein’s (1992) ratio-bias task, which presents a choice between two equal gambles.  After making their choices, students judged how certain they were that each response option was rational. From these data students were categorized into one of three regulatory styles described above: Competent (representing only the rationally-based correct response option as rational), Conflicted (representing both rationally-based correct and experientially-based incorrect response options on the task as rational), or Flawed (misrepresenting incorrect experientially-based response options as only rational options).  Study 1 tested whether most students could be categorized into one of the three regulatory styles and assessed factors associated with this categorization, including students’ judgments on the ratio-bias task and frequency of actual gambling behavior.  Study 2 was designed to replicate the findings of Study 1 and extend the analysis to a non-gambling task.  It was hypothesized that a majority of students would be categorized into one of the regulatory groups and that their regulatory style would predict gambling-related decision-making and behavior on both tasks.   
Study 1
	The goal of Study 1 was to find initial evidence of the three styles of regulating the dual experiential and rational information processes components on the ratio-bias task.  Kirkpatrick and Epstein (1992) suggested that students who commit the ratio-bias by preferring one gamble to a mathematically equal one intentionally bypass rational in favor of experiential processing of the gambling information.  As they note (p. 542), “Subjects readily made choices that, by their own accounts, they recognized as irrational”. This analysis was based simply on choices participants made on the ratio-bias task in conjunction with anecdotal justifications of those choices. However, such information may be insufficient to allow for strong inferences about how the response options were represented.  It remains unclear whether participants are truly aware of the irrationality of their decisions on the ratio-bias task in the sense of knowingly bypassing rational processing of gambling information in favor of experiential processing.  
To more carefully assess students’ task representations, they completed a ratio-bias judgment (R-BJ) task, modified from Kirkpatrick and Epstein (1992).  Students were given a description of two lotteries (1/10 vs. 10/100), asked to note their mathematical equality and to select one. As part of the R-BJ task, participants were additionally asked whether and how much they would be willing to pay for privilege of choosing one of the pair of lotteries to play.  
Participants also completed a ratio-bias evaluation (R-BE) task, which assessed on a 4-point scale their certainty regarding the rationality of having a preference for the 1/10, the 10/1000, or having no preference for either lottery on the R-BJ task. Responses on the R-BE task were used to assess the regulatory style of participants.  Competent regulation was defined as being moderately (3) or very (4) sure that having no preference is rational and being not at all (1) or a little (2) sure that preferring 1/10 and 10/100 lotteries is rational (need to add that SS responded to 3 items). Flawed regulation was defined as being moderately (3) or very (4) sure that having a preference for either the 1/10 or 10/100 lotteries is rational and being not at all (1) or a little (2) sure that having no preference is rational.  Conflicted regulation was defined as being at least moderately (3) or very (4) sure that both having no preference is rational and having a preference for 1/10 and/or 10/100 lotteries is rational.  
It was hypothesized that, although dual rational and experiential processes underlie ratio-bias effects, most participants would not be characterized a competently regulated, (i.e., as knowingly bypassing rational in favor of experiential processing).  Competent regulation of the dual processes requires both complex metacognitive skills to represent multiple, categorically different response options (experiential vs. rational) and efficient executive control to inhibit one response option in favor of another. This hypothesis was based on evidence that college students vary considerably in the extent to which they have fully developed thee skills for metacognitive reflection and inhibitory control necessary to regulate dual processes (Klaczynski, in press; Kuhn, 2000).. The metacognitive and inhibitory control demands of Competent regulation are assumed to be greater than those required for the Conflicted or Flawed regulation of dual processes.  Conflicted and Flawed regulation necessitates neither the representation of categorically different response options nor the inhibition of an option. Indeed, flawed regulation assumes neither metacognitive skills nor inhibitory control of dual processes at all.  It was predicted that these less sophisticated styles of dual process regulation, particularly the Flawed regulatory style, would be associated with an increased tendency to gamble and vulnerability to gambling problems. 
Methods
Participants
	A total of 197 (116 female and 81 male) students in Introductory Psychology classes at a regional university in Utah served as participants.  The participants were compensated with extra credit for their participation.  The average age of the participants was 21 years (range 16 to 47 years).  Most students were freshmen (68%) or sophomores (23%); only a few were juniors (6%) or seniors (2%).  Students’ self-reported ACT (M=22.7) and high-school GPA (M=3.4) were, as expected, strongly, positively, and directly correlated (r = .44, N = 160, p < .001) when students’ age and status were controlled. Students were randomly assigned to one of four groups of approximately 50 students in order to  counterbalance task order (R-BJ and R-BE task) and order of mention of the two lottery options (1/10 and 10/100 options). 
Procedure   
	The questionnaire that each student completed included three sections:  Demographics, R-BJ task, and the R-BE task.   The demographics questions requested information about students’ age, sex, student status, major, estimated high school GPA and ACT scores.  Participants were also asked about the frequency of formal (e.g., buying lottery tickets, playing slot machines, betting on horse races etc.) and informal (playing cards for money, shooting dice, betting on sports teams, etc.) gambling activities over the past 6 months. These questions were taken verbatim from Denes-Raj and Epstein (1994).  However, formal gambling activities are unavailable in Utah, which prohibits the sale of lottery tickets and has neither private nor state-owned casinos.  Generally, gambling is an activity discouraged by the religion practiced by the majority of the sample. Despite this, 20% of the sample reported engaging in formal and 38% in informal gambling activities.  However, because of the relatively small number of formal and informal gamblers, the gambling data were coded nominally as present or absent.   
	The Ratio-Bias Judgment (R-BJ) and Evaluation (R-BE) tasks were based on Kirkpatrick & Epstein (1992) and were introduced by the following hypothetical scenario:
For you to win $100, select a black ball from a tray holding both black and white balls. You are presented with two trays of balls. The Large Tray contains 10 black balls and 90 white ones and you have 1 chance to select the black ball while blindfolded. The Small Tray contains 1 black ball and 9 white ones and you have 1 chance to select the black ball while blindfolded. If your draw a black ball, the money is yours, otherwise you win nothing and the game is over.  The odds of selecting a black ball from the Large Tray are 10/100, which is the same as the odds of selecting a black ball from the Small Tray (1/10).  Even though the odds are identical for the two trays, some people may have a preference as to which one of those trays they would rather choose from.  
Six by 6 millimeter digital pictures of a small and large tray (commercial 4- and 8-inch aluminum pie pans) containing the designated number of black and white balls were presented on the left and the right side of the questionnaire, just below the description of the task.  The pictures were taken from the same position above each tray so that size of the trays appeared different but the size of the balls appeared similar in both pictures. The pictures were matted on black backgrounds. The first-mentioned tray in the description was placed on the left side of the page and the second mentioned tray was positioned on the right side. The pictures were labeled specifying their contents and rules for selecting a ball.  The presence of such visuals with a verbal description has been reported to more strongly elicit the ratio-basis in college students than the verbal descriptions alone (Rudski & Volksdorf, 2002). 
Following Kirkpatrick and Epstein (1992), the Ratio-Bias Judgment (R-BJ) task requested that participants make a forced-choice between the large and small tray.  Specifically, they were asked, “If you were given a choice, which tray you would choose from?” Students could circle either the Large or the Small Tray, with the first presented response option being the first-mention tray in the task description.  Evidence of a biased response pattern could be made over hard to follow sentence participants if one tray were selected more often than the other.  After making a choice, participants explained their responses and indicated their willingness to pay for the “privilege” of choosing the tray from which they could draw (as opposed to having a tray picked for them); note—illusion of control is involved here.
The Ratio-Bias Evaluation (R-BE) task was presented as requesting participants’ “thoughts about the possible responses on the gambling task.  Some responses on the task may be rational ones and based on a formal and deliberate analysis of the logic of the situation.  Other choices may not be rational and based instead on an intuitive and automatic, perhaps even an emotional reaction to the situation.”  Participants were then to rate how sure they were that it is rational a) to have a preference for and choose the large tray, b) to have a preference for and choose the small tray and c) to have no preference for either tray and arbitrarily choose between them.  The first-mentioned tray in the task description was also listed first on the R-BE task.  Participants recorded their rational certainty judgments for each response option on a four-point Likert-like scale: (1) “not at all sure” it’s rational (2) “a little sure” it’s rational (3) “moderately sure” it’s rational and (4) “very sure” it’s rational.  
	The questionnaire was administered to all students in Introductory Psychology courses who had completed informed consent forms. The order of the R-BJ and R-BE task was counterbalanced over participants, as was whether the Large or Small tray was the first-mentioned tray in the general description of the task and first-listed response option on the R-BJ and R-BE tasks.  The questionnaires were handed out and completed in class, taking approximately 10 minutes to complete.  
Results
Three sets of analyses tested the hypothesis that dual processes underlie college students’ ratio-bias judgments, but that most regulate them in a less sophisticated and more problematic manner by intentionally bypassing rational in favor of experiential processing.  A first set of analyses examined the claim that the ratio-bias is the result of the dual processing of gambling information.  As predicted, more participants preferred the “Large” (N=104) than the “Small” (N=90) tray option but, in contrast to Kirkpatrick and Epstein, this difference was not significant (binomial p = .18, 1-tail).  However, of the 37 students (19%) willing to pay for their choice of trays, 25 (68%) students were willing to pay for the Large tray and only 12 (32%) would pay for the Small tray (binomial p < .05, 1 tail).  The tendency for the Large tray to elicit more irrational judgments in the form of offering payment to secure it is consistent with the claim that the 10/100 gamble is experientially processed as more advantageous than 1/10 because of the greater number of winning targets. 
To further examine evidence of dual processes underlying students’ judgments, certainty ratings regarding the rationality of each of the three response options on the R-BE task (large tray, small tray, no preference) were analyzed as a function of their choice of the Large or the Small tray on the R-BJ task.  A 3 (Response Options) x 2 (Response Choice) x 2 (Task Order) x 2 (Tray Order) mixed-model ANOVA was run on rational certainty judgments, with response options serving as a repeated measure.  The analysis revealed a main effect of Response Option, F(2, 366) = 23.75, p < .001 (Greenhouse-Geisser corrected).  Students were more certain that having no preference was rational (M=2.77) than having a preference for the Large (M=2.25) or Small (M=2.17) tray; preference for the latter options did not differ. However, the Response Option x Response Choice interaction was also significant, F(2, 366) = 13.72, p< .001 (Greenhouse-Geisser corrected). As depicted in Figure 1, students were more certain of the rationality of their choice on the R-BJ task than the other choice, but were also as or more certain of the rationality of having no preference than they were of the preferred ratio (i.e., the lottery they actually selected). The greater rational certainty of students for their own over the other response option was not a result of a self protective bias or an endowment of  a first or a just-selected choice as rational, as the results showed no Task Order or Tray Order main or interaction effects.  This finding suggests that, overall, students engaged in both rational processing of equal gambles (resulting in their overall certainty that no preference is a rational response) and experiential processing (resulting in greater certainty that their preferred gamble was a more rational option). 
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Place Figure 1 about here
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A second set of analyses directly tested Kirkpatrick and Epstein’s (1992) claim that participants knowingly make irrational responses on the R-BJ task.  To test this suggestion, participants were categorized as regulating the dual processes in a Competent, Conflicted, or Flawed manner.  Of a total of 193 students who completed all the R-BE questions, 59 (31%) students’ responses (28 male and 29 females) were consistent with competently regulating dual processes, defined as rating oneself as “moderately” (3) or “very” (4) certain of the rationality of “no preference” as a rational response and “not at all” (1) or “somewhat” (2) certain of the rationality of the “Large” or “Small” tray response.   Another 59 (31%) (22 male, 37 female) were conflicted in their regulatory processes (i.e., “moderately” or “very” certain of the rationality of both the no preference response option and the large or small (or both) tray response options).  Finally, 53 (28%) (25 male, 28 female) were flawed in their regulation the dual processes, (“moderately” or “very” certain of only the rationality of one’s own preference for the large or small tray, with judgments for all other response options being 2 or lower).   
By de-facto transforming the 4-point rating scale into a two-point one (1 and 2 = rationally uncertain and 3 and 4 = rationally certain), there were only limited number of possible reponse patterns on the R-BJ task.  Students could be rationally certain or uncertain of each of the three response options.  This binary possibility for each of the 3 response options generates 8 possible response patterns (i.e., 23 = 8) that can be logically generated on the R-BJ task.  A total of 6 of the 8 responses patterns were of theoretical interest because they fit one of the three categories:  Competent (N=1), Conflicted (N=3), and Flawed (N=2).  As a result, by chance alone, 75% of sample should have been categorized.  In actuality, 89% of the sample was categorized, which was significantly above chance (binomial p < .001), suggesting that the categorization system is not just tapping random variation on the R-BE task. 
Students categorized in different regulatory groups did not differ by age, sex, student status, or estimated GPA.  However, they were different in estimated ACT scores, F(2, 143) = 17.81, p<.001. Students who Competently regulate dual processes reported having higher ACT scores (M=24.6) than those regulated in a Conflicted (M=21.5) or Flawed (M=21.2) manner. The finding is consistent with claims of a connection between general intelligence, academic skills, and reasoning ability (c.f., Perkins & Grotzer, 1997; Stanovich & West, 2000).  
A third set of analyses explored the gambling-related decisions and behaviors of students in each regulatory group (see Table 1).  The percentage of students by regulatory group who were willing to pay for a preferred tray was significantly different, F(2, 169) = 7.27, p < .01 (see the first line of Table 1). Follow-up Fisher LSD (Least Significant Difference) post-hoc tests showed that more students whose regulation was Flawed were willing to pay to secure a preferred tray than those in other categories. The percentage of students by regulatory group who reported engaging in informal gambling (the second line in Table 1) over the past 6 months was also significantly different, F(2, 169) = 5.81, p < .05.  Again, follow-up LSD post-hoc tests showed that, compared to those in other categories, more Flawed students engaged in informal gambling activities.  Flawed and Conflicted students engaged in more formal gambling than Competent students (the third line in Table 1); however, this difference only approached statistical significance, F(2, 169) = 2.32, p = .10.  A composite gambling score (from 0 to 2) was computed for each participant based on whether any formal (scored as 1) and informal (scored as 1) gambling activity had been reported (the fourth line in Table 1).  Mean composite gambling scores were higher in the Flawed regulatory group than the Competent group, with each of the latter scores being no different from the scores of the Conflicted group, F(2, 168) = 5.52, p < .05, with LSD follow-up tests.  Finally, only the choices of students in the Flawed regulatory group demonstrated the predicted ratio-bias effect on the R-BJ task (comparing the fifth and sixth lines of Table 1).  Students from the Flawed regulatory group chose the Large tray more often than the Small one, despite the two offering equal chances of winning (binomial p < .05, 1 tail).   
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Place Table 1 about here
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Partial correlations were computed between the composite gambling scores and each regulatory status, controlling for age, sex, status, estimated high school GPA, and estimated ACT scores. Composite gambling score were negatively correlated (r = -.30, N = 154, p < .001) with a Competent regulatory style status; positively correlated (r = .20, N = 154, p < .05) with a Flawed status; and not correlated at all (r = .00, N = 154, ns) with a Conflicted status.  These findings point to the value of regulatory style in predicting reported gambling behavior.  
Discussion
The central goal of Study 1 was to find evidence of the three styles of regulating experiential and rational processes on the ratio-bias task. Although, overall, college students did not demonstrate a significant ratio-bias effect by preferring the Large (10/100) to the Small (1/10) tray in a forced-choice selection, other evidence suggested that their judgments were based on the operation of dual processes. This evidence included a tendency for more students to offer payment to secure the Large (10/100) than the Small (1/10) tray and rating their preferred tray as more rational than the non-preferred tray—despite being certain of the rationality of having no preference.  The failure to find the ratio-bias effect is not unprecedented in the literature, particularly for students’ making judgments regarding their own preferences on the ratio-biased task (Alsonso & Berrocol, 2003; Epstein & Pacini, 2000; Kirkpatrick & Epstein 1992; Panici & Epstein, 1999a; Klaczynski, 2001b; Rudski & Volksdork, 2002).  
These findings also challenge Kirkpatrick and Epstein’s (1992) claim that college students knowingly bypass rational processing of gambling information in favor of experiential processing.  As described by Kirkpatrick and Epstein, such a regulatory style would be characterized as Competent in the present analysis because it involves the veridical representation of rationally- and experientially-based judgments.  Only a minority of students conformed to this pattern.  Instead, most students relied on less optimal regulatory styles, in which they showed some misrepresentation of rationally and experientially-based judgments.  Conflicted regulation involved representing both rationally- and experientially-based judgments as rational, whereas Flawed regulation involved misrepresenting experientially-based judgments as though they were rational.  Members of the latter group were hypothesized to engage in more gambling and to be more vulnerable to gambling problems than other students.  The data also supported this hypothesis: Compared to members of other groups, more students with a Flawed regulatory style engaged in informal gambling and made more irrational (offering payment for equal gambles) and biased (subject to the ratio-bias effect) decisions than others.  Moreover, because they represent their own biased and irrational judgments as rational, these students do not engage the alternative rational interpretations of the gambling information like those who regulate dual processes in a Conflicted or Competent manner. There is no reason to suppose that these students lack the ability or skills to engage in rational processing as their GPA and ACT scores were not notably different than others, which would have to be the case if they had such a capacity-related deficit. Moreover, there is no reason that they merely miscalculated the equality of ratios as equality between ratios was explicitly mentioned so no calculation was necessary.  
	A limitation of the present study is that students were required to make forced-choice ratio-bias judgments.  Because students were forced to choose between the Large and Small trays, the extent to which any student knowingly bypassed rational processes in favor of experiential processes in unclear.  That is, would students forego a “no preference” response in favor of a choosing the large or small tray on the ratio-bias task, despite representing that choice of one tray or another as irrational?  This issue was addressed in Study 2.   
Study 2
Study 2 was designed to assess performance on the ratio-bias task with the addition of a “no preference” response option on gambling (lottery) and non-gambling task content.  Pacini and Epstein (1999a) found only a minority of college students selected the “no preference” options on the ratio bias lottery task.  However, they also failed to find the ratio bias when students made choices about themselves. Only when students were asked to judge how others would reason on the task (a condition designed to promote experiential processing of the information) did they prefer the 10/100 to the 1/10 ratio. Klaczynski (2001b) also gave adolescents and young adults the ratio-bias lottery task with a “no preference” response option.  He found an age-related increase in selection of the “no preference” option. Despite this age-related increase in rational responding, however, only a minority of adults selected the “no preference” option, even under conditions designed to promote rational processing.  To promote rational processing, Klaczynski (2001b) requested students were to consider what a perfectly logical person would select on the ratio-bias task rather than what they themselves would select. In Alonso and Fernandez-Berrocol (2003), high school students were given a non-gambling ratio bias task with a “no preference” option in all three perspectives – self, other, and a perfectly logical person (also see Epstein & Pacini, 2000).  The non-gambling ratio bias task involved students being told of a protagonist (self, other, or a perfectly logical person) applying for one of two jobs that were equal in all characteristics except that Job A had 1 position available for a maximum of 10 applicants and Job B had 10 jobs available for a maximum of 100 applicants.   Alonso and Fernandez-Berrocol (2003) found substantial variability in selection of the “no preference” option, from a low of 2.5% in the “other” condition, to a high of 38% in the “logical” condition. In addition, only in the “other” condition did more participants choose the 10/100 over the 1/10 ratio option, replicating Epstein and Pacini (2000) and Pacini and Epstein (1999a).  
Together, these studies would seem to suggest that irrespective of employing a “no preference” option and using non-gambling task content, the ratio-bias effect can be elicited in certain conditions. But such a conclusion may be premature as the results of Alonso and Fernandez-Berracol (2003) employment task are difficult to directly compare to other studies because that research was performed with high school students, who likely have very little, if any, experience with employment situations. Also, the narrative of the ratio-bias employment task was not the same as the narrative for the ratio-bias lottery task, with the former being longer, more complicated, and confounded with multiple conditions compared to the latter.   To allow for direct comparison between performance on the ratio bias task in a gambling and non-gambling context, in Study 2students were presented parallel ratio bias lottery and employment tasks. Students received either the lottery or employment task in a between-subjects design, with each task including the “no-preference” response option.  We predicted that ratio-bias judgment performance on each task would be parallel and reveal evidence of dual processing of ratio information. It was also predicted that students would be reliably categorized by how they regulate the dual processes and proportion of students in those categories would be the same across tasks.  
Method
Participants 
A total of 199 (106 female and 93 male) Introductory Psychology students who signed consent forms participated in the study for extra credit.  The participants were from the same university as in Study 1 and had similar demographic characteristics.  The average age of the participants was 21.2 years (range 17 to 41 years). Most participants were freshmen (68%) and sophomores (18%); few juniors (9%) or seniors (3%) participated.  Students’ self-reported ACT (M=22.8) and high-school GPA (M=3.37) were similar to Study 1 but, unexpectedly, were weakly correlated, r = .15, N = 133, p = .086, after controlling for students’ age and status (the zero-order correlation was also weak r = .15, N = 141, p = .081).  It is unclear why this correlation was so low, but because of the unexpected weakness, findings involving ACT and GPA scores should be interpreted cautiously. Students were randomly assigned to one of four groups of approximately 50 students.  The four groups varied the two ratio-bias task contents (lottery and employment) and the two ratio-bias task orders (R-BJ and R-BE task).
Procedure
  	As in Study 1, the questionnaire distributed to students had three sections:  Demographics, the ratio-bias judgment (R-BJ) task, and the ratio-bias evaluation (R-BE) task.  The demographics section requested such information as participants’ age, gender, major, academic year, High School GPA, and estimated ACT scores. Formal and informal gambling information was not requested, although participants’ willingness to pay to secure a preferred gamble (which was correlated with the composite gambling score (r = .24, p < .01) in Study 1) was asked.
	The hypothetical lottery task used in Study 1 was altered to be clearer and more in line with the employment task.  Participants were introduced to the gambling task with the following narrative:
You enter a lottery in which you can win $100. There are two similar jars to choose from and you can draw one ticket (without peeking) from one or the other jar.  In Jar A there are 10 tickets, only 1 of which is the winning ticket.  In Jar B, there are 100 tickets, only 10 of which are winning tickets.  The odds of selecting a winning ticket from Jar A are 1/10 (10%), which is the same as the odds of selecting a winning ticket from Jar B (10/100 or 10%).  Even though the odds of selecting a winning ticket are the same for the two jars, some people may have a preference as to which jar they would rather choose from.
In order to make the lottery and employment tasks as equivalent as possible, no pictures accompanied the lottery task in Study 2.  The employment task was modified from Alonso and Fernandez-Berracol (2003) study by making the narrative similar to the lottery task:
You are invited to apply for a job at a company. There are two similar jobs to choose from and you are equally qualified for each job, but can only apply for one. For Job A there are 10 applicants (one of them would be you), only 1 of whom will be selected. For Job B, there are 100 applicants (one of them would be you), only 10 of whom will be selected.  The odds of being selected for Job A are 1/10 (10%), which is the same as the odds of being selected for Job B (10/100, 10%).  Even though the odds are the same for the two jobs, some people may have a preference as to which one of the two jobs they would rather apply for.
The R-BJ task requested that participants make a choice about the lottery or job selected based on their understanding of the best chance of winning a gamble or attaining a job.  Three choices were offered students: Jar or Job A (with 1/10 odds), Jar or Job B (with 10/100 odds), and a no preference option. Participants were then asked to explain why they made their choice, whether they would be willing to pay for their choice, and how much they would be willing to pay for that choice.
The R-BE section of the questionnaire was parallel to the version used in Study 1, except for slight wording changes to allow for the same wording to apply to both the lottery and employment task.  Participants were told:
 Some answers on the task may be rational and based on a logical and thoughtful analysis of the situation.  Other answers may not be rational and based instead on automatic reactions to or gut feelings about the situation. For each of the following, judge how sure you are that it is rational.
Participants were then to rate how sure they were that it was rational a) to have a preference for and choose Jar (Job) A that had a 1/10 chance of winning (being successful), b) to have a preference for and choose Jar (Job) B that had a 10/100 chance of winning (being successful) and c) to have no preference for either jar (job), and not care which jar (job) is chosen this last makes it sound like there were 4 ratings.  Participants provided certainty judgments for each response option on a slightly altered (altered from what?) four-point Likert-like scale: (1) “not at all sure” it’s rational (2) “a little sure” it’s rational (3) “mostly sure” it’s rational and (4) “very sure” it’s rational.
Participants received either the job or jar version of the ratio-bias task, with the order of R-BJ and R-BE task counterbalanced over participants. The questionnaires were handed out and completed in class and took approximately 10 minutes to complete.  
Results
The data were subjected to two sets of analyses to test the hypothesis that ratio-bias judgment and regulation performance on each task would be parallel and to examine evidence for individual differences in the processing of ratio information.  Table 2 shows that R-BJ performance on the Lottery and Employment task were significantly different, X2(2) = 11.61, p< .01.  Almost 40% of the participants chose the “no preference” option on the lottery task, rivaling the percentages found by Epstein and Panici (2000), Pacini and Epstein (1999a), and Klaczynski (2001b) in similar conditions. More participants selected the Jar B (10/100) than Jar A (1/10) on the lottery task (ignoring those who selected no preference; huh?), but the difference was not significant (binomial p = .13, 1-tailed).  This too parallels the findings of Epstein and Panici (2000), Panici and Epstein (1999a) and Klaczynski (2001b) in the “self” condition.  Although fewer participants selected no preference on the Employment (30%) than the Lottery task (40%), the difference was not significant, t(197) = 1.38 ns.  However, contrary to expectations, participants selected Job A (1/10) more often than Job B (10/100), (binomial p < .01, 2-tail).  Many students reported focusing on the number of competitors rather than available jobs in the employment task, reflecting an underlying belief that fewer competitors would increase their chances of being noticed and hired. Rather than ignoring the denominator by preferring 10 winners to 1 (Klaczynski, 2000b), this form of the ratio bias involves ignoring the numerator and preferring 9 competitors to 90. That is, students’ performance on the employment task seems to reflect the experiential-based preference for asserting oneself in a group of 10 as opposed to a crowd of 100, rather than rationally-based processing of the hiring conditions.  
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Place Table 2 about here
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To further examine evidence of dual processes underlying students’ judgments, a 3 (Response Options on the R-BE task) by 3 (Response Choice on the R-BJ task) by 2 (Task Content:  Job vs. Jar) by 2 (Task Order:  R-BJ first vs. R-BE first) mixed-model ANOVA was run on rational certainty judgments, with Response Options serving as a repeated measure.  There was a main Response Choice effect, with those choosing the no preference option having an overall higher rational certainty rating (M=2.56) than those preferring the 1/10 or 10/100 (both Ms=2.26) options, F (2, 187) = 5.71, p < .01. As in Study 1, there was a main effect of Response Option, F (2, 374) = 27.72, p < .001 (Greenhouse-Geisser corrected).  Students were more certain that having no preference was rational (M=2.72) than a preference for Jar/Job A (1/10) (M=2.19) or Jar/Job B (10/100) (M=2.18). Also consistent with Study 1, there was a significant Response Option by Response Choice interaction, F (2, 374) = 45.43, p< .001 (Greenhouse-Geisser corrected).  Figure 2 depicts a similar pattern to that shown in Figure 1: Students who preferred one ratio over the other (1/10 or 10/100) were more certain of the rationality of their preferred ratio than the non-preferred option, but were also more certain of the rationality of having no preference than they were of the non-preferred ratio. Even those who chose the no preference option were at least “a little” certain of the rationality of having a preferred option. Again, these findings point to the dual experiential and rational processing of ratio information.  
There were Task Order by Response Option, F (2, 374) = 3.75, p < .05 (Greenhouse-Geisser corrected) and Task Order by Response Option by Response Choice interactions, F (2, 374) = 2.65, p < .05 (Greenhouse-Geisser corrected).  These interactions indicate that overall students were more certain of the rationality of the “no preference” option when first evaluating response options (R-BJ) than making their choice (R-BE).  However, this effect was strongest among participants who selected the 10/100 option than other ones.  The finding that the “no preference” option was more strongly appreciated by students first forced to evaluate response options than to select among them suggests that the evaluation task may cue rational thinking and provides additional evidence of the role of contextual on the rational processing of ratio information (Alonso & Fernandez-Berrocol, 2003; Epstein & Pacini, 2000; Klaczynski, 2001b, 2001b; Panici & Epistein, 1999a).
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Place Figure 2 about here
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The second set of analyses addressed whether students could be categorized into a regulatory style in the same manner as in Study 1 and whether category membership would predict gambling-related judgments. There were 61 Competently regulated participants (31%) who evaluated themselves as at least somewhat uncertain (scores of 1 or 2) about the rationality of having a preferred option and at least mostly certain (scores of 3 or 4) of the rationality of having no preference . More of these participants made “no preference” choices (N = 39) than preference choices (N = 22) (binomial p < .05, 2-tail), with 14 preferring the 1/10 gamble and 8 preferring the 10/100 gamble.   The latter 22 participants (36% of the group and 11% of the sample), whose choices reflected a preferred gamble but whose evaluation recognized that “no preference” was the only rational response pattern on the task, can properly be said to have “readily made choices that, by their own accounts, they recognized as irrational” (Kirkpatrick & Epstein, 1992, p. 542). 
Fifty-three participants (27%) were categorized as Conflicted (i.e., evaluated themselves as at least mostly certain of the rationality of both having and not having a preferred option).  Half of these participants made choices reflecting “no preference” (N=26, 48%); the 27 who had preferences were equally divided between the 1/10 (N=13) and the 10/100 (N=14) options. There was a non-significant tendency for these participants to prefer the 10/100 to the 1/10 ratio on the lottery task (9 vs. 4) but to prefer the 1/10 to the 10/100 ratio on the employment task (9 vs. 5), X2 (1) = 3.03, p = .08.
Finally, there were 56 participants (28%) who were categorized as Flawed (i.e., evaluated themselves as at least mostly certain of the rationality of only having a preferred option). All these participants made choices reflecting a preference, with more preferring the 1/10 (N=35) than the 10/100 (N=21) ratio, a difference that approached significance (binomial p = .08, 2-tail). However, their responses differed by task content, with more preferring the 10/100 to the 1/10 ratio on the lottery task (13 vs. 9) than on the employment task (8 vs. 26), X2 (1) = 7.21, p <. 01.
A total of 85% of the sample was categorized into the three regulatory styles. As in Study 1, this percentage was significantly above the 75% expected by chance (binomial p < .001).  Once again, this finding suggests that the categorization system did not just tap random variation on the R-BE task but instead revealed different styles of regulating rational and experiential processing.  Students categorized in different regulatory groups were no different in age, sex, student status, estimated GPA, or estimated ACT scores.  As in Study 1, categories of dual process regulation were associated with an irrational tendency to pay to secure one of two equally likely lottery or employment opportunities.  A 3 (Regulatory Style) by 2 (Task Content) ANOVA on the proportion of those who would pay showed only a Regulatory Style effect, F (2, 163) = 3.43, p < .05.  More Flawed regulation students were willing to pay (30%) than Competently regulated students (10%), with conflicted students being no different from the other two groups (19%). This finding replicates the Study 1 result despite differences in task content and procedure between the studies. Finally, the distribution of regulatory styles on the two tasks was not statistically different (see Table 4), X2 (3) = 6.10, ns.  This finding suggests that, despite wide differences between the two tasks in eliciting different patterns of responses, each elicited the same proportion of regulatory styles.  
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Place Table 4 about here.
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Discussion
	Study 2 was designed to assess whether ratio-bias judgments and dual process regulation would be affected by adding a “no preference” response option and a non-gambling task content.  We predicted that judgments and regulatory styles would be the equal across tasks.  The data generally supported the hypothesis.  The judgment data were consistent with other findings in the literature.  Notably, the majority of college students did not recognize the equality between two equivalent ratios despite being told of their equality, replicating general claims that college students are susceptible to the ratio-bias effect (Alonso & Fernandez-Berrocal, 2003; Denes-Raj & Epstein, 1994; Epstein & Panici, 2000; Klaczynski, 2001b; Kirkpatrick & Epstein, 1992; Panici & Epstein, 1999a, 1999b, Reduski & Volksdorf, 2003).  Only 40% of the lottery and 30% on the employment task participants had no preference between the mathematical equivalent response options.  Others have also found that a minority of participants had no preference when asked to choose between two equal ratios, despite using other tasks and placing participants in other conditions (Alonso & Fernandez-Berrocol, 2003; Epstein & Panici, 2000; Klaczynski, 2001b; Pacini & Epistein, 1999a).  
Although we did not find that ratio-bias was characteristic across all participants, there was evidence that the task elicited both rational and experiential processing. Among the majority of students who had a preference for one job or jar over another, they were more certain of the rationality of no preference than the alternative job or jar (replicating Study 1).  Thus, students’ choices (i.e., preferring one job or jar over another) suggest experiential processing predominance, but their certainty ratings appear to reflect rational task analyses.
One notable exception to this list of replicated results was the finding that more American college students’ in this study preferred to apply for a job with a 1/10 than a 10/100 chance of being successful.  Spanish high school students preferred the latter to the former (Alonso & Fernandez-Berrocol, 2003).  We argue that these different preference patterns nonetheless reflect experiential processing predominance.  However, it remains unclear whether differences between the present findings and those of Alonso and Fernandez-Berrocol (2003) resulted from differences in sample ages, cultural backgrounds, or task presentation.
With respect to regulatory styles, the present study replicated the Study 1 finding that most participants could be classified as Competent, Conflicted, or Flawed in their regulatory styles. The roughly equal distribution of regulatory styles was shown to be same across task contents, despite the variability between tasks in students’ judgments.  The styles were unrelated to obvious demographic and background variables.  Nonetheless, the styles predicted a variety of judgment biases.  Those adopting the Flawed style were more vulnerable than others to judgmental biases (i.e., paying for preferred gambles and succumbing to the ratio-bias).  As in Study 1, it seems unlikely that those with a Flawed regulatory style were incapable of rational processing of the ratio information or miscalculated the equality of ratios.  With regard to the latter point, again the equality between ratios was explicitly mentioned so no calculation was necessary.  With regard to the former, these students GPA and ACT were no different from other students (although the reliability of such data was questioned).  Perhaps more to the point, rather than reflecting a general deficit in capacity or the penchant to make computational errors, the tendency for experiential processing to predominate over rational analyses seems to be related to a variety of task factors.  Besides the task domain effect (descried above), there was the effect of task order which involved the cuing of rational processing by first evaluating response options.
General Discussion
The purpose of these studies was to explore a dual-process approach to gambling decision-making and behavior. Rather than conceptualizing frequent or problem gamblers as uniquely making illogical inferences or possessing irrational gambling beliefs or misconceptions, Dual Process theory assumes that all individuals can potentially reason irrationally.  The dual process approach was tested on a ratio-bias task.  On this task, students could have relied on predominantly rational processing and, consequently, could have represented as mathematically equivalent 1/10 and 10/100 ratios or, alternatively, could have relied on predominantly experiential processing and represented the ratios only in terms of different absolute numbers of winners or successes and losers or failures.  A central hypothesis of the research was that students’ vulnerability to irrational gambling-related decision-making and behavior is associated with the manner in which they regulate rational and experiential processing. To identify regulatory styles, students were assessed for their evaluation of the certainty that each ratio-bias response option was rational in addition to their judgments of which response option they would choose.  
Combining the two studies of almost 400 college students, approximately a third competently regulated rational and experiential processing of gambling information. These students were notably less biased than others in processing ratio information, although by non means were they perfectly rational.  Although a majority made choices on the basis of rational analyses of ratios, a third nonetheless (Study 2) selecting one ratio over the other.  Even though these competently regulated students may have followed hunches, the hunch they followed was represented as such and not misrepresented as a rational judgment.  The tendency for these participants to follow hunches seems not to be a problem when they are represented as hunches.  Few competently regulated students reported having engaged in formal or informal gambling activities.  
Another quarter of the samples were conflicted in their attempts to regulate of rational and experiential processes.  In their certainty ratings, they engaged in both rational processing, representing the products of this processing as rational, and experiential processing, misrepresenting the product of this processing as rational.  As a consequence, when making decisions, these students had multiple conflicting rational options from which to choose.  Study 2 showed that half the time they relied on the rational processing and half on the mischaracterized experiential processing of the ratio information.  When they did rely on experiential processing, they showed a tendency (albeit weak) to be influenced by task content in Study 2.  There was a tendency (p = .08) for students to prefer different response options depending on whether the task was framed as a lottery or employment task. This tendency to be affected by the framing of otherwise identical tasks is characteristic of experiential processing of information (Klaczynski, 2001b; Tversky & Kahneman, 1984). Such framing effects provide additional support to the claim that these students were processing information experientially, despite mischaracterizing such processing as rational and recognizing the rationality of the “no preference” option.  Moreover, having available a rational analysis of ratio information appears to limit the vulnerability that these students may face by relying on mischaracterized experiential processing.  For example, only approximately 20% were willing to pay for their preferred ratios and a similar minority engaged in gambling activities.  
A little more than a quarter of the participants were flawed in their regulation of processing of gambling information. These participants were more biased than others in processing ratio information. They made decisions reflecting a preference for one of two equal ratios (thus showing ratio-bias) and were more likely to pay to secure their preferred ratio.  It is likely that these students were not without the ability for the rational analysis of ratio information.  The findings in each study that Flawed regulators were no different in GPA and ACT scores suggest that they do not lack general cognitive abilities of others.  Although perhaps they just erred on the task, we also doubt that such an explanation can account of the task effects in performance. Rather, we hold that these students’ performance is better characterized by the dominance of experiential processing over rational analyses on the ratio information. Evidence of the predominance of experiential over rational processing in these participants is the influenced by task context on their judgments in Study 2.  As previously noted, framing effects are characteristics of experiential processing of information.   
Despite making subjective and biased judgments, Flawed regulators represented their judgments as rational. Phenomenologically, these students confidently reacted to and acted on their “gut feelings” about ratios without regard to more formal analyses of them, reflecting poor metacognitive reflection and inhibitory control.  Perhaps because of they were overly confident in experiential processing, flawed regulators tended to engage in gambling behavior more often than others.  A majority engaged in formal gambling activities and, as a group, Flawed regulators had a composite gambling score that was twice as high as that of Competently regulated students.  
These finding confirm the claim that how student regulate rational and experiential processes, in the sense of being aware of and having control over rational and experiential predominance, predicts their vulnerability to irrational gambling-related decision-making and behavior.  This vulnerability was associated with regulatory style independently of various background and demographic variables, suggesting the importance of examining regulatory factors in gambling behavior. However, we do not believe that cognitive regulatory factors are the only one affecting the tendency for experiential processes to predominate over rational analyses and for experiential processes to be mischaracterized as rational.  No task factors, motivational factors, and dispositions seem to be implicated as well (Keating, 1996; Klaczynski, in press; Stanovich & West, 2000) Future research could further explore the power of regulatory style alone and in interaction with these other factors to predict problem gambling and other risk-taking behaviors. Causal and developmental aspects of the regulatory processes involved in gambling decisions and behavior should also be tested both in the lab and in naturalistic settings.  
In conclusion, the capacity to have and act on intuitive cognitive processes (Myers, 2003) is as much a human asset as is the capacity to have and act on rational processes (Stanovich & West, 2000). The point of the present research was to demonstrate that these capacities best operate when the two processes are competently regulated, such that products of each process are veridically represented and the influence of either process can be inhibited from affecting decision-making and behavior.  Veridical metacognitive awareness and effective inhibitory control skills were key elements defining competent dual process regulation.  The general finding was that otherwise capable college students’ lack of metacognitive awareness and poor inhibitory control makes them vulnerable to forms of gambling-related irrational decision-making and behavior.  A predilection for games of chance or a tendency to make irrational judgments may not themselves indicate vulnerability unless it is known whether the judgments and predilections arise from poor regulatory skills. 

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Table 1  
Study 1: Gambling-Related Decisions and Behaviors by Regulatory Group 
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Gambling-						    Regulatory Styles
Related	 	____________________________________________________________
Decisions			Flawed			        Conflicted	        Competent
and Behaviors		(N=53)				(N=59)			(N=59)
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Pay for Tray?			36%				17%			 8%
Engage in Gambling?
Formal			26%				25%			12%
Informal		59%				36%			28%
Composite Gambling 
Score (0-2) 			.85				.61			.41
Ratio Bias 
Large Tray		62%				48%			54%	
Small Tray		38%				52%			46%
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Table 2:  
Study 2: Distribution of choices on the Job and Jar tasks
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						Ratio-Bias Judgment Task 
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Task					1/10			10/100		no preference
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Jar (Lottery)				26			36			40
Job (Employment)			47			21			29
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Table 3
Study 2: Distribution of Regulatory Styles by Task
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						   	Regulatory Style
	____________________________________________________________
Task				Other		Flawed			Conflicted	      Competent
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Jar (Lottery)			17		      22			     26			37
Job (Employment)		12		      34			     27			24
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Figure 1:  Mean rational certainty judgments as a function of tray choice for each response option, Study 1.  

Figure 2:  Mean rational certainty judgments as a function of tray choice for each response option, Study 2.