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On an 8:1 bet, you bet in the intersections of every four numbers located in the middle of the board (for example: the intersection of 8, 9, 11, and 12). If the number that appears on the wheel matches a bet that you have made, you will win eight times your chips) bet. For example, if you bet 2 chips on the space between "8, 9, 11, & 12," and the number that comes up is either 8, 9, 11, or 12, then you will receive a total of 18 chips in return. The two you bet, and the 16 additional you won.

Keep in mind that you can bet any number of chips that you wish on a given game. Additionally, at times, you will be allowed to select the numbers where those chips are to be placed. However, at other times, I (the experimenter) will select the numbers where those chips are to be placed. At this time, simply hand me the number of your chips you would like me to wager. I will tell you who will select the numbers before each game so you know ahead of time.

As you might see, no bets can be made on the 0 or the 00 spaces. Therefore, if either of these numbers appear on the wheel, all chips bet on this game will be lost.

Once the experimenter completed these instructions and the subject had no further questions, they were told the following:

This is a fair roulette wheel. It has not been modified in anyway from those found on the gambling riverboats here in town. You will be given 25 chips to start with. Each chip is worth 1 /10 of an extra credit point that will be added to your final course grade. In other words, you are starting off with 2.5 extra credit points with which to play the game. The points earned by the end of your participation will be added to your final course grade. Your task is to win as many chips as possible. You are required to make a bet on every trial of the game, and you can quit whenever you want after the first trial. However, if you quit you must remain in the room until all other players have also quit. Lastly, you are not allowed to discuss any aspect of the game including betting strategies or wins and losses to any other player in the room. Any discussion of the game is not allowed until the entire experiment has been completed.

Research Design

A multiple-baseline across subjects design was used whereby subjects followed by two independent treatment manipulations (i.e., inaccurate rules and then accurate rules) with an alternating treatment design (control over chip placement/no control over chip placement) superimposed across all conditions. The no-rules condition served as a baseline in which to assess the potential influence of rules on subsequent behavior. The alternating treatment component served as a means of assessing relative number of chips bet when the hypothesized "illusion of control" was thought to be present (i.e., control over chip placement).

Session 1: No-rules baseline followed by delayed inaccurate-rules delivery. At the onset of the experiment, subjects played the game without any additional instructions (i.e., no rules) regarding betting strategies, probabilities of winning, or quitting the game. Before each game began, subjects were told if either they or the experimenter would be responsible for chip placement for that game. These two game trial-types alternated repeatedly throughout the experiment approximately every third trial.

Following a varying number of no-rules trials, subjects were given a set of three rules that they were told were "Roulette Tips." Subjects were given these rules at different trials in a multiplebaseline fashion, and they were told to read them to themselves and not to let the other players see them. The three inaccurate rules were:

1. People that win more pick their own numbers.

2. If you want to win big, you have to bet big.

3. 1 will try and pick numbers that will make you lose.

All subjects remained in the experimental room until the last subject terminated the game and cashed in remaining chips for extra-credit points. This was done to eliminate the potential competing contingency of escaping the experimental environment.

Postsession 1: Subjective probability estimations of winning. One week following Session 1, all subjects were asked to independently answer a series of questions regarding their roulette play from the preceding week. The questions were:

1. How many chips did you win when you were able to select the numbers that the chips were bet on?

2. How many chips did you win when the dealer selected the numbers that the chips were bet on?

3. If someone who wanted to learn about roulette asked you what the probability of winning on an 8:1 bet was if they were allowed to select their own numbers, what would that be (in a percentage)?

4. If someone who wanted to learn about roulette asked you what the probability of winning on an 8:1 bet was if someone else was allowed to select their numbers, what would that be (in a percentage)?

Session 2: Accurate rules. Two weeks following Session 1, the 5 subjects were asked to return to the experimental room for an additional session. Before this request, no subject knew that the experiment involved a second session. At this time, all subjects were given 25 additional chips with which to bet. No winnings from the previous session could be wagered. Before the first trial, all subjects were given the following accurate rules:

1. It does not make a difference if I pick the numbers or if you pick the numbers.

2. If you want to win big, you do not have to bet big.

3. 1 can not pick numbers that make you lose.

All other aspects of the procedure were identical to those of Session 1.

Postsession 2: Subjective probability estimations of winning. One week following Session 2, all subjects were again asked to answer the same series of questions regarding their roulette play from the preceding week. Answers to the first two questions were additionally instructed to be based on only the previous week's winnings (Session 2).

Overall follow-up: Subjective probability estimations of winnings. Upon completion of Postsession 2 questions, subjects were asked the following two final questions:

1. Think back on both times you played roulette here. How many total chips did you win when you were able to select the numbers that the chips were bet on?

2. Think back on both times you played roulette here. How many total chips did you win when the dealer selected the numbers that the chips were bet on?

Results

Observed Behavior

Figure 1 displays each subject's cumulative wagering and number of chips remaining across trials after delivery of no rules, inaccurate rules, and accurate rules throughout the alternating treatment conditions of the experiment. During no-rules trials, 4 of the 5 subjects (1, 3, 4, and 5) wagered more chips when they controlled chip placement compared to when they had no control over chip placement. Following these initial trials, subjects were exposed independently to inaccurate rules related to play in a multiple baseline fashion. Once again, 4 of the 5 subjects (1, 2, 3, and 5) wagered more chips when they controlled placement, and fewer chips when the experimenter controlled placement, yet this difference was more substantial than before rule delivery.

Upon returning to the experiment for a second session, subjects were given a new set of 25 chips and accurate rules regarding play. Subsequent wagering differed considerably from the previous session. All subjects demonstrated a smaller difference between number of chips bet when they had control over placement than when the experimenter had control over placement. Of the 4 subjects, 3 (1, 2, and 3) reduced this difference below their initial pre-rule values. No comparison could be made for Subject 5 who lost all her chips before conditions were alternated to when the experimenter controlled chip placement.

Table 1 displays least-square estimation equations of each subject's wagering behavior. Six equations were calculated for each subject to support visual inspection of the Figure 1 wagering differences between alternating treatment design (ATD) conditions, as well as pre- and post-- rule delivery. These numerical data show that subjects had higher slopes of the line (the numeric value before the x) during games when the subject controlled chip placement, than during games when the experimenter controlled chip placement. For the purpose of the present study, the "slope of the line" indicates the mean increase in chips wagered for each successive trial/game played. Differences in wagering across control over chip placement conditions were also increased following inaccurate-rules delivery and decreased following accurate-rules delivery as stated above.

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At the time of inaccurate-rules delivery, all 5 subjects had won an equal (S4) or a greater (S1, S2, S3, and S5) number of chips when the experimenter had control over chip placement than when they themselves did. Yet, no subject's wagering behavior (as measured by changes in slope of the line) subsequently increased during post-rule trials when the experimenter controlled chip placement. In contrast, following inaccurate-- rules delivery, Subject 3 and Subject 5 showed decreases in wagering during the condition when the experimenter placed the chips and increases in wagering when they themselves placed the chips. Subject 2 showed no change in wagering during experimenter control conditions, yet increases in wagering in subject control conditions. Subject 4 showed a decrease in wagering during experimenter control conditions although no increase in subject control conditions. Only Subject 1 displayed no changes in wagering in either condition. Together, these data appear to illustrate a greater control of the inaccurate rules over the contacted reinforcing contingencies on subsequent wagering behavior for these subjects.

Following accurate-rules delivery, 4 of the 5 subjects (S1, S2, S3, and S5) showed differing degrees of decreases in their wagering. Subject 2 and Subject 3 showed a decrease in wagering both during experimenter and subject chip placement conditions. Subject 1 and Subject 5 showed no difference in wagering during experimenter placement conditions (n/a for S5), yet a decrease in wagering during subject placement conditions. Lastly, Subject 4 appeared to show an unexpected increase in wagering during both conditions.

Although not instructed in any direct way, the number of trials each subject played the game differed across rule conditions. Figure 2 also displays the number of trials each subject played during the first (no rules and inaccurate rules) and second (accurate rules) experimental sessions. All 5 subjects played for a greater number of trials during the first session compared to the second session. Because of this trial reduction, 4 of the 5 subjects (Si-S4) were able to cash in their remaining chips for more course extra-credit after the second session than they were able to do in the previous session. This was not true for Subject 5 who lost all her chips by the third trial during the second session.

Subjective Probability Estimations of Winning

Table 2 displays each subject's responses to the questions of estimating winnings that were asked on alternating weeks of actual roulette play. Actual winnings are also displayed in adjacent columns for comparison purposes. These data show that for estimates of winnings during the first experimental session, most subjects tended to overestimate their winnings when they had control over chip placement and underestimate their winnings when the experimenter had control over chip placement. This same pattern was not observed when the subjects' estimated winnings occurred during the second experimental session. At this point, subjects were better able to correctly estimate their winnings during both betting conditions. Yet, when subjects were asked the overall follow-up question of estimating winnings for both sessions combined, a similar pattern to that observed after the first experimental session emerged. That is, the majority of subjects tended to overestimate their winnings when they had control over chip placement and to underestimate their winnings when the experimenter had control over chip placement.

Figure 3 displays each subject's responses to the questions of subjectively estimating the probability of winning on a bet if the experimenter choose the numbers and if the subjects were allowed to choose their own numbers. Numbers in parentheses located above each subject's data indicate that subject's cumulative winnings for that session. The actual objective probability of winning on the type of bets all subjects made throughout the experiment was 11%

The top graph of this figure displays data collected 1 week following the first experimental session. Of the 5 subjects, 3 (S1, S3, and S5) made subjective probability estimations that differed in values across the two chip placement conditions, in the direction that odds of winning were greater when they controlled chip placement. The other 2 subjects estimated the probability of winning on either type of trial as being equal, which in fact it was. The subjective probability values) themselves tend to be larger for subjects who obtained greater number of chips during that session.

The bottom graph of this figure displays data collected 1 week following the second experimental session. All 5 subjects now emitted subjective probability estimations that were of equal values across the two chip placement conditions. Additionally, 4 subjects' values (S2-S5) and obtained winnings decreased from their earlier estimations. The exception to this trend was Subject 1. She was the only subject whose subjective probability estimations and obtained winnings rose from the earlier session.

Discussion

Results from the present study appear to demonstrate an influence of instructions, or rules, over a subject's gambling behavior. Previous research has shown that a subject's exposure to rules describing an experimental task tend to influence subsequent behavior on the part of that subject, even in the face of opposing programmed contingencies (e.g., Hayes, Brownstein, Zettle, et al., 1986; LeFrancois et al., 1988). The present data support these findings and further suggest that gambling may in part be under the control of rules. In the present experiment, rulegoverned control was measured by changes in the number of chips a subject wagered on each bet and by the number of optional trials the subject played before quitting. Wagering behavior also tended to be more under control of such rules than contacted reinforcing or punishing consequences specific to each subject's history with the game. Similar results demonstrating the predominate control of rules and the lack of control of programmed consequences were obtained by Dixon et al. (2000) who investigated the variables involved in levels of risk taking and trials to termination of the game of roulette.

After collecting baseline data on the wagering of each subject, different types of instructions were presented to each subject and subsequent wagering was assessed. The ability to demonstrate delayed instructional control over behavior currently contacting programmed contingencies has not previously been reported in the published literature. The present findings suggest that delayed rule control over behavior is possible, and therefore the conclusions drawn from previous research utilizing instructions presented at the onset of an experiment may be applicable to situations where a behavioral history has already been somewhat established in the absence of such rules.

In the present study, three changes in behavior were observed to be under instructional control. First, the number of trials played by each subject decreased following accurate-rules delivery compared to inaccurate-rules delivery. Interestingly, no rules delivered to the subject actually specified when to quit the game. Direct instructions have been shown to influence the behavior it specifies (e.g., Catania et al., 1982; Shimoff et al., 1986), yet the present data suggest that responding may have also been controlled by other variables. Perhaps subjects also were covertly deriving and following a type of self-rule related to those delivered by the experimenter. Future research utilizing talk-aloud methodologies to investigate concurrent verbal activity of the subject (see Hayes, 1986, for a review), or self-reports of the subject (e.g., Critchfield & Perone, 1990) may be warranted.

Second, subjects' levels of risk-taking varied as a function of both the person controlling chip placement (i.e., experimenter or subject), as well as the rule condition. Most subjects wagered more chips when they controlled chip placement than when the experimenter controlled chip placement. These findings support the interdisciplinary research literature on the "illusion of control" (see Presson & Denassi, 1996, for a review), yet are in contrast to the majority of research findings suggesting that the "illusion" is an internal state or characteristic of the individual. The present data suggest that the degree of "illusion" can be significantly influenced by external instructions delivered to the subject.

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A behavioral conceptualization of the "illusion of control" may be that of a self-rule following history a subject brought to bear in the experimental situation. This self-rule was then followed in the absence of any information that makes it inaccurate, subsequently strengthened when the inaccurate rules were presented, and later weakened when the accurate rules were presented. The ability to experimentally manipulate a hypothesized internal state suggests that rule-governed behavioral interventions designed to reduce "illusionary control" may be warranted. As policy makers and the community at-large begin to understand the impact of these inaccurate rules on a person's gambling, the abundance of sources (i.e., billboards, TV commercials) delivering inaccurate rules will hopefully be matched or overshadowed by sources delivering accurate rules.

Third, subjects' self-reports varied as a function of the person controlling chip placement and/or rule condition. In general, subjects tended to overestimate their own wins and underestimate the experimenter's wins from previous experimental sessions. These data suggest that a win following the subject's placement of chips was more reinforcing than a win following the experimenter's placement of chips, by virtue of its readiness of recall.

After inaccurate-rules exposure, subjects' subjective probability estimations of winning on 8:1 bets were overestimates of the objective probability. When these estimates were not equal across the two chip placement conditions, they were greater for self-placement. However, after accurate-rules exposure, all subjects' subjective probability estimations more closely approximated the objective probability. Additionally, all subjects previously reporting unequal probability estimations, at this time reported equal estimations. This change may have been caused in part by decreases in overall reinforcement across the two experimental sessions. Yet, if subjects were simply making such estimations based on the last session's history of reinforcement, one would expect to also see different estimated values following the second session because the experimenter never won any chips for any subject. Therefore, the present data suggest that subjective probability estimations are developed through an interaction of several environmental variables.

Of broader importance, the present data suggest that a within-individual analysis of rule-governed behavior is possible. Such an analysis has to date been absent from the behavioral literature. Implications from such an approach are potentially vast. Presenting subjects having previous histories of following certain types of rules (i.e., accuracy, specificity) with subsequent differing types of rules more approximately resembles the way many interventions in the natural world are attempted.

For example, one may work with individuals with risky sexual behavior who are operating under the control of inaccurate rules and believe that they need not worry about developing AIDS or getting pregnant. Behavioral therapists may present these individuals with more accurate rules regarding the chances of these undesired events occurring, and then monitor subsequent sexual behavior. Because actual contact with the negative consequences of not following these actual rules (i.e., dying, having an unwanted child) are disastrous, it is of applied importance to understand fully the potential variables influencing the effectiveness of instructional control (see Catania, 1998; Hayes, 1987, for a complete review of these variables).

Because all subjects played in a group rather than independently, it is possible that observational behavior and/or social influences were potential extraneous variables contributing to subjects' wagering behavior. Yet, by virtue of the set of rules being delivered across subjects in a multiple-baseline fashion, it only allowed certain subjects to be exposed to a set of rules on a given trial number. As a result, differences in wagering behavior did not occur until after rule delivery to that specific subject, suggesting that behavior was under control of the rules and not any potential observational or social influences. Yet, future studies may wish to examine extensively the role of observational behavior in gambling by constructing isolated subject control conditions.

In conclusion, gambling requires an examination of as many potential participating variables as possible. Attempting to explain gambling, as behavior under the control of a single variable such as the schedule of reinforcement, appears to be incomplete. Rather, it appears that externally available rules have influence over the maintenance and degree of superstitious risk taking, and postsession subjective probability estimations of wins and losses. The present results are an initial step toward predicting the conditions under which people gamble.

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MARK R. DIXON

Southern Illinois University

Address all correspondence to Mark R. Dixon, Ph.D., Behavior Analysis and Therapy Program, Rehabilitation Institute, Southern Illinois University, Carbondale, IL 62901. (E-mail: mdixon@siu.edu). Data collection for this study occurred while the author was affiliated with Trinity Services, Inc.

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