Results of Study 1 provide first evidence for the operation of principles of goal pursuit in chance situations. Study 2 aimed to experimentally replicate the goal-gradient effect and to investigate the moderating role of expectancy of success. We asked participants to play a dice game in which winning a prize was purely chance-determined. While throwing the dice we monitored their hand movements. We predicted that participants would invest an increasing amount of effort (i.e., faster and/or bigger hand movements, longer throwing times) during the dice throw the closer they got to the end of the game. Moreover, we expected this gradient pattern to be stronger for participants who, towards the end of the game, had higher expectations of winning the prize.
Method
Participants and design
Forty-nine students (35 women, two participants did not indicate their gender) took part in this study.Footnote 3 All participants played a dice-game in which they were asked to throw the dice ten times to reach the highest possible sum score.Footnote 4 The prize for throwing the highest sum score was a popular mobile digital audio player.
Materials and procedure
Participants provided informed consent and were introduced to the dice-game and the possibility of winning a prize. The game started with two practice throws.
During the entire game participants’ hand movements were recorded using a 3D position tracker (Polhemus Isotrak II). This tracker uses electro-magnetic fields to monitor and record the relative position of a remote sensor (attached to participants’ middle finger of their dominant hand) in the x, y and z direction at a rate of 60 samples per second. Each throw was introduced with an on screen instruction (“Please throw the dice”), and had a limited response window of 4000 ms, before and after which no data were recorded. Participants were told this sensor monitored their pulse and were asked to keep their hands still after throwing the dice. From the position data, we extracted three types of information per throw. First, the maximum velocity of each throw (in cm/s). Second, the duration of each throw (in s). And last, the largest hand movement the participant made during the throw (i.e., the maximum size of the throw, in cm).
To calculate the maximum velocity, we first calculated the velocity per direction per measurement sample (difference in position between two samples divided by the time separating them; Δp(XN, XN−1)/Δt(XN, XN−1). Then, we merged the velocity estimates across the three directions per sample [sqrt((vX)2 + (vY)2 + (vZ)2)], and selected the maximum value. The duration of a throw was defined as the total amount of time the hand was moving. To estimate that, we first calculated the standard deviation (SD) of the velocity of each individual throw (based on the merged estimates). All samples in which the velocity exceeded the baseline (velocity = 0) by one SD were accumulated and converted into a temporal estimate by multiplying it with time per sample (1 s/60 samples). Finally, to determine the maximum size of each throw, we computed all distances between all positions (240 samples) using the 3D Pythagoras formula, sqrt((Xi − Xj)2 + (Yi − Yj)2 + (Zi − Zj)2), and selected the largest of all distances.
The outcome of each throw was recorded by the experimenter. After ten throws participants learned about their final score, and were asked to rate the likelihood of winning the prize (1 = very unlikely, 9 = very likely). Finally they were asked to provide some demographic information, were debriefed and dismissed.
Results and discussion
We excluded one participant who did not adhere to the procedure. The remaining 48 participants (M
age = 22.72, SD = 6.05; 35 women) were included in the analyses. As in Study 1, we tested the goal-gradient hypothesis by zooming in on the last three throws (8, 9, 10). If motivation increased as a function of the distance to the goal it should be most visible towards the end of the game.
Maximum velocity
We first tested whether the velocity with which participants performed the throw increased across the last three throws. A repeated measures ANOVA revealed no effect, F(1.60, 79.48) = 2.17, p = .129, η
2p
= .04 (Greenhouse-Geisser correction for non-sphericity). Simple contrasts indicated that the maximal velocity in the last, 10th throw (M = 102.09, SD = 57.55) was not significantly larger than in the 9th throw (M = 94.36, SD = 48.04, p = .102, η
2p
= .06) or in the 8th throw (M = 92.16, SD = 50.83, p = .103, η
2p
= .06). To test the moderating role of expectancies, we included their sum score after the 8th throw as an additional continuous factor. That sum-score represented a relatively objective measure of their expectation of winning and was related to their subjective expectancy (assessed after the game, r = .37, p = .010). Neither this analysis, nor a median split (at 28) analysis on the high and low expectation group separately, revealed any effects.
Duration
We then tested whether the duration of participants’ throws increased across the last three throws. Another repeated measures ANOVA revealed a main effect of the repeated factor, F(2, 94) = 6.32, p = .003, η
2p
= .12. Participants spent more time throwing the last dice (M = 1.14, SD = 0.46) compared to the 9th dice (M = 0.96, SD = 0.44, p = .006, η
2p
= .15) and to the 8th dice (M = 0.96, SD = 0.44, p = .003, η
2p
= .17). Adding the sum score after the 8th throw as a continuous factor only showed an additional main effect of score, F(1, 44) = 6.56, p = .014, η
2p
= .13. The higher participants sum score after the 8th throw, the longer their average duration of the last three throws, r = .36, p = .014. When splitting the group on the median sum score and running the analysis for both groups separately, only the group that had high expectations of winning shows the main effect of the repeated factor (p = .014, η
2p
= .16; low expectation group p = .202).
Maximum size
Finally, we tested whether participants’ hands traversed larger distances the closer they got to the end of the game. We ran a similar repeated measures ANOVA and obtained a significant effect of the repeated factor, F(1.62, 76.33) = 10.63, p < .001, η
2p
= .18 (Greenhouse-Geisser correction for non-sphericity). Simple contrasts showed that the maximum size of hand movements was larger in the last (M = 20.33, SD = 13.25) compared to the 9th throw (M = 15.74, SD = 11.16, p = .002, η
2p
= .18), and compared to the 8th throw (M = 15.06, SD = 9.75, p < .001, η
2p
= .25). Adding their sum score after the 8th throw as an additional continuous factor, or splitting the group in high and low expectation subsamples, did not affect their pattern of throwing behavior.
Exploratory analyses
We computed correlations between participants’ hand movements and their explicit chance ratings in order to test whether increased effort investment would lead to increased illusory control. Results suggest that this is not the case. Instead, their explicit chance ratings were largely related to their actual final sum score (r = .39, p = .006).
Finally, we wanted to see whether participants’ level of superstitious beliefs or desire for control could explain the obtained goal-gradient effect. For that purpose, participants filled in two individual difference measures at the very beginning of the experimental session. The first measure was the superstitious belief questionnaire (adapted from Wiseman and Watt 2004). Participants rated 14 statements (e.g., “I believe that the number 13 is unlucky”; α = .84) on Likert-type scale ranging from 1 (completely not true) to 7 (completely true). The second measure was the Desirability of Control scale (translated into Dutch, adapted from Burger and Cooper 1979); participants rated 17 statements (e.g., “I enjoy making my own decisions”; α = .48) on the same scale. Scores were averaged, with higher scores indicating stronger superstitious beliefs and desire for control.
Scores on the superstitious beliefs questionnaire indicated that our sample did not hold very strong superstitious beliefs (M = 2.00, SD = 0.83). Moreover, superstitious beliefs were no moderator in any of the reported results. Due to the low reliability of the second scale, we did not include it in the analysis.