Quarterly Journal of Experimental Psychology The consistency of the subjective concept of randomness Journal: Quarterly Journal of Experimental Psychology Manuscript ID Draft Manuscript Type: Standard Article Date Submitted by the Author: n/a Complete List of Authors: Yu, Ru Qi; University of British Columbia, Department of Psychology Gunn, Jordan; Princeton University Osherson, Dan; Princeton University, Zhao, Jiaying; University of British Columbia, Department of Psychology Keywords: randomness, modalities, alternation bias, anchoring effect, features URL: http:/mc.manuscriptcentral.com/pqje Page 1 of 18 Quarterly Journal of Experimental Psychology RUNNING HEAD: CONSISTENCY OF RANDOMNESS 1 1 2 3 4 5 6 7 8 9 10 11 12 The consistency of the subjective concept of randomness 13 14 15 Ru Qi Yu 1, Jordan Gunn 2, Daniel Osherson 2, and Jiaying Zhao 1,3 16 17 18 1 19 Department of Psychology, University of British Columbia 20 2Department of Psychology, Princeton University 21 3Institute for Resources, Environment and Sustainability, University of British Columbia 22 23 24 25 26 27 28 Manuscript type: Standard article 29 Main text word count: 4439 30 31 Abstract word count: 197/200 32 33 34 35 36 37 38 Please address correspondence to: 39 40 Jiaying Zhao 41 Department of Psychology 42 Institute for Resources, Environment and Sustainability 43 44 University of British Columbia 45 Vancouver, B.C., Canada, V6T 1Z4 46 Telephone: (1)-604-827-2203 47 Fax: (1)-604-827-2203 48 Email: [email protected] 49 50 51 52 53 54 55 56 57 58 59 60 URL: http:/mc.manuscriptcentral.com/pqje Quarterly Journal of Experimental Psychology Page 2 of 18 2 1 2 CONSISTENCY OF RANDOMNESS 3 Abstract 4 5 6 A pervasive bias in the subjective concept of randomness is that people often expect random 7 8 sequences to exhibit more alternations than produced by genuine random processes. Although 9 10 11 this over-alternation bias has received much attention, what is less known is the stability of the 12 13 bias. Here we examine two important aspects of the over-alternation bias: first, whether this bias 14 15 is present in stimuli that vary across feature dimensions, sensory modalities, presentation modes, 16 17 18 and probing methods; and second, how consistent the bias is across these stimulus variations. In 19 20 Experiment 1, participants adjusted sequences until they looked maximally random. The 21 22 sequences were presented as temporal streams of colors, shapes, auditory tones, or tiled as spatial 23 24 25 matrices. In Experiment 2, participants produced random matrices by adjusting the color of each 26 27 cell. We found that participants judged and produced over-alternating stimuli as the most 28 29 random. Importantly, this bias was consistent across presentation modes (temporal vs. spatial), 30 31 32 feature dimensions (color vs. shape), sensory modalities (visual vs. auditory), speed (fast vs. 33 34 slow), stimulus size (small vs. large matrices), and probing methods (adjusting the generating 35 36 37 process vs. individual bits). Overall, the results suggest that the subjective concept of 38 39 randomness is highly stable across stimulus variations. 40 41 42 43 44 Keywords : randomness, modalities, alternation bias, anchoring effect, features 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 URL: http:/mc.manuscriptcentral.com/pqje Page 3 of 18 Quarterly Journal of Experimental Psychology 3 1 2 CONSISTENCY OF RANDOMNESS 3 Introduction 4 5 6 In common parlance the term “random” is applied to sequences of events that appear 7 8 sufficiently disorderly or unstructured. For example, the string hthhtthtttht of heads and tails 9 10 11 from coin tosses might qualify as random, whereas hhhhhhtttttt would not. A contrasting usage, 12 13 adopted here, applies the term to certain mechanisms for generating events, namely, whose 14 15 successive outputs are independent and unbiased. A standard example is a device D that tosses a 16 17 18 fair coin repeatedly (ignoring issues about predictability assuming all forces were known). Any 19 20 sequence of heads and tails produced by such a device is qualified as “randomly generated (by 21 22 D)” regardless of its pattern. Thus, the terminology in this paper follows a “process” rather than 23 24 25 “product” conception of randomness (see Eagle, 2014; Earman, 1986 for an extended discussion 26 27 between the two approaches). Specifically, in our usage a “random” stimulus (or pattern) is an 28 29 object that has been produced by a random process. Non-random stimuli are defined as 30 31 32 productions from a distorted random source. 33 34 The randomness of the coin flipper D can be compromised in various ways, for example, 35 36 37 by making one of its two outcomes more likely than the other. Instead of introducing a bias to 38 39 the probabilities of the two outcomes, here we consider deviations from stochastic independence 40 41 by allowing previous flips to influence the next one, while maintaining the equal frequency of 42 43 44 the two outcomes. This allows us to identify the lay conception of randomness along a 45 46 continuum. Specifically, for each number x in the unit interval (from 0 to 1), let D(x) generate a 47 48 sequence of bits consisting of zeros and ones as follows: 49 50 51 52 53 54 55 56 57 58 59 60 URL: http:/mc.manuscriptcentral.com/pqje Quarterly Journal of Experimental Psychology Page 4 of 18 4 1 2 CONSISTENCY OF RANDOMNESS 3 (1) Sequence generation using the device D(x): An unbiased coin toss determines the first bit. 4 5 th 6 Suppose that the n bit has been constructed (for n ≥ 1). Then with probability x the n + 7 8 1st bit is set equal to the opposite of the n th bit; with probability 1 − x the n + 1 st bit is set 9 10 th 11 equal to the n bit. Repeat this process to generate a sequence of any length. 12 13 14 15 16 This procedure was introduced by Zhao, Hahn, and Osherson (2014). It can be seen that 17 18 D(.5) is a genuinely random device. For x < .5, D(x) tends to repeat itself, resulting in long runs 19 20 whereas for x > .5, D(x) tends to alternate. In particular, D(0) is uniform, either 0000… or 21 22 23 1111…, while D(1) consists of perfectly alternating bits, either 0101… or 1010…. The expected 24 25 proportion of each bit is 50%, for all x ∈ (0, 1). For any sequence produced by D(x), the 26 27 28 expected proportion of alternation — called the “switch rate” — is x. The switch rate of any 29 30 sequence is calculated by the number of switches between two successive bits divided by the 31 32 total number of bits in the sequence minus one. For example, the switch rate of the sequence of 33 34 35 11111 or 00000 is 0, and the switch rate of 01010 is 1. 36 37 A large body of research on randomness perception has revealed a pervasive bias in 38 39 40 people’s concept of “random.” That is, people often expect random sequences to exhibit shorter 41 42 runs (greater alternation) than typically produced by random devices, or in our terms, x > .5 43 44 (Kahneman & Tversky, 1972, Falk & Konold, 1997; Lopes & Oden, 1987; Nickerson & Butler, 45 46 47 2009; Wagenaar, 1972; for reviews on theoretical analysis of major experimental findings see 48 49 Bar-Hillel & Wagenaar, 1991; Oskarsson et al., 2009). 50 51 The over-alternation bias has been typically examined using judgment or production 52 53 54 paradigms (e.g., Kahneman & Tversky, 1972; Wagenaar, 1972), where participants either judged 55 56 how random a sequence appeared, or produced a sequence as if it were generated by a random 57 58 59 60 URL: http:/mc.manuscriptcentral.com/pqje Page 5 of 18 Quarterly Journal of Experimental Psychology 5 1 2 CONSISTENCY OF RANDOMNESS 3 process. The over-alternation bias explains the gambler’s fallacy (Kahneman & Tversky, 1972; 4 5 6 Reuter et al., 2005; Wagenaar, 1988), and can be driven by limitations in working memory 7 8 (Baddeley, 1966; Hahn & Warren, 2009; Kareev, 1992). Despite the overwhelming evidence for 9 10 11 the over-alternation bias, what has been given little attention is whether this bias is consistent 12 13 across sensory modalities, stimulus feature dimensions, presentation modes, and tasks. 14 15 The primary goal of the current study is thus to examine the concept of randomness 16 17 18 across multiple domains. This will offer insights on the stability or consistency of the over- 19 20 alternation bias. In Experiment 1, participants observed a sequence of bits produced by D(x) for 21 22 as long as they wanted. They were asked to adjust the sequence by increasing the alternations or 23 24 25 the repetitions of the bits, in order to make the sequence look maximally random. In other words, 26 27 they manipulated the switch rate x until they were satisfied that the sequence was being 28 29 generated “randomly.” We then measured how close x was to .5 (genuine randomness). Of 30 31 32 principal interest was the stability of the switch rate across variations in the stimuli that represent 33 34 the bits generated by D(x). In Experiment 2, we examined the stability of the switch rate in a 35 36 37 randomness production task. The two experiments investigated the consistency of people’s 38 39 randomness concept across stimulus features, modalities, presentations, and tasks. A summary of 40 41 the stimulus conditions was shown in Table 1 below.