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“ The most exciting phrase to hear in science, the one

“ The most exciting phrase to hear in science, the one that heralds new discoveries, is not ‘Eureka!’ (‘I found it!’) but rather ‘hmm.... That’s funny.’ ” Isaac Asimov. Working Memory Constraints and Biases in Sequential Binary Decision-Making Tasks.

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“ The most exciting phrase to hear in science, the one

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  1. “The most exciting phrase to hear in science, the one that heralds new discoveries, is not ‘Eureka!’ (‘I found it!’) but rather ‘hmm.... That’s funny.’ ” Isaac Asimov

  2. Working Memory Constraints and Biases in Sequential Binary Decision-Making Tasks Alternation bias as a consequence of the small sample (working memory constrained) properties of autocorrelation Kim Kaivanto Department of Economics Lancaster University Management School Encoding of the Past in Economic Agents and Institutions The Scottish Institute for Advanced Studies Friday 9 January 2009 11:30−12:00

  3. Motivation Why is the perception of randomness important?  The basis for subjective prob. reasoning, induction, inference, etc. Why binary sequences?  Basic, fundamental, simplest case. Many problems, evolutionarily important and current, either inherently have a binary structure or can be simplified/reduced to a binary structure. What is alternation bias and why is it important?  ‘Alternation bias’ is a robust and widely-replicated finding...To be described in greater detail…  The – at least ‘an’ – accepted explanation for • the gambler’s fallacy, the ‘hot hand’, overattribution, … • implications for: the disposition effect, escalation of commitment, the St. Petersburg Paradox, …

  4. The idea The individual experiences or ‘sees’ random sequences through the narrow window of working memory. Typical individuals do not store the full history of random sequence in episodic (long term) memory. Instead, let us conjecture that, individuals typically store their subjec- tive characterisations of random sequences in semantic (long term) memory – specifically, a subjective representation/analogue of ‘autocorrelation’.  The ‘wet ware’ analogue of data compression. These subjective autocorrelation characterisations are formed in working memory, and are thereby constrained. The small sample (working memory span) sampling distribution of ρ1, ρ2, ρ3,…become central to perception of randomness. Negative bias in these sampling distributions explains alternation bias.

  5. Alternation Bias Sequences (a) and (b) were each generated by a randomisation device designed to generate an i.i.d. random sequence P(H)=P(T)=½ and P(T|T)=P(H|T)=P(H|H)=P(T|H)=½. (a)HHHTTHHTHHTTHTHHTTTTH (b)HHTTHTHTHHHTTHTTTHHTH One of the two devices is known to operate properly, while the other is known to be defective. Q. Which of the two series is more likely to have been generated by the defective device?

  6. Alternation Bias Sequences (a) and (b) were each generated by a randomisation device designed to generate an i.i.d. random sequence P(H)=P(T)=½ and P(T|T)=P(H|T)=P(H|H)=P(T|H)=½. (a)HHHTTHHTHHTTHTHHTTTTH (b)HHTTHTHTHHHTTHTTTHHTH One of the two devices is known to operate properly, while the other is known to be defective. Q. Which of the two series is more likely to have been generated by the defective device? People tend tochoose (a)

  7. Alternation Bias • Sequences (a) and (b) were each generated by a randomisation • device designed to generate an i.i.d. random sequence • P(H)=P(T)=½ and P(T|T)=P(H|T)=P(H|H)=P(T|H)=½. • (a)HHHTTHHTHHTTHTHHTTTTH • (b)HHTTHTHTHHHTTHTTTHHTH • One of the two devices is known to operate properly, while the • other is known to be defective. • Q. Which of the two series is more likely to have been generated • by the defective device? • P(T|T)=P(H|T)=P(H|H)=P(T|H)=½ • P(T|H)=P(H|T)=.6, P(T|T)=P(H|H)=.4 • People consistently judge series with alteration rates of .6 to be “random”, • whereas an alteration rate of .5 produces runs too long to be thought of • as “random”. People tend tochoose (a) When actually (b) is ‘defective’

  8. Alternation Bias For higher-order effects, consider the conditional probabilities embodying ‘alternation bias’ as reported by Rabin (2002) QJE: P(T|T) = 0.42 so P(H|T) = 0.58 P(T|TT) = 0.38 so P(H|TT) = 0.62 P(T|TTT...) = 0.30 so P(H|TTT…) = 0.70 Thus the literature supplies the following alternation bias estimates: First order Higher order P(H|T) P(H|TT) P(H|TTT) P(H|T) .6 .58 .62 .70 n.b. These estimates are elicited without invoking monetary payoffs. Hence, there can be no confounding of (i) conditional probability distortion with (ii) outcome value weighting.

  9. Alternation Bias Wagenaar (1970) Acta Psychologica; (1972) Psychological Bulletin • “The results showed that sequences with conditional probabilitiesof around .4 [and of alternation of .6] were judged to be most random” Bar-Hillel & Wagenaar (1991) Advances in Applied Mathematics • “Peoples’ perception of randomness is biased in that they see clumps orstreaks in truly random series and expect more alternation, or shorter runs, than are there. Similarly, the produce series with higher than ex-pected alternation rates. …The subjectively ideal random sequenceobeys “local representativeness”; namely, in short segments of it, itrepresents both the relative frequencies (e.g., for a coin, 50%–50%)and the irregularity (avoidance of runs and other patterns).” • most explanations offered are mostly qualitative and ad hoc in nature

  10. Alternation Bias Some explanations: Ross & Levy (1958), Teraoka (1963) – faulty subjective concept of randomness Tune (1964), Baddeley (1966), Weiss (1964, 1965), Machado (1993) – functionallimitations of subjects, e.g. limited capacity of memory, limited attention memory,recall blocking/interference, forgetting, Falk (1981), Kahneman & Tversky (1972) – judgemental heuristic: “representa- tiveness bias” Neuringer (1986) – lack of skill (which can however be learned with appropriatefeedback training)

  11. Rabin (2002) ‘Inference by Believers in the Law of Small Numbers’, QJE, 117(3) R’02→ ‘the law of small numbers’ a.k.a. ‘the local representativeness effect’ is responsible for e.g. the Gambler’s Fallacy and belief in the ‘hot hand’. R’02→ uses an urn analogy (explain/illustrate!) R’02→ ‘overinference’ of e.g. fund manager skill, or of market price trends R’02→ “If a person believes every pair of flips of a fair coin generates one head and one tail,then he believes that two heads in a row indicates a biased coin. If he believes that an average fund manager is successful once every two years, then he believes that a fund manager who is successful two years in a row must be unusually good. I formalize this overinference result by showing that, after two signals, a believer in the law of small numbers always has on average more extreme beliefs than heshould.”

  12. Rapoport & Budescu (1997) Psychological Review People “produce sequences with too few symmetries and long runs and too many alternations among events. The authors propose a psychological theory to ac-count for these findings, which assumes that subjects generate non-random se-quences that locally represent theoretical random series subject to a constraint ontheir short-term memory. Closed-form expressions are then derived for the major statistics that have been used to test for deviations from randomness.”

  13. Limited working memory and ‘Narrow Window Theory’ Human working memory is limited to 7±2 items (e.g. Miller, 1956). However, it can be as low as 4 or as high as 10. Yaakov Kareev (1995, 1997ab, 2000, 2004, 2007): ‘Narrow Window Theory’ “The samples of data that people use in their attempts to detect relationships in the environment are limited in size by working memory capacity. …Pearson's rxy distribution is skewed when the population correlation differs from zero (i.e., when a correlation exists), and the more so, the smaller the sample. As both the median and the mode of the sampling distribution are more extreme than the population value, it follows that samples likely to be encountered indicate a correlation stronger than that in the population. The limited capacity of working memory may serve as an amplifier that helps people to avoid missing strong relationships. As the distribution is more skewed the smaller the sample size, the effect suggests an explanation for the fact that young children detect meaningful covariation fairly rapidly.” Yaakov Kareev (2000) Seven (Indeed, Plus or Minus Two) and the Detection of Correlations, Psychological Review, 107(2).

  14. What about the effect of limited sample size (working memory) on perception of (auto)correlation insequences? How does small-sample bias manifest for (i) fair and (ii) ρ1=0 binarysequences? Huitema & McKean (1991)

  15. What about the effect of limited sample size (working memory) on perception of (auto)correlation insequences? How does small-sample bias manifest for (i) fair and (ii) ρ1=0 binarysequences? + Binarysequences are basic, interesting & fundamental. → Is the empirical small-sample bias negative? → An explanation for Alternation Bias? → An explanation for the Gambler’s Fallacy, Hot Hand / Overattribution? + Explanations based on a documented, robust memory constraint, rather than an ‘urn analogy’ or mere assumption of ‘belief in the law of small numbers’ + Testable hypotheses concerning: • data compression via the narrow window vs. no compression • window ‘width’ and the magnitude of alternation bias

  16. Limited working memory, Alternation Bias Sampling distribution of ρ1 for the sample size 10 drawn 10,000 times from a fair and memoryless Bernoulli process (true, not pseudo-random numbers). Mean = -0.102688 μ2 = 0.0849096 μ3 = 0.00616234 μ4 = 0.0209152

  17. Limited working memory, Alternation Bias Sampling distribution of ρ1 for the sample size 7 drawn 10,000 times from a fair and memoryless Bernoulli process (true, not pseudo-random numbers). Mean = -0.129589μ2 = 0.117093μ3 = 0.0244065μ4 = 0.0495815

  18. Limited working memory, Alternation Bias Sampling distribution of ρ1 for the sample size 5drawn 10,000 times from a fair and memoryless Bernoulli process (true, not pseudo-random numbers). Mean = -0.135855μ2 = 0.173576μ3 = 0.0705881 μ4 = 0.120731

  19. Limited working memory, Alternation Bias Sampling distribution of ρ1 for the sample size 4drawn 10,000 times from a fair and memoryless Bernoulli process (true, not pseudo-random numbers). Mean = -0.1032 μ2 = 0.246817 μ3 = 0.124055 μ4 = 0.207792

  20. Limited working memory, Alternation Bias Sampling distribution of ρ1 for samples of size 10, size 7, and size 5.

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