To a great extent, the normatively “ought” inference criticized by Elqayam & Evans (E&E) is due to the discrepancy between the observed human behavior and the predictions by a normative model. When such a discrepancy is found, researchers either draw a conclusion that humans are irrational, because the discrepancy is deemed an error or a bias (e.g., Tversky & Kahneman Reference Tversky and Kahneman1974), or they adopt an “alternative-norm paradigm” to exonerate a certain human fallacy (e.g., Hahn & Warren Reference Hahn and Warren2009). In general, we disagree with E&E that “Normativism has played out its role in the history of the research on human thinking” (target article, sect. 8, para. 5). In our opinion, neither the “ought” inference nor any particular normative model should be abandoned. Rather, we take that what has biased and hindered research programs on human thinking is the practice of directly comparing any human behavior with any particular product of a normative model and being judgmental on human rationality based on partial comparisons.

We elaborate on the topic of “perception of randomness,” which has been “long perceived as normatively incorrect in the JDM [judgment and decision-making] literature” (target article, sect. 6, para. 10). Interestingly, this normatively “ought” inference resulted primarily from comparing human behaviors with a single normative model, the independent and stationary Bernoulli trials – for example, tossing the same coin repeatedly (for a review, see Oskarsson et al. Reference Oskarsson, Van Boven, McClelland and Hastie2009). However, the process of Bernoulli trials is not as simple as it may seem. It has myriad properties, depending on the variable being measured and the underlying parameters, and some of the properties are fairly novel to the researchers in psychology. Therefore, when the model is used to measure human behaviors, it is critical to distinguish between specific products and products with specific properties (e.g., Lopes & Oden Reference Lopes and Oden1987; Nickerson Reference Nickerson2002).

One particular phenomenon in human perception of randomness is that people pay special attention to streak patterns (e.g., consecutive heads in a row when tossing a fair coin). By contrast, the process of Bernoulli trials predicts that the frequency of streak patterns is the same as non-streak patterns of the same length. This discrepancy has motivated numerous studies and continued to yield new findings on two fronts, the nature of human perception of randomness and the appropriate usage of normative models. (To name a few of these studies: Ayton & Fischer Reference Ayton and Fischer2004; Bar-Eli et al. Reference Bar-Eli, Avugos and Raab2006; Burns Reference Burns2004; Falk et al. Reference Falk, Falk and Ayton2009; Falk & Konold Reference Falk and Konold1997; Gilovich et al. Reference Gilovich, Vallone and Tversky1985; Nickerson & Butler Reference Nickerson and Butler2009; Oppenheimer & Monin Reference Oppenheimer and Monin2009.) In our own research (Y. Sun & Wang Reference Sun and Wang2010a; Reference Sun and Wang2010b), we found that it is critical to distinguish between two types of measurement when Bernoulli trials are used as the normative model: how often a pattern is to occur (frequency, measured by mean time) and when a pattern is to first occur (delay, measured by waiting time). It turns out that despite the equal frequency and mean time among patterns of the same length, streak patterns have the longest delay and waiting time.

Apparently, frequency and delay are two different concepts (for example, think of the experience of waiting for a bus; Gardner Reference Gardner1988, p. 63). Depending on the specific task environment, one measure may be more psychologically relevant than the other. Notably, the statistics of delay may provide normative measures to the descriptive accounts of randomness perception. For example, people judge the frequency of an event on the basis of how it is representative of the underlying population (representativeness), and how easily an example can be brought to mind (availability) (Tversky & Kahneman Reference Tversky and Kahneman1974). Thus, when people think of a random process, a streak pattern might be perceived as the most non-representative – it has the most uneven or clustered distribution over time – and the most unavailable – it is the most delayed pattern and most likely to be preceded by non-streak patterns. Indeed, it has been reported that people's responses to streak patterns are largely consistent with the statistics of waiting time rather than the mean time (Oppenheimer & Monin Reference Oppenheimer and Monin2009).

Since both mean time (frequency) and waiting time (delay) are derived from the same normative model, they provide objective measurements that are invariant across individual human subjects. This leads to another advantage of adhering to the normative models as we can embrace their fast and deep advancement in many different fields. Besides probabilities, the statistics of pattern times literally deal with properties of time and space. Recent neurological studies have drawn attention to some unified theories on representations of space, time, and numbers (e.g., Dehaene & Brannon Reference Dehaene and Brannon2010). It has been reported that people tend to assign a lower probability to an event that is more psychologically distant (Liberman & Trope Reference Liberman and Trope2008; Trope & Liberman Reference Trope and Liberman2010) and are sensitive to the tradeoffs between the delay in time and the probability (Luhmann et al. Reference Luhmann, Chun, Yi, Lee and Wang2008; McClure et al. Reference McClure, Ericson, Laibson, Loewenstein and Cohen2007). In financial theories, the delayed waiting time also means a greater variance of pattern times, and variance has been perceived as an essential component of risk (Markowitz Reference Markowitz1991; Weber et al. Reference Weber, Shafir and Blais2004).

Nevertheless, we agree with E&E that normatively “ought” inferences should be taken with caution, and comparing human behaviors with selected products of a normative model does not immediately lead to conclusions on human rationality. In the case of pattern times, the distinction between the frequency and delay may provide us with a more precise tool to tease apart the task environment from which people's perception of randomness might originate, but it does not necessarily exonerate any human fallacies at the macro level (e.g., the gambler's fallacy, cf., Hahn & Warren Reference Hahn and Warren2009; Reference Hahn and Warren2010; Y. Sun et al. Reference Sun, Tweney and Wang2010a; Reference Sun, Tweney and Wang2010b). In addition, people's perception of randomness may not be reduced to a certain set of statistics. To gain a complete picture, we also need descriptive approaches to address perceptual and cognitive mechanisms that come into play, for example, the working memory capacity (Kareev Reference Kareev1992), perception of proportion and symmetry (Rapoport & Budescu Reference Rapoport and Budescu1997), and, subjective complexity ( Falk & Konold Reference Falk and Konold1997; Falk et al. Reference Falk, Falk and Ayton2009).