Do Bettors Correctly Perceive Odds? Three Studies of How Bettors Interpret Betting Odds as Probabilistic Information

This paper reports on three studies investigating how accurately bettors (=people who regularly bet on sports events) interpret the probabilistic information implied by betting odds. All studies were based on data collected by web surveys prompting a total of 186 experienced bettors to convert sets of representative odds into frequency judgments. Bayesian statistical methods were used to analyze the data. From the results, the following conclusions were made: (i) On the whole, the bettors produced well‐calibrated judgments, indicating that they have realistic perceptions of odds. (ii) Bettors were unable to consciously adjust judgments for different margins. (iii) Although their interval judgments often covered the estimates implied by the odds, the bettors tended to overestimate the variation of expected profitable bets between months. The results are consistent with prior research showing that people tend to make accurate probability judgments when faced with tasks characterized by constant and clear feedback. Copyright © 2015 John Wiley & Sons, Ltd.     

The role of magical thinking in forecasting the future

This article explores the role of magical thinking in the subjective probabilities of future chance events. In five experiments, we show that individuals tend to predict a more lucky future (reflected in probability judgements of lucky and unfortunate chance events) for someone who happened to purchase a product associated with a highly moral person than for someone who unknowingly purchased a product associated with a highly immoral person. In the former case, positive events were considered more likely than negative events, whereas in the latter case, the difference in the likelihood judgement of positive and negative events disappeared or even reversed. Our results indicate that this effect is unlikely to be driven by participants’ immanent justice beliefs, the availability heuristic, or experimenter demand. Finally, we show that individuals rely more heavily on magical thinking when their need for control is threatened, thus suggesting that lack of control represents a factor in driving magical thinking in making predictions about the future.     

The effects of a high-probability instruction sequence and response-independent reinforcer delivery on child compliance

We compared the effects of a high-probability (high-p) instruction sequence and a fixed-time (FT) schedule of reinforcement on the compliance of 2 typically developing children. A multielement experimental design with a reversal component was implemented according to a multiple baseline across participants arrangement. Both the high-p and FT conditions resulted in increased compliance for both participants during the multielement sessions. These results suggest that it may be possible to increase compliance without a response requirement of the type arranged in the high-p instruction sequence.     

Notes on discounting

In general, if a variable can be expressed as a function of its own maximum value, that function may be called a discount function. Delay discounting and probability discounting are commonly studied in psychology, but memory, matching, and economic utility also may be viewed as discounting processes. When they are so viewed, the discount function obtained is hyperbolic in form. In some cases the effective discounting variable is proportional to the physical variable on which it is based. For example, in delay discounting, the physical variable, delay (D), may enter into the hyperbolic equation as kD. In many cases, however, the discounting data are not well described with a single-parameter discount function. A much better fit is obtained when the effective variable is a power function of the physical variable (kDS in the case of delay discounting). This power-function form fits the data of delay, probability, and memory discounting as well as other two-parameter discount functions and is consistent with both the generalized matching law and maximization of a constant-elasticity-of-substitution utility     

Rate, Probability and Matching: Comments on ��The Identities Hidden in the Matching Laws, and Their Uses�� by David Thorne

David Thorne's (2010) article, “The Identities Hidden In The Matching Laws, And Their Uses” performs a valuable service in pointing out alternative expressions of matching. However, some identities tend to obscure rather than illuminate empirical relationships. Three such problematic instances are discussed: interresponse time as a function of interval and ratio schedule parameters; probability equality as implying rate matching; the apparent simplicity of probabilistic functions, as opposed to response rate functions, of reinforcement rate.     

Tracking the Time Course of Word‐Frequency Effects in Auditory Word Recognition With Event‐Related Potentials

Although the word‐frequency effect is one of the most established findings in spoken‐word recognition, the precise processing locus of this effect is still a topic of debate. In this study, we used event‐related potentials (ERPs) to track the time course of the word‐frequency effect. In addition, the neighborhood density effect, which is known to reflect mechanisms involved in word identification, was also examined. The ERP data showed a clear frequency effect as early as 350 ms from word onset on the P350, followed by a later effect at word offset on the late N400. A neighborhood density effect was also found at an early stage of spoken‐word processing on the PMN, and at word offset on the late N400. Overall, our ERP differences for word frequency suggest that frequency affects the core processes of word identification starting from the initial phase of lexical activation and including target word selection. They thus rule out any interpretation of the word frequency effect that is limited to a purely decisional locus after word identification has been completed.     

Imprecise Epistemic Values and Imprecise Credences

A number of recent arguments purport to show that imprecise credences are incompatible with accuracy-first epistemology. If correct, this conclusion suggests a conflict between evidential and alethic epistemic norms. In the first part of the paper, I claim that these arguments fail if we understand imprecise credences as indeterminate credences. In the second part, I explore why agents with entirely alethic epistemic values can end up in an indeterminate credal state. Following William James, I argue that there are many distinct alethic values that a rational agent can have. Furthermore, such an agent is rationally permitted not to have settled on one fully precise value function. This indeterminacy in value will sometimes result in indeterminacy in epistemic behaviour—that is, because the agent’s values aren’t settled, what she believes might not be.     

On the a priori and a posteriori assessment of probabilities

We argue that in spite of their apparent dissimilarity, the methodologies employed in the a priori and a posteriori assessment of probabilities can both be justified by appeal to a single principle of inductive reasoning, viz., the principle of symmetry. The difference between these two methodologies consists in the way in which information about the single-trial probabilities in a repeatable chance process is extracted from the constraints imposed by this principle. In the case of a posteriori reasoning, these constraints inform the analysis by fixing an a posteriori determinant of the probabilities, whereas, in the case of a priori reasoning, they imply certain claims which then serve as the basis for subsequent probabilistic deductions. In a given context of inquiry, the particular form which a priori or a posteriori reason may take depends, in large part, on the strength of the underlying symmetry assumed: the stronger the symmetry, the more information can be acquired a priori and the less information about the long-run behavior of the process is needed for an a posteriori assessment of the probabilities. In the context of this framework, frequency-based reasoning emerges as a limiting case of a posteriori reasoning, and reasoning about simple games of chance, as a limiting case of a priori reasoning. Between these two extremes, both a priori and a posteriori reasoning can take a variety of intermediate forms.