Probability matching and strategy availability

Findings from two experiments indicate that probability matching in sequential choice arises from an asymmetry in strategy availability: The matching strategy comes readily to mind, whereas a superior alternative strategy, maximizing, does not. First, compared with the minority who spontaneously engage in maximizing, the majority of participants endorse maximizing as superior to matching in a direct comparison when both strategies are described. Second, when the maximizing strategy is brought to their attention, more participants subsequently engage in maximizing. Third, matchers are more likely than maximizers to base decisions in other tasks on their initial intuitions, suggesting that they are more inclined to use a choice strategy that comes to mind quickly. These results indicate that a substantial subset of probability matchers are victims of “underthinking” rather than “overthinking”: They fail to engage in sufficient deliberation to generate a superior alternative to the matching strategy that comes so readily to mind.     

Probability matching in the right hemisphere

Previously it has been shown that the left hemisphere, but not the right, of split-brain patients tends to match the frequency of previous occurrences in probability-guessing paradigms (Wolford, Miller, & Gazzaniga, 2000). This phenomenon has been attributed to an "interpreter," a mechanism for making interpretations and forming hypotheses, thought to reside exclusively in the left hemisphere. In this study with a split-brain patient, we had him guess one of two types of faces, stimuli known to be preferentially processed in the right hemisphere of this patient. Unlike previous studies using other kinds of stimuli, the right hemisphere matched the frequency of the previous occurrences of a face-type, but the left hemisphere did not.     

Probability matching in visual selective attention

Jonides (1980, p. 111) suggested a model to account for shifts of spatial attention in visual search tasks with less than 100% cue validity. This two-stage probability matching model was subsequently tested by Jonides (1983). Reaction time distribution characteristics indeed provided some support for the model. In the present paper the relevant data presented in Jonides (1980, 1981, 1983) are further analysed. The results of this analysis strongly support the probability matching hypothesis. Probability matching is apparently a very important strategic characteristic of visual spatial attention that deserves much more study.     

Probability matching: encouraging optimal responding in humans.

Two hundred undergraduate students participated in a repeated-trials binary choice procedure in which choice of one outcome was correct on 75% of trials. Subjects received 192 trials and were divided into five conditions: (1) control; (2) subjects were given the actual probabilities; (3) subjects were told if they did well they could leave early; (4) competition condition; (5) midway through the task subjects were asked to recommend a strategy for another subject. Half of the subjects in each group were told that the best they could do was to be correct on 75% of the trials. This manipulation permitted assessment of the hypothesis that subjects in probability-matching tasks are seeking a strategy that will be correct on 100% of the trials. The results partially confirmed this hypothesis. In addition, two of the variables improved performance significantly (giving probabilities and asking subjects to recommend a strategy). However, while subjects in all groups improved significantly over trials, optimal choice did not occur in this task.     

Probability matching in choice under uncertainty: Intuition versus deliberation

Gaissmaier and Schooler (2008) [Gaissmaier, W., & Schooler, L. J. (2008). The smart potential behind probability matching. Cognition, 109, 416-422] argue that probability matching, which has traditionally been viewed as a decision making error, may instead reflect an adaptive response to environments in which outcomes potentially follow predictable patterns. In choices involving monetary stakes, we find that probability matching persists even when it is not possible to identify or exploit outcome patterns and that many “probability matchers” rate an alternative strategy (maximizing) as superior when it is described to them. Probability matching appears to reflect a mistaken intuition that can be, but often is not, overridden by deliberate consideration of alternative choice strategies.     

Probability matching in risky choice: The interplay of feedback and strategy availability

Probability matching in sequential decision making is a striking violation of rational choice that has been observed in hundreds of experiments. Recent studies have demonstrated that matching persists even in described tasks in which all the information required for identifying a superior alternative strategy—maximizing—is present before the first choice is made. These studies have also indicated that maximizing increases when (1)?the asymmetry in the availability of matching and maximizing strategies is reduced and (2)?normatively irrelevant outcome feedback is provided. In the two experiments reported here, we examined the joint influences of these factors, revealing that strategy availability and outcome feedback operate on different time courses. Both behavioral and modeling results showed that while availability of the maximizing strategy increases the choice of maximizing early during the task, feedback appears to act more slowly to erode misconceptions about the task and to reinforce optimal responding. The results illuminate the interplay between “top-down” identification of choice strategies and “bottom-up” discovery of those strategies via feedback.     

Banking on a bad bet. Probability matching in risky choice is linked to expectation generation

Probability matching is the tendency to match choice probabilities to outcome probabilities in a binary prediction task. This tendency is a long-standing puzzle in the study of decision making under risk and uncertainty, because always predicting the more probable outcome across a series of trials (maximizing) would yield greater predictive accuracy and payoffs. In three experiments, we tied the predominance of probability matching over maximizing to a generally adaptive cognitive operation that generates expectations regarding the aggregate outcomes of an upcoming sequence of events. Under conditions designed to diminish the generation or perceived applicability of such expectations, we found that the frequency of probability-matching behavior dropped substantially and maximizing became the norm.     

Probability matching, accuracy maximization, and a test of the optimal classifier's independence assumption in perceptual categorization

Observers completed perceptual categorization tasks that included 25 base-rate/payoff conditions constructed from the factorial combination of five base-rate ratios (1:3, 1:2, 1:1, 2:1, and 3:1) with five payoff ratios (1:3, 1:2, 1:1, 2:1, and 3:1). This large database allowed an initial comparison of the competition between reward and accuracy maximization (COBRA) hypothesis with a competition between reward maximization and probability matching (COBRM) hypothesis, and an extensive and critical comparison of the flat-maxima hypothesis with the independence assumption of the optimal classifier. Model-based instantiations of the COBRA and COBRM hypotheses provided good accounts of the data, but there was a consistent advantage for the COBRM instantiation early in learning and for the COBRA instantiation later in learning. This pattern held in the present study and in a reanalysis of Bohil and Maddox (2003). Strong support was obtained for the flat-maxima hypothesis over the independence assumption, especially as the observers gained experience with the task. Model parameters indicated that observers' reward-maximizing decision criterion rapidly approaches the optimal value and that more weight is placed on accuracy maximization in separate base-rate/payoff conditions than in simultaneous base-rate/payoff conditions. The superiority of the flat-maxima hypothesis suggests that violations of the independence assumption are to be expected, and are well captured by the flat-maxima hypothesis, with no need for any additional assumptions.     

Probability Theory and Probability Semantics

Book Information Probability Theory and Probability Semantics. By P. Roeper and H. Leblanc. University of Toronto Press. Toronto. 1999. Pp. xii + 240. Hardback, US$65.00.     

Causal Probability

Examples growing out of the Newcomb problem have convinced many people that decision theory should proceed in terms of some kind of causal probability. I endorse this view and define and investigate a variety of causal probability. My definition is related to Skyrms' definition, but proceeds in terms of objective probabilities rather than subjective probabilities and avoids taking causal dependence as a primitive concept.     

Default Probability

A probability may be called “default” if it is neither derived from preestablished probabilities nor based on considerations of frequency or symmetry. Default probabilities presumably arise through reasoning based on causality and similarity. This article advances a model of default probability based on a featural approach to similarity. The accuracy of the model is assessed by comparing its predictions to the probabilities provided by undergraduates asked to reason about mammals.     

Probability kinematics

Probability kinematics is studied in detail within the framework of elementary probability theory. The merits and demerits of Jeffrey's and Field's models are discussed. In particular, the principle of maximum relative entropy and other principles are used in an epistemic justification of generalized conditionals. A representation of conditionals in terms of Bayesian conditionals is worked out in the framework of external kinematics.     

Subjunctive Conditional Probability

There seem to be two ways of supposing a proposition: supposing “indicatively” that Shakespeare didn’t write Hamlet, it is likely that someone else did; supposing “subjunctively” that Shakespeare hadn’t written Hamlet, it is likely that nobody would have written the play. Let P(B//A) be the probability of B on the subjunctive supposition that A. Is P(B//A) equal to the probability of the corresponding counterfactual, A □→B? I review recent triviality arguments against this hypothesis and argue that they do not succeed. On the other hand, I argue that even if we can equate P(B//A) with P(A □→B), we still need an account of how subjunctive conditional probabilities are related to unconditional probabilities. The triviality arguments reveal that the connection is not as straightforward as one might have hoped.