Boundary effects of vague risk information on taxpayer decisions

In compliance decisions, the decision maker usually has only vague or ambiguous knowledge of the probability of being caught and the outcome (amount of penalty). An experiment is reported which extends work on effects of probability ambiguity by manipulating outcome ambiguity as well. When outcomes were limited to a bounded range and probabilities ranged between their natural boundaries [0, 1] in experimental tax decisions, symmetrical boundary effects were found in which vague estimates for both the probability and outcome dimensions caused vagueness aversion (and higher compliance) when the vague estimate was near the more favorable lower boundary of either dimension and vagueness seeking (and lower compliance) when the vague estimate was near the less favorable upper boundary. Probability and outcome vagueness effects were found to be independent of the vagueness of the other dimension, and vagueness effects were not systematically related to the level of the other dimension.The results suggest that a common cognitive process mediates the impact of vagueness on both dimensions. This may be a vagueness-adjustment process in which vague estimates are adjusted toward the middle of the bounded range, or a vagueness-preference process in which vague outcomes, and vague probabilities as well, are evaluated based on utility considerations, as though probability were a tangible commodity. For increasing compliance, the results suggest that risk information should be disseminated only when risks of punishment are relatively high. When risks are low, random enforcement techniques that enhance vagueness become more effective.     

Choice and foraging: the effects of accessibility on acceptability.

Pigeons responded in a successive-encounters choice procedure in which accessibility of the less profitable of two outcomes varied either in terms of probability of encounter or search time to encounter (keeping search time to the more profitable outcome constant). When the less profitable outcome was made more probable its acceptance became more likely. However, when search time to encounter the less profitable outcome was shortened, its acceptance became less likely. Both results are consistent with the delay-reduction hypothesis and with an optimality model developed for application to the successive-encounters choice procedure.     

Multiple Evaluation: A New Testing Paradigm That Exorcizes Guessing

The crux in psychometrics is how to estimate the probability that a respondent answers an item correctly on one occasion out of many. Under the current testing paradigm this probability is estimated using all kinds of statistical techniques and mathematical modeling. Multiple evaluation is a new testing paradigm using the person's own personal estimates of these probabilities as data. It is compared to multiple choice, which appears to be a degenerated form of multiple evaluation. Multiple evaluation has much less measurement error than multiple choice, and this measurement error is not in favor of the examinee. When the test is used for selection purposes as it is with multiple choice, the probability of a Type II error (unjustified passes) is almost negligible. Procedures for statistical item-and-test analyses under the multiple evaluation paradigm are presented. These procedures provide more accurate information in comparison to what is possible under the multiple choice paradigm. A computer program that implements multiple evaluation is also discussed.     

Subjective sampling distributions and the additivity of estimates

Abstract.— Previous studies of sampling distributions have been conducted almost exclusively under the assumption that persons behave in accordance with the "fundamental convention" of probability, i.e. that the sum of all probability estimates will equal 1. When this assumption was tested by asking subjects to give "unrestricted" probability estimates of all possible outcomes of samples from a given population, a general tendency of overestimation made the sum of all probabilities exceed 1 to a considerable extent. The subjective sampling distributions appeared to be unaffected by sample size ( N=5 or 10) and number of outcomes, and were flatter than the corresponding "objective" sampling distributions.     

Subjective probability and everyday life

Abstract.— Recent research on probability judgement indicates that people's ability to estimate probabilities is very limited. It is argued that people may lack the cognitive apparatus necessary for processing probabilistic information, in so far as probability judgements play an unimportant role in everyday life. When probability judgements occasionally are made in everyday life it is argued that they are not based on frequency data but on some more or less well grounded theory.     

How Disadvantaged Group Members Cope With Discrimination When They Perceive That Social Support Is Available1

This experiment examined how disadvantaged group members cope with discrimination when they perceive that social support is available. Women reacted to a failing test grade after ambiguous information about the probability for discrimination. With no social support, women minimized discrimination and attributed their failure to the quality of their answers. Participants were less inclined to minimize discrimination when social support was available. When they perceived that either emotional or informational support was available, women were equally likely to blame their failure on discrimination as on the quality of their answers. The results revealed less minimization of personal discrimination when both emotional and informational support were available, in which case participants blamed their failure more on discrimination and less on themselves.     

Admissible probability measurement procedures

Admissible probability measurement procedures utilize scoring systems with a very special property that guarantees that any student, at whatever level of knowledge or skill, can maximize his expected score if and only if he honestly reflects his degree-of-belief probabilities. Section 1 introduces the notion of a scoring system with the reproducing property and derives the necessary and sufficient condition for the case of a test item with just two possible answers. A method is given for generating a virtually inexhaustible number of scoring systems, both symmetric and asymmetric, with the reproducing property. A negative result concerning the existence of a certain subclass of reproducing scoring systems for the case of more than two possible answers is obtained. Whereas Section 1 is concerned with those instances in which the possible answers to a query are stated in the test itself, Section 2 is concerned with those instances in which the student himself must provide the possible answer(s). In this case, it is shown that a certain minor modification of a scoring system with the reproducing property yields the desired admissible probability measurement procedure.The research reported in this paper was, in part, performed at the Decision Sciences Laboratory in support of Project 4690, Information Processing in Command and Control and, in part, sponsored by the Air Force Systems Command Electronic Systems Division, Decision Sciences Laboratory, under Contract No. AF 19(628)-4304, with ARCON Inc. This report is identified as ESD-TR-65-567. Further reproduction is authorized to satisfy the needs of the U. S. Government.     

Using the Scenario Method to Analyze Cheating Behaviors

Using student self-reported cheating admissions and answers from a hypothetical cheating scenario, this paper analyzes the effects of individual and situational factors on potential cheating behavior. Results confirm several conclusions about student factors that are related to cheating. The probability of cheating is associated with younger students, lower GPAs, alcohol consumption, fraternity/sorority membership, and having cheated in high school. Student perceptions of the certainty and severity of punishment appear to have a negative and significant impact on the probability of cheating on in-class assignments. Students who report a belief that cheating is never acceptable appear to be significantly less likely to cheat in any circumstance. This study illustrates the context-dependent nature of academic dishonesty, and the associated difficulty in understanding the relationships between measurable factors and cheating behavior.     

The influence of reasons on interpretations of probability forecasts

When providing a probability estimate for an event, experts often supply reasons that they expect will clarify and support that estimate. We investigated the possible unintended influence that these reasons might have on a listener's intuitive interpretation of the event's likelihood. Experiments 1 and 2 demonstrated that people who read positive reasons for a doctor's probability estimate regarding a hypothetical surgery were more optimistic than those who read negative reasons for the identical estimate. Experiment 3 tested whether a doctor's failure forecast for a surgery would result in differing levels of pessimism when the potential risk was attributed to one complication that had a probability of 0.30 versus three complications that had a disjunctive probability of 0.30. Overall, the findings are consistent with the argument that a probability estimate, albeit numerically precise, can be flexibly interpreted at an intuitive level depending on the reasons that the forecaster provides as the basis for the estimate. Copyright © 2003 John Wiley & Sons, Ltd.     

Outcome trees and baseball: A study of expertise and list-length effects

When people estimate the probability of an event using a list that includes all or most of the possible events, their estimate of that probability is lower than if the other possible events are not explicitly identified on the list (i.e., are collapsed into an all-other-possibilities category). This list-length (or pruning) effect has been demonstrated to occur even for people who have expertise or considerable knowledge in the event domain. We reasoned that the experts used in previous studies would be unlikely to have probabilistic representations of their problem domains (e.g., auto mechanics, auditors, hospitality managers). We used baseball experts (n= 35) and novices (n = 56) on the assumption that expertise in baseball almost certainly involves mental representations of probability for various baseball events. Subjects estimated the frequency of hits, walks, strikeouts, putouts, and “all other” outcomes for an average major league player in 100 times at bat. Other subjects estimated these event outcome frequencies in a short-list condition (e.g., strikeouts, walks, and “all other”). Strong list-length effects were observed with novices; the frequency estimate for strikeouts, for example, was nearly twice as high in the short-list condition as in the long-list condition. Experts, however, showed no list-length effect and their estimated probabilities were very near the actual (normatively correct) probabilities in all conditions. We argue that the omission effect can be overridden by strong mental representations of the family of possible events and/or a clear knowledge of the probabilities associated with the events. As well, we argue that list-length effects seem to result at least in part from an anchoring-and-adjustment strategy.     

Modeling the Impact of Control on the Attractiveness of Risk in a Prospect Theory Framework

Many decisions involve a degree of personal control over event outcomes, which is exerted through one's knowledge or skill. In three experiments we investigated differences in decision making between prospects based on a) the outcome of random events and b) the outcome of events characterized by control. In Experiment 1, participants estimated certainty equivalents (CEs) for bets based on either random events or the correctness of their answers to U.S. state population questions across the probability spectrum. In Experiment 2, participants estimated CEs for bets based on random events, answers to U.S. state population questions, or answers to questions about 2007 NCAA football game results. Experiment 3 extended the same procedure as Experiment 1 using a within-subjects design. We modeled data from all experiments in a prospect theory framework to establish psychological mechanisms underlying decision behavior. Participants weighted the probabilities associated with bets characterized by control so as to reflect greater risk attractiveness relative to bets based on random events, as evidenced by more elevated weighting functions under conditions of control. This research elucidates possible cognitive mechanisms behind increased risk taking for decisions characterized by control, and implications for various literatures are discussed.     

Judging the accuracy of a likelihood judgment: the case of smoking risk

A standard method for assessing whether people have appropriate internal representations of an event's likelihood is to check whether their subjective probability or frequency estimates for the event correspond with the assumed objective value for that event. When a person's estimate for the event exceeds its assumed objective probability or frequency, the person's expectancy for the event is concluded to be greater than warranted. This paper describes three lines of reasoning as to why conclusions of this sort can be problematic. Recently published findings as well as data from two new experiments are described to support this main thesis. The case of smoking risk is used to illustrate the more general problem, and issues that must be considered to avoid or contend with the problem are discussed. Copyright © 2002 John Wiley & Sons, Ltd.     

Why indeterminate probability is rational

This essay has two objectives: (1) To offer a general motivation for the use of sets of probabilities to represent states of credal probability judgment. Doubt as to which of rival answers to a given question is true is representable by the set of potential answers that have not ruled out as impossible. Doubt concerning probability is representable by the set of probability distributions over potential answers that are not ruled out as impermissible to use in delibration. (2) To defend a more specific thesis that the set of permissible probability distributions ought to be convex.     

A sampling approach to biases in conditional probability judgments: beyond base rate neglect and statistical format

Conditional probability judgments of rare events are often inflated. Early accounts assumed a general deficit in using statistical base rates. More recent approaches predict improvement when problems are presented in frequency format or refer to natural categories. The present theory focuses on sampling processes. Experiment 1 showed that a seeming advantage of frequency over probability formats is due to a confounded factor, the need to mentally transform stimulus samples. An information search paradigm was used in Experiment 2. When sampling by the predictor, the probability to be estimated, p(criterion/predictor), was conserved in the samples and judgments were quite accurate. However, when sampling by the criterion, the low base-rate event was strongly overrepresented, accounting for the entire bias. Judgments were quite sensitive to the sampled data, but failed to take sampling constraints into account, as shown in Experiments 3 and 4.     

Age-trend-related differences in tasks involving conjunctive probabilistic reasoning

Conjunctive probabilistic reasoning has been studied at different ages to ascertain whether the conjunction fallacy is due to a task demand misinterpretation. Such a misinterpretation might occur because a task that requires a comparison between a superordinate class A and a subordinate class A&B is mistakenly interpreted as requiring a comparison between the two complementary subordinate classes of A (i.e., A&B and A¬B). Children (7- and 10-year-olds) and adults were required to make conjunctive probability judgments about problems for which explicit objective probabilities were provided. The total number of A items was kept constant and the frequencies of the A&B and of the A¬B items varied across problems. When the number of A&B items was smaller than the number of A¬B items, the frequency of congruent responses increased with age. When the number of A&B items was greater or equal to that of the A¬B items, the frequency of correct answers decreased. (PsycINFO Database Record (c) 2008 APA, all rights reserved).     

卢卡西维茨逻辑和菲尼蒂融贯性标准的近期发展

真值在最初的时侯只有一个。为了更好地把握“真”这个真值,才引入了真值“假”。在数学推理中,有这两个真值就足够了：因为处理数学句子只需使用一阶逻辑中的二值语义和证明论。然而当我们获得的信息不完全准确、有部分错误甚至被扭曲时,这两个经典的真值就不足以灵活地表达和处理我们的知识。我们将首先分析卢卡西维茨的无穷值逻辑在处理Rényi—Ulam问题时所起的作用。“二十问”游戏中的信息流遵守布尔逻辑的规则,相比起来,该游戏的Rényi—Ulam变异所产生的回答却可能是错误的,对同一个问题重复提问得到的两个相同的回答所产生的信息,要多于仅仅只回答一次时得到的信息：幂等律A＆A=A不再成立。这个游戏中的回答遵守卢卡西维茨的无穷值逻辑。这一逻辑在处理连续事件的融贯概率估计方面具有普遍的意义,它能够推广处理yes—no事件（即在任一可能世界中,要么发生要么不发生的那些事件）的菲尼蒂概率理论。假定事件集E={X1…,Xn}等于某布尔代数F上的一组元素,菲尼蒂证明了,定义在E上的映射p是“融贯的”当且仅当它可以延拓到F的概率测度上。p的融贯性意味着,如果一个赌徒A把概率度p（Xi）分配给每个事件Xi,他的对手B不可能迫使他下注,使得B确保在每个可能世界中都赌赢。菲尼蒂利用他提出的融贯性标准,得出了概率论的柯尔莫哥洛夫公理。使用卢卡西维茨逻辑,我们把菲尼蒂理论推广到测度值为实数区间[0,1]的那些事件上。只要经过恰当的规范化,这类事件的例子就是大量地、或显或隐地存在于我们日常生活的打赌之中。     

Regulation of Teacher Elicitations in the Mathematics Classroom

Using the perspective of instructional conversation, we investigated how one teacher regulated student participation and conceptual reasoning in the middle-school mathematics classroom. We examined the elicitations—questions and provocative statements—made by the teacher over a four-day algebra lesson. Analyses showed how the teacher systematically regulated the level of cognitive complexity of his elicitations in reaction to students' responses. When students gave inaccurate or incomplete answers, the teacher tended to reduce the level of cognitive complexity needed to respond to a subsequent elicitation, with the apparent impact being that he scaffolded participation and reasoning. When students provided responses that were mathematically accurate, the teacher usually increased the elicitation level, which subsequently engaged students in more sophisticated forms of reasoning.     

Statistical judgments are influenced by the implied likelihood that samples represent the same population

When sample information is combined, it is generally considered normative to weight information based on larger samples more heavily than information based on smaller samples. However, if samples appear likely to have been drawn from different subpopulations, it is reasonable to combine estimates of these subpopulation means (typically, the sample means) without weighting these estimates by sample size. This study investigated whether laypeople are influenced by the likelihood of samples coming from the same population when determining how to combine information. In two experiments we show that (1) implied binomial variability affected participants’ judgments of the likelihood that a sample was drawn from a given population, (2) participants' judgments were more affected by sample size when samples were implied to be drawn randomly from a general population, compared to when they were implied to be drawn from different subpopulations, and (3) people higher in numeracy gave more normative responses. We conclude that when determining how to weight and combine samples, laypeople use not only the provided data, but also information about likelihood and sampling processes that these data imply.     

An interactive data-generating and answer-correcting system for problems in statistics

A computer-assisted instructional system is described which provides the student with extensive drill and feedback in the application of statistical methods to computer-simulated data. The system consists of three interactive programs that perform the following functions: (1) FILEMAKE creates “source” data for problems in statistics having whatever population characteristics are desired by the instructor; (2) DATASET generates unique data for each student by performing a random modification, based on sampling theory, of the source data; and (3) CORRECT corrects each student’s unique answers to the problem.