Subjective Conditional Probability Research Articles

How auctioneers set reserve prices in procurement auctions

We introduce a behavioral model that effectively predicts auctioneers’ reserve choice patterns in English clock auctions across varying reserve price formats (whether reserve prices are set ex-ante or ex-post auction events), number of bidders and the distribution of suppliers’ costs. In a two-parameter model, which we call Subjective Conditional Probability (SCP), auctioneers have subjective judgement of conditional probabilities. We theoretically show that the SCP explains two intuitive, but sub-optimal, reserve price setting patterns. It predicts and provides rationalizations for ex-ante reserve prices that decrease in the number of bidders and ex-post reserve prices that increase in the realized auction price. We conduct two experiments; one with a uniform cost distribution and another with a left-skewed cost distribution. We validate the SCP model internally and externally by comparing it to a wide range of models, including reduced form linear regression, risk aversion, anticipated regret model, and subjective probability judgement via in-sample and out-of-sample predictions. We conclude that the SCP has strong external validity across a variety of procurement environments.

The Effects of Job Characteristics on Retirement.

Along with data about actual, desired and anticipated job characteristics, this paper uses a novel data element, the subjective conditional probability of working at age 70, to estimate the causal effects of job characteristics on retirement in the U.S. Having flexible work hours is the most consistent predictor of retirement preferences and expectations: if all current workers had flexible hours, the fraction working at age 70 would be 0.322, but it would be just 0.172 if none had this option. Job stress, physical and cognitive job demands, the option to telecommute, and commuting times were additional predictors of retirement expectations.

A subjective approach to fuzzy events and their coherent probability assessment for decision making under uncertainty

This research starts from the concepts of conditional event and conditional probability introduced by de Finetti, Dubins and Scozzafava and from the concepts on fuzzy probability introduced by Zadeh, Klir, Yuan, and Yager. We propose a formal adjustment (but logically equivalent) to the definition of conditional event and related concepts, which allows us to give a definition of fuzzy event as an extension of a conditional event. Then, we suggest a formal modification (but logically equivalent) of the concept of conditional probability in the sense of de Finetti-Dubins, and we introduce some possible definitions of fuzzy event’s probability as an extension of the conditional probability. Each of these definitions is an extension of the operational de Finetti definition of subjective conditional probability based on a coherent bet. Moreover, it contains a subjective evaluation, which is represented by a score function, that, in appropriate contexts, can be interpreted as a utility function.

A Coherent Aggregation Framework for Stress Testing and Scenario Analysis

We present a methodology to aggregate in a coherent manner conditional stress losses in a trading or banking book. The approach bypasses the specification of unconditional probabilities of the individual stress events and ensures by a linear programming approach so that the (subjective or frequentist) conditional probabilities chosen by the risk manager are internally consistent. The admissibility requirement greatly reduces the degree of arbitrariness in the conditional probability matrix if this is assigned subjectively. The approach can be used to address the requirements of the regulators on the Instantaneous Risk Charge.

RAMSEY’S TEST, ADAMS’ THESIS, AND LEFT-NESTED CONDITIONALS

Adams famously suggested that the acceptability of any indicative conditional whose antecedent and consequent are both factive sentences amounts to the subjective conditional probability of the consequent given the antecedent. The received view has it that this thesis offers an adequate partial explication of Ramsey’s test, which characterizes graded acceptability for conditionals in terms of hypothetical updates on the antecedent. Some results in van Fraassen (1976) may raise hope that this explicatory approach to Ramsey’s test is extendible to left-nested conditionals, that is, conditionals whose antecedent is itself conditional in form. We argue that this interpretation of van Fraassen’s results is to be rejected. Specifically, we provide an argument from material inadequacy against a generalization of Adams’ thesis for left-nested conditionals.

Probability and Instruction Effects in Syllogistic Conditional Reasoning

Abstract — The main aim of this study was to examine whether people understand indicative conditionals on the basis of syntactic factors or on the basis of subjective conditional probability. The second aim was to investigate whether the conditional probability of q given p depends on the antecedent and consequent sizes or derives from inductive processes leading to establish a link of plausible co-occurrence between events semantically or experientially associated. These competing hypotheses have been tested through a 3 x 2 x 2 x 2 mixed design involving the manipulation of four variables: type of instructions (“Consider the following statement to be true”, “Read the following statement” and condition with no conditional statement); antecedent size (high/low); consequent size (high/low); statement probability (high/low). The first variable was between-subjects, the others were within-subjects. The inferences investigated were Modus Ponens and Modus Tollens. Ninety undergraduates of the Second University of Naples, without any prior knowledge of logic or conditional reasoning, participated in this study. follows that “if Results suggest that people understand conditionals in a syntactic way rather than in a probabilistic way, even though the perception of the conditional probability of q given p is at least partially involved in the conditionals’ comprehension. They also showed that, in presence of a conditional syllogism, inferences are not affected by the antecedent or consequent sizes. From a theoretical point of view these findings suggest that it would be inappropriate to abandon the idea that conditionals are naturally understood in a syntactic way for the idea that they are understood in a probabilistic way.

Reasoning with conditionals: Does every counterexample count? It’s frequency that counts

A series of experiments investigated what determines people's degree of belief in conditionals and their readiness to draw inferences from them. Information on the frequency of exceptions to conditional rules was contrasted with information about the number of different disabling conditions causing these exceptions. Experiments 1 and 2, using conditionals with arbitrary contents, revealed a strong effect of frequency information and no effect of disabling information. Experiment 3 established that, in the absence of frequency information, the disabling condition information used in Experiments 1 and 2 affected belief in the conditionals and inference acceptance, as has been found in many previous studies (Byrne, 1989; DeNeys, Schaeken, & d'Ydewalle, 2003b). Experiment 4 extended the results of Experiments 1 and 2 to everyday conditionals. The results show that belief in a conditional, and the confidence in inferences subsequently drawn from it, both depend on the subjective conditional probability of the consequent given the antecedent. This probability is estimated from the relative frequency of exceptions regardless of what causes them.

Causal and noncausal conditionals: An integrated model of interpretation and reasoning

We present an integrated model for the understanding of and the reasoning from conditional statements. Central assumptions from several approaches are integrated into a causal path model. According to the model, the cognitive availability of exceptions to a conditional reduces the subjective conditional probability of the consequent, given the antecedent. This conditional probability determines people's degree of belief in the conditional, which in turn affects their willingness to accept logically valid inferences. In addition to this indirect pathway, the model contains a direct pathway: Availability of exceptional situations directly reduces the endorsement of valid inferences. We tested the integrated model with three experiments using conditional statements embedded in pseudonaturalistic cover stories. An explicitly mentioned causal link between antecedent and consequent was either present (causal conditionals) or absent (arbitrary conditionals). The model was supported for the causal but not for the arbitrary conditional statements.

Against the Ramsey test

The Ramsey test is the idea Frank Ramsey first expressed in 1929, that when people accept a conditional 'if A then C' they are 'fixing their degree of belief in C given A'. Later writers have taken the idea to apply only to indicative conditionals, and have interpreted it as saying that, as Allan Gibbard put it, 'the acceptability, assertability, and the like ... of an indicative conditional ... depend upon the corresponding subjective conditional probability'. It is at the heart of Ernest Adams's account of conditionals, and Vann McGee asserts that Adams's account 'describes what English speakers assert and accept with unfailing accuracy'. Take 'depend on' to mean covariance, that the higher the conditional probability the higher the assertability, and call it Simple Ramsey. Jonathan Bennett (2003: 29) develops the idea as follows:

Conditional Probability and the Cognitive Science of Conditional Reasoning

Abstract: This paper addresses the apparent mismatch between the normative and descriptive literatures in the cognitive science of conditional reasoning. Descriptive psychological theories still regard material implication as the normative theory of the conditional. However, over the last 20 years in the philosophy of language and logic the idea that material implication can account for everyday indicative conditionals has been subject to severe criticism. The majority view is now apparently in favour of a subjective conditional probability interpretation. A comparative model fitting exercise is presented that shows that a conditional probability model can explain as much of the data on abstract indicative conditional reasoning tasks as psychological theories that supplement material implication with various rationally unjustified processing assumptions. Consequently, when people are asked to solve laboratory reasoning tasks, they can be seen as simply generalising their everyday probabilistic reasoning strategies to this novel context.

On absolute and relative performance and the demand for mutual funds—experimental evidence

Empirically, mutual fund flows depend on past performance. It is unclear, however, whether this is rational. Using experiments, we overcome measurement problems of real data. We detect two anomalies: (1) “absolute performance effect”—investors’ tendency to delegate money to a fund increases with performance, even when performance is uninformative; (2) “relative performance effect”—investors’ tendency to delegate money decreases with other funds’ performance, even when their performance is attributed to luck per se. We suggest two descriptive alternatives to expected utility: (1) “subjective conditional probability”—subjective probabilities deviate from Bayesian posteriors; (2) “subjective risk aversion”—a history-dependent utility function.

Computational levels and conditional inference: Reply to Schroyens and Schaeken (2003).

This article is a reply to W. Schroyens and W. Schaeken's (2003; see record 2002-08431-014) critique of M. Oaksford, N. Chater, and J. Larkin's (2000; see record 2000-08540-005) conditional probability model (CP) of conditional inference. It is argued that their meta-analysis does not falsify CP because the evidence may bear on more than one computational level of explanation. Moreover, it is argued that CP provides a rational account of more of the data than W. Schroyens and W. Schaeken's mental models theory. Other points are also addressed. It is suggested that W. Schroyens and W. Schaeken's model and CP converge on the importance of probabilistic prior knowledge in conditional inference. This is consistent with the normative literature, which (like CP) treats conditionals in terms of subjective conditional probabilities.

Inductive Inference: An Axiomatic Approach

A predictor is asked to rank eventualities according to their plausibility, based on past cases. We assume that she can form a ranking given any memory that consists of finitely many past cases. Mild consistency requirements on these rankings imply that they have a numerical representation via a matrix assigning numbers to eventuality-case pairs, as follows. Given a memory, each eventuality is ranked according to the sum of the numbers in its row, over cases in memory. The number attached to an eventuality-case pair can be interpreted as the degree of support that the past case lends to the plausibility of the eventuality. Special instances of this result may be viewed as axiomatizing kernel methods for estimation of densities and for classification problems. Interpreting the same result for rankings of theories or hypotheses, rather than of specific eventualities, it is shown that one may ascribe to the predictor subjective conditional probabilities of cases given theories, such that her rankings of theories agree with rankings by the likelihood functions.

Two kinds of reasoning.

According to one view of reasoning, people can evaluate arguments in at least two qualitatively different ways: in terms of their deductive correctness and in terms of their inductive strength. According to a second view, assessments of both correctness and strength are a function of an argument's position on a single psychological continuum (e.g., subjective conditional probability). A deductively correct argument is one with the maximum value on this continuum; a strong argument is one with a high value. The present experiment tested these theories by asking participants to evaluate the same set of arguments for correctness and strength. The results produced an interaction between type of argument and instructions: In some conditions, participants judged one argument deductively correct more often than a second, but judged the second argument inductively strong more often than the first. This finding supports the view that people have distinct ways to evaluate arguments.

On Absolute and Relative Performance and the Demand for Mutual Funds - Experimental Evidence

Empirically, mutual fund flows depend on past performance. It is unclear, however, whether this behavior is rational. Using the experimental approach we analyze behavior without confronting measurement problems of real data. We detect two anomalies: Absolute Performance Effect -- investors' tendency to delegate money to a fund increases with performance, even when performance is uninformative; and Relative Performance Effect -- investors' tendency to delegate money to the fund decreases with the performance of other funds, even when their performance attributed to luck per-se. We suggest two alternatives to expected utility: Subjective Conditional Probability -- the probabilities assigned by investors to future events differ from the Bayesian posterior; and Subjective Risk Aversion -- the utility function is history dependent.

Comparative probability for conditional events: A new look through coherence

We study, from the standpoint of coherence, comparative probabilities on an arbitrary familyE of conditional events. Given a binary relation ⩽·, coherence conditions on ⩽· are related to de Finetti's coherent betting system: we consider their connections to the usual properties of comparative probability and to the possibility of numerical representations of ⩽·. In this context, the numerical reference frame is that of de Finetti's coherent subjective conditional probability, which is not introduced (as in Kolmogoroff's approach) through a ratio between probability measures. Another relevant feature of our approach is that the family & need not have any particular algebraic structure, so that the ordering can be initially given for a few conditional events of interest and then possibly extended by a step-by-step procedure, preserving coherence.

A parallel distributed processing model of story comprehension and recall

An optimal control theory of story comprehension and recall is proposed within the framework of a “situation”‐state space. A point in situation‐state space is specified by a collection of propositions, each of which can have the values of either “present” or “absent.” A trajectory in situation‐state space is a temporally ordered sequence of situations. A reader's knowledge that the occurrence of one situation is likely to cause the occurrence of another situation is represented by a subjective conditional probability distribution. A multistate probabilistic (MSP) causal chain notation is also introduced for conveniently describing the knowledge structures implicitly represented by the subjective conditional probability distribution. A story is represented as a partially specified trajectory in situation‐state space, and thus, story comprehension is defined as the problem of inferring the most probable missing features of the partially specified story trajectory. The story‐recall process is also viewed as ...

Fertilization under Uncertainty: An Analysis Based on Producer Yield Expectations

AbstractOptimal nitrogen fertilizer levels are computed for a sample of Texas Coastal Bend grain sorghum producers based on their assessed subjective conditional yield and price probabilities, utility measures, and cost information. Producers' mean yield expectations were substantially greater than experimental yield response and their perceptions of the effects of nitrogen on yield variability were contrary to usual characterization of nitrogen as a risk‐increasing input. Optimal fertilizer levels based on expected utility maximization represent actual fertilizer use somewhat better than those based on expected profit maximization. However, both calculated levels were high relative to actual fertilizer use.

Conceptualizations of valence and instrumentality: A fourfold model

Within the framework of valence-instrumentality (VI) theory, a fourfold model is developed to predict the contribution which is made by one outcome of an alternative to the valence of that alternative. The proposed model uses four subjective conditional probabilities which are based on the presence and absence of an alternative, and on the occurrence and nonoccurrence of an outcome. Valence is defined as the difference between the attractiveness values of the occurrence of an alternative (or an outcome) and the nonoccurrence of an alternative (or an outcome). Also, differences between subjective probabilities replace instrumentalities. Alternative valence-instrumentality formulations are analyzed in relation to the proposed fourfold model. The proposed model can be understood as a consistent application of Vroom's model (1964, Work and motivation, New York: Wiley), while other models are judged to be relatively deficient. The fourfold model, in combination with a matrix formulation of expectancy concepts, addresses a number of criticisms of VI theory and makes possible a test of the theory within the limits of its own assumptions.

Subjective conditional probabilities—A new approach

A new technique for estimating the joint probability of a pair of future events is presented. In contrast with the approach previously suggested by Mitchell and Tydeman, the new approach uses a five-outcome model rather than an eight-outcome model and hence requires fewer subjective estimates from the respondents. Conditions for pairwise consistency are discussed.