Subjective Probability Measure Research Articles

Equilibrium multi-agent model with heterogeneous views on fundamental risks

This paper investigates an equilibrium-based multi-agent optimal consumption and portfolio problem incorporating uncertainties on fundamental risks, where multiple agents have heterogeneous (conservative, neutral, aggressive) views on the risks represented by Brownian motions. Each agent maximizes its expected utility on consumption under its subjective probability measure. Specifically, we formulate the individual optimization problem as a sup–sup–inf problem, which is an optimal consumption and portfolio problem with a choice of a probability measure. Moreover, we provide an expression of the state-price density process in a market equilibrium, which derives the representations of the interest rate and the market price of risk. To the best of our knowledge, this is the first attempt to investigate the multi-agent model with heterogeneous views on the risks by considering a market equilibrium and solving sup–sup–inf problems on the choice of a probability measure. We emphasize that the setting, where each agent has heterogeneous views on different risks, incorporates a special case where each agent has only conservative or neutral views on risks with different degrees of conservativeness. Also, the setting includes the case where the agents have aggressive views on risks, commonly observed as bullish sentiments in the financial markets in the monetary easing after the global financial crisis and particularly in the COVID-19 pandemic. Finally, we present numerical examples of the interest rate model, which show how heterogeneous views of the multiple agents on the risks affect the shape of the yield curve.

Bayesian Decision Theory and Stochastic Independence

As stochastic independence is essential to the mathematical development of probability theory, it seems that any foundational work on probability should be able to account for this property. Bayesian decision theory appears to be wanting in this respect. Savage’s postulates on preferences under uncertainty entail a subjective expected utility representation, and this asserts only the existence and uniqueness of a subjective probability measure, regardless of its properties. What is missing is a preference condition corresponding to stochastic independence. To fill this significant gap, the article axiomatizes Bayesian decision theory afresh and proves several representation theorems in this novel framework.

Budget-constrained optimal retention with an upper limit on the retained loss

ABSTRACTUnlike sophisticated institutional insurance buyers, individual insurance seekers often use simple heuristic tools for risk management purposes, such as requiring that an insurance arrangement will not result in a retained loss that exceeds a certain predetermined and fixed level. In this paper, we re-examine the problem of budget-constrained demand for insurance indemnification when the insured and the insurer disagree about the likelihoods associated with the realizations of the insurable loss; but we further impose an additional upper-limit constraint on the retained loss and assume that the insurer distorts his subjective probability measure. We do not impose the no sabotage condition on admissible indemnities. Instead, we impose a state-verification cost that the insurer can incur in order to verify the loss severity, which rules out ex post moral hazard issues that could otherwise arise from possible misreporting of the loss by the insured. We characterize the optimal retention function and show that it has a simple two-part structure: zero retention (full insurance) on an event to which the insurer assigns zero probability, and a retention that could be described as a limited variable deductible on the complement of this event. As an illustration, we examine the case of a distorted Esscher premium principle.

Budget-Constrained Optimal Insurance with an Upper Limit on the Retained Loss

Unlike sophisticated institutional insurance buyers, individual insurance seekers often use simple heuristic tools for risk management purposes, such as requiring that an insurance arrangement will not result in a retained loss that exceeds a certain predetermined and fixed level. In this paper, we re-examine the problem of budget-constrained demand for insurance indemnification when the insured and the insurer disagree about the likelihoods associated with the realizations of the insurable loss; but we further impose an additional upper-limit constraint on the retained loss and assume that the insurer distorts his subjective probability measure. We do not impose the no sabotage condition on admissible indemnities. Instead, we impose a state-verification cost that the insurer can incur in order to verify the loss severity, which rules out ex post moral hazard issues that could otherwise arise from possible misreporting of the loss by the insured. We characterize the optimal retention function and show that it has a simple two-part structure: zero retention (full insurance) on an event to which the insurer assigns zero probability, and a retention that could be described as a limited variable deductible on the complement of this event. As an illustration, we examine the case of a distorted Esscher premium principle.

Probability, coherent belief and coherent belief changes

This paper is about the statics and dynamics of belief states that are represented by pairs consisting of an agent’s credences (represented by a subjective probability measure) and her categorical beliefs (represented by a set of possible worlds). Regarding the static side, we argue that the latter proposition should be coherent with respect to the probability measure and that its probability should reach a certain threshold value. On the dynamic side, we advocate Jeffrey conditionalisation as the principal mode of changing one’s belief state. This updating method fits the idea of the Lockean Thesis better than plain Bayesian conditionalisation, and it affords a flexible method for adding and withdrawing categorical beliefs. We show that it fails to satisfy the traditional principles of Inclusion and Preservation for belief revision and the principle of Recovery for belief withdrawals, as well as the Levi and Harper identities. We take this to be a problem for the latter principles rather than for the idea of coherent belief change.

Asking for frequencies rather than percentages increases the validity of subjective probability measures: Evidence from subjective life expectancy

Survey measures of subjective expectations manifest anomalies in how people report percentages. The current research finds that frequency-based measures deliver more valid subjective probabilities of living to a given age than do questions that elicit a percentage chance.

Assessing the benefits of robbery reduction: The case of a large Brazilian city

Abstract We estimate willingness to pay (WTP) for a first-order stochastic reduction on the risk of robbery at a large city in Brazil. We develop a structural choice model that nests a process of contingent valuation among non-degenerate lotteries and estimate it by both maximum likelihood and geographically weighted regression (GWR) using a dataset from the city of Fortaleza that contains a precise measurement of individual's subjective probability of victimization among detailed socioeconomic measurement, willingness to pay, and police presence variables. Expected loss, gender, age, education, and perception of patrolling explain WTP. Our global model estimated a mean WTP of R$ 19.23 (U$ 10.33) per month. Our local model, estimated by GWR, suggests that there is a reasonable amount of spatial heterogeneity that follows the city's socioeconomic spatial distribution profile. Although the city's northwest periphery presents higher WTP, as long as we go inwards, there is plenty of heterogeneity on its spatial distribution. Our results support a theory of crime with an active role for victim's (costly) precautions influenced by socioeconomic spatial heterogeneity.

Expected Utility in 3D

Consider a subjective expected utility preference relation. It is usually held that the representations which this relation admits differ only in one respect, namely, the possible scales for the measurement of utility. In this paper, I discuss the fact that there are, metaphorically speaking, two additional dimensions along which infinitely many more admissible representations can be found. The first additional dimension is that of state-dependence. The second—and, in this context, much lesser-known—additional dimension is that of act dependence. The simplest implication of their usually neglected existence is that the standard axiomatizations of subjective expected utility fail to provide the measurement of subjective probability with satisfactory behavioral foundations.

An expected utility theory for state-dependent preferences

This note is a generalization and improved interpretation of the main result of Karni and Schmeidler (An Expected utility theory for state-dependent preferences. Working paper no. 48-80 of the Foerder Institute for Economic Research, Faculty of Social Sciences, Tel Aviv University, 1980). A decision-maker is supposed to possess a preference relation on acts and another preference relation on state-prize lotteries, both of which are assumed to satisfy the von Neumann–Morgenstern axioms. In addition, the two preference relations restricted to a state of nature are assumed to agree. We show that these axioms are necessary and sufficient for the existence of subjective expected utility over acts with state-dependent utility functions and a subjective probability measure. This subjective probability measure is unique when conditioned on the set of states of nature in which not all the prizes are equally desirable.

A New Probabilistic Transformation of Belief Mass Assignment

In this paper, we propose in Dezert-Smarandache Theory (DSmT) framework, a new probabilistic transformation, called DSmP, in order to build a subjective probability measure from any basic belief assignment defined on any model of the frame of discernment. Several examples are given to show how the DSmP transformation works and we compare it to main existing transformations proposed in the literature so far. We show the advantages of DSmP over classical transformations in term of Probabilistic Information Content (PIC). The direct extension of this transformation for dealing with qualitative belief assignments is also presented.

A Graded Bayesian Coherence Notion

Coherence is a key concept in many accounts of epistemic justification within ‘traditional’ analytic epistemology. Within formal epistemology, too, there is a substantial body of research on coherence measures. However, there has been surprisingly little interaction between the two bodies of literature. The reason is that the existing formal literature on coherence measure operates with a notion of belief system that is very different from—what we argue is—a natural Bayesian formalisation of the concept of belief system from traditional epistemology. Therefore, formal epistemology has so far only been concerned with one particular—arguably not even very natural—way of formalising coherence of belief systems; it has by no means refuted the viability of coherentism. In contrast to the existing literature, we formalise belief systems as families of assignments of (conditional) degrees of belief (which may be compatible with several subjective probability measures). Within this framework, we propose a Bayesian formalisation of the thrust of BonJour’s coherence concept in The structure of empirical knowledge (Harvard University Press, Cambridge, 1985), using a combination of Bayesian confirmation theory and basic graph theory. In excursions, we introduce graded notions for both logical and probabilistic consistency of belief systems—the latter being based on certain geometrical structures induced by probabilistic belief systems. For illustration, we reconsider BonJour’s “ravens” challenge (op. cit., p. 95f.). Finally, potential objections to our proposed formal coherence notion are explored.

The Measurement of Subjective Probability: Evaluating the Sensitivity and Accuracy of Various Scales

The RISK of an event generally relates to its expected severity and the perceived probability of its occurrence. In RISK research, however, there is no standard measure for subjective probability estimates. In this study, we compared five commonly used measurement formats--two rating scales, a visual analog scale, and two numeric measures--in terms of their ability to assess subjective probability judgments when objective probabilities are available. We varied the probabilities (low vs. moderate) and severity (low vs. high) of the events to be judged as well as the presentation mode of objective probabilities (sequential presentation of singular events vs. graphical presentation of aggregated information). We employed two complementary goodness-of-fit criteria: the correlation between objective and subjective probabilities (sensitivity), and the root mean square deviations of subjective probabilities from objective values (accuracy). The numeric formats generally outperformed all other measures. The severity of events had no effect on the performance. Generally, a rise in probability led to decreases in performance. This effect, however, depended on how the objective probabilities were encoded: pictographs ensured perfect information, which improved goodness of fit for all formats and diminished this negative effect on the performance. Differences in performance between scales are thus caused only in part by characteristics of the scales themselves-they also depend on the process of encoding. Consequently, researchers should take the source of probability information into account before selecting a measure.

Essay Review: The Laws of Belief

Wolfgang Spohn’s book The Laws of Belief: Ranking Theory and Its Philosophical Applications is a detailed presentation of ranking theory and its many and varied applications in philosophy. It is a work in formal or mathematical philosophy, philosophy that uses formal or mathematical tools to tackle philosophical problems. The book uses one simple and elegant yet powerful formal tool, namely, the theory of ranking functions, and applies it fruitfully to a great variety of philosophical problems. These start with problems in epistemology such as the representation of belief and its objects as well as justification and the a priori. They go to problems in metaphysics such as causation and objectification and problems in the philosophy of science such as laws and their confirmation as well as ceteris paribus conditions and dispositions. And they include, at least indirectly, problems in the philosophy of language such as conditionals and problems in the philosophy of mind such as perception and consciousness. The result is a unified picture of the world and our relation to it as epistemic agents: a philosophy. It is arrived at in a constructive attempt to reduce all “natural” or alethic modalities to nonmodal facts on the one hand and doxastic modalities as represented by ranking functions (and subjective probability measures) on the other hand. And it does so by displaying the virtues of using formal tools to tackle philosophical problems. The result is a formal philosophy whose radical nature, however, becomes visible only by working through the details, not something one can do over an extended weekend.

The enduring allure of objective probability

Probability measures are properly considered subjective, if only as a result of the knowledge limitations of human experience. Even if objective probability can be demonstrated as an unambiguously verifiable property of the physical world, our necessarily imperfect access to such aspects of reality would necessitate the construction of subjective probability measures for examinations of human behavior. This paper will explore the distinction between subjective and purportedly objective probability in principle, and demonstrate that actual probabilities can only be subjective, at least ex ante. The paper will next examine the persistent efforts of philosophers, mathematicians, and particularly quantitatively-inclined social scientists, to construct and employ mistaken conceptions of purportedly objective probability. The neo-indeterminist interpretation of sub-atomic particle physics is addressed next, with its attempt to construct an objective-probabilistic account of reality based on the Heisenberg uncertainty principle. An explanation is suggested for the persistent allure of objective probability in the social sciences, which purports to offer a more economical and parsimonious basis for human behavior and the formation of expectations. The view that there are ontologically privileged, objectively-verifiable probabilities and expectations, will be examined and criticized. It appears to be one source of the discredited, though nonetheless widespread and persistent, belief in the efficacy and feasibility of central economic planning.

The term structure of inflation expectations

We use information in the term structure of survey-based forecasts of inflation to estimate a factor hidden in the nominal yield curve. We construct a model that accommodates forecasts over multiple horizons from multiple surveys and Treasury real and nominal yields by allowing for differences between risk-neutral, subjective, and objective probability measures. We establish that model-based inflation expectations are driven by inflation, output, and one latent factor. We find that this factor affects inflation expectations at all horizons but has almost no effect on the nominal yields; that is, the latent factor is hidden. We show that this hidden factor is not related to either current and past inflation or the standard set of macro variables studied in the literature. Consistent with the theoretical property of a hidden factor, our model outperforms a standard macro-finance model in its forecasting of inflation and yields.

A Note on Confidence Momentum and Term Structure of Confidence with Applications to Financial Markets

This note is based on a recent confidence index introduced in the context of compensating probability factors for deviations of subjective probability measures from equivalent martingale measures. The index is adjusted for loss gain probability spreads, and it explains momentum in confidence. We use the index to introduce a confidence matrix operator which shows how a subject transforms gain domain into fear of loss. So she is loss averse or risk averse. By contrast, the adjoint confidence matrix operator is an Euclidean motion which rotates and reverses loss domain into hope of gain. Thus, signifying risk seeking over loss domains in hope of gain. Simulation of the model shows that the distribution of prior loss [gain] probabilities is a predictor of confidence momentum and fields of confidence. Moreover, our field theory of confidence mimics a sample of Gallup Monthly Economic Confidence Index, and depicts a term structure of confidence for hope and fear. It plainly shows that the growth rate of Gallup Economic Confidence Index -- which is highly correlated with popular confidence indexes like UBS/Gallup Investor Optimism Index; Michigan Consumer Confidence Index; and Yale Investor Confidence Index -- predict bubbles and crashes.

Coherent risk measure for derivatives under Black‐Scholes economy with regime switching

PurposeThe purpose of this paper is to provide a scenario‐based risk measure for a portfolio of European‐style derivative securities over a fixed time horizon under the regime switching Black‐Scholes economy.Design/methodology/approachThe risk measure is constructed by using the risk‐neutral probability, the physical probability and a family of subjective probability measures. The subjective probabilities can be interpreted as risk managers or regulators' risk preferences and/or subjective beliefs.FindingsThe authors provide closed form expressions for the European option and barrier option.Research limitations/implicationsThe results are difficult to apply to a portfolio with many different kinds of options.Practical implicationsThe results provide some insights on risk management of portfolios with derivatives.Originality/valueThe paper presents a scenario‐based risk measure for a portfolio of European‐style derivative securities over a fixed time horizon under the regime switching Black‐Scholes economy. Risk management is the most important task for almost all financial industries, although it cannot be claimed that the method and results of this paper solve the problem, it is believed to provide some insights to the problem, albeit theoretical. For vanilla European options and barrier options, the authors obtained a closed form expression for the risk measure. The idea of this paper can be applied to some other exotic options. For portfolios containing different kinds of derivatives, the results of this paper provide some guideline and insights.

A New Probabilistic Transformation in Generalized Power Space

The mapping from the belief to the probability domain is a controversial issue, whose original purpose is to make (hard) decision, but for contrariwise to erroneous widespread idea/claim, this is not the only interest for using such mappings nowadays. Actually the probabilistic transformations of belief mass assignments are very useful in modern multitarget multisensor tracking systems where one deals with soft decisions, especially when precise belief structures are not always available due to the existence of uncertainty in human being's subjective judgments. Therefore, a new probabilistic transformation of interval-valued belief structure is put forward in the generalized power space, in order to build a subjective probability measure from any basic belief assignment defined on any model of the frame of discernment. Several examples are given to show how the new transformation works and we compare it to the main existing transformations proposed in the literature so far. Results are provided to illustrate the rationality and efficiency of this new proposed method making the decision problem simpler.

A simplified axiomatic approach to ambiguity aversion

This paper takes the Anscombe-Aumann framework with horse and roulette lotteries, and applies the Savage axioms to the horse lotteries and the von Neumann-Morgenstern axioms to the roulette lotteries. The resulting representation of preferences yields a subjective probability measure over states and two utility functions, one governing risk attitudes and one governing ambiguity attitudes. The model is able to accommodate the Ellsberg paradox and preferences for reductions in ambiguity.

Probability Logic of Finitely Additive Beliefs

Probability logics have been an active topic of investigation of beliefs in type spaces in game theoretical economics. Beliefs are expressed as subjective probability measures. Savage's postulates in decision theory imply that subjective probability measures are not necessarily countably additive but finitely additive. In this paper, we formulate a probability logic Σ+ that is strongly complete with respect to this class of type spaces with finitely additive probability measures, i.e. a set of formulas is consistent in Σ+ iff it is satisfied in a finitely additive type space. Although we can characterize Σ+-theories satisfiable in the class as maximally consistent sets of formulas, we prove that any canonical model of maximally consistent sets is not universal in the class of type spaces with finitely additive measures, and, moreover, it is not a type space. At the end of this paper, we show that even a minimal use of probability indices causes the failure of compactness in probability logics.