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.     

Subjective Expected Utility Theory with State-Dependent Preferences

This paper extends Savage′s subjective expected utility theory to include state-dependent preferences. The dependence of the decision maker′s preferences over consequences on the states of nature is represented by state-specific mappings of the set of consequences onto itself. Within this framework Savage′s postulates are reformulated and it is shown that there exist subjective expected utility representations of the preference relation over acts with unique, nonatomic, probability measure on the algebra of all events, and a state-dependent utility function over the set of consequences. Journal of Economic Literature Classification Number: D81.     

A theory of anticipated utility

A new theory of cardinal utility, with an associated set of axioms, is presented. It is a generalization of the von Neumann-Morgenstern expected utility theory, which permits the analysis of phenomena associated with the distortion of subjective probability.     

A new integral for capacities

A new integral for capacities is introduced and characterized. It differs from the Choquet integral on non-convex capacities. The main feature of the new integral is concavity, which might be interpreted as uncertainty aversion. The integral is extended to fuzzy capacities, which assign subjective expected values to random variables (e.g., portfolios) and may assign subjective probability only to a partial set of events. An equivalence between the minimum over sets of additive capacities (not necessarily probability distributions) and the integral w.r.t. fuzzy capacities is demonstrated. The extension to fuzzy capacities enables one to calculate the integral also in cases where the information available is limited to a few events. I wish to thank Eran Hanany, David Schmeidler, Eilon Solan and especially Yaron Azrieli and the anonymous referee of Economic Theory for their helpful comments.     

Subjective evaluations of n-person games

Using an axiomatization of subjective expected utility due to Fishburn, we characterize a class of utility functions over a set of n-person games in characteristic-function form. A probabilistic value is defined as the expectation of some player's marginal contribution with respect to some probability measure on the set of coalitions of other players. We decribe conditions under which a utility function on the set of n-person games is a probabilistic value; we prove as well an analogous result for simple games. We present additional axioms that characterize the semivalues and, in turn, the Shapley and Banzhaf values.     

On the uniqueness of subjective probabilities

Summary The purpose of this paper is twofold: First, within the framework of Savage (1954), we suggest axiomatic foundations for the representation of event-dependent preference relations over acts. This representation has the form of expectation of event-debendent utility with respect to non-unique subjective probabilities on the set of states. Second, we give an economic-theoretic motivation for selecting a unique probability distribution as an appropriate concept of subjective probabilities. However, unlike in Savage's theory, this notion of subjective probabilities does not necessarily represent the decisions-maker's belief regarding the likelihood of events.Our approach involves a departure from Savage's postulate P4, which guarantees the completeness of Savage's likelihood relation on the set of all events. Instead, we assume the existence of a finite partition of the set of states, {S ₁,...S _n}, such that, for events within each element of this partition P4 is satisfied. This weakening of Savage's axioms suffices for the existence of an expected event-dependent utility representation, but not for the uniqueness of the subjective probabilities.In many economic problems involving decision-making under uncertainty the existence of a unique probability is presumed and, in fact, is essential for the statement of the result. An example is Arrow's (1965) finding that all risk averse decision-makers will invest in a risky asset provided its expected rate of return exceeds that of an alternative risk-free asset. We show that a unique probability distribution can be chosen so as to render such results meaningful. Namely, any risk averse decision-maker will hold a positive position in the risky asset if and only if its expected rate of return with respect to the chosen probability exceeds that of the riskless asset.The research described in this paper began while the authors visited the Mathematisches Forschungsinstitut Oberwolfach, Germany. It was carried out in part while the second author visited the Santa Fe Institute in Sante Fe, New Mexico, USA and the Autonomous University of Barcelona, Spain. The second author would also like to acknowledge the financial support by NSF grant 911873.     

Proper scoring rules for general decision models

If a decision maker whose behavior conforms to the max-min expected utility model is faced with a scoring rule for a subjective expected utility decision maker, she will always announce a probability belonging to her set of priors; moreover, for any prior in the set, there is a scoring rule inducing the agent to announce that prior. We also show that on the domain of Choquet expected utility preferences with risk neutral lottery evaluation and totally monotone capacities, proper scoring rules do not exist. This implies the non-existence of proper scoring rules for any larger class of preferences (CEU with convex capacities, multiple priors).     

Logic,rationality and knowledge in Ramsey's thought: reassessing ‘human logic’

This paper reconsiders Frank Ramsey's essay on subjective probability (1926) as a consistent way to articulate logic, rationality and knowledge. The first part of the essay builds an axiomatic theory of subjective probability based on ‘formal logic’, defining rationality as choice-consistency. The second part seems to open up different horizons: the evaluation of degrees of belief by ‘human logic’. Because of the interest Keynes (1931) had taken in ‘human logic’, it was considered to be a possible alternative to the formal logic underlying the neoclassical theory of individual behaviour. The analysis of Ramsey's method in the entire paper, the relation between logic and rationality it constructs and the conception of uncertainty it reveals, lead me to note on the contrary that Ramsey's human logic was a complementary logic rather than an alternative to formal logic. Defining a standard to evaluate beliefs formation according to a frequentist criterion, it completes a normative representation of rationality which supports an original theory of knowledge that appears more in line with further developments of neoclassical methodology than with Keynesian economics.     

Subjective probabilities on “small” domains

The Savagian choice-theoretic construction of subjective probability does not apply to preferences, like those in the Ellsberg Paradox, that reflect a distinction between risk and ambiguity. We formulate two representation results—one for expected utility, the other for probabilistic sophistication—that derive subjective probabilities but only on a “small” domain of risky events. Risky events can be either specified exogenously or in terms of choice behavior; in the latter case, both the values and the domain of probability are subjective. The analysis identifies a mathematical structure—called a mosaic—that is intuitive for both exogenous and behavioral specifications of risky events. This structure is weaker than an algebra or even a λ-system.     

An analysis of credit terms in the eurodollar market

A model is presented incorporating various elements involved in the determination of theinterest rate in a capital market that is not purely competitive. It yields an equilibrium relation between the interest rate and the probability of default such that higher probability implies higher interest. Other factors affecting the rate of interest are the elasticity of demand, the perceived rate of loss related to default, and an extra premium due to risk aversion. The equilibrium relation is used as an econometric model which, under appropriate specifications, generates estimates of the weights attached to subjective risk indicators. The data cover transactions in the Eurocurrency market and deal only with publicly guaranteed loans to developing countries. Several economic indicators are identified as significantly affecting the subjective probability. These can be used to generate estimates of the subjective probabilities themselves.     

The impact of religiosity on self- assessments of health and happiness: evidence from the US Southwest

Using an age stratified random sample from an ongoing population-based study of Mexican Americans 45 years of age or older living in the Southwest this study fexamines the relationship between religiosity and self-rated indices of physical health, subjective health status and happiness. After estimating a set of binary logit models and controlling for individual characteristics such as age and gender, findings indicate that religiously involved respondents have a lower probability of reporting a health problem than those less or not involved. Further, those respondents who attend religious services regularly are more likely to assess themselves healthier and happier than those reporting sporadic or no attendance. However, when the religious variable is factored into six constructs, as the frequency of religious attendance increases the happiness measure initially increases to an inflection point then it continues to increase but at a slower rate. This result is consistent with the argument that those individuals who, on average, attend religious services once a week appear to reap the greatest incremental rewards in terms of assessments of subjective health and overall happiness.     

Home Bias in International Stock Return Expectations

Despite the advantages of international portfolio diversification, actual equity portfolio holdings reveal a strong bias towards domestic stocks. One hypothesis is that this bias can be explained by stock return expectations expressed in probability judgments. To test that hypothesis and to analyze the underlying effects that might cause distortions in investors' expectations, we conducted a cross country study in Germany and the U.S. comparing participants' judgments about an identical set of German and U.S. stocks.

Results show that both test groups feel more competent about domestic stocks. The asymmetric perception of competence is connected with an asymmetric assessment of probabilities. For both test groups subjective probability distributions of stock returns are significantly less dispersed and more optimistic for stocks associated with high competence levels than for stocks associated with low competence levels.     

城乡分割、就业状况与主观幸福感差异

本文讨论了不同人群主观幸福感的影响因素，并在对概率模型进行分解分析的基础上，对城乡之间以及城镇失业与就业居民之间的主观幸福感差异进行了比较分析。本文的结论发现，农村居民的主观幸福感高于城镇居民主要是由于预期的满足程度、收入变化预期、对生活状态改善的评价等差异所引起的；城镇失业与就业居民的主观幸福感差异则主要是由收入效应导致的。     

Survival expectations,subjective health and smoking: evidence from SHARE

This article aims to assess how the risk perceptions of smokers affect survival expectations and subjective health. Data from the Survey of Health, Ageing and Retirement in Europe, which include a numerical measure of subjective survival probability, are used to estimate a joint recursive system of equations that describe the relationships among survival expectations, subjective health status and smoking duration. A finite mixture model is used to address endogeneity and unobservable heterogeneity. This approach allows for two types of individuals with different observable characteristics to be identified in the examined population. We find that only in the population of the first type, current and former smokers incorporate the effects of smoking duration into their assessments of survival probabilities. For both types, quitting smoking affects current perceptions of smoking risks, causing the overestimation of both survival probability and subjective health.     

Unanimous subjective probabilities

This note shows that if the space of events is sufficiently rich and the subjective probability function of each individual is non-atomic, then there is a σ-algebra of events over which everyone will have the same probability function, and moreover, the range of this common probability is the entire unit interval.      

基于信息修正的非期望效用模型

人们对事件发生的可能性存在着主观判断。在不同的概率区间,人们对概率变化的敏感度是不一样的。传统的期望效用理论忽视了决策者对概率的主观反应,无法准确描述风险决策行为。基于信息修正的非期望效用模型,将客观概率转换成主观决策权重,可以弥补期望效用模型在捕捉决策者对概率主观反应方面的缺陷;同时,利用基于信息修正的非期望效用模型,通过量化人们在购买保险或股票时对风险的主观概率判断,可以对人们的保险需求和证券投资行为作出更好的解释或预测。     

Option smiling when investors’ estimates of asset volatility disagree

This study sheds light on why heterogeneous beliefs in volatility manifest the smile effect of options and how the degree of belief disagreement influences option-implied volatility. It is found that when investors’ level of heterogeneous beliefs increases, agents who over estimate volatility raise their subjective probability about those outer dividend states and hence increase more consumption to those states. As a result, an increase of consumption causes a decrease of their marginal rate of transformation in outer states. The raising subjective probability increases the prices of outer states, while the lower marginal rate of transformation decreases the prices of that. However, the influence of the former dominates the latter, causing a fatter tail of the state-price density. As a consequence, the values increase for those call options with high strike prices and put options with low strike prices, leading to a U-shaped implied volatility and hence causes a smell effect.     

Risk and decentralization in the theory of economic policy

In this note, uncertainty is incorporated into Tinbergen's model of economic policy. If the Central Planning Authority's preferences amomg subjective probability distributions can be described by a von Neumann-Morgenstern utility function, and if certain restrictions are imposed on these probability distributions, then the Authority can delegate control over partitions of the instrument vector to its Agencies and so decentralize its decision process.     

城乡分割、就业状况与主观幸福感差异

本文讨论了不同人群主观幸福感的影响因素,并在对概率模型进行分解分析的基础上,对城乡之间以及城镇失业与就业居民之间的主观幸福感差异进行了比较分析。本文的结论发现,农村居民的主观幸福感高于城镇居民主要是由于预期的满足程度、收入变化预期、对生活状态改善的评价等差异所引起的;城镇失业与就业居民的主观幸福感差异则主要是由收入效应导致的。     

Revisiting Cheerful Jane and Miserable John: the impact of income,good health,social contacts and education declines with increasing subjective well-being

This short note seeks to replicate the quantile regression analysis in Binder and Coad (2011), but taking into account individual-specific fixed effects (FE; using the British Household Panel Survey data set). It finds declining effects of the four main variables of interest (health, social life, income, education) over the quantiles of the subjective well-being distribution, with attenuated effect sizes for the FE model. Equivalized log income has a negative impact on subjective well-being throughout the distribution. Apart from a number of robustness checks, existing research is extended by looking into the quantile effects of the above variables on a set of domain satisfactions.