Three pseudo-paradoxes in ‘quantum’ decision theory: Apparent effects of observation on probability and utility

In quantum domains, the measurement (or observation) of one of a pair of complementary variables introduces an unavoidable uncertainty in the value of that variable's complement. Such uncertainties are negligible in Newtonian worlds, where observations can be made without appreciably disturbing the observed system. Hence, one would not expect that an observation of a non-quantum probabilistic outcome could affect a probability distribution over subsequently possible states, in a way that would conflict with classical probability calculations. This paper examines three problems in which observations appear to affect the probabilities and expected utilities of subsequent outcomes, in ways which may appear paradoxical. Deeper analysis of these problems reveals that the anomalies arise, not from paradox, but rather from faulty inferences drawn from the observations themselves. Thus the notion of quantum decision theory is disparaged.     

A deterministic event tree approach to uncertainty,randomness and probability in individual chance processes

Both Popper and Good have noted that a deterministic microscopic physical approach to probability requires subjective assumptions about the statistical distribution of initial conditions. However, they did not use such a fact for defining an a priori probability, but rather recurred to the standard observation of repetitive events. This observational probability may be hard to assess for real-life decision problems under uncertainty that very often are - strictly speaking - non-repetitive, one-time events. This may be a reason for the popularity of subjective probability in decision models. Unfortunately, such subjective probabilities often merely reflect attitudes towards risk, and not the underlying physical processes.In order to get as objective as possible a definition of probability for one-time events, this paper identifies the origin of randomness in individual chance processes. By focusing on the dynamics of the process, rather than on the (static) device, it is found that any process contains two components: observer-independent (= objective) and observer-dependent (= subjective). Randomness, if present, arises from the subjective definition of the rules of the game, and is not - as in Popper's propensity - a physical property of the chance device. In this way, the classical definition of probability is no longer a primitive notion based upon equally possible cases, but is derived from the underlying microscopic processes, plus a subjective, clearly identified, estimate of the branching ratios in an event tree. That is, equipossibility is not an intrinsic property of the system object/subject but is forced upon the system via the rules of the game/measurement.Also, the typically undefined concept of symmetry in games of chance is broken down into objective and subjective components. It is found that macroscopic symmetry may hold under microscopic asymmetry. A similar analysis of urn drawings shows no conceptual difference with other games of chance (contrary to Allais' opinion). Finally, the randomness in Lande's knife problem is not due to objective fortuity (as in Popper's view) but to the rules of the game (the theoretical difficulties arise from intermingling microscopic trajectories and macroscopic events).Dedicated to Professor Maurice Allais on the occasion of the Nobel Prize in Economics awarded December, 1988.     

A Characterization for the Spherical Scoring Rule

Strictly proper scoring rules have been studied widely in statistical decision theory and recently in experimental economics because of their ability to encourage assessors to honestly provide their true subjective probabilities. In this article, we study the spherical scoring rule by analytically examining some of its properties and providing some new geometric interpretations for this rule. Moreover, we state a theorem which provides an axiomatic characterization for the spherical scoring rule. The objective of this analysis is to provide a better understanding of one of the most commonly available scoring rules, which could aid decision makers in the selection of an appropriate tool for evaluating and assessing probabilistic forecasts.      

Coherent decision analysis with inseparable probabilities and utilities

This article explores the extent to which a decision maker's probabilities can be measured separately from his/her utilities by observing his/her acceptance of small monetary gambles. Only a partial separation is achieved: the acceptable gambles are partitioned into a set of belief gambles, which reveals probabilities distorted by marginal utilities for money, and a set of preference gambles, which reveals utilities reciprocally distorted by marginal utilities for money. However, the information in these gambles still enables us to solve the decision maker's problem: his/her utility-maximizing decision is the one that avoids arbitrage (i.e., incoherence or Dutch books).     

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 foundation of Bayesian statistics (How to deal with fears)

A rational statistical decision maker whose preferences satisfy Savage's axioms will minimize a Bayesian risk function: the expectation with respect to a revealed (or subjective) probability distribution of a loss (or negative utility) function over the consequences of the statistical decision problem. However, the nice expected utility form of the Bayesian risk criterion is nothing but a representation of special preferences. The subjective probability is defined together with the utility (or loss) function and it is not possible, in general, to use a given loss function - say a quadratic loss - and to elicit independently a subjective distribution.I construct the Bayesian risk criterion with a set of five axioms, each with a simple mathematical implication. This construction clearly shows that the subjective probability that is revealed by a decider's preferences is nothing but a (Radon) measure equivalent to a linear functional (the criterion). The functions on which the criterion operates are expected utilities in the von Neumann-Morgenstern sense. It then becomes clear that the subjective distribution cannot be eliciteda priori, independently of the utility function on consequences.However, if one considers a statistical decision problem by itself, losses, defined by a given loss function, become the consequences of the decisions. It can be imagined that experienced statisticians are used to dealing with different losses and are able to compare them (i.e. have preferences, or fears over a set of possible losses). Using suitable axioms over these preferences, one can represent them by a (linear) criterion: this criterion is the expectation of losses with respect to a (revealed) distribution. It must be noted that such a distribution is a measure and need not be a probability distribution.     

Stochastic Dominance and Prospect Dominance with Subjective Weighting Functions

Laboratory experiments with and without real money repeatedly reveal that even if all subjects observe the same pair of cumulative distributions F and G, they act as if they were other cumulative probability functions F* and G* different for different investors. Namely, the subjects assign (subjective) weights to the various probabilities. In their breakthrough article Kahneman and Tversky [1979] suggest that in making decisions under uncertainty, the subjects apply a monotonic transformation (p) where p are the probabilities, and investors make decisions by comparing (p) corresponding to the two distributions under consideration rather than by comparing the true probabilities, p, themselves.     

Known, Unknown, and Unknowable Uncertainties

In normative decision theory, the weight of an uncertain event in a decision is governed solely by the probability of the event. A large body of empirical research suggests that a single notion of probability does not accurately capture peoples' reactions to uncertainty. As early as the 1920s, Knight made the distinction between cases where probabilities are known and where probabilities are unknown. We distinguish another case –- the unknowable uncertainty –- where the missing information is unavailable to all. We propose that missing information influences the attractiveness of a bet contingent upon an uncertain event, especially when the information is available to someone else. We demonstrate that the unknowable uncertainty –- falls in preference somewhere in between the known and the known uncertainty.     

An Experimental Comparison of Induced and Elicited Beliefs

Understanding choice under risk requires knowledge of beliefs and preferences. A variety of methods have been proposed to elicit peoples beliefs. The efficacy of alternative methods, however, has not been rigorously documented. Herein we use an experiment to test whether an induced probability can be recovered using an elicitation mechanism based on peoples predictions about a random event. We are unable to recover the induced belief. Instead, the estimated belief is systematically biased in a way that is consistent with anecdotal evidence in the economics, psychology, and statistics literature: people seem to overestimate low and underestimate high probabilities.     

The Optimality of the Expert and Majority Rules Under Exponentially Distributed Competence

We study the uncertain dichotomous choice model. In this model a set of decision makers is required to select one of two alternatives, say support or reject a certain proposal. Applications of this model are relevant to many areas, such as political science, economics, business and management. The purpose of this paper is to estimate and compare the probabilities that different decision rules may be optimal. We consider the expert rule, the majority rule and a few in-between rules. The information on the decisional skills is incomplete, and these skills arise from an exponential distribution. It turns out that the probability that the expert rule is optimal far exceeds the probability that the majority rule is optimal, especially as the number of the decision makers becomes large.     

Second-order probabilities and belief functions

A second-order probability Q(P) may be understood as the probability that the true probability of something has the value P. True may be interpreted as the value that would be assigned if certain information were available, including information from reflection, calculation, other people, or ordinary evidence. A rule for combining evidence from two independent sources may be derived, if each source i provides a function Q _i (P). Belief functions of the sort proposed by Shafer (1976) also provide a formula for combining independent evidence, Dempster's rule, and a way of representing ignorance of the sort that makes us unsure about the value of P. Dempster's rule is shown to be at best a special case of the rule derived in connection with second-order probabilities. Belief functions thus represent a restriction of a full Bayesian analysis.     

Risk,uncertainty and hidden information

People are less willing to accept bets about an event when they do not know the true probability of that event. Such uncertainty aversion has been used to explain certain economic phenomena. This paper considers how far standard private information explanations (with strategic decisions to accept bets) can go in explaining phenomena attributed to uncertainty aversion. This paper shows that if two individuals have different prior beliefs about some event, and two sided private information, then each individuals willingness to bet will exhibit a bid ask spread property. Each individual is prepared to bet for the event, at sufficiently favorable odds, and against, at sufficiently favorable odds, but there is an intermediate range of odds where each individual is not prepared to bet either way. This is only true if signals are distributed continuously and sufficiently smoothly. It is not true, for example, in a finite signal model.     

Subjectively weighted linear utility

An axiomatized theory of nonlinear utility and subjective probability is presented in which assessed probabilities are allowed to depend on the consequences associated with events. The representation includes the expected utility model as a special case, but can accommodate the Ellsberg paradox and other types of ambiguity sensitive behavior, while retaining familiar properties of subjective probability, such as additivity for disjoint events and multiplication of conditional probabilities. It is an extension, to the states model of decision making under uncertainty, of Chew's weighted linear utility representation for decision making under risk.     

Violations of betweenness and choice shifts in groups

In decision theory, the betweenness axiom postulates that a decision maker who chooses an alternative A over another alternative B must also choose any probability mixture of A and B over B itself and can never choose a probability mixture of A and B over A itself. The betweenness axiom is a weaker version of the independence axiom of expected utility theory. Numerous empirical studies documented systematic violations of the betweenness axiom in revealed individual choice under uncertainty. This paper shows that these systematic violations can be linked to another behavioral regularity—choice shifts in a group decision making. Choice shifts are observed if an individual faces the same decision problem but makes a different choice when deciding alone and in a group.     

Tempered Regrets Under Total Ignorance

Several decision rules, including the minimax regret rule, have been posited to suggest optimizing strategies for an individual when neither objective nor subjective probabilities can be associated to the various states of the world. These all share the shortcoming of focusing only on extreme outcomes. This paper suggests an alternative approach of tempered regrets which may more closely replicate the decision process of individuals in those situations in which avoiding the worst outcome tempers the loss from not achieving the best outcome. The assumption of total ignorance of the probabilities associated with the various states is maintained. Applications and illustrations from standard neoclassical theory are discussed.     

E-Capacities and the Ellsberg Paradox

Ellsberg's (1961) famous paradox shows that decision-makers give events with known probabilities a higher weight in their outcome evaluation. In the same article, Ellsberg suggests a preference representation which has intuitive appeal but lacks an axiomatic foundation. Schmeidler (1989) and Gilboa (1987) provide an axiomatisation for expected utility with non-additive probabilities. This paper introduces E-capacities as a representation of beliefs which incorporates objective information about the probability of events. It can be shown that the Choquet integral of an E-capacity is the Ellsberg representation. The paper further explores properties of this representation of beliefs and provides an axiomatisation for them.     

Impersonal probability as an ideal assessment based on accessible evidence: A viable construct?

When risk analysts and others refer to the true probability of an event, it is not easy to give it a meaning which is sound and useful as a communication device for regulatory, research planning, and related purposes. An interpretation is herein offered which, unlike Bayesian probability, is impersonal and does not depend on a particular assessor; unlike Carnap's logical probability, it does not depend on information actually to hand. It is a generalization of frequency and propensity interpretations of impersonal probability applicable to unique events: an ideal assessment based on currently accessible (not in general perfect) evidence. The argument is illustrated from decision-aiding experience which motivated the enquiry.This work was supported by the National Science Foundation, Division of Social and Economic Sciences. The author thanks John Pratt, Marvin Cohen, Dennis Lindley, Jon Baron, Kathy Laskey, and Stephen Watson for their most helpful review. They do not necessarily share his views.     

Empirical evidence of two-attribute utility dependence on probability

We investigate utility dependence on probability using a new methodology that examines how indifference statements vary with the probability of obtaining times and costs of individual trips. Of 127 subjects, 8 supplied 3 (out of 3) sets of indifference statements consistent with probability independence. Those subjects with 2 or more sets of indifference statements violating probability independence exhibited a systematic dependence, in that knowing the direction of a subject's violation in one set of indifference statements would increase the likelihood of his or her violating other sets of indifference statements in the same direction. Data show that this systematic violation of dependence should not be attributed to artifacts of the experiment.     

Endogenous risk and protection premiums

Endogenous risk implies an individual perceives he can influence the likelihood that a state of nature will occur. To add structure to endogenous risk models, I define a protection premium for reduced uncertainty about protection efficiency when a stochastic variable enters the probability functionp(x) rather than the utility function. For a binary lottery, a measure of aversion of uncertain protection efficiency(x) =-p(x)/p(x) is defined to unambiguously determine the effects of increased risk on an individual's voluntary contribution to public good supply earmarked to reduce the probability of an undesirable state. Finally, I examine the protection premium in ann-state discrete lottery and when uncertainty exists in both the probability and utility function.