Conditional Desirability

Conditional attitudes are not the attitudes an agent is disposed to acquire in event of learning that a condition holds. Rather they are the components of agent's current attitudes that derive from the consideration they give to the possibility that the condition is true. Jeffrey's decision theory can be extended to include quantitative representation of the strength of these components. A conditional desirability measure for degrees of conditional desire is proposed and shown to imply that an agent's degrees of conditional belief are conditional probabilities. Rational conditional preference is axiomatised and by application of Bolker's representation theorem for rational preferences it is shown that conditional preference rankings determine the existence of probability and desirability measures that agree with them. It is then proven that every conditional desirability function agrees with an agent's conditional preferences and, under certain assumptions, every desirability function agreeing with an agent's conditional preferences is a conditional desirability function agreeing with her unconditional preferences.     

A Rule For Updating Ambiguous Beliefs

When preferences are such that there is no unique additive prior, the issue of which updating rule to use is of extreme importance. This paper presents an axiomatization of the rule which requires updating of all the priors by Bayes rule. The decision maker has conditional preferences over acts. It is assumed that preferences over acts conditional on event E happening, do not depend on lotteries received on E ^c, obey axioms which lead to maxmin expected utility representation with multiple priors, and have common induced preferences over lotteries. The paper shows that when all priors give positive probability to an event E, a certain coherence property between conditional and unconditional preferences is satisfied if and only if the set of subjective probability measures considered by the agent given E is obtained by updating all subjective prior probability measures using Bayes rule.     

Products of non-additive measures: a Fubini-like theorem

For non-additive set functions, the independent product, in general, is not unique and the Fubini theorem is restricted to slice-comonotonic functions. In this paper, we use the representation theorem of Gilboa and Schmeidler (Math Oper Res 20:197?C212, 1995) to extend the M?bius product for non-additive set functions to non-finite spaces. We extend the uniqueness result of Ghirardato (J Econ Theory 73:261?C291, 1997) for products of two belief functions and weaken the requirements on the marginals necessary to obtain the Fubini property in the product. More importantly, we show that for the M?bius product one side of the Fubini theorem holds for all integrable functions if one of the marginals either is a probability or a convex combination of a chain of unanimity games, i.e., we relax the requirement of slice-comonotonicity and enrich the set of possible applications.     

A challenge to the compound lottery axiom: A two-stage normative structure and comparison to other theories

This paper examines preferences among uncertain prospects when the decision maker is uneasy about his assignment of subjective probabilities. It proposes a two-stage lottery framework for the analysis of such prospects, where the first stage represents an assessment of the vagueness (ambiguity) in defining the problem's randomness and the second stage represents an assessment of the problem for each hypothesized randomness condition. Standard axioms of rationality are prescribed for each stage, including weak ordering, continuity, and strong independence. The Reduction of Compound Lotteries' axiom is weakened, however, so that the two lottery stages have consistent, but not collapsible, preference structures. The paper derives a representation theorem from the primitive preference axioms, and the theorem asserts that preference-consistent decisions are made as if the decision maker is maximizing a modified expected utility functional. This representation and its implications are compared to alternative decision models. Criteria for assigning the relative empirical power of the alternative models are suggested.     

Two-parameter decision models and rank-dependent expected utility

This paper extends the existing literature concerning the relationship between two parameter decision models and those based on expected utility in two main directions. The first relaxes Meyer's location and scale (or Sinn's linear class) condition and shows that a two-parameter representation of preferences over uncertain prospects and the expected utility representation yield consistent rankings of random variables when the decision maker's choice set is restricted to random variables differing by mean shifts and monotone meanpreserving spreads. The second shows that the rank-dependent expected utility model is also consistent with two-parameter ranking methods if the probability transform satisfies certain dominance conditions. The main implication of these results is that the simple two-parameter model can be used to analyze the comparative statics properties of a wide variety of economic models, including those with multiple sources of uncertainty when the random variables are comonotonic. To illustrate this point, we apply our results to the problem of optimal portfolio investment with random initial wealth. We find that it is relatively easy to obtain strong global comparative statics results even if preferences do not satisfy the independence axiom.     

The incoherence of agreeing to disagree

The agreeing-to-disagree theorem of Aumann and the no-expected-gain-from-trade theorem of Milgrom and Stokey are reformulated under an operational definition of Bayesian rationality. Common knowledge of beliefs and preferences is achieved through transactions in a contingent claims market, and mutual expectations of Bayesian rationality are defined by the condition of joint coherence,i.e., the collective avoidance of arbitrage opportunities. The existence of a common prior distribution and the impossibility of agreeing to disagree follow from the joint coherence requirement, but the prior must be interpreted as a risk-neutral distribution: a product of probabilities and marginal utilities for money. The failure of heterogenous information to create disagreements or incentives to trade is shown to be an artifact of overlooking the potential role of trade in constructing the initial state of common knowledge.     

Simple methods for evaluating and comparing binary experiments

We consider a confidence parametrization of binary information sources in terms of appropriate likelihood ratios. This parametrization is used for Bayesian belief updates and for the equivalent comparison of binary experiments. In contrast to the standard parametrization of a binary information source in terms of its specificity and its sensitivity, one of the two confidence parameters is sufficient for a Bayesian belief update conditional on a signal realization. We introduce a confidence-augmented receiver operating characteristic for comparisons of binary experiments for a class of “balanced” decision problems, relative to which the confidence order offers a higher resolution than Blackwell’s informativeness order.

Where observation is concerned, Chance favors only the prepared mind. —Louis Pasteur (1822–1895).

     

Ramsey's Foundations Extended to Desirabilities

In his Truth and Probability (1926), Frank Ramsey provides foundations for measures of degrees of belief in propositions and preferences for worlds. Nonquantitative conditions on preferences for worlds, and gambles for worlds and certain near-worlds, are formulated which he says insure that a subject's preferences for worlds are represented by numbers, world values. Numbers, for his degrees of belief in propositions, probabilities, are then defined in terms of his world values. Ramsey does not also propose definitions of desirabilities for propositions, though he is in a position to do this. Given his measures for probabilities of propositions and values of worlds, he can frame natural definitions for both evidential and causal desirabilities that would measure respectively the welcomeness of propositions as items of news, and as facts. His theory is neutral with respect to the evidential/causal division. In the present paper, as Ramsey's foundations are explained, several problems and limitations are noted. Their distinctive virtue – their evidential/causal neutrality – is demonstrated. Comparisons are made with other foundational schemes, and a perspective is recommended from which nonquantitative foundations are not the be all for quantitative theories of ideal preferences and credences.     

Bayesian decisions with ambiguous belief aversion

This study provides an empirical perspective on the effect of ambiguous environmental risk information on lottery preferences using a sample of 646 adults. The learning process follows a Bayesian expected utility model in terms of the overall magnitude and sign of the weights that respondents place on the risk information. Significant ambiguous belief aversion that is consistent with the Ellsberg paradox is also evident. The extent of this aversion increases with the size of the risk spread, but at a decreasing rate. These results are consistent with both probability-based and preference-based models of ambiguous probabilities. The findings also indicate the presence of cognitive limitations in the processing of risk information, but lead to rejection of more extreme models in which individuals respond in alarmist fashion or do not learn at all.     

Conditional expected,extensive utility

Luce and Krantz (1971) presented an axiom system for conditional expected utility. In this theory a conditional decision is a function whose domain is a non-null subevent and whose range is a subset of a set of consequences. Given a family of conditional decisions that is closed under unions of decisions whose domains are disjoint and under restrictions to non-null subevents, the second major primitive is an ordering of the family. Axioms were given that are adequate to construct a numerical utility function over decisions and a probability function over events for which the conditional expectation of the utility is order preserving. Several of the axioms are quite complex and seem a bit artificial, and the proof is very long. Here the structure is modified by adding to the set of outcomes a concatenation operation, and the representation theorem is modified by requiring that the utility function be additive over this binary operation as well as exhibiting the expected utility property. The advantages of this pair of changes are, first, it exploits the obvious fact that the union of consequences is itself a consequence; second, it reduces the mathematical burden carried by the set theoretic structure of conditional decisions and, as a result, the axioms can be made much easier to understand; and third, it permits a considerably shorter proof because one can draw more readily on known results. The major drawback of this approach is, of course, that it is inconsistent with the evidence that utility is not additive over consequences - at least, not over increasing amounts of a single good (diminishing marginal utility).This work was supported by a grant from the Alfred P. Sloan Foundation to the Institute for Advanced Study. I wish to thank P. C. Fishburn and F. S. Roberts for their comments.     

A Model of Minimal Probabilistic Belief Revision

In the literature there are at least two models for probabilistic belief revision: Bayesian updating and imaging [Lewis, D. K. (1973), Counterfactuals, Blackwell, Oxford; Gärdenfors, P. (1988), Knowledge in flux: modeling the dynamics of epistemic states, MIT Press, Cambridge, MA]. In this paper we focus on imaging rules that can be described by the following procedure: (1) Identify every state with some real valued vector of characteristics, and accordingly identify every probabilistic belief with an expected vector of characteristics; (2) For every initial belief and every piece of information, choose the revised belief which is compatible with this information and for which the expected vector of characteristics has minimal Euclidean distance to the expected vector of characteristics of the initial belief. This class of rules thus satisfies an intuitive notion of minimal belief revision. The main result in this paper is to provide an axiomatic characterization of this class of imaging rules.     

A possibilistic hierarchical model for behaviour under uncertainty

Hierarchical models are commonly used for modelling uncertainty. They arise whenever there is a `correct' or `ideal' uncertainty model but the modeller is uncertain about what it is. Hierarchical models which involve probability distributions are widely used in Bayesian inference. Alternative models which involve possibility distributions have been proposed by several authors, but these models do not have a clear operational meaning. This paper describes a new hierarchical model which is mathematically equivalent to some of the earlier, possibilistic models and also has a simple behavioural interpretation, in terms of betting rates concerning whether or not a decision maker will agree to buy or sell a risky investment for a specified price. We give a representation theorem which shows that any consistent model of this kind can be interpreted as a model for uncertainty about the behaviour of a Bayesian decision maker. We describe how the model can be used to generate buying and selling prices and to make decisions.     

A Theory of Rational Choice under Ignorance

This paper contributes to a theory of rational choice for decision-makers with incomplete preferences due to partial ignorance, whose beliefs are representable as sets of acceptable priors. We focus on the limiting case of `Complete Ignorance' which can be viewed as reduced form of the general case of partial ignorance. Rationality is conceptualized in terms of a `Principle of Preference-Basedness', according to which rational choice should be isomorphic to asserted preference. The main result characterizes axiomatically a new choice-rule called `Simultaneous Expected Utility Maximization'. It can be interpreted as agreement in a bargaining game (Kalai-Smorodinsky solution) whose players correspond to the (extremal) `acceptable priors' among which the decision maker has suspended judgment. An essential but non-standard feature of Simultaneous Expected Utility choices is their dependence on the entire choice set. This is justified by the conception of optimality as compromise rather than as superiority in pairwise comparisons.     

Subjective probabilities and utility with event-dependent preferences

This article generalizes Savage's theory to include event-dependent preferences. The state space is partitioned into finitely many events. The induced preferences over consequences are assumed independent of the underlying states within, but not across, these events. This results in an additively separable representation of preferences over acts. The dependence of the preference relation over consequences on the events is represented by event-dependent mappings of the set of consequences onto itself. Given these mappings, the preferences on acts are represented by the expectation of event-dependent utilities on the consequences with respect to unique subjective probabilities on the states.Helpful discussions with David Schmeidler are gratefully acknowledged.     

A No-Trade Theorem under Knightian Uncertainty with General Preferences

This paper derives a no-trade theorem under Knightian uncertainty, which generalizes the theorem of Milgrom and Stokey (1982, Journal of Economic Theory 26, 17) by allowing general preference relations. It is shown that the no-trade theorem holds true as long as agents' preferences are dynamically consistent in the sense of Machina and Schmeidler (1991, Econometrica 60, 745), and satisfies the so-called piece-wise monotonicity axiom. A preference satisfying the piece-wise monotonicity axiom does not necessarily imply the additive utility representation, nor is necessarily based on beliefs. This revised version was published online in June 2006 with corrections to the Cover Date.     

On Robust Constitution Design

We study a class of representation mechanisms, based on reports made by a random subset of agents, called representatives, in a collective choice problem with quasi-linear utilities. We do not assume the existence of a common prior probability describing the distribution of preference types. In addition, there is no benevolent planner. Decisions will be carried out by an individual who cannot be assumed impartial, a self-interested executive. These assumptions impose new constraints on Mechanism Design. A robust mechanism is defined as maximizing expected welfare under a vague prior probability distribution, and over a set of mechanisms which is at the same time immune from opportunistic manipulations by the executive, and compatible with truthful revelation of preferences by representatives. Robust mechanisms are characterized and their existence is shown. Sampling Groves mechanisms are shown to be robust.      

Conditioning Capacities and Choquet Integrals: The Role of Comonotony

Choquet integrals and capacities play a crucial role in modern decision theory. Comonotony is a central concept for these theories because the main property of a Choquet integral is its additivity for comonotone functions. We consider a Choquet integral representation of preferences showing uncertainty aversion (pessimism) and propose axioms on time consistency which yield a candidate for conditional Choquet integrals. An other axiom characterizes the role of comonotony in the use of information. We obtain two conditioning rules for capacities which amount to the well-known Bayes' and Dempster–Schafer's updating rules. We are allowed to interpret both of them as a lack of confidence in information in a dynamic extension of pessimism. This revised version was published online in June 2006 with corrections to the Cover Date.     

Mean utility preserving increases in risk for state dependent utility functions

This paper defines the concept of a mean utility preserving spread across states (MUPSAS) for state dependent utility functions and analyzes the behavioural impact of shifts in the probability distribution of wealth across states such that overall mean utility is preserved. The main result provides an alternative way of ranking state dependent utility functions according to their degree of risk aversion (thus extending Kami's theorem of comparative risk aversion) and establishes a link between increases in risk and risk aversion for state dependent preferences. In a portfolio problem where preferences and the rate of return of the risky venture are state dependent, we find sufficient conditions to determine the impact of a MUPSAS on the optimal share of the portfolio invested in the risky asset.

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Dynamically Consistent Preferences with Quadratic Beliefs

This article characterizes a family of preference relations over uncertain prospects that (a) are dynamically consistent in the Machina sense and, moreover, for which the updated preferences are also members of this family and (b) can simultaneously accommodate Ellsberg- and Allais-type paradoxes.Replacing the "mixture independence" axiom by "mixture symmetry" proposed by Chew, Epstein, and Segal (1991) for decision making under objective risk, and requiring that for some partition of the state space the agent perceives ambiguity and so prefers a randomization over outcomes across that partition (proper uncertainty aversion), preferences can be represented by a (proper) quadratic functional. This representation may be further refined to allow a separation between the quantification of beliefs and risk preferences that is closed under dynamically consistent updating.     

How to deal with partially analyzable acts?

In some situations, a decision is best represented by an incompletely analyzed act: conditionally on a given event A, the consequences of the decision on sub-events are perfectly known and uncertainty becomes probabilizable, whereas the plausibility of this event itself remains vague and the decision outcome on the complementary event [`(A)]{\bar{A}} is imprecisely known. In this framework, we study an axiomatic decision model and prove a representation theorem. Resulting decision criteria aggregate partial evaluations consisting of (i) the conditional expected utility associated with the analyzed part of the decision, and (ii) the best and worst consequences of its non-analyzed part. The representation theorem is consistent with a wide variety of decision criteria, which allows for expressing various degrees of knowledge on (A, [`(A)]{A, \bar{A}}) and various types of attitude toward ambiguity and uncertainty. This diversity is taken into account by specific models already existing in the literature. We exploit this fact and propose some particular forms of our model incorporating these models as sub-models and moreover expressing various types of beliefs concerning the relative plausibility of the analyzed and the non-analyzed events ranging from probabilities to complete ignorance that include capacities.