Exactly what happens after the Anscombe–Aumann race?

This paper derives a representation of preferences for a choice theory with vague environments; vague in the sense that the agent does not know the precise lotteries over outcomes conditional on states. Instead, he knows only a possible set of these lotteries for each state. Thus, this paper’s main departure from the standard subjective expected utility model is to relax an assumption about the environment, rather than weakening the axiomatic structure. My model is consistent with the behavior observed in the Ellsberg experiment. It can capture the same type of behavior as the multiple priors models, but can also result in behavior that is different from both the behavior implied by standard subjective expected utility models and the behavior implied by the multiple priors models. This paper is a revised chapter of my Ph.D. dissertation at Cornell University. I am very grateful to David Easley for extensive comments and suggestions. I also thank Larry Blume, Ani Guerdjikova, Edi Karni, Ted O’Donoghue, Maureen O’Hara, Jacob Sagi, seminar participants at the University of Copenhagen, Cornell University, ESEM 2006, Fuqua—Duke University, FUR 2006, Johns Hopkins University, Purdue University, Queen’s University, the University of Warwick, the associate editor, and an anonymous referee for comments and suggestions.     

Testable implications of subjective expected utility theory

I show that the predictive content of the hypothesis of subjective expected utility maximization critically depends on what the analyst knows about the details of the problem a particular decision maker faces. When the analyst does not know anything about the agent's payoffs or beliefs and can only observe the sequence of actions taken by the decision maker any arbitrary sequence of actions can be implemented as the choice of an agent that solves some intertemporal utility maximization problem under uncertainty.     

A new approach to modeling decision-making under uncertainty

This paper presents two axiomatic models of decision making under uncertainty that avoid the use of a state space. The first is a subjective expected utility model with action-dependent subjective probabilities and effect-dependent preferences (the case of effect-independent preferences is obtained as a special instance). The second is a nonexpected utility model involving well-defined families of action-dependent subjective probabilities on effects and a utility representation that is not necessarily linear in these probabilities (a probabilistic sophistication version of this model, with action-dependent subjective probabilities is obtained as a special case). The hospitality of EUREQua, University of Paris 1, and financial support by the National Science Foundation grant SES-0314249 are gratefully acknowledged.     

Testable implications of subjective expected utility theory

I show that the predictive content of the hypothesis of subjective expected utility maximization critically depends on what the analyst knows about the details of the problem a particular decision maker faces. When the analyst does not know anything about the agent's payoffs or beliefs and can only observe the sequence of actions taken by the decision maker any arbitrary sequence of actions can be implemented as the choice of an agent that solves some intertemporal utility maximization problem under uncertainty.     

Sequential decisions with several agents

Summary. We consider k agents who have different subjective probabilities and are utility maximizers. A planner, who knows the beliefs of the agents, maximizes the social expected utility, which is increasing and symmetric in the utilities of the agents. She does that by optimally stopping the flow of information released to the agents. The explicit form of the optimal stopping time is given. Received: October 23, 1995; revised version: June 25, 1997     

Internal rationality, imperfect market knowledge and asset prices

We present a decision theoretic framework in which agents are learning about market behavior and that provides microfoundations for models of adaptive learning. Agents are ‘internally rational’, i.e., maximize discounted expected utility under uncertainty given dynamically consistent subjective beliefs about the future, but agents may not be ‘externally rational’, i.e., may not know the true stochastic process for payoff relevant variables beyond their control. This includes future market outcomes and fundamentals. We apply this approach to a simple asset pricing model and show that the equilibrium stock price is then determined by investors? expectations of the price and dividend in the next period, rather than by expectations of the discounted sum of dividends. As a result, learning about price behavior affects market outcomes, while learning about the discounted sum of dividends is irrelevant for equilibrium prices. Stock prices equal the discounted sum of dividends only after making very strong assumptions about agents? market knowledge.     

Subjective recursive expected utility

We axiomatize a subjective version of the recursive expected utility model. This development extends the seminal results of Kreps and Porteus (Econometrica 46:185–200 (1978)) to a subjective framework and provides foundations that are easy to relate to axioms familiar from timeless models of decision making under uncertainty. Our analysis also clarifies what is needed in going from a represention that applies within a single filtration to an across filtration representation.Part of this research was conducted when Ozdenoren visited MEDS in Fall 2003. We thank Tapas Kundu, Costis Skiadas, Jean-Marc Tallon and Tan Wang for helpful discussions and also thank audiences at Koc University, Northwestern University, the CERMSEM conference “ Mathematical Models in Decision Theory” at Universite Paris I, and the FUR XI conference on foundations and applications of utility, risk and decision theory     

Ordinal aggregation and quantiles

Consider the problem of aggregating a profile of interpersonally comparable utilities into a social utility. We require that the units of measurement of utility used for agents is the same as the units of measurement for society (ordinal covariance) and a mild Pareto condition (monotonicity). We provide several representations of such social aggregation operators: a canonical representation, a Choquet expectation representation, a minimax representation, and a quantile representation (with respect to a possibly non-additive set function on the agents). We also isolate an additional condition that gives us a quantile representation with respect to a probability measure, in both the finite and infinite agents case.     

Subjective expected utility theory with costly actions

This paper explores alternative axiomatizations of subjective expected utility theory for decision makers with direct preferences over actions; including a general subjective expected utility representation with action-dependent utility, and separately additive representations. In the context of the state-space formulation of agency theory, the results of this paper constitute axiomatic foundations of the agent's behavior.     

Self is Never Neutral: Why Economic Agents Behave Irrationally

Modern economics understands utility from the concept of decision utility inferred from individual choice making. It explains agents' decisions or choices in turn by the paradigm of utility maximizing. From our perspective, however, this is a fatal mistake because economic agents do not always choose what they really want in order to maintain their "self-value." In fact, subjects are never neutral. When agents are not able to obtain something they want, they downplay its desirability in order to get psychological satisfaction. But when they are forced to accept what they do not want, they try to rationalize that they really did want it, again in order to save face. Although such "irrational" behavior may decrease economic utility, it gives agents psychological satisfaction and subjective comfort, thus increasing their immaterial utility. In this sense, agents remain rational when conducting such behaviors, even though they run directly contrary to neoclassical rationality concepts.     

ARBITRATION,INDIVISIBLE ASSETS AND THE POSSIBILITY OF MALICE

Many real‐life contexts involve allocating an indivisible prize between two claimants. One (or both) claimants may derive ‘malice utility’ from depriving the other claimant. I study an arbitrator who does not know if malice is involved in a particular dispute, but wishes the prize to be allocated to the party with the higher intrinsic valuation for it and discuss two simple mechanisms that achieve his purpose regardless of whether malice is actually present. I examine several extensions.     

Dynamic beliefs

This paper studies the temporal path of subjective probability assessments. A reference-dependent agent who experiences utility from anticipation and from changes in this anticipatory emotion makes utility-maximizing assessments about his likelihood of success in a future lottery. Consistent with the empirical evidence, the model predicts that if the lottery is sufficiently valuable, optimism decreases as the payoff date approaches. Intuitively, as time goes by, last-period expected disappointment becomes increasingly important relative to the joy of anticipating a favorable outcome. Applying the model to the optimal timing of productivity bonuses, I find that a decreasing path of beliefs reduces the cost of providing incentives. Thus, optimal bonuses are sizable and are not frequently offered.     

Recursive utility and preferences for information

Summary. This paper presents an axiomatic foundation for recursive utility that captures the role of the timing of resolution of uncertainty without relying on exogenously specified objective beliefs. Two main representation results are proved. In the first one, future utility enters the recursion through the type of general aggregators considered in Skiadas (1997a), and as a result the formulation is purely ordinal and free of any probabilities. In the second representation these aggregators are conditional expectations relative to subjective beliefs. A new recursive representation incorporating disappointment aversion is also suggested. The main methodological innovation of the paper derives from the fact that the basic objects of choice are taken to be pairs of state-contingent consumption plans and information filtrations, rather than the temporal (objective) lotteries of the existing literature. It is shown that this approach has the additional benefit of being directly applicable to the continuous-time version of recursive utility developed by Duffie and Epstein (1992). Received: February 18, 1997; revised version: July 18, 1997     

TESTING POVERTY LINES

In theory, a poverty line can be defined as the cost of a common (inter‐personally comparable) utility level across a population. But how can one know if this holds in practice? For groups sharing common consumption needs but facing different prices, the theory of revealed preference can be used to derive testable implications of utility consistency knowing only the “poverty bundles” and their prices. Heterogeneity in needs calls for extra information. We argue that subjective welfare data offer a credible means of testing utility consistency across different needs groups. A case study of Russia's official poverty lines shows how revealed preference tests can be used in conjunction with qualitative information on needs heterogeneity. The results lead us to question the utility consistency of Russia's official poverty lines.     

Ambiguity Without a State Space

Many decisions involve both imprecise probabilities and intractable states of the world. Objective expected utility assumes unambiguous probabilities; subjective expected utility assumes a completely specified state space. This paper analyses a third domain of preference: sets of consequential lotteries. Using this domain, we develop a theory of objective ambiguity without explicit reference to any state space. We characterize a representation that integrates a non-linear transformation of first-order expected utility with respect to a second-order measure. The concavity of the transformation and the weighting of the measure capture ambiguity aversion. We propose a definition for comparative ambiguity aversion.     

Decision making and trade without probabilities

This paper studies trade in a first-price sealed-bid auction where agents know only a range of possible payoffs. The setting is one in which a lemons problem arises, so that if agents have common risk preferences and common priors, then expected utility theory leads to a prediction of no trade. In contrast, we develop a model of rational non-probabilistic decision making, under which trade can occur because not bidding is a weakly dominated strategy. We use a laboratory experiment to test the predictions of both models, and also of models of expected utility with heterogeneous priors and risk preferences. We find strong support for the rational non-probabilistic model.     

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.     

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.     

Underestimation of probabilities modifications: characterization and economic implications

The aim of this paper is to propose a behavioral characterization of individuals who underestimate probabilities modifications and to characterize this behavior in the standard preferences representation models under risk (expected utility, dual theory, rank dependent utility theory and MaxMin expected utility). Our main results are the following. Underreaction to probabilities modifications is in general independent from standard risk aversion and prudence. In models involving probabilities transformation functions, it is characterized by the slope of the probability transformation function. In the MaxMin expected utility model under risk, it is related to the weights of the maximal and minimal consequences in the preferences representation function. Considering a simple prevention decision, consisting in the reduction in the probability of a monetary loss, we show that individuals who underreact to probabilities modifications, invest less in prevention than individuals who objectively evaluate these modifications. Underreaction to probabilities modification is thus a possible explanation for low investment in prevention.     

On preferences with infinitely many subjective states

Models with subjective state spaces have been extremely useful in capturing novel psychological phenomena that consist of both a preference for flexibility and for commitment. Interpreting the utility representations of preferences as capturing these phenomena requires one to use the notion of a sign of a state. For linear preferences, we completely characterise the sign of a state in terms of its analytic representation as an integral with respect to a signed measure. In models with finitely many states, a state is either positive or negative, but never both. We show that in models with infinitely many states, a state can be both positive and negative. Thus, models with finitely many states may not capture all the behavioural features of an infinite model. Our methods are also useful in constructing utility functionals over menus with desired local properties.