Differentiating ambiguity and ambiguity attitude

The objective of this paper is to show how ambiguity, and a decision maker (DM)'s response to it, can be modelled formally in the context of a general decision model. We introduce a relation derived from the DM's preferences, called “unambiguous preference”, and show that it can be represented by a set of probabilities. We provide such set with a simple differential characterization, and argue that it is a behavioral representation of the “ambiguity” that the DM may perceive. Given such revealed ambiguity, we provide a representation of ambiguity attitudes. We also characterize axiomatically a special case of our decision model, the “α-maxmin” expected utility model.     

Imprecise probabilistic beliefs as a context for decision-making under ambiguity

Coherent imprecise probabilistic beliefs are modeled as incomplete comparative likelihood relations admitting a multiple-prior representation. Under a structural assumption of Equidivisibility, we provide an axiomatization of such relations and show uniqueness of the representation. In the second part of the paper, we formulate a behaviorally general “Likelihood Compatibility” axiom relating preferences and probabilistic beliefs and characterize its implications for the class of “invariant biseparable” preferences that includes the MEU and CEU models among others.     

Attitude toward imprecise information

This paper presents an axiomatic model of decision making under uncertainty which incorporates objective but imprecise information. Information is assumed to take the form of a probability–possibility set, that is, a set P of probability measures on the state space. The decision maker is told that the true probability law lies in P and is assumed to rank pairs of the form (P,f) where f is an act mapping states into outcomes. The key representation result delivers maxmin expected utility (MEU) where the min operator ranges over a set of probability priors—just as in the MEU representation result of Gilboa and Schmeidler [Maxmin expected utility with a non-unique prior, J. Math. Econ. 18 (1989) 141–153]. However, unlike the MEU representation, the representation here also delivers a mapping, , which links the probability–possibility set, describing the available information, to the set of revealed priors. The mapping is shown to represent the decision maker's attitude to imprecise information: under our axioms, the set of representation priors is constituted as a selection from the probability–possibility set. This allows both expected utility when the selected set is a singleton and extreme pessimism when the selected set is the same as the probability–possibility set, i.e., is the identity mapping. We define a notion of comparative imprecision aversion and show it is characterized by inclusion of the sets of revealed probability distributions, irrespective of the utility functions that capture risk attitude. We also identify an explicit attitude toward imprecision that underlies usual hedging axioms. Finally, we characterize, under extra axioms, a more specific functional form, in which the set of selected probability distributions is obtained by (i) solving for the “mean value” of the probability–possibility set, and (ii) shrinking the probability–possibility set toward the mean value to a degree determined by preferences.     

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.     

Transitive measurable utility

This paper continues a study of theories of preferences under risk that do not use the independence axiom of the von Neumann-Morgenstern theory. Unlike its predecessor, it assumes that preferences are transitive. The effects of transitivity are noted in two representations of preferences. The first, which also uses continuity and dominance axioms, involves a function u on a set P of probability measures for which u(p) > u(q) if and only if p is preferred to q. Although u might be nonlinear, it has other features of a von Neumann-Morgenstern linear utility function. The second representation has linear functions u and w on P, with w strictly positive except perhaps at preference-extreme measures—where it might vanish, such that u(p) w(q) > u(q) w(p) if and only if p is preferred to q. A symmetry axiom along with the axioms for the first representation are necessary and sufficient for the second representation.     

On S-curves and tipping points

In his discussion in this journal of Kurzweil's The Singularity is Near, Modis criticizes Kurzweil's loose characterization of the “knee” of a growth curve. Likewise, the “tipping points” described by Gladwell (The Tipping Point) are clearly relevant to forecasting systems, but Gladwell did not mathematically identify such points. Both concepts refer to a point on the curve where growth accelerates dramatically and sustains itself. What can be said in a rigorous way about knees and tipping points in growth systems?The answer has to do with the number of parameters of the growth curve, and the (probabilistic) model underlying the growth curve. Using probability theory and computational experiments, this paper clarifies these points for the logistic and Bass curves (identifying an unambiguous tipping point for the latter), and explores the merits of a 3-parameter model of innovation adoption. It concludes that if forecasters are to deal scientifically with the now-established management notion of “tipping points,” a 3-parameter model is needed. The paper also resolves four minor but annoying paradoxes in the growth curve literature.     

Dynamic decision trees for building resilience into future eco-cities

Intensifying global urbanization and environmental changes bring about the imperative of sustainable urban development and decisions upon inescapable pressures and risks, but knowledge integration between disciplines is a limiting contextual challenge. This paper proposes a reformulation, in terms of urban risk management, of an earlier developed ontological scenario generation method. The procedure consists of several steps: (i) identification-and-prioritization of main pressures, (ii) paired discussion of pressures using four-cell matrices, (iii) re-visit of the pressures' priority order, (iv) articulation of short-listed pressures as decision-making questions, and (v) generation of scenarios via “yes/no” responses to each question, in their order of priority. In this article, the method feeds upon the general context described in recent multi-disciplinary urban studies and public strategic plans in the city of Iasi (Romania), to propose a formal procedure for enabling the acceleration of productive decision making towards city sustainability. Answering three top priority questions, namely “Implement a business-friendly and efficient governance system?”, “Develop a resource management system?”, and “Carry out a human capital accelerator strategy?” results in a 4-scenario set: Receding City, Wanting City, Promising City, Inspiring City. The scenarios are discussed in terms of systemic risks at the end of post-communist transition and beginning of the socio-economic convergence with Western Europe.     

Factor Saving Innovation

It has been argued that concave models exhibit less “endogeneity of growth” than models with increasing returns to scale. Here we study a simple model of factor saving technological improvement in a concave framework. Capital can be used either to reproduce itself, or, at some additional cost, to produce a higher quality of capital, which requires less labor input. If better quality capital can be produced quickly, we get a model of exogenous balanced growth as a special case of ours. If, however, better quality capital can be produced slowly, we get a model of “endogenous growth” in which the growth rate of the economy and the rate of adoption of new technologies is determined by preferences, technology and initial conditions. Moreover, in the latter case, the process of growth is necessarily uneven, exhibiting a natural cycle with alternating periods of high and slow growth. Growth paths and technological innovations also exhibit dependence upon initial conditions. The model provides a step toward a theory of endogenous innovation under conditions of perfect competition. Journal of Economic Literature Classification Numbers: O30, O40, D92, C61, D24, D41.     

Strategy-Proof Probabilistic Decision Schemes for One-Dimensional Single-Peaked Preferences

Collective decision problems are considered with a finite number of agents who have single-peaked preferences on the real line. A probabilistic decision scheme assigns a probability distribution over the real line to every profile of reported preferences. The main result of the paper is a characterization of the class of unanimous and strategy-proof probabilistic schemes with the aid of fixed probability distributions that play a role similar to that of the phantom voters in H. Moulin (Public Choice35 (1980), 437-455). Thereby, the work of Moulin (1980) is extended to the probabilistic framework. Journal of Economic Literature Classification Numbers: D71, D81.     

The veil of public ignorance

A theory of cooperative choice under incomplete information is developed in which agents possess private information at the time of contracting and have agreed on a utilitarian “standard of evaluation” governing choices under complete information. The task is to extend this standard to situations of incomplete information. Our first main result generalizes Harsanyi's (J. Polit. Econ. 63 (1955) 309) classical result to situations of incomplete information, assuming that group preferences satisfy Bayesian Coherence and Interim Pareto Dominance. These axioms are mutually compatible if and only if a common prior exists. We argue that this result partly resolves the impossibility of Bayesian preference aggregation under complete information.     

Priorities in the location of multiple public facilities

A collective decision problem is described by a set of agents, a profile of single-peaked preferences over the real line and a number of public facilities to be located. We consider public facilities that do not suffer from congestion and are non-excludable. We characterize the class of rules satisfying Pareto-efficiency, object-population monotonicity and sovereignty. Each rule in the class is a priority rule that selects locations according to a predetermined priority ordering among “interest groups”. We characterize the subclasses of priority rules that respectively satisfy anonymity, avoid the no-show paradox, strategy-proofness and population-monotonicity. In particular, we prove that a priority rule is strategy-proof if and only if it partitions the set of agents into a fixed hierarchy. Any such rule can also be viewed as a collection of generalized peak-selection median rules, that are linked across populations, in a way that we describe.     

Justifiable preferences

We characterize preferences over acts that can be represented by a utility function and a multiple-prior, such that an act f is preferred to act g if there is a prior under which the expected utility induced by f is higher than that induced by g. These preferences are referred to as justifiable preferences. We further introduce a generalized model of ambiguity that involves a collection of multiple-priors, namely, multiple multiple-priors and incorporate Bewley?s Knightian model in justifiability: f is preferred to g if, according to at least one set of priors, f is unanimously preferred to g.     

A characterization of consistent collective choice rules

We characterize a class of collective choice rules such that collective preference relations are consistent. Consistency is a weakening of transitivity and a strengthening of acyclicity requiring that there be no cycles with at least one strict preference, which excludes the possibility of a “money pump.” The properties of collective choice rules used in our characterization are unrestricted domain, strong Pareto, anonymity and neutrality. If there are at most as many individuals as there are alternatives, the axioms provide an alternative characterization of the Pareto rule. If there are more individuals than alternatives, however, further rules become available.     

Interdependent preferences and segregating equilibria

This paper shows that models where preferences of individuals depend not only on their allocations, but also on the well being of other persons, can produce both large and testable effects. We study the allocation of workers with heterogeneous productivities to firms. We show that even small deviations from purely “selfish” preferences leads to widespread workplace skill segregation. That is, workers of different abilities tend to work in different firms, as long as they care somewhat more about the utilities of workers who are “close”. This result holds for a broad class and distribution of social preferences.     

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).     

Non-addictive habits: optimal consumption-portfolio policies

We formulate a model of preferences with non-addictive habits, where consumption is required to be non-negative at all times, but can fall below a “standard of living” index that aggregates past consumption. We study the consumption-portfolio problem taking account of the non-negativity constraint on consumption, and provide a constructive proof for the existence of an optimal policy on a finite time-horizon [0,T]. We show that the consumption constraint binds up to an endogenous stopping time τ∗∈[0,T], after which it remains slack until T. A decomposition of constrained consumption involving an Asian average-strike capped call-option is demonstrated.     

Group decision-making in the shadow of disagreement

A model of group decision-making is studied, in which one of two alternatives must be chosen. While agents differ in their preferences over alternatives, everybody prefers agreement to disagreement. Our model is distinguished by three features: private information regarding valuations, differing intensities in preferences, and the option to declare neutrality to avoid disagreement. There is always an equilibrium in which the majority is more aggressive in pushing its alternative, thus enforcing their will via both numbers and voice. However, under general conditions an aggressive minority equilibrium inevitably makes an appearance, provided that the group is large enough. Such equilibria invariably display a “tyranny of the minority”: the increased aggression of the minority always outweighs their smaller number, leading to the minority outcome being implemented with larger probability than the majority alternative. We fully characterize the asymptotic behavior of this model as group size becomes large, and show that all equilibria must converge to one of three possible limit outcomes.     

Strategy-proof assignment on the full preference domain

We consider the problem of efficiently allocating several indivisible objects between agents who are to receive at most one object and whose preferences are private information. We examine this standard “assignment” problem from the perspective of mechanism design giving up the usual assumption of linear preferences and instead using a full preference domain (with indifferences permitted). We characterize two classes of mechanisms: (i) Bi-polar Serially Dictatorial Rules by Essential Single-Valuedness, Pareto Indifference, Strategy-Proofness and Non-Bossiness; and (ii) all selections from Bi-polar Serially Dictatorial Rules by Single-Valuedness, Efficiency, Strategy-Proofness and Weak Non-Bossiness. We compare the outcomes from the (Bi-polar) Serially Dictatorial Rules with the outcomes obtained using a market based approach, namely the “core” of the market. We show that all strongly efficient outcomes in the core can be generated using Serially Dictatorial Rules. Moreover, we argue that Serially Dictatorial Rules have an advantage over the market based approach in that they yield strongly efficient solutions for all preference profiles, making it possible to use randomization to restore equity. When preferences are private information, this type of ex ante equity cannot be implemented using the market based approach.     

Redistributive policies with heterogeneous social preferences of voters

There is growing evidence on the roles of fairness and other-regarding preferences as fundamental human motives. Call voters with fair preferences, as in Fehr and Schmidt (1999), fair-voters. By contrast, traditional political economy models are based on selfish-voters who derive utility solely from “own” payoff. In a general equilibrium model with endogenous labor supply, a mixture of fair and selfish voters choose optimal policy through majority voting. First, we show that majority voting produces a unique winner in pairwise contests over feasible policies (the Condorcet winner). Second, we show that a preference for greater fairness leads to greater redistribution. An increase in the number of fair voters can also lead to greater redistribution. Third, we show that in economies where the majority are selfish-voters, the decisive policy could be chosen by fair-voters, and vice versa. Fourth, while choosing labor supply, even fair voters behave exactly like selfish voters. We show how this apparently inconsistent behavior in different domains (voting and labor supply) can be rationalized within the model.