Subjective random discounting and intertemporal choice

This paper provides an axiomatic foundation for a particular type of preference shock model called the random discounting representation where a decision maker believes that her discount factors change randomly over time. For this purpose, we formulate an infinite horizon extension of [E. Dekel, B. Lipman, A. Rustichini, Representing preferences with a unique subjective state space, Econometrica 69 (2001) 891-934], and identify the behavior that reduces all subjective uncertainties to those about future discount factors. We also show uniqueness of subjective belief about discount factors. Moreover, a behavioral comparison about preference for flexibility characterizes the condition that one's subjective belief second-order stochastically dominates the other. Finally, the resulting model is applied to a consumption-savings problem.     

Subjective probability over a subjective decision tree

A state space has been assumed as a primitive for modeling uncertainty, which presumes that the analyst knows all the uncertainties a decision maker (DM) perceives. This is problematic because states are private information of the DM, and hence are not directly observable to the analyst. Dekel et al. [Representing preferences with a unique subjective state space, Econometrica 69 (2001) 891-934] derive, rather than assume, the subjective state space from preference over suitable choice objects.In a dynamic setting, a decision tree, that is, a pair consisting of a state space and a filtration, has been taken as a primitive. This assumption is also problematic—a decision tree should be derived rather than assumed as a primitive. We formulate a three-stage extension of the above literature in order to model a DM who anticipates subjective uncertainty to be resolved gradually over time. We identify also subjective beliefs on the subjective state space.     

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.     

The value of a stochastic information structure

Upon observing a signal, a Bayesian decision maker updates her probability distribution over the state space, chooses an action, and receives a payoff that depends on the state and the action taken. An information structure determines the set of possible signals and the probability of each signal given a state. For a fixed decision problem, the value of an information structure is the maximal expected utility that the decision maker can get when the observed signals are governed by this structure. Thus, every decision problem induces a preference order over information structures according to their value. We characterize preference orders that can be obtained in this way. We also characterize the functions defined over information structures that measure their value.     

Rationalizability and the savage axioms

Summary. Suppose there is a finite set of acts defined on a finite state space and a decision maker chooses an act from the set. In this setting, the subjective expected utility model is observationally indistinguishable from all models of preference that satisfy Savage's axiom P3. The result has implications also for rationalizability in strategic games. Received: September 18, 1998; revised version: January 29, 1999     

Robust estimation and control under commitment

In a Markov decision problem with hidden state variables, a decision maker expresses fear that his model is misspecified by surrounding it with a set of alternatives that are nearby as measured by their expected log likelihood ratios (entropies). Sets of martingales represent alternative models. Within a two-player zero-sum game under commitment, a minimizing player chooses a martingale at time 0. Probability distributions that solve distorted filtering problems serve as state variables, much like the posterior in problems without concerns about misspecification. We state conditions under which an equilibrium of the zero-sum game with commitment has a recursive representation that can be cast in terms of two risk-sensitivity operators. We apply our results to a linear quadratic example that makes contact with findings of T. Ba?ar and P. Bernhard [H^∞-Optimal Control and Related Minimax Design Problems, second ed., Birkhauser, Basel, 1995] and P. Whittle [Risk-sensitive Optimal Control, Wiley, New York, 1990].     

The subjective expected utility hypothesis and revealed preference

Summary A choice behavior of a decision maker is said to satisfy the subjective expected utility hypothesis if there exist a utility and a subjective probability such that a decision maker chooses acts with the highest expected utility. We present a revealed preference characterization of choice behavior that is consistent with the subjective expected utility hypothesis. Our result applies to any state space and compact sets of prizes and observations (either finite or infinite).     

Ambiguity,data and preferences for information – A case-based approach

We model decision making under ambiguity based on available data. Decision makers express preferences over actions and data sets. We derive an α-max–min representation of preferences, in which beliefs combine objective characteristics of the data (number and frequency of observations) with subjective features of the decision maker (similarity of observations and perceived ambiguity). We identify the subjectively perceived ambiguity and separate it into ambiguity due to a limited number of observations and ambiguity due to data heterogeneity. The special case of no ambiguity provides a behavioral foundation for beliefs as similarity-weighted frequencies as in Billot et al. (2005) [3].     

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.     

Bayesian consistent belief selection

A subjective expected utility agent is given information about the state of the world in the form of a set of possible priors. She is assumed to form her beliefs given this information. A set of priors may be updated according to Bayes' rule, prior-by-prior, upon learning that some state of the world has not obtained. In a model in which information is completely summarized by this set of priors, we show that there exists no decision maker who obeys Bayes' rule, conditions her prior only on the available information (by selecting a belief in the announced set), and who updates the information prior-by-prior using Bayes' rule.     

The rationality and efficacy of decisions under uncertainty and the value of an experiment

Summary A decision maker faces a known prior distribution over payoff relevant states. We compare the expected utility of this individual under two scenarios. In the first, the decision maker makes a choice without further information. In the second, the decision maker has access to an experiment before choosing an action. However, the decision maker does not know the true joint distribution over states and messages. The value of the experiment as measured by the difference in the two utility levels can be negative as well as positive. We give a condition which is necessary and sufficient for the experiment to be valuable in our sense, for any decision problem.An earlier version of this paper was circulated under the title Noisy Bayes Updating and the Value of Information. We have gained from the comments of Stephen Coate, John Geanakoplos, Larry Samuelson, Timothy Van Zandt and seminar participants at Harvard Business School, Princeton, Boston University, the international conference on game theory at Stony Brook 1992 and the Winter meeting of the Econometric Society at Anaheim 1993. The first author received support for this project from NSF grant #SES-9308515 and a University of Pennsylvania Research Foundation Grant.     

Completing the State Space with Subjective States

We define an opportunity act as a mapping from an exogenously given objective state space to a set of lotteries over prizes, and consider preferences over opportunity acts. We allow the preferences to be possibly uncertainty averse. Our main theorem provides an axiomatization of the maxmin expected utility model. In the theorem we construct subjective states to complete the objective state space. As in E. Dekel et al. (Econometrica, in press), we obtain a unique subjective state space. We also allow for preference for flexibility in some of the subjective states and commitment in others. Journal of Economic Literature Classification Number: D81.     

Minimax regret with imperfect ex-post knowledge of the state

We consider decision problems under complete ignorance and extend the minimax regret principle to situations where, after taking an action, the decision maker does not necessarily learn the state of the world. For example, if the decision maker only learns what the outcome is, then all she knows is that the actual state is one of the (possibly several) states that yield the observed outcome under the chosen action. We refer to this situation as imperfect ex-post information. We show that, given a choice between more information and less information, the decision maker prefers the latter. We also extend the framework to encompass the possibility of less than the extreme degree of pessimism that characterizes the minimax regret criterion.     

Aggregation of multiple prior opinions

Experts are asked to provide their advice in a situation of uncertainty. They adopt the decision maker?s utility function, but each has a potentially different set of prior probabilities, and so does the decision maker. The decision maker and the experts maximize the minimal expected utility with respect to their sets of priors. We show that a natural Pareto condition is equivalent to the existence of a set Λ of probability vectors over the experts, interpreted as possible allocations of weights to the experts, such that (i) the decision maker?s set of priors is precisely all the weighted-averages of priors, where an expert?s prior is taken from her set and the weight vector is taken from Λ; (ii) the decision maker?s valuation of an act is the minimal weighted valuation, over all weight vectors in Λ, of the experts? valuations.     

A behavioral characterization of plausible priors

Recent decision theories represent ambiguity via multiple priors, interpreted as alternative probabilistic models of the relevant uncertainty. This paper provides a robust behavioral foundation for this interpretation. A prior P is “plausible” if preferences over some subset of acts admit an expected utility representation with prior P, but not with any other prior Q≠P. Under suitable axioms, plausible priors can be elicited from preferences, and fully characterize them; also, probabilistic sophistication implies that there exists only one plausible prior; finally, “plausible posteriors” can be derived via Bayesian updating. Several familiar decision models are consistent with the proposed axioms.     

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.     

Coping with ignorance: unforeseen contingencies and non-additive uncertainty

Summary. In real-life decision problems, decision makers are never provided with the necessary background structure: the set of states of the world, the outcome space, the set of actions. They have to devise all these by themselves. I model the (static) choice problem of a decision maker (DM) who is aware that her perception of the decision problem is too coarse, as for instance when there might be unforeseen contingencies. I make a “bounded rationality' assumption on the way the DM deals with this difficulty, and then I show that imposing standard subjective expected utility axioms on her preferences only implies that they can be represented by a (generalized) expectation with respect to a non-additive measure, called a belief function. However, the axioms do have strong implications for how the DM copes with the type of ignorance described above. Finally, I show that some decision rules that have been studied in the literature can be obtained as a special case of the model presented here (though they have to be interpreted differently). Received: December 16, 1999; revised version: March 22, 2000     

What is an ‘endogenous state space’?

Summary. In the literature on choice under unforeseen contingencies, the decision maker behaves as if she aggregates possible instances of future rankings indexed by a set S. The set S is interpreted as a subjective state space even though subsequent rankings need not conform to any one of the aggregated utilities. This paper proposes a definition for a subjective state space under unforeseen contingencies that is topologically unique, derives its existence from preference primitives as opposed to the representation of preferences, and does not commit to an interpretation in which states correspond to future realized rankings. The definition topologically concurs with and extends the identification of the essentially unique subjective state space due to Dekel, Lipman and Rustichini [4].Received: 28 October 2003, Revised: 13 October 2004, JEL Classification Numbers: D11, D81, D91.I thank Eddie Dekel, Alan Kraus, Bart Lipman, Chris Shannon, and the referee for some helpful remarks.     

Information structures with unawareness

I construct a state space model with unawareness following [R.J. Aumann, Agreeing to disagree, Ann. Stat. 76 (1976) 1236-1239]. Dekel et al. [E. Dekel, B.L. Lipman, A. Rustichini, Standard state-space models preclude unawareness, Econometrica 66 (1998) 159-173] show that standard state space models are incapable of representing unawareness. The model circumvents the impossibility result by endowing the agent with a subjective state space that differs from the full state space when he has the unawareness problem. Information is modeled as a pair, consisting of both factual information and awareness information. The model preserves the central properties of the standard information partition model.     

Matching through position auctions

Using a mechanism design framework, we characterize how a profit-maximizing intermediary can design matching markets when each agent is privately informed about his quality as a partner. Sufficient conditions are provided that ensure a version of positive assortative matching (what we call truncated positive assortative matching) maximizes profits. Under these conditions, all-pay position auctions always implement the profit-maximizing allocation. Winners-pay position auctions, however, only do so in sufficiently large markets.