A theory of subjective compound lotteries

We develop a Savage-type model of choice under uncertainty in which agents identify uncertain prospects with subjective compound lotteries. Our theory permits issue preference; that is, agents may not be indifferent among gambles that yield the same probability distribution if they depend on different issues. Hence, we establish subjective foundations for the Anscombe-Aumann framework and other models with two different types of probabilities. We define second-order risk as risk that resolves in the first stage of the compound lottery and show that uncertainty aversion implies aversion to second-order risk which implies issue preference and behavior consistent with the Ellsberg paradox.     

Bayesian learning and convergence to Nash equilibria without common priors

Summary. Consider an infinitely repeated game where each player is characterized by a “type” which may be unknown to the other players in the game. Suppose further that each player's belief about others is independent of that player's type. Impose an absolute continuity condition on the ex ante beliefs of players (weaker than mutual absolute continuity). Then any limit point of beliefs of players about the future of the game conditional on the past lies in the set of Nash or Subjective equilibria. Our assumption does not require common priors so is weaker than Jordan (1991); however our conclusion is weaker, we obtain convergence to subjective and not necessarily Nash equilibria. Our model is a generalization of the Kalai and Lehrer (1993) model. Our assumption is weaker than theirs. However, our conclusion is also weaker, and shows that limit points of beliefs, and not actual play, are subjective equilibria. Received: March 3, 1995; revised version: February 17, 1997     

Preference for Flexibility and Dynamic Consistency

Dekel, Lipman, and Rustichini [3] characterize preferences over menus of lotteries that can be represented by the use of a unique subjective state space and a prior. We investigate what would be the appropriate version of Dynamic Consistency in such a setup. The condition we find, which we call Flexibility Consistency, is linked to a comparative theory of preference for flexibility. When the subjective state space is finite, we show that Flexibility Consistency is equivalent to a subjective version of Dynamic Consistency and that it implies that the decision maker is a subjective state space Bayesian updater. Later we characterize when a collection of signals can be interpreted as a partition of the subjective state space of the decision maker.     

Foundations of Bayesian theory

This paper states necessary and sufficient conditions for the existence, uniqueness, and updating according to Bayes’ rule, of subjective probabilities representing individuals’ beliefs. The approach is preference based, and the result is an axiomatic subjective expected utility model of Bayesian decision making under uncertainty with state-dependent preferences. The theory provides foundations for the existence of prior probabilities representing decision makers’ beliefs about the likely realization of events and for the updating of these probabilities according to Bayes’ rule.     

Discounting and divergence of opinion

Agents impatience rate and their anticipations about the future of the economy, are two essential determinants of the equilibrium discount rate, as illustrated by the Ramsey formula. Heterogeneity in time preference rates and in anticipations is widely acknowledged. Our objective is to determine the equilibrium discount rate when this heterogeneity is taken into account. Among others we tackle the following questions: As an additional risk or uncertainty, can dispersion in agents' characteristics lead to lower discount rates? What is the asymptotic behavior of the discount rate in such a setting? More generally, what is the shape of the yield curve?     

THE RISK‐FREE RATE IN A FINITE HORIZON MODEL WITH BEQUESTS

This paper studies the risk‐free rate in an overlapping generations economy with bequests. It is shown that the risk‐free rate depends on risk aversion, the elasticity of intertemporal substitution, the share of wealth invested in human wealth, life expectancy, and the preference for bequests. In a standard life‐cycle context, mortality increases the subjective time rate of discount, and thus increases the compensation required to postpone consumption. This latter effect is offset in a bequest‐driven model of the type considered here, leading to much more powerful income effects. In this sense, the model provides a bequest‐motive explanation for the risk‐free rate puzzle put forward by Weil in 1989.     

Non-parametric test of time consistency: Present bias and future bias

This paper reports the elicited time preference of human subjects in a laboratory setting. The model allows for non-linear utility functions, non-separability between delay and reward, and time inconsistency including future bias in addition to present bias. In particular, the experiment (1) runs a non-parametric test of time consistency and (2) estimates the form of time discount function independently of instantaneous utility functions, and then (3) the result suggests that many subjects exhibiting future bias, indicating an inverse S-curve time discount function.     

Discounting Nordhaus

This paper evaluates Nordhaus's neoclassical complaints about the Stern Review from the vantage point of classical growth theory. Nordhaus criticizes the Stern Review because it uses a discount rate that is well below the market rate of return on capital. From the perspective of classical growth theory, Nordhaus's belief in choosing preference parameters for the social planner based on observed market rates of return is equivalent to assigning the preferences of the capitalist agents to the social planner. This equivalence is an implication of the Cambridge Theorem, which interprets the Ramsey equation as the saving function of the capitalist agents.     

On the size and structure of group cooperation

When time preferences are heterogeneous and bounded away from one, how “much” cooperation can be achieved by an ongoing group? How does group cooperation vary with the group's size and structure? This paper examines characteristics of cooperative behavior in the class of symmetric, repeated games of collective action. These are games characterized by “free rider problems” in the level of cooperation achieved. The Repeated Prisoner's Dilemma games is a special case.We characterize the level of maximal average cooperation (MAC), the highest average level of cooperation, over all stationary subgame perfect equilibrium paths, that the group can achieve. The MAC is shown to be increasing in monotone shifts, and decreasing in mean preserving spreads of the distribution of discount factors. The latter suggests that more heterogeneous groups are less cooperative on average. Finally, in a class of Prisoner's Dilemma games, we show under weak conditions that the MAC exhibits increasing returns to scale in a range of heterogeneous discount factors. That is, larger groups are more cooperative, on average, than smaller ones. By contrast, when the group has a common discount factor, the MAC is invariant to group size.     

HETEROGENEOUS RISK ATTITUDES IN A CONTINUOUS‐TIME MODEL*

We prove that every continuous‐time model in which all consumers have time‐homogeneous and time‐additive utility functions and share a common probabilistic belief and a common discount rate can be reduced to a static model. This result allows us to extend some of the existing results of the representative consumer and risk‐sharing rules in static models to continuous‐time models. We show that the equilibrium interest rate is lower and more volatile than in the standard representative consumer economy, and that the individual consumption growth rates are more dispersed than in the absence of uncertainty.     

Managerial incentives and collusive behavior

I characterize the effects of empirically observed managerial incentives on long-run oligopolistic competition. When managers have a preference for smooth time-paths of profits - as revealed by the empirical literature on “income smoothing” - manager-led firms can sustain collusive agreements at lower discount factors. Capped bonus plans and incumbency rents with termination threats make collusion supportable at any discount factor, independent of contracts' duration. When managers have these preferences/incentives and demand fluctuates, “price wars during booms” need not occur: the most collusive price may then be pro-cyclical.     

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.     

Decreasing impatience and the magnitude effect jointly contradict exponential discounting

The experimental literature on time preference finds that the manner in which subjects discount money (as opposed to utility) exhibits properties known as Decreasing Impatience and the Magnitude Effect. While these findings are often referred to as anomalies for the Exponential Discounting model, several authors have demonstrated that each of these qualitative findings can be explained by the curvature of utility and thus are not anomalies. We prove that, under basic regularity conditions, the two findings jointly imply the existence of Preference Reversals, and thus jointly contradict the Exponential Discounting model.     

Social Discount Rates from Stratospheric Ozone Control*

We develop a model of optimal project timing to infer the discount rate US residents attach to investments in stratospheric ozone preservation. In this model, the calendar date at which preservation-related benefits offset costs is shown to be an increasing function of the real social rate of time preference. Estimates of US costs and benefits associated with the original (1987) Montreal Protocol are used to determine a range of discount rates from the US control of stratospheric ozone depletion. From this example the range is about 3.0 to 8.77 per cent.     

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.     

A theory of (relative) discounting

We derive a representation theorem for time preferences (on the prize-time space) which identifies a novel notion of relative discounting as the key ingredient. This representation covers a variety of time preference models, including the standard exponential and hyperbolic discounting models and certain non-transitive time preferences, such as the similarity-based and subadditive discounting models. Our axiomatic work thus unifies a number of seemingly disparate time preference structures, thereby providing a tractable mathematical format that allows for investigating certain economic environments without subscribing to a particular time preference model. This point is illustrated by means of an application to sequential bargaining theory.     

A subjective model of experimentation

Typically an experiment is defined by a set of possible signals and a likelihood function, and both are specified exogenously—they are taken to be observable by the analyst. This paper renders them subjective by showing that they may be derived from suitable choice behavior. This is done in the context of an axiomatic representation theorem for preference on a suitable domain.     

Proper belief revision and equilibrium in dynamic games

We present a theory of rationality in dynamic games in which players, during the course of the game, may revise their beliefs about the opponents’ utility functions. The theory is based upon the following three principles: (1) the players’ initial beliefs about the opponents’ utilities should agree on some profile u of utility functions, (2) every player should believe, at each of his information sets, that his opponents are carrying out optimal strategies and (3) a player at information set h should not change his belief about an opponent's ranking of strategies a and b if both a and b could have led to h. Scenarios with these properties are called preference conjecture equilibria for the profile u of utility functions. We show that every normal form proper equilibrium for u induces a preference conjecture equilibrium for u, thus implying existence of preference conjecture equilibrium.     

Reference-dependent subjective expected utility

A reference-dependent generalisation of subjective expected utility theory is presented. In this theory, preferences between acts depend both on final outcomes and on reference points (which may be uncertain acts). It is characterised by a set of axioms in a Savage-style framework. A restricted form of the theory separates attitudes to end states (encoded in a ‘satisfaction function’) from attitudes to gains and losses of satisfaction. Given weak additional assumptions, the restricted theory excludes cycles of choice, explains observed disparities between willingness-to-pay and willingness-to-accept valuations of lotteries, and predicts preference reversal.     

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.