This paper considers decision problems involving partial information, and presents a conception of commodification of information under which a buyer with limited probabilistic sophistication can reduce any resulting ambiguity through purchases of new probabilistic information. In the specific formulation articulated herein, a lottery of information batches helps overcome the induced holdup during the contracting process. Under the proposed formulation, additional noise in the lottery leads to a reduction in the randomness of the buyer’s ambiguity by the introduction of an expectation operator for random sets of beliefs. We show that the buyer’s utility function for information, formalized as Hurwicz expected utility, is monotone, continuous and linear in information. We then discuss the value of information within the corresponding decision problem and contrast it with the influential formulation of Radner–Stiglitz.

This paper examines the impact of introducing a Rank-Dependent Utility (RDU) agent into a von Neumann-Morgenstern (vNM) pure-exchange economy with no aggregate uncertainty. In the absence of the RDU agent, the classical theory predicts that Pareto-optimal allocations are full-insurance, or no-betting, allocations. We show how the probability weighting function of the RDU agent, seen as a proxy for probabilistic risk aversion that is not captured by marginal utility of wealth, can lead to Pareto optima characterized by endogenous betting, despite common baseline beliefs. Such endogenous betting at an optimum leads to uncertainty-generating trade arising purely from heterogeneity in the perception of risk, rather than in beliefs. Our results formalize the intuitive understanding that probability weighting can act as an endogenous source of belief heterogeneity, and provide a new behavioral foundation for the coexistence of common beliefs and speculative behavior, in an environment with no initial aggregate uncertainty. Interpreting the RDU agent's nonlinear weighting function as an ``internality'' prompts the question of whether a social planner should intervene. We show how a benevolent social planner can nudge the RDU agent to behave closer to a vNM agent, through costly statistical or financial education, thereby (partially) restoring the optimality of full-insurance allocations.

In a pure-exchange economy with no aggregate uncertainty, we characterize in closed form and full generality Pareto-optimal allocations between two agents who maximize (nonconcave) rank-dependent utilities (RDU). We then derive a necessary and sufficient condition for Pareto optima to be no-betting allocations (i.e., deterministic allocations or full insurance allocations). This condition depends only on the probability weighting functions of the two agents and not on their (concave) utility of wealth. Hence, with RDU preferences, it is the difference in probabilistic risk attitudes given common beliefs rather than heterogeneity or ambiguity in beliefs that is a driver of betting behavior. As by-product of our analysis, we answer the question of when sunspots matter in this economy. Funding: M. Ghossoub acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada [Grant 2018-03961].

The analysis of optimal risk sharing has been thus far largely restricted to nonexpected utility models with concave utility functions, where concavity is an expression of ambiguity aversion and/or risk aversion. This paper extends the analysis to α‐maxmin expected utility, Choquet expected utility, and cumulative prospect theory, which accommodate ambiguity seeking and risk seeking attitudes. We introduce a novel methodology of quasidifferential calculus of Demyanov and Rubinov (1986, 1992) and argue that it is particularly well suited for the analysis of these three classes of utility functions, which are neither concave nor differentiable. We provide characterizations of quasidifferentials of these utility functions, derive first‐order conditions for Pareto optimal allocations under uncertainty, and analyze implications of these conditions for risk sharing with and without aggregate risk.

We study a general class of utility processes V(c)=(V_{t}(c)), where V_{t}(c), a dynamic utility operator, is a decision criterion that quantifies a decision maker's evaluation of uncertain consumption streams c. We call this dynamic utility operator robust and its distinctiveness is that it features the diffusion of the process V(c), i.e., the utility is affected by its own variability. A main result of this paper is to identify a general class of robust dynamic utility operators that are monotone and, yet, irreducibly depend on the utility variability. A principal motivation for studying such robust dynamic operators is that, by incorporating utility variability into the decision criterion, they bring a facility required to adapt models of ambiguity sensitive preferences to Brownian environments. In particular, those preference models which permit flexibility in ambiguity attitudes. We demonstrate this facility by obtaining continuous-time extensions of two prominent ambiguity aversion frameworks which incorporate variable ambiguity attitude, the smooth ambiguity model and the α-maxmin expected utility.

Motivated by financial and empirical arguments and in order to introduce a more flexible methodology of pricing, we provide a new approach to asset pricing based on Backward Volterra equations. The approach relies on an arbitrage-free and incomplete market setting in continuous time by choosing non-unique pricing measures depending either on the time of evaluation or on the maturity of payoffs. We show that in the latter case the dynamics can be captured by a time-delayed backward stochastic Volterra integral equation here introduced which, to the best of our knowledge, has not yet been studied. We then prove an existence and uniqueness result for time-delayed backward stochastic Volterra integral equations. Finally, we present a Lucas-type consumption-based asset pricing model that justifies the emergence of stochastic discount factors matching the term structure of Sharpe ratios.

We consider a general framework of optimal mechanism design under adverse selection and ambiguity about the type distribution of agents. We prove the existence of optimal mechanisms under minimal assumptions on the contract space and prove that centralized contracting implemented via mechanisms is equivalent to delegated contracting implemented via a contract menu under these assumptions. Our abstract existence results are applied to a series of applications that include models of optimal risk sharing and of optimal portfolio delegation.

The α-maxmin model is a prominent example of preferences under Knightian uncertainty as it allows to distinguish ambiguity and ambiguity attitude. These preferences are dynamically inconsistent for nontrivial versions of α. In this paper, we derive a recursive, dynamically consistent version of the α-maxmin model. In the continuous-time limit, the resulting dynamic utility function can be represented as a convex mixture between worst and best case, but now at the local, infinitesimal level. We study the properties of the utility function and provide an Arrow- Pratt approximation of the static and dynamic certainty equivalent. We derive a consumption-based capital asset pricing formula and study the implications for derivative valuation under indifference pricing.

This paper considers fundamental questions of arbitrage pricing that arises when the uncertainty model incorporates ambiguity about risk. This additional ambiguity motivates a new principle of risk- and ambiguity-neutral valuation as an extension of the paper by Ross (1976) (Ross, Stephen A. 1976. The arbitrage theory of capital asset pricing. Journal of Economic Theory 13: 341–60). In the spirit of Harrison and Kreps (1979) (Harrison, J. Michael, and David M. Kreps. 1979. Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory 20: 381–408), the paper establishes a micro-economic foundation of viability in which ambiguity-neutrality imposes a fair-pricing principle via symmetric multiple prior martingales. The resulting equivalent symmetric martingale measure set exists if the uncertain volatility in asset prices is driven by an ambiguous Brownian motion.

We consider finite-player simultaneous-play games of private information in which a player has no prior belief concerning the information under which the other players take their decisions, and which he therefore cannot discern. This dissonance leads us to develop the notion of Hurwicz–Nash equilibria of non-Bayesian games, and to present a theorem on the existence of such an equilibrium in a finite-action setting. Our pure-strategy equilibrium is based on non-expected utility under ambiguity as developed in Gul and Pesendorfer (2015). We do not assume a linear structure on the individual action sets, but do assume private information to be “diffused” and “dispersed.” The proof involves a multi-valued extension of an individual's prior to the join of the finest σ-algebra F of the information of the other players, and hinges on an absolute-continuity assumption on an individual's belief with respect to the extended beliefs on F.

We study economies with Knightian uncertainty about state prices. We introduce an equilibrium concept with sublinear prices and prove that equilibria exist under weak conditions. In general, such equilibria lead to inefficient allocations; they coincide with Arrow-Debreu equilibria if and only if the values of net trades are ambiguity-free in mean. In economies without aggregate uncertainty, inefficiencies are generic. Equilibrium allocations under price uncertainty are efficient in a constrained sense that we call uncertainty-neutral efficient. Arrow-Debreu equilibria turn out to be non-robust with respect to the introduction of Knightian uncertainty.

Any dynamic or stochastic notion of a general equilibrium relies on the underlying commodity space. Under sole risk and without multiple-prior uncertainty, the usual choice is a Lebesgue space from standard measure theory. In the case of volatility uncertainty it turns out that such a type of function space is no longer appropriate. For this reason we introduce and discuss a new natural commodity space, which can be constructed in three independent and equivalent ways. Each approach departs from one possible way to construct Lebesgue spaces. Moreover, we give a complete representation of the resulting topological dual space. This extends the classic Riesz representation in a natural way. Elements therein are the candidates for a linear equilibrium price system. This representation result has direct implications for the microeconomic foundation of finance under Knightian uncertainty.

In diffusion models, a few suitably chosen financial securities allow to complete the market. As a consequence, the efficient allocations of static Arrow–Debreu equilibria can be attained in Radner equilibria by dynamic trading. We show that this celebrated result generically fails if there is Knightian uncertainty about volatility. A Radner equilibrium with the same efficient allocation as in an Arrow–Debreu equilibrium exists if and only if the discounted net trades of the equilibrium allocation display no ambiguity in the mean. This property is violated generically in endowments, and thus Arrow–Debreu equilibrium allocations are generically unattainable by dynamically trading a few long-lived assets.

Knightian uncertainty leads naturally to nonlinear expectations. We introduce a corresponding equilibrium concept with sublinear prices and establish their existence. In general, such equilibria lead to Pareto inefficiency and coincide with Arrow--Debreu equilibria only if the values of net trades are ambiguity--free in the mean. Without aggregate uncertainty, inefficiencies arise generically. We introduce a constrained efficiency concept, uncertainty--neutral efficiency and show that Knight--Walras equilibrium allocations are efficient in this constrained sense. Arrow--Debreu equilibria turn out to be non--robust with respect to the introduction of Knightian uncertainty.

This article considers general equilibrium economies with a primitive uncertainty model that features ambiguity about continuous-time volatility. For the resulting non-equivalence of priors, an appropriate commodity-price space is introduced. Agents are heterogeneous in the size of captured ambiguity, endowment and preference for risk and ambiguity. Preferences are of variational type à la Maccheroni et al. (Econometrica 74(6):1447–1498, 2006). One important implication involves a problematic aspect of linear equilibrium price systems. Positive payoffs are for free on events outside the domain of the representing equilibrium pricing measure. Moreover, when aggregate risk is present and aggregate ambiguity is absent, the insurance properties of optimal allocations depend on the notion of ambiguity-free payoffs.

We establish a class of fully nonlinear conditional expectations. Similarly to the usage of linear expectations when a probabilistic description of uncertainty is present, we observe analogue quantitative and qualitative properties. The type of nonlinearity captures the agents sentiments of optimism and pessimism in an ambiguous environment. We then introduce an expected utility under a nonlinear expectation, and show monotonicity and continuity of utility. Risk aversion is characterized, and the properties of the certainty equivalent are discussed. Finally, we derive an Arrow-Pratt approximation of the static certainty equivalent and investigate the dynamic version via recursive equations.

The α-maxmin model is a prominent example of preferences under Knightian uncertainty as it allows to distinguish ambiguity and ambiguity attitude. These preferences are dynamically inconsistent for nontrivial versions of α. In this paper, we derive a recursive, dynamically consistent version of the α-maxmin model. In the continuous-time limit, the resulting dynamic utility function can be represented as a convex mixture between worst and best case, but now at the local, infinitesimal level.

This paper establishes, in the setting of Brownian information, a general equilibrium existence result in a heterogeneous agent economy. The existence is generic among income distributions. Agents differ moreover in their stochastic differential formulation of intertemporal recursive utility. The present class of utility functionals is generated by a recursive integral equation and incorporates preference for the local risk of the stochastic utility process. The setting contains models in which Knightian uncertainty is represented in terms of maxmin preferences of Chen and Epstein (Econometrica 70:1403–1443, 2002 ). Alternatively, Knightian decision making in terms of an inertia formulation from Bewley (Decis. Econ. Financ. 25:79–110, 2002 ) can be modeled as well.

Under risk, Arrow-Debreu equilibria can be implemented as Radner equilibria by continuous trading of few long-lived securities. We show that this result generically fails if there is Knightian uncertainty in the volatility. Implementation is only possible if all discounted net trades of the equilibrium allocation are mean ambiguity-free.