Simone Cerreia-Vioglio

• Bocconi University

We introduce Cautious Utility, a new model based on the idea that individuals are unsure of trade‐offs between goods and apply caution. The model yields an endowment effect, even when gains and losses are treated symmetrically. Moreover, it implies either loss aversion or loss neutrality for risk, but in a way unrelated to the endowment effect, and...

We show that a normalized capacity $\nu: \mathcal{P}(\mathbf{N})\to \mathbf{R}$ is invariant with respect to an ideal $\mathcal{I}$ on $\mathbf{N}$ if and only if it can be represented as a Choquet average of $\{0,1\}$-valued finitely additive probability measures corresponding to the ultrafilters containing the dual filter of $\mathcal{I}$. This i...

This article proposes a model of non-Bayesian social learning in networks that accounts for heuristics and biases in opinion aggregation. The updating rules are represented by non-linear opinion aggregators from which we extract two extreme networks capturing strong and weak links. We provide graph-theoretic conditions for these networks that chara...

Orthogonal decompositions are essential tools for the study of weakly stationary time series. Some examples are given by the classical Wold decomposition of Wold (A study in the analysis of stationary time series, Almqvist & Wiksells Boktryckeri, Uppsala, 1938) and the extended Wold decomposition of Ortu et al. (Quant Econ 11(1):203–230, 2020), whi...

We provide two characterizations, one axiomatic and the other neuro-computational, of the dependence of choice probabilities on deadlines, within the widely used softmax representation $$ \begin{align} p_{t}\left(a,A\right) =\dfrac{e^{\frac{u\left( a\right) }{\lambda\left( t\right) }+\alpha\left( a\right) }}{\sum_{b\in A}e^{\frac{u\left( b\right) }...

We analyze the problem of constructing multiple buy-and-hold mean-variance portfolios over increasing investment horizons in continuous-time arbitrage-free stochastic interest rate markets. The orthogonal approach to the one-period mean-variance optimization of Hansen and Richard (Econometrica 55(3):587–613, 1987) requires the replication of a risk...

We define a new notion of equilibrium for nonatomic anonymous games, termed ε-estimated equilibrium, and prove its existence for any positive ε. This notion encompasses and brings to nonatomic games recent concepts of equilibrium such as self-confirming, peer-confirming, and Berk-Nash. This augmented scope is our main motivation. Our approach also...

We consider random choice rules that, by satisfying a weak form of Luce’s choice axiom, embody a form probabilistic rationality. We show that for this important class of stochastic choices, the law of demand for normal goods—arguably the main result of traditional consumer theory—continues to hold on average when strictly dominated alternatives are...

This paper provides a general framework for analyzing self-confirming policies. We study self-confirming equilibria in recurrent decision problems with incomplete information about the true stochastic model. We characterize stationary monetary policies in a linear-quadratic setting.

Given a probability measure space (X, Σ, µ), it is well known that the Riesz space L 0 (µ) of equivalence classes of measurable functions f : X → R is universally complete and the constant function 1 is a weak order unit. Moreover, the linear functional L ∞ (µ) → R defined by f → f dµ is strictly positive and order continuous. Here we show, in part...

Given a probability measure space $(X,\Sigma,\mu)$, it is well known that the Riesz space $L^0(\mu)$ of equivalence classes of measurable functions $f: X \to \mathbf{R}$ is universally complete and the constant function $\mathbf{1}$ is a weak order unit. Moreover, the linear functional $L^\infty(\mu)\to \mathbf{R}$ defined by $f \mapsto \int f\,\ma...

We prove that a random choice rule satisfies Luce's Choice Axiom if and only if its support is a choice correspondence that satisfies the Weak Axiom of Revealed Preference, thus it consists of alternatives that are optimal according to some preference, and random choice then occurs according to a tie-breaking mechanism among such alternatives that...

The Boltzmann distribution family describes a single parameter (temperature) class of probability distributions over a state space; at any given temperature, the ratio of probabilities of two states depends exponentially on their difference in energy. Beyond physics, this distribution family is very popular in many important disciplines, under diff...

One of the most well known models of non‐expected utility is Gul's (1991) model of disappointment aversion. This model, however, is defined implicitly, as the solution to a functional equation; its explicit utility representation is unknown, which may limit its applicability. We show that an explicit representation can be easily constructed, using...

We use decision theory to confront uncertainty that is sufficiently broad to incorporate "models as approximations." We presume the existence of a featured collection of what we call "structured models" that have explicit substantive motivations. The decision maker confronts uncertainty through the lens of these models, but also views these models...

We prove that a random choice rule satisfies Luce's Choice Axiom if and only if its support, the set of "alternatives that can be chosen," is a choice correspondence that satisfies the Weak Axiom of Revealed Preference, and random choice occurs according to a stochastic tie breaking among optimizers that satisfies Renyi's Conditioning Axiom. Our re...

The Boltzmann distribution describes a single parameter (temperature) family of probability distributions over a state space; at any given temperature, the ratio of probabilities of two states depends on their difference in energy. The same family is known in other disciplines (economics, psychology, computer science) with different names and inter...

We develop a general framework to study source-dependent preferences in economic contexts. We behaviorally identify two key features. First, we drop the assumption of uniform uncertainty attitudes and allow for source-dependent attitudes. Second, we introduce subjective prices to compare outcomes across different sources. Our model evaluates profil...

In this paper, we provide an axiomatic foundation for the value-based version of the drift diffusion model (DDM) of Ratcliff, a successful model that describes two-alternative speeded decisions between consumer goods. Our axioms present a test for model misspecification and connect the externally observable properties of choice with an important ne...

We add here another layer to the literature on nonatomic anonymous games started with the 1973 paper by Schmeidler. More specifically, we define a new notion of equilibrium which we call $\varepsilon$-estimated equilibrium and prove its existence for any positive $\varepsilon$. This notion encompasses and brings to nonatomic games recent concepts o...

We introduce an algorithmic decision process for multialternative choice that combines binary comparisons and Markovian exploration. We show that a functional property, transitivity, makes it testable.

We provide two characterizations, one axiomatic and the other neuro-computational, of the dependence of choice probabilities on deadlines, within the widely used softmax representation \[ p_{t}\left( a,A\right) =\dfrac{e^{\frac{u\left( a\right) }{\lambda \left( t\right) }+\alpha \left( a\right) }}{\sum_{b\in A}e^{\frac{u\left( b\right) }{\lambda \l...

We study a decision maker characterized by two binary relations. The first reflects his judgments about well-being, his mental preferences. The second describes the decision maker’s choice behavior, his behavioral preferences. We propose axioms that describe a relation between these two preferences, so between mind and behavior, thus disentangling...

We study stochastic choice as the outcome of deliberate randomization. We derive a general representation of a stochastic choice function where stochasticity allows the agent to achieve from any set the maximal element according to her underlying preferences over lotteries. We show that in this model stochasticity in choice captures complementarity...

We study how changes in wealth affect ambiguity attitudes. We define a decision maker as decreasing (resp., increasing) absolute ambiguity averse if he becomes less (resp., more) ambiguity averse as he becomes richer. Our definition is behavioral. We provide different characterizations of these attitudes for a large class of preferences: monotone a...

We show that a probability measure on a metric space has full support, if, and only if, the set of all probability measures, that are absolutely continuous with respect to it, is dense in the set of all Borel probability measures. We illustrate the result through a general version of Laplace’s method, which in turn leads to general stochastic conve...

Pre-Hilbert A-modules are a natural generalization of inner product spaces in which the scalars are allowed to be from an arbitrary algebra. In this perspective, submodules are the generalization of vector subspaces. The notion of orthogonality generalizes in an obvious way too. In this paper, we focus on f-algebras A that are either of L∞ or of L⁰...

In this work, we propose a definition of comonotonicity for elements of B(H)sa, i.e. bounded self-adjoint operators defined over a complex Hilbert space H. We show that this notion of comonotonicity coincides with a form of commutativity. Intuitively, comonotonicity is to commutativity as monotonicity is to bounded variation. We also define a notio...

In a decision problem under uncertainty, a decision maker considers a set of alternative actions whose consequences depend on uncertain factors beyond his control. Following Luce and Raiffa (Games and decisions: introduction and critical survey. Wiley, New York, 1957), we adopt a natural representation of such a situation which takes as primitives...

We consider real pre-Hilbert modules H on Archimedean f-algebras A with unit e. We provide conditions on A and H such that a Riesz representation theorem for bounded/continuous A-linear operators holds.

Recent research emphasizes the importance of information feedback in situations of recurrent decisions and strategic interaction, showing how it affects the uncertainty that underlies selfconfirming equilibrium (e.g., Battigalli et al., 2015, Fudenberg and Kamada, 2015). Here, we discuss in detail several properties of this key feature of recurrent...

Motivated by dynamic asset pricing, we extend the dual pairs’ theory of Dieudonné (1942) and Mackey (1945) to pairs of modules over a Dedekind complete f-algebra with multiplicative unit. The main tools are: • a Hahn–Banach Theorem for modules of this kind;• a topology on the f-algebra that has the special feature of coinciding with the norm topolo...

We characterize the consistency of a large class of nonexpected utility preferences (including mean-variance preferences and prospect theory preferences) with stochastic orders (for example, stochastic dominances of different degrees). Our characterization rests on a novel decision theoretic result that provides a behavioral interpretation of the s...

This paper provides a general framework for the analysis of self-confirming policies. We first study self-confirming equilibria in recurrent decision problems with incomplete information about the true stochastic model. Next we illustrate the theory with a characterization of stationary monetary policies in a linear-quadratic setting. Finally we pr...

An action is justifiable if it is a best reply to some belief. We show that higher ambiguity aversion expands the set of justifiable actions.

Recent research emphasizes the importance of information feedback in situations of recurrent decisions and strategic interaction, showing how it affects the uncertainty that underlies selfconfirming equilibrium (e.g., Battigalli et al., American Economic Review, 2015, Fudenberg and Kamada, Theoretical Economics, 2015). Here, we discuss in detail se...

We establish an Ergodic Theorem for lower probabilities, a generalization of standard probabilities widely used in applications. As a by-product, we provide a version for lower probabilities of the Strong Law of Large Numbers.

We analyze a notion of self-confirming equilibrium with non-neutral ambiguity attitudes that generalizes the traditional concept. We show that the set of equilibria expands as ambiguity aversion increases. The intuition is quite simple: by playing the same strategy in a stationary environment, an agent learns the implied distribution of payoffs, bu...

Maccheroni, Marinacci, and Rustichini (2006), in an Anscombe–Aumann framework, axiomatically characterize preferences that are represented by the variational utility functional V(f)=minp∈Δ{∫u(f)dp+c(p)}∀f∈F, where u is a utility function on outcomes and c is an index of uncertainty aversion. In this paper, for a given variational preference, we stu...

Given a functional defined on a nonempty subset of an Archimedean Riesz space with unit, necessary and sufficient conditions are obtained for the existence of a (convex or concave) niveloid that extends the functional to the entire space. In the language of mathematical finance, this problem is equivalent to the one of verifying if the policy adopt...

Many violations of the Independence axiom of Expected Utility can be traced to subjects' attraction to risk-free prospects. The key axiom in this paper, Negative Certainty Independence (Dillenberger, 2010), formalizes this tendency. Our main result is a utility representation of all preferences over monetary lotteries that satisfy Negative Certaint...

Our recent research emphasizes the importance of information feedback in situations of recurrent decisions and strategic interaction, showing how it affects the uncertainty that underlies selfconfirming equilibrium. Here, we discuss in detail the properties of this key feature of recurrent interaction. This allows us to elucidate our notion of Maxm...

Starting with the seminal paper of Gilboa and Schmeidler (1989) an analogy between the maxmin approach of Decision Theory under Ambiguity and the minimax approach of Robust Statistics -- e.g. Huber and Strassen (1973) -- has been hinted at. The present paper formally clarifies this relation by showing the conditions under which the two approaches a...

We consider decision makers who know that payoff-relevant observations are generated by a process that belongs to a given class M, as postulated in Wald [Wald A (1950) Statistical Decision Functions (Wiley, New York)]. We incorporate this Waldean piece of objective information within an otherwise subjective setting à la Savage [Savage LJ (1954) The...

Many violations of the Independence axiom of Expected Utility can be traced to subjects’ attraction to risk-free prospects. The key axiom in this paper, Negative Certainty Independence (Dillenberger, 2010), formalizes this tendency. Our main result is a utility representation of all preferences over monetary lotteries that satisfy Negative Certaint...

We study the interplay of probabilistic sophistication, second order stochastic dominance and uncertainty aversion, three fundamental notions in choice under uncertainty. In particular, our main result. Theorem 2, characterizes uncertainty averse preferences that are probabilistically sophisticated, as well as uncertainty averse preferences that sa...

We establish integral representation results for suitably pointwise continuous and comonotonic addi- tive functionals of bounded variation de?ned on Stone lattices.

We give a general integral representation theorem (Theorem 6) for nonadditive functionals de?ned on an Archimedean Riesz space X with order unit. Additivity is replaced by a weak form of modularity, or equivalently dual comonotonic additivity, and integrals are Choquet integrals. Those integrals are de?ned through the Kakutani [8] isometric identi?...

We extend the Fundamental Theorem of Finance and the Pricing Rule Representation Theorem of Cox and Ross (see Ross [29] and [31] and Cox and Ross [8]) to the case in which market frictions are taken into account but the Put?Call Parity is still assumed to hold. In turn, we obtain a representation of the pricing rule as a discounted expectation with...

This paper analyzes preferences in the presence of ambiguity that are rational in the sense of satisfying the classical ordering condition as well as monotonicity. Under technical conditions that are natural in an Anscombe–Aumann environment, we show that even for such a general preference model, it is possible to identify a set of priors, as first...

We study uncertainty averse preferences, that is, complete and transitive preferences that are convex and monotone. We establish a representation result, which is at the same time general and rich in structure. Many objective functions commonly used in applications are special cases of this representation.

As in [Gilboa et al., 2010], we consider a decision maker characterized by two binary relations: ≽* and ≽∧. The first binary relation is a Bewley preference. It models the rankings for which the decision maker is sure. The second binary relation is an uncertainty averse preference, as defined by [Cerreia-Vioglio et al., 2011c]. It models the rankin...

We propose to bring together two conceptually complementary ideas: (1) selfconfi?rming equilibrium (SCE): at a rest point of learning dynamics in a game played recurrently, agents best respond to confi?rmed beliefs, i.e. beliefs consistent with the evidence they accumulate, and (2) ambiguity aversion: agents, coeteris paribus, prefer to bet on even...

When there is uncertainty about interest rates (typically due to either illiquidity or defaultability of zero coupon bonds) the cash-additivity assumption on risk measures becomes problematic. When this assumption is weakened, to cash-subadditivity for example, the equivalence between convexity and the diversification principle no longer holds. In...

We study convex preferences over lotteries and over menus of lotteries. We consider a set of consequences C and we characterize complete, transitive, and convex binary relations over lotteries on the set C. We prove that convex preferences correspond to a decision criterion in which the Decision Maker reveals pessimism and a lack of con…dence in th...

We introduce a notion of complete monotone quasiconcave duality and we show that it holds for important classes of quasiconcave functions.