Fabio Maccheroni

Bocconi University | Bocconi · Carlo F. Dondena Centre for Research on Social Dynamics

The drift diffusion model (DDM) is a prominent account of how people make decisions. Many of these decisions involve comparing two alternatives based on differences of perceived stimulus magnitudes, such as economic values. Here, we propose a consistent estimator for the parameters of a DDM in such cases. This estimator allows us to derive decision...

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...

We develop a full-fledged analysis of an algorithmic decision process that, in a multialternative choice problem, produces computable choice probabilities and expected decision times.

Our paper contributes to the theory of conditional risk measures and conditional certainty equivalents. We adopt a random modular approach which proved to be effective in the study of modular convex analysis and conditional risk measures. In particular, we study the conditional counterpart of optimized certainty equivalents. In the process, we prov...

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) }...

One of the mantras of risk measurement is the avoidance of risk concentration. However, most formal approaches to the topic actually require more than this. In “Star-Shaped Risk Measures,” Castagnoli, Cattelan, Maccheroni, Tebaldi, and Wang study this property “in purity” for monetary risk measures. They show that it unites value at risk and convex...

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 establish a simple condition, based on the willingness to bet on events, under which two biseparable preferences have cardinally equivalent utilities

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...

This paper provides a robust epistemic foundation for predicting and implementing collective actions when only the proportions that take specific actions in the population matter. We apply $\Delta$-rationalizability to analyze strategic sophistication entailed in (structural) quantal response equilibrium (QRE); the former is called $\Delta(p)$-rati...

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...

In this paper monetary risk measures that are positively superhomogeneous, called star-shaped risk measures, are characterized and their properties studied. The measures in this class, which arise when the controversial subadditivity property of coherent risk measures is dispensed with and positive homogeneity is weakened, include all practically u...

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...

Simulated Annealing is the crowning glory of Markov Chain Monte Carlo Methods for the solution of NP-hard optimization problems in which the cost function is known. Here, by replacing the Metropolis engine of Simulated Annealing with a reinforcement learning variation -- that we call Macau Algorithm -- we show that the Simulated Annealing heuristic...

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...

Two extensive game structures with imperfect information are said to be behaviorally equivalent if they share the same map (up to relabelings) from profiles of structurally reduced strategies to induced terminal paths. We show that this is the case if and only if one can be transformed into the other through a composition of two elementary transfor...

Two extensive game structures with imperfect information are said to be behav-iorally equivalent if they share the same map (up to relabelings) from profiles of structurally reduced strategies to induced terminal paths. We show that this is the case if and only if one can be transformed into the other through a composition of two elementary transfo...

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...

Given a topological space $X$ and an ideal $\mathcal{I}$ on the positive integers, we say that $\ell$ is a cluster point of a sequence $(x_n)$ provided that $\{n: x_n \in U\}\notin \mathcal{I}$ for each neighborhood $U$ of $\ell$. It is shown that, if $X$ is sequential Lindel\"{o}f, then $X$ is compact if and only if every sequence $(x_n)$ admits a...

Given an ideal $\mathcal{I}$ on $\omega$, we prove that a sequence in a topological space $X$ is $\mathcal{I}$-convergent if and only if there exists a ``big'' $\mathcal{I}$-convergent subsequence. Then, we study several properties and show two characterizations of the set of $\mathcal{I}$-cluster points as classical cluster points of a filters on...

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...

This chapter reviews developments in the theory of decision making under risk and uncertainty, focusing on models that, over the last 40 years, dominated the theoretical discussions. It also surveys some implications of the departures from the "linearity in the probabilities" aspect of expected utility theory to game theory. The chapter consists of...

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...

We study a two-period economy in which agents' preferences take into account their relative economic position. The study builds on a decision theoretic analysis of the social emotions that underlie these relative concerns. These emotions, envy and pride, respond to social losses and gains, respectively. Our main result is that envy leads to conform...

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...

We study a two-period economy in which agents' preferences take into account their relative economic position. The study builds on a decision theoretic analysis of the social emotions that underlie these relative concerns. These emotions, envy and pride, respond to social losses and gains, respectively. Our main result is that envy leads to conform...

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...

We reconcile Kaplan and Garrick's seminal definition of risk with classical subjective expected utility, filling in the relevant gaps and providing a framework that is ready-to-use in applications. We show that Kaplan and Garrick's "frequency" format can be set in one-to-one correspondence with [26]'s utility theory. Kaplan and Garrick's "probabili...

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 derive the analogue of the classic Arrow–Pratt approximation of the certainty equivalent under model uncertainty as described by the smooth model of decision mak-ing under ambiguity of Klibanoff, Marinacci, and Mukerji (2005). We study its scope by deriving a tractable mean-variance model adjusted for ambiguity and solving the cor-responding por...

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...

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.

We study orders of risk and model uncertainty aversion in the smooth ambiguity model proposed by Klibanoff, Marinacci, and Mukerji (2005). We consider a quadratic approximation of their model and we show that both risk and model uncertainty attitudes have at most a second order effect. Specifically, the order depends on the properties of the suppor...

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...

We derive the analogue of the classic Arrow-Pratt approximation of the certainty equivalent under model uncertainty as defined by the smooth model of decision making under ambiguity of Klibanoff, Marinacci and Mukerji (2005). We study its scope via a portfolio allocation exercise that delivers a tractable mean-variance model adjusted for model unce...

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...

A decision maker (DM) is characterized by two binary relations. The first reflects choices that are rational in an " objective" sense: the DM can convince others that she is right in making them. The second relation models choices that are rational in a " subjective" sense: the DM cannot be convinced that she is wrong in making them.In the context...

This paper develops and empirically confirms a theory that explains why media content predicts takeover outcomes. It shows why target shareholders pay attention to the news, despite the risk of distorted reporting. To test the model's prediction, this paper constructs a novel empirical measure that quantifies text-based media content pertaining to...

We propose a portfolio selection model based on a class of monotone preferences that coincide with mean-variance preferences on their domain of monotonicity, but differ where mean-variance preferences fail to be monotone and are therefore not economically meaningful. The functional associated with this new class of preferences is the best approxima...

In this paper we consider the classical problem of dividing a land among many agents so that everybody is satisfied with the parcel she receives. In the literature, it is usually assumed that all the agents are endowed with cardinally comparable, additive, and monotone utility functions. In many economic and political situations violations of these...

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 foundati...

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

Dynamic consistency is a fundamental property in dynamic choice models. It requires that if a decision maker plans to take some action at some juncture in the future, he should consistently take that action when finding himself at that juncture, and vice versa if he plans to take a certain action at a certain juncture, he should take that plan in m...

Given falling birth rates, ageing baby boomers approaching retirement age as well as a pension crisis in most advanced economies, understanding the characteristics of the labour supply function of the elderly have taken on a new significance. Even in developing countries, with labour surplus economies, this is a major issue as these poor countries...

Given falling birth rates, ageing baby boomers approaching retirement age as well as a pension crisis in most advanced economies, understanding the characteristics of the labour supply function of the elderly have taken on a new significance. Even in developing countries, with labour surplus economies, this is a major issue as these poor countries...

For (S,Σ) a measurable space, let C 1 and C 2 be convex, weak* closed sets of probability measures on Σ. We show that if C 1 ∪ C 2 satisfies the Lyapunov property, then there exists a set A ∈ Σ such that \( \min _{\mu _1 \in \mathcal{C}_1 } \mu _1 \left( A \right) \) > \( \min _{\mu _2 \in \mathcal{C}_2 } \mu _2 \left( A \right) \) . We give applic...

Given falling birth rates, ageing baby boomers approaching retirement age as well as a pension crisis in most advanced economies, understanding the characteristics of the labour supply function of the elderly have taken on a new significance. Even in developing countries, with labour surplus economies, this is a major issue as these poor countries...

We characterize, in the Anscombe-Aumann framework, the preferences for which there are a utility functionu on outcomes and an ambiguity indexc on the set of probabilities on the states of the world such that, for all acts f and g, [GRAPHICS] The function u represents the decision maker's risk attitudes, while the index c captures his ambiguity atti...

Given falling birth rates, ageing baby boomers approaching retirement age as well as a pension crisis in most advanced economies, understanding the characteristics of the labour supply function of the elderly have taken on a new significance. Even in developing countries, with labour surplus economies, this is a major issue as these poor countries...

For (S, Σ) a measurable space, let \({\cal C}_1\) and \({\cal C}_2\) be convex, weak* closed sets of probability measures on Σ. We show that if \({\cal C}_1\) ∪ \({\cal C}_2\) satisfies the Lyapunov property , then there exists a set A ∈ Σ such that minμ1∈\({\cal C}_1\) μ1(A) > maxμ2 ∈ \({\cal C}_2\)(A). We give applications to Maxmin Expected Util...

We study the cores of non-atomic market games, a class of transferable utility cooperative games introduced by Aumann and Shapley (Values of non-atomic games, 1974), and, more in general, of those games that admit a na-continuous and concave extension to the set of ideal coalitions, studied by Einy etal. (Int J Game Theory 28:1–14, 1999). We show t...

We consider a totally monotone capacity on a Polish space and a sequence of bounded p.i.i.d. random variables. We show that, on a full set, any cluster point of empirical averages lies between the lower and the upper Choquet integrals of the random variables, provided either the random variables or the capacity are continuous.

In a multiple priors model á la Gilboa and Schmeidler (1989), we provide necessary and sufficient behavioral conditions ensuring the countable additivity and non-atomicity of all priors. Copyright Springer-Verlag Berlin/Heidelberg 2005

We investigate the class of stochastic orders induced by Generalized Gini Functionals (GGF) [Yaari (1987) dual functionals] and identify the maximal classes of functionals associated with these orders. Our results are inspired by Marshall (1991) and are dual to those obtained for additive representations in Müller (1997) and in Castagnoli and Macch...

Economists often operate under an implicit assumption that the tastes of a decision maker are quite stable, while his beliefs change with the availability of new information. We show that for a general class of preferences, a separation of a key component of tastes, the utility function, from the other components of the representation is possible o...

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 s...

In the classic Anscombe and Aumann decision setting, we give necessary and sufficient conditions that guarantee the existence of a utility function u on outcomes and an ambiguity index c on the set of all probabilities on the states of the world such that acts are ranked according to the criterion V(f)=min{E(u(f),p)+c(p)} where p ranges over all pr...

Let # be a preference relation on a convex set F . Necessary and sufficient conditions are given that guarantee the existence of a set of affine utility functions on F such that # is represented by U ( f ) = u l ( f ) if f F l ; where each F l is a convex subset of F . The interpretation is simple: facing a "non-homogeneous" set F of alternatives,...

This note shows how Yaari (1987)'s dual theory of choice under risk naturally extends to the case of incomplete preferences. This also provides an axiomatic characterization of a large and widely studied class of stochastic orders used to rank the riskiness of random variables or the dispersion of income distributions (including, e.g., second order...

We show that, if prices in a market are Choquet expectations, the existence of one frictionless asset may force the whole market to be frictionless. Any risky asset will cause this collapse if prices depend only on the distribution with respect to a given nonatomic probability measure; the frictionless asset has to be fully revealing if such depend...

For (S, S) a measurable space, let C1 and C2 and be convex, weak* closed sets of probability measures on S. We show that if C1 C2 satisfies the Lyapunov property, then there exists a set A S such that min C1 (A) > max C2 (A). We give applications to Maxmin Expected Utility and to the core of a lower probability.

This paper explores the consequences of cognitive dissonance, coupled with time-inconsistent preferences, in an intertemporal decision problem with two distinct goals: acting decisively on early information (vision) and adjusting flexibly to late information (flexibility). The decision maker considered here is capable of manipulating information to...

We show that the monotone continuity condition introduced by Villegas (1964) and Arrow (1970) is the behavioral counterpart of countable additivity (and relative weak compactness) in a multiple priors model. This generalizes their original result, in which the special case of a singleton set of priors is considered. Further extending their results,...

The Dutch book argument is a coherence condition for the existence of subjective probabilities. This paper gives a general framework of analysis for this argument in a nonadditive probability setting. Particular cases are given by comonotonic and affinely related Dutch books that lead to Choquet expectations and Min expectations.

Existential and constructive solutions to the classic problems of fair division are known for individuals with constant marginal evaluations. By considering nonatomic concave capacities instead of nonatomic probability measures, we extend some of these results to the case of individuals with decreasing marginal evaluations. Copyright Springer-Verla...

Let % be a preference relation on a convex set F. Necessary and sufcient conditions are given that guarantee the existence of a set fulg of afne util- ity functions on F such that % is represented by U ( f ) = ul ( f ) if f 2 Fl; where each Fl is a convex subset of F. The interpretation is simple: facing a ìnon-homogeneousî set F of alternatives, a...

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 very general decision model. In the first part of the paper we introduce an "unambiguous preference" relation derived from the DM's preferences, and show that it can be represented by a set of probability me...

Let $\succsim $ be a continuous and convex weak order on the set of lotteries defined over a set Z of outcomes. Necessary and sufficient conditions are given to guarantee the existence of a set $\mathcal{U}$ of utility functions defined on Z such that, for any lotteries p and q, \[ p\succsim q \Leftrightarrow \min_{u\in{\mathcal U}}{\Bbb E} _p\lef...