arXiv subjects

Fabio Maccheroni

Publications and source records attributed to Fabio Maccheroni.

At least 19 recordsLinked to original sources

arXiv2025

Disappointment Aversion and Expectiles

This paper recasts Gul's (1991) theory of disappointment aversion in a Savage framework, with general outcomes, new explicit axioms of disappointment aversion, and novel explicit representations. These permit broader applications of the theory and a better understanding of its decision-theoretic foundations. Our results exploit an unexpected connection between Gul's model and the econometric framework of Newey and Powell (1987) of asymmetric least squares estimation. Our main axiomatization result shows that a preference relation over Savage acts is probabilistically sophisticated, invariant biseparable, and disappointment hedging if and only if it admits a representation \emph{\`a la} Gul, and hence all explicit equivalent representations that we present in the paper. We also derive a neurocomputational foundation of the theory based on recent neuroscience findings and a novel reinforcement learning result.

econ.TH↗

arXiv2024

Affine Gateaux Differentials and the von Mises Statistical Calculus

This paper presents a general study of one-dimensional differentiability for functionals defined on convex domains that are not necessarily open. The local approximation is carried out using affine functionals, as opposed to linear functionals typically employed in standard Gateaux differentiability. This affine notion of differentiability naturally arises in certain applications and has been utilized by some authors in the statistics literature. We aim to offer a unified and comprehensive perspective on this concept.

math.FA↗

arXiv2023

Risk Aversion and Insurance Propensity

We provide a new foundation of risk aversion by showing that this attitude is fully captured by the propensity to seize insurance opportunities. Our foundation, which applies to all probabilistically sophisticated preferences, well accords with the commonly held prudential interpretation of risk aversion that dates back to the seminal works of Arrow (1963) and Pratt (1964). In our main results, we first characterize the Arrow-Pratt risk aversion in terms of propensity to full insurance and the stronger notion of risk aversion of Rothschild and Stiglitz (1970) in terms of propensity to partial insurance. We then extend the analysis to comparative risk aversion by showing that the notion of Yaari (1969) corresponds to comparative propensity to full insurance, while the stronger notion of Ross (1981) corresponds to comparative propensity to partial insurance.

econ.TH↗

arXiv2023

Capacities and Choquet Averages of Ultrafilters

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 is obtained as a consequence of an abstract analogue in the context of Archimedean Riesz spaces.

math.FA↗

arXiv2023

Algorithmic Decision Processes

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.

econ.TH↗

arXiv2022

Conditional divergence risk measures

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 provide representation results for niveloids in the conditional $L^{\infty}$-space. By employing such representation results we retrieve a conditional version of the variational formula for optimized certainty equivalents. In conclusion, we apply this formula to provide a variational representation of the conditional entropic risk measure.

q-fin.MF↗

arXiv2022

A Characterization of the Vector Lattice of Measurable Functions

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 $f \mapsto \int f\,\mathrm{d}\mu$ is strictly positive and order continuous. Here we show, in particular, that the converse holds true, i.e., any universally complete Riesz space $E$ with a weak order unit $e>0$ which admits a strictly positive order continuous linear functional on the principal ideal generated by $e$ is lattice isomorphic onto $L^0(\mu)$, for some probability measure space $(X,\Sigma,\mu)$.

math.FA↗

arXiv2021

Quantal Response Equilibrium and Rationalizability: Inside the Black Box

This paper aims to connect epistemic and behavioral game theory by examining the epistemic foundations of quantal response equilibrium (QRE) in static games. We focus on how much information agents possess about the probability distributions of idiosyncratic payoff shocks, in addition to the standard assumptions of rationality and common belief in rationality. When these distributions are transparent, we obtain a solution concept called $\Delta^p$-rationalizability, which includes action distributions derived from QRE; we also give a condition under which this relationship holds true in reverse. When agents only have common belief in the monotonicity of these distributions (for example, extreme value distributions), we obtain another solution concept called $\Delta^M$-rationalizability, which includes action distributions derived from rank-dependent choice equilibrium, a parameter-free variant of QRE. Our solution concepts also provide insights for interpreting experimental and empirical data.

econ.TH↗

arXiv2021

Multialternative Neural Decision Processes

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

cs.AI↗

arXiv2021

A Canon of Probabilistic Rationality

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 among such alternatives that satisfies Renyi's Conditioning Axiom. Our result shows that the Choice Axiom is, in a precise formal sense, a probabilistic version of the Weak Axiom. It thus supports Luce's view of his own axiom as a "canon of probabilistic rationality."

econ.TH↗

arXiv2021

Star-shaped Risk Measures

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 used risk measures, in particular, both convex risk measures and Value-at-Risk. From a financial viewpoint, our relaxation of convexity is necessary to quantify the capital requirements for risk exposure in the presence of liquidity risk, competitive delegation, or robust aggregation mechanisms. From a decision theoretical perspective, star-shaped risk measures emerge from variational preferences when risk mitigation strategies can be adopted by a rational decision maker.

econ.TH↗

arXiv2021

Multinomial logit processes and preference discovery: inside and outside the black box

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 \[ p_{t}\left( a,A\right) =\dfrac{e^{\frac{u\left( a\right) }{λ\left( t\right) }+α\left( a\right) }}{\sum_{b\in A}e^{\frac{u\left( b\right) }{λ\left( t\right) }+α\left( b\right) }}% \] where $p_{t}\left( a,A\right) $ is the probability that alternative $a$ is selected from the set $A$ of feasible alternatives if $t$ is the time available to decide, $λ$ is a time dependent noise parameter measuring the unit cost of information, $u$ is a time independent utility function, and $α$ is an alternative-specific bias that determines the initial choice probabilities reflecting prior information and memory anchoring. Our axiomatic analysis provides a behavioral foundation of softmax (also known as Multinomial Logit Model when $α$ is constant). Our neuro-computational derivation provides a biologically inspired algorithm that may explain the emergence of softmax in choice behavior. Jointly, the two approaches provide a thorough understanding of soft-maximization in terms of internal causes (neurophysiological mechanisms) and external effects (testable implications).

econ.TH↗

arXiv2020

Ergodic Annealing

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 can be very effective also when the cost function is unknown and has to be learned by an artificial agent.

cs.AI↗

arXiv2020

Making Decisions under Model Misspecification

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 as simplifications, and hence, as misspecified. We extend the max-min analysis under model ambiguity to incorporate the uncertainty induced by acknowledging that the models used in decision-making are simplified approximations. Formally, we provide an axiomatic rationale for a decision criterion that incorporates model misspecification concerns.

econ.TH↗

arXiv2020

Axiomatic Tests for the Boltzmann Distribution

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 interpretations. Such widespread use in very diverse fields suggests a common conceptual structure. We identify it on the basis of few natural axioms. Checking whether observables satisfy these axioms is easy, so our characterization provides a simple empirical test of the Boltzmannian modeling theories.

math.PR↗

arXiv2020

Equilibria of nonatomic anonymous games

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 of equilibrium such as self-confirming, peer-confirming, and Berk--Nash. This augmented scope is our main motivation. At the same time, our approach also resolves some conceptual problems present in Schmeidler (1973), pointed out by Shapley. In that paper\ the existence of pure-strategy Nash equilibria has been proved for any nonatomic game with a continuum of players, endowed with an atomless countably additive probability. But, requiring Borel measurability of strategy profiles may impose some limitation on players' choices and introduce an exogenous dependence among\ players' actions, which clashes with the nature of noncooperative game theory. Our suggested solution is to consider every subset of players as measurable. This leads to a nontrivial purely finitely additive component which might prevent the existence of equilibria and requires a novel mathematical approach to prove the existence of $\varepsilon$-equilibria.

econ.TH↗

arXiv2019

Behavioral Equivalence of Extensive Game Structures

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 transformations, commonly known as \textquotedblleft Interchanging of Simultaneous Moves\textquotedblright\ and \textquotedblleft Coalescing Moves/Sequential Agent Splitting.\textquotedblright

econ.TH↗

arXiv2019

Characterizations of Ideal Cluster Points

Given an ideal $\mathcal{I}$ on $ω$, 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 $X$ and as the smallest closed set containing ``almost all'' the sequence. As a consequence, we obtain that the underlying topology $τ$ coincides with the topology generated by the pair $(τ,\mathcal{I})$.

math.CA↗