arXiv subjects

M. Ali Khan

Publications and source records attributed to M. Ali Khan.

At least 19 recordsLinked to original sources

arXiv2026

Almost-Uniform Bayesian Convergence to the Truth Is Not Characterized by Countable Additivity on Conditional Hitting Times

We revisit the analysis of Bayesian convergence to the truth under finite additivity in a recent paper by Nielsen (J. Philos. Logic, 2021, doi:10.1007/s10992-020-09569-2) Its principal theorem proves that the posteriors of a probability function converge to the truth almost uniformly if and only if the function has two properties: an approximation property, and so-called countable additivity on conditional hitting times. We show that the two properties are necessary but not sufficient, so that the theorem is false, and we locate the error in its published proof. We construct a merely finitely additive probability function that has both properties and whose posteriors converge to the truth almost surely but not almost uniformly. Three of the paper's four remaining theorems, and its corollary, lose their published proofs as well, two with the failed implication and two to a separate defect that we also identify. Two of the four results we reprove and one we leave undecided; the last is the corollary, which our counterexample does not refute, and which we establish for a family including the counterexample and leave open in general. We also show that almost-sure convergence of posteriors to the truth for every event does not characterize countable additivity. We close with what a repaired characterization of almost-uniform convergence would have to add.

math.PR↗

arXiv2026

All Games Have Equilibria

Research on Nash equilibrium existence for infinite games has grown into a patchwork of technical preconditions and counterexamples. This paper presents a unified program in equilibrium theory by revising the predominant model of mixed strategies based on countable additivity. A game is specified by a nonempty set of players and, for each player, a nonempty action set and a bounded von Neumann-Morgenstern utility function. Every such game is shown to admit a Nash equilibrium in finitely additive mixed strategies. In addition, the equilibrium correspondence for any such game is shown to be nonempty, compact-valued, and upper hemicontinuous, and the same is true for equilibria obtained as limits of finite approximations. Techniques developed in this paper show that infinite games long treated as intractable become amenable to direct equilibrium analysis.

econ.TH↗

arXiv2026

On Continuity of Separately Convex Preferences and Correspondences

We study separate convexity for preferences and correspondences, and show that this weakening of the usual convexity postulate is strong enough to recover standard equivalences among continuity assumptions. For complete and transitive preferences, we establish equivalence theorems linking separate continuity, mixture continuity, Archimedean-type postulates, solvability and graph continuity, successively on product mixture sets and on Euclidean spaces. The results highlight the role of weaker axiomatic assumptions by yielding representations for multilinear cardinal utility, continuous separately quasiconcave ordinal utility in $n$-person decision problems, and a scalar Anscombe--Aumann setting. For non-ordered preferences, formulated as correspondences, we characterize the open graph property under separate convexity and weak section-continuity, generalizing results of Schmeidler, Shafer, and Bergstrom-Parks-Rader. Examples identify the boundaries of our results.

econ.TH↗

arXiv2026

Existence of Equilibria in Large Competitive Markets with Bads, Production and Comprehensive Externalities

This paper establishes existence of equilibrium in a measure-theoretic general equilibrium (MGE) model with production, bads, and comprehensive externalities. These features are jointly essential for modeling perfect competition in which emissions of production byproducts impose harm on agents. We show that, when bads and externalities are modeled in an economically natural way, equilibrium exists. This yields the first existence theorem with bads for MGE models, the benchmark for perfect competition, overcoming Hara (2005)'s nonexistence example. The proof uses nonstandard analysis, which provides a systematic technique to extend results for finite to infinite models.

econ.TH↗

arXiv2026

The Effects of Social Pressure on Fundamental Choices: Indecisiveness and Deferral

In mainstream neoclassical economics, utility maximization is the only engine of individual action, and the other or the social, if it is modeled for decisions deemed fundamental, it is done as a tacit externality parameter affecting an agent's maximized payoff. And even when hitched to a social reference point, a fully decisive and immediate response is invariably assumed. In this paper, we propose a non-standard articulation of the trade-off between personal utility and social distance, one motivated by experimental evidence from psychology, management science, and economics. Our approach deconstructs non-recurrent consumer choice to two stages: a non-decisive first stage in which a binary relation, called one-many ordering, yields an interval, the consideration set, to which the deferred choice is confined; a decisive second stage in which the distance from the average social choice, and future social expectations, are taken into account in present utility. Finally, we embed this indecisive consumer in an exploratory game-theoretic setting, and show that indecisiveness and choice deferral may cause social loss.

econ.TH↗

arXiv2024

Two-Person Adversarial Games are Zero-Sum: An Elaboration of a Folk Theorem

The observation that every two-person adversarial game is an affine transformation of a zero-sum game is traceable to Luce & Raiffa (1957) and made explicit in Aumann (1987). Recent work of (ADP) Adler et al. (2009), and of Raimondo (2023) in increasing generality, proves what has so far remained a conjecture. We present two proofs of an even more general formulation: the first draws on multilinear utility theory developed by Fishburn & Roberts (1978); the second is a consequence of the ADP proof itself for a special case of a two-player game with a set of three actions.

econ.TH↗

arXiv2023

On Existence of Berk-Nash Equilibria in Misspecified Markov Decision Processes with Infinite Spaces

Model misspecification is a critical issue in many areas of theoretical and empirical economics. In the specific context of misspecified Markov Decision Processes, Esponda and Pouzo (2021) defined the notion of Berk-Nash equilibrium and established its existence in the setting of finite state and action spaces. However, many substantive applications (including two of the three motivating examples presented by Esponda and Pouzo, as well as Gaussian and log-normal distributions, and CARA, CRRA and mean-variance preferences) involve continuous state or action spaces, and are thus not covered by the Esponda-Pouzo existence theorem. We extend the existence of Berk-Nash equilibrium to compact action spaces and sigma-compact state spaces, with possibly unbounded payoff functions. A complication arises because the Berk-Nash equilibrium notion depends critically on Radon-Nikodym derivatives, which are necessarily bounded in the finite case but typically unbounded in misspecified continuous models. The proofs rely on nonstandard analysis and, relative to previous applications of nonstandard analysis in economic theory, draw on novel argumentation traceable to work of the second author on nonstandard representations of Markov processes.

econ.TH↗

arXiv2022

Continuity Postulates and Solvability Axioms in Economic Theory and in Mathematical Psychology: A Consolidation of the Theory of Individual Choice

This paper presents four theorems that connect continuity postulates in mathematical economics to solvability axioms in mathematical psychology, and ranks them under alternative supplementary assumptions. Theorem 1 connects notions of continuity (full, separate, Wold, weak Wold, Archimedean, mixture) with those of solvability (restricted, unrestricted) under the completeness and transitivity of a binary relation. Theorem 2 uses the primitive notion of a separately-continuous function to answer the question when an analogous property on a relation is fully continuous. Theorem 3 provides a portmanteau theorem on the equivalence between restricted solvability and various notions of continuity under weak monotonicity. Finally, Theorem 4 presents a variant of Theorem 3 that follows Theorem 1 in dispensing with the dimensionality requirement and in providing partial equivalences between solvability and continuity notions. These theorems are motivated for their potential use in representation theorems.

econ.TH↗

arXiv2022

On Sustainability and Survivability in the Matchbox Two-Sector Model: A Complete Characterization of Optimal Extinction

We provide a complete characterization of optimal extinction in a two-sector model of economic growth through three results, surprising in both their simplicity and intricacy. (i) When the discount factor is below a threshold identified by the well-known $δ$-normality condition for the existence of a stationary optimal stock, the economy's capital becomes extinct in the long run. (ii) This extinction may be staggered if and only if the investment-good sector is capital intensive. (iii) We uncover a sequence of thresholds of the discount factor, identified by a family of rational functions, that represent bifurcations for optimal postponements on the path to extinction. We also report various special cases of the model having to do with unsustainable technologies and equal capital intensities that showcase long-term optimal growth, all of topical interest and all neglected in the antecedent literature.

econ.TH↗

arXiv2022

The Continuity Postulate in Economic Theory: A Deconstruction and an Integration

This paper presents six theorems and ten propositions that can be read as deconstructing and integrating the continuity postulate under the rubric of pioneering work of Eilenberg, Wold, von Neumann-Morgenstern, Herstein-Milnor and Debreu. Its point of departure is the fact that the adjective continuous applied to a function or a binary relation does not acknowledge the many meanings that can be given to the concept it names, and that under a variety of technical mathematical structures, its many meanings can be whittled down to novel and unexpected equivalences that have been missed in the theory of choice. Specifically, it provides a systematic investigation of the two-way relation between restricted and full continuity of a function and a binary relation that, under convex, monotonic and differentiable structures, draws out the behavioral implications of the postulate.

econ.TH↗

arXiv2021

Fuzzy Core Equivalence in Large Economies: A Role for the Infinite-Dimensional Lyapunov Theorem

We present the equivalence between the fuzzy core and the core under minimal assumptions. Due to the exact version of the Lyapunov convexity theorem in Banach spaces, we clarify that the additional structure of commodity spaces and preferences is unnecessary whenever the measure space of agents is "saturated". As a spin-off of the above equivalence, we obtain the coincidence of the core, the fuzzy core, and the Schmeidler's restricted core under minimal assumptions. The coincidence of the fuzzy core and the restricted core has not been articulated anywhere.

econ.TH↗

arXiv2020

Binary Relations in Mathematical Economics: On the Continuity, Additivity and Monotonicity Postulates in Eilenberg, Villegas and DeGroot

This chapter examines how positivity and order play out in two important questions in mathematical economics, and in so doing, subjects the postulates of continuity, additivity and monotonicity to closer scrutiny. Two sets of results are offered: the first departs from Eilenberg's (1941) necessary and sufficient conditions on the topology under which an anti-symmetric, complete, transitive and continuous binary relation exists on a topologically connected space; and the second, from DeGroot's (1970) result concerning an additivity postulate that ensures a complete binary relation on a σ-algebra to be transitive. These results are framed in the registers of order, topology, algebra and measure-theory; and also beyond mathematics in economics: the exploitation of Villegas' notion of monotonic continuity by Arrow-Chichilnisky in the context of Savage's theorem in decision theory, and the extension of Diamond's impossibility result in social choice theory by Basu-Mitra. As such, this chapter has a synthetic and expository motivation, and can be read as a plea for inter-disciplinary conversations, connections and collaboration.

econ.TH↗

arXiv2020

The Yannelis-Prabhakar Theorem on Upper Semi-Continuous Selections in Paracompact Spaces: Extensions and Applications

In a 1983 paper, Yannelis-Prabhakar rely on Michael's selection theorem to guarantee a continuous selection in the context of the existence of maximal elements and equilibria in abstract economies. In this tribute to Nicholas Yannelis, we root this paper in Chapter II of Yannelis' 1983 Rochester Ph.D. dissertation, and identify its pioneering application of the paracompactness condition to current and ongoing work of Yannelis and his co-authors, and to mathematical economics more generally. We move beyond the literature to provide a necessary and sufficient condition for upper semi-continuous local and global selections of correspondences, and to provide application to five domains of Yannelis' interests: Berge's maximum theorem, the Gale-Nikaido-Debreu lemma, the Gale-McKenzie survival assumption, Shafer's non-transitive setting, and the Anderson-Khan-Rashid approximate existence theorem. The last resonates with Chapter VI of the Yannelis' dissertation.

econ.TH↗

arXiv2019

On an Extension of a Theorem of Eilenberg and a Characterization of Topological Connectedness

On taking a non-trivial and semi-transitive bi-relation constituted by two (hard and soft) binary relations, we report a (i) p-continuity assumption that guarantees the completeness and transitivity of its soft part, and a (ii) characterization of a connected topological space in terms of its attendant properties on the space. Our work generalizes antecedent results in applied mathematics, all following Eilenberg (1941), and now framed in the context of a parametrized-topological space. This re-framing is directly inspired by the continuity assumption in Wold (1943-44) and the mixture-space structure proposed in Herstein and Milnor (1953), and the unifying synthesis of these pioneering but neglected papers that it affords may have independent interest.

econ.TH↗

arXiv2018

Topological Connectedness and Behavioral Assumptions on Preferences: A Two-Way Relationship

This paper offers a comprehensive treatment of the question as to whether a binary relation can be consistent (transitive) without being decisive (complete), or decisive without being consistent, or simultaneously inconsistent or indecisive, in the presence of a continuity hypothesis that is, in principle, non-testable. It identifies topological connectedness of the (choice) set over which the continuous binary relation is defined as being crucial to this question. Referring to the two-way relationship as the Eilenberg-Sonnenschein (ES) research program, it presents four synthetic, and complete, characterizations of connectedness, and its natural extensions; and two consequences that only stem from it. The six theorems are novel to both the economic and the mathematical literature: they generalize pioneering results of Eilenberg (1941), Sonnenschein (1965), Schmeidler (1971) and Sen (1969), and are relevant to several applied contexts, as well as to ongoing theoretical work.

econ.TH↗

arXiv2018

Completeness and Transitivity of Preferences on Mixture Sets

In this paper, we show that the presence of the Archimedean and the mixture-continuity properties of a binary relation, both empirically non-falsifiable in principle, foreclose the possibility of consistency (transitivity) without decisiveness (completeness), or decisiveness without consistency, or in the presence of a weak consistency condition, neither. The basic result can be sharpened when specialized from the context of a generalized mixture set to that of a mixture set in the sense of Herstein-Milnor (1953). We relate the results to the antecedent literature, and view them as part of an investigation into the interplay of the structure of the choice space and the behavioral assumptions on the binary relation defined on it; the ES research program due to Eilenberg (1941) and Sonnenschein (1965), and one to which Schmeidler (1971) is an especially influential contribution.

econ.TH↗

arXiv2017

Fatou's Lemma, Galerkin Approximations and the Existence of Walrasian Equilibria in Infinite Dimensions

This essay has three objectives: (i) to report recent generalizations of Fatou's lemma to multi-functions taking values in a Banach space, and framed in terms of both Bochner and Gelfand integration; (ii) to delineate the importance of Galerkin approximations in Walrasian general equilibrium theory with a continuum of agents and commodities; and thereby (iii) to present two new results on the existence of a Walrasian equilibrium in economies where the continuum of agents is formalized as a saturated measure space.

math.FA↗

arXiv2016

Relaxed Large Economies with Infinite-Dimensional Commodity Spaces: The Existence of Walrasian Equilibria

Whereas "convexification by aggregation" is a well-understood procedure in mathematical economics, "convexification by randomization" has largely been limited to theories of statistical decision-making, optimal control and non-cooperative games. In this paper, in the context of classical Walrasian general equilibrium theory, we offer a comprehensive treatment of {\it relaxed economies} and their {\it relaxed Walrasian equilibria}: our results pertain to a setting with a finite or a continuum of agents, and a continuum of commodities modeled either as an ordered separable Banach space or as an $L^\infty$-space. As a substantive consequence, we demonstrate that the convexity hypothesis can be removed from the original large economy under the saturation hypothesis, and that existing results in the antecedent literature can be effortlessly recovered.

math.OC↗