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On Continuity of Separately Convex Preferences and Correspondences

This paper demonstrates that the weaker assumption of separate convexity is sufficient to establish standard equivalences among various continuity postulates for both preferences and correspondences, thereby enabling the derivation of multilinear and continuous utility representations in diverse decision-making contexts.

Original authors: Metin Uyanik, Aniruddha Ghosh, M. Ali Khan

Published 2026-07-23

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CC0 1.0

Original authors: Metin Uyanik, Aniruddha Ghosh, M. Ali Khan

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to organize a massive, chaotic library. You want to find a way to sort books so that if you like a sci-fi novel and a mystery novel, you also like a book that is a perfect mix of both. In the world of economics and decision-making, this idea is called convexity. It's the mathematical rule that says "the middle ground is always safe." If you prefer outcome A over outcome B, convexity suggests you should also prefer a 50-50 mix of A and B over just B. This rule is the backbone of most modern economic theories because it makes the math work, allowing economists to predict how people behave in markets, games, and under risk.

However, real life is messy. Sometimes, the "middle ground" isn't safe. Maybe you love spicy food and you love sweet food, but a mix of the two tastes terrible. In these cases, the strict rule of convexity breaks down. For decades, economists have struggled with this: if people don't follow the "safe middle" rule, can we still use our powerful mathematical tools to understand them? This paper enters that messy corner of the library, looking for a new way to sort the books when the old rules don't fit. It asks a simple question: Can we relax the rules just enough to handle these weird, non-convex preferences without losing the ability to make predictions?

The authors of this paper, Aniruddha Ghosh, M. Ali Khan, and Metin Uyanik, propose a clever workaround they call separate convexity. Think of it like navigating a city grid. The old, strict rule of convexity demands that you can walk in a straight line in any direction and stay on the "good" path. The new, "separate" rule is much more relaxed: it only asks that you can walk straight North-South and stay on the good path, and separately, that you can walk straight East-West and stay on the good path. You don't have to be able to walk diagonally!

The paper proves that this "grid-walking" rule is surprisingly powerful. Even though it's a weaker requirement than the old strict rule, it turns out to be strong enough to recover all the standard mathematical equivalences that economists rely on. In plain terms, the authors show that if your preferences are "grid-friendly" (separately convex), you can still use the same trusted mathematical shortcuts to figure out utility, continuity, and decision-making as you could with the strict rules. They demonstrate this through a series of mathematical proofs that link different ways of describing "smoothness" in preferences. For example, they show that if a preference is smooth when you look at it one direction at a time, and it follows the grid rule, then it is smooth in the big picture too.

The paper also tackles some famous open problems in the field. It explicitly rules out the idea that these new rules work in every possible mathematical space. The authors provide a specific counter-example (using infinite-dimensional spaces, which are like grids with infinitely many directions) to show that their "grid-walking" magic stops working when the space gets too big and complex. They prove that the "finite-dimensional" nature of our everyday world (like the $n$ -person games they study) is essential for their results to hold.

Furthermore, the paper applies these findings to real-world scenarios like multi-person games and subjective decision-making under uncertainty. They show that by using separate convexity, we can derive complex utility functions (mathematical formulas that represent happiness or value) for groups of people without needing the strict, often unrealistic, assumption that everyone's preferences are perfectly convex. They even show how to recover the "completeness" of a decision (the idea that you can always compare any two options) just by looking at how you compare options one dimension at a time.

In short, this paper doesn't just say "convexity is hard, let's give up." Instead, it says, "Convexity is hard, but if we look at it one direction at a time, we can still do the math." It provides a new, more flexible toolkit for economists and mathematicians to model human behavior when the world doesn't follow the neat, straight lines of traditional theory. The results are presented as rigorous mathematical proofs, offering a solid foundation for understanding preferences that are complex, non-convex, and deeply human.

Technical Summary: On Continuity of Separately Convex Preferences and Correspondences

Problem Statement
The paper addresses the foundational role of convexity in modern economic theory, specifically within the contexts of resource allocation, competitive markets, and decision theory. While the standard convexity postulate is essential for many equilibrium and representation results, it is often stronger than necessary for specific applications, such as those involving risk, ambiguity, or multi-attribute decision-making. The authors investigate "separate convexity," a weaker algebraic notion lying between full convexity and non-convexity. The central problem is to determine whether this weaker assumption is sufficient to recover the standard equivalences among various continuity postulates (e.g., mixture continuity, section continuity, graph continuity, and Archimedean properties) that are typically derived under full convexity. Furthermore, the paper seeks to extend these results to non-ordered preferences formulated as correspondences, addressing open problems regarding the relationship between section continuity and the open graph property in general domains.

Methodology
The authors employ a rigorous axiomatic approach, utilizing tools from mathematical economics, topology, and convex analysis. The methodology proceeds in three stages:

Definitions and Framework: The paper defines separate convexity for binary relations and correspondences. A relation is separately convex if its upper and lower sections are convex when restricted to lines parallel to coordinate axes (or coordinate sections in mixture sets).
Equivalence Theorems: The authors establish equivalence theorems linking different continuity postulates under the assumption of separate convexity. These proofs rely on "coordinate-release" arguments, where properties are upgraded from separate to joint continuity by iteratively freeing coordinates, often requiring specific geometric properties of the domain (e.g., Property A for product sets or Property B for polyhedra).
Counterexamples and Boundary Analysis: To delineate the limits of their results, the authors construct specific counterexamples (e.g., Example 4, Example 10) demonstrating that finite-dimensional structures are essential for certain equivalences and that results do not automatically extend to infinite-dimensional spaces or arbitrary convex sets without additional assumptions.

Key Contributions and Results

Equivalence of Continuity Postulates for Ordered Preferences:
- Mixture Sets: Theorem 1 establishes that for complete and transitive preferences on product mixture sets, separate convexity is sufficient to equate mixture continuity with separate mixture continuity and Archimedean-type postulates. Specifically, upper separate convexity pairs with lower mixture continuity, and vice versa.
- Euclidean Spaces: Theorems 2 and 3 extend these results to convex subsets of $\mathbb{R}^n$ . Theorem 2 shows that under separate convexity, separate section-continuity upgrades to ordinary section-continuity. Theorem 3 links graph continuity to restricted solvability under separate convexity and order denseness.
- Dimensionality Constraints: The paper highlights that these equivalences are intrinsically finite-dimensional. Example 10 demonstrates that on products of infinite-dimensional factors (specifically Hilbert spaces with weak topology), separate convexity and separate continuity do not imply joint continuity, even for bilinear forms.
Utility Representations in $n$ -Person Games:
- Multilinear Utility: Proposition 1 provides a new axiomatization for multilinear utility in strategic form games. It replaces the standard separate Archimedean $^*$ assumption with separate independence and separate intermediate value property (IVP), recovering the multilinear representation.
- Separately Quasiconcave Utility: Proposition 2 axiomatizes continuous, separately quasiconcave utility representations. It shows that upper continuity can be weakened to upper separate continuity if the lower sections satisfy separate convexity in $n-1$ coordinates.
- Subjective Expected Utility (SEU): Proposition 3 and Corollary 1 demonstrate that in a scalar Anscombe–Aumann environment, completeness and full mixture-continuity are "hidden" under separate continuity and weak independence assumptions. This allows for the derivation of SEU representations without positing completeness as a primitive.
Continuity of Correspondences (Non-Ordered Preferences):
- Open Graph Property: Theorems 4 and 5 characterize the open graph property for correspondences under separate convexity. Theorem 4 shows that for correspondences with separately convex upper sections on domains satisfying Property A (product sets or open sets), the open graph property is equivalent to having open lower sections and separately open upper sections.
- Generalization of Classical Results: These results generalize and weaken the assumptions of Schmeidler (1969), Shafer (1974), and Bergstrom, Parks, and Rader (1976). Specifically, they replace strong convexity or monotonicity assumptions with separate convexity and allow for more general domains.
- Resolution of Open Problems: Example 4 resolves a question left open by Bergstrom, Parks, and Rader (1976) by showing that their result regarding the equivalence of open sections and open graphs cannot be extended to arbitrary convex sets in $\mathbb{R}^n$ without additional structural assumptions (like Property A).

Significance and Claims
The paper claims three primary contributions:

Theoretical Consolidation: It demonstrates that the widely used continuity postulates in economics and mathematical psychology are equivalent under the weaker assumption of separate convexity, thereby consolidating existing results and showing that full convexity is often not required for these equivalences.
New Axiomatizations: It provides new axiomatizations for utility representations in $n$ -person games (both cardinal and ordinal) and for subjective expected utility, utilizing separate continuity to recover joint properties like completeness.
Extension to Correspondences: It extends the analysis of continuity to non-ordered preferences (correspondences), characterizing the open graph property under weak convexity and section-continuity assumptions, thereby generalizing seminal results by Schmeidler, Shafer, and Bergstrom-Parks-Rader.

The authors emphasize that their work highlights the role of weaker axiomatic assumptions in capturing the "essentials" of economic theory (referencing Koopmans, 1957) and clarifies the boundaries of these results, particularly the necessity of finite-dimensional structures for upgrading separate continuity to joint continuity. The paper does not claim to solve equilibrium existence problems directly but focuses on the axiomatization literature, showing how weaker assumptions can yield standard representation theorems.

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