[Paper Review] A polarity theory for sets of desirable gambles

Alessio Benavoli, Alessandro Facchini+2 more|arXiv (Cornell University)|May 26, 2017

Decision-Making and Behavioral Economics11 references10 citations

TL;DR

This paper establishes a duality between coherent sets of desirable gambles and convex sets of lexicographic probabilities using a novel polarity theory for convex cones. By leveraging orthogonal matrices and lexicographic ordering, it proves that these two models are isomorphic, enabling bidirectional transformation while preserving key operations like conditioning.

ABSTRACT

Coherent sets of almost desirable gambles and credal sets are known to be equivalent models. That is, there exists a bijection between the two collections of sets preserving the usual operations, e.g. conditioning. Such a correspondence is based on the polarity theory for closed convex cones. Learning from this simple observation, in this paper we introduce a new (lexicographic) polarity theory for general convex cones and then we apply it in order to establish an analogous correspondence between coherent sets of desirable gambles and convex sets of lexicographic probabilities.

Motivation & Objective

• To establish a formal isomorphism between coherent sets of desirable gambles and convex sets of lexicographic probabilities.
• To extend polarity theory from closed convex cones to general convex cones, enabling duality in uncertainty modeling.
• To preserve fundamental operations such as conditioning under the duality transformation.
• To provide a geometric and algebraic framework for transferring constructions between gambles and lexicographic probabilities.
• To enable deeper analysis of lexicographic probabilities through duality with the more intuitive gamble-based framework.

Proposed method

• Develop a new lexicographic polarity theory for general convex cones, generalizing classical polarity for closed convex cones.
• Define a duality transformation using orthogonal matrices that maps coherent sets of desirable gambles to convex sets of lexicographic probabilities.
• Introduce a conditioning operation on stochastic matrices via a reduction rule (R) that preserves matrix structure and stochasticity.
• Construct the transformation $\mathbf{G}(\mathcal{K})$ from a coherent set of desirable gambles $\mathcal{K}$ to a set of lexicographic probabilities using Gram-Schmidt orthogonalization.
• Prove that the duality preserves conditioning: $\mathbf{G}(\mathcal{K} \rfloor_{\Pi}) = (\mathbf{G}(\mathcal{K})) \rfloor_{\Pi}$ for any subset $\Pi \subset \Omega$.
• Use lexicographic ordering ($<_{L}$) to define positivity in the dual space, ensuring compatibility with lexicographic probability models.

Experimental results

Research questions

• RQ1 Can a duality be established between coherent sets of desirable gambles and convex sets of lexicographic probabilities?
• RQ2 How can polarity theory be extended to general convex cones to support such a duality?
• RQ3 Does the duality preserve fundamental operations like conditioning on events?
• RQ4 What is the role of orthogonal matrices in constructing the duality transformation between the two models?
• RQ5 Can structural properties such as independence be better understood through this duality?

Key findings

• A coherent set of desirable gambles $\mathcal{K}$ is isomorphic to a convex set of lexicographic probabilities via a duality transformation $\mathbf{G}$.
• The duality transformation preserves conditioning: $\mathbf{G}(\mathcal{K} \rfloor_{\Pi}) = (\mathbf{G}(\mathcal{K})) \rfloor_{\Pi}$, ensuring operational consistency.
• The transformation is constructed using orthogonal matrices and Gram-Schmidt orthogonalization, ensuring that lexicographic positivity is preserved.
• The set of lexicographic probabilities $\mathbf{G}(\mathcal{K})$ is convex and closed under the operations of the original gamble model.
• The duality allows for transferring constructions and insights between the gamble-based and probability-based frameworks.
• The theory extends to conditional models, with conditioning defined via a reduction rule (R) that maintains matrix structure and stochasticity.