Symmetric Splendor: Unraveling Universally Closest Refinements and Fisher Market Equilibrium through Density-Friendly Decomposition

Abstract

We present a comprehensive framework that unifies several research areas within the context of vertex-weighted bipartite graphs, providing deeper insights and improved solutions. The fundamental solution concept for each problem involves refinement, where vertex weights on one side are distributed among incident edges. The primary objective is to identify a refinement pair with specific optimality conditions that can be verified locally. This framework connects existing and new problems that are traditionally studied in different contexts. We explore three main problems: (1) density-friendly hypergraph decomposition, (2) universally closest distribution refinements problem, and (3) symmetric Fisher Market equilibrium. Our framework presents a symmetric view of density-friendly hypergraph decomposition, wherein hyperedges and nodes play symmetric roles. This symmetric decomposition serves as a tool for deriving precise characterizations of optimal solutions for other problems and enables the application of algorithms from one problem to another.

Figures (5)

• Figure 1: Counterexample
• Figure 2: Consider a sample space $\Omega$ with 6 elements indicated with different rainbow colors. Two distributions on $\Omega$ are given by the arrays: $P = [0, 0.1, 0.14, 0.11, 0.41, 0.24]$ and $Q = [0.15, 0.3, 0.2, 0.1, 0.25, 0]$, where the elements $\omega \in \Omega$ are sorted in increasing order of the ratio $\frac{Q(\omega)}{P(\omega)}$. The left figure shows the power curve $g = \mathsf{Pow}(P||Q)$, and the right figure shows $\widehat{g} = \mathsf{Pow}(Q \| P)$ which is the reflection of $g$ about the line $y = 1 - x$.
• Figure 3: Using the same example in Figure \ref{['fig:power-func']}, consider a line segment with slope $\gamma = 1.2$ touching the curve $g$ at $x$. In this case, the red, orange, and yellow elements $\omega$ to the left of point $x$ satisfy $\frac{Q(\omega)}{P(\omega)} > 1.2$.
• Figure 4: We use the same example in Figure \ref{['fig:power-func']} with the power curves $g$ and $\widehat{g} = \mathfrak {R}(g)$, with stretching factor $\gamma = 1.25$. In each subfigure, the location of the black arrow denotes the orientation, where an arrow next to the point $(\iota, \iota)$ indicates orientation $\iota \in \{0,1\}$. The direction of the arrow indicates the direction of the stretching. The solid curve indicates the original power function, and the dotted curve indicates the stretched power function. Observe that (a) and (c) are reflections of each other, and (b) and (d) are reflections of each other.
• Figure 5: We consider the same $g = \mathsf{Pow}(P||Q)$ from Figure \ref{['fig:power-func']}. The dotted curve in left figure shows $\mathfrak {S}^{(1, {\mathsf {tan}})}_\gamma(g)$. Taking the minimum with $\mathfrak {S}^{(0, {\mathsf {tan}})}$ from Figure \ref{['fig:0tan']}, the dotted curve on the right figure shows shows $\mathfrak {S}_\gamma(g)$.

Theorems & Definitions (61)

• Definition 1.1: Closest Distribution Refinement Problem
• Theorem 1.2: Universally Closest Refinement Pair
• Theorem 1.4: Universal Maximum Refinement Matching
• Definition 2.1: Input Instance
• Remark 2.2
• Definition 2.3: Allocation Refinement and Payload
• Definition 2.4: Proportional Response
• Definition 2.5: Local Maximin Condition
• Remark 2.6
• Definition 2.7: Notions of Error