A Link Diagram Visualizing Relations between Two Ordered Sets

TL;DR

This paper introduces a link diagram for visualizing relations between two ordered sets, aiding in understanding rankings and Pareto optimal solutions in strategic decision problems.

Contribution

The paper presents a novel link diagram method for visualizing relations between two ordered sets, including rankings and game solutions, emphasizing splittability of loops as a key feature.

Findings

The diagram visualizes whether rankings satisfy Condorcet's principle.

It helps identify Pareto optimal solutions in the prisoners' dilemma.

Loop splittability indicates key relational properties.

Abstract

This article provides a link diagram to visualize relations between two ordered sets representing precedences on decision-making options or solutions to strategic form games. The diagram consists of floating loops whose any two loops cross just twice each other. As problems formulated by relations between two ordered sets, I give two examples: visualizing rankings from pairwise comparisons on the diagram and finding Pareto optimal solutions to a game of prisoners' dilemma. At visualizing rankings, we can see whether a ranking satisfies Condorcet's principle or not by checking whether the top loop is splittable or not. And at finding solutions to the game, when a solution of the game of prisoners' dilemma is Pareto optimal, the loop corresponding to the solution has no splittable loop above it. Throughout the article, I point out that checking the splittability of loops is an essence. I also mention that the diagram can visualize natural transformations between two functors on free construction categories.