A Link Diagram Visualizing Relations between Two Ordered Sets
AA Link Diagram Visualizing Relations betweenTwo Ordered Sets
T. Mizuno * February 8, 2021
Abstract
This article provides a link diagram to visualize relations between two orderedsets representing precedences on decision-making options or solutions to strategicform games. The diagram consists of floating loops whose any two loops crossjust twice each other. As problems formulated by relations between two orderedsets, I give two examples: visualizing rankings from pairwise comparisons on thediagram and finding Pareto optimal solutions to a game of prisoners’ dilemma. Atvisualizing rankings, we can see whether a ranking satisfies Condorcet’s principleor not by checking whether the top loop is splittable or not. And at finding solutionsto 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. Throughoutthe article, I point out that checking the splittability of loops is an essence. Ialso mention that the diagram can visualize natural transformations between twofunctors on free construction categories.
Keywords: link diagram, ranking, pairwise comparison, game theory, prison-ers’ dilemma, category theory
Order is the essential structure of mathematical models. We often draw directedgraphs to visualize ordered sets. Each node of the graphs represents an option ofdecision-making problems or a choice of strategic games, and each directed arcbetween a pair of nodes represents which node precedes another node.However, drawing nodes and arcs is not suitable to represent local relationsand global relations between two ordered sets. Local relations are orders amongelements in each set. Global relations are correspondences between elements of aset and elements of another set. When we use directed graphs, we will use arcsto represent the two relations; it makes us confused. The two relations need twokinds of notations, respectively.This article provides a link diagram that can represent local relations and globalrelations between two ordered sets. By using the diagram, we can see that rankings * Meijo University, eMail: [email protected] a r X i v : . [ c s . G T ] F e b igure 1: A loop of the link diagram. from pairwise comparisons whether the rankings satisfy Condorcet’s principle ornot. Then I describe how to represent games of the prisoners’ dilemma on thelink diagram. We can find Pareto optimal solutions to the games with the diagram.They are two types of abstract data visualization that represent relations betweentwo ordered sets. The former represents how an ordered set changes to another,and the latter represents how two people see a set. The diagram also can representa natural transformation among functors of categories. Notice
In this article, I write x (cid:31) y when an element x precedes another element y , or x is preferred to y , or x beats y .Generally, elements of ordered sets may have the following properties:• (antisymmetry) for any elements x and y , x (cid:31) y and y (cid:31) x iff x = y .• (comparability) for every pair of elements x and y , x (cid:31) y or y (cid:31) x .• (transitivity) for any elements x , y , and z , if x (cid:31) y and y (cid:31) z then x (cid:31) z .Ordered sets treated in this article have antisymmetry and comparability but maynot have transitivity. A link diagram provided in this article consists of floating loops. Each loop, suchas Figure 1, represents an element of a set.Figure 2 is an example of link diagram. Loops are arranged as every pairof loops intersect just two times on the left and the right sides of the diagram.Each block on each side consists of crosses of loops represents an ordered set; thediagram represents two ordered sets. How loops cross in each block representslocal relations, which are precedence relations between elements in each orderset. When an element i precedes j in a set, the loop i is above the loop j at theirintersection in the diagram (Figure 3). ≻ 1 1 ≻ 33 ≻ 2 Figure 2: A link diagram. The left block represents a cycle of preferences, and the rightrepresents a total order. ! ≻
Figure 3: Which loop is above means which element precedes.3 n the diagram in Figure 2, there are three elements 1, 2, and 3. Crosses on theleft block represent a cycles of preference; 2 (cid:31)
1, 1 (cid:31)
3, and 3 (cid:31)
2. While crosseson the right block represents a total order; 2 (cid:31) (cid:31) We often face to decide priorities among options in social choices, in private shop-pings, and so on. When we are hard to give ranks whole options at once, we willtry pairwise comparisons. For every pair of options, we compare them and declarewhich option is preferable to another, and then a ranking is aggregated from thecomparisons.There are many ways to aggregate the comparisons. In this section, I deal withthe geometric mean method that aggregate pairwise comparisons. The method isused in AHP (the Analytic hierarchy process) (Saaty, 1980) which is a decision-making process. AHP consists of three phases: construction of hierarchy structure,weighting options in each criterion, and synthesis by the sum of products multi-plied the weights by criteria’s importance. In the second phase, weighting options,pairwise comparisons are used in AHP. For each criterion, decision-makers regardthe criterion and show how an option is preferable to another one for every pair ofoptions. The preference is represented in a ratio greater than zero.For example, in the case of three options { , , } , in a criterion, ratios areshown by decision-makers and arranged into the following pairwise comparisonmatrix. A = ( a i j ) =
12 19 . (1)The element a = i is preferred to j , i (cid:31) j , then a i j >
1, otherwise 0 < a i j ≤ i as the geometricmean of elements in i -th row of the matrix, and the weights in the example are ( w , w , w ) = ( √ , (cid:112) / , (cid:112) / ) , where w i is the weight of i . It means 2 (cid:31) (cid:31)
3, and the winner is the option 2. Now, let us look again at the pairwise comparisons. The option 1 precedes allof the others; 1 (cid:31) a = > ) and 1 (cid:31) a = > i wins j in the result of aggregation, then we expect that i precedes j in their pairwise comparison. The example shows that the geometric mean methodmay not satisfy the requirement.In the 18th century, Condorcet mentioned the requirement as now known asCondorcet’s principle (Condorcet, 1785). He claimed that an option that beats allof the others in pairwise comparisons must be the winner in the result of aggrega-tion when such option exists. The principal eigenvector method is also widely used to calculate weights from pairwise comparisonsin AHP. In three options cases, its result is equivalent to the geometric mean method by ignoring positiveconstant multiple. (cid:31) (cid:31) I introduce naturality that is a modified condition of Condorcet’s principle.Naturality is the condition that an option i beating j in aggregation result mustprecede j directly or indirectly in their pairwise comparison.An option beating all others has no option preceding it either directly or indi-rectly. So, if naturality is satisfied, Condorcet’s principle is satisfied. And more, anoption beaten by all of the others in pairwise comparisons is the lowest in rankingsatisfying naturality when such option exists.We can easily see whether a ranking satisfies the naturality by visualizing theranking on the link diagram. Let us put the ordered set representing pairwise com-parisons on the left block and put the ordered set of the aggregation result on theright. The ranking of the above example aggregated by the geometric mean methodis in Figure 4. While a ranking satisfying naturality for the example is in Figure 5.If an option beats all other options in pairwise comparisons, its loop is the topon the link diagram’s left block. It is clear that if such an option exists and theranking satisfying naturality, its loop is splittable from all other loops. Similarly, ifan option beaten from all other options exists and the ranking satisfying naturally,its loop is splittable from all other loops. So, we can check whether a rankingsatisfies naturality or not is whether loops are splittable or not in the link diagram.Then, you may have a question: Are there such a ranking way always? Theanswer is yes. An ordered set treated in this article is equivalent to a tournamentgraph. A tournament graph is a kind of directed graph that must have at leastone Hamilton path (Wilson, 2012). The path is a directed path that visits all nodesexactly once. That is, extracting the path is the way of ranking satisfying naturality. The game of prisoners’ dilemma is a model of interaction of multiple playerswhose choices affect each other’s outcomes.Let us consider a criminal trial in that two prisoners face a choice to confess (cid:31) (cid:31)
3. The top loop and the bottom loop are splittable.Table 1: Outcoms of palyer AA’s imprisonment B confesses B keeps silentA confesses 4 years 1 yearA keeps silent 5 years 2 years their crime or keep silent. If they both keep silent, their both imprisonments willbe short term, two years, because of insufficient evidence of the crime. If onlyone of them confesses, his/her imprisonment will be reduced to one year, and theconfession will be a witness against the other; the other’s imprisonment will befive years. If they both confess, their imprisonment will be four years less than fiveyears because of cooperating with the authorities.We can model the criminal trial as a strategic game. Two prisoners are player Aand player B, and they choose simultaneously one from their strategies: to confessor keep silent. Outcomes of players A and B by their choices are in Table 1 andTable 2, respectively.A pair of their choices are referred to as a solution to the game. By usingnotations C and S for confession and for keeping silent, respectively, there are foursolutions CC, CS, SC, and SS. For example, CS means player A confesses, andplayer B keeps silent. Players, of course, prefer shorter imprisonment to longer,so player A’s precedence of the solutions is CS (cid:31) SS (cid:31) CC (cid:31) SC, while player B’s isSC (cid:31) SS (cid:31) CC (cid:31) CS.We can represent the game on a link diagram by arranging player A’s prece-
Table 2: Outcoms of player BB’s imprisonment B confesses B keeps silentA confesses 4 years 5 yearsA keeps silent 1 year 2 years6igure 6: A link diagram that represents the game of prisoners’ dilemma of Table 1and Table 2.Figure 7: The loop CC and SS are splittable from each other. dence in the diagram’s left and player B’s in its right (Figure 6).In such strategic games, which solution must players choose? There are someconcepts to decide on the choices. In this article, I focus on Pareto optimality.If one player takes a strategy, and no strategy shrinks his/her imprisonmentwithout extending other’s imprisonment, and another player also takes such strat-egy, then the solution that consists of the strategies is Pareto optimal. In otherwords, if they can shrink both imprisonments by changing both strategies from asolution, the solution is not Pareto optimal.In the example, the solution CC is not Pareto optimal. Both players can shrinktheir imprisonments by changing their strategies from C to S. Existing the loop SSabove CC represents it, and both loops are splittable from each other (Figure 7).In the link diagram, a Pareto optimal solution has no splittable loop above it;CS, SC, and SS are Pareto optimal. (cid:2)(cid:1) (cid:3) (cid:3) → → → → → Figure 8: Examples of arrows and a composited arrow in the link diagram.
We can build a free construction category from an ordered set (Leinster, 2014).Category is an abstract mathematical structure that consists of objects and arrows.A category is built from an ordered set as follows. Each object of the categorycorresponds to each element of the ordered set. If an element i precedes j , i (cid:31) j , inthe set, then an arrow i → j exists in the category. Composited arrows are also inthe category. An arrow i → j and an arrow k → l can be composited as i → j → l when j = k . Category made as above is often referred to as the free constructioncategory.Under the construction, the existence of an arrow from i to j means that theelement i precedes j directly or indirectly. If the element i precedes j directly, then i → j exists. If i precedes j indirectly, then there is a composited arrow from i to j . If there is the composited arrow is i → p → q → j , then i may be defeated by j directly, but i beats j indirectly; i (cid:31) p and p (cid:31) q and q (cid:31) j .In the link diagram, an arrow i → j of the category is represented as jump fromthe upper loop i to the lower loop j at crossing of the two loops. In Figure 8, thearrow 1 → → → L (local), the latter category as G (global), and a correspon-dence, referred to as functor, from G to L as F . Categories in this article, objectsare same in both category; F ( i ) = i , where i is an object of G . In particularly, thefunctor is called covariant functor when an arrow i → j is corresponded to a arrowfrom i to j ; F ( i → j ) = i → ··· → j . So, ranking satisfies naturality is building acategory G and a covariant functor F from G to L .And also, naturality is represented as a commutative diagram of a natural trans-formation η from the identity functor id , which changes nothing, to the covariant unctor F . F ( i ) = i η ←−−−−− i = id ( i ) F ( i → j )= i →···→ j (cid:121) (cid:121) i → j = id ( i → j ) F ( j ) = j η ←−−−−− j = id ( j ) (2)It means that the link diagram represents a natural transformation. In this article, I provided a link diagram to visualize a relation between two orderedsets.I demonstrated that the link diagram could visualize rankings from pairwisecomparisons. We can grasp whether a ranking satisfies naturality, a generalizationof Condorcet’s principle, or not by checking whether the top loop of the diagramis splittable or not.I also visualized a game of prisoners’ dilemma with the diagram. I pointed outthat a solution to the game is Pareto optimal when the solution’s loop does not havesplittable loops above it. Of course, the link diagram can represent other strategicform games such as the battle of sexes and matching pennies.After them, I mentioned that the ordered sets are considered categories, andthe link diagram also represents categories and, in particular cases, their naturaltransformations.In the field of knot theory, the diagram in this article is a link whose arbitrarytwo loops have just two intersections. Splitability of the link can represents twoconcepts: naturality and Pareto optimality. I am searching other usefull conceptsvisualized on the link.
References
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Basic Category Theory . Cambridge University Press.Saaty, T. L. (1980).
The Analytic Hierarchy Process . McGraw Hill, New York.Wilson, R. (2012).
Introduction to Graph Theory . Pearson.. Pearson.