15 research outputs found
An Iterative Procedure for Evaluating Digraph Competitions
A competition which is based on the results of (partial) pairwise comparisons can be modelled by means of a directed graph.Given initial weights on the nodes in such digraph competitions, we view the measurement of the importance (i.e., the cardinal ranking) of the nodes as an allocation problem where we redistribute the initial weights on the basis of insights from cooperative game theory.After describing the resulting procedure of redistributing the initial weights, we describe an iterative process is described which repeats this procedure: at each step the allocation obtained in the previous step determines the new input weights.Existence and uniqueness of the limit is established for arbitrary digraphs.Applications to the evaluation of e.g. sport competitions and paired comparison experiments are discussed
The Consensus Value: A New Solution Concept for Cooperative Games
By generalizing the standard solution for 2-person games into n-person cases, this paper develops a new solution concept for cooperative games: the consensus value.We characterize the consensus value as the unique function that satisfies efficiency, symmetry, the quasi dummy property and additivity.By means of the transfer property, a second characterization is provided.By defining the stand-alone reduced game, a recursive formula for the value is established.We also show that this value is the average of the Shapley value and the equal surplus solution.Furthermore, we discuss a possible generalization.cooperative games;game theory
A network-based dynamical ranking system for competitive sports
From the viewpoint of networks, a ranking system for players or teams in
sports is equivalent to a centrality measure for sports networks, whereby a
directed link represents the result of a single game. Previously proposed
network-based ranking systems are derived from static networks, i.e.,
aggregation of the results of games over time. However, the score of a player
(or team) fluctuates over time. Defeating a renowned player in the peak
performance is intuitively more rewarding than defeating the same player in
other periods. To account for this factor, we propose a dynamic variant of such
a network-based ranking system and apply it to professional men's tennis data.
We derive a set of linear online update equations for the score of each player.
The proposed ranking system predicts the outcome of the future games with a
higher accuracy than the static counterparts.Comment: 6 figure
A graph interpretation of the least squares ranking method
The paper aims at analyzing the least squares ranking method for generalized
tournaments with possible missing and multiple paired comparisons. The
bilateral relationships may reflect the outcomes of a sport competition,
product comparisons, or evaluation of political candidates and policies. It is
shown that the rating vector can be obtained as a limit point of an iterative
process based on the scores in almost all cases. The calculation is interpreted
on an undirected graph with loops attached to some nodes, revealing that the
procedure takes into account not only the given object's results but also the
strength of objects compared with it. We explore the connection between this
method and another procedure defined for ranking the nodes in a digraph, the
positional power measure. The decomposition of the least squares solution
offers a number of ways to modify the method
On the additivity of preference aggregation methods
The paper reviews some axioms of additivity concerning ranking methods used
for generalized tournaments with possible missing values and multiple
comparisons. It is shown that one of the most natural properties, called
consistency, has strong links to independence of irrelevant comparisons, an
axiom judged unfavourable when players have different opponents. Therefore some
directions of weakening consistency are suggested, and several ranking methods,
the score, generalized row sum and least squares as well as fair bets and its
two variants (one of them entirely new) are analysed whether they satisfy the
properties discussed. It turns out that least squares and generalized row sum
with an appropriate parameter choice preserve the relative ranking of two
objects if the ranking problems added have the same comparison structure.Comment: 24 pages, 9 figure
Additive and multiplicative properties of scoring methods for preference aggregation
The paper reviews some additive and multiplicative properties of ranking procedures
used for generalized tournaments with missing values and multiple comparisons. The methods analysed are the score, generalised row sum and least squares as well as fair bets and its variants. It is argued that generalised row sum should be applied not with a fixed parameter, but a variable one proportional to the number of known comparisons. It is shown that a natural additive property has strong links to independence of irrelevant matches, an axiom judged unfavourable when players have different opponents