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Unlike parametric methods, nonparametric statistics are ideal for nominal and ordinal data, requiring fewer assumptions about the population's nature or distribution. This makes nonparametric methods easier to apply and interpret, as they do not depend on parameters like mean or standard deviation. One common approach in nonparametric analysis is to sort data according to a specific criterion. For instance, we might arrange weather data from hottest to coldest days in a month or rank cities...
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The graph of the equation where y equals x squared forms a curve known as a parabola. This curve acts as a boundary in the coordinate plane, dividing it into distinct regions based on the relative position of points.When the equality sign in the equation is replaced with an inequality—such as greater than, less than, greater than or equal to, or less than or equal to—the graphical representation changes from a single curve into a broader shaded area that signifies the set of all...
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Lexicographic solutions for coalitional rankings.

Encarnación Algaba1, Stefano Moretti2, Eric Rémila3

  • 1Matemática Aplicada II and Instituto de Matemáticas de la Universidad de Sevilla (IMUS), Escuela Superior de Ingenieros, Camino de los Descubrimientos, s/n , 41092 Sevilla, Spain.

Social Choice and Welfare
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Summary

This study introduces new social ranking methods that consider group sizes, revising the lexicographic excellence solution for ranking agents or items within coalitions. These methods are applied to analyze football player performance in the UEFA Champions League.

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Area of Science:

  • Social Choice Theory
  • Game Theory
  • Decision Analysis

Background:

  • Ranking agents or items from group rankings is crucial in various real-world scenarios.
  • Existing methods like the lexicographic excellence solution have limitations in accounting for group structures.

Purpose of the Study:

  • To revise the lexicographic excellence solution for social rankings.
  • To introduce two novel social ranking solutions that incorporate group sizes.
  • To axiomatize and compare these three ranking methods.

Main Methods:

  • Axiomatic approach to define and compare social ranking solutions.
  • Revision of the lexicographic excellence solution.
  • Development of two new group-size-aware ranking algorithms.

Main Results:

  • Three distinct social ranking solutions are presented and axiomatized.
  • The novel solutions offer different perspectives on ranking compared to the lexicographic excellence solution.
  • A comparative analysis highlights the differences and strengths of each method.

Conclusions:

  • The proposed social ranking methods provide valuable tools for ranking individuals within groups, considering group size.
  • These methods offer a more nuanced approach to social ranking problems.
  • The application to football player performance demonstrates practical utility in real-world scenarios.