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Polynomial representation of TU-games

Ulrich Faigle, Michel Grabisch

Abstract

We propose in this paper a polynomial representation of TU-games, fuzzy measures, capacities, and more generally set functions. Our representation needs a countably infinite set of players and the natural ordering of finite sets of $\mathbb{N}$, defined recursively. For a given basis of the vector space of games, we associate to each game $v$ a formal polynomial of degree at most $2^n-1$ whose coefficients are the coordinates of $v$ in the given basis. By the fundamental theorem of algebra, $v$ can be represented by the roots of the polynomial. We present some new families of games stemming from this polynomial context, like the irreducible games, the multiplicative games and the cyclotomic games.

Polynomial representation of TU-games

Abstract

We propose in this paper a polynomial representation of TU-games, fuzzy measures, capacities, and more generally set functions. Our representation needs a countably infinite set of players and the natural ordering of finite sets of , defined recursively. For a given basis of the vector space of games, we associate to each game a formal polynomial of degree at most whose coefficients are the coordinates of in the given basis. By the fundamental theorem of algebra, can be represented by the roots of the polynomial. We present some new families of games stemming from this polynomial context, like the irreducible games, the multiplicative games and the cyclotomic games.
Paper Structure (16 sections, 8 theorems, 44 equations, 1 table)

This paper contains 16 sections, 8 theorems, 44 equations, 1 table.

Table of Contents

  1. Introduction
  2. The framework
  3. Games and their bases
  4. Cardinality polynomials
  5. The natural ordering of finite subsets of ${\mathbb N}$
  6. Equivalence classes of games
  7. Polynomial and algebraic representations of games
  8. Factorization and irreducible games
  9. Factorization of polynomials
  10. Products of games
  11. Factorization of games
  12. Multiplicative games
  13. Cyclotomic games
  14. Cyclotomic polynomials
  15. Cyclotomic games
  16. ...and 1 more sections

Key Result

Lemma 1

Given a transform $\Psi$, each game $v$ in $\mathscr{G}$ has a (unique) algebraic representation $\mathfrak{a}(v,\Psi)$. The algebraic representation corresponds to a unique element of $\mathscr{G}/_{\equiv^\Psi}$, i.e., all games $v'\in[v]^\Psi$ have the same algebraic representation as $v$, and on

Theorems & Definitions (15)

  • Example 1
  • Lemma 1
  • Remark 1
  • Example 2: Example \ref{['ex:1']} ct'd
  • Theorem 1
  • Theorem 2: uniqueness of factorization
  • Theorem 3: Eisenstein irreducibility criterion
  • Example 3
  • Theorem 4
  • Example 4: Ex. \ref{['ex:1']} cont'd
  • ...and 5 more