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2016 | OriginalPaper | Chapter

A New Look at the Study of Solutions for Games in Partition Function Form

Author : Joss Sánchez-Pérez

Published in: Recent Advances in Game Theory and Applications

Publisher: Springer International Publishing

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Abstract

This chapter studies the structure of games in partition function and according to an axiomatic point of view, we provide a global description of linear symmetric solutions by means of a decomposition of the set of such games (as well as of a decomposition of the space of payoff vectors). The exhibition of relevant subspaces in such decomposition and based on the idea that every permutation of the set of players may be thought of as a linear map, allow for a new look at linear symmetric solutions.

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The precise statement will be provided in Sect. 3.

Notice that G can be identified with the set of real vectors indexed on the elements of EC, and so, \(\dim G = \left \vert EC\right \vert\).

So, θ ⋅ w is the partition function game which would result from w if we relabeled the players by permutation θ.

The formal statement will be found at the end of this section.

Formally, if Y be a subspace of a vector space X, then Y is invariant (for the action of S _n) if for every y ∈ Y and every θ ∈ S _n, we have that θ ⋅ y ∈ Y.

That is, a subspace Y is irreducible if Y itself has no invariant subspaces other than {0} and Y itself.

Here, \(\mathbf{0} = (0, 0,\ldots, 0) \in \mathbb{R}^{n}\).

This seems like the natural inner product to consider, since intuitively G can be identified with \(\mathbb{R}^{EC}\).

This type of games may be thought of as its counterpart for symmetric games in TU games.

That is, r(S, Q) = 0 for every (S, Q) ∈ EC.

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Title: A New Look at the Study of Solutions for Games in Partition Function Form
Author: Joss Sánchez-Pérez
Publisher: Springer International Publishing
Book: Recent Advances in Game Theory and Applications
Print ISBN: 978-3-319-43837-5

Electronic ISBN: 978-3-319-43838-2

Copyright Year: 2016
DOI: https://doi.org/10.1007/978-3-319-43838-2_12