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2016 | OriginalPaper | Buchkapitel

The Selten–Szidarovszky Technique: The Transformation Part

verfasst von : Pierre von Mouche

Erschienen in: Recent Advances in Game Theory and Applications

Verlag: Springer International Publishing

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Abstract

A technique due to Selten and Szidarovszky for the analysis of Nash equilibria of games with an aggregative structure is reconsidered. Among other things it is shown that the transformation part of this technique can be extended to abstract games with co-strategy mappings and allows for a purely algebraic setting.

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Vorheriges Kapitel A Dynamic Model of a Decision Making Body Where the Power of Veto Can Be Invoked

Nächstes Kapitel Generalized Nucleoli and Generalized Bargaining Sets for Games with Restricted Cooperation

However, it may be good to mention here also [9].

That is, that there exists at most one equilibrium.

Some recent contributions concerning this technique are [1, 3, 7, 8, 16, 17].

Also see [3] and [18].

It should be said that in most articles dealing with the Selten–Szidarovszky technique there is no clear distinction between the two parts and that each article uses the technique in its own way. Also there is no standard terminology for the objects in this technique.

The notion of “abstract game” belongs to Vives [15].

I like to call this proof a “Proof from the Book” when these 12 lines were easier to understand.

Also see [16] where sufficient conditions are provided for an aggregative game to have a unique Nash equilibrium.

The way of obtaining the results in [7] supports my observation on the algebraic nature of the transformation part.

However, see [17].

For example, in the case of homogeneous Cournot oligopolies where the price function has a market satiation point v, under weak conditions one can take Y = [0, v] (see, for instance, [16]).

I use the following convention for a correspondence C: A ⊸ B. If a ∈ A and C(a) = { b}, then I also write C(a) = b. And when C is single-valued, I also consider it as a mapping C: A → B.

For example, the following result holds in the case where X _i are proper intervals of \(\mathbb{R}\) and φ _i = γ: if R _i is interior (that is, R _i(z) ⊆ X _i ^∘ for each \(\mathbf{z} \in \mathbf{X}_{\hat{\imath}}\)), then B _i(y _i) = ∅ for all y _i ∈ Y _i ∖ Y _i ^∘.

Indeed: Suppose B _i(y _i) ≠ ∅ and x _i ∈ B _i(y _i). Let \(\mathbf{z} \in \mathbf{X}_{\hat{\imath}}\) such that x _i = R _i(z) and y _i = x _i + ∑ _l z _l. As x _i ∈ X _i ^∘, it follows that y _i ∈ Y _i ^∘.

G ⁿ has in a natural way the structure of a left G module: for A ⊆ G and g = (g ₁, …, g _n) ∈ G ⁿ the set A g is defined as {(ag ₁, …, ag _n) | a ∈ A}.

effdom ( B ) denotes the effective domain of B, i.e., the set of points in the domain of the correspondence B at which B is not the empty set.

Indeed: as G is an integral domain and α _i ≠ 0, the set α _i σ _i(B _i(y _i)) has the same number of elements as the set σ _i(B _i(y _i)). And, as σ _i is injective, the set σ _i(B _i(y _i)) has the same number of elements as the set B _i(y _i).

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Titel: The Selten–Szidarovszky Technique: The Transformation Part
verfasst von: Pierre von Mouche
Verlag: Springer International Publishing
Buch: Recent Advances in Game Theory and Applications
Print ISBN: 978-3-319-43837-5

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

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-3-319-43838-2_8