The meta-theorem for the existence of equilibrium in secure strategies

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Abstract

The paper is a continuation of the cycle of papers in 2018-2023 devoted to the theoretical justification of equilibrium in secure strategies (EinSS) as a concept of solving non-cooperative games in pure strategies.A method for constructing EinSS existence theorems from known Nash equilibrium (NE) existence theorems is presented. In particular, theorems for the existence of Nash equilibria are formulated in a standard form and are inserted as a condition into the meta-theorem for the existence of EinSS. According to this method, two theorems for the existence of EinSS are derived and proven based on the theorem of Reny (1999) on the existence of Nash equilibria. The general scheme of deriving existence theorems is as follows. Section 2 summarizes the theorems published in the author's previous papers. Section 3 presents two original theorems from Reny's paper. Section 4 gives a detailed interpretation of the conditions of Reni's theorems, compared to the conditions of Debre's theorem. Section 5 gives a detailed analysis of Reni's theorem. Using a number of examples, the condition of the theorem is interpreted as the condition that there are no jump points or points that guarantee the best answer. Section 6 constructs formally, by the method of meta-theorem, two existence criteria for EinSS that use the original NE existence theorems. In Sections 7 and 8, two theorems are formulated and proved, which are specifically refined for solving applied problems (Hotelling's spatial competition, Tullock's rent competition, Bertrand – Edgeworth oligopoly). All considered theorems are summarized in a final table.

About the authors

Mikhail Borisovich Iskakov

V.A. Trapeznikov Institute of Control Sciences of RAS

Email: mih_iskakov@mail.ru
Moscow

Alexey Borisovich Iskakov

V.A. Trapeznikov Institute of Control Sciences of RAS

Email: isk_alex@mail.ru
Moscow

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