Alternative characterizations of Nash equilibrium
Creators: name: Fatighenti, Dario
Publisher: name: IMT Alti Studi Lucca
Year: 2014
Subjects: HB Economic Theory

description: In this thesis I investigate the solution concept of Nash equilibrium. This thesis is composed of three articles, 
that cover pure and mixed strategies Nash equilibrium. The aim of the first article is to provide an alternative characterization of Nash equilibrium in pure 
actions; The second article provides elementary methods to prove the existence of mixed symmetric Nash equilibria in symmetric bimatrix games; While the third 
article focuses on mixed actions Nash equilibrium of general finite games. In the first article ”Semi-Connected Games” we will briefly present two relevant 
approaches to the study of Nash equilibrium: one approach consists in the search of progressive relaxation of the sufficient conditions established by Nash, 
the other approach consists in the application to the problem of existence conditions of the results of the algorithmic game theory [finite improvement property,
weakly acyclic games ...]. Contributions from these two fields stimulated the discovery of conditions that are not only sufficient but also necessary to the 
existence of Nash equilibria in pure actions [Tian (2009)]. We will reverse the line of investigation from previous literature, looking for the largest class 
of games which do not possess any equilibrium in pure actions, establishing the class of semi-connected games we prove that no games in that class can have a 
Nash equilibrium, moreover, we will show that only semi-connected games do not possess any Nash equilibrium in pure strategy. Unfortunately this first result 
is not immediately applicable to the context of equilibria in mixed strategies, for which the problem of developing polynomial time algorithms is still an open
question. As Gilboa and Zemel point out in their 1989 article, this may be imputed to the fact that no elementary proof of existence has been proposed. 
To try filling this gap we investigate both general games and symmetric games [in the third and in the second article respectively]. In the second article 
”Notes on Symmetric Bimatrix games”, we investigate the existence of symmetric equilibria in symmetric games without assuming the presence of Nash equilibria. 
In particular, we propose two new methodologies to prove the existence of symmetric equilibria in symmetric bimatrix games, which is a finite symmetric game 
with only two players. The first one, based on the elementary proof of Sion’s theorem, is applicable only to a special class of symmetric bimatrix games. 
The second one, based on Nessah and Tian’s elementary proof of Fan’s minimax inequality [in their article of 2013, revised in 2014 by Tian], covers all 
symmetric equilibria of all symmetric bimatrix games. Both methodologies are elementary and not purely existential. In the last article ”Mixed strategy Nash 
equilibria in strategic form games” we provide a proof of the existence of mixed strategy Nash equilibria in finite normal form games applying results 
proposed in the second article. Thus we proposes a novel methodology, showing how Nash equilibria of general games coincide with symmetric equilibria of appropriate symmetric games.