Suppose we have a set of players, i.e., {p1, p2, ..., pn} and each player has three different strategies, i.e., {s1, s2, s3}. They play m number of games. In each game, each player seeks to maximize its profit by selecting a strategy with highest playoff. The profit associated with each strategy is as follows.
1) Payoff for selecting strategy s1 is zero
2) Payoff for selecting strategy s2 is a real number, which is calculated using some formula f1
3) Payoff for selecting strategy s3 is also a real number, however, it is calculated using another formula f2
I want to prove the existence of Nash equilibrium when all the players select one of the available strategies.
I have searched on web and found several documents, however, I couldn't get a clear idea to prove it mathematically.
Any help is deeply appreciated. Please let me know if I have missed any information. Thank you in advance.
Dear Amar,
I suggest you to see links and attached files in topic.
- Existence theorems for generalized Nash equilibrium problems - Hal
https://hal.archives-ouvertes.fr/hal-00828948v2/document
- Lecture Notes on Non-Cooperative Game Theory - Hamilton Institute
www.hamilton.ie/ollie/Downloads/Game.pdf
- Notes on Equilibria in Symmetric Games - Institutional Knowledge at ...
ink.library.smu.edu.sg/cgi/viewcontent.cgi?article=2212...sis...
- oundations of non-cooperative games - Ron Holzman
holzman.technion.ac.il/files/2012/09/mine.pdf
- Nash's (1951) non-cooperative equilibrium, in ... - ScienceDirect.com
www.sciencedirect.com/science/article/pii/.../pdf?md5...pid...
Best regards
Do you have what exactly are formulas f1 and f2? because the existence or non existence (in pure strategies) will strongly depend on that. Particularly it could lead to some asymptotic behavior (for example, in the long run n_1 players have to play s1, n_2 play s_2 and n_3 play s_3, with n=n_1+n_2+n_3)
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