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## On Maximal Vector Spaces of Finite Noncooperative Games

We consider finite noncooperative (Formula presented.) person games with fixed numbers (Formula presented.), (Formula presented.), of pure strategies of Player (Formula presented.). We propose the following question: is it possible to extend the vector space of finite noncooperative (Formula presented.)-games in mixed strategies such that all games of a broader vector space of noncooperative (Formula presented.) person games on the product of unit (Formula presented.)-dimensional simplices have Nash equilibrium points? We get a necessary and sufficient condition for the negative answer. This condition consists of a relation between the numbers of pure strategies of the players. For two-person games the condition is that the numbers of pure strategies of the both players are equal.

Authors of Word2Vec claimed that their technology could solve the word analogy problem using the vector transformation in the introduced vector space. However, the practice demonstrates that it is not always true. In this paper, we investigate several Word2Vec and FastText model trained for the Russian language and find out reasons of such inconsistency. We found out that different types of words are demonstrating different behavior in the semantic space. FastText vectors are tending to find phonological analogies, while Word2Vec vectors are better in finding relations in geographical proper names. However, we found out that just four out of fifteen selected domains are demonstrating accuracy more that 0.8. We also draw a conclusion that in a common case, the task of word analogies could not be solved using a random word pair taken from two investigated categories. Our experiments have demonstrated that in some cases the length of the vectors could differ more than twice. Calculation of an average vector leads to a better solution here since it closer to more vectors.

Three management problems that a state (or a public administration acting on its behalf) faces in procuring goods and/or services are considered: a) choosing the type of a contract to be awarded and the type of a competitive bidding to determine the winning bid, b) setting the initial price for a contract being the subject of the bidding, and c) designing (or choosing) a set of rules for determining the winning bid by means of the chosen competitive bidding. Mathematical models and decision procedures for analyzing and solving these problems are discussed.