Multi-player End-Nim Games And Its Stochastic Models

"Two players End Nim game" is a classical game of many conbinatorial games.The term game theory can be described as:there are N piles of counters,the number of piles is limited.The two players take turns to move legally.Each removing any positive number of counters from the first pile or the last pile.Under normal play convention,the first player who cannot make a legal move loses.Respectively,under misere play convention,this player wins.This paper makes study on the extensions of two player End-Nim.We have two kinds of extensions.(1)The number of players is extended from two to n>2,and "Multi-player End-Nim"(MEN)is obtained.(2)And on the basis of the MEN game,the decision made by the people who participate in the End-Nim game is divided into rational and irrational(random),and a new game model is obtained,which is called MREN game.This paper is divided into three chapter:The first chapter is the introduction which introduces the development of impartial combinatorial game,it also elaborates the research situation of End-Nim game.The second chapter studies the End-Nim game with N heaps for n-player.Under the misere play convention,we present the game values and the optimal strategy of MEN(N,n)for n>N +1,n = N + 1,n = N and n<N,respectively.In the third chapter,we study RMEN game with exactly 1 random player.And under the misere play convention,we present the game values and the optimal strategy of RMEN game.