N-Player Nim Games With Random Players

Two-player’s game is more common in the combinatorial game theory.Krawec gen-eralized the number of the game players in 2012,and studied the multi-player impartial combinatorial game.After that,more than a large number of scholars began to study multi-player game.In 2015,Krawec added the concept of randomness to multi-player game,and developed a method of analyzing multi-player impartial combinatorial game with random players.In the game,the non-random players choose the optimal strategy,while the random players play randomly.In the multi-player impartial combinatorial game with random players,the game value g reflects the probability distribution of each player’s victory,so the key of the game is to find the game value g.In this paper,we use the method developed by Krawec to analyze a class of n-player N-pile Nim game with arbitrary l random players,which is denoted by Nim(N,n,l).In the Nim(N,n,l)game,we consider the number of players,the structure of the random players,the number of the game piles,and the type of alliance matrix respectively,and then study the influence of the above factors on the Nim(N,n,l)game.Finally,we have the following conclusions:(1)The game value of n-player and 1-pile Nim game with one random player under the standard alliance matrix;(2)The game value of n-player and 1-pile Nim game with arbitrary l random players under the standard alliance matrix;(3)The game value of n-player and N-pile Nim game with one random player under the standard alliance matrix;(4)The game value of n-player and 1-pile Nim game with l random players under other types of alliance matrix.Firstly,under the standard alliance matrix,we analyze the n-player and 1-pile Nim game with one random player,which is defined by the Nim(1,n,1)game.And then we present the game values for n=3 and for any integer n≥ 4,respectively.Secondly,we study the influence of the structure of random players on the game value.Under the standard alliance matrix,we analyze the n-player and 1-pile Nim game with l random players,which is defined by the Nim(1,n,l)game.In this paper,we come up with the general structure of the random players,and then respectively study the game values of l non-adjacent random players and l adjacent random players and eventually get the game values of Nim(1,n,l)game under the arbitrary structure of random players.Thirdly,we study the influence of the number of game piles N on the game value.Under the standard alliance matrix,we analyze the n-player and N-pile Nim game with one random player,which is defined by the Nim(N,n,1)game.And then for any integer N≥ 2,we present the game values of Nim(N,n,l)game with one random player.Finally,we study the influence of the alliance matrix on the game value.We define two new alliance matrices,and analyze the Nim(1,n,l)game under new alliance matrices.And we compare the game values under the new alliance matrices and the standard alliance matrix,then we present the game values of Nim(1,n,l)game under the alliance matrix As and the game values of Nim(1,n,1)game under the alliance matrix A*.