Multi-player Last Nim with Passes

We introduce a class of impartial combinatorial games, Multi-player Last Nim with Passes, denoted by MLNim(^{(s)}(N,n)): there are N piles of counters which are linearly ordered. In turn, each of n players either removes any positive integer of counters from the last pile, or makes a choice ‘pass’. Once a ‘pass’ option is used, the total number s of passes decreases by 1. When all s passes are used, no player may ever ‘pass’ again. A pass option can be used at any time, up to the penultimate move, but cannot be used at the end of the game. The player who cannot make a move wins the game. The aim is to determine the game values of the positions of MLNim(^{(s)}(N,n)) for all integers (Nge 1) and (nge 3) and (sge 1). For (n>N+1) or (n=N+1ge 3), the game values are completely determined for any (sge 1). For (3le nle N), the game values are determined for infinitely many triplets (N, n, s). We also present a possible explanation why determining the game values becomes more complicated if (nle N).