Winning Ways of Weighted Poset Games

摘要

In this paper, we introduce the weighted poset game, which is defined as an extension of the poset (partially ordered set) game by adding a weight on every element of the poset. Each player has their own non-negative number of lives, and loses as many lives as the sum of the element weights they took. The player whose lives become negative first is the loser. We consider winning ways of this problem. First, for the problem with {0, 1}-weights, we find that (1) if the number of lives are different, then the player who has the large number of lives is the winner, (2) if the number of lives are the same and all maximal elements have positive weights, then the second player is a winner, and (3) otherwise, the game is reduced to an (unweighted) poset game. Next, for general weights, we consider the case where the partial order is a total order, and derive a polynomial-time algorithm for calculating who is the winner and the winning way for the winner.