Grim Under a Compensation Variant

Games on graphs are a well studied subset of combinatorial games. When analyzing a game, heuristics and strategies for winning are often at the forefront of the discussion. One such combinatorial graph game that can be considered is Grim. In Grim there are winning strategies for a variety of well-known families of graphs, many of which favor the first player. Hoping to develop a fairer Grim, we look at Grim played under a slightly different rule set and develop winning strategies for this modified version of the game. Throughout, we compare our new results to those previously known and discuss whether our altered Grim is a fairer game than the original.