I have heard different opinions on playing games on Shabbat. Are games using money (ie Monopoly) mukza? What about games without money. (obviously games with writing necessary are muksa and need not be addressed.)
There isn't a problem with playing regular board games, such as chess or checkers, on shabbos. Games that may lead to writing the score down may be a problem. Games that involve fake money are forbidden by many poskim.
Btw, once you can't use something, it often becomes mukza (forbidden to move), but the question should be can one play games on shabbos. If that is forbidden, one can discuss whether they are also mukza.