I "feel" I know what is going on, here, but it's hard to put into words.
You likely have a fairly "tense" workplace and the senior management is looking for a "quick-fix" to loosen everyone up and get them all to get along. You're a young, generally happy person, and management gave you the task because you must know something about happiness, right?
What you are looking for is a complete shift in the GroupThink, and you think the game will do it. The problem is that the game will only work if you've fixed the problem, first. It's not a chicken/egg issue. The game will help break down some bitterness and disengagement, but only if you fix the problems that created those issues, first.
In my experience, factors that can cause these feelings to build up are:
Conflicting goals from management. Nothing will make two employees despise each other like incompatible objectives and limited resources. Company and departmental goals should be given to groups, not individuals, and preferably in writing.
Nepotism / Cronyism. Hiring the boss's son as a college intern is one thing, but putting the former roommate of the CFO in charge of development when he is clearly unqualified will start an all-out mutiny. (Been there, done that, got the T-Shirt.)
Misplaced recognition. If executive management is recognizing the manager of a group for the accomplishments of the group, teeth will be gnashed. Senior management needs to recognize and reward entire teams, not just the team leaders. This is especially important with monetary awards. Nothing will make your key employees more upset than seeing their manager get a 4 or 5 figure bonus for the sweat and toil of someone else. "Rock Stars" will appreciate a private, "I saw what you did, there. Great job!" talk much more than a bonus (usually) anyway. A team victory needs a team award.
Obvious pay disparity. This kind of goes hand-in-hand with Nepotism/Cronyism, but seeing someone who doesn't pull their own weight bring in serious money just kills motivation and fosters contempt. It makes employees feel that "The game is rigged!" and that no matter what their own contribution, they aren't in "the Club" and it won't matter. If there is serious pay disparity, those on the high side of it need to be STRONGLY encouraged to not show it off at the office. No fancy watch, leave the Audi at home and drive the Chevy, etc.
I can't speak for your office, but if you're trying to do "something fun" and getting cool responses or even outright resistance, I'd bet you've got a problem with at least one of the above issues.