What makes a "game" a Game?

Yesterday I read an interesting conceptual description for an educational "game", however it seemed to be missing all of the gaming elements, and the reason why a game has been selected as the structure for the solution eluded me altogether. I believe however that the author had selected the game concept because they has heard about how gaming can be an effective communication tool for children. But, just because you want to solve your problem with a game, it doesn't mean that a game is the best solution to the problem. that lead me to consider, if we look at something that we will all agree is a game, what are the critical few identifying characteristics that make us all agree that yes, we are looking at a game? lets use Monopoly as a our case study. What can we say about it:
- it is perceived as fun
- it is often considered a source of entertainment
- at "play time" it gets presented as an option
- it involves a group of people
- there is an objective(*)
- there are rules(*)
- players are subject to risk that has consequences
- players are subject to chance encounters with rewards
- uncertainty creates excitement
- there is a winner and a loser
- players accumulate posessions(*)
- playes achieve public status(*)
- players receive micro-rewards(*)

there are probably more ways to describe the attributes of this game, but if we want to select just the essential "gaming elements", that could be universaly applied to any game, maybe they would be:
- there is an objective(*)
- there are rules(*)

and what has made this game successful?
- players accumulate posessions(*)
- playes achieve public status(*)
- players receive micro-rewards(*)

I know this is not exhaustive, but, I am after a rudimentary litmus test. Is it a game, or not?