Direct Consequences of Games
A game is only a game if its direct consequences are divorced from relevance. The game's indirect consequences may be relevant; in fact they are inseparable from relevance.
This aspect of the definition of "game" is essential in order to separate out games from other activities that merely follow game-like patterns. A classic example is a war.
War is often called a game, despite the dire consequences of its being played out. This is because in war, like in games, the people who control the maneuvers rely on strategy, luck, tactics, and many other elements also found within games. In fact, a great many people would have a hard time trying to explain why war is not a game while Chess is.
The answer is the relevance of the play and outcome of a game versus the play and outcome of a non-game.
In a game, the actions taken within the game are not directly consequential to achieving any real-world goal. Moving a piece on a Chess board may have indirect consequences, but the direct consequence must be irrelevant for it to be a game. You may learn from it, you may impress someone from it, you may even signal someone by doing it. But picking up the physical piece off of one square of the board, and then placing it back down on another square of the board is, in and of itself, an irrelevant act of no consequence.
Similarly to the actions within the game, achieving the goal of a game is entirely inconsequential as a direct act. The physical act of knocking over your opponent's King is entirely inconsequential, in and of itself. Winning the game may result in someone committing suicide, but that would be an indirect result of the game's conclusion.
In a war, on the other hand, an "act" such as moving a platoon from one place to another place holds real-world significance and consequence. So is the act of forcing an surrender from an opponent. Therefore, according to my requirement, war cannot be considered simply a game.
(Ah-ha! What if you take a Chess game and wire up the pieces so that pressing a piece onto a board space blows up a building? Is this no longer a "game"? In my opinion, it is still a game; the act of placing the piece may blow up a building, but that is an indirect result of the game play; a very nearly direct result, but still indirect.)
Therefore, regardless of any other ways you have to define games by competition, interaction, goals, rules, and so on, I suggest adding that a game is abstracted from reality; that playing a game is, by definition, a step outside of direct consequence.
Indirect Consequences of Games
It is trivially easy to think of the indirect consequences of games. So trivial, that it becomes difficult to think of games that don't have indirect consequences.
Games raise of lower self-esteem. They teach logic, patience, math, pattern matching, and manners, and sometimes history, geography, science, and a host of other lessons.
Games create social connection and foster family and friend relationships. Games facilitate communication and community.
Some games result in small or vast amounts of money changing hands, make or break fortunes or lives, swell national pride, start wars, end wars, unite or divide lovers, inspire or drive away congregations, and so on.
From Rock Paper Scissors to a Marathon, games give us focus, challenge, inspiration, perspiration, joy, deep thought, laughter and fortitude.
Indirect consequences are the stuff that games are made of. That's why we play them.
It is impossible to play most sports without building muscle and health. People often play sports for this very reason.
Similarly, it is nearly impossible to play many mind games, such as classic abstracts, word games, negotiation games, and the like, if played regularly and with some seriousness, without building some mental muscle and health. Again, some people play these games for this very reason.
A great number of games are played for stakes: money, beer, or prestige. A great many other games are played to avoid loneliness.
In many situations, the game is played primarily for its indirect consequence. If these consequence were absent, the person would not play the game. In many of these situations, the consequence is an unavoidable result of playing the game. Still, no specific move in the game is directly related to the consequence, nor is the outcome of the game, nor is the consequence relevant to the rules of the game.
Still, considering indirect consequences, games matter a great deal.