There have been many different definitions and attempts at defining the term "game" but for the purposes of dog training, here is a definition that fits in the instructional setting as well as for life experiences.
Just so we are all on the same page, I will also define the terms used in the definition.
Cooperation-To work together toward a common goal, justly and honestly. Cooperation implies the ability to engage in communication with understanding; to be honest about intentions and purposes and the rely of information; to do one’s rightful share of the work; to effectively perform one’s job and assist in the survival of the players until the goal is reached.
Players-As an adjective, to be "game" is someone who is eager and willing to do something new or challenging with the purpose of reaching a goal or creating something new. This person is a "player". A game can have one player or multiples. These players can be working cooperatively or separately, even in conflict as a means to deter the other players from reaching the stated goal.
Quantifiable-Capable of being measured. For instance, if you were playing the game of 1-2-3-break which has the goal of the dog learning to stay in the position asked for until released, you could measure your progress with duration of stay, level of distractions, how many different environments the behavior is perfected in and the distance the handler can move away from the dog.
Challenge-A test of one's abilities or resources in a demanding but stimulating undertaking. A challenge also invites the player to learn new things, refine already known behaviors and abilities, and create new pathways and new worlds thus guaranteeing survival.
Rules-A rule is a statement explaining what someone can or cannot do in a particular system, game, or situation. The rules of the game are the structure that allows the goal to be reached. Rules create the boundaries, set the tone and make cooperation inevitable.
Goal-The end toward which an endeavor is directed; an objective. Having a goal is often what differentiates between play and a game. A game can have more than one goal. For instance, coming back to the 1-2-3-Break game, there are several goals - 1) learning to stay in one place 2) waiting for a release 3) creating self-control in the face of distractions.
Purpose-The reason for which something is done or created or for which something exists. Each teaching game has a stated purpose, a reason why the game is being played. Sometimes there can be many different purposes for one game. For instance, crate games teach the release word, to sit at doors, stay, patience, how to earn rewards and drive.
Intention- a determination to act in a certain way; to have in mind a purpose or plan, to direct the mind, to aim. Without intention, we go nowhere. It is determination in the fullest sense, almost the need to move forward.
Focus-A central point, as of attraction, attention, or activity.