A game mechanics designer...
     ...creates the design of a game using game mechanics.
     ...Game mechanics are rules.
     ...games are devices for learning.
Much  like a writer uses the words of their language to construct a  novel,  the game designer uses game mechanics to assemble a game.
Unlike  disciplines like engineering, game mechanics are abstract ideas whose  forms are not bound by the laws of physics but instead by the boundaries  of the game designer's imagination.
A  game's design exists independently of the medium it is used to express   it. A game's medium can be digital or physical. Prototypes of even some   videogames can be physical so the designer may freely manipulate the   pieces during the design process to solidify the ideas.
The goal of a game designer is to make a fun game. If the game is not fun then the designer has failed in their task.
Therefore  the purpose of game design is to create a fun way to learn (and whether  the information learned through play is of value outside of play is a  separate matter, deciding whether the game is entertainment or  edutainment).
Much of what a game mechanics designer  makes are mathematical patterns; in video games, these are algorithms  that will be translated into a programming language so they can be  expressed to the player through the digital medium. So even if a game  mechanics designer is not a programmer they should familiarize  themselves with how programming works in order to draft patterns that  translate well into existing languages.
Example of a mathematical pattern used to create a mechanic
{rnd  (minATK, maxATK) x ItemModifier x Skill Modifier x (1-DEF / 100) - STR -  VIT/2} x Elemental Modifier = Stereotypical  "Attack" command damage  formula in an RPG video game
Even the design of a board  game like Monopoly requires mathematical calculations to determine the  odds of dice rolls (2d6 has bell curves, resulting in certain numbers  more probable to occur during play) to make predictions about the value  of places on the board, which is essential to designing a balanced game (we'll get to this in a'bit).
Qualities of a game designer
1) Spatial intelligence:  The ability to visualize with the mind's eye. Required in order to  understand and create ordered relationships, such as flowcharts and  algorithms.It is also required so one can imagine transformations in the  design, as well as the consequences of making these changes to an  existing game's design based on prior knowledge of player behavior.
2) Interpersonal skills:  High interpersonal skills allow one to be sensitive to others moods,  feelings, temperaments and motivations. This allows one to guess what  players will do and design to accommodate their actions; to predict emergent  gameplay; that is, the actions players will take during game that may  not be openly stated in the instructions but are allowed by the rules of  the game. (Ex. A MMO may not have trade windows to allow secure trading  between players, or a currency, but a meta-system of bartering may  emerge around the 'Drop item' and 'Pickup item' commands).
3) Intrapersonal skills:  This is one's ability to be self-reflective. It requires one to have a  deep understanding of yourself and predict reactions and emotions of  yourself in various situations. Knowing what makes you tick is going to  help you understand what makes other's tick, and how you can design  content for other groups of gamers you don't belong to.
4) Logical-Mathematical Intelligence:  This is your ability to use logic, abstractions, reasoning and using  numbers. This is not necessarily all about math; it is more about  understanding logic and being rational. It is needed to perform accurate  and unbias investigations into raw data to make ego-free decisions. A  game designer must be able to think logically to solve problems in a  novel way independent of any acquired knowledge, as well as identify the  patterns and relationships that underpin those problems. It is the  ability to use inductive reasoning and deductive reasoning.
5) Excellent Memory:  Your ability to do all of the above is going to depend greatly on your  own experiences and acquired knowledge from other sources, such as books  and postmortem reports from completed games-- as well playing many  different types of games, especially the bad ones.
Your  ability to use logic to evaluate problems and use spatial intelligence  to imagine all the ways a design might go wrong rely heavily on your  ability to identify situations you'd already experienced or that you've  heard about before.
6) A love of learning:  Mechanics take form from your imagination. You need stimulation from  books, movies, places you vacation, dates you go on, etc. I've come up  with ideas for elements in my designs by trying to model things I've  seen in everything from history channel specials to comics I read as a  child. Game mechanics are an artistic expression the same as painting  and story-telling are. Game designing is a creative endeavor and at its  highest level it comes from within the same melting pot of history,  legend and culture that Tolkien's Cauldron of Story does; both simmer  over the fire of human imagination.
A love of learning  is crucial to obtain true mastery of mechanics rather than simply  imitate the mechanics of others; that is, to create entirely new ones or  utilize old ones effectively.
Most important elements of a game's design
1) Synergy: All mechanics in the game working together as opposed to having some mechanics be combative with other mechanics.
Example:  Talent trees in World of Warcraft from The Burning Crusade. You had  three tanking classes of Paladins, Druids and Warriors but only Warriors  were able to generate enough threat to maintain aggro on end-game raid  bosses, and Warriors had superior damage mitigation cooldowns, as well  as the ability to break fear, which many bosses used often. Paladins and  Druids didn't have these parameters yet had tanking trees, sending the  message to players that they could do the tanking role in raids when the  design of their trees simply did not allow this to occur in any  effective way. Thus the design of raid encounters did not have synergy  with Paladin and Druid tanking tree design and this resulted in a  problem.
2) Balance: The difficulty of a game.  Many players and even some game designers think this means the player  and opponents must have equal chance to win. This is false. If this were  true then a perfectly balanced game would result in the player losing  half the time, all the time and that is not fun!
Once  system mastery is obtained the player should rarely lose; losing should  punish lack of system mastery, and a player should never lose for any  other reason.
Even in cases of PVP in an MMORPG, the  player and/or team with higher system mastery should defeat the player /  team of lower system mastery. Game balance is essentially the pacing of  the player's learning of the system, on the road to system mastery. A  game where system mastery is obtained too quickly and effortlessly is a  game that is too easy and a game where mastery is slow and difficult to  comprehend is hard.
3) Clearly defined target audience:  Games are devices of learning. A game designer is the architect of the  game and does not need to learn the rules; only players can learn rules.  Games are thus intended for others to play.
It is  extraordinarily difficult to design a game that can be all things to all  people; the designer must decide who they want to play their game so  they can design the mechanics to appeal to that demographic.
You  cannot please everyone but you can please your target audience. That is  an achievable and realistic goal. By having a target audience you will  also generate a cohesive and focused design, and be able to more easily  decide which elements need changing or deletion in order for the game  mechanics to all have synergy with one another.
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