They have been called achievements, badges, challenges, medals and so forth. Their primary purpose is to extend the life of a game by providing goals beyond completion of the game. They are a reward for completing a non-standard task such as finding a secret, collecting in game items, defeating a foe within specific parameters, defeating a predetermined amount of foes, completing a task within a specific time frame, multiple play throughs, playing with specific difficulty settings or simply reaching certain way points.
A bit of history on the subject.
While these varied achievements now are now an expected part of games they were originally nothing more complex than a high score in an arcade machine. But as video game culture expanded so too did the achievements.
In the early 80's Activision began offering iron on patches through mail order for the games you played at home. It was largely a gimmick. The idea was that you mailed in a photo of a TV showing your high score with a small fee and they would send you the 'Badge' so you can prove to everyone you were a master at the game. In reality and not surprisingly, they sent the badges to most anyone who sent in for them.
This was not a huge success for Activision but it did cause other game companies to offer their own versions of rewards for gaming prowess. Atari even had an entire unfinished game series based around the notion of obtaining high scores to unlock clues that you would enter into a contest. Some developers even included what is known as 'Easter Eggs' in their games such as the hidden message in 'Adventure'.
After the North American video game crash in '83-'84, these special rewards were obviously much more uncommon. Then in '85 Nintendo released the NES system in North America and with it a massive marketing campaign. This included ads, books, guides, contests and give-aways. In '88 Nintendo started publishing 'Nintendo Power'. A magazine devoted specifically to the Nintendo fan base. Almost instantly Nintendo had readers send in photos of their high scores or achievements from the NES library to be immortalized in it's pages to huge success.
These contest prizes and recognition in their magazine was the closest to what we consider achievements currently. But their games were also evolving to include unlockable rewards that were not required to complete the games. Games like the Final Fantasy series introduced unlockable characters that could only be obtained through a series of specific actions or Super Mario Brothers' warp pipes. Coincidentally, the NES library is also the source for the evolution of mini-games.
As time progressed, more and more games included their own versions of unlockable content continuing well into the next millennium. With games offering more varied challenges and a way to view them in game.
Mid 00's X-Box created the 'Gamerscore' as a way for players to track their achievements over multiple games. This proved so popular that both Steam and Playstation quickly followed suit.
Almost overnight a new gaming sub-culture was created. The internet became inundated with guides, blogs, videos and even entire websites devoted to the singular goal of these rewards. So ravenous and competitive was this sub-culture that Microsoft had to strictly govern the entire Gamerscore system. They instituted strict rules that games had to follow with their achievements and they cracked down on players who artificially inflated their scores. Both Steam and Playstation also have rule systems that guard against 'cheating' and game companies that abuse these types of rewards.
Since then these rules and heavy handed punishments for infractions have been lessened but they do still exist.
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