In Real Life, human brain acquires new knowledge and puts it to use via a spectacular cascade of electrochemical reactions taking place across a heavily networked biological cell grid. Learning and improving one's skills is a continuous, life-long process, while expertise in one field often proves unexpectedly useful in others, seemingly unrelated domains. But complex neuroscience has no place in video games, because a) it violates the Rule of Fun, and b) it runs a high risk of spawning a murderous AI on your computer.
Instead, tabletop and video games--particularly Role Playing Games and games with RPG Elements--narrow down the areas of human knowledge to a handful of quantifiable "skills" that are relevant to the plot and whose advancement and effects can be defined in terms of gameplay mechanics. At the start of the game, the players are handed a list of these skills and a few "skill points" to assign to their characters, with more skills rewarded later on for completing the game's objectives.
There are two common ways to define character skills in the Game System terms: skill scores (a.k.a. "skill levels") and perks (a.k.a. "feats" and "traits"). Following table outlines their main differences:
Improve basic in-game actions
Unlock new moves and unique bonuses
Can be leveled up multiple times (between 3 and 100)
Usually cannot be leveled up, although other perks may confer bonuses
Skill levels have index numbers or generic labels (novice, trained, master)
Higher skill levels may cost more than the lower ones
More advanced perks may cost more than the basic ones
Unlocked level-by-level; may be capped by a character stat
May have prerequisite perks, character levels, skill and stat scores
Skill scores and perks often coexist side-by-side, mixed-and-matched in a myriad of ways, and many games (especially contemporary RPGs) even blur the line between the two.
See also The Six Stats, Skill Point Reset.
Tabletop game examples:
Dungeons & Dragons is likely the Trope Codifier, if not the outright Trope Maker. The specifics vary a little by edition, but D&D uses both skill scores, purchased points that act as a modifier on skill rolls, and "feats," purchased perks that may do everything from modifying rolls further to allowing the character to perform special actions or be immune to certain things (and that's just the tip of the iceberg, especially with the vast expanses of 3.X Edition). D&D-based video games usually use some or all of these mechanics.
Both the Storyteller and the Storytelling System are based around "traits", which are basically a hybrid mash-up of stats ("attributes"), skill scores ("abilities" in oWoD, "skills" in nWoD), and upgradable perks ("advantages"). The latter include both storyline perks (like background) and active abilities (like the vampiric Disciplines). Attribute scores range from 1 to 5, abilities/skills and advantages from 0 to 5. Occasionally, levels up to 10 may be allowed. Trait levels are acquired by spending character points.
Video game examples:
RPG -- Action
Diablo II featured a hybrid skill score/perk system, wherein each class had three unique skill trees consisting of several tiers of perks. Individual tiers were unlocked one by one at certain character levels, after which any number of skill points (gained at each level and from some quests) could be invested into any unlocked perk, increasing its efficiency and often giving bonuses to more advanced perks derived from it. On a side note, Diablo II had one of the first popular implementations of the aura-type perks (with its Paladin class).
Mass Effect 1 had a hybrid skill score/perk system wherein "talents" encompassed both passive (armor, weapon) and active (biotic attacks) skills. Talents could be leveled up multiple times, increasing their efficiency and unlocking additional perks and even further upgradeable talents at certain levels (forming an implicit skill tree). Mass Effect 2 and Mass Effect 3 replaced them with "powers"--active combat skills, which were essentially perks with up to five (mutually exclusive at level 4+) upgrade perks available for each of them. Which talents and powers were available depended on the character's class and although most were so from the start, some were only unlocked after certain story events.
The original Deus Ex featured both skill scores and perks (dubbed "augmentations"). Skill scores were upgraded using skill points acquired by completing missions, had four levels (Untrained, Trained, Advanced, Master), and improved your performance with different types of weapons, or non-combat actions (lock-picking, electronics, medicine, etc.). Augmentations were picked up as items on missions and implanted into Denton's body slots, giving him new abilities. They also could be upgraded. Deus Ex: Human Revolution did away with skill scores and instead translates experience points into Praxis Points, which in turn can be invested into any augmentation unlocked in the perk tree (justified by that all augs are already built into Jensen and only need to be activated).
Path of Exile has an enormous perk tree consisting of 1300 nodes available to each class, although all nodes are passive bonuses (active skills are instead imbued into items). Furthermore, a majority of nodes are unspectacular permanent bonuses to one of the character attributes (strength, dexterity, intelligence), but buying them is required to get to the juicy, named perks that provide significant combat advantages.
RPG -- Eastern
Uncharted Waters: New Horizons features a combination of seven skill scores (ranging from Leadership, through Knowledge, to Swordsmanship) which increase your performance as a fleet captain, and five perks: Celestial Navigation lets you auto-sail to any known port, Cartography allows you to sell maps, Gunnery drastically improves combat performance, Accounting tells you best places to sell your goods, and Negotiation lets you haggle for better prices.
The Crystarium system Final Fantasy XIII consisted of class-and-character-specific (mostly linear with minor sideways branching) perk trees, whose nodes had increasing "crystogen point" costs. Most nodes provided bonuses to either Strength, Magic, or Health, but some unlocked class/role-specific moves. In Final Fantasy XIII-2, each character has only one linear "perk tree", but its nodes are technically empty slots where the player can place perks (bonuses or abilities) of any of the six available classes (up to 99 per class) by investing crystogen points.
RPG -- Western
The underlying game system of The Elder Scrolls series is based mainly around skill scores, which are increased by using them (which also indirectly increases their governing character stats), but Oblivion added a number perks, which were unlocked by reaching certain levels in skills: e.g. a Power Attack at Blade 25, no shield wear-out at Shield 50, jumping off water surface at Acrobatics 100, etc. Skyrim expanded the available perks to an entire tree (one per skill score); perks now have prerequisite perks and corresponding skill score levels and can be bought and upgraded with perk points (gained with every level up) once unlocked.
In Dragon Age: Origins skill scores ("skills") and perks ("talents" and spells) existed in parallel, although the latter were much more prevalent. Eight (11 in Awakening) skills could be leveled up four times, providing passive bonuses to anything from persuasiveness, through pickpocketing, to combat efficiency. Talents, meanwhile, were distinct moves, auras, passives, and upgrades grouped into "trees" of three (four in Awakening) linear four item-long branches (mages got spells instead of talents but they worked the same way). There were also class- and specialization-specific perk trees. In addition to being part of a skill tree, new perks had stat and (in case of weapon talents) skill level requirements. Talent points were gained at every level, and skill points, every three (two for rogues) levels. Dragon Age II did away with the skills but expanded the talent trees to branch out more.
Alpha Protocol uses a system similar to Mass Effect 1 in that your skill points both improve your proficiency directly and, at predefined points along the tree, grant access to special actions such as Bullet Time or HUD indicators as to enemies' location, disposition, and facing.
No One Lives Forever 2 included a skill score system that represented various aspects of being a super-spy (stealth, marksmanship, gadgets, etc.). Each skill score could be upgraded four times, giving various passive bonuses to the respective basic action (hiding, shooting, breaking codes, etc.). Each level cost progressively more skill points, which were obtained from completing missions and finding manuals scattered throughout the game.
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