A home stage is the stage that is most commonly associated with a character, by the game and/or by players. The subject of home stages can at times be somewhat confusing in the Super Smash Bros. series. In all the Super Smash Bros. games, there are playable characters who do not have a stage belonging to their universe. In addition, certain characters in Melee have more than one stage. As secret characters are fought on stages specific to each of those characters, and Melee's All-Star Mode also gives each character a unique "home" stage, it's possible to determine which characters "own" which stages, even though the results may occasionally cause confusion.
Super Smash Bros.
In Super Smash Bros., each of the eight starter characters has their own stage. Of the four unlockable characters, two of them (Jigglypuff and Luigi) share their stages with the starters Pikachu and Mario; sharing Saffron City and Peach's Castle respectively. The remaining two unlockable characters (Captain Falcon and Ness) have their home stages defined by where they are fought when they are unlocked, as the F-Zero and EarthBound series do not have stages in this game. Instead, Planet Zebes and Dream Land are considered Captain Falcon's and Ness's home stages. There is another stage from the Mario series, Mushroom Kingdom, which is sometimes considered both Mario's and Luigi's stage, but on 1P Game they cannot be fought there.
Super Smash Bros. Melee
In Super Smash Bros. Melee, the situation is somewhat more complicated. While secret characters all have a stage of their own for the purpose of unlocking them, all characters have a stage associated with them for the purposes of All-Star Mode, as listed below. All series with playable characters have at least one stage with the exception of Fire Emblem, meaning that the Fire Emblem representatives Marth and Roy are always encountered on other series' stages. However, since the song Fire Emblem plays on the stage Temple as alternate music; that stage is considered Marth and Roy's. Also of note is that the All-Star Mode stage assignments use every non-Past Stage barring Big Blue.
Super Smash Bros. Brawl
In Super Smash Bros. Brawl, stages and characters are not matched one-to-one like they are in Melee. What follows is a list of home stages based on where characters are fought in Classic Mode and All-Star Mode, almost always based on a character's universe. The only exception is R.O.B., who has no stage from his universe. Because of this, he is given exclusive usage of the stage Mario Bros.; which is considered R.O.B's as it is not used in Classic Mode and All-Star Mode for any characters from the Mario universe.
As was the case with the Past Stages in Melee, Melee stages returning from Super Smash Bros. Melee are never seen in Classic Mode or All-Star Mode. This leads to the fact that Halberd is always seen in Classic Mode and All Star Mode no matter what (since that is the Kirby universe's only Brawl stage); and the Kirby series is always fought in Classic Mode and All-Star Mode.
*R.O.B.'s home stage is Delfino Plaza in Classic Mode and All-Star Mode when Mario Bros. is not unlocked.
Super Smash Bros. 4
Unlike in previous games, Past Stages also act as home stages, as some of these familiar stages are the only stages available for their respective universes. Characters from universes without stages are assigned home stages whose universes lack characters. The Mii Fighters' home stage is considered to be Battlefield, as they are part of the Super Smash Bros. universe and are only fought in Classic Mode as enemy teams. Characters are fought on their home stages in Classic Mode for Super Smash Bros. for 3DS.
Theory of "home advantage"
There is no clear or consistent benefit to a character fighting on their "home stage." Any advantages that certain characters have over others on stages happen across the entire stage list and are situational; they are not affected by the universes in which the characters originate from (for example, Fox and Pikachu dominate on Temple, even though they do not originate from the Zelda universe.) In some cases, it may be to a character's disadvantage to be fighting on their home stage; for example, Ness is unable to use his recovery move when knocked between buildings in Fourside. Additionally, Final Destination, which acts as the home stage for Roy, is disadvantageous for him against nearly every character (except for arguably Fox and Falco) as he is much more vulnerable to combos without any platforms to escape to. In Brawl, the ledges of the Lylat Cruise stage are thinner than most other stages, causing Fox, Falco, and most notably Wolf to curve underneath the stage more frequently when using their respective recovery moves instead of directly grabbing the edge. Additionally, in Skyworld, Pit's Centurions in his Final Smash will crash into the breakable platforms when they aim for opponents, quickly destroying the stage and temporarily removing all ledges; this consequently removes Pit's ability to plank and makes him much easier to gimp.
However, in contrast, some stages do offer advantages to some characters; as mentioned before, any such advantages are non-circumstantial, but these certain stages may have been the origination for the "home advantage" theory. In Melee, Fountain of Dreams serves as Marth's home stage, and is frequently cited as one of his best stages in the entire game, even at high-level tournament play. The stage's small size allows him to control space more efficiently, and the extended bottom blast line allows him more room to recover with Dolphin Slash, which travels purely vertically. In Brawl, on Bridge of Eldin, Zelda and Sheik's Light Arrow, Ganondorf's Beast Ganon, and Toon Link and Link's Triforce Slash are much easier to use, and harder to dodge; the frequent absence of a lower blast-line also acts as a benefit to Link, Ganondorf and Sheik, who normally have relatively poor recoveries. The low blastline of Shadow Moses Island acts as a benefit to Snake, as most of his attacks (especially his special moves) have mostly upward knockback; his high weight and falling speed make him more resistant to vertical KOs, so he is not as affected.
On other stages, character can receive neither a clear advantage nor disadvantage. On Summit, for instance, all the characters slide around, except for the Ice Climbers, who experience no change in their friction when walking on the stage. Like other characters, however, they are still unable to grab onto the ledges in the stage; this trait causes Belay to more frequently cause Nana to be KO'd, thanks to Belay giving Popo more height than Nana when there is no ledge to tether to. Another example is Little Mac on Boxing Ring: the stage gives no access to the bottom blast line, which remedies Mac's poor recovery, but also has plenty of verticality, making it difficult for him to traverse the stage. The Ω form of Boxing Ring takes this even further in Mac's favor - while being based off of Final Destination means it has a lower blast line, the walls of the actual stage reach all the way down to it, allowing Mac to make full use of his uncharacteristically superb wall jump.
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