Those who posses the Wild Card are often symbolized by multiple masks.
"The power of the Wild Card doesn't stem just from one's skill in a fight. His strength comes from his knowledge that he cannot avoid fighting to gain his desire. That strength happens to manifest as power in battle."
The Wild Card is an ability in the Persona universe. Specifically, the ability appears in Persona 3, Persona 3 FES and Persona 4, however, theoretically, as all five current Persona games take place in the same universe, it would suggest that the ability has always existed. Those who possess the Wild Card can summon and carry multiple Personas and switch them in battle. Wild Card is also associated with the capabilities of Fusion Spells, or Fusion Raids in the Japanese version, where multiple Personas are summoned at once in the same time to execute a powerful attack.
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Early Persona titles
The Wild Card did not exist prior to Persona 3.
In Megami Ibunroku Persona, Persona 2: Innocent Sin and Persona 2: Eternal Punishment, all playable characters were able to switch Personas in combat and had access to the Velvet Room. As mentioned above, all playable characters are able to access the Fusion Spell system, which requires two or more respective characters with the appropriate Personas to execute.
Persona 3
The Protagonist and the Female Protagonist are the only character in Persona 3 to possess the Wild Card. While they have never shown to have a direct interaction with Philemon, they are visited by a blue butterfly. Philemon often took the guise of a butterfly in the waking world, so it is suggested to be him. It should be noted that both Protagonists are also visited by a child that is not visible to others, who prompts them to sign a contract, which later reveals to be an important source behind the Protagonists' Wild Card. Having signed the contract, the Protagonists are granted access to the Velvet Room. Thus, both are allowed to fuse and create new Personas.
The Protagonists's Wild Card ability appears to be significantly superior to that of all other known users. Unlike prior users, they are compatible with all known Arcana, with the exception of The World and Pierrot/Lust. This could be due to their natural Arcana being that of the Fool, which can supposedly become anything in its awakening. On the other hand, they have no high-compatible Persona either, as any Persona of any Arcana they use do average performance (no more/less SP cost or faster/slower Level Up). This is on contrast with Persona and Persona 2 characters whose each of them uses Personas more or less effectively (or even, cannot use) based on their Arcana compatibility. Further more, they are also the only known Persona-users capable of using Fusion Spells on their own, without the assistance of others.
In events of Persona 3, it is later revealed that the reason behind both Protagonists' superior Wild Card ability are tied to their past. During their childhood, Death was sealed within their body, in which Death's powers are eventually imbued to the Protagonists, amplifying their talents in summoning Personas, and thus their Wild Card as well, though both Protagonists were exceptionally talented to begin with.
Persona 3 FES
In the playable epilogue of Persona 3 FES, titled The Answer, Aigis is the only character to gain the ability of Wild Card. Like the Protagonist before her, she was approached by a blue butterfly before being allowed into The Velvet Room.
Similar to the Protagonist before her, Aigis is also compatible with all Arcana, with the exception of The World and Pierrot/Lust. However, unlike the Protagonist, her Wild Card is not boosted by Death's powers, and such, she does not gain the ability to use Fusion Spells like her predecessor.
Persona 4
Protagonist's Wild Card, as seen when first using a Persona.
The Protagonist of Persona 4 stands unique in contrast to the characters in the previous installments, as he was automatically welcomed to the Velvet Room. Furthermore, the Protagonist was not shown to have any encounter with Philemon nor the butterfly prior his appearance in the Velvet Room (save points appear as the butterfly, but only appear afterwards). It is hinted that when Izanami gave him the power to enter the Midnight Channel, she inadvertently awakened his wild card ability, but that does not explain how he was able to enter the Velvet Room without it.
A small note should be added that the protagonist had already signed the contract by the time he is approached by Igor, implying that he may have been approached beforehand. Also nothing is said in the game about the butterfly savepoints so that may be implied that he had already seen it by the time of the beginning of the game.
Similar to his predecessor, the Protagonist of Persona 4 also has the accessibility to all of the Arcana. Similarly, the Protagonist of Persona 4 has access to the Aeon Arcana as well in Persona 4: Golden. His skills with the Wild Card seem to match that of Aigis, his predecessor, rather than that of the Protagonists of Persona 3, because like her, he also has access to most Arcana without any particular preferences, but since his skills also weren't enhanced by Death, he is also unable to use Fusion Spells on his own.
Persona 4 Arena
The Wild Card is an important story element in Elizabeth's story mode, and a minor story element to its current wielders, Aigis and Yu Narukami, though they do not use it in the game. Much of her story is centered on her understanding the true nature of the Wild Card, through challenging its last known wielders, Aigis and Yu Narukami, in battle. Despite their inability to defeat Elizabeth, they manage to succeed in helping her understand the full potential of the Wild Card, enabling her to receive the Fool Arcana.
Yu and Aigis are mostly unable to change Personas during battles and they are only able to do so when they successfully launch their Instant Kills, where Izanagi and Athena will change to Izanagi-no-Okami and Pallas Athena in order to execute their final attacks.
Persona Q: Shadow of the Labyrinth
All playable characters originating from Persona 3 and Persona 4 have the ability of the Wild Card and can switch their Personas.