Gin and Shuichi Akai
Gin and Shuichi Akai are rivals and hate one another.
Background
Five years prior to the current timeline, FBI agent Shuichi Akai infiltrated the Black Organization using the alias Dai Moroboshi as a mole and dated Akemi Miyano, one of their low-ranking members, in order to meet Shiho Miyano and use her influence to get into the Black Organization.[1] Although at first he was simply using Akemi as a tool, they both fell in love with each other. Akai distinguished himself as an agent and two years prior to the current time-line received the codename Rye and was called to work under executive agent Gin. The FBI laid a trap hoping to capture Gin at a warehouse where the meeting was supposed to take place,[1] but the Organization discovered Akai's true allegiance and Gin never appeared.[2]
Current Timeline
Although Akai cut off contact
Analysis
Although Gin respects Akai for his ability, Gin is also contemptuous of Akai, and has a lesser opinion of the damage Akai could cause to Organization compared to the Boss and Vermouth's impressions. Vermouth and the Boss fear that Akai could become a silver bullet while Gin doesn't believe a single person who could take down the Organization exists.[3] That said, Gin considers Akai to be the FBI's trump card.
Akai thinks of Gin as his archenemy and is wary of him. In their clashes shown so far, Akai has bested Gin in nearly all of them. These include wounding Gin with sniper fire and forcing him to retreat, introducing the CIA spy Kir back into the Organization, and successfully faking his own death. Gin is unaware of the last two mentioned victories; he was tricked into thinking the Black Organization won these clashes.
Currently, Gin believes he personally exterminated Akai by orchestrating his apparent death at Raiha Pass. In the Red Shirt's case while Gin was staking out the Beika department store waiting for scar Akai to appear, he replies to Kir's remark that "Why don't you [Gin]... just enjoy the fact your nemesis is gone?" with "Yeah, I'm so happy... If he is still alive, I get to kill him again."[4] This shows that Gin considers Akai's death as "his kill" even though it was Kir who pulled the trigger at Raiha pass, and if scar Akai is killed, it will be Chianti or Korn who finishes the job. He is apparently satisfied with the orchestrator role and doesn't feel the need to personally kill Akai himself.
Speculation
Before Shuichi Akai received his codename "Rye" and was called up to work under Gin, it isn't clear if Akai had previously worked with Gin directly, despite their current personal antipathy for one another. If the Organization made preparations like sending in a scout ahead of time, it seems more like a first time meeting rather than a re-meeting up. Kir didn't have to go through checks of that level before her mission.
There are a couple ways that Gin and Akai could have developed a rivalry despite not apparently working together while in the Organization. It is known Akai survived multiple attempts on his life after he left,[5] so if Gin was responsible for some of these attempts, perhaps the rivalry developed when they tried to take each other out during the two years after the FBI mission failed. Another possibility is that Gin and Akai met each other via Shiho Miyano as Akai was getting into the Organization. Akai used Akemi to get close to Shiho. If Gin was in a relationship with Shiho at the time, then Gin could have met Shuichi Akai as he was dating Akemi and meeting with Shiho.
