This is the most typical story base used in many of the earlier, now-critically acclaimed games of the genre. Street Fighter, Tekken, The King of Fighters are the most well-known examples of how this kind of story is executed. In a formalized tournament, there is an organizer, most often the head of a large operation (usually a criminal syndicate or powerful corporation) The organizer creates a tournament using a singular grand prize, something that we all want for different reasons. Typically, the grand prize is money, or something that would facilitate money and notoriety. Along with this direct prize is an indirect prize in the form of bragging rights, the ability to say "I took on the best and I'm the last one standing."
The main strength of the type of story base is that it makes the task of creating a story would serve multiple character more doable. All of the characters' stories and motivations can easily be explained through the tournament's grand prize. Gen Fu has a sick granddaughter, so he enters the Dead or Alive Tournament to help with the hospital bills. Marshall Law enters Tekken because wants to open a restaurant. Others like Bruce Irvin and Balrog are hired by their tournament's organizer and compete in the tournament as part of the job of bodyguard. Obviously not everyone fights for the money and that's fine, too. Ryu and Paul Phoenix enter their respective tournaments seeking worthy challenges. Characters like Chun-Li and Terry Bogard seek revenge against the organizer of their respective tournaments for past wrongdoings.
The primary weakness of the formalized tournament is that it limits the kind of stories you can tell, and sometimes it can create unrealized scenarios to otherwise plausible story lines. With all due respect to Ryu and Paul Phoenix, fighting simply to prove you're the best is insipid and boring. If Marshall Law wanting to open a restaurant is admirable, but why fight in a tournament? Go apply for a business loan. Start a Kickstarter fund. Knock over a bank. Why fight? Gen Fu's granddaughter has a "terrible disease" and needs the money for medical bills. Again, admirable, but shallow and highly implausible. If you're that desperate, use your awesome fighting skills and pull a John Q. Why fight? Where's the motivation to fight coming from? If it's for revenge, why participate in a formalized tournament that has rules that may restrict you from exacting that revenge? If your an assassin like Nina Williams, why would you out yourself as a professional killer by participating in the very tournament that you target is hosting?
I read a book called 20 Master Plots and How To Build Them by Ronald Tobias. Of the list of plots in the book, one of them is called the Pursuit Plot, which the book describes as "the literary version of hide-and-seek" (Tobias, page 79).
For lack of a better term, The Hunt is "yin" to the Formalized Tournament's "yang". Whereas a tournament would lend itself to structure and organization, The Hunt deals more in the chaotic. There is a grand prize. However, there's no need to fighter through all of the tough opponents in order to reach it. If you want it, you better find it before the others do. The best examples of the Hunt are Skullgirls and Soul Caliber. While there are examples to the contrary, most fighting game stories that use The Hunt as its story base don't have a true antagonist pulling the strings. This isn't to say that the bad guys don't exist. They do. But they're goal isn't fighters. It's the prize. BlazBlue is unique to Hunt stories in that its main character, Ragna, also serves as the grand prize everyone is hunting for.
The characters and their stories aren't unified by a tournament, but by the prize itself, creating a scavenger hunt of sorts.
Save The World
When discussing the Formalized Tournament story base, some of you might have noticed that I didn't include Mortal Kombat in the list of examples that best fight the category. The reason is simple: Mortal Kombat is not a tournament story at all. It masquerades as a tournament story when, in fact, it's the same type of action/adventure story that we've had since the beginning of the storytelling craft. The MK series is about saving the world. Simply put, this kind of tournament story is used to facilitate the central character's objective of stopping the primary villain from destroying or ruling the planet (or humanity or the universe). Examples include the aforementioned Mortal Kombat series and, to a smaller extent, several incarnations of the King of Fighters series.