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So what kinds of stories can we put into big open worlds that the player can explore in any order, and in which travel is an integral part of the experience? Well, here's a handful:
Find the buried treasure. This is a low urgency game good for kids because the treasure's not going anywhere. The player runs all over town assembling clues. The clues don't necessarily have to be found in a specific order; they might be scattered pieces of a treasure map.
Find the buried treasure before somebody else does. Same story, but there's some pressure on the player. You can make this a simple race, or you can raise and lower the tension by having the enemy team try to sabotage the player.
If they succeed, then the player is under more pressure; if the player sabotages them instead, some of the pressure is off. (Be sure this doesn't create too much positive feedback, though.)
Find the time bomb. Obviously the most pressure of all, and tricky to pull off in open worlds, but not impossible. One way to give the player a little control over the pace is to endow the avatar with a limited amount of super-speed, like Superman or the Flash or Neo from The Matrix. The player can use the power at his discretion to buy himself a little more time.
Find people rather than objects, and of course this is made more complicated by the fact that people move around. Players can be bounty hunters looking for fugitives, private detectives looking for missing family members, counter-intelligence officers looking for spies, and so on.
Police procedural. This goes all the way back to the old Sierra On-Line Police Quest games. To build a watertight case, the police spend a great deal of time traveling around to visit witnesses, question the known associates of suspects, and look for physical evidence. Some evidence can be found in any order, while other evidence appears only after following a chain of clues.
You might need to keep your town small, since although travel is an intrinsic part of the job it's not terribly interesting. You could spice it up a bit by letting your cops deal with street crime or spot witnesses or suspects walking around. To keep it low-pressure, have players search for evidence with a suspect already in custody. To add pressure, let the suspect escape.
Infiltrate a large open area from any direction. Too many shooters put the player on a rail. It's cheaper that way, but it's less interesting for the player. In the current Afghan war, the NATO allies have air supremacy and helicopters, so they can put troops down anywhere outside a combat zone and let them walk into it from any direction they prefer. Mission planning involves examining aerial photographs and choosing an approach that looks good.
Escape through hostile territory from somewhere in the middle to the edge. In my game design workshops I often challenge one team to make a game about the Underground Railroad, the network of abolitionists that helped escaping slaves to freedom before the American civil war. Some of the real Underground Railroad routes were hundreds of miles long, and thus not convenient for a video game, but you get the idea.
Smuggling is about not just infiltrating or escaping, but doing both -- and often with an awkward cargo. During the long, long wars between Britain and France in the 18th and 19th centuries, trade between the two nations was technically cut off -- yet there was still plenty of brandy in Britain. The coastlines of both nations have numerous inlets that a warship can't get into, and the smugglers knew them all.
Like police, reporters do a lot of traveling, usually under deadline pressure, to gather information. So do spies.
Hunters naturally move around searching for game. Hunting doesn't ordinarily generate much of a story unless something unusual happens, but occasionally it does. As a kid I read two different stories in which wealthy (and therefore evil, apparently) big game hunters indulged a secret passion for hunting human beings -- specifically, the guides they had hired.
Root out the criminal gang. Professor Moriarty's tentacles are everywhere. No matter where Sherlock Holmes goes in London, he encounters evidence of Moriarty's dastardly deeds. But where are Moriarty's senior lieutenants, where is his headquarters, and where is the man himself?
Resistance. Another game idea I give out in the workshops requires the team to design a game about a resistance movement in an occupied country. They're not allowed to make it a shooter; the game has to be about sabotage and making the occupier's lives miserable without getting caught. This, too, can be spread over a wide space, with soldiers of the occupying army constantly searching for the resistance fighters and keeping the player under pressure.
There are plenty of other kinds of jobs or hobbies that routinely involve travel: fire fighters, electricity linemen, tornado hunters, restaurant critics... not all of these are necessarily suitable for video games, but it only takes a little imagination. I'm sure you can think of more.
To make an experience story-like, you have to avoid too many repetitive or random (unrelated) events. (See my column Dramatic Novelty in Games and Stories for more about that.) If you read a thriller set in World War II, it doesn't consist of shooting an endless parade of identical Nazis; every situation is unique. This means that your sandbox has to be full of all different kinds of things, not just a lot of the same thing.
This is probably the strongest argument against sandbox storytelling: it's expensive and a lot of work. But unlike rail games, if you construct the world carefully enough, the game will be highly replayable. Different paths through the world will offer different experiences. Nor do they need to have the same objective or ending.
For several hundred years the people of Rome gave their allegiance to one of four factions that supported chariot racing. The drivers wore colored clothing so people could tell them apart, and the factions were named the Reds, Whites, Blues, and Greens.
Suppose the player has just moved into Rome. He can join any group of supporters, just as we can support any sports team today -- but with a difference: the factions often rioted, and there were bloody fights in the streets. What this means in practice is that an NPC who belongs to the faction that the player chooses is an ally, but if the player replays the game and chooses a different faction, the same person is an enemy. No need to write two stories or design the character twice; drama naturally emerges from the situation itself.
In short I think sandbox storytelling both possible and fun. You'll need to fill the sandbox pretty full so as to offer plenty of dramatic opportunities (many sandboxes feel rather empty and sterile), and you'll have to decide how much pressure you want to put on the player and how you'll apply it. This may include using some time-dependent, player-independent plot events to keep things moving forward. The environment itself is also critical -- it has to be a place that the player really enjoys being, because he's going to spend a lot of time there.
There was a famous film noir called The Naked City that was later adapted into a TV show. At the end of the film, and every episode of the TV show, the narrator said, "There are 8 million stories in the Naked City. This has been one of them." Try building your own Naked City, and see how many stories you can get in.