An interesting thing that's been happening with local multiplayer videogames is the exploration of interactions that happen outside of the computer. I wrote a bit about this before when I released O. Wide range of stuff, from J.S. Joust to Fingle. I'm particularly interested in touchscreen multiplayer - devices like the ipad are ideal for playing games together, being portable enough to take places but large enough for more than one player to comfortably interact at the same time.
The key idea is that the software doesn't know whose finger is whose. This rules out some possible games you might want to make; often we would like to be able to tell. If you're to do a single-screen genre-RTS with each player dragging to move their own units, you're probably going to want to network it - if not you'll be getting in each other's way and interfering with each other's controls (whether by accident or intent). (Possibly you could constrain the design to work with each player only touching one side of the screen while sending units over to the other side, Shot Shot Shoot-style, but in this case beware of the "wandering hands" effect.) But we can embrace this interference in our designs, it enables a new type of game.
Now, I suspect the natural response of many game developers to this is to view it as a "gimmick", a cute but shallow feature. I believe that it's not a gimmick. Several fundamentally different games have already been made using this concept, and I suspect that more possibilities exist. This is a potentially rich vein of design to explore further.
Chicanery is one of the simplest possible games to use this concept. The digital side of the game offers no way to win, only a way to lose: letting go of your button. So players must somehow persuade their opponents to let go. I'm honestly pretty scared of playing it.
A lot of potential designs in this space end up feeling like "essentially the same game as Chicanery": the game is decided by a physical wrestling match, and victory in the digital realm is purely a formality. I would like to avoid this for the simple reason that we already have Chicanery.
Finger Battle doesn't explicitly use this concept. Each player presses their side of the screen as many times as possible; whoever presses the most times wins. There's no overlap in inputs, no benefit in touching your opponent's side of the screen - but of course you can convert it into Chicanery by trying to prevent your opponent touching the screen while still pressing yours. Again this is a design sink - a lot of potential games essentially reduce to "press stuff as fast as possible"; let's try to avoid it.
Ready Steady Bang also doesn't use this concept, but is worth mentioning because its idea shows up later as a component of larger systems. It's a quick-draw western-themed game; players compete to be the first to touch the screen after a signal "ready, steady, bang!". The important difference here is that it demands attention; players must react to something that happens on the screen in a way that they didn't in the previous two examples. What this means is that if you Chicanerise it there's still a chance that the physically weaker player can win - by paying closer attention and managing to sneak through and press at the right time despite their opponent restraining them. This is a significant enough effect that most players will not attempt to wrestle at all.
Next in the complexity hierarchy is a hypothetical game that I don't believe anyone has released (though I briefly prototyped it during O's development). Call it "Touchscreen Snap". Pieces appear on the screen (demanding attention), and players compete to be the first to touch them (indicated by dragging them to their side of the screen, since the program doesn't know who owns a finger). This results in some physical conflict, because even if players don't actively fight each other their hands will often collide as they race to grab the same pieces. Essentially this is Ready Steady Bang with overlapping input domains.
In Circulets, pieces appear in each player's colour on their side of the screen, and players compete to collect them as fast as possible (by dragging). It is mostly a variation on Finger Battle, with a small amount of extra attention required to drag pieces from random positions rather than blindly pressing. Additionally, an occasional green piece appears in the centre which is valuable to either player - like a momentary round of Touchscreen Snap.
(There appears to have been some attempt to deepen it by making pieces worth -1 point when collected by the wrong player, but since this is an effective loss of one point for both sides there is no incentive for either player to ever do it. I'm not sure what the intent here was, but it seems poorly thought out.)
Bloop is similar to Circulets, with each player tapping pieces of their own colour. Since pieces are colour-coded it's sufficient to just touch them - no need to drag to tell the program who you are. It's greatly enriched over Circulets by the pieces being randomly intermingled across the screen rather than reliably sorted on each side: significant attention is required to touch as much of your own colour as possible while avoiding touching the others. (Note that Bloop preceded Circulets chronologically; I've just ordered them here to suit my exposition.)
Bam fu is kind of a variation on Bloop, with each player pressing pieces of a particular colour across the screen. But rather than appearing randomly, the pieces are fixed in place and just change colour when pressed according to a predictable cyclic sequence. This makes the attention element much deeper - you can predict where your colour will show up by watching for when other players press theirs. It really comes into its own with more than two players, subtly strategising to manipulate the flow of colours towards you while preventing anyone else from winning.
O has the attention element of collecting randomly-appearing coloured pieces. But rather than the colours matching the players, they feed into a set-collection scoring system: each piece of the same colour you collect in sequence is worth an extra point. Collecting a different colour breaks your sequence, which demands extra attention and avoids some of the weaknesses of Touchscreen Snap, and also creates an incentive for your opponent to sometimes force pieces onto your side of the screen. O also gains some depth from basic physics - flinging balls across the screen, bouncing them off each other, holding them in place to block others.
Slamjet Stadium has a ball that both players want to collect, but avoids Touchscreen Snap-style hand-collisions by making that something you can only indirectly interact with. There's a second kind of piece which you fling at the ball to push it around, and very rarely do you have a first-order desire to touch the same piece, because you want to push the ball from opposite sides (there may be a second-order desire to prevent your opponent doing what they want, but that's a lesser effect).
Greedy Bankers starts with an already-complex single-player game and encourages just a bit of cross-screen interaction by valuing pieces higher if they've been moved across the centre of the screen.
Centrifeud has separated inputs most of the time, but is occasionally spiced up with a powerup asking players to touch their piece - like a momentary round of Bloop.
A Bastard (touchscreen version currently unreleased) has players navigating a maze with tank controls that creep across the screen and sometimes overlap - encouraging players to press their opponent's buttons as well as their own. It feels a lot like Glitch Tank (which was an inspiration) in that you must strike a balance between planning carefully and acting quickly - but your plans may involve moving your opponent as well as yourself.
I played an unreleased prototype by @grapefrukt of a game where players tap Pipe Mania-style tiles to rotate them, attempting to join up a network of pipes in a way advantageous to them and disrupt their opponent's network. Feels very different to any of these other games.
(Fingle deserves a mention too though it's not competitive. Physical interactions obstruct digital goals; the challenge comes from your fingers getting in each other's way. The game intentionally provokes physical contact.)
I hope it's clear from these examples that very different games already exist using the "players interact on the same screen" concept, and that there's likely to be room for more that we haven't found yet. They share some common challenges and constraints - in particular, a need for a strong attention cost if they're not to degenerate into Chicanery - and many of them share some common ideas, but they have very different characters. But even the most complex of these are very minimal: what happens if you try to build a more intricate strategic game along these lines? Possibly the inherent messiness puts an upper bound on complexity, but it's worth finding out.