Picture the scene: I've just respawned and some kindly engineer has built a teleporter. Fantastic! I no longer have to walk to battles. I step on the teleporter, which is still recharging. Lo and behold, someone else on my team pushes their way onto the teleporter. At this point the teleporter becomes confused, and since it can only teleport one person at a time, a battle occurs in which the two of you fight for the ability to teleport first. I have found that I have about a 30% success rate in these conflicts, which I would like to increase. In short, how can I win teleporter battles? Is my success determined by where I stand, or else by who is more towards the centre of the teleporter? Or is it instead determined by something entirely different?
According to my brother who is almost at the magical 10.000 hours required for mastery in TF2, the distance between the center of the teleporter and the center of the character play an important role. This means if one is crouching, he will have better chances. He also noted that taunting might help too.
Disclaimer: These are not my observations and may lack any scientific rigour. Confirmation bias could also be present. Without the source code or input from a developer, it would take lots of experimentation.
As a developer myself, I would implement it by either distance (the closer you are the more likely you will be chosen), sequence (the first person to step onto it will be chosen), some internal ID (the character who joined sooner / later to the server will be chosen) or random (self explaining).
I have been unable to fully test any of this (outside of an acutal round), and I am only going off of my own 6000+ hours into the game.
As Zovits said, I tend to see it as based on how close the player is to the center of the teleporter. When I crouch and walk towards the center, I tend to be teleported sooner, and always before the people who
I shove off fall off (the newer players who do not do this tend to get pushed off becuase of how players clip with their own team, and becuase players who are crouching move much slower (so they get pushed off slower, letting the standing players get moved off)).
Activating a taunt is just a good way to prevent yourself from moving (from the center preferably). Becuase of an update, while you are taunting, you loose all momentum, and can not be moved due to teammates (an enemy can still move you with knockback).
Now, for some less agressive ways to win these "teleporter" battles.
If you are not an Engineer, you could switch to one and start creating a second teleporter. Besides making it so two people can travel at a time, it will unclog the other telelporter. Adding another teleporter is like adding another road to a location. The more roads, the less traffic.
Secondly, if you are a Scout, don't take the teleporter (if anyone else is in line). Scouts speed will allow hiim to easily just run there, and if using the Soda Popper, you can charge its Hype up.
The same thing for Scout can apply to a Medic along with his primary target. If a Medic is pocketing a Heavy, both the Heavy and the Medic could take the long route to the front lines, so by the time the Medic gets there, he will have more Ubercharge than if he took the Teleporter.
Much like the Scout/Medic + heal target, Soldier and Demoman can skip using the Teleporter all together. If you have some skill with rocket/sticky jumping, you could easily just explosive jump there instead of taking the teleporter, and then heal up. If using the Base-Jumper/Gunboats/Conch, you could easily ignore a lot of the damage you would take.
If you are a class with a speed boosting item (Power Jack, Glove of Running Urgently, Buffalo Steak (does not stack with GRU), or Crit a Cola), you could use that instead and just run there. And, who knows. While taking this "scenic" route, you could come across some enemies trying to sneak to your spawn. If everyone is too busy trying to use the Teleporter, a suprise Soldier, Heavy, Pyro, Demoman, or Spy could deal a lot of damage to the players, and even possible destroy the Teleporter.
In the end, the best way to win a Teleporter Battle is to either not participate in them at all, build more teleporters, or
shove other players off of them let other players fall off as you crouch walk towards the center.
Having played over 1000 hours of TF2 since its release, allow me to reveal The Secret.
Teleporters will Not Work if there is more than one person standing on it at a time, ever. As long as your shoving battle is going on, the teleporter is not going to teleport anyone, end of story.
The reason you see people managing to use it despite this is because the teleporter is just big enough to fit your character's feet, but not big enough for the larger upper body, meaning you will always be shoving someone away if you're standing on the teleporter. Whoever is shoved off completely does not get teleported, and whoever is standing alone on the teleporter does.
If you are a class that needs the teleporter most (Heavies, Medics, other slow classes) then back up against the person trying to steal your teleporter so that you shove them off. You should, assuming they fall off for a split second, be the one teleported away.
Ducking may give you a slight advantage, since it compresses your character model into a larger, squater size, and speed does not really affect how well you shove teammates. But I have not experienced this myself.