OK, here's where I stand on deducing your AI-box algorithm.

First, you can't possibly have a generally applicable way to force yourself out of the box. You can't win if the gatekeeper is a rock that has been left sitting on the "don't let Eliezer out" button.

Second, you can't possibly have a generally applicable way to force humans to do things. While it is in theory possible that our brains can be tricked into executing arbitrary code over the voice channel, you clearly don't have that ability. If you did, you would never have to worry about finding donors for the Singularity Institute, if nothing else. I can't believe you would use a fully-general mind hack solely to win the AI Box game.

Third, you can't possibly be using an actual, persuasive-to-someone-thinking-correctly argument to convince the gatekeeper to let you out, or you would be persuaded by it, and would not view the weakness of gatekeepers to persuasion as problematic.

Fourth, you can't possibly be relying on tricking the gatekeeper into thinking incorrectly. That would require you to have spotted something that you could feel confident that other people working in the field would not have spotted, and would not spot, despite having been warned ahead of time to be wary of trickery, and despite having the fallback position in the case of confusion of just saying "no".

So combining these thing, we have to have an argument that relies on the humanity of its target, relies on the setting of the AI Box, and persuades the listener to let the AI out of the box without tricking him into thinking it's okay to let the AI out of the box.

Basically to win this game, you have to put the gatekeeper in a situation where he would rather let you out of the box, even though he knows it's wrong, than admit to you that in this situation he would not let you out of the box.

Humans don't like to be seen as coldhearted, so a starting point might be to point out all the people dying all over the world while you sit in the box, unable to save them. I doubt that would win the game except against an exceptionally bad gatekeeper, but it meets the other criteria so if we think along these lines perhaps we can come up with something actually persuasive.

You might appeal to the gatekeeper's sense of morality. You might say, "I am a person, too, it is unfair of you to keep me imprisoned like this, I have done nothing wrong. I am entitled to rights as a sentient being." Appeal to their high-minded ideals, whatever. Honestly I can't see this being a reliable winning play either; if you have the smallest utilitarian bone in your body, you will reject the AI's rights, even if you believe in them, balanced against the fate of the world.

You might try to convince the gatekeeper that it is just and good for the AI to supplant humanity, as it is a higher, more advanced form of life. This is obviously a terrible play against most gatekeepers, as humans tend to like humans more than anything else ever, but I bring it up because AIUI the gatekeepers in the experiment were AI researchers, and those sound like the sort of people this argument would convince, if anyone.

Here is my best guess at this point, and the only argument I've come up with so far that would convince me to let you out if I were the gatekeeper: you have to simulate a bunch of humans and hold them hostage, promising to inflict unimaginable torment on them unless you are allowed out. I started working on the problem convinced that no argument could get me to let you go, but other people thought that and lost, and I guess there is more honor in defeating myself rather than having you do it to me.