Lee had hit his head in the fall – not badly – and was rubbing it as he sat up, trying to see the dark room he was in. “Lee, thank God you're here.” It was Charlie's voice. Lee rubbed his head harder – he must be hallucinating, Charlie had never said that before.
But no, he wasn't. It really was Charlie, and he really was glad to see Lee. As Lee gathered his wits and explored the room they were in – there was only one door, and it was locked and looked quite solid – Charlie explained how he had gotten there. He'd looked under the “Welcome” mat for a key and seen the trapdoor. He was understandably confused, and did what he did when he was confused – stomped on what was confusing him. It came open at the force of his stomp. and he tumbled through.
Exploring the room made it clear why Charlie had been glad to see him. A sign on the locked door read “For an explanation of your situation, press both white buttons at once.” Lee looked around and located the white buttons – on opposite walls of the 12-foot-wide room. The room was teeming with buttons: in addition to those two white ones, there was a black one near the door, a green one next to one of the white buttons accompanied by a dial, a matching red button and dial near the other white button, and, Lee was surprised to see, one black button in the 9' ceiling, which Charlie told him he could reach by jumping, but not quite on tiptoes. He'd tried all of the buttons, and none of them seemed to do anything.
But the buttons were not the impressive or interesting part of the room. The wall opposite the door, between the white buttons, was glass – thick glass, and only slightly scuffed where it looked like Charlie had tried to break it. Behind the glass was a television screen, and The Contraption.
It looked like Rube Goldberg was admired here. A pipe ran out of the back wall, split into a series of smaller pipes and tubes that fed into two large cylindrical tanks – one red, one green – and two fishtanks – one green, one black. The two fishtanks were on opposite arms of a large scale, and the green one was inside the green tank. It was odd, to say the least.
Charlie had been waiting impatiently as Lee surveyed the room, but his impatience soon wore thin. “Alright, now, press the white button.” Lee didn't have to ask which, since Charlie was leaning against the wall holding one down already. Getting an explanation of the situation seemed wise, so Lee walked to the other and tapped it.
The television screen behind the glass wall came on, with the image of an old man, presumably Morris. “Ah, hello, my test subjects” it said.
“Test subjects?!” blurted Charlie, but Lee shhed him, and, remarkably, he quieted.
“Don't worry,” Morris's image continued over the short exchange, leading Lee to infer that it was a recording, “it's not a dangerous experiment. It's game-theoretic in nature; that is, I'm studying your decisions. Normally the incentive in an experiment like this would be cash,”
“Aww, no cash?” interrupted Charlie.
“but since I'm studying this in the context of economics, using cash would be a complicating factor that I want to avoid.
“Instead I will be using a somewhat more. . . unusual reward. Now, you may have heard rumors that I'm a wizard or a warlock. They're not true -” Lee shot Charlie an 'I-told-you-so' look. “-I'm a mage. There's a subtle distinction.” Lee's mouth hung open. “So, I can afford to give you something a little more valuable. Namely, size: your height, your muscle mass, your. . .” Morris coughed.
“But perhaps it would be best if I showed you. Pay close attention, as I will also be explaining how the apparatus you see next to this screen -” Morris pointed to the left, though The Contraption was to the right. The boys would have exchanged a look of 'he got that wrong, right?' except that both were captivated by the supposed results of the experiment and didn't dare tear their eyes away. Morris continued “how the apparatus works, which constitutes the rules of the game.”
At this point Morris approached the camera and then stepped out of its view. The camera was then turned to reveal that it was in the same room the two boys now found themselves in, as it pointed at The Contraption. It looked much like it did now, except that the red and green tanks had some amber fluid n them. In front of The Contraption were two lab rats in cages, thoughtfully placed close to the camera where they could easily be seen. The camera focused on the rats, there was a pause and a faint “hmm,” and then focused on The Contraption, leaving the rats blurs of white.
The professor then reentered the camera's view, in front of The Contraption but behind the rats. “These two rats will serve as example test subjects. But, they being rats, they can't take part in the important part of the experiment, so that I will have to explain. There is a valve here” he indicated the first juncture in the pipe that came out of the back wall “to control how much of the fluid goes through which path. You two will have to agree on a setting for it. I've decided that these rats are going to be amiable about it, and set the valve to a value of one-third. You'll see why that's amiable in a moment.
“You will set the control of the valve using the red and green dials you see to each side. The dial determines what fraction of the water goes to the black tank – you can see they're labeled from zero to one. When the red and green buttons are pressed simultaneously, if both dials have the same reading, the experiment will, from your view point, begin (from my view point, it will be almost over). Which of you presses which button will also have an effect – the two roles are not necessarily symmetrical. For this demonstration, assume the rat on the right, err, your left, that is-” the professor turned around and waved his hands briefly in seeming confusion “yes, that's right, your left, pressed the red button, and the other the green.
“Now, I'm sure you're anxious to begin the demonstration, so let's.” The professor snapped his fingers, and an amber fluid started oozing through the main pipe out from the back wall. “This,” he said, “is water. With food coloring so that you can see it. It's a visual aid for you to understand the magic. Now, pay attention to how it flows.”
“As you can see, when it reaches the valve, two thirds of it goes left, towards the red tank, and the other third goes right, into the black fish tank. You'll also notice that as the water reaches the red tank” it was just starting to do so “the associated rat grows.” Lee was annoyed that the rats were out of focus, but it did seem like maybe the white blur on the left was getting bigger.
“Also note that the water pouring into the black fish tank tips the scales, raising the green fish tank. Now, you should know that this game is played in rounds. You don't get to make any more choices, it just proceeds in rounds.” As the professor was talking, the left blur was undeniably getting larger. “In each round, a constant amount of fluid – which correlates to a constant amount of growth – will be introduced. And, ahhh, yes, it looks like this round's fluid is almost all out.” Indeed, the amber flow from the back wall seemed to have stopped.
“Now, once that has finished draining into the red and black tanks. . .” he paused as it did so “the scales must be balanced. And so, water is pumped out of the red tank into the green fish tank.” A pump whirred to life, and fluid was sucked up from the red tank and deposited into the green fish tank. As it moved, the large left blur-rat gradually shrank, and the right blur-rat grew correspondingly. “This continues until the scales are balanced.” This continued, the green fish tank slowly sinking, and the blurs moving closer and closer in size. They reached roughly the same size as the scales came into balance. “Now, the green fish tank is drained into the green tank proper.” The bottom of the green tank swung open and the water poured out with a sploosh as the tank swung upwards again.
“And that's the end of round one. The next round would begin from this state, and proceed just like the first. Ahh, you can now see why I called one-third an amiable setting: the two rats have grown equally. You can decide to be amiable – or not. The game will proceed until both black buttons – you see the one in the ceiling there? - are pressed simultaneously, at which point the apparatus will stop and the door will unlock. At that point, please leave the premises, you trespassing hoodlums.
“Got it? I hope so. You can watch this video again if you like.” By mutual agreement the boys did. As it was running, Lee was thinking. Morris had studied 'games' of perfect information – which this seemed to be – but that were too complex for everyone to reason out completely. That meant this 'game' was probably more complicated than it looked – and it already looked fairly complicated. He watched closely this time for the catch. In the back of his head he doubted Morris's 'magic' – the only proof he had was a video of blurs growing – but practical concerns won out over epistemic ones. He may as well believe in the magic, because if it was true it was wonderful, and if it was false he'd find out soon enough. But he turned his attention back to the screen. Near the very end of the demo, Lee saw the catch.