And both prisoners had to make their decision without knowing the other one's choice, and neither would be given a chance to change their decision afterward.
Draco had observed that if the two prisoners had been Death Eaters during the Wizarding War, the Dark Lord would have killed any traitors.
Harry had nodded and said that was one way to resolve the Prisoner's Dilemma - and in fact both Death Eaters would want there to be a Dark Lord for exactly that reason.
(Draco had asked Harry to stop and let him to think about this for a while before they continued. It had explained a lot about why Father and his friends had agreed to serve under a Dark Lord who often wasn't nice to them...)
In fact, Harry had said, this was pretty much the reason why people had governments - you might be better off if you stole from someone else, just like each prisoner would be individually better off if they defected in the Prisoner's Dilemma. But if everyone thought like that, the country would fall into chaos and everyone would be worse off, like what would happen if both prisoners defected. So people let themselves be ruled by governments, just like the Death Eaters had let themselves be ruled by the Dark Lord.
(Draco had asked Harry to stop again. Draco had always taken for granted that ambitious wizards put themselves in power because they wanted to rule, and people let themselves be ruled because they were scared little Hufflepuffs. And this, on reflection, still seemed true; but Harry's perspective was fascinating even if it was wrong.)
But, Harry had continued afterward, the fear of a third party punishing you was not the only possible reason to cooperate in the Prisoner's Dilemma.
Suppose, Harry had said, you were playing the game against a magically produced identical copy of yourself.
Draco had said that if there were two Dracos, of course neither Draco would want anything bad to happen to the other one, not to mention that no Malfoy would let himself become known as a traitor.
Harry had nodded again, and said that this was yet another solution to the Prisoner's Dilemma - people might cooperate because they cared about each other, or because they had senses of honor, or because they wanted to preserve their reputation. Indeed, Harry had said, it was rather difficult to construct a true Prisoner's Dilemma - in real life, people usually cared about the other person, or their honor or their reputation or a Dark Lord's punishment or something besides the prison sentences. But suppose the copy had been of someone completely selfish -
(Pansy Parkinson had been the example they'd used)
- so each Pansy only cared what happened to her and not to the other Pansy.
Given that this was all Pansy cared about... and that there was no Dark Lord... and Pansy wasn't worried about her reputation... and Pansy either had no sense of honor or didn't consider herself obligated to the other prisoner... then would the rational thing be for Pansy to cooperate, or defect?
Some people, Harry said, claimed that the rational thing to do was for Pansy to defect against her copy, but Harry, plus someone named Douglas Hofstadter, thought these people were wrong. Because, Harry had said, if Pansy defected - not at random, but for what seemed to her like rational reasons - then the other Pansy would think exactly the same way. Two identical copies wouldn't decide different things. So Pansy had to choose between a world in which both Pansies cooperated, or a world in which both Pansies defected, and she was better off if both copies cooperated. And if Harry had thought 'rational' people did defect in the Prisoner's Dilemma, then he wouldn't have done anything to spread that kind of 'rationality', because a country or a conspiracy full of 'rational' people would dissolve into chaos. You would tell your enemies about 'rationality'.
Which had all sounded reasonable at the time, but now the thought was occurring to Draco that...
"You said," Draco said, "that the rational solution to the Prisoner's Dilemma is to cooperate. But of course you would want me to believe that, wouldn't you?" And if Draco was fooled into cooperating, Harry would just say, Ha ha, betrayed you again! and laugh at him about it afterward.
"I wouldn't fake your lessons," Harry said seriously. "But I have to remind you, Draco, that I didn't say you should just automatically cooperate. Not on a true Prisoner's Dilemma like this one. What I said was that when you choose, you shouldn't think like you're choosing for just yourself, or like you're choosing for everyone. You should think like you're choosing for all the people who are similar enough to you that they'll probably do the same thing you do for the same reasons. And also choosing the predictions made by anyone who knows you well enough to predict you accurately, so that you never have to regret being rational because of the correct predictions that other people make about you - remind me to explain about Newcomb's Problem at some point. So the question you and I have to ask, Draco, is this: are we similar enough that we'll probably do the same thing whatever it is, making our decisions in mostly the same way? Or do we know each other well enough to predict each other, so that I can predict whether you'll cooperate or defect, and you can predict that I've decided to do the same thing I predict you'll do, because I know that you can predict me deciding that?"
...and Draco could not help but think that since he had to strain just to understand half of that, the answer was obviously 'No'.
"Yes," said Draco.
There was a pause.
"I see," said Harry, sounding disappointed. "Oh, well. I guess we'll have to think of some other way, then."
Draco hadn't thought that was going to work.
Draco and Harry talked about it back and forth. They had both agreed much earlier that what they did on the battlefield would not count as broken promises in real life - though Draco was a little angry about what Harry had done in Professor Quirrell's office, and said so.
But if the two of them couldn't rely on honor or friendship, that did leave the question of how to get their armies to work together on beating Sunshine, despite everything Granger might try to break them up. Professor Quirrell's rules didn't make it tempting to let Sunshine kill the other army's soldiers - that just increased the bar you had to pass yourself - but it did tempt each side to steal kills instead of acting like a single army would, or to shoot some of the other side's soldiers during the confusion of battle...
Hermione was walking back to Ravenclaw not really looking where she was going, her mind preoccupied with war and treachery and other age-inappropriate concepts, and she turned a corner and bumped straight into a grownup.
"Sorry," she said automatically, and then, entirely without thinking, "Eeeeek!"
"Don't worry, Miss Granger," said the cheerful smile, set beneath the twinkling eyes, and above the silver beard, of the HEADMASTER OF HOGWARTS. "You are quite forgiven."
Her gaze was helplessly locked on the kindly face of the most powerful wizard in the world, who was also the Chief Warlock, who was also the Supreme Mugwump, who had gone insane years ago from the stress of fighting the Dark Lord, and numerous other facts that were popping up into her mind one after the other while her throat went on making little embarrassing squeaks.
"In fact, Miss Granger," said Albus Percival Wulfric Brian Dumbledore, "it is quite lucky that we bumped into each other. Why, I was just now wondering curiously what the three of you were thinking of asking for your wishes..."
Saturday dawned bright and clear and with the students speaking in hushed voices, as though the first to shout might set off the explosion.
Draco had hoped that they would be fighting in the upper levels of Hogwarts again. Professor Quirrell had said that real fights were more likely to take place in cities than forests, and fighting inside schoolrooms and corridors was supposed to simulate that, with ribbons to mark the allowed areas. Dragon Army had done well in those fights.
Instead, just as Draco had feared, Professor Quirrell had come up with something special for this battle.
The battleground was the Hogwarts Lake.
And not in boats, either.
They were fighting underwater.
; The Giant Squid had been temporarily paralyzed; spells had been set in place to keep away the grindylows; Professor Quirrell had gone and talked to the merfolk; and all the soldiers had been issued potions of underwater action that allowed them to breathe, see clearly, talk to each other, and swim not quite as fast as a fast walk by kicking their legs.
A huge silver sphere hung in the center of the battleground, shining like a small underwater moon. It would help to provide a sense of direction - at first. The moon would slowly go into eclipse as the battle went on, and when it had gone entirely dark, the battle would end if it hadn't already.
War in water. You couldn't defend a perimeter, attackers could come at you from any direction, and even with the potion you couldn't see very far in the darkness of the lake.
And if you swam too far away from the action, you would start to glow after a while, and be easy to hunt down - ordinarily if an army scattered and ran instead of fighting, Professor Quirrell would just declare them defeated; but today they were working on a points system. Of course you still had some time before you started to glow, if you wanted to play assassin.
Dragon Army had been set low in the water at the start of the game; above and far away, the distant underwater moon shone. The murky water was mostly lit by Lumos Charms, though his soldiers would extinguish the lights as soon as they began maneuvers. There was no point in letting the enemy see you before you saw them.
Draco kicked his legs a few times, propelling him to a higher position from which he could gaze down at where his soldiers hovered in the water.
The conversations died down almost at once under Draco's icy glare, his soldiers looking up at him with gratifying expressions of fear and worry.
"Listen to me very carefully," said General Malfoy. His voice came out a little lower, a little burbly with bubbles, libsten to me vebwy caerbfully, but the sound traveled clearly. "There's only one way we can win this. We've got to march on Sunshine together with Chaos, and beat Sunshine. Then we fight it out with Potter and win. That's got to happen, understand? No matter what else goes on, that part has to happen that way -"
And Draco explained the plan he and Harry had come up with.
Astonished looks were exchanged among the soldiers.
"- and if any of your plots get in the way of that," finished Draco, "after we are out of the water, I will set you on fire."
There was a nervous chorus of yessirs.
"And everyone with secret orders, make sure you carry them out to the letter," said Draco.
Around half his soldiers openly nodded, and Draco marked them for death after he rose to power.
Of course all the private orders were fake, like one Dragon being told to offer a false traitor's commission to another Dragon, and the second Dragon being told in hushed confidence to report anything said by the first Dragon. Draco had told each Dragon that the whole war could depend on that one thing, and that he hoped they understood it was more important than the plans they'd previously made. With luck that would keep all the idiots happy, and maybe flush out a few spies to boot, if the reports didn't match the instructions.
Draco's real plan for winning against Chaos... well, it was simpler than the one he'd burned, but Father still wouldn't have liked it. Despite trying, though, Draco hadn't been able to think of anything better. It was a plot that couldn't possibly have worked against anyone except Harry Potter. In fact it had been Harry's plan originally, according to the traitor, though Draco had guessed that without being told. Draco and the traitor had just modified it a little...
Harry took a deep breath, feeling the water gurgle harmlessly in his lungs.
They'd fought in the forest, and he hadn't gotten a chance to say it.
They'd fought in the corridors of Hogwarts, and he hadn't gotten a chance to say it.
They'd fought in the air, broomsticks issued to every soldier, and it still hadn't made sense to say it.
Harry had thought he wouldn't ever get to say those words, not while he was still young enough for them to be real...
The Chaos Legionnaires were looking at Harry in puzzlement, as their general swam with his feet pointing up toward the distant light of the surface, and his head pointed down toward the murky depths.
"Why are you upside down?" the young commander shouted at his army, and began to explain how to fight after you abandoned the privileged orientation of gravity.
A hollow, booming bell echoed through the water, and on the instant, Zabini and Anthony and five other soldiers struck out downward, into the murky depths of the lake. Parvati Patil, the only Gryffindor in the group, turned her head back for a moment and gave them all a cheery wave as she dived; and after a moment, Scott and Matt did the same. The rest just sank and vanished.
General Granger swallowed a lump in her throat as she watched them go. She was risking everything on this, dividing her army instead of just trying to take as many enemy soldiers with them as possible.
The thing to realize, Zabini had told her, was that no army would move until they had a plan that let them expect victory. Sunshine couldn't just plan to win themselves, they had to make both other armies think they would win until it was too late.
Ernie and Ron still looked like they were in shock. Susan was gazing after the disappearing soldiers with a calculating look. Her army, what was left of it, just looked bewildered, traceries of light dappling on their uniforms as they all drifted just below the sunlit surface of the lake.
"Now what?" said Ron.
"Now we wait," said Hermione, loudly enough for all the soldiers to hear. It felt odd to talk with her mouth full of water, she kept feeling like she was committing some sort of horrible impoliteness at the dinner table and was about to drool all over herself. "All of us left here are going to get zapped, but that was going to happen anyway with Dragon and Chaos ganging up on us. We've just got to take as many of them with us as we can."
"I've got a plan," said one of her Sunshine Soldiers... Hannah, her voice had been a little hard to recognize at first. "It's like all complicated, but I know how we can get Dragon and Chaos to start fighting each other -"
"Me too!" said Fay. "I've got a plan too! See, Neville Longbottom is secretly on our side -"
"You were talking to Neville?" said Ernie. "That's not right, I was the one who -"
Daphne Greengrass and a couple of other Slytherins who hadn't gone with Zabini were giggling helplessly as the cries of "No, wait, I was the one who got Longbottom" erupted from one soldier after another.
Hermione just looked at them all wearily.
"Okay," said Hermione when it had all died down, "does everyone get it? All your plots were faked by the Chaos Legion, or maybe some by Dragon. Anyone who really wanted to betray Harry or Malfoy went straight to me or Zabini, not you. Just go ahead and compare notes on all your secret plots and you'll see it for yourselves." She might not be as good at plotting as Zabini, but she could always understand what all her officers told her, that was why Professor Quirrell had made her the general. "So don't bother trying to do any plots when the other armies get here. Just fight, okay? Please?"
"But," said Ernie with shock on his face, "Neville is in Hufflepuff! You're saying he lied to us?"
Daphne was laughing so hard and so helplessly that the exhalations had turned her upside down in the water.
"I'm not sure what Longbottom is," said Ron darkly, "but I don't think he's a Hufflepuff any more. Not now that Harry Potter's got to him."
"Do you know," said Susan, "I asked him that, and Neville told me he had become a Chaos Hufflepuff?"
"Anyway," said Hermione in a loud voice. "Zabini took with everyone who we thought was a spy, so in our army we can stop watching each other quite so hard now, I hope."
"Anthony was a spy?" yelled Ron.
"Parvati was a spy?" gasped Hannah.
"Parvati was totally a spy," said Daphne. "She shopped at the spy shoe store and wore spy lipstick, and someday she's going to marry a nice spy husband and have a
lot of little spies."
And then a gong sound echoed through the water, indicating that Sunshine had just scored two points.
This was shortly followed by the triple gong of Dragon losing a single point.
Traitors weren't allowed to kill generals, not after the disaster of the first battle in December when all three generals had been shot in the first minute. But with any luck...
"Aw," said Hermione. "It sounds like Mr. Crabbe is taking a little nap."
Like two shoals of fish, the armies swam along.
Neville Longbottom kicked his feet in slow, measured motions. Diving, always diving in whatever direction you happened to be moving. You wanted to show the enemy the smallest profile, present them with your head or your feet. So you were always diving, downward and head-first, and the enemy was always down.
Like every Chaos Legionnaire in the army, Neville's head was constantly rotating as he swam, looking up, down, around, to every side. Not just watching for Sunshine Soldiers, but watching for any sign that a Chaos Legionnaire had drawn their wand and was about to betray them. Usually traitors waited until the confusion of battle to make their move, but that early gong had put them all on guard.
...the truth was, Neville was feeling sad about that. In November he'd been a soldier in a united army, all of them pulling together and helping each other, and now they were all watching each other constantly for the first signs of betrayal. It might have been more fun for General Chaos, but it wasn't nearly as much fun for Neville.
Harry Potter and the Methods of Rationality Page 59