Riddles are Fiction, actually

in which we once again confuse the map with the territory

Photo by Beth Macdonald on Unsplash

I.

You are a prisoner. One morning, a guard {Carlos} leads you out of your cell and down a hallway. At the end of the hallway there are two more guards {Alice, Bob} guarding two doors {A, B}. Carlos informs you that one of {Alice, Bob} always speaks the truth while the other one always lies. Also: behind one of the doors there is freedom, behind the other there waits the hangman. You are allowed to ask exactly one question, and pass through exactly one door.

You have seen this riddle before. You have seen the genre before. You understand that this riddle wants you to do a logic. You think about it and come up with something like:

Ask Alice if Bob would say that Door A leads to freedom

Does that work? You tell me. In any case you do not:

• Ask why you should trust Carlos to represent the situation accurately

• Ask how you ended up in prison

• Ask what kind of society this is that apparently is willing to pardon prisoners based on their ability to solve trivial logic puzzles …

• … or are they willing to hang otherwise innocents for the crime of not being able to solve trivial logic puzzles?

You think of all of those things, surely, but you understand that by questioning the setup, you are not playing along, not engaging with the riddle on its own terms.

Your free spirit gets the better of you. Boldly you open door A without having asked any question. You can see the sun in the open sky; smell the freedom. Alice pulls out a gun and shoots you. “Should have played by the rules.”

II.

You are a prisoner. One morning, a guard {Carlos} brings a pencil and a piece of paper to your cell. Carlos informs you that if you can draw a square with three lines, you will be free.

“Impossible”, you exclaim. Everybody knows that a square is made of four lines. “You are a prisoner of your own mind”, says Carlos (who prides himself on being a philosopher), “think better.”

You look at your cell door. At eye-level it has a small square window. The window is interrupted by three vertical bars. You have an epiphany. This riddle is a different genre. It is disguising as a geometry puzzle but actually it wants you to do a language game.

You draw a square the normal way, then draw three additional vertical lines. Coincidentally, the picture resembles the small window in your cell door. You can barely suppress your smug grin. “A square with three lines, as requested”, you blurt out. Carlos just stares at you. “Don’t you get it?” you ask. “That’s not a square”, says Carlos, “should have asked for a compass”.

You are brought in front of a judge. You are sentenced to a life-time of working in the corporate world where “thinking outside the box” in totally predictable ways amounting to language games is often confused with cleverness.

III.

You are a prisoner. One morning, a guard {Carlos} leads you out of your cell and down a hallway. At the end of the hallway there is a door and a carton of cigarettes. Very valuable on the prison’s black-market. The cigarettes that is. Carlos informs you that this particular prison has access to an infallible Oracle {Omega}. The door, by the way, leads to freedom.

While you were sleeping, Carlos and Omega had a chat. Carlos asked Omega to predict which of the following two options you would choose:

(a) Grab the cigarettes, then try the door.

(b) Try the door without first taking the cigarettes.

If Omega predicted you would choose (a), Carlos would lock the door.

If Omega predicted that you would choose (b), Carlos would leave the door unlocked.

You are not allowed to grab additional items after trying the door but you are allowed to keep any items you were holding when trying the door no matter what else happens.

If you find the door unlocked you may pass through it and be free. If you find the door locked, you will stay imprisoned forever.

You definitely have seen this one before. Veritasium did a video on it and it was a whole thing.

If Omega is indeed infallible, as claimed by Carlos, then whatever you are going to do now is, by construction, what Omega already predicted. So if you ignore the cigarettes and try the door that must have been what Omega predicted, Carlos will have left the door unlocked and you walk out to your freedom. You would start a new life and never think about the carton of cigarettes you left behind.

Your free spirit gets the better of you. The door already is locked or unlocked. Nothing you do now can possibly change the state of the door. Boldly you grab the cigarettes. Cigarettes in hand and smug grin on your face you try the door, which of course is locked. Which part of “infallible Oracle” did you not understand?

IV.

Much has been written about Newcomb’s paradox and any attempt to replicate all of that here would be besides the point. I recommend this review by Benford and Wolpert. Let g be Omega’s prediction and y be your choice, the question according to Benford and Wolpert is whether Omega chooses the conditional distribution P(g | y) or the unconditional distribution P(g). We might say, still according to Benford and Wolpert, that Omega being infallible (they say: “perfectly accurate”) means that Omega chooses P(g|y) = 1 iff g = y, zero otherwise. If Omega gets to choose P(g|y) that is. If Omega gets to choose P(g) what it being infallible means is anybody’s guess.

The thing is: the text of the riddle explicitly states that Omega is an infallible Oracle. And what else could it mean for Omega to be infallible? As I have hinted at in my own narrative, if you try the door first without taking the cigarettes (one-box in Newcomb lingo), the door will be open. Because if it would not be open that would be a contradiction to the assumption that Omega correctly predicted your action. By a symmetric argument, if you take the cigarettes first (two-box in Newcomb lingo), the door will be closed. The alternative would be an incorrect prediction.

It is noteworthy that this interpretation of “infallible” that is both intuitive and supported by the text implies one-boxing to be the correct strategy.

If we follow any of the many arguments for two-boxing, we choose to emphasise other aspects of the riddle over this interpretation of “infallible” — this seems to be Benford and Wolpert’s implicit stance. Or we choose to emphasise high level concepts like “free will” and “causality” — which Benford and Wolpert reject but other authors explicitly do. We might learn something interesting from analysing where that leads us. But we learn that at the price of softening the premise of the text.

I can see many reasons why it would not be plausible to assume an infallible oracle as outlined above. But, I cannot help myself, the text says what the text says. Strange logical prisons where guards hang or pardon people based on an undergrad understanding of logic also are not remotely plausible. Yet we understand that we are asked to follow the implausible premise to its logical conclusion. Conversely we must understand that by following a riddle to its logical conclusion we do not endorse its narrative as realistically plausible.

V.

You are a prisoner. One morning, a guard {Carlos} leads you out of your cell and down a hallway. At the end of the hallway there are two more guards {Alice, Bob} guarding two doors {A, B}. Carlos informs you that one of {Alice, Bob} always speaks the truth while the other one always lies. Also: behind one of the doors there is freedom, behind the other there waits the hangman. You are allowed to ask exactly one question, and pass through exactly one door.

Glad for a second chance — what is this? A recurring nightmare? — you ask Alice: “Would Bob say that door A leads to freedom?”

Alice politely replies: “The moon is made of cheese”. That sure is a lie. You wake up in your cell.

VI.

You are a prisoner. One morning, a guard {Carlos} brings a pencil and a piece of paper to your cell. Carlos informs you that the prison is engaging in a program of mass amnesty. You have to write “red” or “blue” on the piece of paper. All the other prisoners will be offered the exact same deal. The outcome will be determined as follows:

(a) everyone who wrote “red” will be allowed to walk freely no matter what

(b) the “blue” notes will be tallied

(ba) more than 50% of all prisoners wrote “blue” → everyone goes free

(bb) 50% or less of all prisoners wrote “blue” → everyone who wrote “blue” will be hanged, everyone who wrote “red” will walk free as per (a).

You have seen this riddle before. You have seen the genre before. You understand that this riddle wants you to do a decision theory.

Assume everyone votes red. If you vote blue, you will be hanged, accomplishing nothing. If you vote red, you will harm nobody, because everyone else will have come to the same conclusion, will also be voting red and will be safe no matter what you do. You should vote red, everybody should vote red, everybody is going to be safe. There is literally no downside to everybody voting red. All is going to be well.

“Red”, you say with the aura of a prisoner fully expecting to be set free any moment from now.

“You are a prisoner of your own mind”, says Carlos (who prides himself on being a philosopher), “think better.” He shows you a piece of paper, presumably from another prisoner, with the word “blue” written on it. “Should have shown some compassion”.

You are brought in front of a judge. You are sentenced to a life-time of fruitless internet debate.

VII.

We have seen that such riddles work within the conventions of their genre. As mentioned, team red wants the genre of the last riddle to be decision theory. Accepting that convention, the equilibrium where everybody votes red is supported by the text and inherently stable.

The riddle has been engineered to abstract away every condition under which cooperation would be required. There is no scarcity (red works for everyone, unboundedly). There is no information asymmetry. There is no power differential. There is no ongoing relationship between prisoners. There is no history of mutual obligation. The choice is one-shot, anonymous, costless to the red voter, and uncoordinated. In a world built like that, red is correct — trivially, almost tautologically. In this sense everybody voting red is the logical conclusion of the riddle.

Team red tends to notice this construction but treat it as a feature. The payoff structure looks like a coordination game but is actually constructed such that the individually dominant option is also a social equilibrium. From the red point-of-view this is neat — and the blues are to be ridiculed for confusing it with the coordination problem it superficially resembles.

But outside of the red POV, this looks like motivated reasoning. Team red wants to believe that if everyone saves themselves everyone will be safe. The implication: If you don’t save yourself, that’s on you. Could have voted red. And so the fiction of the riddle is chosen to justify what team red wants to believe anyway. We might call such a fiction red-coded. It breaks down at contact with reality. In the real world, there is no simple good option that works for everyone.

The blue argument starts, stands, and falls with the observation that some people will vote blue. Those people, it follows, will die if not enough people vote blue. This observation deserves our compassion. We are morally obliged to vote blue to save the other blue voters. If we do not vote blue, by definition this will increase the chance that the quorum fails and people will die. Therefore, voting red equals being complicit in the death of those people. It is evil. Therefore, we can learn what kind of person someone is, by observing the way they vote.

And here we observe a strange gap in the discourse. The fiction of the riddle does not support the claim that some people will vote blue. Why would they? Everyone votes red is a stable equilibrium within the fiction. The claim instead has to be imported from the real world, analytically or empirically. Doing so does not invalidate the red analysis within the original fiction. It does not make it immoral to follow the original fiction to conclusion. Instead it puts into question the fiction itself.

I am desperately waiting for team blue to make this point. To call out team red on the utter naivite of their premises. To dismiss the riddle as so far from reality that nothing of value can be learned from it about the real world. There is a difference, as we have shown, between following the premise of a riddle to its logical conclusion and endorsing the premise as a realistic narrative. Stop fighting the conclusions (“immoral!”) and start questioning the premises.

But this is not happening. Team blue insists that it can win this within the framework of the red fiction — now there’s a parable after all.

VIII.

You are a prisoner. One morning, a guard {Carlos} leads you out of your cell and down a hallway. At the end of the hallway there are two more guards {Alice, Bob} guarding two doors {A, B}. Also there is five other prisoners waiting in line at door A. Carlos informs you, that those five prisoners were unjustly held captive and are now scheduled for release. But, he explains, there has been a mixup and somehow your name has also made it onto the release list. Would you please stand in front of door B?

You try door B but it is locked. There is a, ahem catch. Door A and door B are coupled in such a way that you can flip a switch, lock door A and unlock door B. Flip the switch and the five other prisoners will not be able to leave, but you will allowed to exit through door B and into freedom. Don’t flip the switch and go back to your cell.

You have seen the genre before. You understand that this riddle wants you to do an ethics. No riddle has ever been easier. All is going to be well.

Fade to Black.