The Blind Dice-Thrower

Imagine a centipede game is being played by perfectly rational agents. As has been well-established, the first player reasons using backward induction, and also reasons that their opponent will use backward induction, and therefore to maximise their own utility they are forced to end the game immediately, taking 1 util for themselves and none for their opponent, even when it was possible that they receive (about) 100 utils each had they co-operated.

Here's the question. If the first player didn't end the game immediately, the second player would have good evidence that the first player plans to co-operate for some time to come. If the first player made the co-operative decision on the first turn, then they will on most of the rest since they are identical scenarios. Therefore the second player ought also to co-operate. Even if the other player decides to end the game prematurely in the future for no discernible reason, you still make more than you would by playing an ultra-defensive strategy. However there still is some kind of paradox looming, as the end-game will still be in the minds of the players. I haven't worked this one out yet.

Centipede could be one of those rare games that turns up in game theory where it pains the player to be rational, just as Lewis thought the Newcomb's game paid off those less rational 1-boxers. One problem with this game is about knowledge. In the centipede game, your opponent knows that you are "rational", therefore you cannot choose to play irrationally because it is a given fact already how you will play. Hence decision theory of this sort becomes ironically indecisive and determined. We need to use the concept of common belief rather than common knowledge. In Centipede, if you play the co-operative move you also falsify the other player's belief, which will cause them to play differently. Not irrationally. It just happens that their behaviour is identical to the behaviour of an irrational agent, but not their thought process. This is a phenomenon that is observed very frequently in the world of professional poker.

Suppose CBR, common belief in rationality, and suppose that Backwards Induction is essential for rationality. My earliest instinct then is that, given both players have CBR, if the first chooses to play co-operatively, the second player no longer has CBR, i.e. he comes to believe that the first player cannot be using backwards-induction and is therefore irrational. However, this cannot be the case. There are players who do use backwards induction but who still co-operate because they expect or suspect their opponent of being able to sympathise with them. So when the first player co-operates, essentially the other player still cannot tell whether the first is using BI or not. Like in poker, when an opponent makes a play you can only tell that they are not an intermediate, they could still either be an expert or a beginner.

I'm just reading about this supposed paradox that apparently results from Jeffrey's logic of decision. It's a lot like the Newcomb's Paradox in that it raises the issue of whether causal back-tracking is metaphysically supported.

The problem is this. You are in the cinema and want some popcorn. What you prefer most is for you to go to the lobby, there to be popcorn, you buy some, return to your seat and enjoy it, even though you will miss some of the film. The next best option is if you decide not to get popcorn and there really wasn't any there, so at least you didn't miss the film. The next best alternative is if you go down to the lobby and there is no popcorn so you miss some of the film. The worst option is to miss out on popcorn by deciding not to leave the cinema, and find on the way out that there was popcorn there all along.

At the same time, you are fairly sure that there is no popcorn out there, because there very rarely is. Also, you're certain that if there was popcorn out there, the owners of the cinema would project onto the screen a subliminal message, reading "POPCORN!!!" every few seconds. You consider yourself to be suggestible, so if this were occurring you would definitely choose to go and look in the lobby for popcorn.

Sobel argues that of all the alternatives, the probability that popcorn being in the lobby given that you look for it, and the probability that there will be no popcorn given that you don't look for it, are both close to 1. As the first of these alternatives has the highest expected utility, using Jeffrey's conditional decision theory we ought to go and look for the popcorn, which would otherwise seem irrational given that we already think the probability of popcorn being there is suitably low.

This is similar to the Newcomb's problem because in both cases we suppose that our future actions could causally determine a past action to come to pass or be reversed. We know that the popcorn is either already out there or not. Our deciding to go and look for it should not affect it's being there. This is my conclusion and that of the 2-boxers, a group I am proud to share with the great David Lewis. I reject Sobel's argument for the following reason:

We are told simply that if they had popcorn then they would be displaying the subliminal message, and we would decide to look for popcorn. But presumably we could decide to look for popcorn independently of the subliminal message, which it seems is a lot like what's happening now to the 1-boxers. Can we be sure that we are in an identical mental state now to how we would have been had the subliminal message been aired, i.e. that it would cause us to invoke decision theory and make the reasoning we are currently using? Let's just say yes for the purpose of the problem. Now, if it were also established that these thoughts of decision theory would not have entered my head unless the subliminal message was displayed, then I would be certain (or at least highly suspect) that there is popcorn outside and go looking for it. If it isn't impossible, i.e. if I could independently think of that reasoning without the message, then I would need to know the frequency with which my thought is caused by the message and the frequency with which my thought is caused by my own curiosity or urge for popcorn. If I don't know the frequencies then I probably end up making a principle of indifference between them, assigning them equally 1/2. I therefore only go outside to get the popcorn if my utility for "leave and get popcorn" is more than double my utility for "stay and miss out on popcorn".

That is my current conclusion. I'll red what Sobel has to say about it and possibly edit this post later to account for belief revision.

I think my favourite story during my childhood was that depicted in the Iliad, of the Greek invasion of Troy, focusing on Odysseus, Agamemmnon, Menelaus, Achilles and Diomedes, and Hector and Paris. When I was young we used to listen to a story in the car when we were driving on holiday. There were two audiobooks, "Odysseus: The Hero Of Them All" and "Odysseus: The Journey Through Hell", co-written and narrated by Tony Robinson. For about 5 years I tried to find these on the internet as we had had them from the library, and I finally managed to track them down and buy two copies of them, although I haven't re-listened to them yet because I'm waiting until I can convert them to a digital format.

But these stories were one of the best parts of my childhood, the story of how Odysseus was called away from Ithica to fight in a war he didn't want to fight in, and the events of that war and his unbelievable journey to get home after the war. Tony Robinson managed to express comedy and tragedy perfectly, expressing deep sorrow for the deaths of Odysseus' comrades, the predicament of Penelope and the suitors, and Odysseus' unceasing fury when he returns.

Since then I've read or watched at least three other depictions of the Trojan war. Firstly, there is the Iliad, wrriten by Homer, which the others are mostly based on. It's a tough read, mostly about constant warfare, talking about one hero killing another and very hard to follow given the number of names involved. There's one section close to the start that simply lists all the names of the Greek heroes and contingents sailing for Troy which lasts several pages. Secondly, there's Shakespeare's play "Troilus and Cressida", which puts forward some strange differences to the original. For example, Pandarus is not an archer but a merchant, and Troilus is considered to be a good fighter who at one point fights off both Diomedes and Menelaus (I think).

Thirdly, there is the quite new film called Troy, starring Brad Pitt and Eric Bana. I've actually seen several film and TV adaptations of the Trojan war, but that one is by far the best. I now cannot even imagine the character of Achilles without thinking of Brad Pitt. Achilles was thought of as being the most beautiful man alive, and so is Brad Pitt. What I particularly like about the film is how you are never sure which side you prefer to win. The Iliad is extremely pro-Greek (and "Hellenocentric"?). You only ever see Hector dressed for full battle. Quite often you only hear about him on some other side of the battlefield cutting swathes through the Greeks. But the worse thing about the Iliad depiction, I believe, is that when Achilles challenges Hector to fight, Hector runs away from him leading to a humiliating chase around the walls of the city before he finally gets caught. And then he's killed with one stroke of Achilles' sword. I much prefer the depiction of the fight from the film Troy, where Hector is shown to know that he will die fighting Achilles but goes anyway as a matter of duty. Troy is one of the few Hollywood films with a fight where the audience honestly does not know who to root for. As far as the audience is concerned both Hector and Achilles are virtuous and good. That's extremely rare for Hollywood as it's almost always "Goodie vs. Baddie". In Troy there is a baddie, the despotic king, Agamemmnon, but the general attitude is that both the Greeks and Trojans are good (or they're all equally bad?), and that this war is an inevitable and tragic result of politics.

The film does have some ridiculous alterations to the plot which ruins it though, for example Agammemmnon being killed by Briseis, and also Hector killing Menelaus during his duel with Paris. There's also a suggestion that Hector kills Ajax, and there's a complete lack of the other important characters like Diomedes, Antilochus, Idomeneus, Aeneas, Sarpedon, "little Ajax", Cassandra and a few others.

When I read the Iliad a few years ago, I was quite disappointed about the ending, Hector's burial. I quickly began reading the Odyssey and was saddened to see a huge gap in the events that I had learned about growing up. Where was Achilles' death? Where was Ajax's suicide? Where was Paris' death, killed by an arrow shot by the bow of Hercules? (tipped presumably with the blood of the hydra) And of course, the famous wooden horse, the murders of Priam and Little Ajax's rape of Cassandra? (I think he's referred to as Oilean Ajax but I prefer "little Ajax"). In the film, Troy, I believe they had Briseis kill Agammemmnon to introduce some feminine influence, but they could have included the Amazonian Queen Penthesilea, who Achilles killed. They could have had the Ethiopian king, Memnon, who killed Antilochus and was in turn slain by Achilles. So much happens between the Iliad and the Odyssey which Homer only vaguely refers to during the Odyssey. Where did all this stuff even come from? Who wrote about those other stories?

What I'd quite like to do is collect all the sources, all of the stories, and collect them into one good novel, probably about as long as The Lord Of The Rings. Absolutely every part of the story could be included, even Shakespeare's version, everything that doesn't contradict with some other bit anyway. Plus, like the film, Troy, the metaphysic of the novel would be atheistic, it would even resemble our best current scientifically influenced metaphysical theories as best as possible. For instance, Achilles would not be immortal, but purely an excellent fighter. Many others might believe him to be immortal, although he probably doesn't believe it himself. The novel could be written based on the notion that the Iliad represents an exaggerated and mystical history of the original event. Everybody believed Achilles was immortal, then he was shot in the heel with a poison-tipped arrow, and suddenly the story would become "He was dipped in the River Styx as a child and was immortal everywhere except his heel." There would be no gods fighting on the battlefield, guiding arrows to their targets, or arguing on Mount Olympus between scenes, although the characters might reasonably believe it.

The toughest part of writing such a novel would be how to treat the Odyssey. How do you explain a cyclops or Scylla, a journey through Hell or sirens with a naturalistic explanation? It's possible that there would actually be a good opportunity for creativity to re-write the Iliad, explaining that most of the widely-known story is simply story, and that what really happened was actually completely different but somehow still related closely enough to make the plot intriguing. There's even the possibility you could develop the character of Odysseus to make him more sinister and cowardly than he was thought to be. If you think about it, if all that's left of an army is its general, who's washed up on the Ithican beach, you'd think there's more likely than not some foul play going on there. We already know that he was very quick-thinking and a good problem-solver, so the story could include his fabricating the whole story of the Odyssey. There is definitely a great potential for a brilliant story here. Possibly the story could be developed to include the stories of Aeneas, Diomedes, Menelaus and Agamemmnon after the Trojan War to make it a true epic tale that ties up every loose end.

I was reading Peter van Inwagen's introduction to metaphysics recently, and he uses an argument against Anti-Realism which I've seen all too often and which I think badly misrepresents the Anti-Realist. I've seen the same sort of argument used against the Sceptic. First, let's just throw some definitions out there:

Global Scepticism (GS): The proposition that knowledge about any proposition is impossible.
Global Anti-Realism (GAR): The proposition that there is no objective reality, and therefore no true or false propositions.

The van Inwagens of this world use the following sort of argument:

GS is self-contradictory because if you believe that there is no knowledge, then how can you know that? It's simply self-contradictory to assert that all propositions are unknown because to assert that statement is to assert that that statement is known, and that statement is a member of the set of all propositions. Therefore even asserting Global Scepticism is as meaningless to your audience as asserting "This statement is false".

Likewise with GAR. If you argue that no propositions have the property of being true, you automatically assert that that statement is true.

Now in the cases of both GS and GAR there is one consistent position that van Inwagen may allow. Call these WGS and WGAR respectively ("W" for "Weak"). WGS states that "The only known proposition is that no other propositions are known." This is a lot like the famous saying of Socrates which went something like, "The only true wisdom is in knowing you know nothing", but don't confuse the two as the latter is a kind of aphorism, whereas the former is truly believed by an advocate of WGS. It could be argued that WGS is also contradictory because once you know that you can't know anything, you may also know a few more things, like "Either nothing is known, or nothing is known." But I disagree. The whole point of Global Scepticism is to deny knowledge of everything, including that there are axioms of logic which entail that "A or A" follows from "A". I think WGS and WGAR are consistent positions, but they are a little arbitrary.

van Inwagen argues that a GARist is forced to express something like "It seems to me that there is no objective truth", which is not incompatible with "It seems to you that there is an objective truth.", and that therefore the GARists are forced to just shut up and let the realists carry on with their lives.

I believe that both GS and GAR are consistent propositions, and this is why. They should not be expressed as I have expressed them in my original definitions, which are quite similar to how most people define them. Instead they should be defined like this:

Global Scepticism: If knowledge is defined as XYZ, then it is not the case that knowledge exists. (Or there exists no proposition which has the property of being known).

Global Anti-Realism: If objective truth is defined as ABC, then there is no proposition with the property of truth.

These sentences are not self-contradictory, because they do not self-refer. These positions actually assert that the opposing views (let's call them Realism and Epistemic Realism) are false, at least given the definitions of truth and knowledge available and agreed upon. Global Sceptics are not asserting that they know that nothing is knowable. Instead they are asserting that if we adopt a certain definition of knowledge, then nothing in the world corresponds to that definition.

This position is sometimes referred to as Pyrrhonism, and distinguished from regular Scepticism. Why it is distinguished I have no idea, as it's immediately obvious that Scepticism, understood as how I defined Global Scepticism at the beginning of this post, is self-contradictory, whereas Pyrrhonism is quite sensible. I know of no similar distinction with regard to Anti-Realism though. All the GARist is really pointing out is that you cannot reasonably accept some definitions of "truth" and also believe that anything is true, because actually nothing has the property that you define as truth. That is all that is going on. I think that most Anti-Realists are merely saying that truth is an impossible thing, and what is important is some other thing that they may try to analyse. Likewise most Sceptics argue that knowledge, defined as justified true belief, does not exist, but that there is some other epistemic notion which is important and worth looking for and worth having.

It occurred to me recently that petitions are almost completely pointless. If I were the Prime Minister and I received a petition signed by 10 million people that called for a stop to X, I would infer that probably 95% of those 10 million people really don't care much about X and whether it continues or not. Because it's just far too easy to sign a petition. There's nothing to lose. In their programme, "Bullshit!", Penn and Teller demonstrated as much when they persuaded people to endorse banning water and other such ridiculous notions because they just didn't think about what they were signing and inferred that it was easier to sign than not.

I have a solution! In every town and city we should have a public building devoted to solitary reflection. When you sign a petition endorsing some action occur you must also agree to attend one of these centres for one hour and sit there on your own and reflect for an hour upon the cause that you think is so important. Well, of course you wouldn't be forced to think about that, but you are essentially giving up one hour of your time to show some tiny degree of dedication to the cause. You can't read, do crosswords, or even talk to other people, but just sit, in relative comfort, and think about something. You're entitled to meditate but you cannot sleep, otherwise it would be too easy to appear more concerned about a cause than you really are, by just sleeping through the hour. Only when you have attended one of these events is your vote on the petition made official. If I were the prime minister and was told that 10 million people had individually given an hour of their time as an expression that they believed in some cause, that a total of 10 million hours had been sacrificed, then I would take immediate notice of the cause, and do something to satisfy the people. Until this state of affairs has become actual, petitions will continue to be meaningless.