For a community that endorses Bayesian epistemology we have had surprisingly few discussions about the most famous Bayesian contribution to epistemology: the Dutch Book arguments. In this post I present the arguments, but it is far from clear yet what the right way to interpret them is or even if they prove what they set out to. The Dutch Book arguments attempt to justify the Bayesian approach to science and belief; I will also suggest that any successful Dutch Book defense of Bayesianism cannot be disentangled from decision theory. But mostly this post is to introduce people to the argument and to get people thinking about a solution. The literature is scant enough that it is plausible people here could actually make genuine progress, especially since the problem is related to decision theory.¹

Bayesianism fits together. Like a well-tailored jacket it feels comfortable and looks good. It's an appealing, functional aesthetic for those with cultivated epistemic taste. But sleekness is not a rigourous justification and so we should ask: why must the rational agent adopt the axioms of probability as conditions for her degrees of belief? Further, why should agents accept the principle conditionalization as a rule of inference? These are the questions the Dutch Book arguments try to answer.

The arguments begin with an assumption about the connection between degrees of belief and willingness to wager. An agent with degree of belief b in hypothesis h is assumed to be willing to buy wager up to and including $b in a unit wager on h and sell a unit wager on h down to and including $b. For example, if my degree of belief that I can drink ten eggnogs without passing out is .3 I am willing to bet $0.30 on the proposition that I can drink the nog without passing out when the stakes of the bet are $1. Call this the Will-to-wager Assumption. As we will see it is problematic.

The trolley problem is one of the more famous thought experiments in moral philosophy, and studies by psychologists and anthropologists suggest that the response distributions to its major permutations remain roughly the same throughout all human cultures. Most people will permit pulling the lever to redirect the trolley so that it will kill one person rather than five, but will balk at pushing one fat person in front of the trolley to save the five if that is the only available option of stopping it.

However, in informal settings, where the dilemma is posed by a peer rather than a teacher or researcher, it has been my observation that there is another major category which accounts for a significant proportion of respondents' answers. Rather than choosing to flip the switch, push the fat man, or remain passive, many people will reject the question outright. They will attack the improbability of the premise, attempt to invent third options, or appeal to their emotional state in the provided scenario ("I would be too panicked to do anything",) or some combination of the above, in order to opt out of answering the question on its own terms.

Related to Belief In Belief

Suppose that a neighbor comes to you one day and tells you “There’s a dragon in my garage!” Since all of us have been through this before at some point or another, you may be inclined to save time and ask “Is the dragon by any chance invisible, inaudible, intangible, and does it convert oxygen to carbon dioxide when it breathes?”

The neighbor, however, is a scientific minded fellow and responds “Yes, yes, no, and maybe, I haven’t checked. This is an idea with testable consequences. If I try to touch the dragon it gets out of the way, but it leaves footprints in flour when I sprinkle it on the garage floor, and whenever it gets hungry, it comes out of my garage and eats a nearby animal. It always chooses something weighing over thirty pounds, and you can see the animals get snatched up and mangled to a pulp in its invisible jaws. It’s actually pretty horrible. You may have noticed that there have been fewer dogs around the neighborhood lately.”

This triggers a tremendous number of your skepticism filters, and so the only thing you can think of to say is “I think I’m going to need to see this.”

“Of course,” replies the neighbor, and he sets off across the street, opens the garage door, and is promptly eaten by the invisible dragon.

For a very long time, philosophy has presented us with two straw men in combat with one another and we are expected to take sides. Both straw men appear to have been proved true and also proved false. The straw men are Determinism and Free Will. I believe that both, in any useful sense, are false. Let me tell a little story.

Mary's story

Mary is walking down the street, just for a walk, without a firm destination. She comes to a T where she must go left or right and she looks down each street finding them about the same. She decides to go left. She feels she has, like a free little birdie, exercised her will without constraint. As she crosses the next intersection she is struck by a car and suffers serious injury. Now she spends much time thinking about how she could have avoided being exactly where she was, when she was. She believes that things have causes and she tries to figure out where a different decision would have given a different outcome and how she could have known to make the alternative decision. 'If only..' ideas crowd into her thoughts. She believes simultaneously that her actions have causes and that there are valid alternatives to her actions. She is using both deterministic logic and free will logic, neither alone leads to 'If only..' scenarios – it takes both. If only she had noticed that the next intersection on the right had traffic lights but on the left didn't. If only she had not noticed the shoe store on the left. What is more she is doing this in order to change some aspect of her decision making so that it will be less likely to put her in hospital, again this is not in keeping with either logic. But really both forms of logic are deeply flawed. What Mary is actually attempting is to do maintenance on her decision making processes so that they can learn whatever is available to be learned from her unfortunate experience.

What is useless about determinism

There is a big difference between being 'in principle' determined and being determined in any useful way. If I accept that all is caused by the laws of physics (and we know these laws – a big if) this does not accomplish much. I still cannot predict events except trivially: in general but not in full detail, in simple not complex situations, extremely shortly into the future rather than longer term, etc. To predict anything really sizable, like for instance, how the earth came to be as it is, or even how little-old-me became what I am, or even why I did a particular thing a moment ago, would take more resources and time than can be found in the life of our universe. Being determined does not mean being predictable. It does not help us to know that our decisions are determined because we still have to actually make the decisions. We cannot just predict what the outcomes of our decisions will be, we really, really have to go through the whole process of making them. We cannot even pretend that decisions are determined until after we have finish making them.

What is useless about freewill

There is a big difference between being free in the legal, political, human rights type of freedom. To be free from particular, named restraints is something we all understand. But the free in 'free will' is a freedom from the cause and effect of the material world. This sort of freedom has to be magical, supernatural, spiritual or the like. That in itself is not a problem for a belief system. It is the idea that something that is not material can act on the material world that is problematic. Unless you have everything spiritual or everything material, you have the problem of interaction. What is the 'lever' that the non-material uses to move the material, or vice versa. It is practically impossible to explain how free will can affect the brain and body. If you say God does it, you have raised a personal problem to a cosmic one but the problem remains – how can the non-physical interact with the physical? Free will is of little use in explaining our decision process. We make our decisions rather than having them dictated to us but it is physical processes in the brain that really do the decision making, not magic. And we want our decisions to be relevant, effective and in contact with the physical world, not ineffective. We actually want a 'lever' on the material world. Decisions taken in some sort of causal vacuum are of no use to us.

The question we want answered

Just because philosophers pose questions and argue various answers does not mean that they are finding answers. No, they are make clear the logical ramifications of questions and each answer. This is a useful function and not to be undervalued, but it is not a process that gives robust answers. As an example, we have Zeno's paradox about the arrow that can never landing because its distance to landing can always be divided in half, but on the other hand, the knowledge that it does actually land. Philosophers used to argue about how to treat this paradox, but they never solved it. It lost its power when mathematics developed the concept of the sum of a infinite series. When the distance is cut in half, so is the time. When the infinite series of remaining distance reaches zero so does the series of time remaining. We do not know how to end an infinite series but we know where it ends and when it ends – on the ground the moment the arrow hits it. The sum of an infinite series can still be considered somewhat paradoxical but as an obscure mathematical question. Generally, philosophers are no longer very interested in the Zeno paradox, certainly not its answer. Philosophy is useful but not because it supplies consensus answers. Mathematics, science and their cousins, like history, supply answers. Philosophy has set up a dichotomy between free will and determinism and explored each idea to exhaustion but not with any consensus about which is correct. That is not the point of philosophy. Science has to rephrase the problem as, 'how exactly are decisions made?' That is the question we need an answer to, a robust consensus answer.

But here is the rub

This move to a scientific answer is disturbing to very many people because the answer is assumed to have effects on our notions of morals, responsibility and identity. Civilization as we know it may fall apart. Exactly how we think we make decisions once we study the question without reference to determinism or freewill seems OK. But if the answer robs us of morals, responsibility or identity, than it is definitely not OK. Some people have the notion that what we should do is just pretend that we have free will, while knowing that our actions are determined. To me this is silly: believe two incompatible and flawed ideas at the same time rather than believe a better, single idea. It reminds me of the solution proposed to deal with Copernicus – use the new calculations while believing that the earth does not revolve. Of course, we do not yet have the scientific answer (far from it) although we think we can see the general gist of it. So we cannot say how it will affect society. I personally feel that it will not affect us negatively but that is just a personal opinion. Neuroscience will continue to grow and we will soon have a very good idea of how we actually make decisions, whether this knowledge is welcomed or not. It is time we stopped worrying about determinism and free will and started preparing ourselves to live with ourselves and others in a new framework.

Identity, Responsibility, Morals

We need to start thinking of ourselves as whole beings, one entity from head to toe: brain and body, past and future, from birth to death. Forgot the ancient religious idea of a mind imprisoned in a body. We have to stop the separation of me and my body, me and my brain. Me has to be all my parts together, working together. Me cannot equate to consciousness alone.

Of course I am responsible for absolutely everything I do including something I do while sleep walking. Further a rock that falls from a cliff is responsible for blocking the road. It is what we do about responsibility that differs. We remove the rock but we do not blame or punish it. We try to help the sleep walker overcome the dangers of sleep walking to himself and others. But if I as a normal person hit someone in the face, my responsibility is not greater than the rock or the sleep walker but my treatment will be much, much different. I am expected to maintain my decision-making apparatus in good working order. The way the legal system will work might be a little different from now, but not much. People will be expected to know and follow the rules of society.

I think of moral questions as those for which there is no good answer. All courses of action and of inaction are bad in a moral question. Often because the possible answers pit the good of the individual against the good of the group, but also pit different groups and their interests against each other. No matter what we believe about how decisions are made, we are still forced to make them and that includes moral ones. The more we know about decisions, the more likely we are to make moral decisions we are proud of (or least guilty or ashamed of), but there is no guarantee. There is still a likelihood that we will just muddle along trying to find the lesser of two evils with no more success than at present.

Why should we believe that being closer to the truth or having a more accurate understanding is going to make things worst rather than better? Shouldn't we welcome having a map that is closer to the territory? It is time to be open to ideas outside the artificial determinism/freewill dichotomy.

This is a variant built on Gary Drescher's xor problem for timeless decision theory.

You get an envelope from your good friend Alpha, and are about to open it, when Omega appears in a puff of logic.

Being completely trustworthy as usual (don't you just hate that?), he explains that Alpha flipped a coin (or looked at the parity of a sufficiently high digit of pi), to decide whether to put £1000 000 in your envelope, or put nothing.

He, Omega, knows what Alpha decided, has also predicted your own actions, and you know these facts. He hands you a £10 note and says:

"(I predicted that you will refuse this £10) if and only if (there is £1000 000 in Alpha's envelope)."

What to do?

EDIT: to clarify, Alpha will send you the envelope anyway, and Omega may choose to appear or not appear as he and his logic deem fit. Nor is Omega stating a mathematical theorem: that one can deduce from the first premise the truth of the second. He is using XNOR, but using 'if and only if' seems a more understandable formulation. You get to keep the envelope whatever happens, in case that wasn't clear.

This example (and the whole method for modelling blackmail) are due to Eliezer. I have just recast them in my own words.

We join our friends, the Countess of Rectitude and Baron Chastity, in bed together. Having surmounted their recent difficulties (she paid him, by the way), they decide to relax with a good old game of prisoner's dilemma. The payoff matrix is as usual:

Were they both standard game theorists, they would both defect, and the payoff would be (1,1). But recall that the baron occupies an epistemic vantage over the countess. While the countess only gets to choose her own action, he can choose from among four more general tactics:

1. (Countess C, Countess D)→(Baron D, Baron C)   "contrarian" : do the opposite of what she does
   
2. (Countess C, Countess D)→(Baron C, Baron C)   "trusting soul" : always cooperate
   
3. (Countess C, Countess D)→(Baron D, Baron D)   "bastard" : always defect
   
4. (Countess C, Countess D)→(Baron C, Baron D)   "copycat" : do whatever she does

Recall that he counterfactually considers what the countess would do in each case, while assuming that the countess considers his decision a fixed fact about the universe. Were he to adopt the contrarian tactic, she would maximise her utility by defecting, giving a payoff of (0,5). Similarly, she would defect in both trusting soul and bastard, giving payoffs of (0,5) and (1,1) respectively. If he goes for copycat, on the other hand, she will cooperate, giving a payoff of (3,3).

Thus when one player occupies a superior epistemic vantage over the other, they can do better than standard game theorists, and manage to both cooperate.

"Isn't it wonderful," gushed the Countess, pocketing her 3 utilitons and lighting a cigarette, "how we can do such marvellously unexpected things when your position is over mine?"

This is Eliezer's model of blackmail in decision theory at the recent workshop at SIAI, filtered through my own understanding. Eliezer help and advice were much appreciated; any errors here-in are my own.

The mysterious stranger blackmailing the Countess of Rectitude over her extra-marital affair with Baron Chastity doesn't have to run a complicated algorithm. He simply has to credibly commit to the course of action:

"If you don't give me money, I will reveal your affair."

And then, generally, the Countess forks over the cash. Which means the blackmailer never does reveal the details of the affair, so that threat remains entirely counterfactual/hypothetical. Even if the blackmailer is Baron Chastity, and the revelation would be devastating for him as well, this makes no difference at all, as long as he can credibly commit to Z. In the world of perfect decision makers, there is no risk to doing so, because the Countess will hand over the money, so the Baron will not take the hit from the revelation.

Indeed, the baron could replace "I will reveal our affair" with Z="I will reveal our affair, then sell my children into slavery, kill my dogs, burn my palace, and donate my organs to medical science while boiling myself in burning tar" or even "I will reveal our affair, then turn on an unfriendly AI", and it would only matter if this changed his pre-commitment to Z. If the Baron can commit to counterfactually doing Z, then he never has to do Z (as the countess will pay him the hush money), so it doesn't matter how horrible the consequences of Z are to himself.

To get some numbers in this model, assume the countess can either pay up or not do so, and the baron can reveal the affair or keep silent. The payoff matrix could look something like this:

Would you prefer a 50% chance of gaining €10, one chance in a million off gaining €5 million, or a guaranteed €5? The standard position on Less Wrong is that the answer depends solely on the difference between cash and utility. If your utility scales less-than-linearly with money, you are risk averse and should choose the last option; if it scales more-than-linearly, you are risk-loving and should choose the second one. If we replaced €’s with utils in the example above, then it would simply be irrational to prefer one option over the others.

There are mathematical proofs of that result, but there are also strong intuitive arguments for it. What’s the best way of seeing this? Imagine that X₁ and X₂ were two probability distributions, with mean u₁ and u₂ and variances v₁ and v₂. If the two distributions are independent, then the sum X₁ + X₂ has mean u₁ + u₂, and variance v₁ + v₂.

Now if we multiply the returns of any distribution by a constant r, the mean scales by r and variance scales by r2. Consequently if we have n probability distributions X₁, X₂, ... , X_n representing n equally expensive investments, the expected average return is (Σⁿ_i=1 u_i)/n, while the variance of this average is (Σⁿ_i=1 v_i)/n². If the v_n are bounded, then once we make n large enough, that variance must tend to zero. So if you have many investments, your averaged actual returns will be, with high probability, very close to your expected returns.