A simple example of non-VNM total orders

It occurred to me yesterday that there is an extremely common and natural example of a total ordering (modulo indifference. i.e. it’s really a total pre-ordering) of distributions which doesn’t satisfy The VNM Axioms.

(Note: We are talking about distributions rather than random variables in this, so we will follow the VNM notation of using pA + (1-p)B to mean “choose A with probability p or B with probability (1 – p)”. This has the effect of a similar combination on the distributions. It’s not addition of random variables).

Specifically, ordering distributions by their lower median value (assuming a total ordering over outcomes). More generally, by any percentile value.

This does not satisfy continuity: To see that it does not satisfy continuity, suppose our outcomes are 1, 2 and 3. Let A, B, C be lotteries choosing each of these with probability 1. Then A < B < C. But the lower median of \(pA + (1-p)B\) is 1 if \(p \geq \frac{1}{2}\) or 3 if \(p < \frac{1}{2}\). The median of \(B\) is always 2, so we are never indifferent between them. You could argue that the problem is that at \(p = \frac{1}{2}\) we should take an averaging of the two values and the median should be 2 there, but that doesn't save you. Instead consider where we have 5 values and A, B, C choose 1, 2 and 5. Then at \(p = \frac{1}{2}\) the median will be 3, so we're still not indifferent to it at any point. It also does not satisfy independence. Here's an example: Suppose we have two possible outcomes, 0 and 1. Consider distributions \(A = [0.35, 0.65]\), \(B = [0.4, 0.6]\), \(C = [0.9, 0.1]\). Let \(p = 0.75\). Then the median of \(A\) and \(B\) are both 1, while the median of \(C\) is 0. Further the median of \(pA + (1-p)C\) is 1, because the probability of it being 0 is \(0.75 * 0.35 + 0.25 * 0.9 \approx 0.49 < 0.5\). The median of \(pB + (1-p)C\) however is 0 because the probability of it being 0 is \(0.525\). So \(A \preceq B\) but \(pA + (1-p)C \succ pB + (1-p)C\) as desired. (Disclaimer: I didn't work this out by inspection, I totally just ran a computer program to find examples for me because I was pretty sure they must exist)

A theorem on dominance of random variables

I wrote previously about a dominance relation amongst \(\mathbb{N}\) valued random variables. I’ve realised a nice characterisation of the relationship, which is what this post is about.

We’ll change the setting slightly to \(\mathbb{R}\) valued random variables, as proving this theorem works more nicely in this context, but the result will still hold for \(\mathbb{N}\) as a subset of it.

Definition: Let \(X, Y\) be real valued random variables. Say \(x \preceq y\) if \(\forall t. P(X \geq t) \leq P(Y \geq t)\).

Theorem: \(X \preceq Y\) iff for every monotone function \(h : \mathbb{R} \to \mathbb{R}\), \(E(h(X)) \leq E(h(y))\).

Proof of this will need a version of a lemma from the last post:

Lemma: Let \(X\) be a real-valued random variable. Then \(E(X) = \int\limits_{-\infty}^\infty P(X \geq t) dt\).

I’m going to skip proving this for now. I have proofs, but they’re a little fiddly and I got lost in the details when trying to write this up pre-lunch (it’s easy to prove given some stronger continuity assumptions if you use integration by parts). So IOU one proof.

Proof of theorem:

If \(X \preceq Y\) and \(h\) is monotone, then \(E(h(X)) = \int\limits_{-\infty}^\infty P(h(X) \geq t) dt\) and similarly for \(y\).

But \(h\) is monotone, so \(H_t = \{x : h(x) \geq t\}\) is either \([y, \infty)\) or \((y, \infty)\) for some \(y\). The latter can be written as \(\bigcup [y_n, \infty)\) for some sequence \(y_n\). We know that \(P(X \geq y) \leq P(Y \geq y)\), so we know that \(P(X \in H_t) \leq P(Y \in H_t)\) due to this characterisation. But these probabilities are respectively \(P(h(X) \geq t)\) and \(P(h(Y) \geq t)\).

So \[\begin{align*}
E(h(X)) & = \int\limits_{-\infty}^\infty P(h(X) \geq t) dt \\
& \leq \int\limits_{-\infty}^\infty P(h(Y) \geq t) dt \\
& = E(Y) \\
\end{align*}\]

as desired.

Now for the converse:

Let \(t \in \mathbb{R}\) with \(P(X \geq t) > P(Y \geq t)\). Let \(h(x) = 0\) if \(x < t\) and \(h(x) = 1\) if \(x \geq t\). Then \(h\) is monotone increasing and \(E(h(X)) = P(X \geq t) > P(Y \geq t) = E(h(Y))\) as desired.

QED

A sketchy and overly simplistic theory of moral change

This is another post inspired by a conversation with Paul Crowley.

Up front warning: The morality described herein is a very hippy left wing morality. If you subscribe to any form of consequentialism you’re probably going to at least find it compatible with your own. If you think Some Victimless Crimes Are Just Plain Wrong Dammit you’re probably not. Or rather, you may agree with most of what I have to say but think there are other highly important things too.

I hate trolley problems.

Or rather, I think peoples’ responses to trolley problems are an interesting thing to study empirically. I just think they’re a lousy way to approach morality.

Why? Well, fundamentally, I don’t think most failures of morality are failures of moral reasoning. I think morality is fundamentally much less complex than we want to believe it to be, and I think most reasonable moral commandments can reasonably be summed up as “You’re hurting people. Do less of that”.

That’s not to say that this is the be all and the end all of morality, or that there are no tricky moral dilemmas. Obviously it’s not and there are. I just think that they are tricky because they are unusual, and that most failures of morality happen long before we reach anything that complicated, and simply boil down to the fact that you are hurting people and should do less of that. I also think that trying to convince ourself that morality is a complex thing which we don’t understand is more of an excuse to fail to act morally (“Look! It’s hard! What would you do with this trolley?!”) than it is a true attempt to understand how we should act.

If you honestly find yourself in a situation where the rule doesn’t apply, then apply your complicated theory of moral philosophy. In the meantime: You’re hurting people. Do less of that.

Generally speaking, I feel people are pretty good at understanding this rule, and that if they don’t understand this rule then it is very unlikely that after a period of careful reflection and self-contemplation they will go “Oh! Right! I’m being a bad person. I should not do that, huh?”. A carefully argued case for why they should be a good person is also rather unlikely to work.

And yet people can clearly change their morality and become better people. If not individuals, at least societies can – many things we once did we now recognise as morally awful. Many things we currently do the next generation will probably recognise as awful.

So given that I believe self-reflection and argument don’t work, what does actually work?

I think most moral failings boil down to three basic issues:

I don’t understand that I am hurting people
I don’t believe that I am hurting people
I don’t care that I am hurting people

And I think there is a fairly easy litmus test to see which of the three categories you find yourself in.

If someone says “When you do X, it hurts me because Y”, how do you respond?

If you say “Oh, shit. Sorry. I had no idea. I’ll stop doing X then!” then you did not understand.

If you say “Yeah, right. You obviously made that up” then you do not believe you are hurting people.

If you say “Oh well. I’m going to keep doing X” then you do not care that you are hurting people.

Let me set something straight right now: These are all acceptable answers.

I’ll take it as read that an apology and a promise to do better is acceptable.

“When you support gay rights, it disrupts my connection to god and makes my inner angel cry” – “Yeah, right”

“When you support the government taxing me, it makes me sad” “Oh well. I’m going to keep supporting the government taxing you”

I don’t intend to defend these points. Only to point out that these are cases where I will react that way, and I think it is OK to do so.

The interesting thing about these three is that the forces which change them are all different.

In particular, only the first is amenable to reason. You can present evidence, you can present arguments, and at the end of it they will have a new understanding of the world and realise that their previous behaviour hurt people and hopefully will fix it. This is what I referred to previously as the moral argument for rationality.

How do you change the third? In a word: diversity. You know that thing that sometimes happens where some politician’s child comes out as gay and all of a sudden they’re all “Oh, right! Gays are people!” and they about face on gay marriage? That’s moral change brought about by a change of caring. Suddenly the group of people you are hurting has a human face and you actually have to care about them.

How do you affect change of belief? I don’t know. From the inside, my approach is to simply bias towards believing people. I’m not saying I always believe people when they say I’m hurting them (I pretty much apply a “No, you’re just being a bit of a dick and exploiting the rules I’ve precommitted to” get out clause for all rules of social interaction), but I’m far more likely to than not. From the outside? I think it’s much the same as caring: People will believe when people they have reason to trust put forth the argument.

In short, I believe that arguments don’t change morals, people do, and I think that sitting around contemplating trolley problems will achieve much less moral change than exposing yourself to a variety of different people and seeing how your actions affect them.

Comment moderation

Just as a note, I’ve turned comment moderation on. All comments will be held in a queue. Sorry about that.

The reason for this is that for some reason Akismet is utterly failing to do an adequate job of spam detection recently, and I’ve been getting an ever larger amount of it, so I’ve decided that rather than have my blog be a haven for spammers I’d rather just do slightly more work and whitelist comments.

I will likely approve your comment very rapidly – The only comments that will not make it through moderation are spam and anything I consider truly beyond the pale (which has never happened to me so far. Please don’t take that as a challenge).

Another model for bribing MPs

As you might have noticed, I have Flattr attached to my blog posts. It’s a system which gives me entire pennies in income and is mostly there as a vanity project.

Terence kindly Flattred my post on bribing MPs, which got me to thinking about another strategy for crowd-funding subverting the course of the democratic process by making our elected officials actually pay attention to the people they’re representing: We adopt the Flattr model. Let’s call it Bribr.

Here’s how I imagine it would work:

Every time a bill, debate or other significant event happened in parliament, we would have an electronic representation of which MPs took which sides. You could then choose to cast your vote of approval with a particular position. Every MP who took that position gets one bribry point from you (a lot of stuff goes through parliament. I imagine you could filter by some significant metrics in order to not get overwhelmed).

At the end of the month you are asked how happy you were with parliament this month. Happiness is a quantity measured in pounds sterling. That quantity is then divided by MPs according to how much you bribred each of them: If you only engaged in bribry over one issue and 10 MPs voted on that issue in accordance with your favour, they’d each get 10% of your monthly allocation.

So each MP ends the month with a certain amount of money, spread over bribs from many different people. Rather than giving them money directly there are a bunch of reward tiers – ideally you want a hundred or so different reward possibilities just to keep things varied – and the MP gets a reward appropriate to the amount of money they have assigned to them, with any not spent rolling over to the next month. Any MP who ends up with a truly ridiculous amount of money (say, multi thousands of pounds) gets one of the most lavish rewards plus a note saying “Hey. We really love what you’re doing. Is there anything you’d like that’s worth about this much?”

Moral status? I’m actually more comfortable with this then I am with the other one. You’re essentially just rewarding MPs for doing their jobs well. I’d still have to hold my nose a bit to run something like this, but I don’t think I’d have a problem using it. Legal status? Not sure. I don’t know enough about the bribery statutes to say one way or another. It’s clearly much closer to bribery than the other version, because people are being directly rewarded for their votes. Again, would have to talk to an actual lawyer.