Author Archives: david

On the Von Neumann Morgenstern Utility Theorem, Part 1

A question I occasionally get asked is “What do you have against the Von Neumann Morgenstern Utility Theorem, David?”. Or the more specific variant “Which of its premises do you disagree with?”.

Well, it would be faster for me to list what aspects of the theorem I do agree with.

Which is, err.

Um.

Well I haven’t come up with a convincing objection to the independence hypothesis. That’s not to say that I necessarily think it’s right, only that I haven’t got a strong opinion on it either way.

On top of its explicit premises, I also reject the implicit ones that this is a sensible way to reason about peoples’ behaviours, and the conclusion would say anything useful even if I were to grant all of the other premises.

So, yeah, I have a lot of objections. Far too many to write only one article about it. So instead I’m going to break this up and write it as and when I feel like it. Advance warning: When I write series of articles, they tend not to be finished. I will try to make each of these self contained so you don’t need to hold your breath waiting for the next one.

I will start with the most obvious place to begin this series: Somewhere in the middle picked almost at random.

Preferences over lotteries are discontinuous

Amongst the “reasonable hypotheses” of the VNM theorem is a continuity assumption. It can be formulated as this:

Suppose we have lotteries A, B, C with A < B < C. There is some probability p such that the lottery \(L_p = pA + (1-p)C\) is equivalently preferable to B. There are other, weaker, formulations of this but that doesn't matter very much: If continuity is not a hypothesis of your variant of the VNM theorem, it's still a conclusion of it, so its truth value is relevant.

I think continuity is simply false. Consider the following example: C is “I have a sandwich”, B is “I have a kitten to play with” and A is “The world is destroyed”.

We have \(L_0 = A < B < C = L_p\), certainly. We also have that for \(p < p'\), \(L_p > L_{p’}\) (I will always prefer to increase the probability of my getting a sandwich versus destroying the world).

There is no probability with which I will accept a sandwich as a fair exchange for some probability of destroying the world. No matter how tasty the sandwich is, or how cute the kitten I’d get in the non world destroying chance.

But I certainly prefer kittens to sandwiches (not to eat of course. To cuddle). If the probability of destroying the world is functionally indistinct from 0, I will pick \(L_p\) – give me a kitten with probability 1. If it is functionally distinct from 0, I will always pick \(B\) – the possibility of the kitten isn’t worth it, give me the sandwich instead.

You might argue that there is some true tiny probability with which I would consider them equal, but I’m skeptical. Further, I think that even if there were such a probability in theory (which I am not convinced of), it is so small as to not be plausible to estimate or work with.

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A magic system with consistency and depth

Today I am thinking about magic systems, inspired by a link from Dave and a conversation on Twitter with @palfrey.

It occurred to me there are basically two characteristics I really like in a magic system: The first is very well thought out magic systems which feel like they actually make sense and don’t completely disrupt the world they live in when you think through the implications. A good example of an author who does these very well is Brandon Sanderson – he’s got magic down to a science. For the sake of having a good word for this I will call this “Consistency”.

There’s another characteristic which I really like – the feeling that magic is deep and significant and contains powers beyond the ken of mere mortals. The feeling that magic being in the world makes it a fundamentally different and more mysterious place. I’ll refer to this as “Depth”. It is often associated with heavily ritualized magic, though I suppose there’s no intrinsic reason why it needs to be. An example of this being done well is Susanna Clarke’s Jonathan Strange and Mr Norrel.

These two seem to be fundamentally in tension – the more the magic makes sense, the less depth it seems to add.

One way I’ve seen this tension handled well is systems where magic is mostly fairly well understood and mechanistic but gives you access to higher powers and realms, and the more you draw on those the more mysterious and murky things get. For example, the Deryni books by Katherine Kurtz handle it this way: At the low end, Deryni magic is practically mundane, or at the very least looks like psychic powers. You can chat mentally, you can summon little balls of light, you can read letters without opening it. As the scope of the magic you’re performing increases, the amount of ritual required increases. At the grander scales of ritual you are invoking higher powers, summoning the archangels to watch over you and bless your working, and they… start to take notice of you.

Fundamentally though the two are still in tension here – it’s not that you’ve got the two side by side, it’s that you can trade off between the two of them in-universe.

So I started thinking about how you might go about getting both depth and consistency working together in harmony. Here’s what I came up with.

On the fundamentals of magic and the skills of mages

There is another realm, distinct from the world in which we live. Different cultures and traditions call it different things, and perceive it in differently. We shall call it Faerie.

Faerie is not a land of grass and trees and elves and unicorns frolicking in the sunlight. Those things are there, but Faerie is much more than that.

All our minds are reflected in Faerie, and joined to it. They shape it, and it shapes them. Faerie is a realm where our unconscious thoughts and imagination take form.

But it does not stop there. The ideas which are born in our mind linger long after we have stopped thinking them, whether through inattention or death. Some of them fade away, but some find new homes in others’ minds, or join with ideas spawned in other minds and take on a life of their own.

Many of these things which live in Faerie are wonderful. Many are terrible. Some are both.

Faerie is also extremely mutable. As it is spawned from our minds, so may it be shaped by them. When we perceive a segment of Faerie we may also shape it to our desires.

The skills required to perform magic are threefold.

The first skill is that of perception. The better you understand the world around you, the easier it is to spell cast in there. This includes both skills of observation with your mundane senses and additional ways of feeling the impression the world leaves on your mind and what it tells about you. An experienced mage can see in the dark, through walls and inside closed boxes. They can hear a pin drop and feel the heat of a candle from across the room.

The second skill is that of shaping. You must find or create a region of Faerie that is as alike to where you wish to cast the spell as possible, and differs only in the manner you wish it to differ. If you wish to summon a dragon, you must first find or create the region in which you wish to summon the dragon within Faerie. You must then shape this region, placing a dragon within it. Other minds will fight you – their perception of the world will impinge upon yours, and they will deny your dragon. You must persuade or overrule them and their wishes. This will be easier if they believe in dragons, and it will be harder if you are trying to shape something they have no familiarity with, or cannot believe would be here. In the desert it is far easier to shape Behemoth than Leviathan.

The third skill is that of summoning. You have created or found a region of Faerie which has what you desire. Now you must bring it through. This is at once the most straightforward and the most dangerous of skills. The ease with which you can do this depends crucially on how well you have succeeded at the first two steps – the more closely your region of Faerie matches the world where you want to bring it forth, the easier you will find this process, but you may not know how well you have achieved this until you try to bring the two into alignment. It is certainly possible to overcome a poorly matched shaping through brute force, but it is extremely taxing on the body and has on occasion proven fatal in its own right. Further, a failed summoning may have highly unpredictable consequences – you may lose control of Faerie as you bring it into alignment, and may bring through something you did not expect or change the world in wild and dangerous ways.

Not all magecraft involves all three skills. The skill of perception is highly useful in its own right, and the skill of shaping is the sole root of the mind skills. Summoning is rarely used on its own except by untrained wild talents, as it is so dependent on the other two, but it is not unknown for desperate or mad mages to bring things through from Faerie without proper preparation. We would strongly caution you against this, but the mere threat of it is sufficient to make most people think twice about giving a well trained mage no route of escape.

On summoned creatures

Many summoned things operate on principles that simply do not work in the world, and require active maintenance from a conscious mind to keep them in alignment with Faerie. Levitation will not sustain itself without the mage’s constant intervention, nor will magical fire burn eternal.

Many summoned creatures however are conscious entities in their own right, and many of them have a sufficient grasp of summoning that they can maintain their own link with Faerie and enforce the rules themselves. Thus a summoned dragon is not dependent on the mage for their powers of flight or fire, and a summoned demon is entirely capable of causing mischief without the mage’s consent.

On the reading and shaping of minds

Minds are directly linked to Faerie. A mage well trained in shaping can see into Faerie and use this sight to perceive other peoples’ thoughts and feelings. Further, they can manipulate these thoughts as the connection goes both ways. This sort of magic requires a great deal of skill to perform deftly and without damage to the other’s mind, but many mages specialize in just this sort of manipulation as it allows them to focus primarily on shaping to the exclusion of other skills.

This can be protected against. It is much easier to defend your mind than it is to attack it – a mage trained in shaping can erect walls and barriers around where their mind impinges on Faerie. Although they cannot cut off the link to Faerie itself, they can make it very difficult for others to affect. For this reason many people who are not otherwise interested in the study of magic learn enough shaping to guard themselves.

On healing

Fundamentally healing is magically simple – you match a region of Faerie to the body, you shape it healthy and you bring the change through to the world.

A word of caution though: Many things can sustain themselves in Faerie which are not viable in the world. The human body is good at adapting, but it has only so much capability to do so. If you do not understand the workings of the human body most thoroughly, there is a significant danger that you will merely make the problem worse.

It is much easier for a mage to heal themselves, because they are able to continually maintain the summoning and adjust it. They can perceive if they have created a problem and fix it, in the same way that a summoned creature sustains their link to Faerie themselves.

This is where the rumours of a mage’s immortality come from – it is untrue that a mage will live forever, but with enough training in the healing arts they can sustain themselves for a long time. As time goes on however an increasingly large amount of time and effort must be spent on maintaining themselves, and it is common for sufficiently old mages to voluntarily end their life knowing they will continue to manifest within Faerie.

On the raising of the dead

In many ways, the raising of the dead is the most trivial of acts. When a person’s body dies, the presence of their mind in Faerie remains. No longer routed to the world, it is now adrift in Faerie and may grow or fade or merge in a way that a living mind can not. As a result the older a mind is the less it resembles its human self, and it may indeed no longer exist in any recognisable form, but the mind of one recently deceased is present and may be readily conversed with. It is even possible to shape it to a new body and bring it through to the real world.

A word of caution: There is a question as to whether the person brought back is truly the one that died. Many religions believe that it is not, and that the person in question is an undead abomination. Many report that their resurrected friends lack some vital “spark” that they once had.

As further evidence that this is not a true raising, the same person may be raised many times, and indeed it is entirely possible to raise the living in the same manner.

On the denizens of Faerie

Many minds and beings live within Faerie. A mage trained in shaping may converse with them, and a mage trained in summoning may further enable them to manifest in the real world.

It is unclear whether these creatures are all the product of human imagination, or whether some of them arose spontaneously and indeed predate the human race (many of them claim to be this, but of course they would).

In particular many creatures we would recognise as gods or demons live within Faerie. They are unable to influence the world directly without a human being to bring them through, but they will be ever so grateful if you do so.

You should be extremely careful about granting these requests.

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An interesting experiment that no one seems to have performed

A thing I am interested in is group problem solving and group intelligence. When pondering this, the following interesting experiment occurred to me. I can’t find any evidence of it ever having been done unfortunately (yes, I could run it, but I’m not really in the right career for that…):

The basic idea is simple. You take some sort of untimed intelligence test – e.g. Raven’s Progressive Matrices might be a good choice (I don’t know enough about psychometrics to say for sure if it is or not).

You now take a largish sample of people. Each of these people takes the test.

Now you do something different. You pair the people off, and they take the test collaboratively. i.e. two people are put together in a room with the test and are asked to solve it together.

The question is this: How does the score of pairs relate to the score of individuals? Does this depend on any priming on how to collaborate you give them? (e.g. if one person has absolute deciding factor versus if they flip a coin on disagreements). Is it larger than the maximum of their two scores? Smaller?

You’d have to do some careful experiment design to watch for training effects – e.g. do half of the individual tests after the paired tests rather than before and see if it makes a difference – but I think most of these problems can be overcome with careful experiment design.

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An announcement and a brief retrospective

You may have noticed that I rapidly went quiet about the whole interviewing companies thing. This is actually because I found a new job rather than just because I got bored of writing about it.

I will be starting at Lumi on Monday. It’s not quite the job I claimed I was looking for, but it’s an opportunity to work on interesting research problems at scale with a great team who have done a successful startup before. How could I say no?

I mean, I totally could have said no if they’d failed my interview. It’s inappropriate for me to comment on specific questions, but suffice it to say they did OK. There were a few problems, but they were largely of the “We know this is a problem, but we’re either in the process of fixing it” or “If you feel very strongly this is a problem and want to take fixing it in hand, we’d be very up for that”, which I felt was a good enough response.

In general I’m not sure how useful the questions actually turned out to be. It was much harder to fit them into the interview than I expected it to be, and I’m not sure it really changed my mind about any of the companies I interviewed at – no one did amazingly, and the companies that did badly were companies I was already worried about. I think part of the problem is that the questions focused more on eliciting whether I should run away very fast rather than whether it would be a great place to work, and in general companies which are dysfunctional enough to be worth avoiding are so obviously so that the questions are maybe not needed.

I still think they’re useful to have, but I think maybe next time I find I’m looking for work I’ll throw out about half of them (I have no idea which half) and add in a few ones that probe whether it’s somewhere I should actually be excited about working at.

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The winning strategy for random ballot is not what I thought it was

In a previous post I used a normal distribution to show that if you didn’t have a majority of the vote then your probability of winning a majority of seats in a random ballot was maximized if you had an equal representation in every constituency.

Without the normality assumption, this turns out to be false. What is true is the following:

Theorem: Let \(W_i \sim \mathrm{Bernoulli}(q_i)\) be independent. Let \(q\) be a vector which maximizes \(P(\sum W_i > t)\) subject to \(\sum q_i = \mu\). Then if \(q_i \neq 0, 1\) and \(q_j \neq 0, 1\) then \(q_i = q_j\).

Proof:

Fix \(i, j\). Let \(u = \frac{q_i + q_j}{2}\), \(v = \frac{q_i – q_j}{2}\).

Then doing some algebra which I can’t currently be bothered to replicate in LaTeX (it’s fiddly but easy) we get \(P(W > t) = A + B v^2\) for some constants A, B. We are free to vary \(v\) without changing \(\sum q_i\), so we may choose \(v\) freely to maximize this value.

This expression only has a local maximum at \(v = 0\), but it may also be maximized by the end-points. These end-points occur when at least one of \(q_i, q_j\) is \(1\) or \(0\).

So for any pair \(i, j\) and any vector \(q\), if \(q_i \neq q_j\) and neither are 1 or 0 then we can strictly increase the probability \(P(W > t)\) by redistributing the two coordinates so that either \(q_i = q_j\) or one of \(q_i, q_j\) is \(1\) or \(0\).

This now proves our result almost immediately:

Let \(q\) be a vector that maximizes \(P(W > t)\). Suppose there are coordinates \(i, j\) such that neither \(q_i, q_j\) are \(1\) or \(0\). Then if they are not equal we can find some other \(q\) which strictly increases the probability, contradicting that \(q\) was a maximum.

QED

The question now becomes: What combination of zeroes and ones maximizes this probability?

In general, I don’t know. However some simulation suggests that the answer is quite different than I expected: If \(\mu > t\) then it’s easy, you just set at least \(t\) of the \(q_i\) to \(1\) and you’re done. However it appears to be the case that if \(mu < t\) what you want to do is have \(q_i\) non-zero for exactly \(t + 1\) values of i.

Here are some tables of brute force calculated optimum strategies. It’s all done with floating point numbers so some of these numbers are probably wrong where there’s not much in it, but should give a decent idea of the picture.

100 Constituencies

Popular vote Full constituencies Partial constituencies Victory probability
1% 0 49 0
2% 1 98 0
3% 2 97 0
4% 0 50 3.33067e-16
5% 1 50 3.33067e-16
6% 2 50 3.33067e-16
7% 0 50 1.44329e-15
8% 1 50 1.44329e-15
9% 2 50 1.44329e-15
10% 3 50 1.44329e-15
11% 4 50 1.44329e-15
12% 0 50 3.10862e-15
13% 1 50 3.10862e-15
14% 2 50 1.9984e-15
15% 3 50 3.10862e-15
16% 4 50 3.10862e-15
17% 1 49 6.66134e-15
18% 1 49 6.9611e-14
19% 1 49 6.14397e-13
20% 1 49 4.68026e-12
21% 1 49 3.11595e-11
22% 1 49 1.83468e-10
23% 1 49 9.65132e-10
24% 1 49 4.57618e-09
25% 1 49 1.9709e-08
26% 1 49 7.76285e-08
27% 1 49 2.81309e-07
28% 1 49 9.42896e-07
29% 1 49 2.93716e-06
30% 1 49 8.53927e-06
31% 1 49 2.32595e-05
32% 1 49 5.95608e-05
33% 1 49 0.000143831
34% 1 49 0.000328471
35% 1 49 0.000711237
36% 1 49 0.0014636
37% 1 49 0.00286849
38% 1 49 0.00536505
39% 1 49 0.00959356
40% 1 49 0.0164292
41% 1 49 0.0269891
42% 1 49 0.0425947
43% 1 49 0.0646781
44% 1 49 0.0946258
45% 1 49 0.133574
46% 1 49 0.182179
47% 1 49 0.240411
48% 1 49 0.307415
49% 1 49 0.381478

650 Constituencies

Popular vote Full constituencies Partial constituencies Victory probability
2% 3 325 3.77476e-15
3% 7 325 4.88498e-15
4% 2 325 7.88258e-15
5% 10 325 6.21725e-15
6% 15 325 7.88258e-15
7% 4 325 1.38778e-14
8% 10 325 1.04361e-14
9% 9 325 8.99281e-15
10% 23 325 1.04361e-14
11% 22 325 8.99281e-15
12% 1 325 1.14353e-14
13% 10 325 1.53211e-14
14% 10 325 2.67564e-14
15% 23 325 1.53211e-14
16% 4 325 1.86517e-14
17% 17 325 2.65343e-14
18% 8 325 1.96509e-14
19% 16 325 1.94289e-14
20% 8 325 2.19824e-14
21% 9 325 2.17604e-14
22% 21 325 2.19824e-14
23% 22 325 2.17604e-14
24% 34 325 2.19824e-14
25% 35 325 2.17604e-14
26% 12 325 2.88658e-14
27% 5 325 2.28706e-14
28% 3 325 2.94209e-14
29% 8 325 2.77556e-14
30% 0 325 3.25295e-14
31% 4 325 3.64153e-14
32% 11 325 3.34177e-14
33% 17 325 3.64153e-14
34% 23 325 4.21885e-14
35% 6 325 4.54081e-14
36% 1 324 1.14242e-13
37% 1 324 5.57576e-12
38% 1 324 1.98633e-10
39% 1 324 5.21813e-09
40% 1 324 1.02063e-07
41% 1 324 1.49951e-06
42% 1 324 1.66858e-05
43% 1 324 0.00014173
44% 1 324 0.000926012
45% 1 324 0.00468993
46% 1 324 0.0185649
47% 1 324 0.0579747
48% 1 324 0.144439
49% 1 324 0.291241

(1% value omitted because it gave buggy results, I think due to underflow).

So if we take 0.01% as the “this is kinda plausible” chance (i.e. we expect it to happen about once every 100 elections) and \(10^-7\) as the “You don’t need to worry about this chance”, then with 100 constituencies you’re not going to win if you’ve got 29% or less of the vote and you become plausible at 40%. With 650 constituencies on the other hand you’ve no chance below 40% and become plausible somewhere between 47% and 48%.

So the strategy I had was wrong, but the basic result that this isn’t something we need to worry about seems to hold up.

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