Applications of Statistics

Lars Syll has commented on a book by David Salsburg, criticising workaday applications of statistics. Lars has this quote:

Kolmogorov established the mathematical meaning of probability: Probability is a measure of sets in an abstract space of events.

This is not quite right.

  • Kolmogorov established a possible meaning, not ‘the’ meaning. (Actually Wittgenstein anticipated him.)
  • Even taking this theory, it is not clear why the space should be ‘measurable‘. More generally one has ‘upper’ and ‘lower’ measures, which need not be equal. One can extend the more familiar notions of probability, entropy, information and statistics to such measures. Such extended notions seem more credible.
  • In practice one often has some ‘given data’ which is at least slightly distant from the ‘real’ ‘events’ of interest. The data space is typically rather a rather tame ‘space’, so that a careful use of statistics is appropriate. But one still has the problem of ‘lifting’ the results to the ‘real events’.

These remarks seem to cover the criques of Syll and Salsburg, but are more nuanced. Statistical results, like any mathematics, need to be interpreted with care. But, depending on which of the above remarks apply, the results may be more or less easy to interpret: not all naive statistics are equally dubious!

Dave Marsay

AI pros and cons

Henry A. Kissinger, Eric Schmidt, Daniel Huttenlocher The Metamorphosis Atlantic August 2019.

AI will bring many wonders. It may also destabilize everything from nuclear détente to human friendships. We need to think much harder about how to adapt.

The authors are looking for comments. My initial reaction is here. I hope to say more. Meanwhile, I’d appreciate your reactions.

 

Dave Marsay

What logical term or concept ought to be more widely known?

Various What scientific term or concept ought to be more widely known? Edge, 2017.

INTRODUCTION: SCIENTIA

Science—that is, reliable methods for obtaining knowledge—is an essential part of psychology and the social sciences, especially economics, geography, history, and political science. …

Science is nothing more nor less than the most reliable way of gaining knowledge about anything, whether it be the human spirit, the role of great figures in history, or the structure of DNA.

Contributions

As against others on:

(This is as far as I’ve got.)

Comment

I’ve grouped the contributions according to whether or not I think they give due weight to the notion of uncertainty as expressed in my blog. Interestingly Steven Pinker seems not to give due weight in his article, whereas he is credited by Nicholas G. Carr with some profound insights (in the first of the second batch). So maybe I am not reading them right.

My own thinking

Misplaced Concreteness

Whitehead’s fallacy of misplaced concerteness, also known as the reification fallacy, “holds when one mistakes an abstract belief, opinion, or concept about the way things are for a physical or “concrete” reality.” Most of what we think of as knowledge is ‘known about a theory” rather than truly “known about reality”. The difference seems to matter in psychology, sociology, economics and physics. This is not a term or concept of any particular science, but rather a seeming ‘brute fact’ of ‘the theory of science’ that perhaps ought to have been called attention to in the above article.

Morphogenesis

My own speciifc suggestion, to illustrate the above fallacy, would be Turing’s theory of ‘Morphogenesis’. The particular predictions seem to have been confirmed ‘scientifically’, but it is essentially a logical / mathematical theory. If, as the introduction to the Edge article suggests, science is “reliable methods for obtaining knowledge” then it seems to me that logic and mathematics are more reliable than empirical methods, and deserve some special recognition. Although, I must concede that it may be hard to tell logic from pseudo-logic, and that unless you can do so my distinction is potentially dangerous.

The second law of thermodynamics, and much common sense rationality,  assumes a situation in which the law of large numbers applies. But Turing adds to the second law’s notion of random dissipation a notion of relative structuring (as in gravity) to show that ‘critical instabilities’ are inevitable. These are inconsistent with the law of large numbers, so the assumptions of the second law of thermodynamics (and much else) cannot be true. The universe cannot be ‘closed’ in its sense.

Implications

If the assumptions of the second law seem to leave no room for free will and hence no reason to believe in our agency and hence no point in any of the contributions to Edge: they are what they are and we do what we do. But Pinker does not go so far: he simply notes that if things inevitably degrade we do not need to beat ourselves up, or look for scape-goats when things go wrong. But this can be true even if the second law does not apply. If we take Turing seriously then a seeming permanent status quo can contain the reasons for its own destruction, so that turning a blind eye and doing nothing can mean sleep-walking to disaster. Where Pinker concludes:

[An] underappreciation of the Second Law lures people into seeing every unsolved social problem as a sign that their country is being driven off a cliff. It’s in the very nature of the universe that life has problems. But it’s better to figure out how to solve them—to apply information and energy to expand our refuge of beneficial order—than to start a conflagration and hope for the best.

This would seem to follow more clearly from the theory of morphogenesis than the second law. Turing’s theory also goes some way to suggesting or even explaining the items in the second batch. So, I commend it.

 

Dave Marsay

 

 

Probability as a guide to life

Probability is the very guide to life.’

Cicero may have been right, but ‘probability’ means something quite different nowadays to what it did millennia ago. So what kind of probability is a suitable guide to life, and when?

Suppose that we are told that ‘P(X) = p’. Often there is some implied real or virtual population, P, a proportion ‘p’ of which has the property ‘X’. To interpret such a probability statement we need to know what the relevant population is. Such statements are then normally reliable. More controversial are conditional probabilities, such as ‘P(X|Y) = p’. If you satisfy Y, does P(X)=p ‘for you’?

Suppose that:

  1. All the properties of interest (such as X and Y) can be expressed as union of some disjoint basis, B.
  2. For all such basis properties, B, P(X|B) is known.
  3. That the conditional probabilities of interest are derived from the basis properties in the usual way. (E..g. P(X|B1ÈB2) = P(B1).P(X|B1)+P(B2).P(X|B2)/P(B1ÈB2).)

The conditional probabilities constructed in this way are meaningful, but if we are interested in some other set, Z, the conditional probability P(X|Z) could take a range of values. But then we need to reconsider decision making. Instead of maximising a probability (or utility), the following heuristics that may apply:

  • If the range makes significant difference, try to get more precise data. This may be by taking more samples, or by refining the properties considered.
  • Consider the best outcome for the worst-case probabilities.
  • If the above is not acceptable, make some reasonable assumptions until there is an acceptable result possible.

For example, suppose that some urn, each contain a mix of balls, some of which are white. We can choose an urn and then pick a ball at random. We want white balls. What should we do. The conventional rule consists of assessing the proportion of white balls in each, and picking an urn with the most. This is uncontroversial if our assessments are reliable. But suppose we are faced with an urn with an unknown mix? Conventionally our assessment should not depend on whether we want to obtain or avoid a white ball. But if we want white balls the worst-case proportion is no white balls, and we avoid this urn, whereas if we want to avoid white balls the worst-case proportion is all white balls, and we again avoid this urn.

If our assessments are not biased then we would expect to do better with the conventional rule most of the time and in the long-run. For example, if the non-white balls are black, and urns are equally likely to be filled with black as white balls, then assessing that an urn with unknown contents has half white balls is justified. But in other cases we just don’t know, and choosing this urn we could do consistently badly. There is a difference between an urn whose contents are unknown, but for which you have good grounds for estimating proportion, and an urn where you have no grounds for assessing proportion.

If precise probabilities are to be the very guide to life, it had better be a dull life. For more interesting lives imprecise probabilities can be used to reduce the possibilities. It is often informative to identify worst-case options, but one can be left with genuine choices. Conventional rationality is the only way to reduce living to a formula: but is it such a good idea?

Dave Marsay

Uncertainty is not just probability

I have just had published my paper, based on the discussion paper referred to in a previous post. In Facebook it is described as:

An understanding of Keynesian uncertainties can be relevant to many contemporary challenges. Keynes was arguably the first person to put probability theory on a sound mathematical footing. …

So it is not just for economists. I could be tempted to discuss the wider implications.

Comments are welcome here, at the publisher’s web site or on Facebook. I’m told that it is also discussed on Google+, Twitter and LinkedIn, but I couldn’t find it – maybe I’ll try again later.

Dave Marsay

Instrumental Probabilities

Reflecting on my recent contribution to the economics ejournal special issue on uncertainty (comments invited), I realised that from a purely mathematical point of view, the current mainstream mathematical view, as expressed by Dawid, could be seen as a very much more accessible version of Keynes’. But there is a difference in expression that can be crucial.

In Keynes’ view ‘probability’ is a very general term, so that it always legitimate to ask about the probability of something. The challenge is to determine the probability, and in particular whether it is just a number. In some usages, as in Kolmogorov, the term probability is reserved for those cases where certain axioms hold. In such cases the answer to a request for a probability might be to say that there isn’t one. This seems safe even if it conflicts with the questioner’s presuppositions about the universality of probabilities. In the instrumentalist view of Dawid, however, suggests that probabilistic methods are tools that can always be used. Thus the probability may exist even if it does not have the significance that one might think and, in particular, it is not appropriate to use it for ‘rational decision making’.

I have often come across seemingly sensible people who use ‘sophisticated mathematics’ in strange ways. I think perhaps they take an instrumentalist view of mathematics as a whole, and not just probability theory. This instrumentalist mathematics reminds me of Keynes’ ‘pseudo-mathematics’. But the key difference is that mathematicians, such as Dawid, know that the usage is only instrumentalist and that there are other questions to be asked. The problem is not the instrumentalist view as such, but the dogma (of at last some) that it is heretical to question widely used instruments.

The financial crises of 2007/8 were partly attributed by Lord Turner to the use of ‘sophisticated mathematics’. From Keynes’ perspective it was the use of pseudo-mathematics. My view is that if it is all you have then even pseudo-mathematics can be quite informative, and hence worthwhile. One just has to remember that it is not ‘proper’ mathematics. In Dawid’s terminology  the problem seems to be that the instrumental use of mathematics without any obvious concern for its empirical validity. Indeed, since his notion of validity concerns limiting frequencies, one might say that the problem was the use of an instrument that was stunningly inappropriate to the question at issue.

It has long seemed  to me that a similar issue arises with many miscarriages of justice, intelligence blunders and significant policy mis-steps. In Keynes’ terms people are relying on a theory that simply does not apply. In Dawid’s terms one can put it blunter: Decision-takers were relying on the fact that something had a very high probability when they ought to have been paying more attention to the evidence in the actual situation, which showed that the probability was – in Dawid’s terms – empirically invalid. It could even be that the thing with a high instrumental probability was very unlikely, all things considered.

Artificial Intelligence?

The subject of ‘Artificial Intelligence’ (AI) has long provided ample scope for long and inconclusive debates. Wikipedia seems to have settled on a view, that we may take as straw-man:

Every aspect of learning or any other feature of intelligence can be so precisely described that a machine can be made to simulate it. [Dartmouth Conference, 1956] The appropriately programmed computer with the right inputs and outputs would thereby have a mind in exactly the same sense human beings have minds. [John Searle’s straw-man hypothesis]

Readers of my blog will realise that I agree with Searle that his hypothesis is wrong, but for different reasons. It seems to me that mainstream AI (mAI) is about being able to take instruction. This is a part of learning, but by no means all. Thus – I claim – mAI is about a sub-set of intelligence. In many organisational settings it may be that sub-set which the organisation values. It may even be that an AI that ‘thought for itself’ would be a danger. For example, in old discussions about whether or not some type of AI could ever act as a G.P. (General Practitioner – first line doctor) the underlying issue has been whether G.P.s ‘should’ think for themselves, or just apply their trained responses. My own experience is that sometimes G.P.s doubt the applicability of what they have been taught, and that sometimes this is ‘a good thing’. In effect, we sometimes want to train people, or otherwise arrange for them to react in predictable ways, as if they were machines. mAI can create better machines, and thus has many key roles to play. But between mAI and ‘superhuman intelligence’  there seems to be an important gap: the kind of intelligence that makes us human. Can machines display such intelligence? (Can people, in organisations that treat them like machines?)

One successful mainstream approach to AI is to work with probabilities, such a P(A|B) (‘the probability of A given B’), making extensive use of Bayes’ rule, and such an approach is sometimes thought to be ‘logical’, ‘mathematical, ‘statistical’ and ‘scientific’. But, mathematically, we can generalise the approach by taking account of some context, C, using Jack Good’s notation P(A|B:C) (‘the probability of A given B, in the context C’). AI that is explicitly or implicitly statistical is more successful when it operates within a definite fixed context, C, for which the appropriate probabilities are (at least approximately) well-defined and stable. For example, training within an organisation will typically seek to enable staff (or machines) to characterise their job sufficiently well for it to become routine. In practice ‘AI’-based machines often show a little intelligence beyond that described above: they will monitor the situation and ‘raise an exception’ when the situation is too far outside what it ‘expects’. But this just points to the need for a superior intelligence to resolve the situation. Here I present some thoughts.

When we state ‘P(A|B)=p’ we are often not just asserting the probability relationship: it is usually implicit that ‘B’ is the appropriate condition to consider if we are interested in ‘A’. Contemporary mAI usually takes the conditions a given, and computes ‘target’ probabilities from given probabilities. Whilst this requires a kind of intelligence, it seems to me that humans will sometimes also revise the conditions being considered, and this requires a different type of intelligence (not just the ability to apply Bayes’ rule). For example, astronomers who refine the value of relevant parameters are displaying some intelligence and are ‘doing science’, but those first in the field, who determined which parameters are relevant employed a different kind of intelligence and were doing a different kind of science. What we need, at least, is an appropriate way of interpreting and computing ‘probability’ to support this enhanced intelligence.

The notions of Whitehead, Keynes, Russell, Turing and Good seem to me a good start, albeit they need explaining better – hence this blog. Maybe an example is economics. The notion of probability routinely used would be appropriate if we were certain about some fundamental assumptions. But are we? At least we should realise that it is not logical to attempt to justify those assumptions by reasoning using concepts that implicitly rely on them.

Dave Marsay

Traffic bunching

In heavy traffic, such as on motorways in rush-hour, there is often oscillation in speed and there can even be mysterious ’emergent’ halts. The use of variable speed limits can result in everyone getting along a given stretch of road quicker.

Soros (worth reading) has written an article that suggests that this is all to do with the humanity and ‘thinking’ of the drivers, and that something similar is the case for economic and financial booms and busts. This might seem to indicate that ‘mathematical models’ were a part of our problems, not solutions. So I suggest the following thought experiment:

Suppose a huge number of  identical driverless cars with deterministic control functions all try to go along the same road, seeking to optimise performance in terms of ‘progress’ and fuel economy. Will they necessarily succeed, or might there be some ‘tragedy of the commons’ that can only be resolved by some overall regulation? What are the critical factors? Is the nature of the ‘brains’ one of them?

Are these problems the preserve of psychologists, or does mathematics have anything useful to say?

Dave Marsay

Who thinks probability is just a number? A plea.

Many people think – perhaps they were taught it – that it is meaningful to talk about the unconditional probability of ‘Heads’ (I.e. P(Heads)) for a real coin, and even that there are logical or mathematical arguments to this effect. I have been collecting and commenting on works which have been – too widely – interpreted in this way, and quoting their authors in contradiction. De Finetti seemed to be the only example of a respected person who seemed to think that he had provided such an argument. But a friendly economist has just forwarded a link to a recent work that debunks this notion, based on wider  reading of his work.

So, am I done? Does anyone have any seeming mathematical sources for the view that ‘probability is just a number’ for me to consider?

I have already covered:

There are some more modern authors who make strong claims about probability, but – unless you know different – they rely on the above, and hence do not need to be addressed separately. I do also opine on a few less well known sources: you can search my blog to check.

Dave Marsay

Unbirthday Paradox

Another puzzle, courtesy of a mathematics lecture I attended last night. It is  variant of the Birthday ‘Paradox‘. The original ‘paradox’ is that a typical group of people is much more likely to contain two people that share a birthday than most people would think. The variant was where 20 people were asked to pick an integer between 1 and 100 and it was found that two had picked ’42’. The mathematics is the same as for the birthday problem. But is it right?

.

.

.

.

Are there any unwarranted assumptions?

.

.

.

.

The ‘official’ (Wikipedia) answer to the birthday paradox would be correct if people were randomly selected from a population whose birthdays were uniformly distributed through the year. But are they? This is not a mathematical question, so it cannot be that the answer provided is ‘mathematically’ correct, can it? But one could perhaps say is that the answer provided would be correct if the assumptions were true, and would be approximately correct if they were approximately true – but a sensitivity analysis would be revealing.

The variant brings in greater uncertainties. For example, before the experiment we all guessed the probability. We did much better than the previous audience. Could this be relevant? In any case, why should we expect the guesses to be evenly distributed? Might there not be lucky numbers or other special numbers – such as 42 – that would be chosen?

If numbers were clumped for any reason, the probability of a match significantly increases. I can imagine lots of reasons why numbers should be clumped, but none why they should be anti-clumped, so it seems to me that the ‘official’ probability is actually at the lower end of a range of possible probabilities. Thus if the official probability is 83% I would consider 91% (= (83%+100%)/2) a better guess, and [83%,100%] better still.

The calculations are simpler if we consider the possibility of a match when we toss a coin twice. If P(Heads) = 0.5+e then

P(Match) = (0.5+e)2 + (0.5-e)2 = 0.5 + 2.e2. Thus 0.5 is a lower bound on the probability of a match, provided that coin tosses are independent.

See Also

More Puzzles.

….Dave Marsay

JIC, Syria and Uncertainty

This page considers the case that the Assad regime used gas against the rebels on 21st August 2013 from a theory of evidence perspective. For a broader account, see Wikipedia.

The JIC Assessment

The JIC concluded on 27th that it was:

highly likely that the Syrian regime was responsible.

In the covering letter (29th) the chair said:

Against that background, the JIC concluded that it is highly likely that the regime was responsible for the CW attacks on 21 August. The JIC had high confidence in all of its assessments except in relation to the regime’s precise motivation for carrying out an attack of this scale at this time – though intelligence may increase our confidence in the future.

A cynic or pedant might note the caveat:

The paper’s key judgements, based on the information and intelligence available to us as of 25 August, are attached.

Mathematically-based analysis

From a mathematical point of view, the JIC report is an ‘utterance’, and one needs to consider the context in which it was produced. Hopefully, best practice would include identifying the key steps in the conclusion and seeking out and hastening any possible contrary reports. Thus one might reasonably suppose that the letter on the 29th reflected all obviously relevant information available up to the ends of the 28th, but perhaps not some other inputs, such as ‘big data’, that only yield intelligence after extensive processing and analysis.

But what is the chain of reasoning (29th)?

It is being claimed, including by the regime, that the attacks were either faked or undertaken by the Syrian Armed Opposition. We have tested this assertion using a wide range of intelligence and open sources, and invited HMG and outside experts to help us establish whether such a thing is possible. There is no credible intelligence or other evidence to substantiate the claims or the possession of CW by the opposition. The JIC has therefore concluded that there are no plausible alternative scenarios to regime responsibility.

The JIC had high confidence in all of its assessments except in relation to the regime’s precise motivation for carrying out an attack of this scale at this time – though intelligence may increase our confidence in the future.

The report of the 27th is more nuanced:

There is no credible evidence that any opposition group has used CW. A number continue to seek a CW capability, but none currently has the capability to conduct a CW attack on this scale.

Russia claims to have a ‘good degree of confidence’ that the attack was an ‘opposition provocation’ but has announced that they support an investigation into the incident. …

In contrast, concerning Iraqi WMD, we were told that “lack of evidence is not evidence of lack”. But mathematics is not so rigid: it depends on one’s intelligence sources and analysis. Presumably in 2003 we lacked the means to detect Iraqi CW, but now – having learnt the lesson – we would know almost as soon as any one of a number of disparate groups acquires CW.  Many outside the intelligence community might not find this credible, leading to a lack of confidence in the report. Others would take the JIC’s word for it. But while the JIC may have evidence that supports their rating, it seems to me that they have not even alluded to a key part of it.

Often, of course, an argument may be technically flawed but still lead to a correct conclusion. To fix the argument one would want a much greater understanding of the situation. For example, the Russians seem to suggest that one opposition group would be prepared to gas another, presumably to draw the US and others into the war. Is the JIC saying that this is not plausible, or simply that no such group (yet) has the means? Without clarity, it is difficult for an outsider to asses the report and draw their own conclusion.

Finally, it is notable that regime responsibility for the attack of the 21st is rated ‘highly likely’, the same as their responsibility for previous attacks. Yet mathematically the rating should depend on what is called ‘the likelihood’, which one would normally expect to increase with time. Hence one would expect the rating to increase from possible (in the immediate aftermath) through likely to highly likely, as the kind of issues described above are dealt with. This unexpectedly high rating calls for an explanation, which would need to address the most relevant factors.

Anticipating the UN Inspectors

The UN weapons inspectors are expected to produce much relevant evidence. For example, it may be that even if an opposition group had CW an attack would necessarily lack some key signatures. But, from a mathematical point of view, one cannot claim that one explanation is ‘highly likely’ without considering all the alternatives and taking full account of how the evidence was obtained. It is quite true, as the PM argued, that there will always be gaps that require judgement to span. But we should strive to make the gap as slight as possible, and to be clear about what it is. While one would not want a JIC report to be phrased in terms of mathematics, it would seem that appropriate mathematics could be a valuable aid to critical thinking. Hopefully we shall soon have an assessment that genuinely rates ‘highly likely’ independently of any esoteric expertise, whether intelligence or mathematics.

Updates

30th August: US

The US assessment concludes that the attack was by Assad’s troops, using rockets to deliver a nerve agent, following their usual procedures. This ought to be confirmed or disconfirmed by the inspectors, with reasonable confidence. Further, the US claim ‘high confidence’ in their assessment, rather than very high confidence. Overall, the US assessment appears to be about what one would expect if Assad’s troops were responsible.

31st August: Blog

There is a good private-enterprise analysis of the open-source material. It makes a good case that the rockets’ payloads were not very dense, and probably a chemical gas. However, it points out that only the UN inspectors could determine if the payload was a prohibited substance, or some other substance such as is routinely used by respectable armies and police forces.

It makes no attribution of the rockets. The source material is clearly intended to show them being used by the Assad regime, but there is no discussion of whether or not any rebel groups could have made, captured or otherwise acquired them.

2nd September: France

The French have declassified a dossier. Again, it presents assertion and argumentation rather than evidence. The key points seem to be:

  • A ‘large’ amount of gas was used.
  • Rockets were probably used (presumably many).
  • No rebel group has the ability to fire rockets (unlike the Vietcong in Vietnam).

This falls short of a conclusive argument. Nothing seems to rule out the possibility of an anti-Assad outside agency loading up an ISO container (or a mule train) with CW (perhaps in rockets), and delivering them to an opposition group along with an adviser. (Not all the opposition groups all are allies.)

4th September: Germany

A German report includes:

  • Conjecture that the CW mix was stronger than intended, and hence lethal rather than temporarily disabling.
  • That a Hezbollah official said that Assad had ‘lost his nerve’ and ordered the attack.

It is not clear if the Hezbollah utterance was based on good grounds or was just speculation.

4th September: Experts

Some independent experts have given an analysis of the rockets that is similar in detail to that provided by Colin Powell to the UN in 2003, providing some support for the official dossiers. They asses that each warhead contained 50 litres (13 gallons) of agent. The assess that the rebels could have constructed the rockets, but not produced the large quantity of agents.

No figure is given for the number of rockets, but I have seen a figure of 100, which seems the right order of magnitude. This would imply 5,000 litres or 1,300 gallons, if all held the agent. A large tanker truck has a capacity of about 7 times this, so it does not seem impossible that such an amount could have been smuggled in.

This report essentially puts a little more detail on the blog of 31st August, and is seen as being more authoritative.

5th September: G20

The UK has confirmed that Sarin was used, but seems not to have commented on whether it was of typical ‘military quality’, or more home-made.

Russia has given the UN a 100 page dossier of its own, and I have yet to see a credible debunking (early days, and I haven’t found it on-line).

The squabbles continue. The UN wants to wait for its inspectors.

6th September: Veteran Intelligence Professionals for Sanity

An alternative, unofficial narrative. Can this be shown to be incredible? Will it be countered?

9th September: German

German secret sources indicate that Assad had no involvement in the CW attack (although others in the regime might have).

9th September: FCO news conference

John Kerry, at a UK FCO news conference, gives very convincing account of the evidenced for CW use, but without indicating any evidence that the chemicals were delivered by rocket. He is asked about Assad’s involvement, but notes that all that is claimed is senior regime culpability.

UN Inspectors’ Report

21st September. The long-awaited report concludes that rockets were used to deliver Sarin. The report, at first read, seems professional and credible. It is similar in character to the evidence that Colin Powell presented to the UN in 2003, but without the questionable ‘judgments’. It provides some key details (type of rocket, trajectory) which – one hopes – could be tied to the Assad regime, especially given US claims to have monitored rocket launches. Otherwise, they appear to be of  type that the rebels could have used.

The report does not discuss the possibility, raised by the regime, that conventional rockets had accidentally hit a rebel chemical store, but the technical details do seem to rule it out. There is an interesting point here. Psychologically, the fact that the regime raised a possibility in their defence which has been shown to be false increases our scepticism about them. But mathematically, if they are innocent then we would not expect them to know what happened, and hence we would not expect their conjectures to be correct. Such a false conjecture could even be counted as evidence in their favour, particularly if we thought them competent enough to realise that such an invention would easily be falsified by the inspectors.

Reaction

Initial formal reactions

Initial reactions from the US, UK and French are that the technical details, including the trajectory, rule out rebel responsibility. They appear to be a good position to make such a determination, and it would normally be a conclusion that I would take at face value. But given the experience of Iraq and their previous dossiers, it seems quite possible that they would say what they said even without any specific evidence. A typical response, from US ambassador to the UN Samantha Power was:

The technical details of the UN report make clear that only the regime could have carried out this large-scale chemical weapons attack.”

Being just a little pedantic, this statement is literally false: one would at least have to take the technical details to a map showing rebel and regime positions, and have some idea of the range of the rockets. From the Russian comments, it would seem they have not been convinced.

Media reaction

A Telegraph report includes:

Whether the rebels have captured these delivery systems – along with sarin gas – from government armouries is unknown. Even if they have, experts said that operating these weapons successfully would be exceptionally difficult.

”It’s hard to say with certainty that the rebels don’t have access to these delivery systems. But even if they do, using them in such a way as to ensure that the attack was successful is the bit the rebels won’t know how to do,” said Dina Esfandiary, an expert on chemical weapons at the International Institute for Strategic Studies.

The investigators had enough evidence to trace the trajectories followed by two of the five rockets. If the data they provide is enough to pinpoint the locations from which the weapons were launched, this should help to settle the question of responsibility.

John Kerry, the US secretary of state, says the rockets were fired from areas of Damascus under the regime’s control, a claim that strongly implicates Mr Assad’s forces.

This suggests that there might be a strong case against the regime. But it is not clear that the government would be the only source of weapons for the rebels, that the rebels would need sophisticated launchers (rather than sticks) or that they would lack advice. Next, given the information on type, timing and bearing it should be possible to identify the rockets, if the US was monitoring their trajectories at the time, and hence it might be possible to determine where they came from, in which case the evidence trail would lead strongly to the regime. (Elsewhere it has been asserted that one of the rockets was fired from within the main Syrian Army base, in which case one would have thought they would have noticed a rebel group firing out.)

17 September: Human Rights Watch

Human Rights Watch has marked the UN estimate of the trajectories on a map, clearly showing tha- they could have been fired from the Republican Guard 104 Brigade area.

Connecting the dots provided by these numbers allows us to see for ourselves where the rockets were likely launched from and who was responsible.

This isn’t conclusive, given the limited data available to the UN team, but it is highly suggestive and another piece of the puzzle.

This seems a reasonable analysis. The BBC has said of it:

Human Rights Watch says the document reveals details of the attack that strongly
suggest government forces were behind the attack.

But this seems to exaggerate the strength of the evidence. One would at least want to see if the trajectories are consistent with the rockets having been launched from rebel held areas (map, anyone?) It also seems a little odd that a salvo of M14 rockets appear to have been fired over the presidential palace. Was the Syrian Army that desperate? Depending on the view that one takes of these questions, the evidence could favour the rebel hypothesis. On the other hand, if the US could confirm that the only rockets fired at that time to those sites came from government areas, that would seem conclusive.

(Wikipedia gives technical details of rockets. It notes use by the Taliban, and quotes its normal maximum range as 9.8km. The Human Rights Watch analysis seems to be assuming that this will not be significantly reduced by the ad-hoc adaptation to carry gas. Is this credible? My point here is that the lack of explicit discussion of such aspects in the official dossiers leaves room for doubt, which could be dispelled if their ‘very high confidence’ is justified.)

18 September: Syrian “proof”

The BBC has reported that the Syrians have provide what they consider proof to the Russia that the rebels were responsible for the CW attack, and that the Russians are evaluating it. I doubt that this will be proof, but perhaps it will reduce our confidence in  the ‘very high’ likelihood that the regime was responsible. (Probably not!) It may, though, flush out more conclusive evidence, either way.

19 September: Forgery?

Assad has claimed that the materials recovered by the UN inspectors were forged. The report talks about rebels moving material, and it is not immediately clear, as the official dossiers claim, that this hypothesis is not credible, particularly if the rebels had technical support.

Putin has confirmed that the rockets used were obsolete Soviet-era ones, no longer in use by the Syrian Army.

December: US Intelligence?

Hersh claims that US had intelligence that the Syrian rebels had chemical weapons, and that the US administration  deliberately ‘adjusted’ the intelligence to make it appear much more damning of the Syrian regime. (This is disputed.)

Comment

The UN Inspectors report is clear about what it has found. It is careful not to make deductive leaps, but provides ample material to support further analysis. For example, while it finds that Sarin was delivered by rockets that could have been launched from a regime area, it does not rule out rebel responsibility. But it does give details of type, time and direction, such that if – as appears to be the case from their dossier – the US were monitoring the area, it should be possible to conclude that the rocket was actually fired by the regime. Maybe someone will assemble the pieces for us.

My own view is not that Assad did not do it or that we should not attack, but that any attack based on the grounds that Assad used CW should be supported by clear, specific evidence, which the dossiers prior to the UN report did not provide. Even now, we lack a complete case. Maybe the UN should have its own intelligence capability? Or could we attack on purely humanitarian grounds, not basing the justification on the possible events on 21 Aug? Or share our intelligence with the Russians and Chinese?

Maybe no-one is interested any more?

See Also

Telegraph on anti-spy cynicism. Letters. More controversially: inconclusive allegations. and an attempted debunking.

Discussion of weakness of case that Assad was personally involved. Speculation on UN findings.

A feature of the debate seems to be that those who think that ‘something must be done’ tend to be critical of those who question the various dossiers, and those who object to military action tend to throw mud at the dossiers, justified or not. So maybe my main point should be that, irrespective of the validity of the JIC assessment, we need a much better quality of debate, engaging the public and those countries with different views, not just our traditional allies.

A notable exception was a private blog, which looked very credible, but fell short claiming “high likelihood”. It gives details of two candidate delivery rockets, and hoped that the UN inspectors will have got evidence from them, as they did. Neither rocket was known to have been used, but neither do they appear to be beyond the ability of rebel groups to use (with support). The comments are also interesting, e.g.:

There is compelling evidence that the Saudi terrorists operating in Syria, some having had training from an SAS mercenary working out of Dubai who is reporting back to me, are responsible for the chemical attack in the Ghouta area of Damascus.

The AIPAC derived ‘red line’ little game and frame-up was orchestrated at the highest levels of the American administration and liquid sarin binary precursors mainly DMMP were supplied by Israeli handled Saudi terrorists to a Jabhat al-Nusra Front chemist and fabricator.

Israel received supplies of the controlled substance DMMP from Solkatronic Chemicals of Morrisville, Pa.

This at least has some detail, although not such as can be easily checked.

Finally, I am beginning to get annoyed by the media’s use of scare quotes around Russian “evidence”.

Dave Marsay

Are more intelligent people more biased?

It has been claimed that:

U.S. intelligence agents may be more prone to irrational inconsistencies in decision making compared to college students and post-college adults … .

This is scary, if unsurprising to many. Perhaps more surprisingly:

Participants who had graduated college seemed to occupy a middle ground between college students and the intelligence agents, suggesting that people with more “advanced” reasoning skills are also more likely to show reasoning biases.

It seems as if there is some serious  mis-education in the US. But what is it?

The above conclusions are based on responses to the following two questions:

1. The U.S. is preparing for the outbreak of an unusual disease, which is expected to kill 600 people. Do you: (a) Save 200 people for sure, or (b) choose the option with 1/3 probability that 600 will be saved and a 2/3 probability no one will be saved?

2. In the same scenario, do you (a) pick the option where 400 will surely die, or instead (b) a 2/3 probability that all 600 will die and a 1/3 probability no one dies?

You might like to think about your answers to the above, before reading on.

.

.

.

.

.

The paper claims that:

Notably, the different scenarios resulted in the same potential outcomes — the first option in both scenarios, for example, has a net result of saving 200 people and losing 400.

Is this what you thought? You might like to re-read the questions and reconsider your answer, before reading on.

.

.

.

.

.

The questions may appear to contain statements of fact, that we are entitled to treat as ‘given’. But in real-life situations we should treat such questions as utterances, and use the appropriate logics. This may give the same result as taking them at face value – or it may not.

It is (sadly) probably true that if this were a UK school examination question then the appropriate logic would be (1) to treat the statements ‘at face value’ (2) assume that if 200 people will be saved ‘for sure’ then exactly 200 people will be saved, no more. On the other hand, this is just the kind of question that I ask mathematics graduates to check that they have an adequate understanding of the issues before advising decision-takers. In the questions as set, the (b) options are the same, but (1a) is preferable to (2a), unless one is in the very rare situation of knowing exactly how many will die. With this interpretation, the more education and the more experience, the better the decisions – even in the US 😉

It would be interesting to repeat the experiment with less ambiguous wording. Meanwhile, I hope that intelligence agents are not being re-educated. Or have I missed something?

Also

Kahneman’s Thinking, fast and slow has a similar example, in which we are given ‘exact scientific estimates’ of probable outcomes, avoiding the above ambiguity. This might be a good candidate experimental question.

Kahneman’s question is not without its own subtleties, though. It concerns the efficacy of ‘programs to combat disease’. It seems to me that if I was told that a vaccine would save 1/3 of the lives, I would suppose that it had been widely tested, and that the ‘scientific’ estimate was well founded. On the other hand, if I was told that there was a 2/3 chance of the vaccine being ineffective I would suppose that it hadn’t been tested adequately, and the ‘scientific’ estimate was really just an informed guess. In this case, I would expect the estimate of efficacy to be revised in the light of new information. It could even be that while some scientist has made an honest estimate based on the information that they have, some other scientist (or technician) already knows that the vaccine is ineffective. A program based on such a vaccine would be more complicated and ‘risky’ than one based on a well-founded estimate, and so I would be reluctant to recommend it. (Ideally, I would want to know a lot more about how the estimates were arrived at, but if pressed for a quick decision, this is what I would do.)

Could the framing make a difference? In one case, we are told that ‘scientifically’, 200 people will be saved. But scientific conclusions always depend on assumptions, so really one should say ‘if …. then 200 will be saved’. My experience is that otherwise the outcome should not be expected, and that saving 200 is the best that should be expected. In the other case we are told that ‘400 will die’. This seems to me to be a very odd thing to say. From a logical perspective one would like to understand the circumstances in which someone would put it like this. I would be suspicious, and might well (‘irrationally’) avoid a program described in that way.

Addenda

The example also shows a common failing, in assuming that the utility is proportional to lives lost. Suppose that when we are told that lives will be ‘saved’ we assume that we will get credit, then we might take the utility from saving lives to be number of lives saved, but with a limit of ‘kudos’ at 250 lives saved. In this case, it is rational to save 200 ‘for sure’, as the expected credit from taking a risk is very much lower. On the other hand, if we are told that 400 lives will be ‘lost’ we might assume that we will be blamed, and take the utility to be minus the lives lost, limited at -10. In this case it is rational to take a risk, as we have some chance of avoiding the worst case utility, whereas if we went for the sure option we would be certain to suffer the worst case.

These kind of asymmetric utilities may be just the kind that experts experience. More study required?

 

Dave Marsay

Mathematics, psychology, decisions

I attended a conference on the mathematics of finance last week. It seems that things would have gone better in 2007/8 if only policy makers had employed some mathematicians to critique the then dominant dogmas. But I am not so sure. I think one would need to understand why people went along with the dogmas. Psychology, such as behavioural economics, doesn’t seem to help much, since although it challenges some aspects of the dogmas it fails to challenge (and perhaps even promotes) other aspects, so that it is not at all clear how it could have helped.

Here I speculate on an answer.

Finance and economics are either empirical subjects or they are quasi-religious, based on dogmas. The problems seem to arise when they are the latter but we mistake them for the former. If they are empirical then they have models whose justification is based on evidence.

Naïve inductivism boils down to the view that whatever has always (never) been the case will continue always (never) to be the case. Logically it is untenable, because one often gets clashes, where two different applications of naïve induction are incompatible. But pragmatically, it is attractive.

According to naïve inductivism we might suppose that if the evidence has always fitted the models, then actions based on the supposition that they will continue to do so will be justified. (Hence, ‘it is rational to act as if the model is true’). But for something as complex as an economy the models are necessarily incomplete, so that one can only say that the evidence fitted the models within the context as it was at the time. Thus all that naïve inductivism could tell you is that ‘it is rational’ to act as if the  model is true, unless and until the context should change. But many of the papers at the mathematics of finance conference were pointing out specific cases in which the actions ‘obviously’ changed the context, so that naïve inductivism should not have been applied.

It seems to me that one could take a number of attitudes:

  1. It is always rational to act on naïve inductivism.
  2. It is always rational to act on naïve inductivism, unless there is some clear reason why not.
  3. It is always rational to act on naïve inductivism, as long as one has made a reasonable effort to rule out any contra-indications (e.g., by considering ‘the whole’).
  4. It is only reasonable to act on naïve inductivism when one has ruled out any possible changes to the context, particularly reactions to our actions, by considering an adequate experience base.

In addition, one might regard the models as conditionally valid, and hedge accordingly. (‘Unless and until there is a reaction’.) Current psychology seems to suppose (1) and hence has little to help us understand why people tend to lean too strongly on naïve inductivism. It may be that a belief in (1) is not really psychological, but simply a consequence of education (i.e., cultural).

See Also

Russell’s Human Knowledge. My media for the conference.

Dave Marsay

Coin toss puzzle

This is intended as a counter-example to the view, such as Savage’s, that uncertainty can, in practice, be treated as numeric probability.

You have a coin that you know is fair. A known trickster (me?) shows you what looks like an ordinary coin and offers you a choice of the following bets:

  1. You both toss your own coins. You win if they match, otherwise they win.
  2. They toss their coin while you call ‘heads’ or ‘tails’.

Do you have any preference between the two bets? Why? And …

In each case, what is the probability that their coin will come up heads?

Dave Marsay

Clarification

In (1) suppose that you can arrange things so that the trickster cannot tell how your coin will land in time to influence their coin, so that the probability of a match is definitely 0.5, with no uncertainty. The situation in (2) can be similar, except that your call replaces the toss of a fair coin.

See Also

Other uncertainty puzzles .

Risks to scientists from mis-predictions

The recent conviction of six seismologists and a public official for reassuring the public about the risk of an earthquake when there turned out to be one raises many issues, mostly legal, but I want to focus on the scientific aspects, specifically the assessment and communication of uncertainty.

A recent paper by O’Hagan  notes that there is “wide recognition that the appropriate representation for expert judgements of uncertainty is as a probability distribution for the unknown quantity of interest …”.  This conflicts with UK best practice, as described by Spiegelhalter at understanding uncertainty. My own views have been formed by experience of potential and actual crises where evaluation of uncertainty played a key role.

From a mathematical perspective, probability theory is a well-grounded theory depending on certain axioms. There are plausible arguments that these axioms are often satisfied, but these arguments are empirical and hence should be considered at best as scientific rather than mathematical or ‘universally true’.  O’Hagan’s arguments, for example, start from the assumption that uncertainty is nothing but a number, ignoring Spiegelhalter’s ‘Knightian uncertainty‘.

Thus, it seems to me, that where there are rare critical decisions with a lack of evidence to support a belief in the axioms, one should recognize the attendant non-probabilistic uncertainty, and that failure to do so is a serious error, meriting some censure. In practice, one needs relevant guidance such as the UK is developing, interpreted for specific areas such as seismology. This should provide both guidance (such as that at understanding uncertainty) to scientists and material to be used in communicating risk to the public, preferably with some legal status. But what should such guidance be? Spiegelhalter’s is a good start, but needs developing.

My own view is that one should have standard techniques that can put reasonable bounds on probabilities, so that one has something that is relatively well peer-reviewed, ‘authorised’ and ‘scientific’ to inform critical decisions. But in applying any methods one should recognize any assumptions that have been made to support the use of those methods, and highlight them. Thus one may say that according to the usual methods, ‘the probability is p’, but that there are various named factors that lead you to suppose that the ‘true risk’ may be significantly higher (or lower). But is this enough?

Some involved in crisis management have noted that scientists generally seem to underestimate risk. If so, then even the above approach (and the similar approach of understanding uncertainty) could tend to understate risk. So do scientists tend to understate the risks pertaining to crises, and why?

It seems to me that one cannot be definitive about this, since there are, from a statistical perspective – thankfully – very few crises or even near-crises. But my impression is that could be something in it. Why?

As at Aquila, human and organisational factors seem to play a role, so that some answers seem to need more justification that others. Any ‘standard techniques’ would need take account of these tendancies. For example, I have often said that the key to good advice is to have a good customer, who desires an adequate answer – whatever it is – who fully appreciates the dangers of misunderstanding arising, and is prepared to invest the time in ensuring adequate communication. This often requires debate and perhaps role-playing, prior to any crisis. This was not achieved at Aquila. But is even this enough?

Here I speculate even more. In my own work, it seems to me that where a quantity such as P(A|B) is required and scientists/statisticians only have a good estimate of P(A|B’) for some B’ that is more general than B, then P(A|B’) will be taken as ‘the scientific’ estimate for P(A|B). This is so common that it seems to be a ‘rule of pragmatic inference’, albeit one that seems to be unsupported by the kind of arguments that O’Hagan supports. My own experience is that it can seriously underestimate P(A|B).

The facts of the Aquila case are not clear to me, but I suppose that the scientists made their assessment based on the best available scientific data. To put it another way, they would not have taken account of ad-hoc observations, such as amateur observations of radon gas fluctuations. Part of the Aquila problem seems to be that the amateur observations provided a warning which the population were led to discount on the basis of ‘scientific’ analysis. More generally, in a crisis, one often has a conflict between a scientific analysis based on sound data and non-scientific views verging on divination. How should these diverse views inform the overall assessment?

In most cases one can make a reasonable scientific analysis based on sound data and ‘authorised assumptions’, taking account of recognized factors. I think that one should always strive to do so, and to communicate the results. But if that is all that one does then one is inevitably ignoring the particulars of the case, which may substantially increase the risk. One may also want to take a broader decision-theoretic view. For example, if the peaks in radon gas levels were unusual then taking them as a portent might be prudent, even in the absence of any relevant theory. The only reason for not doing so would be if the underlying mechanisms were well understood and the gas levels were known to be simply consequent on the scientific data, thus providing no additional information. Such an approach is particularly indicated where – as I think is the case in seismology – even the best scientific analysis has a poor track record.

The bottom line, then, is that I think that one should always provide ‘the best scientific analysis’ in the sense of an analysis that gives a numeric probability (or probability range etc) but one needs to establish a best practice that takes a broader view of the issue in question, and in particular the limitations and potential biases of ‘best practice’.

The O’Hagan paper quoted at the start says – of conventional probability theory – that  “Alternative, but similarly compelling, axiomatic or rational arguments do not appear to have been advanced for other ways of representing uncertainty.” This overlooks Boole, Keynes , Russell and Good, for example. It may be timely to reconsider the adequacy of the conventional assumptions. It might also be that ‘best scientific practice’ needs to be adapted to cope with messy real-world situations. Aquila was not a laboratory.

See Also

My notes on uncertainty and on current debates.

Dave Marsay

Haldane’s The dog and the Frisbee

Andrew Haldane The dog and the Frisbee

Haldane argues in favour of simplified regulation. I find the conclusions reasonable, but have some quibbles about the details of the argument. My own view is that much of our financial problems have been due – at least in part – to a misrepresentation of the associated mathematics, and so I am keen to ensure that we avoid similar misunderstandings in the future. I see this as a primary responsibility of ‘regulators’, viewed in the round.

The paper starts with a variation of Ashby’s ball-catching observation, involving dog and a Frisbee instead of a man and a ball: you don’t need to estimate the position of the Frisbee or be an expert in aerodynamics: a simple, natural, heuristic will do. He applies this analogy to financial regulation, but it is somewhat flawed. When catching a Frisbee one relies on the Frisbee behaving normally, but in financial regulation one is concerned with what had seemed to be abnormal, such as the crisis period of 2007/8.

It is noted of Game theory that

John von Neumann and Oskar Morgenstern established that optimal decision-making involved probabilistically-weighting all possible future outcomes.

In apparent contrast

Many of the dominant figures in 20th century economics – from Keynes to Hayek, from Simon to Friedman – placed imperfections in information and knowledge centre-stage. Uncertainty was for them the normal state of decision-making affairs.

“It is not what we know, but what we do not know which we must always address, to avoid major failures, catastrophes and panics.”

The Game Theory thinking is characterised as ignoring the possibility of uncertainty, which – from a mathematical point of view – seems an absurd misreading. Theories can only ever have conditional conclusions: any unconditional misinterpretation goes beyond the proper bounds. The paper – rightly – rejects the conclusions of two-player zero-sum static game theory. But its critique of such a theory is much less thorough than von Neumann and Morgenstern’s own (e.g. their 4.3.3) and fails to identify which conditions are violated by economics. More worryingly, it seems to invite the reader to accept them, as here:

The choice of optimal decision-making strategy depends importantly on the degree of uncertainty about the environment – in statistical terms, model uncertainty. A key factor determining that uncertainty is the length of the sample over which the model is estimated. Other things equal, the smaller the sample, the greater the model uncertainty and the better the performance of simple, heuristic strategies.

This seems to suggest that – contra game theory – we could ‘in principle’ establish a sound model, if only we had enough data. Yet:

Einstein wrote that: “The problems that exist in the world today cannot be solved by the level of thinking that created them”.

There seems a non-sequitur here: if new thinking is repeatedly being applied then surely the nature of the system will continually be changing? Or is it proposed that the ‘new thinking’ will yield a final solution, eliminating uncertainty? If it is the case that ‘new thinking’ is repeatedly being applied then the regularity conditions of basic game theory (e.g. at 4.6.3 and 11.1.1) are not met (as discussed at 2.2.3). It is certainly not an unconditional conclusion that the methods of game theory apply to economies beyond the short-run, and experience would seem to show that such an assumption would be false.

The paper recommends the use of heuristics, by which it presumably means what Gigernezer means: methods that ignore some of the data. Thus, for example, all formal methods are heuristics since they ignore intuition.  But a dog catching a Frisbeee only has its own experience, which it is using, and so presumably – by this definition – is not actually using a heuristic either. In 2006 most financial and economics methods were heuristics in the sense that they ignored the lessons identified by von Neumann and Morgenstern. Gigerenzer’s definition seems hardly helpful. The dictionary definition relates to learning on one’s own, ignoring others. The economic problem, it seems to me, was of paying too much atention to the wrong people, and too little to those such as von Neumann and Morgenstern – and Keynes.   

The implication of the paper and Gigerenzer is, I think, that a heuristic is a set method that is used, rather than solving a problem from first principles. This is clearly a good idea, provided that the method incorporates a check that whatever principles that it relies upon do in fact hold in the case at hand. (This is what economists have often neglecte to do.) If set methods are used as meta-heuristics to identify the appropriate heuristics for particular cases, then one has something like recognition-primed decision-making. It could be argued that the financial community had such meta-heuristics, which led to the crash: the adoption of heuristics as such seems not to be a solution. Instead one needs to appreciate what kind of heuristic are appropriate when. Game theory shows us that the probabilistic heuristics are ill-founded when there is significant innovation, as there was both prior, through and immediately after 2007/8. In so far as economics and finance are games, some events are game-changers. The problem is not the proper application of mathematical game theory, but the ‘pragmatic’ application of a simplistic version: playing the game as it appears to be unless and until it changes. An unstated possible deduction from the paper is surely that such ‘pragmatic’ approaches are inadequate. For mutable games, strategy needs to take place at a higher level than it does for fixed games: it is not just that different strategies are required, but that ‘strategy’ has a different meaning: it should at least recognize the possibility of a change to a seemingly established status quo.

If we take an analogy with a dog and a Frisbee, and consider Frisbee catching to be a statistically regular problem, then the conditions of simple game theory may be met, and it is also possible to establish statistically that a heuristic (method) is adequate. But if there is innovation in the situation then we cannot rely on any simplistic theory or on any learnt methods. Instead we need a more principled approach, such as that of Keynes or Ashby,  considering the conditionality and looking out for potential game-changers. The key is not just simpler regulation, but regulation that is less reliant on conditions that we expect to hold but for which, on maturer reflection, are not totally reliable. In practice this may necessitate a mature on-going debate to adjust the regime to potential game-changers as they emerge.

See Also

Ariel Rubinstein opines that:

classical game theory deals with situations where people are fully rational.

Yet von Neumann and Morgenstern (4.1.2) note that:

the rules of rational behaviour must provide definitely for the possibility of irrational conduct on the part of others.

Indeed, in a paradigmatic zero-sum two person game, if the other person players rationally (according to game theory) then your expected return is the same irrespective of how you play. Thus it is of the essence that you consider potential non-rational plays. I take it, then, that game theory as reflected in economics is a very simplified – indeed an over-simplified – version. It is presumably this distorted version that Haldane’s criticism’s properly apply to.

Dave Marsay