Mathematical modelling

I had the good fortune to attend a public talk on mathematical modelling, organised by the University of Birmingham (UK). The speaker, Dr Nira Chamberlain CMath FIMA CSci, is a council member of the appropriate institution, and so may reasonably be thought to be speaking for mathematicians generally.

He observed that there were many professional areas that used mathematics as a tool, and that they generally failed to see the need for professional mathematicians as such. He thought that mathematical modelling was one area where – at least for the more important problems – mathematicians ought to be involved. He gave examples of modelling, including one of the financial crisis.

The main conclusion seemed very reasonable, and in line with the beliefs of most ‘right thinking’ mathematicians. But on reflection, I wonder if my non-mathematician professional colleagues would accept it. In 19th century professional mathematicians were proclaiming it a mathematical fact that the physical world conformed to classical geometry. On this basis, mathematicians do not seem to have any special ability to produce valid models. Indeed, in the run up to the financial crash there were too many professional mathematicians who were advocating some mainstream mathematical models of finance and economies in which the crash was impossible.

In Dr Chamberlain’s own model of the crash, it seems that deregulation and competition led to excessive risk taking, which risks eventually materialised. A colleague who is a professional scientist but not a professional mathematician has advised me that this general model was recognised by the UK at the time of our deregulation, but that it was assumed (as Greenspan did) that somehow some institution would step in to foreclose this excessive risk taking. To me, the key thing to note is that the risks being taken were systemic and not necessarily recognised by those taking them. To me, the virtue of a model does not just depend on it being correct in some abstract sense, but also that ‘has traction’ with relevant policy and decision makers and takers. Thus, reflecting on the talk, I am left accepting the view of many of my colleagues that some mathematical models are too important to be left to mathematicians.

If we have a thesis and antithesis, then the synthesis that I and my colleagues have long come to is that important mathematical model needs to be a collaborative endeavour, including mathematicians as having a special role in challenging, interpret and (potentially) developing the model, including developing (as Dr C said) new mathematics where necessary. A modelling team will often need mathematicians ‘on tap’ to apply various methods and theories, and this is common. But what is also needed is a mathematical insight into the appropriateness of these tools and the meaning of the results. This requires people who are more concerned with their mathematical integrity than in satisfying their non-mathematical pay-masters. It seems to me that these are a sub-set of those that are generally regarded as ‘professional’. How do we identify such people?

Dave Marsay 

 

The limits of (atomistic) mathematics

Lars Syll draws attention to a recent seminar on ‘Confronting economics’ by Tony Lawson, as part of the Bloomsbury Confrontations at UCLU.

If you replace his every use of the term ‘mathematics’ by something like ‘atomistic mathematics’ then I would regard this talk as not only very important, but true. Tony approving quotes Whitehead on challenging implicit assumptions. Is his implicit assumption that mathematics is ‘atomistic’? What about Whitehead’s own mathematics, or that of Russell, Keynes and Turing? He (Tony) seems to suppose that mathematics can’t deal with emergent properities. So What is Whitehead’s work on Process, Keynes’ work on uncertainty, Russell’s work on knowledge or Turing’s work on morphogenesis all about?

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

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.

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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.

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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

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

Assessing and Communicating Risks and Uncertainty

David Spielgelhalter Assessing and Communicating Risks and Uncertainty Science in Parliament vol 69, no. 2, pp. 21-26. This is part of the IMA’s Mathematics Matters: A Crucial Contribution to the Country’s Economy.

This starts with a Harvard study showing that “a daily portion of red meat was associated with an increase in the annual risk of death by 13% over the period of the study”. Does this mean, as the Daily Express claimed, that “10% of all deaths could be avoided”?

David S uses ‘survival analysis’ to show that “a 40 year-old  man who eats a quarter-pound burger for his working lunch each day can expect, on average, to live to 79, while his mate who avoids the burger can expect to live to 80.” He goes on: “over a lifetime habit, each daily portion of red meat is associated with about 30 minutes off your life expectancy .. ” (my emphasis.)

As a mathematician advising politicians and other decision-makers, I would not be comfortable that policy-makers understood this, and would act appropriately. They might, for example, assume that we should all be discouraged from eating too much red meat.

Even some numerate colleagues with some exposure to statistics might, I think, suppose that their life expectancy was being reduced by eating red meat. But all that is being said is that if a random person were selected from the population as a whole then – knowing nothing about them – a statistician would ‘expect’ them to have a shorter life if they eat red meat. But every actual individual ‘you’ has a family history and many by 40 will have had cholesterol tests. It is not clear what the relevance to them is of the statistician’s ‘averaged’ figures.

Generally speaking, statistics gathered for one set of factors cannot be used to draw precise conclusions about  other sets of factors, much less about individuals. David S’s previous advice at Don’t Know, Can’t Know applies. In my experience, it is not safe to assume that the audience will appreciate these finer points. All that I would take from the Harvard study is that if you eat red meat most days it might be a good idea to consult your doctor. I would also hope that there was research going on into the factors in the apparent dangers.

See Also

I would appreciate a link to the original study.

Dave Marsay