The advice theorem says the following :

"All advice is good advice, because there is some course of action , between the advised course and its complete opposite, which must with mathematical certainty lead to the best possible outcome".

An imaginary example shows how application of the advice theorem could save your life.

Suppose you are suffering from scurvy because you are eating only trace fruit and veg and are completely ignorant of the requirement for dietary vitamin C, and a well meaning but (as) ignorant friend advises you that your ill health is caused by the presence in your diet of small amounts of fruit and veg - you will be fine, your friend advises, if you switch to a diet consisting solely of corned beef from a tin.

Luckily you are in possession of the advice theorem. Applying the theorem to the advice you have received, you appreciate that whatever the merits of this advice, the fact is that the best possible outcome will be achieved with a diet somewhere between all corned beef, and the complete opposite of that - say, all fruit and veg - i.e. somewhere along the new dietary axis implied by your friend's advice.

Unfortunately the theorem is unable to help with the problem of choosing a point along that axis - but commonsense suggests that the optimum point is more likely to be an interior one, rather than at either end - there are vastly more interior points than there are boundary points (just two) - so rather than a diet of all corned beef or all fruit and veg, you decide to introduce a moderate amount of fruit and veg into your diet so as to operate at an interior point of the new advisory axis, rather than at the "all corned beef" end point suggested by your friend. Within days your health is improving - your friend's incorrect advice, moderated by the advice theorem, has saved your life.

The only prerequisites for application of the advice theorem, are that :

1. You are able to define one or more axes of action implied by the advice - so that you can identify the two extremes within which both the actual course of action you take, and the optimum outcome, must occur - i.e. you can identify a course of action which is in some sense the complete opposite of the one advised. Obviously there will generally be no unique "completely opposite" course of action - but it doesn't matter , there will be some optimum point on whatever axis you choose. Of course , some axes will be more productive than others - but any axis you choose must contain some course of action which will result in some zero or greater improvement to your current situation.

2. You are able to rank at least notionally the outcomes of actions on a numeric scale such that the concept of a maximum is meaningful.

Provided these two conditions are met, then the advice theorem may be pictured as a graph of outcomes, with the vertical axis being how good the outcome, and the horizontal axis being the course of action taken intermediate between that recommended and its opposite.

Then - if for example the graph is a horizontal straight line, it does not matter which course you take. And there will be some graphs where one of the end points *is* the best outcome. And some with a hump - the optimum in the middle , and some a wiggly line and the optimum is just somewhere along there. However - for all possible graphs , it is the case that at least one of the courses of action along the axis *must* achieve the maximum possible outcome.

I discovered the theorem late last year, while driving back from Omarama to Dunedin after a weekend away with the kids in a tent, and swimming in the Ahuriri river and having a look at the World Gliding Grand Prix. As I drove back down towards Kurow, I reflected somewhat soberly, amid that somewhat sober landscape, on a bunch of advice I had dished out to a colleague a week before, and wondered whether in fact the complete opposite of my recommendations might not be the best course.

Then it hit me - whatever the true situation, my advice had at least some value in that it created for my colleague a new axis of action - consisting of all courses of action between what I recommended and its complete opposite, and that somewhere along that axis there must surely be a point which would achieve the best possible outcome. On my return to Dunedin I communicated by email the exciting news to my colleague - my advice could be shown mathematically to guarantee the best possible outcome....though it may need a bit of titration to find the optimal point, between following it to the letter, and doing the complete opposite. (Shortly afterwards - on Boxing day actually - my entire family came down with Salmonella, which made for a miserable Christmas and New Year. )

I was just reading an interesting article about something called "hormesis" in the New Scientist magazine today (9 August issue). I experienced an odd sensation of anti-deja-vu....I have never seen this before ! Which is indeed odd for such an apparently basic idea. So - this *is* homeopathy , right, under a different name ? (Not that I have anything against homeopathy - I learned from Mandy's blog that she consults a homeopath, and she's a really clever bastard so it can't be complete bollocks !)

But also - I *have* seen this before - its nothing other than my advice theorem : almost anything is good for you , its just the dose that you have to get right - but that is at least in part a simple mathematical tautology, rather than being biologically meaningful.

So my advice is - always apply the advice theorem to any advice you receive. (Unfortunately this leads to a still to be resolved paradox - "the advice paradox" : should we apply the advice theorem to the advice to always apply the advice theorem ? If we choose not to apply the advice theorem to this advice, then this implies we accept without qualification the advice to always apply the advice theorem, which contradicts the assumption that we did not apply it. This suggests that it is impossible not to apply the advice theorem to this advice - yet in that case it is impossible to apply the theorem, which requires us to be able to not apply the theorem)

But....pointless paradoxes aside - the advice theorem is a wonderfully liberating thing for advice-giving busy-bodies like myself - we can go forth and dish out our hot air with promiscuous abandon - just so long as we also hand out the antidote - a pamphlet describing the advice theorem (with suitable warnings not to try applying the theorem to advice relating to the theorem itself as serious injury may result)

And we should always every minute of our lives try to find novel axes of action and titrate our way up to the optimal point along them. Since - its a simple mathematical fact that its good to try new things.

## Friday, August 1, 2008

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