Saturday, 20 April 2013

The Problem of Points - Symbols

Christian Kramp invented the usage of the symbol ! to represent factorial.  The idea of $n(n-1)(n-2)..2.1$ as an interesting product which represents all the ways that n unique and completely distinct objects can be permuted dates to at least the 12th century.  To show this is the case imagine the following thought experiment.

Imagine that a man walks into one of the world's most sumptuous brothels.  His reward was to have a sexual encounter with every available woman in the brothel.  He enters a room filled with seven of the most desirable women he has ever seen.  One has jet black hair, is nearly six feet tall and has a full and firm ass.  Two is smaller with short black hair and a wicked twinkle in here eye.  Three is central Asian looking, small perfectly formed breasts.  Four is from south east Asia, an imperturbable and fiercely independent soul.  Five is from central Europe, innocently cheeky.  Six is from the Caribbean  tall and rangy.  He knows that he will fuck them all.  He picks number one, goes with her, comes back, picks number three this time.  Repeats until he is done.  This is a permutation.  A single permutation.  How many ways could he have chosen?  The answer is 6!, which   is 720.  He could have had 720 chronologically different sexual experiences.

Imagine you are him.  Imagine his decision moments.  His first decision was to pick the tall jet black haired girl.  This was a so-called free choice.  A choice of one in six.  Immediately start imagining a total of six parallel worlds.  In each world, he started with one or another of the six girls.  OK.  Focus on his reality again.  In the real world, he now has to chose which one of the five remaining girls to fuck.  Now, in each of the 6 parallel worlds, he'll similarly need to make a 1 in 5 choice.  So for each and every one of the 6 imagined worlds, a further multiplication of 5 occurs.  There are now 6 times  5 or thirty worlds which capture all the possibilities he has to chose from.  Repeat and you will find 720 different worlds.  In making his particular ordering choice,  he in essence picked one of 720 different worlds.

Without knowing the man, his sexual preferences, etc, if we had to guess his selection we could imagine each of the 720 worlds was equally likely to be chosen.  That is, you might say that the likelihood of choosing the fucking order of <tall-dark then short dark then central Asian then SE Asian then central European and finally Caribbean> was $\frac{1}{6!}$. 

Insofar as every object is uniquely distinguishable from every other object, then this idea of permutations is quite core to the most basic choosing activity - their ordering.  Visually you can see this as lining them up in a particular way, touching them in a particular, order, placing them sequentially on a particular chronology.

Imagine you got to know that some kind of selection process has 1,000,000 different distinct possibilities, the final result being a collection of 6 uniquely distinguishable objects.  Well, if you didn't actually care about the ordering your set of possibilities would be reduced down to $\frac{1000000}{6!}$.  Imagine you needed to pick 6 women to come on a journey with you.  Someone informs you that the selection process imposed on you allowed you to pick a particular order-important set of 6 women with equal chances $\frac{1}{1000000}$.  But you know that it doesn't matter to you what order they got picked.  They're coming with you all the same.  In this case, you'd divide your million possibilities by 6!.  Thus $\frac{6!}{1000000}$ is the likelihood of picking those 6 women (or indeed any 6 women).  Here you're using the permutation formula to make order irrelevant.  By knowing the set of choices implied in a strict ordering, you can use that knowledge to throw away the importance of ordering.

Klamp chose the word 'factorial' because it sounded more French than its rivals.

Sunday, 14 April 2013

The road not built

My main criticism of Hayek's book was his weakness in dealing with the possibility of political mixtures.  This also extends to the titular metaphor of the book itself - the road metaphor, one leading to serfdom and the other to liberalism nicely captures the starkness of the choice Hayek wants to present.  A traveller faces a choice of selecting which road to travel down.  But in the realm of ideas, the metaphor seems to restrictive.  Ideas can be freely constructed, built, developed.    Whereas it makes no sense for a traveller to build a road which is somehow a mixture of the collectivist and the liberal road - if your destination choices were the city of liberalism and the city of collectivism, for ideas, this kind of constraint is unjustified.  We can and do build as many alternative roads as we like.

His characterisation of the totalitarian destination is a great critique.  But it leaves out the pleasure some might find in becoming infantalised by the state.  I have heard even young Russians talk favourably about Stalin, which I find deeply worrying.  Likewise, his characterisation of the ideal of a nineteenth century liberalism is intellectually bracing but it becomes hard to see how a man with cystic fibrosis, just to take an example, couldn't help feeling even in a liberal polity, somehow infaltalised and dependent on the largesse of others, and also perhaps somehow defeated by that culture.  And finally his refusal to talk about the endless variety of middle ways, those unspoken roads, unimagined destinations remains not a logical consequence of some line of reasoning but as a rhetorical device in a well-intentioned struggle to place as much distance between the society he respected and the totalitarian regime he despised.

Saturday, 13 April 2013

Problem of Points - The Solution

The solution to the problem of points tells you how you would divide up the stakes in a fair game (fair in the sense of each step outcome being equally likely to favour any player) between two players A and B if A needed $n_A$ more wins and B needs $n_B$.  Pascal and Fermat both end up counting the set of all possibilities and comparing the respective counts to each other and come up with the ratio 

$\sum_{k=0}^{n_B-1}\frac{(n_A+n_B-1)!}{k!(n_A+n_B-1-k)!}$ to $\sum_{k=n_B}^{n_A+n_B-1}\frac{(n_A+n_B-1)!}{k!(n_A+n_B-1-k)!}$.  

This rather ugly looking formulation is something I'll be looking at over the next couple of posts, in a way mathematicians usually don't.  Enamoured of Euclid, they think interesting maths involves proof and concise statement.  That does not work for me.  I want to unpack it and see some examples with real numbers.  Get a feel for it in use.  And after those posts, I'll be doing the same for gambler's ruin, which I personally think has caused me to think a lot more generally than this solution to the problem of points.

Before I finish on this short post, I'd like to say that this solution to the problem of the division of stakes, if you think about it, is the price of the seat if somebody wanted to buy you out of the game.  This is the fair price of your seat at that moment, or the fair price of your position in the same.  And given that the moment in question can analysed at any point in the game, including the moment before the game starts, it also represents an algorithm for working out the fair price of the game for both players, at all points.  That is, it tells you fully at all moments in the game the expected value of each hand.

If the stakes are a value of S then the expected value of one player is 

$S\frac{\sum_{k=0}^{n_B-1}\frac{(n_A+n_B-1)!}{k!(n_A+n_B-1-k)!}}{\sum_{k=0}^{n_B-1}\frac{(n_A+n_B-1)!}{k!(n_A+n_B-1-k)!} + \sum_{k=n_B}^{n_A+n_B-1}\frac{(n_A+n_B-1)!}{k!(n_A+n_B-1-k)!}}$

and for the other just has the other sum on the numerator.

Sunday, 7 April 2013

Mongrel ascendancy - how democracy can enshadow Hayekian freedom

Some more thoughts of Hayek's position on the extreme desirability of freedom.  Having read the Road to Serfdom I decided that the biggest weakness was in dealing with the possibility of mixed political systems.  Clearly he ought to have been aware that collectivist systems could survive for potentially quite long times, at the very least.  That is, he accepts that they're not inherently and immediately unstable.  Given this, why is he so weak on the possibilities inherent in mixed styles, somewhat Hayekian, somewhat collectivist?  I pointed out he makes a William Paley-like blunder in alleging the necessary starkness of this choice by the populace, assuming of course the populace have been afforded a voice.

I'd like to take a leaf out of Hayek's book on argumentative style and describe a possible society where collectivism wins every time at the ballot box, and Hayek would need to decide whether democracy is allowed to win in that case, or whether he can find a way for his form of freedom to win.

The set up actually contains two independent societies and in each case the electorate votes always for collectivism.  Society one is what I'll call the post-liberal society.  Imagine a society where a large majority people support a welfare state, where the government runs large fractions of industry, where a persistent and recalcitrant underclass are politically alienated, dulled, and distracted by other parts of their culture.  Where even those who are doing well depend on the state for education, health of poorer family members, support for the many many times in their careers that they lose their job.  For the purposes of the argument, I'm agnostic on how the culture got itself into that state (i.e. whether this was the effect, as Hayek might argue, of creeping collectivism's insipid effect on human culture, whether this was the effect of a cruel and unequal capitalism driven by corporate cabals expropriating increasing fractions of output in the name of capital, leaving labour less secure) or even whether this state is necessarily a bad thing.  That is, despite how I described it, I am happy to remain morally neutral on the desirability of this culture for the purposes of this argument.  Now imagine a range of political parties which vie for the votes of the masses.  One subset of those parties are pro-Hayek and another subset pro-collectivism.  It is entirely possible that for centuries, perhaps even for millennia, the populace would vote consistently for a party from the collectivist subset.  Despite all arguments made by the pro-Hayekian subset.  This was likely on Hayek's mind when he wrote the Road to Serfdom.  Democracy will have defeated that form of nineteenth century freedom Hayek prefers, and if, by hypothesis, those parties fail to persuade, the idea of Hayekian freedom is indefinitely defeated.  

OK.  Now society two.  A society of Hayekian tactical voters.   Let's imagine that a majority of them are in a similar position of government dependence as society one, but that they're all politically engaged.  And, by hypothesis, they all thought about it and decided that, in their own heads, they accepted Hayek's point about the corrosive effect on freedom that this kind of collectivism imposes.  With the extra wrinkle that, from a purely self-interested point of view, they continue indefinitely to vote for one or other collectivist parties.  The Hayekians made all the right arguments, won over all the hearts and minds, and still they chose to vote for the status quo due to a particular brand of rationalist calculation which weighed the cost of the transition to the new order too expensive relative to their present state.  Perhaps their utility function has a very steep discount curve which dramatically devalues future worth steeply, who knows.  This is either logically possible to imagine, or not.  I guess the Hayekian would argue that it was logically impossible, but I'm not so sure.  In any case, what else could the committed Hayekian do - he's won all of the arguments but finds self-interest of the mass producing a series of collectivist variants of political parties which, for generations, retain power and replicate the status quo.
Real Hayekians cannot ban collectivist parties.  Imagine Norway, year 2200.  Several generations of superb sovereign wealth fund investment decisions followed by a series of further discoveries of commodity resources just offshore have effectively allowed Norwegians to idle in relative comfort at the support of the state.  The state in turn has amassed enough capital to ensure a fairly decent standard of living for non-workers.  They lock down their borders. Those who don't like it either leave or somehow manage to run or work in private enterprises. The population growth is manageable.  Norwegian industry becomes hopelessly unproductive when compared with other nations.  They become, in effect, a rent seeking nation living off the diversified capital owned by the state.  Every Norwegian party which gets to power promises to maintain this status quo, among other things.  All the Hayekian Norwegian parties win over the populace in theory concerning the long term benefits of freedom.  But Norwegians become Augustinian in their love of it - endless deferring to a time in the future when they feel ready to switch, but never switching.

Another point missed by Hayek in his insistence that no political regime can be somewhat Hayekian, somewhat collectivist.  That in multi-party democracies where the current incumbents on occasion get ejected produces a series of striped Hayekian, then collectivist, then Hayekian, then collectivist political regimes.  This is also, in a sense, a mixed political regime.  The legislature will be similarly striped by both forms of policy approach.  The various government departments likewise will exhibit battle scars indicating the endless flip-flopping into and out of collectivist and minimal government regimes.  Assume Hayek is right. Then this regime switching is necessarily less efficient than a permanently Hayekian one.  Assume the social democrats are right.  Likewise this regime switching is necessarily less efficient than a permanently social democratic one.  The desired pattern of behaviour of democracy, namely that incumbents occasionally get defenestrated, would then, no matter who was right, result in less efficient societies.  Unless, of course, the most efficient regimes were mixed regimes.

Anyway, where do political parties come from anyway?  In particular, their policy variability?  Imagine a Hayek victory.  A party in power which implements Hayekian hegemony.  Collectivist parties lose the voter base.  For democracy to survive, the likely outcome is a series of similar, but somewhat different generally Hayekian parties.  But I would argue that you can rank all the broadly Hayekian parties which thrive and survive in this hypothetical society with respect to the degree of collectivism implicit in their manifestos.  Hayek himself saw a wide and important role for the state.  A family of Hayekian parties surely would vary in ways which could be characterised as more or less government controlled.  More or less collectivist.  Even in Hayek's wildest dreams, surely he has to face up to mixed regimes and the bare possibility that they may be better?

Thursday, 4 April 2013

The road not taken

I've been reading Hayek's Road to Serfdom this Easter and hugely enjoyed it, despite a couple of significant holes in the argumentative structure.

He does a great job bringing out the positive aspects of maintaining a liberal political structure in the face of totalitarian regimes.  And I mean great.  But he is incredibly weak on the possibility of what he refers to as a Middle Way between a liberal and a planned political economy.  He gives a corking definition of a liberal social order as follows : " the ordering of our affairs we should make as much use as possible of the spontaneous forces of society, and resort as little as possible to coercion".  This is a great definition and works really well if you are confident in the constancy and benign effects of those spontaneous forces (benign in the sense that their benefits outweigh the benefits of benign planning).

He traces the intellectual roots of this liberal view of his through Cicero, Tacitus, Montaigne, Hobbes (implicitly), Hume, Locke, Henry Sidgwick, John Stuart Mill.  But I also read him as a pessimist, in line with Schopenhauer, who lines up in direct opposition to all things Hegelian.  This is somewhat ironic given his view that liberalism died when it hit Germany.  The other ironic point here is that he tries to come up with a reason why the collectivist socialism which was born, he claims, in Germany should have had such an effect.  He says that the ideas were ".. supported .. by the great material progress of Germany .." among other things.  But he doesn't chase this potential connection any further - perhaps the ideas themselves stimulated a successful form of post-liberal social democratic capitalism which worked quite well.  This is all apposite in the current (2013) European crisis, I thing, insofar as that particular form of German capitalism which took such a hit in the 1940s and 1950s is once again demonstrating its ongoing robustness.  At some point, if not now already, this evidence of economic success may justifiably cause a re-valuation of the forms of post-liberal social democracy and stakeholder capitalism they represent.

Whereas Hayek tries his best to demolish the alleged benefits of centralised planning by criticising even idealised forms of it, you can see how Buchanan later comes in to point out the detail of the painful reality of central planning with public choice theory.  But Hayek's generalised arguments against the planning function are quite rich and varied.

He resists a movement in the meaning of 'freedom' away from the freedom from "..the arbitrary power of
other men" to "freedom from necessity".  This is the essence of his resistance and pits him as Schopenhauer against Bismarckian Hegelian Prussia.  The social democrats see it as an evolution up the hierarchy of wants, but Hayek sees it as a wrong turn.  This is a loss of power for you - the power for an other to order a re-distribution of your wealth for an end decided externally to you.  His is a literal and metaphorical refusal to travel down a Hegelian road which he thinks always (inevitably, I'm tempted to say only it would sound too Hegelian) leads to a bad destination.

His key line: "..competition ...cannot be combined with planning to any extent we like without ceasing to operate as an effective guide to production".  This is how he argues against the 'middle way'.    Just for now, leave out 'to any extent we like' and you have a statement which is plain wrong.  How did liberalism evolve in the first place?  And in the dozens of places where it did evolve, surely it managed precisely this combination.  It kind of reminds me of William Paley's famously wrong argument against evolution with its metaphor of the perfection of the eye's design.  I think Hayek sees  liberalism a bit like this.  Now the phrase I left on the side 'to any extent we like'.  Well, when you add that back in, the claim becomes incredibly weaker, merely stating that some limits must exist for the combination of liberalism and central planning you're now considering.

On that same page as the above quote, he suggests the two approaches are, when combined "..poor and efficient tools if they are complete", a classic Paley-like comment.  He goes on: "they are alternative principles used to solve the same problem, and a mixture of the two means that neither will really work and that the result will be worse than if either system had been consistently relied on".  This is, it seems to me, utterly unjustified by him anywhere.

The book makes wonderful contrasts between this artificially black and white choice and does so at precisely a moment in recent world history where the blackness and whiteness seemed most justified, mid-way through the second world war.

In chapter 4, he does a decent job of knocking out collectivist arguments by debunking the so-called inevitability of the breakdown of competition as capitalism evolves through accumulated technological progress.  I think it is one of his strongest chapters.

He's on shakier grounds in dealing with planning and democracy, in chapter 5.  He bemoans the vagueness of collectivist political ends whilst failing to notice precisely the same vagueness of a Sidgwickian utilitarianism.  He thinks he can see a withering or failure of a complete ethical code, something he says the collectivists need, but one argument for this withering is a reduction in the number and generality of ethical rules.  But their number and generality are neither here nor there.  What matters surely is the quantity of their real effect.  His pessimism on our own natural tendency to understand and favour our own kind, our own community leads him to happily abandon hope of any kind of collectivist planning, but he's happy to see the possibility of a liberal-inspired international federated political order.  Though, to be fair to him, he does distinguish between the kind of economic planning collectivists seek and the liberal 'negative' Rule of Law based planning which sees the creation of international federated political institutions.

For him the Rule of Law  ".. means that government in all its actions is bound by rules fixed and announced beforehand".  I could imagine middle way social democrats as happy to sign up to this with no fear of logical contradiction.  However when he claims that collectivist planning ".. necessarily involves deliberate discrimination  between particular needs of different people, and allowing one man to do what another must be prevented from doing" my first thought was that this is what a market also does.

He's back on form in the chapter on economic control and totalitarianism which a large number of on-target criticisms of some of the collectivist's favourite arguments.

A key point in his chapter 'Who, whom?' is the choice he thinks which we have to make in two systems around who does the planning of whom.  Either a small set of planners, versus a combination of our own individual enterprise together with a large dose of randomness.  But the success of individual enterprise is itself a function, partly, of capital, so we're right back to a small set of lucky planners.  And surely, at the macroeconomic level, if you actively chose option 2, namely randomness and enterprise, then institutionalising randomness like this we're forcing it to remain forever random.  Spelled out, it is like saying: do not try to fathom the business cycle as any attempt to understand it with a view to mitigating it is necessarily impossible.  This stationary position didn't win him too many friends going into the great depression, it is true, but chimes with his generally much more pessimistic view of the limits of human reasoning.  He spells out the likely psychological and economic consequences of a planned collectivist society well here too.  However, having argued in favour of randomness and enterprise picking winners, by chapter 9, on security, he's arguing that mitigating business cycles via the good kind of planning is something worth doing.  However he's withering on other forms of economic security or protectionism, and he tackles these by showing their unintended consequences which are, he believes, self-defeating in the long run.  This Schumpeterian view does strike the modern reader as harsh.  Ignatief's 'The needs of strangers' is an old favourite of mine and makes a decent case for a degree of economic security for citizens.

As I've pointed out, Hayek doesn't deliver on knock out arguments against a third way between liberalism and a splash of collectivist planning, something which seems a priori worth much deeper investigation by him.  

Secondly, his anti-Hegelian spirit may have left him with too static a view of the ideal of liberalism.  Surely he ought to be more pragmatic in the Rortian sense of avoiding such a historically rooted final vocabulary?  

Third, he appears to be peculiarly anti-Fractal in his implicit raising of the nation state to a level of importance which he doesn't justify. He clearly loves the individual and sees societies of individuals working best when there's no centralisation or concentration of power.  And this reasoning he extends to nation states, then finally to supra-national federations.  But why those boundaries?  Couldn't he work 'inwards' to fiefdoms and cliques below the level of nation states?  He talks often of collections of 'small nations' and perhaps this is implicit in his position - the deconstruction of the larger states into many more mini-states and state-lets.  But he would prefer his aggregated collections of humans to reflect precisely the same kind of liberalism?  Or variants?  Or other forms, including totalitarianism?  How does the variance of political structure compare to the variance amongst real individuals' motivations?  And on a galactic scale?  What if planet earth was one instance of a form of political life, against a whole universe of alternative forms?  Wouldn't nation state-hood seem somewhat parochial in this context?  You certainly need to see nineteenth century British liberalism in a historical context.  Seeing it as such of course doesn't force you to abandon liberal principles, but you may need to argue harder for them.

But for all that, Hayek's message is powerful - we are often tempted by the social utopia of collectivism but it pays us to look beyond our emotional response to the desirability of these utopias to the unintended consequences they may bring.  This scepticism we must keep, but balanced with optimism and what Rorty called called solidarity.