Stock Market Distribution

Well a year late it seems I have my answer. I've already shown that a log/log graph of FTSE day changes is mostly linear. This indicates a scale invariant power law distribution!

Such distributions exist where the more you have the more you get. It implies, as suspected, that the size of a days change is liable to snow ball. Large losses encouraging more and more people to become bearish. Likewise large gains encouraging more and more people to become bullish.

This supports the "speculative" side of the market: that people trade not for long term fundamentals but try to trade in the short term and take advantage, or protect themselves, from current trends.

The question then is the distribution of bear/bull thresholds in the average human population. By which I mean if the number of people turning bearish increases with the size of the down turn there must be a distribution of thresholds beyond which people decide to sell. If there was just one threshold at say 1% then were the market ever to turn below 1% then everyone would sell. This doesn't happen.

Obviously the down turn is different from different people. People buying on that day will see a 1% drop. People who were perhaps 100% up will fall to 98% on a 1% drop. People who bought in after the 1% drop would only see a 1% drop where the market drop further to 2% down. There is a distribution of people going into the market, coming out and the existence of thresholds will need to be wrestled from them.

2Do!

Nature of Flagger/Coward Distribution

A linear function was chosen to describe the number of flaggers in a group, such that the total number of flaggers across all the groups was equal to the total number of cowards.

f(x) = a + x (n-2a)/g , where n is the total number of rabbits in each group, g is the number of groups and a is the number of flaggers in the 1st group. This integrates to ng/2 i.e. half the rabbits. The gradient of the function is given by grad=(n-2a)/g.
Using this as f(x) the expression Gc – Gf (from before) was calculated. This gives the average predator pressure experienced by flaggers relative to cowards for populations of heterogenous groups with varying numbers of flaggers and cowards as described by ‘a’.
Gf – Gc =


Substituting the expression a = (n-g*grad)/2 into the expression enables a clearer expression of heterogeneity.

Results
p=0.5, n=10, g=10

Once the gradient is steeper then 60% then flaggers experience rapidly decreasing predator pressure. This is a distribution with group one having 8 flaggers and 2 cowards, and group 10 having 2 flaggers and 8 cowards a range of {2,8}
For 50 groups (g=50) the graph has the same shape but the gradient only needs to be 12% before flaggers benefit more than cowards. The general rule is that the cut-off is proportional to 6/g.
p=0.5, n=20, g=10


For groups of large size there is little extra benefit to cowards until the groups have a larger range than {5,15}. After which there is considerable advantage to flaggers.
p=0.1, n=10, g=10

If the probability of not seeing the predator is very low (10%) there is range of distributions which benefit cowards very much. Suddenly when the range reaches {0.5,9.5} is benefits flaggers. 0.5 isn’t possible so there are no distributions which favour flaggers in small groups.
p=0.9, n=10, g=10

A simple curve where ranges {2.5,7.5} and steeper benefit flaggers. Flagging strategy is an anti-predator strategy and obviously only benefits where predators are skillful and not easily seen by individuals.
ConclusionFlagging as a strategy depends upon strong heterogeneity in flagger/coward ratios. Low dispersal of offspring i.e. children remaining in the warren will create such a heterogeneity. Sexual selection for flaggers is vulnerable to the evolution of a “slag” gene but may provide a mechanism for heterogeneity also.
Another avenue of investigation is in free markets. The local marginal strategy is always to defect and become a coward. However under certain situations strategies that benefit the group will lead to selection for groups of individuals who do not do this, i.e.e those who abandon local marginal thinking.

Co-operative genetics

On holiday while looking at rabbits was wondering how genes for particular sexual attractions spread in populations. What if there was a gene that made brave individuals attractive. Decided to look at the old area I did at college that of tail flagging in rabbits to warn of predators (assume it’s a warning signal rather than predator visual confusion). If this is in doubt then alarm calls will do as another model. Let there be two genes F and C which cause their bearer to F=flag (or give an alarm call) or C=just run away. Let there be two other genes that cause their bearer to find this behaviour attractive Lf and Lc. I’ll ignore a ‘Slag’ gene that just breeds with anything for now (for good reason since it messes this whole argument up, unless there is another gene which doesn’t like Slags etc). I assume a simple haploid genetics to make like easy.

Cowards and Braves

F = brave flagger
C= coward who saves himself
Lf= find flagger attractive
Lc=find coward attractive

These parents: flag+likes cowards x coward+likes flaggers give the following children:

FLc x CLf –> FLc,FLf,CLc,CLf (in equal proportion)

It can be seen that only 1/4 of FLc and CLf matings are like the parent and 1/4 are FLf and CLc.

FLf and CLc parents however only produce themselves.

The other combinations won’t happen since the very nature of the Lf and Lc genes is to make the bearer look to mate with only flaggers and cowards respectively. So if you are Lc you will only look for cowards to mate with, and if you are a flagger yourself only Lf will mate with you so FLc must mate with only CLf.

As a result the population of FLc and CLf both drop by 75% each generation (assuming that all individuals mate freely). If low density means that it is harder to find a mate then this will be larger.

By contrast FLf and CLc populations replace themselves each generation and get added to by the 25% of the FLc and CLf populations.

CONCLUSION: (without the Slag gene) flagging genes rapidly become associated with genes that mate with flaggers at the exclusion of cowards, and vice versa.

The Population Models

Let p=the probability that an individual does not see a predator. This is the probability the individual will be eaten in some time frame.

Take a group n numbering f flaggers and c cowards (n=f+c). The group comes under attack from an ambush predator looking for 1 individual. To succeed the predator needs only catch the prey unawares, after which it is a certain kill. The probability of being taken P is:

P = 1 / Number of unaware individuals

However the flaggers alert the group so they all need to be unaware at once. If all the flaggers are unaware (prob = p^f) then the total number of unaware individuals is f (since all the flaggers are unaware) + pc (the unaware cowards). The probability of being taken as a flagger is then:

Pf = p^f / (f + pc)

The cowards benefit from the flaggers flagging but also the probability that they are looking too or conversely the probability they are taken is p^(f+1). The probability of being taken as a coward is thus:

Pc = p^(f +1) / (f + pc)

CONCLUSION: Thus as a coward your probability of being taken is always p less than being a flagger. Flaggers because of their relative generosity will decline in number relative to cowards in a given population.

However note that Pc has a maximum value when c==n and a minimum value when c==0.

CONCLUSION: Thus having flaggers around instead of cowards is beneficial to cowards. It is better for cowards to move towards areas with higher proportions of flaggers. However as will be seen this leads to mating problems if there are genes that don’t like cowards.

Model 1

Let the population be spread out in g groups. The predation pressure on the population as a whole can be estimated by looking at the individual average, that is the sum of f*Pf for each group divided by the total number . As a continuous function we therefore need (given that f+c=n so c=n-f),

  for flaggers and

  for cowards

If f[x] is a constant f (as before) then the equations decompose to the equations before i.e. Gf = Pf and Gc = Pc

If f[x]=x i.e. it linearly increases through all the groups so that groups have increasing numbers of flaggers and let there be n groups so that flaggers range from 0 to n, then the integrals to calculate are (where g=n):

Which numerically integrated for p=1, n=10 gives

Gf = 0.1
Gc = 0.1

As expected since the average predation in any number of groups of 10 individuals if they are blind (unable to detect the predator) will be 1/10. i.e. Gf/Gc = 1. For the following probabilities where group size is n=10, Gf/Gc gives:

p=0.25   0.264687
p=0.5     0.295375
p=0.75   0.496825

That is the average predation of flaggers is around 30% that of cowards for lower probabilities and rises relatively as both groups become less vigilant. For larger groups (and a larger range of groups) the effect is ever more apparent so that with groups of size n=30 predation of flaggers is 10% that of cowards.

CONCLUSION
When the ratio of flaggers to cowards varies linearly across all the groups it remains beneficial to be a coward as an individual in any one group. However the flaggers in mostly cooperative groups do better than the cheats in mostly cheating groups so the average globally benefits the flaggers. So while cowards will locally do better than their neighbouring flaggers, globally their population will drop significantly faster.

Model 2

In all groups the rate of reduction of cowards is p less than that of flaggers. That is cowards will therefore come to dominate all static groups and if the group size is limited (as by limiting resources) then eventually flaggers will become extinct in these groups. However it is beneficial for cowards to move to areas of high flagging density.

In reality offspring, likely to carry the parents flagging or coward gene, will disperse around the parents creating pockets of high density. Infiltrating individuals may be excluded by territorial behaviour. It would be in the interests of the cowards to let flaggers into their territory, but not vice-versa.

Sexual selection may also provide a barrier to migration as described above.

2Do
Model that looks at changing rate of populations in cells (with genetics) dy/dx = ry(genetic factors) – pred(y)
Model that looks at changing rate of populations globally (with genetics)

Model that looks at changing rate of populations in a cell (with steady influx of individuals) i.e. like mandelbrot… look to see if there are critical influxes that lead to changes in the cell ratios… then we know any cell with neighb\ours that cause that level of migration will turn themselves into such a cell. i.e. if their outflux goes to the level that needed to promote them.

Examine the shape of distribution of ratios across groups. Population heterogeneity can create incongruent local versus global properties! Apply to markets where marginal action (i.e. to become a coward) is offset by global benefits of being a flagger. Relevance of markets? Conditions where “Invisible Hand of Markets” no longer operates… consider information dispersal through a population as one means to create heterogeneity.

4 Points on Anti-Capitalism

1) Anti-Capitalism doesn't exist. If someone described me as an anti-Nazi I'd attack the term because obv. they were trying to frame the discussion in Nazi terms. Anti-Nazism doesn't exist precisiely because we reject Nazism. No one supports "anti-unicorn believers" because we think there aren't any unicorns! 1st up: rejection of anti-Capitalism and all debate surrounding it (including the illegally biased news coverage). The correct thinking is: there is Capitalism AND everything else. Once the debate is allowed to start "anti-Capitalism" becomes a vast array of "other" views. Science and God-religions offer two huge subsets of "anti-Capitalism"... most "anti-Capitalists" are primarily attacking this fascism that refuses any debate whatsoever on the subject.

2) The problem with Capitalism for me is not itself, but how a society adopts it. Our society is making it the model of all human organisation. That is why scientists should be anti-Capitalist: because it is being adopted by prejudice and not research or knowledge. This is the main criticism of Capitalism. Final 2 points are why it is unsuitable.

3) Capitalism uses "exchange" in a market to value things. It has no other model of value. In a society where value is determined by exchange there can be no friendship, no family and no love. The reason is that when anyone interacts in such a society they keep an account book open of debt and credit. I help a "friend" only because I expect them to help me back. I have a child only because I think they will be useful to me. This does happen--and so does murder--but it is important it is not the only basis for human relationship. This is exacty what religions are teaching against in fact and why religions are "anti-Capitalism". "Real" wealth cannot be counted in account books they argue and the opposite behaviour of deleting account books is recommended (as the EU is being forced to discuss now ironically).

4) Who owns the wealth? This is the most famous "anti-Capitalist" issue. Mr Wheel and Mr Plough (the respective inventors of their namesakes) have brought humans literally untold riches. The human population rose on the invention of the latter alone from 1million to 100s of millions. If they somehow owned the rights to that then 1 family would literally own the world. Likewise if someone invented a single machine that could do all human labour more cheaply (a day that is approaching fast) would they really own everything, and more importantly how can an economy function when all humans are finally put out of work? It is clear to me that Capitalism isn't a complete, or even very well thought out system, all the time old problems remain to be solved... and the elite who are happy with the status quo will do everything to stop them being solved (like ever lasting light bulbs, and cancer cures, and DVD players that work longer than the warranty etc)... which is the greatest truth.