Transcript Anatomy of success, hierarchy, and inequality

Anatomy of success, hierarchy, and
inequality I.
Domains with a lot of uncertainty
have the highest
likelihood of skilled people failing
Success
Success comes with perseverance and improvements,
opposing luck, over which we have zero control.
Success
While the chances of getting lucky might be as small as
p0 ≪ 1 – success results from the subsequent deliberate
actions and skill acquisition with the aim to work on the
opportune occasion for boosting its chances of recurrence in
the future,
The positive probability increment δp > 0 in can describe the
effectiveness of learning, or the gain of advantageous skill
contributing toward a favorable outcome.
Enhancing success chances by
persistent learning and skill acquisition
We suppose that the efficiency of learning process can be
described by the probability gain of getting success in the
future, after every successful trial,
pn = 0 if pn−1+ω < 0.
The positive probability increment ω > 0 describes a
positive feedback on the motivation to perform further
trials after the previous success.
Enhancing success chances by
persistent learning and skill acquisition
Enhancing success chances by
persistent learning and skill acquisition
With no effect of learning, pi = p, all ui are independent identically
distributed random variables and PN is given by the binomial
distribution
Enhancing success chances by
persistent learning and skill acquisition
The effect of positive feedback
for the bimodal model
is revealed by the geometric distribution of distances Di between sequent success
events,
Enhancing success chances by
persistent learning and skill acquisition
This marginal distribution satisfies an intuitively plausible Pascal type recurrence
relation for the probabilities PN(n), expressing the simple idea of that n successes in
N trials can be reached either from n successes in N − 1 trials plus a final failure, or
from n − 1 or from n − 1 successes in N − 1 trials and a final success:
Enhancing success chances by
persistent learning and skill acquisition
This marginal distribution satisfies an intuitively plausible Pascal type recurrence
relation for the probabilities PN(n), expressing the simple idea of that n successes in
N trials can be reached either from n successes in N − 1 trials plus a final failure, or
from n − 1 or from n − 1 successes in N − 1 trials and a final success:
Multiplying it by xn and summing over all n = 0, . . . ,∞,
Enhancing success chances by
persistent learning and skill acquisition
Enhancing success chances by
persistent learning and skill acquisition
Enhancing success chances by
persistent learning and skill acquisition
Enhancing success chances by
persistent learning and skill acquisition
p = 1 − e−ωt, and r = p0/ω.
Enhancing success chances by
persistent learning and skill acquisition
• The probability density plot for the
number of successful trials in the process
with persistent learning, in which the
initial probability of success is p0 = 0.1
and the probability increment ω = 0.02.
• When learning matters, the number of
tries is attributed to skill.
• The probability gradients shown by
arrows ”worsen” the chances for success
if the number of trials is small, but
”enhance” these chances for longer trial
sequences.
Driving down cycle time in trials allows for more experiments, which
can produce better results for those with early luck compounded
Like a squirrel in a wheel.
Freud's repetition compulsion
When cause and effect are not well understood, or the environment is
permanently changing even those who do everything ”right”, will fail
more likely than not, as acquired skill does not necessarily pay off over
time.
The natural mechanism that can help to improve the long-term performance under
uncertainty is diversification of activities. By getting involved in many different
projects, e.g. by making a portfolio of many investments, or by bearing and raising
many children, one can dramatically increase the chance of that advantageous skill
and persistent efforts will redeem over time.
Like a squirrel in a wheel.
Freud's repetition compulsion
Let us consider a large group of individuals engaged in many different activities, each
being characterized by some probability of initial success p0 > 0 and by some
probability increment ω > 0, after every successful trial. The state of getting precisely n
< nmax successful outcomes after t < tmax trials in every activity is then characterized by
the probability
where nmax and tmax are precisely determined by p0 and ω given.
Like a squirrel in a wheel.
Freud's repetition compulsion
Let us consider a large group of individuals engaged in many different activities, each
being characterized by some probability of initial success p0 > 0 and by some
probability increment ω > 0, after every successful trial. The state of getting precisely n
< nmax successful outcomes after t < tmax trials in every activity is then characterized by
the probability
where nmax and tmax are precisely determined by p0 and ω given.
The equilibrium state of such a group striving for success in a variety of activates
can be determined as a state of maximum entropy:
Like a squirrel in a wheel.
Freud's repetition compulsion
The equilibrium state of such a group striving for success in a variety of activates
can be determined as a state of maximum entropy:
It is worth a mention that the number of
different (n, t)–states of having precisely
n successful outcomes after t trials
grows unboundedly when the
probability increment tends to zero ω →
0 and therefore entropy of success does
either.
Like a squirrel in a wheel.
Freud's repetition compulsion
The equilibrium state of such a group striving for success in a variety of activates
can be determined as a state of maximum entropy:
The value of entropy increases as the
number of states grows, as
nmax, tmax ∼ ω−1
and tends to infinity for reducing
probability increments ω → 0.
The maximum entropy gradient is
observed for the initial probability
of success p0 ≈ 1/3.
Like a squirrel in a wheel.
Freud's repetition compulsion
The equilibrium state of such a group striving for success in a variety of activates
can be determined as a state of maximum entropy:
Therefore, when success might breed
success, but initial success was random,
the most likely behavior to be
observed over the large enough group
of individuals engaged in a variety of
activities is to make things into a
matter of routine, by continuously
repeating actions over and over again,
without searching for any improvement
of the future chances for success.
Repetition compulsion
Being in a group in the face of uncertainty, we are forced to repeat always the same
behavior pattern, without any improvement – as no lesson can been learnt from the
old experience.
The endless repetition of behavior, or life patterns, which were difficult (or
distressing) in previous life was a key concept in Freud’s understanding of mental
life – repetition compulsion.
Repetition compulsion
Being in a group in the face of uncertainty, we are forced to repeat always the same
behavior pattern, without any improvement – as no lesson can been learnt from the
old experience.
The endless repetition of behavior, or life patterns, which were difficult (or
distressing) in previous life was a key concept in Freud’s understanding of mental
life – repetition compulsion.
The essential character-traits which remain always the same and which are
compelled to find expression in a repetition of the same experience appeared to
Freud as ultimately contradicting with the organism’s search for pleasure:
”hypothesis of a compulsion to repeat - something that seems more primitive,
more elementary, more instinctual than the pleasure principle which it overrides”.
In the later editions of his work, Freud had extended this point, by stating that
”such the repetitions are of course the activities of instincts intended to lead
to satisfaction; but no lesson has been learnt from the old experience of these
activities having led only to unpleasure”.
The rich get richer. Pareto principle
The self-reinforcing behavior of certain probability distributions and stochastic
processes are known since the early works of Gibrat and Yule:
A stochastic urn process
discrete units of wealth, usually called ”balls” ◦ , are added continuously as
an increasing function of the number of balls already present in a set of
cells, usually called ”urns” | |, arranged in linear order
The rich get richer. Pareto principle
At each round of the urn process, either a bar or a ball is selected with probability
α and 1 − α, respectively. If a ball is selected, it is thrown in such a way that each
space in all cells has an equal chance of receiving it. If a bar is selected, it is placed
next to an existing bar, so that the new cell of unit size emerges at a rate α.
The rich get richer. Pareto principle
The size of the k-th cell, tk, be the number of balls in this cell plus one, i.e.
the number of spaces existing in the cell: between two balls, or between
two bars, or between a ball and a bar. The aggregate size of all cells
is increased steadily by one at the end of the round, regardless of whether
a bar or a ball is selected at any given round, either because the size of
one of the cells is increased by one or because a new cell of size 1 is
added.
The rich get richer. Pareto principle
Let p(x, t) be the expected value of the number of cells with size x when the
aggregate size of all cells is t. Then, for x = 1, we have
where α is the probability that p(1, t) is increased by one and (1 − α)p(1, t)/t is the
probability that p(1, t) is decreased by one as a result of a ball falling in one of the
unit-sized cells.
The rich get richer. Pareto principle
Let p(x, t) be the expected value of the number of cells with size x when the
aggregate size of all cells is t. Then, for x = 1, we have
where α is the probability that p(1, t) is increased by one and (1 − α)p(1, t)/t is the
probability that p(1, t) is decreased by one as a result of a ball falling in one of the
unit-sized cells.
At the steady state, for all x = 1, 2, ..., it should be
The rich get richer. Pareto principle
Let p(x, t) be the expected value of the number of cells with size x when the
aggregate size of all cells is t. Then, for x = 1, we have
The rich get richer. Pareto principle
Let p(x, t) be the expected value of the number of cells with size x when the
aggregate size of all cells is t. Then, for x = 1, we have
The rich get richer. Pareto principle
for any time t, and therefore, for the stationary distribution, it will be also true that
The rich get richer. Pareto principle
for any time t, and therefore, for the stationary distribution, it will be also true that
The Yule distribution,
The rich get richer. Pareto principle
The Yule
distribution,
The cumulative distribution function for the Yule distribution
x
The exponent ρ is the inverse probability
to add a ball (a unit of wealth) at a round
that is nothing else but the average
wealth per cell in the urn model.
The processes of accumulated
advantage lead to the skewed,
heavy-tailed (Pareto) distributions
of wealth.
Inequality rising from risk taking
under uncertainty
”Anyone who bet any part of his fortune, however small, on a mathematically
fair game of chance acts irrationally,” wrote Daniel Bernoulli in 1738.
People’s preferences with regard to choices that have uncertain outcomes
are described by the expected utility hypothesis. This hypothesis states that
under the quite general conditions the subjective value associated with an
uncertain outcome is the statistical expectation of the individual’s
valuations over all outcomes.
In particular, a decision maker could use the expected value criterion as a
rule of choice in the presence of risky outcomes.
Inequality rising from risk taking
under uncertainty
The individual’s risk aversion is accounted by a mathematical function called the
utility function
Utility refers to the perceived value of a good (or wealth), and the utility
function (viewed as a continuous function of actual wealth) describes the
attitudes towards risky projects of a ”rational trader”, whose objective
is to maximize growth of his wealth in the long term. Such a trader would
attach greater weight to losses than he would do to gains of equal magnitude.
Inequality rising from risk taking
under uncertainty
The risk aversion implies that the utility functions of interest are concave.
The plausible example of utility functions is given by
where 0 < λ < 1 is the risk tolerance
parameter – as λ decreases, traders
become more risk-averse and vice versa.
Inequality rising from risk taking
under uncertainty
The risk aversion implies that the utility functions of interest are concave.
The plausible example of utility functions is given by
where 0 < λ < 1 is the risk tolerance
parameter – as λ decreases, traders
become more risk-averse and vice versa.
In the limit of maximum risk avoidance, λ → 0,
Inequality rising from risk taking
under uncertainty
Inequality rising from risk taking
under uncertainty
Accordingly the maximum entropy principle, the system would evolve toward
the state of maximum entropy characterized by the probability distribution
which can be achieved in the largest number of ways, being the most likely
distribution to be observed.
Inequality rising from risk taking
under uncertainty
Accordingly the maximum entropy principle, the system would evolve toward
the state of maximum entropy characterized by the probability distribution
which can be achieved in the largest number of ways, being the most likely
distribution to be observed.
Inequality rising from risk taking
under uncertainty
We are interested in the probability distribution of wealth over the population pw
with maximum entropy under the condition of maximum risk avoidance.
Inequality rising from risk taking
under uncertainty
We are interested in the probability distribution of wealth over the population pw
with maximum entropy under the condition of maximum risk avoidance.
The Pareto
distribution of wealth
Wealth inequality can be viewed as a direct statistical consequence of making
decisions under uncertainty, under the condition of zero risk tolerance
Inequality rising from risk taking
under uncertainty
We are interested in the probability distribution of wealth over the population pw
with maximum entropy under the condition of maximum risk avoidance.
Inequality rising from risk taking
under uncertainty
We are interested in the probability distribution of wealth over the population pw
with maximum entropy under the condition of maximum risk avoidance.
The more risk is taken by traders investing under uncertainty, the more unequal
distribution of assets among them is likely to be observed in the long term.
... the more adventurous traders, the more their fortune, the less the number of
lucky ones.
Conclusions
• We have introduced and studied the probability model of success. The
probability increments of getting success in the future can be maximized
over time when the consistent efforts are made in a direction of highest
positive impact and when the personal role of an actor within a team is
increasingly important.
• We have also demonstrated that being in a group facing uncertainty, we
may be trapped within the repetition compulsion and forced to repeat
always the same behavior pattern, without any improvement – as no
lesson can been learnt from the old experience.
• Wealth inequality among the population rises from taking risky decisions
under uncertainty by the vital few: the more adventurous traders, the
more their fortune, the less the number of lucky ones.