Transcript Survival under uncertainty

ディマ
D.Sc.(habil.) Dimitri Volchenkov
Survival Under Uncertainty
An Introduction to Probability
Models of Social Evolution
Survival Under Uncertainty
An Introduction to Probability
Models of Social Evolution
State secession in 1800-2014
•The mean lifetime
of a state is 122
years;
•Half–life of a state
84.53 years
Survival Under Uncertainty
An Introduction to Probability
Models of Social Evolution
State secession in 1800-2014
•The mean lifetime
of
a state isdo
122the states decay exponentially fast?
• Why
years;
• There
•Half–life
of awill
state be a lot of migrants everywhere ...
84.53 years
 Uncertainty about efficiency of the societal
institutions ...
 Survival under uncertainty...
Structure of uncertainty
Interplay of
many random
factors
May evolve on
the different
time scales
Uncertainty
objective
uncertainty
subjective
uncertainty
A sudden volcanic eruption and an occasional fail to
pass a test manifest the different types of uncertainty
that we may face.
A species subsists as long as the means available
suffice to maintain it
•The minimal level of subsistence
(demand) for a given species during a
certain period of time can be quantified
by a real number d ∈ [0, 1].
•Another real number, s ∈ [0, 1],
appraises the amount of nourishment
(supply) available during the same
period of time.
We assume that the species survives as
long as d < s, but dies out immediately
after the carrying capacity of the
habitat is exceeded, d ≥ s.
Survival occurs at a threshold of instability, with the
precarious levels of supply and demand in the fragile
balance.
A species subsists as long as the means available
suffice to maintain it
• with probability h , the level of
demand d is drawn anew from the
probability distribution function F
while the level of supply s keeps the
value it had at time t − 1, or
• with probability 1 − η, the level of
demand d is updated anew from the
probability distribution function F, and
the level of supply s is updated either,
with respect to the probability
distribution function G.
Factors of objective and subjective uncertainty may
challenge our survival across different time scales.
s  [0, 1], the available
amount of resources (supply)
s  0,1: Prs  u  Gu 
d  [0, 1], the needs for
subsistence (demand)
d  0,1: Prd  u F u
s revaluated
Prrenewed s  1  h
h  0,1
t  t 1
s unchanged
Prs retained   h
s ≥ d, satisfaction
d >< s
d > s, dissatisfaction
The expected sequence of survival events is longer in the case of a stable
supply level (under incoherent uncertainty) than in the case of the coherent
random updates of supply and demand (under dual uncertainty).
Time is measured by the number of random updates of the demand level.
Probability of subsistence under uncertainty
Transitory subsistence under dual uncertainty
A species has a fairly regular rate of extinction, in line with the
observations of Leigh Van Valen on that all groups of species go
extinct (in million years) at a rate that is constant for a given group.
Extraordinary longevity under incoherent
uncertainty
The actual life duration of the species remains indeed finite.
The absence of a characteristic time scale indicates that virtually all ages
may present in a population subsisting under incoherent uncertainty,
including individuals of extraordinarily long life spans (”centenarians”).
When the factors responsible for the objective and
subjective types of uncertainty evolve on the
incomparable time scales, there are no mass extinctions in
the population.
Zipf’s longevity in a land of plenty
Probability
What are the best possible chances for survival under
uncertainty?
The best possible chances for survival under uncertainty abide the
Zipf law: a lifespan twice as long occurs half as often.
On the Optimal Strategy of Subsistence
under Uncertainty
1. Initial destabilization of the environment at
each time step, with the consistent updates of
demand and supply, is required in order to
boost the chances for survival during the initial
stage.
2. Intermediate stabilization of the
environment by keeping the level of supply
unchanged.
3. A “safe haven in a land of plenty” is
required in order to enjoy extraordinary
longevity.
Interaction statistics in organizations
The radio-frequency identification sensors reported
on occasions of physical proximity
Duration of intervals between
sequent communications (min)
The distributions are remarkably skewed, indicating a significant
proportion of the abnormally long periods of activity/ inactivity.

exp  t 2 22.05
2


exp  t  2  21.78
2 2.05
2
2
2

2 1.78
2
t   t 3 , t  1
1 t  1t  2   1
1
t
1
t2
,   103
 t 1 0.09
1. Casual interaction: Short time intervals largely remain unmanaged
and unregulated. Short occasional interactions/breaks in
communication are tolerated;
2. Spontaneous interaction: Time intervals of intermediate durations
are thoroughly managed by individuals demonstrating the high
propensity to keep the current interaction going while filtering out
the potentially unimportant forthcoming communications;
3. Institutional interaction: where the Zipf's Law manifests itself, the
top-down, almost mandatory interaction occurs.
Ordering of demographic pyramids
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.
Ordering of demographic pyramids
Planning under uncertainty:
Divide and conquer strategy
The divide and conquer strategy is useful for
uncertainty management as even though an
immediate short-term decision is deemed
imperfect later, it has the benefit of reducing
uncertainty in the near future. An incremental
approach that can help us to manage strategic
uncertainty avoiding to take on too much risk that
may come with large scale sweeping decisions.
How many “cuts”/intermediary decissions do we
need to perform in average?
Survival under uncertainty
In a discrete time model, every single act of survival
up to time T can be represented by an integer
partition,
are the sequent times of reproduction.
The survival strategies in the probability tree are
ranging from
the r-selection strategy of reproducing as quickly as
possible, to
the K-selection strategy of doing time T in just a few
generations, to
the marginal child-free strategy of promoting personal
longevity till time T while neglecting reproduction.
Combinatorics
Mortality
The Stirling partition number (the Stirling number of the second kind)
When all partitions are equiprobable under
uncertainty, configurations with mT present
overwhelmingly.
The most likely strategy under uncertainty for a long enough
period of time T ≫ 1 would consist of approximately T/ log T
short-term segments
It is natural to assume that
• reproduction does affect the local ecology and the current carrying capacity of the
habitat of a biological species – instigating environmental changes;
• every intermediary decision reducing strategic uncertainty would also change
environments;
The degree of environmental stability η(m) - the rate of
behavioral strategy attuned to cues of home ecology.
η(m) → 0: m → T
reproduction at each step/ as quick as possible / ”faster”
behavioral strategies/ having more children, risk taking, and
impulsivity -> ecologies of the desperate end that exert physical
strain on the individual and are characterized by the high degree
of random fluctuation in environmental events;
η(m) → 1: m → 1
slower” behavioral strategies; hopeful ecologies;
Empirical facts:
Individuals from desperation ecologies tend to reproduce earlier and faster, clearly
emphasizing offspring quantity over quality. Girls whose fathers are absent from home exhibit
earlier age of menarche, first sex, and first child, as father absence might signal high male
mortality rates and unstable pair bonds; individuals become more risk taking and presentoriented.
It is natural to assume that
• reproduction does affect the local ecology and the current carrying capacity of the
habitat of a biological species – instigating environmental changes;
• every intermediary decision reducing strategic uncertainty would also change
environments;
The degree of environmental stability η(m) - the rate of
behavioral strategy attuned to cues of home ecology.
η(m) → 0: m → T
reproduction at each step/ as quick as possible / ”faster”
behavioral strategies/ having more children, risk taking, and
impulsivity -> ecologies of the desperate end that exert physical
strain on the individual and are characterized by the high degree
of random fluctuation in environmental events;
η(m) → 1: m → 1
slower” behavioral strategies; hopeful ecologies;
It is natural to assume that
• reproduction does affect the local ecology and the current carrying capacity of the
habitat of a biological species – instigating environmental changes;
• every intermediary decision reducing strategic uncertainty would also change
environments;
The degree of environmental stability η(m) - the rate of
behavioral strategy attuned to cues of home ecology.
to the progressive decay of reproduction
rate with time in our model.
For T < Tc, the survival process can be viewed as
adaptation of the species to the relatively stable
conditions of their habitat;
For T > Tc the survival success of the species is
threatened by an evolutionary trap, since the species
(presumably well adapted) produces not enough
offspring.
When adaptations become liabilities
If the rate of environmental changes is higher than the rate of
adaptation to the variable environment, the adaptations once
enhanced fitness of the species become rather its liabilities, so that
the species is fall into the evolutionary trap.
Evolutionary trap
Survival by endurance
running
The evolution of certain human characteristics can be viewed as an
evidence for selection for endurance running. Human endurance
running capabilities either match or exceed those of mammals
adapted for running, including dogs and equids.
Survival by endurance running
The evolution of a successfully surviving
species (essentially that one presumably
dominating the planet) should be
imprinted, first of all, on the specific
adaptations for evading the evolutionary
traps by a rapid and permanent change of
the precarious environments, either by
running through them, at the early stages
of the survival process, or by fostering the
radical changes in them (destroying), at
the later stages of the process.
Harsh and perfectly stable ecologies alike equally threaten
the survival success. The optimal survival strategy in any
foreseeable time period consists of a regular change of
scenery by innovation and migration to other environments.
Logarithmic usefulness (utility) of time and hyperbolic
discounting of the future under uncertainty
The characteristic time interval of stability
can be estimated by
In a situation of planning actions over the time
horizon T in the face of uncertainty, any change
in the operational situation gives rise to the
need for deciding and acting immediately - the
expected duration of stable operation would
also lasts for u(T) ≈ log(T)
Logarithmic usefulness (utility) of time and hyperbolic
discounting of the future under uncertainty
The characteristic time interval of stability
can be estimated by
In a situation of planning actions over the time
horizon T in the face of uncertainty, any change
in the operational situation gives rise to the
need for deciding and acting immediately - the
expected duration of stable operation would
also lasts for u(T) ≈ log(T)
The measure of risk aversion (Arrow–Pratt)
describe how much time the individual is
willing to spend in the best case scenario (for
example, playing sports in order to stay in
shape) for securing the longer lifespan in the
worst case scenario.
uT  1
RT   

uT  T
The characteristic of prudence quantifies how
an increase in uncertainty about future survival
(say, under distress following a chronic disease
diagnosis) will affect current time spending.
T   
uT  2

uT  T
Hyperbolic discount of time: When offered a choice between getting $20 today or $25 a month from now,
most people would take $20 today. However, if $20 is offered six months from now and $25 seven month
from now, most people would take $25 - a rational person should choose the higher amount. But … Real
individuals are most likely to be ”present-biased”.
Inequality rising from risk taking under
uncertainty
Utility refers to the perceived value of a good (or
wealth), and the utility describes the attitudes towards
risky projects of a ”rational trader”, who would attach
greater weight to losses than he would do to gains of
equal magnitude -- The risk aversion implies that the
utility functions of interest are concave.
u  log w
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.
The Pareto
distribution of
wealth
Wealth inequality can be viewed as a direct statistical consequence of
making decisions under uncertainty
Cross-database analysis suggests the
worldwide growth-inequality relation (Ucurve)
The American economist S. Kuznets had
suggested that as an economy develops, market
forces first increase and then decrease economic
inequality. Kuznets demonstrated this
relationship using cross-sectional data about
income inequality collected in the middle of the
20-th century, in the United States.
As a country develops, more capital is
accumulated by the owners of industry,
introducing inequality. However, more
developed countries move then back to lower
levels of inequality through various
redistribution mechanisms, such as social
welfare programs.
We have used all existent data series (1870 - 2014) in the World Top
Incomes Database -- F. Alvaredo, T. Atkinson, T. Piketty and E. Saez
– Paris School of Economics.
The GDP historical database of the Maddison Project on the Gross
domestic product (GDP) per capita (per person) as the main source
of data on economic development and evolving living standards.
the mean GDP
level of the World
(US$14,402 for
2013)
Inequality is closely correlated with low growth, yet with high growth
either.
the mean GDP
level of the World
(US$14,402 for
2013)
As the economic growth is a
global worldwide
phenomenon, rising
inequality accompanying
that is a global worldwide
phenomenon either,
occurring once the national
economy is out of step with
the World average.
Inequality is closely correlated with low growth, yet with high growth
either.
A stagnant planned/command
economy aiming at the complete
elimination of risks (by rationing food,
for example) engenders scarcity of
virtually everything in the society.
Scarcity fuels ultimate inequality, as
just a few redistributing people get
the unfettered access to the scarce
resources while the shares of others
dwindle continuously.
Industrial economy /
Planning is possible
As economy had developed,
inequality were mostly down
due to the generosity of wealth
redistribution.
Industrial economy /
Planning is possible
The economic and social policy
regimes, providing important
social protection for millions of
industrial workers in developed
countries, had proven to be
adequate for a globally
dominant industrial economy,
underlying three decades of
widely shared economic
growth.
Then the ever rising need to
propel growth through risky
entrepreneurship and
innovation has bent
incentives toward the
short-term maximization of
share prices rather than
planning for long-term
growth.
Industrial economy /
Planning is possible
Rampant inequality in a
”winner-take-all economy”
breeds political polarization
and boosts social mistrust.
The sharp picks of
inequality level visible
synchronously for many
countries at once do
announce the periods of
global conflicts and
uncertainty of
international relations a
good deal in advance.
Rational country leaders facing
common dissatisfaction with
domestic policies and aggravating
economic conditions that prompt
their removal from office are
likely to gamble on a risky
diversionary war aiming to divert
attention of the public away from
domestic issues, increasing that
available time the government
has to address the internal
troubles.
Rampant inequality may transform uncertainty of national
economic development into uncertainty of international
relations.
Certainty
(the ensemble of
microstates is kept fixed):
• Principle of least action
Nature always finds the most
efficient course from one point
to another.
Uncertainty
(the ensemble of microstates
may be broader than initial):
•
Principle of maximum entropy
(over the maximal possible
ensemble of states)
A shattered vase is more likely
than intact one, as smithereens
allow for more disordered
”microscopic” states than a
single state of intact vase.
Certainty
(the ensemble of
microstates is kept fixed):
• Principle of least action
Uncertainty
(the ensemble of microstates
is broader):
•
Principle of maximum entropy
(over the maximal possible
ensemble of states)
A crowd is also more likely to occur under uncertainty than a well coordinated team of
individuals, as once individual identity disappears crowd members are unable to resist any
passing idea or emotion.
Certainty
(the ensemble of
microstates is kept fixed):
• Principle of least action
Uncertainty
(the ensemble of microstates
is broader):
•
Principle of maximum entropy
(over the maximal possible
ensemble of states)
While the survival of the fittest rewards the talented under certainty, the
community members experiencing uncertainty may cut ”tall poppies” down, as
their talents and achievements distinguish them from their peers.
Evolution has not stopped
yet!
Now we are selected by our own
choice: to be or not to be, to go or
not go, to do or not to do.
In the face of upcoming
uncertainty, we are as much
responsible for everything that
happens to us in this life as we
were once upon a time in
savannah, at the early dawn of our
history. We had managed to survive
at the time, as being the most
enduring in race among all other
living species.
So … let’s keep jogging!