The Theory of Human Behavior
Suppose that a certain group of people is called to take a binary
decision on some problem such as electing the president, voting a law or even choosing a
phone company among two.
These people interact with themselves at random with a certain frequency nu by exchanging
their ideas and so modifying the relative probability
of the two possible choices of being selected.
These relative probabilities define the state vector for the stochastic
system in question
In more complex problems, the state vector may be done of more than
two components of course. For example this happens if one wants to study the behavior of
consumers in the choice of a product among different n’s which are in competition.
In that case the state vector becomes
but nothing of the forthcoming theory changes except for the number
of items on which to sum.
Here phi1 is defined by being the probability that a person selected at random in the
group is inclined to decision 1. The same for decision 2.
It is easy to understand that in every moment we will have
which expresses the conservation of probability.
Now, in the act of exchanging ideas everyone has half of the chances to convince a person
of the opposite idea while nothing happens in the encounter between two persons who agree
on the choice, of course.
Therefore in the act of meeting between an unknown and a person oriented to decision 1 the
relative probabilities for the unknown after the meeting are given by
while in the meeting event of an unknown with a decision-2-oriented
person the situation is
This defines the two operators of decision-influence
which act together at the same time in modifying the probabilities.
Since it is also unknown who an unknown person will meet the two operators will have to be
weighted with the corresponding probabilities when writing the equations for the two
states:
 
that already begins to sound familiar to those expert in tensor
calculus . To ease notation, let us now write
which defines T as being a symmetric tensor
called decision-influence tensor and done up of two operators.
Now, using compact tensor notation, the equations for the stochastic process can be
grouped and rewritten as
which is a tensor (multi-linear) equation describing the evolution of
this process.
As it is easily seen, the equation is not non-linear, as it is custom to call Boltzmann equations in mathematics, but multi-linear as tensors
correctly express. This is also true for nonlinear Boltzmann equations with integral
dependencies where you have a number of integrations (here summations) equal to the degree
of appearance of the unknown function.
Moreover, here tensor formalism is not pure ease of notation since it brings all its
properties. You can change the base of representation from yes/no to any linear
combination of these and the tensor equation will not change as long as you follow the
appropriate transformation rules.
The equation reported, as it is easily understood, tends to make one of the two states
prevail during time and exactly that one who is prevalent at the beginning. There is
therefore no need to study asymptotic solutions in two-state problems. This result does
not hold in multi-state problems however where there is also a need for a more precise
definition of the decision-influence tensor.
The two-state problem is easy and may seem purely academic however it can be very helpful
if you have to make predictions on decisions that need qualified majorities or if you want
to forecast the market share of a certain product at a certain future time.
In an uncertain world the only way of managing is to accept some non-vital errors and make
the best possible forecasts.
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