The Natural Evolution of Opinions
by Bernard Beauzamy
SCM SA
111 Faubourg Saint Honoré,
75008 Paris, France
I.Introduction
A general belief is that, at least in modern democracies, there is a natural tendency to stability, both political and economical. In simple words, it would mean that, if the information was totally open, if the markets worked perfectly, then the repartition of opinions would remain roughly constant, and the economy would tend to an equilibrium (for instance between the amount of work and the number of workers). Many economists even tried to prove this, and of course they failed.
What we show here is that this belief is totally wrong. In general, Nature does not look for stability, but, on the contrary, imposes fluctuations, that tend to become higher and higher in amplitude.
We concentrate here on the evolution of opinions with time ; similar models may be built for economic factors. We build a simple model, in which we have only two groups of people and , between which a total population of size splits (Democrats and Republicans, left and right, and so on). Each year, some people quit each group and they join the opposite camp. We consider these departures as random. This does not mean at all that people take their decision tossing a coin ! It means that we do not try to investigate their reasons : they may be due to some dissatisfaction with their own politicians, or some attractiveness of the adverse camp : we do not know and we do not care. In our book [1] we described the reasons why a "real life" procedure should be considered as random ; usually, as this is the case here, it means that we do not want to make precise investigation.
So, we have a very simple and robust model. Our conclusions are striking : fluctuations will constantly occur, and each group, in its turn, will become dominant, that is will absorb almost all the population (and even all the population, if you wait long enough !). These fluctuations are independent of the original size of the groups.
In other words, only by the laws of probabilities, a group, initially extremely small (say for instance less than 1 % of the total size of the population) will become some day so large that it will contain 99 % of the population, but will come back later to its original size of 1 %.
These fluctuations are extremely slow : we give a quantitative estimate.
The same conclusion holds for species evolution, in a given territory. Contrarily to what people usually think, if several species coexist in a territory, no equilibrium happens. On the contrary, the same laws of probabilities show that we have fluctuations, and each species becomes dominant at its turn.
The simplest example of a model in which fluctuations necessarily occur is given in [1]. It concerns two players, who toss a coin. The winner receives 1 Euro from the loser, each time. Then one can show that the fortune of each player becomes arbitrarily large, positive and negative : there will be a time when has won a billion Euros, and there will be a time when he has lost a billion Euros. So these oscillations are unbounded : this is the biggest difference with the present situation, where the total size of the population is an obvious bound for the size of both groups.
II.A mathematical model
The total population has elements. It is divided into two groups, respectively denoted by andOriginally, the group has elements and the grouphas elements, with.
Both groups are considered as “opinion groups”. We look at the following question : how do these opinions change with time, between both groups ? Is one of the groups dominant? Do we have stability ?
We consider that transitions from one group to the other are random. Each member makes his own decision, once a year, independently of the other members. Our time unit is the year. These assumptions are not absurd ; they are not completely realistic either.
At each time step :
Each member of the group has a probability to pass to group and thus a probability to stay in group
Each member of the grouphas a probabilityto pass to groupand thus aprobabilityto stay ingroup
We denote by
theprobabilitythat a personinitiallyinis still in at time ;
theprobabilitythat a personinitiallyinis in attime ;
and the same way :
theprobabilitythat a personinitiallyinis in attime ;
theprobabilitythat a personinitially inis in at time ;
These quantities are linked by the recurrence formula :
(1)
Indeed, we have two situations, leading to the fact that a person is in at time : either he was already in attime, or he was in attime.
Sinceequation (1) becomes :
,
or :
, (2)
with.
Set. We obtain the induction formula :
. (3)
The explicit computation of each is immediate:
Set; we get; with, this gives , and so, with. So finally:
. (4)
Since and, we have, and sowhen . So.
We obtained :
Lemma 1.–When :
;
;
and, permuting and:
;
.
We see that these probabilities of transition become constant, independent of the time and independent of the initial size of the sets. The convergence in (4) is extremely quick ; so we may consider that these probabilities of transition are constant.
Let be the random variable indicating the number of people, initially in , that are still in at time. The same way,represents the number of people, initiallyin , that are in at time. The laws of these random variables are known:
(5)
(6)
Let be the random variable indicating the number of elements in at timeIf we have elements in at time some of them were initially in and the rest initiallyin So we have the formula:
(7)
which can be written :
(8)
and finally :
(9)
So we get :
Lemma 2. – The variable follows a binomial law of parameters : this law is independent of .
This simple remark will imply the fluctuations we mentioned earlier, in the size of the groups. Indeed :
Theorem 1. – LetThere exists a timesuch that, withprobability, all the population is in the group no later than time.
The precise meaning of this statement is as follows. Let us fix a threshold (say 99 %). If we wait long enough, we haveprobability 0.99 that all the population is in no later than time . Of course, the population will not remain in , this group will shrink again. And some time will come (perhaps sooner, perhaps later) where all the population will be in and so on : this gives the oscillations we mentioned.
Proof of Theorem 1
We have a population of persons, with initial state 0 or 1 (0 is for , 1 is for ). We want to show that, with probability, there is a time where all are in
After one time step, the probabilitythat all people are in is : : people who are in must stay in it, and people who are in must leave. We denote by the number of elements in and by that of
Set . We have :
Let be the event "absorbed the whole population attime" and be the converse event. Then the event :
is characterized by the fact that, up to time included, the set never absorbed the whole population.
We have:
,
But, whereare respectively the number of elements in and after one time step. Therefore :
.
Assume we have shown :
(10)
Then :
In order to compute , we need to distinguish according to all possible compositions of after the k-th time step. Let be the event : at the k-th time step, the sethas elements (). Then:
and so :
and finally
,
and :
. (11)
which shows (10) by induction.
We deduce that when. Let us fix a threshold. We have
as soon as, or. But if, then . But : this means that the sethas absorbed the whole population, at least once, before time This proves the theorem.
For instance, if , . Ifpersons, and if we take , we get time steps : this is obviously quite slow.
We may look for the time such that, withprobability, the size of will be larger than no later than time:
Theorem 2.- If, , where:
represents a partial sum of the binomial law.
Proof of Theorem 2.
As we did previously, let us denote by the random variable representing the number of elements in at time. We are going to compute, in the case where .
We have :
and so on, until :
(12)
The condition impliesif.
The condition implies that loses at most elements. So we have :
This last quantity will be denoted by . It represents the probabilitythat a random variable following a binomial law with parameters satisfies . This probabilityissince.
So we get, from (12):
(13)
The conditionis satisfied as soon as :
. (14)
But the converse to is that went above the value no later than time. This concludes the proof of Theorem 2.
With the numerical values given above (, ) et , we find .
We saw that the situation did not tend to some equilibrium, but was constantly fluctuating. A law may be constructed for these fluctuations, as we now see.
As before, we denote by the number of elements in attime, the number of those who leave at time and the number of those who leave We have the relation :
. (15)
We denote by, , the variables , , , assuming known.
We get :
, (16)
and therefore, since the conditional laws of andare known :
. (17)
So we will have if and only if :
. (18)
We obtained :
Theorem 3.– If the number of people in group is smaller than the threshold value , this number will have a tendency to increase (this can be characterized by the condition ); if this number is larger than the threshold, it will have a tendency to decrease.
The word “tendency” means "on average". This statement does not contradict the fact that each population will become dominating. There is no paradox between theorems 1 and 3 : the first one describes actual fluctuations and the last one average fluctuations. Both are interesting in practice.
Reference
[1]Bernard Beauzamy : Méthodes probabilistes pour l’étude des phénomènes réels. Société de Calcul Mathématique SA, 2004.
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