Many particle interaction in the kinetic exchange models

Another idea, which may be used to improve kinetic exchange models, might be the introduction of the many particle interactions. In the ideal gasses these interactions do not occur, but in the social systems they might have interesting consequences. Namely they might be used to explain why certain voting events (elections, referendums) yield unpredictable results.

Impact of many particle interactions on the wealth (energy) distribution

Let us start with a modification of elementary kinetic exchange model. In this case let us assume that only \( N \)-particle interactions occur and the total wealth (energy) is uniformly distributed among them:

\begin{equation} w_i(t+1) = \frac{\varepsilon_i}{E(I)} W(t, I) , \quad\forall i \in I , \end{equation}

here \( I \) is a set of particles selected for this interaction, \( E(I) = \sum_{i \in I} \varepsilon_i \) is a random number normalization factor, while \( W(t, I) = \sum_{i \in I} w_i(t) \) is a total energy.

It should be evident that the energy (wealth) distribution will approach uniform distribution then \( N \) approaches a number of particles in the system. In the applet below you can see that in case \( N=2 \) we have exponential distribution, while with larger \( N \) values the uniform part becomes more prominent.

Referendum model

How do people make their decisions? Evidently not all of us are skilled and informed enough to make decisions on our own in all possible everyday scenarios, thus in such case we have to rely on our social contacts (acquaintances, co-workers, family members and etc.) to help us reach a decision. To keep things simple we can assume that after discussion the people will follow the opinion of the majority of their discussion group. Everything remains simple if the question (or task) is simple and may be answered with "yes" or "no". If odd number of people is interacting, then we will have a clear majority. While if even number of people is interacting, then we might have a situation with no clear majority (both "yes" and "no" getting same number of "votes"). In this case S. Galam notes that people tend to doubt and thus in case of a tie the more conservative option (for the sake of simplicity let us assume that it is a "no" option) will be chosen. Interestingly enough this simple assumption may cause the society to shift from positive (\( \xi(t)>0 \)) attitude towards the negative (\( \xi(t)<0 \)) attitude.

We would like to draw your attention to a fact that \( \xi \) here represents the mean opinion. If you prefer a raw fraction of individuals choosing "yes" option, then you may obtain it this way:

\begin{equation} x_{+} = \frac{1 + \xi }{2} . \end{equation}

In the applet below you can tune \( p[N] \) parameters, which define the probability that \( N \) individuals will meet.