Spontaneous Coalition Forming. Why Some Are Stable? - Cellular Automata - page 6

A: mixed A: happy

ε 1 =1

+J: maximum -J: minimum +J-C: minimum -J-C: minimum

3 3

1 2 1 2

A: mixed B: happy A: happy B: happy

ε 2 =1 ε 3 =−1

-J: minimum -J-C: minimum

Initial configuration
Unstable

New stable configuration

Fig.2. Starting from one possible unstable configuration of alliances (left) with coun-
tries 1 and 2 in A and country 3 in B, the stabilization is shown to result from the
existence of the various with 1 = 2 = − 3 .

belong to a world level coalition produces more advantages than purely local un-
proper relationship. Local bond propensities are neutralized since overwhelmed
by the two block site exchanges. The overall system is very stable. There exists
one stable distribution between both competing alliances.
We consider first the coherent case in which cultural and economical trends
go along the same coalition, i.e., β i = i . Then from Eq. (8) the minimum of H is
unique with all country propensities satisfied. Each country chooses its coalition
according to its natural belonging, i.e., η i = i . This result is readily proven via
the variable change τ ≡ i η i which turns the energy to,

X

N X

H 1 = − 1
2

C ij τ i τ j − b i τ i . (9)

<i,j> i =1

Above Hamiltonian representd a ferromagnetic Ising Hamiltonian in positive
symmetry breaking fields b i . Indeed it has one unique minimum with all τ i =+1.
The remarkable result here is that the existence of two apriori world level
coalitions is identical to the case of a unique coalition with every country in it. It
shed light on the stability of the Cold War situation where each country satisfies
its proper relationship. Differences and conflicts appear to be part of an overall
cooperation within this scenario.
The dynamics for one unique coalition including every country, or two com-
peting alliances, is the same since what matters is the existence of a well defined
stable configuration. However there exists a difference which is not relevant at
this stage of the model since we assumed J i,j = 0. In reality J i,j 6 = 0 makes the
existence of two coalitions to produce a lower “energy” than a unique coalition
since then, more J i,j can also be satisfied.