Policy and Complex Systems
tor separated by ”/” are given in the square brackets . The example thus illustrates a series of transitions suggesting an infinite cycling as follows .
The figure illustrates a possible branch of configuration transitions ( from the left to the right ) suggesting an infinite cycling , as follows . In the first configuration , where actor 1 has the maximal gain (+ 5 ), the unsatisfied actor 3 makes a change expecting for its maximal gain in a later step . As a result , 1 loses its maximum down at (+ 1 ) and 2 gains its own (+ 4 ). Prospecting to get back its maximal gain , actor 5 makes a change . This new configuration is the one that yields the expected maximum (+ 3 ) for actor 3 . At this configuration , actor 2 makes a change to gain an immediate improvement bringing the system back to the initial state with its symmetrical reversal equivalent ( the reverse state-colors in the figure ). It is interesting to observe that in the case of limited one-step rationality actors , the above system is stable in the third configuration : no actor can observe an immediate improvement of its gain .
Theoretically , the instability of the system of rational actors is defined as the situation when in any configuration of the actors ’ states there is an actor which is able to forecast an improvement of its gain . The well-defined geometrical terms of the instabilities read as follows . Denote a circle of actors by C and the actors composing the circle by 1 , 2 , …., k .
If there is a closed circle of actors on which the product of total propensities is negative ,
Π i , j
Ω p ij < 0
∈
( 2 ) then the system is unstable .
The negative product on a circle implies an unpaired negative coupling where two neighbors are found to be connected both through positive and negative branches in the circle . This fact creates an everlasting competition between the neighbors for the exclusive arrangement to ally with the positive branch . The actors thereby continuously shift their respective choices producing the instability .
2.2 Global Alliance Model of the Coalition Forming
The global alliance model starts from a global concept , which represents an external field polarizing the interests of the countries . This leads to the emergence of two opposing global alliances . The countries attach themselves to one or to the other based on their pragmatic interests with respect to the global principle . The new interactions , while favoring either cooperation or conflict , stimulate contributions to the countries ’ mutual propensities . The new prospects unify or separate the countries based on the pragmatic motivations , which in combination with the historical concerns allow other distributions of coalitions .
We denote the two global opposing alliances by M and C , where M unifies the countries that support the global concept and C unifies its opponents . Actor I ’ s individual disposition to the alliances , which is determined by the countries ’ cultural and historical experiences , is represented by the rational actor ’ s parameter of natural belonging εi , where εi =+ 1 if the actor has
natural attraction toward alliance M and εi = −1 for C .
By making a choice among the two possible state values S i
=+ 1 and S i
= −1 , actor i chooses to belong to either alliance M or C . Countries i and j ’ s choices of one or the other alliance creates new exchanges that define additional propensity between the countries . The propensity is determined by the amplitude G ij of the exchanges in the
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