Dissolution of a Global Alliance
be defined as follows . It considers a system of N individual actors whose historical interactions have defined propensity bonds between them , which are either positive or negative . Each actor I , ranging from 1 to N is associated with its state variables S i which can assume one of the two values S i
=+ 1 or
S i
= −1 . The values correspond to the actor ’ s choice between the two possible coalitions . The same choice allies two actors to the same coalition , while different choices separate them into the opposite coalitions .
The configuration of states of all the actors S ={ S 1
, S 2 , S 3
, …, S N
} defines an allocation of coalitions , where by symmetry both configuration S and its inverse −S define the same coalitions .
Bilateral propensities Jij emerged from the actors i and j ’ s mutual historical experience measure the amplitude and the direction of the exchange between the two
actors . The propensity is symmetric with
Jij = Jji and is zero when there are no direct exchanges between the actors . The product JijS i
S j measures the benefit from interactions between both actors as a function of the actors ’ choices . Aimed to maximize their benefit , the actors seek to ally to the same coalition when Jij is positive and to the opposing ones , otherwise . Thus , depending on the direction of the primary propensity , the conflict can be beneficial to the same extent as the cooperation .
The sum of the benefits from all the interactions of actor I for a configuration
S makes up the net gain of the actor :
H i
( S ) = S i
∑ j≠i
J ij
S j
( 1 )
Thus , the configuration S , which maximizes the gain function defines the actor ’ s most beneficial coalition setting .
We depict the system of actors through a weighted connected graph with actors at the nodes and bilateral propensities as the weights of their respective edges ( see Figure 1 ). We take red ( dark ) color for the + 1 choice and blue ( light ) color for the −1 choice .
Figure 1 : Triangle of three conflicting actors 1 , 2 , 3 with negative mutual bonds and different amplitudes .
Within the confines of gain maximization , the two cases of limited and complete rationality of actors must be distinguished . Actors with limited rationality , for example one-step actors such as spins , are able to foresee either the immediate improvements only ( spin-like actors ) or the improvements in very limited amount of intermediate steps . Actors with complete rationality , in contrast , possess the complete-step visibility to foresee a worth case of improvement in intermediate steps .
When the most beneficial coalition configurations of different actors do not coincide , the maximization of individual gains induces competitions for the beneficial associations . Among the actors with complete rationality , those competing interactions cause endless instability in the system . However , the system may remain stable when some actors have limited rationality — not being aware of attainability of a better configuration , they are satisfied having reached a local maximum .
Below is an example of rational instability — instability in the system of actors with complete rationality ( Figure 2 ). The actual gain and the maximal gain of each ac-
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