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Natural Coalition Forming
The next section formally presents the natural model of coalition forming, followed by a detailed analysis of the instability rooted in the coalition forming as a decentralized maximization processes driven by each country’s objective to attain its best coalition allocations. We study the terms of existence of the optimal and stable coalitions, and the ways to reach them in the ranges of extensive or limited rationality. Within these results, we investigate the interesting and little explored phenomena of information manipulation and non-optimal stability peculiar to any network of selfish actors. The theoretical framework is accompanied by analysis of real cases which justify the practical use of the model.
Natural Model of Coalition Forming
Present policies require the countries to make a choice, to agree or disagree, and thus to ally with one side or the other. Within absence of external incentives, this is based on their historical and geographical background and experience that the countries decide weather to agree or disagree on the policies, for the sake of gain which results from the actors beneficial (or non-beneficial)existence. The agreement turns to be unfavorable to countries which went through a conflict in their past, while in the same way, the disagreement is unfavorable to friendly countries. Those primary mutual propensities between countries, which are the issues of their historical experience, can hardly be changed at a will and affect all the subsequent interactions and exchanges between the countries. The propensities do evolve with time, yet much more slowly than the coalition dynamics and therefore, as a first approximation, they are assumed to be static. We keep the discussion with relation to countries, however it applies to any type of rational actors, either economical such as companies, political such as parties, or social such as organizations, people.
Natural Model Denition
Let us briefly remind the definition the natural model of coalition forming. Consider a group of N independent countries which had experienced geographic, cultural or economic interactions during their history. The countries are respectively denoted by characters or indicators ranging from 1 to N. Each country thus makes its choice between two competing options +1 and −1, corresponding to two possible coalitions. The same choice allies the countries to the same coalition, while different choices separate them into the opposite ones. According to the postulate of minimum conflict, being part of the same coalition benefits the countries with the propensities to cooperate, while countries inclined to conflict, bear loss from cooperation. A country is represented by a discrete variable that can assume one of the two state values S = +1 and S = −1. The combination S = {S1, S2, S3, . . . , SN} makes up the configuration of the choices of the countries. The configuration of choices, as well as it’s inverse S = {−S1,−S2,−S3, . . . ,−SN} by symmetry, represents a particular configuration of coalitions in the system. Consider any two countries i and j, and denote by Jij the value that measures the degree and the direction of the historical exchanges between the countries. Jij represents the bond of original propensity between the countries, which is symmetric, Jij = Jji, and may vary for each pair of countries. Jij = 0 when no mutual bond exists between the countries, which represents an absence of a direct exchange. We shall describe by JijSiSj the measure of the interaction between the countries as a function of their choices. In case the countries agree, i.e., when Si = Sj , a positive value of Jij results in the positive effect for both the countries. When Jij is negative, the cooperation between the countries results in a negative effect on both the countries. This describes a system of countries maintaining short range interactions guided by primary mutual propensities. Thus, the propensities either favor the cooperation (Jij > 0), the conflict (Jij < 0), or signify the ignorance (Jij = 0) where neither the agreement nor the disagreement influences the outcome of the countries. For the sake of visualization, we represent the system of countries as a connected weighted graph with the countries in the nodes and the bilateral propensities as the weights of the respective edges, (see Figure (13)). We take red (dark) color for +1 choice and blue (light) color for −1 choice. ESF conflicting triangle. Note that in the representation of neighbors in the model, we leave the spin glass-standard square lattices and move to the star-like 3d shapes. Sum of the measures of all the interactions of a country i in the system is the net gain of the country Hi(S) = Si Σ j̸=i JijSj . (8) The total gain (or the energy) of the system in configuration S is measured by the total of the contributions in the configuration H(S) = 1 2 Σ i Hi(S). (9)
Illustration Through an Historical Example
In order to illustrate a natural model, we suggest to consider historical example of the conflicting triangle of Spain, England and France. The countries were alternatively enemies and allies during a long period of time. For these countries, the period of 1521 – 1604 has been marked by a series of land and sea wars, driven by the historical background of colonization, religion, and by progress in navel technology. The commercial rivalry between England and Spain, and political and religious ambitions of France were the major forces that pushed Europe into wars. The conflicts were usually initiated between any two of the countries, with the third one joining one or other of the sides. Accordingly, the historical background of the countries during this period has defined the particular distribution of mutual negative propensities.
The natural model is shown schematically in the ESF conflicting triangle in choices of the countries S, E and F are represented by the state variables SE,SS and SF respectively. Different colors attached to the state nodes of the countries correspond to the different coalitions of the countries. The state nodes are linked by the following primary propensities JSE = −3, JSF = −2, JEF = −1. The
value of propensities are illustrative only, and these are their relative magnitudes of importance: the conflict between Spain and France is less deep than the one between Spain and England, yet it is deeper than the one between England and France.
Rationality of Countries
During the coalition forming, at any particular configuration, a country may observe another configuration where it can reach a higher gain. When it happens, the country shall take appropriate changes in order to take advantage of this opportunity. The sequence of such changes by all the countries constitutes the process of maximization of the countries’ gains.
The depth of observation of a country’s benefit improvements depends on its capacity of rationality, which associates to the ability not only evaluate the immediate benefit change, but also to envision a way up to a better configuration, which may emerge through intermediate loosing states. Rationality may be extensive, or long horizon, and limited, or short horizon rationality. The shortest horizon corresponds to the immediate improvement, which is peculiar to spin-like actors. It may seem irrational and risky to make a change opposite to the one of an immediate best configuration, however this is the rationality in the long horizon strategy. The risk is hidden in the fact that an opponent country may possess the equivalent capacity of rationality, as well as that the changes may be performed simultaneously. Which result in causal ambiguity in the process of maximization of the countries’ gains.
Extensive rationality, is a new approach to this class of models is peculiar in many type of actors in reality – countries, individuals, economical or political organizations. The feature of long horizon rationality allows to explain the advantage a country may gain by possessing advanced technologies and knowledge in collection of system’s information, such as others’ strategies and possible transitions scenarios. It also provides a hint on why in some cases a country having the technological leap could gain interest in sharing its private information with a few others – knowing of existence of a common optimum benefits each involved country. Limited rationality, on the contrary, bounds the strategic horizon of countries. In the simplest case of limited rationality, spin-like actors, this is a closest local maximum. Figure (5) shows a case of system where for the spin-like actors the process of maximization stabilizes in the local maximum SM = +1, +1,−1,−1,−1. Here, the gain of the triangle’s actors decreases immediately at a change which keeps the actors from fluctuating. In such system, rational actors would loop in between all the individual maximal configurations.
Table of contents :
Acknowledgments
Introduction
Incipit
Spin Glass Models
Coalition Forming Model
Historical Overview
Outline of the Thesis
I Coalition Forming and Instability
1 Comparison of Models
2 Natural Coalition Forming
2.1 Natural Model of Coalition Forming
2.1.1 Natural Model Definition
2.1.2 Illustration Through an Historical Example
2.2 Rationality of Countries
2.2.1 Geometric Terms of Instability
2.2.2 Analytical Terms of Instability
2.2.3 Optimal Configurations
2.2.4 Illustration of Stable and Unstable Systems
2.3 Rational Maximization of Countries
2.3.1 Maximization As a Sequence of Individual Choices .
2.3.2 Rationality Tree
2.4 A Formal Implementation of Rational Maximization
2.5 Non-optimal Stability
2.5.1 Rationality Limitations
2.5.2 Formal Implementation of Non-optimal Stability .
3 Global Alliance Model of Coalition Forming
3.1 Global Alliance Model Formal Settings
3.2 Stabilization By Additional Factors
3.2.1 The Uni-Factor Stabilization
3.2.2 The Case of The England-Spain-France Triangle
3.2.3 The Multi-Factor Stabilization
3.2.4 Multi-factor Stabilization of the England-Spain-France Triangle
3.3 Physical Interpretation of the Multi-Factor Stabilization
4 Real Historical Cases Illustration and Analysis
4.1 Multi-factor Stabilization in Western Europe
4.2 The Stability of the Eurozone Coalition
4.2.1 Eurozone Coalition within Natural Model
4.3 The Multi-factor Stability of the Eurozone Coalition
5 Dissolution of Global Alliance
5.1 Formal Definition of Dissolution
5.2 The Two Cases of Dissolution
5.3 Dissolution of The Global Alliances in Syria – Unstable System
5.4 Dissolution of The Soviet Global Alliance – Semi-Stable System
5.5 Modeling of Dissolution
5.6 Remarks on Applications of Dissolution
6 Conclusion and Remarks
II Simulation of Coalition Forming, Instability and Sta- bilization
7 Implementation of Coalition Forming Computer Simulation
7.1 Computer Implementation of Actors and Coalition Forming .
7.2 Technical Details of the Computer Implementation
7.3 Simulation of Coalition Forming in NM
7.4 Simulation of Coalition Forming in GAM
8 Appendix: Computer Code of Coalition Forming Simulation
III Viability Correction of Dynamic Networks
9 Dynamic Network – Viability and Decentralized Correction
9.1 Dynamic Network Definition
9.2 Dynamic Network of a Coalition Forming Model
9.3 Viability Correction Introduction
9.3.1 Historical Overview
9.3.2 Correction by Viability Multipliers
9.4 Prerequisites From Viability Theory
9.4.1 The Viability Theorem and Viability Multipliers
9.4.2 Restoring Viability
9.5 Network
9.5.1 Network’s Viability
9.5.2 Restoring the Network’s Viability
9.5.3 Viability Correction in Coalition Forming Model
9.5.4 Proof of the Theorem on Restoring the Network’s Viability
9.6 Economic Interpretation of the Dynamic Network
10 Appendix: Duality and Tensor in a Vector Space
Conclusions
Publications
References
