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A Landscape Theory of Aggregation

Published online by Cambridge University Press:  03 March 2009

Robert Axelrod  and
D. Scott Bennett
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Abstract

Aggregation means the organization of elements of a system into patterns that tend to put highly compatible elements together and less compatible elements apart. Landscape theory Predicts how aggregation will lead to alignments among actors (such as nations), whose leaders are myopic in their assessments and incremental in their actions. The predicted configurations are based upon the attempts of actors to minimize their frustration based upon their pairwise Propensities to align with some actors and oppose others. These attempts lead to a local minimum in the energy landscape of the entire system. The theory is supported by the results of two cases: the alignment of seventeen European nations in the Second World War and membership in competing alliances of nine computer companies to set standards for Unix computer operating systems. The theory has potential for application to coalitions of political Parties in parliaments, social networks, social cleavages in democracies and organizational structures.

Type
Articles
Information
British Journal of Political Science , Volume 23 , Issue 2 , April 1993 , pp. 211 - 233
Copyright
Copyright © Cambridge University Press 1993

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References

1 For example, the distance between two jobs in a hierarchical organizational structure can be regarded as the number of layers of the organization that have to be ascended to reach a common boss. See p. 232 below.

2 The symmetry of propensities guarantees that if one nation reduces its frustration by switching sides then the energy for the whole system will be reduced. Here is the proof. Without loss of generality, let X = A′ versus B where A′ = A∪{k}, and let Y = A versus B′ where B′ = B∪{k}. To shorten the notation let K = {k} and r ij = stsjpij. E(X) = ΣA′ΣBrij + ΣBΣA′rij since d ij(X) = 0 for i∈A′, j∈A′ or i∈B, j∈B; and d ij = 1 for i∈A, j∈B′ or i∈B, j∈A′. Likewise E(Y) = ΣAΣB′rij + ΣB′ΣArij. So E(X) − E(Y) = ΣA′ΣBrij − ΣAΣB′rij + ΣBΣA′rij − ΣB′ΣArij = ΣKΣBrij − ΣAΣKrij + ΣBΣKrij − ΣKΣArij since ΣA′ΣBrij = ΣAΣBrij + ΣKΣBrij. But ΣKΣBrij = ΣBΣKrij and ΣAΣKrij = ΣKΣArij since pij = pij. So E(X) − E(Y) = 2(ΣKΣBrij − ΣKΣA′rij) = 2(SkΣBSjpkj − SkΣASjpkj) = 2S k(F k{X) − F k(Y)), since d kj(X) = 0 for jΣA, dkj(X) = 1 for jΣB, dkj(Y) = 0 for jΣB, and dkj(Y) = 1 for jΣA. But Sk > 0. So for adjacent configurations, X and Y, differing only by nation k, if F k(X) − F k(Y) > 0 then E(X) − E(Y) > 0.

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29 This is 217/2. Each country can be in one of two possible sides, but which side is listed first is arbitrary.

30 For example, Britain declared war on Germany in 1939. Poland was first invaded by Germany and hence is counted as being aligned opposite to Germany. Hungary and Romania were allied with Germany and in 1941 assisted in the invasion of the Soviet Union.

31 In Configuration 2, Greece and Yugoslavia join the Soviet Union largely to avoid aligning with Germany, with whom both have a history of war.

32 There are 154 configurations that are as accurate or more so than the configuration that had two mistakes among the seventeen predicted nations. Since two different predictions are made and there are 217/2 = 65,536 configurations, the chance that one of them would be this good is 2 x (154/65,536) = 0.0047.

33 Steepest descent in the energy landscape is used to calculate basin size.

34 For example, as late as 1939 when the Soviet Union invaded Finland, there were some active voices in Britain and France calling for intervention against the Soviet Union, despite the growing consensus that Germany was the major threat. Had Germany not blocked access by invading Norway, such action against the Soviets would not have been out of the question. Incidentally, the main reason that Yugoslavia and Greece side with the Soviet Union in Configuration 2 is that they both have a war history with Germany, but no serious problems with the Soviet Union.

35 The error of placing Poland on the anti-German side occurred because Poland disliked the Soviet Union even more than it disliked Germany. This in turn was largely due to the Soviet Union's greater size (national capabilities) in 1936. As discussed below, this error was eliminated by 1939 as Germany mobilized its strength faster than the Soviet Union did. Portugal, which was actually neutral, was incorrectly placed on the German side because that side was more favourable for Portugal's Catholic religious propensity.

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35 The only change in the factors that went into the propensities from 1936 to 1939 was that Romania switched from a democratic government to an authoritarian government in 1938. Thus the changes in the landscape from 1936 to 1939 were almost entirely due to changes in the national capabilities of the various countries as they mobilized for war.

38 Note that six of these countries were not destined to enter the war on either side for another year or two. In 1940 Hungary and Romania allied with Germany, and Denmark and Greece were invaded. In 1941, Yugoslavia and the Soviet Union were invaded.

39 Since there are 65,536 different configurations, and only eighteen of them are off by zero or one country, the probability of a result this good happening by chance is 18/65,536 = 0.00027.

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