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A Landscape Theory of Aggregation

  • Robert Axelrod and D. Scott Bennett


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.



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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.

3 Since every allowable change lowers the energy of the system, the system can never return to a previous configuration.

4 David, Paul, ‘Clio and the Economics of QWERTY’, American Economics Review Proceedings, 75 (1985), 332–7.

5 Arthur, Brian W., ‘Self-Reinforcing Mechanisms in Economies’, in Anderson, P. W., Arrow, K. J. and Pines, D., eds, The Economy as an Evolving Complex System (Reading, Calif.: Addison-Wesley, 1988).

6 See Arnol, Vladimir Igorevich'd, Mathematical Methods of Classical Mechanics (New York: Springer, 1978 (translated from the Russian)); Abraham, Ralph and Shaw, Christopher, Dynamics - The Geometry of Behavior (Santa Cruz, Calif.: Aerial Press, 1983); Nicolis, Gregoire and Prigogine, Ilya, Exploring Complexity, An Introduction (New York: Freeman, 1989).

7 Wright, Sewell, ‘The Roles of Mutation, Inbreeding, Crossbreeding and Selection in Evolution’, Proceedings of the International Congress of Genetics, 1 (1932), 356–66; Kauffman, Stuart A., ‘Adaptation of Rugged Fitness Landscapes’, in Stein, Daniel L., ed., Lectures on the Sciences of Complexity, vol. I (Redwood City, Calif.: Addison-Wesley, 1989).

8 See Hopfield, John J., ‘Neutral Networks and Physical Systems with Emergent Computational Abilities’, Proceedings of the National Academy of Sciences (USA), 79 (1982), 2554–8. In biology and artificial intelligence, the polarity of the landscape is reversed so that the improvement is thought of as hill-climbing rather than descent into valleys.

9 Pines, David, ed., Emerging Synthesis in Science (Santa Fe, N. Mex.: Santa Fe Institute, 1985); Chowdhury, Debashish, Spin Glasses and Other Frustrated Systems (Princeton, NJ: Princeton University Press, 1986); Mezard, Marc, Parisi, Giorgio and Virasoro, Miguel Angel, Spin Glass Theory and Beyond (Singapore: World Scientific, 1987).

10 Weidlick, W., ‘Statistical Description of Polarization Phenomena in Society’, British Journal of Mathematical Statistical Psychology, 24 (1971), 251–66; Stein, Daniel L., ‘Disordered Systems: Mostly Spin Glasses’, in Stein, Daniel L., ed., Lectures on the Sciences of Complexity, vol. 1 (Redwood City, Calif.: Addison-Wesley, 1989), pp. 301–53.

11 Nicolis, and Prigogine, , Exploring Complexity, An Introduction.

12 Thorn, Rene, Structural Stability and Morphogenesis (Reading, Mass.: W. A. Benjamin, 1975 (translated from French)); Zeeman, E. C., Catastrophe Theory (Reading, Mass.: Addison-Wesley, 1977).

13 Aldenderfer, Mark S. and Blashfield, Roger K., Cluster Analysis (Beverly Hills, Calif.: Sage. 1984); Duran, Benjamin and Odell, Patrick L., Cluster Analysis (Berlin: Springer-Verlag, 1974); Kaufman, Leonard and Rousseeuw, Peter J., Finding Groups in Data: An Introduction to Cluster Analysis (New York: Wiley, 1990).

14 Kaufman, and Rousseeuw, , Finding Groups in Data, p. 1.

15 Kaufman, and Rousseeuw, , Finding Groups in Data, p. 37.

16 Other techniques which measure how good particular configurations are according to specific static criteria based on pairwise relationships include blockmodelling for which see Baker, Wayne, ‘Three-Dimensional Blockmodels’, Journal of Mathematical Sociology, 12 (1986), 191223; for simplicial decomposition, see Hearn, D. W., Lawphongpanich, S., and Ventura, J. A., ‘Finiteness in Restricted Simplicial Decomposition’, Operations Research Letters, 4 (1985), 125–30; for correspondence and canonical analysis, see Wasserman, Stanley, Faust, Katherine and Galaskiewicz, Joseph. Correspondence and Canonical Analysis of Relational Data’, Journal of Mathematical Sociology, 11 (1989), 1164, and for a variety of techniques based on factor analysis including smallest space analysis and non-linear mapping, see Everitt, Brian, Graphical Techniques for Multivariate Data (London: Heineman Educational Books, 1978). There are also econometric techniques to analyse how variables in dynamic systems aggregate from nearly decomposable subsystems: see Simon, Herbert A. and Ando, Albert, ‘Aggregation of Variables in Dynamic Systems’, Econometrica, 29 (1961), 111–38; Simon, Herbert A. and Iwasaki, Yuma, ‘Causal Ordering, Comparative Statics, and Near Decomposability’, Journal of Econometrics, 39 (1988), 149–73; Kydland, Finn, ‘Hierarchical Decomposition in Linear Economic Models’, Management Science, 21 (1975), 1029–39.

17 Morgenthau, Hans J., Politics Among Nations (New York: Alfred A. Knopf, 1956); Waltz, Kenneth N., Theory of International Politics (Reading, Mass.: Addison-Wesley, 1979).

18 Walt, Stephen M., The Origins of Alliances (Ithaca, NY: Cornell University Press, 1987).

19 Snyder, Glenn H., ‘The Security Dilemma in Alliance Polities’, World Politics, 36 (1984), 461–95.

20 Snyder, , ‘The Security Dilemma in Alliance Polities’, p. 464.

21 Liska, George, Nations in Alliance (Baltimore, Md.: Johns Hopkins University Press, 1962), p. 27.

22 See Holsti, Ole R., Hopmann, Terence and Sullivan, John D., Unity and Disintegration in International Alliances: Comparative Studies (New York: Wiley, 1973), pp. 263–7 for a listing of hypotheses on alliance formation that go beyond power; also see Morrow, James D., ‘Social Choice and System Structure in World Polities’, World Politics, 41 (1988), 7597.

23 For example, Altfield, Michael F. and de Mesquita, Bruce Bueno, ‘Choosing Sides in War’, International Studies Quarterly, 23 (1979), 87112; Altfield, Michael F., ‘The Decision to Ally: A Theory and Test’, Western Political Quarterly, 37 (1984), 523–44; Morrow, James D., ‘On the Theoretical Basis of a Measure of National Risk Attitudes’, International Studies Quarterly, 31 (1987), 423–38; Walt, Stephen M., ‘Testing Theories of Alliance Formation: The Case of Southwest Asia’, International Organization, 42 (1988), 275316; and The Origins of Alliances.

24 Waltz, , Theory of International Politics, p. 167.

25 Specifically, the countries selected are the five major ropean powers (Britain, France, Germany, Italy and the Soviet Union) and the twelve countries which had a formal defence or neutrality pact with any of them. Turkey was not considered to be in Europe. Two European countries were excluded: Albania because it was not independent of Italy, and Belgium because it withdrew from its defence agreement with France in 1936. Information about which was allied with which was not used in the analysis. The sources of alliance data are Singer, J. David and Small, Melvin, ‘Formal Alliances, 1815–1939’, Journal of Peace Research, 3 (1966), 131 and Small, Melvin and Singer, J. David, ‘Formal Alliances, 1815–1965: An Extension of the Basic Data’, Journal of Peace Research, 6 (1969), 257–82.

26 Singer, David J., Bremer, Stuart and Stuckey, John, ‘Capability Distribution, Uncertainty and Major Power War, 1920–1965’, in Russett, Bruce, ed., Peace, War, and Numbers (Beverly Hills, Calif.: Sage, 1972).

27 Due to the limitations of available data, we have not been able to operationalize economic issues and the level of economic interdependence in all dyads. Hence, we have simply omitted this category when calculating propensity.

28 With n countries, there are n(n - 1)/2 pairwise propensities. For n = 17, there are 136 pairwise propensities. Propensities are estimated as follows: ethnic conflict, a border disagreement or a recent history of war between two nations counted as -1 each for their propensity. Similarity of religion was counted as +1 within categories (Catholic, Protestant, Orthodox, Muslim and Atheist), and – 1 across major categories (Christian, Muslim, Atheist), all calculated according to proportions of each religion in each country. Similarity or difference of government type was considered for two countries with democratic, fascist or communist governments: +1 if they were the same type and – 1 if they were of different types. The source for ethnic conflict, border disagreement, history of war, and government type is Kinder, Hermann and Hilgemann, Werner, The Anchor Alias of World History, vol. II (New York: Anchor Press, 1978). Religion is given in the Correlates of War Project's Cultural Data Set for 1930 (version of 7/90 prepared by Phil Schaefer). Selecting equal weights for the five propensity factors is the least arbitrary way of combining them.

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.

34 Kaufman, and Rousseeuw, , Finding Groups in Data, pp. 47–8.

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.

40 Altfeld, and de Mesquita, Bueno, ‘Choosing Sides in War’.

41 Thus mutual membership in the EEC added one point to the propensities of a pair of such countries. Because of limited data availability from the former Eastern bloc countries, a more precise measure of economic interdependence is unavailable. War history was based on the Second World War; Italy was considered to have a war history with no one since it fought on both sides. An additional source for coding ethnic conflicts is Larrabee, Stephen F., ‘Long Memories and Short Fuses, Change and Instability in the Balkans’, International Security, 15 (1990/1991), 5891. Size data was available as of 1985. To simplify the calculations Benelux was treated as one country and Spain/Portugal as another.

42 New York Times, 11 11 1991 and 11 January 1992.

43 Axelrod, Robert, Mitchell, Will, Thomas, Robert E., Bennett, D. Scott and Bruderer, Erhard, ‘A Landscape Theory of Alliances with Application to Standards Setting’ (University of Michigan, Graduate School of Business Administration, Working Paper No. 666, 1991).

44 Owen, Guillermo, ‘Values of Games with a Priori Unions’, in Hein, R. and Moeschlin, O., eds, Essays in Mathematical Economics and Game Theory (New York: Springer-Verlag, 1977), pp. 7788.

45 See Katz, Michael and Shapiro, Carl, ‘Network Externalities, Competition, and Compatibility’, American Economic Review, 75 (1985), 400–24; Shapiro, Michael and Shapiro, Carl, ‘Technology Adoption in the Presence of Network Externalities’, Journal of Political Economy, 94 (1986), 822–41; Farrell, Joseph and Saloner, Garth, ‘Co-ordination Through Committees and Markets’, Rand Journal of Economics, 19 (1988), 235–52; Besen, Stanley M. and Saloner, Garth, ‘The Economics of Telecommunications Standards’, in Crandall, R. W. and Flamm, K., eds, Changing the Rules: Technological Change, International Competition, and Regulation in Communication (Washington, DC: The Brookings Institution, 1989); Hamel, Gary, Doz, Yves L. and Prahalad, C. K., ‘Collaborate With Your Competitors – and Win’, Harvard Business Review, 67 (1989), 133–9; Jorde, Thomas M. and Teece, David J., ‘Innovation and Co-operation: Implications for Competition and Antitrust’, Journal of Economic Perspectives, 4 (1990), 7596. For a review of the role of compatibility standards see David, Paul and Grecnstein, Shane, ‘Selected Bibliography on the Economics of Compatibility Standards and Standardization’, Economics of Innovation and New Technology, 1 (1991), 341.

46 Here is another derivation of the same propensity values. Firms prefer to be aligned with each other to increase the size of their alliance, but they prefer not to align with rivals. For distant rivals, these two considerations counterbalance, making p ij= 0. For close rivals, the rivalry consideration dominates, makingp ij = – 1.

47 As in the international cases, it was assumed that there would be at most two groupings.

48 There are 27= 128 ways of assigning the remaining seven firms to two alliances. Of these, one is completely correct and seven are off by one firm. Therefore, the probability of getting six or more of seven right by chance is (7 + 1)/128 = 1/16.

49 DeSwaan, Abraham, Coalition Theories and Cabinet Formation (San Francisco: Josscy-Bass, 1973).

50 Morgan, Michael-John, ‘The Modeling of Governmental Coalition Formation’ (doctoral thesis, Political Science Department, University of Michigan); Laver, Michael and Schofield, Norman, Multiparty Government: The Politics of Europe (Oxford: Oxford University Press, 1990); Laver, Michael and Hunt, W. Ben, Policy and Party Competition (New York: Routledge, forthcoming).

51 Homans, George C., The Human Group (New York: Harcourt Brace, 1950); Carrington, Peter and Heil, Greg H., ‘Coblock: A Hierarchical Method for Blocking Network Data’, Journal of Mathematical Sociology, 8 (1981), 103–31.

52 Carley, Kathleen, ‘An Approach for Relating Social Structure to Cognitive Structure’, Journal of Mathematical Sociology, 12(1986), 137–89.

53 Newcomb, Theodore M., The Acquaintance Process (New York: Holt, Rinehart and Winston, 1961).

54 Ross, Edward Alsworth, The Principles of Sociology (New York: Century, 1920); Dahl, Robert A., Pluralist Democracy in the United States (Chicago: Rand McNally, 1967).

55 Burnham, Walter Dean, Critical Elections (New York: Norton, 1970).

56 For example Lipset, Seymour M. and Rokkan, Stein, Parly Systems and Voter Alignments: Cross-National Perspectives (New York: The Free Press, 1967).

57 Thompson, James D., Organizations in Action (New York: McGraw-Hill, 1967).

* Institute of Public Policy Studies and Department of Political Science, respectively, University of Michigan, Ann Arbor. For helpful suggestions the authors thank Erhard Bruderer, Michael Cohen, John Holland, Will Mitchell, David Sanders, Carl Simon, Robert Thomas and anonymous reviewers. This work was supported by National Science Foundation grants SES 8808459 and SES 9106371.

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A Landscape Theory of Aggregation

  • Robert Axelrod and D. Scott Bennett


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