For the purposes of this survey, I will define the “critical mass“ model as having the following characteristics:
The primary result of interest is the equilibrium (for a static model) or steady state (for a dynamic model) behavior of the average choice. Standard results include multiple equilibria/steady states, nonlinear response to the distribution of preferences, and “tipping“ dynamics.
The literature review below confines attention to articles published in economics journals, articles published in the top 5 journals of other fields, and/or articles that had a significant influence on subsequent work.
Models outside of the social sciences: Many of the terms and metaphors associated with this class of models, such as “critical mass,“ “epidemics,“ and “contagion“ are drawn from basic models in either particle physics or epidemiology which have some of the same structure. For example the SIR (susceptibility/infection/recovery) model in mathematical epidemiology dates to 1927 (Kermack WO, McKendrick AG. 1927. "A contribution to the mathematical theory of epidemics." Proc. R. Soc. London Ser. A 115:700-21).
Early economic treatments of social influence in choice: Although it had yet to take the familiar form of the critical mass model, the idea that behavior is subject to social influences has never been wholly absent from economics. James S. Dusenberry (Income, saving, and the theory of consumer behavior, 1949) and Harvey Leibenstein (“Bandwagon, snob, and Veblen effects in the theory of consumers' demand,“ Quarterly Journal of Economics 64(2) 1950, 183-207) were the first to formalize social influences in consumption; their models only share some of the features of the critical mass model. Hans Föllmer (“Random economies with many interacting agents,“ Journal of Mathematical Economics 1, 1974 51-62) includes social influences on consumption in a more modern general equilibrium model of a pure exchange economy.
Simon's model of voting: One of the first appearances in the social sciences of an identifiable critical mass model is Herbert Simon's treatment of voting (“Bandwagon and underdog effects and the possibility of election predictions“ in Public Opinion Quarterly, Fall 1954 p 239-253). In this paper, Simon assumes that some voters may prefer to vote for the underdog and some may prefer to vote for the winner (in addition to their preferences over candidates). The motivating question in this paper is whether a social scientist could make accurate predictions on this basis, given that the predictions may themselves influence choices. He finds that they can, though there may be multiple self-fulfilling predictions (i.e., equilibria), potentially giving the social scientist the ability to choose the election's winner if his or her predictions are believed by the public. These results were later extended and formalized by Timur Kuran (“Chameleon voters and public choice,“ Public Choice 53(1) 53-78 1987).
During the 1970's game theorist Thomas Schelling and sociologist Mark Granovetter formulated and popularized the critical mass model as a generalized description for a wide variety of phenomena. Schelling's Micromotives and Macrobehavior has 775 citations in the SSCI (some of which are in relation to his discussion of segregation), Granovetter's AJS article has 272.
Schelling: Schelling's influential early 1970's analysis of racial segregation (“Dynamic models of segregation,“ Journal of Mathematical Sociology July 1971; “A process of residential segregation: Neighborhood tipping“ in A. Pascal ed. Racial Discrimination in Economic Life, 1972) developed a set of models with the basic features of a critical mass model, and which Schelling later identified as a special case of the critical mass model. In his segregation models, households with different preferences for integrated neighborhoods decide whether to remain or leave a neighborhood with a given racial composition. Schelling shows that these models exhibit “tipping,“ a dynamic analogue to multiple equilibria in which the neighborhood composition can shift rapidly between apparent steady states. Schelling also provides a brief treatment of critical mass models in the course of a more general analysis of socially-influenced binary choice (“Hockey helmets, concealed weapons, and daylight saving: A study of binary choices with externalities,“ Journal of Conflict Resolution 1973).
Schelling's 1978 book Micromotives and Macrobehavior is a key event in this literature, as it provides one of the earliest and best-known discussions of the critical-mass model as a general model applicable to a variety of social phenomena. Schelling identifies both his own earlier work on segregation and Akerlof's well-known “lemons“ model (“The market for 'lemons': Quality uncertainty and the market mechanism,“ Quarterly Journal of Economics 1970) as prior examples of critical mass models, and suggests that other applications of the model include various examples of collective action (attendance at faculty seminars, pick-up volleyball games, jaywalking, applause), neighborhood deterioration, segregation, prejudice, bank runs, voting, political revolutions, traffic signals, and daylight savings. In his analysis, Schelling notes explicitly that he “idealizes“ the frequency distribution of preferences with a smooth CDF and solves for static equilibria. His primary finding is that there may be multiple static equilibria.
Granovetter: At about the same time, Mark Granovetter (“Threshold models of collective behavior,“ American Journal of Sociology 1978) provides another early general treatment of the critical mass model. His treatment is inspired by the literature in sociology on the genesis of riots and revolutions, and so rather than focusing on Nash equilibria considers the characteristics of the steady state which would be reached from an initial condition of zero. This consideration of dynamics yields a feature of the critical mass model which was not emphasized in Schelling's treatment: the equilibrium correspondence (or the lowest steady state) is nonlinear or discontinuous in group composition, i.e., a slight shift in the group's level of heterogeneity or in its average preferences may lead to a large shift in overall behavior. Granovetter suggests that the model could be applied to diffusion of innovations, rumors, diseases, strikes, voting, educational attainment, leaving social occasions, migrations, and intervention of bystanders; and provides a detailed analysis of rioting. With coauthor Roland Soong, Granovetter later applied this model to diffusion of technology (“Threshold models of diffusion and collective behavior,“ Journal of Mathematical Sociology 1983), consumption (“Threshold models of interpersonal effects in consumer demand,“ Journal of Economic Behavior and Organization 1986), and diversity (“Threshold models of diversity: Chinese restaurants, residential segregation, and the spiral of silence,“ Sociological Methodology 1988).
Critical mass models of collective action: Following Granovetter, critical mass models became a major stream of the theoretical sociology literature on crowd behavior and collective action. This literature begins with pre-Granovetter work by Richard A. Berk (“A gaming approach to crowd behavior,“ American Sociological Review, June 1974).
Together with various coauthors, Pamela Oliver and Gerald Marwell explored the implications of the critical mass model in a series of papers (Oliver, Marwell, and Ruy Teixeira, “A theory of critical mass I: Interdependence, group heterogeneity, and the production of collective action“, American Journal of Sociology November 1985; Oliver and Marwell, “The paradox of group size in collective action: A theory of the critical mass II,“ American Sociological Review 1988; Marwell, Oliver, and Ralph Prahl, “Social networks and collective action: A theory of the critical mass III, American Journal of Sociology 1988; Oliver and Marwell, “A theory of the critical mass IV: Selectivity, group size, organizing cost, and heterogeneity in mobilizing for collective action,“ mimeo 1991; Prahl, Marwell, and Oliver, “Reach and selectivity as strategies for recruitment: A theory of the critical mass V“, Journal of Mathematical Sociology 1991), and these ideas were further developed by Michael Macy (“Learning theory and the logic of critical mass,“ American Sociological Review 1990; “Chains of cooperation: Threshold effects in collective action,“ American Sociological Review 1991) and Karen Rasler (“Concessions, repression, and political protest in the Iranian revolution,“ American Sociological Review Feb 1996 132-153). Pamela Oliver (“Formal models of collective action,“ Annual Review of Sociology 1993) provides an overview of the literature through the early 1990's. More recent and more formal treatments include work by Chien-Chung Yin (“Equilibria of collective action in different distributions of protest thresholds,“ Public Choice 97 535-567 1998) and Michael S. Chwe (“Structure and strategy in collective action,“ American Journal of Sociology 1999).
Additional applications in sociology: Sociologists have also found the critical mass model is useful in understanding diffusion of ideas and innovations: Peter Hedström's analysis of unionization (“Contagious collectivities: On the spatial diffusion of Swedish trade unions, 1890-1940,“ American Journal of Sociology 99(5): 1157-79, 1994), and Kenneth C. Land, Glenn Deane, and Judith R. Blau's empirical analysis of religion (“Religious pluralism and church membership: A spatial diffusion model,“ American Sociological Review 1991) both use critical mass models to motivate their applications. Other papers using the critical mass model include Steven E. Clayman's ethnographic analysis of booing during the 1988 US election debates (“Booing: The Anatomy of a Disaffiliative Response“ American Sociological Review, Vol. 58, No. 1. Feb., 1993, pp. 110-130), and Lauren Krivo and Ruth Peterson's analysis of racial differences in homicide rates (“The structural context of homicide: Accounting for racial differences in process,“ American Sociological Review August 2000)
Applications in political science: Beth A Simmons and Zachary Elkins discussion of the diffusion of market-oriented policies (“The Globalization of Liberalization: Policy Diffusion in the International Political Economy,“ American Political Science Review (2004), 98:171-189), Timur Kuran's analysis of the 1979 Iranian revolution (“Sparks and prairie fires: A theory of unanticipated political revolution,“ Public Choice 61(1) 41-74, 1989), B. Dan Wood and Alesha Doan's use of the critical mass model to explain the effect of the Anita Hill-Clarence Thomas hearings on public opinion on sexual harassment laws (“The politics of problem definition: Applying and testing threshold models,“ American Journal of Political Science 46(4) 2003, 640-653), William H. Kaempfer and Anton D. Lowenberg discuss applications of the critical mass model to several issues in international relations (“Using threshold models to explain international relations,“ Public Choice 73 419-443 1992)
Applications in criminology: Criminologist Gary LaFree argues that the critical mass model is useful in explaining the tendency for crime rates to experience “booms“ and “busts“ over time (“Declining violent crime rates in the 1990's: Predicting Crime Booms and Busts“ Annual Review of Sociology, 25:145-68, 1999.) and applies the model to cross-country data (Gary LaFree and Kriss A. Drass, “Counting crime booms among nations: Evidence for homicide victimization rates, 1956-1998,“ Criminology, 0011-1384, November 1, 2002, Vol. 40, Issue 4).
Applications in economics: In economics, critical mass models were used to explain social customs and social norms over the next 10-15 years after Granovetter and Schelling. The most well-known of these applications are by George Akerlof (“A theory of social custom, of which unemployment may be one consequence,“ Quarterly Journal of Economics 94(4): 749-775, 1980; note that he explicitly identifies his model as a critical mass model and ties it to Schelling's work in footnote 8, page 759). A particularly frequent application is to the effect of social welfare programs on social norms for labor supply, illegitimacy, etc. (Assar Lindbeck, “Incentives and social norms in household behavior,“ American Economic Review 2:370-377, 1997; Assar Lindbeck, Sten Nyberg, and Jorgen W. Weibull, “Social norms and economic incentives in the welfare state,“ Quarterly Journal of Economics 114:1-35, 1999; Thomas J. Nechyba (“Social approval, values, and the AFDC: A reexamination of the illegitimacy debate,“ Journal of Political Economy 2001)
Timur Kuran (“Preference falsification, policy continuity and collective conservatism,“ Economic Journal 97 1987 642-665) uses the critical mass model to explain the resistance of populations to policy changes.
Related work in economics: A number of well-known papers in economics cover some of the same ground, but deviate from the critical mass model in that they have more “microfoundations,“ including H. Peyton Young (“The evolution of conventions“ Econometrica 1993), Abhijit Banerjee (“A simple model of herd behavior,“ Quarterly Journal of Economics 107(3):797-817, 1992), Sushil Bikhchandani, David Hirshleifer, and Ivo Welch (“A Theory of Fads, Fashion, Custom, and Cultural Change as Informational Cascades,“ Journal of Political Economy 100:5, 992-1026, 1992), and Douglas Bernheim (“A theory of conformity,“ Journal of Political Economy 102(5): 841-877 1994).
Brock and Durlauf: Since the early to mid 1990's, social interactions have been an active subject of research. In this context, William Brock, Steven Durlauf, and various coauthors have argued for the relevance of their formalization of Schelling/Granovetter critical mass model. Most of their key theoretical results are provided in “Discrete choice with social interactions“, Review of Economic Studies 2001. Durlauf (“Statistical mechanics approaches to socioeconomic behavior,“ in W.Brian Arthur, Steven N. Durlauf, and David Lane, eds. The Economy as an Evolving Complex System II 1997) generalizes the BD model to an arbitrary network structure and connects it to the literature on interacting particle systems in physics. Brock and Durlauf (“A multinomial choice model of neighborhood effects,“ American Economic Review 92(2) 2002 298-303) extend their results from binary choice to multinomial choice, and show (Theorem 1) that the condition on parameter values for multiplicity of equilibria is made stronger as the number of choices increases. Blume and Durlauf (“Equilibrium concepts for social interaction models,“ International Game Theory Review 5(3) 2003 193-209) add evolutionary dynamics to the BD model, and find some results about equilibrium selection. The journal Sociological Methodology featured an invited paper by Durlauf on applications of this model to sociology (Durlauf, Steven N. “A framework for the study of individual behavior and social interactions,“ Sociological Methodology 2001), as well as comments by several other researchers including Samuel Bowles, Aimee Dechter, Lin Tao, and Christopher Winship.
Additional Theoretical treatments: Alberto Bisin, Ulrich Horst, and Onur Özgür (“Rational expectations equilibria of economies with local interactions,“ Journal of Economic Theory, forthcoming) characterize equilibria for a generalized choice set under one-sided local interactions. Ulrich Horst and Jose Scheinkman (“Equilibria in systems of social interactions,“ Princeton University working paper) outlines a number of conditions that insure existence and/or uniqueness of equilibria under a wide class of network structures. Edward Glaeser and Jose Scheinkman (“Non-market interactions,“ Advances in Economics and Econometrics: Theory and Applications, Eight World Congress, M. Dewatripont, L.P. Hansen, and S. Turnovsky (eds.), Cambridge University Press, 2002) analyze a continuous action space and an arbitrary network; they define the MSI condition ensuring uniqueness, and tie these results to BD's results.
Direct applications: Several recent papers directly apply the Brock-Durlauf model to empirical and/or policy work. Applications include fertility transitions (Hans-Peter Kohler, “Fertility decline as a coordination problem,“ Journal of Development Economics 63:231-263, 2000; Pramila Krishnan, “Cultural norms, social interactions, and the fertility transition in India“, University of Cambridge working paper), trading patterns in the Marseilles fish market (Gerard Weisbuch, Alan Kirman, and Dorothea Herreiner, “Market organization and trading relationships,“ Economic Journal 110:411-436, 2000), the adoption of new theories in science (Brock and Durlauf, “A formal model of theory choice in science,“ Economic Theory 14:113-130, 1999), the timing of commercial breaks by radio stations (Andrew Sweeting, “Coordination games, multiple equilibria, and the timing of radio commercials,“ Northwestern University working paper 2004), deforestation in Costa Rica (Juan A. Robalino and Alexander S.P. Pfaff, “Spatial Interactions in Forest Clearing: Deforestation and Fragmentation in Costa Rica,“ Columbia University working paper), the spread of foot-and-mouth disease (Karl M. Rich and Alex Winter-Nelson, “Regionalization and foot-and-mouth disease control in South America: lessons from a spatial model of regulatory coordination and interactions,“ University of Illinois Department of Agricultural Economics working paper), recycling behavior (Gorm Kipperberg, “Discrete choices with social interactions: An application to consumer recycling“ University of California - Davis Agricultural Economics PhD thesis), asset prices (Joao Amaro de Matos, “Information Flow, Social Interactions and the Fluctuations of Prices in Financial Markets,“ Universidade Nova de Lisboa working paper), and punctual vs. late payment of wages in Russia (John S. Earle and Klara S. Peter, “Contract violations, neighborhood effects, and wage arrears in Russia“, Upjohn Institute staff working paper 04-101, July 2004).
Applications with other network structures: A number of applications are based on versions of the BD model with alternative network structures. Edward Glaeser, Bruce Sacerdote, and Jose Scheinkman (“Crime and social interactions,“ Quarterly Journal of Economics 1996) use a model with a simple one-sided linear network to estimate the extent of social influences on the decision to commit crimes. Timothy Conley and Giorgio Topa (“Dynamic properties of local interaction models,“ NYU working paper) use a random network model to infer social influences on job search from the spatial distribution of unemployment in Los Angeles census tracts.
Related work: The literature on customs and social norms has continued, including work on cropsharing contracts (H. Peyton Young and Mary A. Burke, “Competition and custom in economic contracts: A case study of Illinois agriculture,“ American Economic Review 91(3):559-573, 2001)