Psychological Review © 2012 American Psychological Association 2012, Vol. 119, No. 2, 345–372 0033-295X/12/$12.00 DOI: 10.1037/a0027121 The Burden of Social Proof: Shared Thresholds and Social Influence Robert J. MacCoun University of California at Berkeley Social influence rises with the number of influence sources, but the proposed relationship varies across theories, situations, and research paradigms. To clarify this relationship, I argue that people share some sense of where the “burden of social proof” lies in situations where opinions or choices are in conflict. This suggests a family of models sharing 2 key parameters, one corresponding to the location of the influence threshold, and the other reflecting its clarity—a factor that explains why discrete “tipping points” are not observed more frequently. The plausibility and implications of this account are examined using Monte Carlo and cellular automata simulations and the relative fit of competing models across classic data sets in the conformity, group deliberation, and social diffusion literatures. Keywords: conformity, deliberation, jury, influence, threshold Supplemental materials: http://dx.doi.org/10.1037/a0027121.supp Regimes topple, neighborhoods gentrify, financial bubbles col- 1968; Milgram, Bickman, & Berkowitz, 1969), an “s-curve” (ob- lapse, and fads burst onto the scene. These stark discontinuities of served in the classic Asch study and in research on small group social life galvanize our attention, marked by many labels, includ- decision processes), or even a checkmark pattern (Cialdini, Reno, ing critical mass, information cascades, bandwagons, domino ef- & Kallgren, 1990). While many constructs and processes in the fects, the “hundredth monkey phenomenon,” and most famously, influence literature can be deployed to verbally explain such tipping points—a phrase attributed to Grodzins (1958), formalized qualitative discrepancies, our existing formal models fail to ac- by Schelling (1969, 1978), and popularized by Gladwell (2000). count for them. And a third puzzle is why models that aptly Tipping points surely exist, but they are far from ubiquitous in describe behavior in one influence paradigm (e.g., conformity, social life. Change often happens slowly, linearly, and gradually. group deliberation, or diffusion of responsibility) fare more poorly One reason might be that we are often situated near a critical point in others. These paradigms differ in psychologically meaningful of inflection. But there are many domains in which we observe ways, and they were developed with the intention of capturing considerable shifts in social movement—crowds assembling at an different social influence situations. But none of the current formal event, new products gaining market share, politicians mustering models successfully reconciles these paradigms or clarifies their popular support—without seeing anything as starkly discontinuous differences. The models I propose here help to resolve all three as a tipping point. Thus, a Google search (March 9, 2011) turned puzzles using a common integrative framework. The models serve up about 3.8 million references to a “tipping point” and 5 million a predictive and explanatory purpose but also provide an empirical to a “critical mass” (some of which may involve nonsocial phe- tool for estimating psychologically meaningful parameters across nomena)—but there are over a billion entries matching the search social settings and research paradigms. .”popular ءstring “increasing The theoretical account offered here attempts to clarify several Threshold Concepts in Psychology unresolved issues in the literature. One puzzle is how to reconcile seemingly knife-edged “tipping points” with the more gradual Kimble (1996) identified the concept of thresholds as one of a change that is the norm. Ideally, a single theory should explain handful of bedrock principles belonging at the core of a unified both types of response patterns and clarify when each should be scientific psychology. In some areas of psychology, we make observed. Another puzzle is why the impact of the first few routine use of thresholds to interpret data; examples include sta- influence sources qualitatively differs across settings and research tistical significance testing (e.g., Cohen, 1994), signal detection paradigms: a concave “r-curve” (Gerard, Wilhelmy, & Conolley, theory (e.g., Swets, 1988), dose-response functions in pharmacol- ogy (Berkson, 1944), and item-response theory in testing (e.g., Reise, Ainsworth, & Haviland, 2005). But psychologists have made more use of threshold concepts to explain perception and This article was published Online First February 20, 2012. judgment than to explain social behavior. The arguments developed here owe much to my long collaboration with Many social behaviors involve continuously varying responses—how Norbert Kerr. I am grateful for his generosity, patience, and enthusiasm as a mentor and friend. fast one walks down a hallway, how much one contributes to a Correspondence concerning this article should be addressed to Robert J. public good, how generously one tips a waiter. And for method- MacCoun, Goldman School of Public Policy, University of California at ological reasons, psychologists often prefer continuous or at least Berkeley, 2607 Hearst Avenue, Berkeley, CA 94720. E-mail: interval-level dependent measures. But much social action takes a [email protected] more binary, “digital” form, and this type of action is often rich in 345 346 MACCOUN social meaning (see Harre´, Clarke, & De Carlo, 1985; Watzlawick, the location of the threshold varies considerably from domain to Bavelas, & Jackson, 1967)—voting for a candidate, joining an domain.1 organization, choosing a university, deciding to drink at a party, deciding to use a condom, proposing marriage, quitting one’s job. The BOP Framework Nonlinear threshold concepts are an important tool for understand- ing how continuous latent preferences get converted into overt This article examines and compares several alternative threshold categorical choices and actions. and nonthreshold models of social influence, but the focus is on a Threshold models conventionally assume that the probability of family of closely related logistic threshold models; for conve- a response varies as a nonlinear, sigmoidal function of some latent nience, I refer to these as BOP models, where BOP can stand for propensity. Examples include models of neuronal firing (Rumel- “burden of (social) proof” or “balance of pressures.” hart, Hinton, & McClelland, 1986), declarative and procedural People are acutely sensitive to social consensus information memory activation (J. R. Anderson et al., 2004), and the behavior (Asch, 1951, 1955, 1956; Chudek & Henrich, 2011; Festinger, of social insects (Camazine et al., 2001). With this kind of sigmoid 1954; Fiedler & Juslin, 2006; Leary & Baumeister, 2000), a function, the linear assumption may be a reasonable approxima- trait we appear to share with many other organisms (Couzin, tion, but only for part of the range of the latent propensity, and 2009). For a dichotomous issue, social consensus judgments only when the slope of the function is fairly shallow. require us to track at least two frequencies, those who share our The logistic threshold models I propose in this article share position and those who favor an alternative position. Following features with item-response models in psychometrics (Birn- Latane´’s (1981) notation, I label these frequencies T (for tar- baum, 1968; De Boeck, Wilson, & Acton, 2005; Reise et al., gets) and S (for sources). I operationalize outside influence in 2005), as well as discrete choice models in psychology (Au- two different ways—as a ratio (S/T) or as a proportion (S/N, gustin, 2005; Luce, 1959), economics (McFadden, 1974, 2001), where N ϭ S ϩ T). The social pressure may be either active and and sociology (Macy, 1991). But not all logistical models are intended by its sources or passive and unintended by the threshold models, and the threshold models presented here do sources. Consensus information, alone, is sometimes sufficient not require the logistic function. I show that another threshold to bring about opinion change (or at least stated opinion model, derived by Norbert Kerr (Kerr, MacCoun, & Kramer, change) even in the absence of any contact with or arguments 1996, Appendix B) bears little resemblance to a logistic equa- provided by other people (Asch, 1956; Kerr, MacCoun, Hansen, tion but produces similar behavior. I began this project by & Hymes, 1987; Mutz, 1998). trying to extend that model beyond its original domain (group Whether or not an actor changes positions in response to social deliberation), but I concluded that a logistic threshold model influence will of course be influenced by a great many factors, was more suitable as a bridge across paradigms and disciplines. including informational versus normative influence (Deutsch & Ge- Still, it is important to emphasize that the Kerr model can be rard, 1955); compliance, identification, and internalization (Kelman, successfully applied well beyond its original application. 1958); or coercive power, reward power, reference power, legitimate power, and expert power (French & Raven, 1960). The motivation to maintain one’s position in the face of opposition is also influenced by Previous Threshold Accounts of Social Phenomena attitude importance, knowledge, elaboration, certainty, extremity, and The two most famous threshold models of collective behavior accessibility