Cost heterogeneity and peak prediction in collective actions
Introduction
The collective action refers to the social phenomenon that people gather and act together to achieve specific collective goals (Olson, 1965, Willer, 2009), such as strikes, protests, occupying, demonstrating, etc. For societal impacts and political implications (Meyer, 2004, Polletta and Jasper, 2001, Willer, 2009), collective action is the foundation of the society as human beings often cooperate with each other (Meyer, 2004, Olson, 1965). Collective actions are investigated via three clusters of methods: (a) The qualitative approach refers to theoretical analysis based on observations and case study (Benford and Snow, 2000, Chant, 2007, Goldstone, 1980, Hardin, 1968, Jenkins, 1983, Marx and Wood, 1975, McCarthy and Zald, 1977, Semann, 2009, Tarrow, 1988, Voss and Williams, 2012, Wright, 2009, Zhou, 1993, Zomeren and Spears, 2009); (b) the empirical approach refers to statistically evaluating empirical data (Bennett and Segerberg, 2011, Eisinger, 1973, Ellmers and Barreto, 2009, Hornsey et al., 2006, Mannarini and Talo, 2011, Qiu et al., 2015, Stroebe, 2013, Willer, 2009, Yu and Zhao, 2006, Zaal et al., 2012, Fernandez and McAdam, 1988); and (c) the mathematical approach refers to mathematical models and simulations to discover features and properties of collective action (Centola, 2010, Centola, 2013, Granovetter, 1978, Hu et al., 2014, Myatt and Wallace, 2009, Oliver, 1993, Ostrom, 2003, Siegal, 2009, Sigmund et al., 2010, Jin et al., 2014). Although the mechanisms and processes are heavily investigated, the distribution of participants is paid less attention, especially the phenomenon of peaks. The collective action follows a common regularity that it owns the peak of participants that can be mobilized. The peak varies depending on different cases.
The number of mobilized participants measures the success of collective actions like strikes, protests, and revolutions (Centola, 2013, Granovetter, 1978, Polletta and Jasper, 2001). The peak number of participants reflects the maximum power, influence, and impact of collective actions and therefore becomes a key indicator to predict the success rate. Despite the importance and potential applications, little attention has been paid to the peaks. Fortunately, related mathematical models or formal models of collective actions indirectly pave the way for the exploration and even prediction of the peak, such as the threshold model (Granovetter, 1978, Granovetter, 1986), standing ovation model (Miller & Page, 2004), network model (Alba, 1981, Gould R, 1993, Snow et al., 1980, Fernandez and McAdam, 1988), stochastic learning model (Macy, 1991a, Macy, 1991b, Macy and Flache, 1998, Macy and Flache, 2002, Mannarini and Talo, 2011, Flache and Macy, 2002), critical mass model (Marwell et al., 1988, Oliver et al., 1985, Pamela et al., 1988) or freezing period model (Wang, Liu, Wang, Zhang, & Wang, 2014), and game theory model (Heckathorn, 1988, Heckathorn, 1990, Wang et al., 2014b, Wang et al., 2015, Jin et al., 2014). It suggests in these researches that there are two factors influencing the peak: (a) The Jointness of supply (J) measuring how the size of group affects the individual payoffs. In general, the growing of J facilitates the payoff of individuals. Zero jointness of supply (J = 0) reduces participants and results into the group size paradox in large groups (Hardin, 1988, Olson, 1965). The paradox can be fixed by the “pure jointness of supply” (J = 1) where the size does not reduce individual payoffs and will not undermine participations (Pamela et al., 1988). Binary values of J (0 or 1) cannot capture features of all collective actions. Macy, 1990, Macy, 1991a, Macy, 1991b) expands J to the unit interval [0,1] and makes it continuous. It shows that J raises the number of participants and therefore facilitates the emergence of critical mass (Macy, 1990, Oliver et al., 1985); (b) the heterogeneity (Centola, 2013, Granovetter, 1986, Miller and Page, 2004, Oliver et al., 1985, Yu and Zhao, 2006) among members makes it harder to organize collective actions, playing a negative role in mobilizing individuals to participate (Centola, 2013, Schelling, 2005). However, the homogeneity increases participations of collective actions (Centola, 2011, Centola, 2013).
Following the mathematical approach, this paper focuses on the dynamic process and therefore prediction of peak, constructing a formal model based on the existing finding that both J and heterogeneity have negative effects on participations. Besides of the utility heterogeneity (Marwell et al., 1988, Oliver et al., 1985, Pamela et al., 1988), the cost heterogeneity should be considered as well because cost varies across different people. Therefore, this paper includes three factors such as cost heterogeneity, utility heterogeneity, and Jointness of supply to explore and predict the peaks. The cost heterogeneity refers to that participants have different costs to take part in the same collective action, and the cost homogeneity means that they share the same cost; Similarly, the utility heterogeneity refers to that agents have different subject scorings or feelings of their incomes, and the utility homogeneity means the same score of them. Under the joint homogeneity of cost and utility, ideal peaks can be obtained. Real peaks will be evaluated under cost heterogeneity or utility heterogeneity. Why we study the peak in collective actions? As more participants raise the success probability (Centola, 2011, Centola, 2013, Granovetter, 1978, Pamela et al., 1988), the dynamics and success rate of a certain collective action can be forecasted in advance, which is meaningful for both the organizers and opponents of the collective action. Our target is to estimate and predict the distribution traits (mean and SD) of real peaks. Via comparison effects of utility and cost heterogeneity, a higher fitness of estimation is achieved.
Section snippets
Jointness of supply
The collective goods is the aim or pursuit of collective actions perceived by all members, including participants and nonparticipants. The collective goods take on different forms and meanings in reality, such as justice (Tallman & Ihinger-Tallman, 1979), equality (Sarah, Soule & Olzak, 2004), human or civil rights (Luders, 2006, Suárez and Bromley, 2012), anti–genetic (Schurman & Munro, 2009), environmental issues (Roser-Renouf, Maibach, Leiserowitz, & Zhao, 2014), resent and grievances (Opp,
Effects of heterogeneity on peaks
In real societies, heterogeneity among agents, individuals or participants seems widespread and overwhelming. Heterogeneity usually plays a negative role in cooperation and collective actions (Centola, 2010, Centola, 2011, 2013), Therefore, we should focus on measuring the effects of heterogeneities on peaks as well as the dynamic processes. Fig. 1 depicts the evolutionary dynamics and peak points of collective actions under different heterogeneities of cost and utility. Circle points indicate
Calculating ideal peaks
Homophily is the ideal state facilitating collective action (Centola, 2013), so the ideal peak Pi is calculated under homogeneity of relevant factors. Ideal peaks Pi is obtained under the combination of utility homogeneity and cost homogeneity, where each one has the same utility v and cost c. Eq. (5) provides the form of ideal peaks function, which is determined by four factors, J, v, Vg, and c. Eq. (6) is the reduced form of the ideal peaks function using the ratio R to substitute the rate vi
Predicting peak's fluctuations
We run each simulation 100 times to get more accurate values of peaks, and the average values are taken as expected values of real peaks (Centola, 2013). Given the expected value or mean of real peaks have been estimated and predicted with the accuracy of over 96.46% in Table 3, the stability of peaks should be inspected. The standard deviation of peaks σP is applied to measure the fluctuation of real peaks.
Conclusions and discussions
Heterogeneity strategies should be taken to promote the outbreak of collective actions, while homogeneity strategies are needed to increase the participants, raise the peak, and eventually improve the success chance of collective actions. On facilitating the outbreak of collective actions, heterogeneity's effect is positive as it forms chains of actions (Granovetter, 1978, Granovetter, 1986, Macy, 1991a, Miller and Page, 2004, Oliver, 1993, Oliver et al., 1985). So before the collective action
Acknowledgement
This work is supported by National Natural Science Foundation of China (Grant No. 71673159), Beijing Social Science Foundation (Grant No. 16SRB014), Beijing Natural Science Foundation (Grant No. 9164029), and National Social Science
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