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Social network design for inducing effort

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Abstract

Many companies create and manage communities where consumers observe and exchange information about the effort exerted by other consumers. Such communities are especially popular in the areas of fitness, education, dieting, and financial savings. We study how to optimally structure such consumer communities when the objective is to maximize the total or average amount of effort expended. Using network modeling and assuming peer influence through conformity, we find that the optimal community design consists of a set of disconnected or very loosely connected sub-communities, each of which is very densely connected within. Also, each sub-community in the optimal design consists of consumers selected such that their “standalone” propensity to exert effort correlates negatively with their propensity to conform and correlates positively with their propensity to influence others.

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Notes

  1. Another difference is that in Goyal et al. action is binary, whereas in the present study it is continuous.

  2. Similar results are obtained by Bojanowski and Buskens (2011), Neary (2012), Advani and Reich (2015), and Ellwardt et al. (2016).

  3. Community design is also an important area of study in organizational behavior and management literatures. There are, however, key differences with our work. First and most importantly, while these studies investigate links between network structure and outcomes, they do not attempt to optimize structure. Second, social conformity is not the root motivation for employee behavior. Financial incentives influence employee behavior. Finally, in optimization of collective employee effort, there is a concern for free-riding because firm objectives can be reached even when an individual is not exerting effort. In the context that we study, individuals’ goals are not tied to an overall objective and they do not gain utility from others’ progress unless they also exert effort.

  4. We use the terms effort and activity interchangeably.

  5. In a “larger” model, one may want to consider how individuals react to the choices of the community network designer, taking into account that the latter has connected him or her to a non-representative group of individuals. We abstract away from such considerations. In a model where individuals are fully aware of the non-representativeness of their ties and prefer to conform to the population average rather than their distinct peers, \(\{\alpha _{i},\lambda _{i},\chi _{i}\}_{i=1}^{n}\) are common knowledge, and individuals act fully rationally accordingly, each individual should be able to infer and act towards an unbiased estimate of the population average effort. This seems to render the problem of network design much less relevant. Hence, the effectiveness of community network design relies on the premise that individuals will conform to their ties, even when they might be aware that these ties are non-representative.

  6. This number is 64 for n = 4, larger than 32,000 for n = 6, and is considerably larger for any greater sized population (e.g., n > 100).

  7. Similarly, theoretical network science studying equilibrium behavior and identifying theoretical properties of graphs and processes evolving on graphs typically does so for \(n \to \infty \) (e.g., Acemoglu et al. 2011; Vincent and Ismael 2017; Iijima and Kamada 2017).

  8. Specifically, when α,λ or χ− 1 are assumed to vary, we draw each from a Beta(2, 2) distribution featuring values bounded between 0 and 1. When either λ or χ− 1 is assumed not to vary, we set its value to 1. In scenario 4, we induce a positive correlation of 0.5 between λ and χ− 1 using the method of Plackett (1965) . In scenario 5, we draw α and χ− 1 each from a Beta(2, 2) distribution, and then set \(\lambda _{i}=0.6-2(\chi _{i}^{-1}-0.5)^{2}\).

  9. The simulated annealing algorithm is implemented closely following Gastner and Newman (2006). In each step, one or multiple network links can be added, removed, or re-wired. We use a very slow cooling schedule; we run the algorithm for 107 steps and the final temperature is 10− 7 of the initial temperature. The objective function is \(\frac {1}{n}{\sum }_{i=1}^{n}(y_{i}^{*}-\alpha _{i})\) and the initial temperature is set fairly high at 0.01.

  10. Such networks are colloquially referred to as caveman graphs (e.g., Watts 2004, pp. 43-44 and 103), an extreme case in the small-world network literature. There is also a clear parallel between synchrony among oscillators (e.g., chirping crickets) studied in that literature and the social conformity among people we consider. However, small-world networks have not been studied as resulting from a centralized optimization.

  11. In fact, a stronger result also holds: \(q({\Phi },n+1)>\frac {{\sum }_{k=1}^{n}k\cdot q({\Phi },k)}{{\sum }_{k=1}^{n}k}\). The right hand side of this inequality is a weighted average and assigns more weight to the larger communities.

  12. In the Figure we assume that \((\alpha _{i},\lambda _{i})\sim \text {Beta}(2,2)\times \text {Beta}(2,2)\) and Beta(1, 1) ×Beta(1, 1). Note that Beta(2, 2) is a symmetric bell shaped distribution over the unit interval centered around 0.5, appropriate for populations where there is a greater mass of individuals carrying close to median characteristics in the importance they give to their goals and the degree of susceptibility they are subject to. In contrast, Beta(1, 1) is the uniform distribution, suggesting a more homogeneous spread of individual types.

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  1. The Wharton School, University of Pennsylvania, Philadelphia, PA, USA

    Pinar Yildirim & Christophe Van den Bulte

  2. Marshall School of Business, University of Southern California, Los Angeles, CA, USA

    Yanhao Wei

  3. Tepper School of Business, Carnegie Mellon University, Pittsburgh, PA, USA

    Joy Lu

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Yildirim: Assistant Professor of Marketing, The Wharton School, University of Pennsylvania, email: pyild@wharton.upenn.edu. Wei: Assistant Professor of Marketing, Marshall School of Business, University of Southern California. Van den Bulte: Gayfryd Steinberg Professor and Professor of Marketing, The Wharton School, University of Pennsylvania. Lu: Assistant Professor of Marketing, Tepper School of Business, Carnegie Mellon University. All correspondence about the manuscript can be directed to the first author. We thank Anthony Dukes, Upender Subramanian, Olivier Toubia, and the reviewer team for comments on an earlier draft.

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Yildirim, P., Wei, Y., Van den Bulte, C. et al. Social network design for inducing effort. Quant Mark Econ 18, 381–417 (2020). https://doi.org/10.1007/s11129-020-09227-6

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