Pricing and referrals in diffusion on networks

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From theory to application
Matt V. Leduc, Matthew O. Jackson, Ramesh Johari
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From Theory to Application

When a new product or technology is introduced, there is uncertainty about its quality. The decision to adopt a new technology is a difficult one and countless examples illustrate how consumers or firms can be slow to adopt such new technologies, ranging from smart phones to electric cars.

This uncertainty leads to informational free-riding: a potential consumer may wish to delay adoption in order to let other agents bear the risks of experimenting with the technology and learn from their experiences. This can considerably delay the diffusion of a new technology. This problem occurs in many settings: not only do consumers benefit from the research of friends and relatives into new products, but farmers benefit from the experience of other farmers with a new crop. Likewise in industry, research spills over to other firms. People benefit from the experience of their friends and relatives regarding a vaccine of unknown side effects. In developing countries, villagers may learn about a new microcredit program from the experience of other members of the community.

This complicates the problem of new product or technology adoption and can lead to inefficiencies in diffusion processes, as there are risks from being an early adopter and externalities in early adoption decisions. Indeed, the possibility of free-riding induces a specific form of social inefficiency: agents who interact the least with the rest of the population have the greatest incentives to try the product since they have the least opportunity to observe others' choices. Given the risks of experimentation, it would be more socially efficient to have agents who interact the most with the rest of the population experiment since they are observed by many others, thus lowering the number of experimenters needed to achieve a given level of information in the society. Thus, firms and governments must carefully optimize information diffusion when trying to maximize the adoption of a new product, program, or technology deemed to be socially beneficial.

Strategic interactions: game theoretic setting

To analyze this problem, Matt V. Leduc, Matthew Jackson, and Ramesh Johari provide (Leduc et al. 2017) a game theoretic analysis of agents connected in a social network who choose either to adopt early or to delay adoption of a product of unknown quality in order to observe their friends’ actions. In particular, if an agent adopts early, she tells her friends about the product’s quality. This is similar to a word-of-mouth process, as studied in Campbell (2013) and enables social learning via free-riding. A monopolist seller (or social planner) can induce people to experiment with the product via two types of incentives: (i) price discounts offered to early adopters and (ii) referral rewards. The latter are payments to an agent who tries the product early based on how many of his friends later adopt the product.

Studying the decisions of agents in such a context poses challenges since the decisions of all agents depend on those of their friends, which depend on others, etc. However this problem can be studied tractably using methods developed for analyzing network games, as for example in Jackson and Yariv (2007), Manshadi and Johari (2009), Galeotti et al. (2010), Leduc (2014) or Leduc and Momot (2017).

Incentivizing informational diffusion: price discounting versus referral incentives

Referral incentives are seen in many settings with new products or technologies. For instance, in July of 2015, Tesla Motors announced a program by which an owner of a Model S Sedan would receive a 1000 dollar benefit if the owner referred a friend who also buys a Model S Sedan[1]. Dropbox rapidly grew from around one hundred thousand users in the fall of 2008 to over four million by the spring of 2010, with more than a third of the signups coming through its official referral program that offered free storage to both referrer and referree[2].  Such programs have been used by many new companies from Airbnb to Uber, and also by large existing companies when introducing new products (e.g., Amazon's Prime).

Referral incentives induce agents who have more connections to try the product early since they have more friends to refer and thus expect greater referral rewards. In contrast, price discounts given to early adopters mostly induce agents with few connections to try the product early. Indeed, the discount is given equally to any early adopter, irrespectively of his number of friends. Agents with few connections are the ones with the greatest incentives to try early in any case since they are unlikely to collect information from their friends.

Referral incentives are effectively a screening device that can be used to induce highly-connected individuals to adopt early and thus take advantage of their popularity, solving an informational inefficiency at the same time as increasing profits. The monopolist (or social planner) does not need to have any precise information about the structure of the social network to implement it. Incentivizing agents with price discrimination (e.g., offering lower prices to agents based on how connected they are) is generally more complex and requires detailed knowledge of the connectivity structure, which is rarely available.

Homogenous versus heterogeneous interaction structure

A profit-maximizing monopolist ideally wishes to minimize the number of early adopters, since it is costly to induce experimentation - either price discounts or referral incentives must be offered and the monopolist would like to minimize such payments and maximize the number of eventually informed high-paying adopters. A social planner wishing to spread the adoption of some socially beneficial program or technology also wishes to do that since she wants to minimize the number of agents who take the risk of experimenting. The optimal incentivizing strategy, however, depends on social network structure via the relative numbers of agents with different numbers of connections. If the social network presents enough heterogeneity in the sense that a small fraction of agents have a disproportionately larger number of connections, then referral incentives are more efficient. Indeed, such agents are more likely to be observed by others and referral incentives will entice them to adopt early. This achieves high information diffusion for a small number of experimenters.

If instead the social network is homogenous (or regular) in the sense that all agents have roughly the same number of connections, then referral incentives are less effective and price discounts are a useful tool to maximize profits. Indeed, on such a social network, the above-mentioned effects cannot be leveraged since there are no agents who are substantially more connected and thus referral incentives have less bite. Moreover, referrals also tend to be costlier to the monopolist, not only because she must pay them to consumers, but also since they are only given if the product turns out to be of good quality. Indeed, if the product turns out to be of bad quality, an early adopter will not be able to reap many referral rewards since few or none of her friends will ultimately adopt it. The monopolist must thus inflate the value of the referral reward to compensate consumers for that uncertainty. A price discount, on the other hand, is given to early adopters independently of whether the product turns out to be of good or bad quality.

This work should help develop more sophisticated models of technology adoption taking into account uncertainty about product quality as well as informational free-riding and the role of referral incentives. Such models can in turn lead to the development of better dynamic pricing mechanisms that optimize the spread of information in large social systems.  This can also inform policy, both in regulating monopolists and also promoting other diffusion processes. Examples may include government or other non-profit programs such as health, education, sanitation or more sustainable forms of energy.


[1] Bloomberg Business News, “Musk Takes Page From PayPal With Tesla Referral Incentive,” August 31, 2015

[2] Forbes,  “Learn The Growth Strategy That Helped Airbnb And Dropbox Build Billion-Dollar Businesses,” Feb. 15, 2015



Leduc, M.V., Jackson, M.O. and Johari, R. 2017. Pricing and Referrals in Diffusion on Networks. Games and Economic Behavior 104, 568-594.

Campbell, A. 2013. Word-of-mouth communication and percolation in social networks. American Economic Review 103, 6, 2466-2498.

Jackson, M. O. and Yariv, L. 2007. Diffusion of behavior and equilibrium properties in network games. American Economic Review 97, 2, 92-98.

Manshadi, V. and Johari, R. 2009. Supermodular network games. Allerton Conference for Communication, Control, and Computing.

Galeotti, A., Goyal, S., Jackson, M. O., Vega-Redondo, F., and Yariv, L. 2010. Network games. Review of Economic Studies. 77, 218-244.

Leduc, M. V. 2014. Mean-field models in network game theory. Ph.D. thesis, Stanford University.

Leduc, M. V. and Momot, R. 2017. Strategic investment in protection in networked systems. Network Science 5, 1, 108-139.


About the Authors


Matt V. Leduc is currently a postdoctoral fellow at the Aix-Marseille School of Economics (AMSE) and the GREQAM (France). He is also an associate research scholar at IIASA (Austria). He obtained his PhD from Stanford University, where he worked on network game theory. He has also been a visiting research scholar at Cambridge University (Department of Economics/INET Institute). Dr. Leduc's research focuses mainly on microeconomics, game theory, the economics of networks and contract theory. 



Matthew O. Jackson is the William D. Eberle Professor of Economics at Stanford University and an external faculty member of the Santa Fe Institute and a senior fellow of CIFAR. Jackson's research interests include game theory, microeconomic theory, and the study of social and economic networks, on which he has published many articles and the book Social and Economic Networks.  He also teaches an online course on networks and co-teaches two others on game theory.  Jackson is a Member of the National Academy of Sciences, a Fellow of the American Academy of Arts and Sciences,  a Fellow of the Econometric Society, and an Economic Theory Fellow, and his other honors include the von Neumann Award, a Guggenheim Fellowship, the Social Choice and Welfare Prize, the B.E.Press Arrow Prize for Senior Economists.  


Ramesh Johari is an Associate Professor at Stanford University, with a full-time appointment in the Department of Management Science and Engineering (MS&E), and courtesy appointments in the Departments of Computer Science (CS) and Electrical Engineering (EE). He is a member of the Operations Research group and the Social Algorithms Lab (SOAL) in MS&E, the Information Systems Laboratory in EE, the Institute for Computational and Mathematical Engineering, the steering committee of the Stanford Cyber Initiative, and the Stanford Bits and Watts Initiative. He received an A.B. in Mathematics from Harvard, a Certificate of Advanced Study in Mathematics from Cambridge, and a Ph.D. in Electrical Engineering and Computer Science from MIT.
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