Social Choice and Social Networks

CHAPTER 1 Social Choice and Social Networks Umberto Grandi 1.1 Introduction [[TODO. when a group of people takes a decision, the structure of the group needs to be taken into consideration.]] Take the example of a vote in a small committee, like a department meeting or a company board. Each member typically has a partial knowledge of other members preferences, and before a decision is taken they will typically take strategic decisions on what information to disclosure and to whom. There are important variables that they take into account: They know which people they can count on; They know whose opinion they trust; They also know whether their opinion will be influential and on which other committee member. That is, they reason about the structure of different networks that relate the individuals around them, and are able to devise and play complex strategies to achieve their goals and influence the result of the collective decision. But this is not a property that pertains to small groups of people. Recent elections have shown how echo-chamber effects seem to polarise the opinions of societies, and how viral content can rapidly shift the public view, making traditional polls rather unstable. Moreover, emerging technologies in the field of e-democracy present new challenges for designing trustworthy mechanisms for collective decisions on social media or on the Internet in general. Social network analysis (Jackson, 2008; Easley and Kleinberg, 2010) is a burgeoning area which provides tools to analyse networks. It has proven very successful in many diverse fields, e.g., in the study of virus diffusion in biology, job markets analyses, targeted marketing. Well-established economic frameworks such as game theory and the study of markets have already started considering more complex models of society based on the study of networks. Social choice theory has been left relatively untouched by these recent developments, and it is only recently that researchers have started to study the interplay between the networks relating members of a community and the collective decisions they take. These problems have been attacked under several angles, and this chapter gives an overview of the main results obtained so far. First, we can start from a standard situation of social choice like an election or a coalitional game, and explore the effects of taking into consideration a network structure that relates the decision-makers. Second, once information on a social network is available,

2 U. Grandi we can design decision mechanisms that exploit this additonal structure, for instance to prevent strategic actions by individuals or simply to obtain a better collective decision. Third, ideas from social choice theory have been used to devise novel models of opinion diffusion on social networks, based on the aggregation of the opinions of an agent s influencers. 1.2 Basic definitions Some general notation, if useful, will be added here. 1.3 Effects Of Social Networks on Collective Choices Maximum likelyhood estimators with a (possibly unknown) network: Tsang et al. (2015) Conitzer (2012) Procaccia et al. (2015) Iterative voting with plurality, with agents observing both a social poll and their neighbours votes: Sina et al. (2015) Tsang and Larson (2016) Coalitional games on social networks Elkind (2014) based on work by: Aumann and Dreze (1974) Demange (2004) 1.4 Social Choice Mechanisms Designed for Networks Proxy voting (adapting PageRank to committee selection): Boldi et al. (2009) Representing influence and preferences in a compact way: Maran et al. (2013) Trust-based recommender systems: Andersen et al. (2008) (axiomatic, trust defined from network) Brill et al. (2016a) (new mechanism based on network structure) Rating systems, similar idea to Brill et al but game-theoretic analysis: Grandi and Turrini (2016)

Social Choice and Social Networks 3 Social polls (like doodle) with individuals voting sequentially: Gaspers et al. (2013) Voting rules that take into account the intrinsic utility gained by voting like one s neighbours: Salehi-Abari and Boutilier (2014) 1.5 Opinion Diffusion [TOBEUPDATED] The problem of diffusion on networks has been extensively studied in the field of social network analysis, be it diffusion of diseases, information, or opinions. Often, these models adapted techniques from physics or biology, such as flocking models, cellular automata, or Ising models. Novel frameworks were also developed to deal with the diffusion of individual opinions, in which individual views are updated by averaging the views of neighboring individuals. Two classical such examples are threshold models (Granovetter, 1978), with its more recent generalisations Kempe et al. (2003, 2005), and the De Groot or Lehrer-Wagner model (de Groot, 1974; Lehrer and Wagner, 1981), which are however based on a simple representation of opinions as a binary view on a single issue, or a real-valued view in the interval [0, 1]. Building on this latter literature, a recent stream of papers have adapted averaging models to more complex and realistic representations of opinions: knowledge bases (Schwind et al., 2015, 2016; Cholvy, 2016) preferences over alternatives (Ghosh and Velázquez-Quesada, 2015; Brill et al., 2016b) binary evaluations (Grandi et al., 2015; Slavkovik and Jamroga, 2016) While the use of (propositional) logic in this paper is confined to the representation of constraints, we mention that a logical perspective on diffusion in social networks has been explored in a number of papers (see, e.g., Baltag et al. (2013); Christoff and Hansen (2015)). [ADD] Talk about: Grabisch and Rusinowska (2010) [ADD selection of results] 1.6 Conclusions Write some open problems, directions of research. [note: political science references still missing] Topics I did not mention: network elicitation from votes: Fish et al. (2016) preference based network formation: Salehi-Abari and Boutilier (2015)

4 U. Grandi Bibliography R. Andersen, C. Borgs, J. Chayes, U. Feige, A. Flaxman, A. Kalai, V. Mirrokni, and M. Tennenholtz. Trust-based recommendation systems: An axiomatic approach. In Proceedings of the 17th International Conference on World Wide Web (WWW), 2008. R. J. Aumann and J. H. Dreze. Cooperative games with coalition structures. International Journal of Game Theory, 3(4):217 237, 1974. A. Baltag, Z. Christoff, J. U. Hansen, and S. Smets. Logical models of informational cascades. Studies in Logic, 47:405 432, 2013. P. Boldi, F. Bonchi, C. Castillo, and S. Vigna. Voting in social networks. In Proceedings of the 18th ACM Conference on Information and Knowledge Management (CIKM), 2009. M. Brill, V. Conitzer, R. Freeman, and N. Shah. False-name-proof recommendations in social networks. In Proceedings of the 2016 International Conference on Autonomous Agents & Multiagent Systems (AAMAS), 2016a. M. Brill, E. Elkind, U. Endriss, and U. Grandi. Pairwise diffusion of preference rankings in social networks. In Proceedings of the 25th International Joint Conference on Artificial Intelligence (IJCAI), 2016b. L. Cholvy. Diffusion of opinion and influence. In 10th International Conference on Scalable Uncertainty Management (SUM), 2016. Z. Christoff and J. U. Hansen. A logic for diffusion in social networks. Journal of Applied Logic, 13(1):48 77, 2015. V. Conitzer. Should social network structure be taken into account in elections? Mathematical Social Sciences, 64(1):100 102, 2012. M. H. de Groot. Reaching a consensus. Journal of the American Statistical Association, 69(345):118 121, 1974. G. Demange. On group stability in hierarchies and networks. Journal of Political Economy, 112(4):754 778, 2004. D. Easley and J. Kleinberg. Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, New York, NY, USA, 2010. E. Elkind. Coalitional games on sparse social networks. In Proceedings of the 10th International Conference on Web and Internet Economics (WINE), 2014. B. Fish, Y. Huang, and L. Reyzin. Recovering social networks by observing votes. In Proceedings of the 2016 International Conference on Autonomous Agents & Multiagent Systems, 2016.

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6 U. Grandi N. Schwind, K. Inoue, G. Bourgne, S. Konieczny, and P. Marquis. Belief revision games. In Proceedings of the Twenty-Ninth AAAI Conference on Artificial Intelligence (AAAI), 2015. N. Schwind, K. Inoue, G. Bourgne, S. Konieczny, and P. Marquis. Is promoting beliefs useful to make them accepted in networks of agents? In Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence (IJCAI), 2016. S. Sina, N. Hazon, A. Hassidim, and S. Kraus. Adapting the social network to affect elections. In Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems (AAMAS), 2015. M. Slavkovik and W. Jamroga. Iterative judgment aggregation. In Proceedings of the 22nd European Conference on Artificial Intelligence (ECAI), ECAI, 2016. A. Tsang and K. Larson. The echo chamber: Strategic voting and homophily in social networks. In Proceedings of the International Conference on Autonomous Agents & Multiagent Systems (AAMAS), 2016. A. Tsang, J. A. Doucette, and H. Hosseini. Voting with social influence: Using arguments to uncover ground truth (extended abstract). In Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems (AAMAS), 2015.