We study a setting where individuals prefer to coordinate with others but they differ on their preferred action. Our interest is in understanding the role of link formation with others in shaping behavior. So we consider the situation in which interactions are exogenous and a situation where individuals choose links that determine the interactions. Theory is permissive in both settings: conformity (on either of the actions) and diversity (with different groups choosing their preferred actions) are both sustainable in equilibrium. We conduct an experiment to understand how link formation affects equilibrium selection. Our experiment reveals the powerful effect of linking on equilibrium selection: with an exogenous complete network, subjects choose to conform on the majority’s preferred action. By contrast, with endogenous linking—irrespective of the costs of linking—subjects always opt for diversity of actions.
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A similar tension also arises in the context of markets with network externalities in which consumers prefer a particular standard/platform, but there is social value of everyone being on the same standard.
For concreteness, in the two person game, suppose individuals of type U earn 4 from coordinating on action up, and they earn 2 from coordinating on action down. The payoffs of the type D go the other way: D types earn 4 from coordinating on down and only earn 2 from coordinating on up. Finally, both players earn zero if they miscoordinate.
We refer to this as the exogenous complete network.
Specifically, we assume that the cost of linking is smaller than the payoff from successful coordination on the less preferred action.
We also considered an experimental treatment with a minority of 3 members, and a majority of 12: when the minority is so small we find that the freedom to form links makes no difference. Subjects choose to conform with the majority’s preferred action both in the exogenous complete network as well as when links are endogenous. This treatment is presented in the Supplementary Material.
Majority group subjects choose their preferred action and persist with that action from early on, in both treatments.
The present paper reports an experiment with human subjects; there is also a literature that studies simulations of complex network dynamics. For a recent paper in this line of work, that studies network linking and segregation, see Lipari et al. (2019).
Kearns et al. (2012) and Kearns et al. (2009) study voting behaviour by biased voters. In this game, players must coordinate on the same vote to earn a payoff. Individuals differ on their preferred outcome. Kearns et al. (2009) show that with exogenous networks subjects are quite successful in achieving coordination. By contrast, Kearns et al. (2012) show that with endogenous linking, subjects form rich networks but fail to reach coordination. This finding is in the same spirit as our work: with conflicting preferences, endogenous linking can lead to a decrease in welfare.
The detailed proof is provided in Appendix A.
It is worth noting that this argument holds for arbitrary values of \(\alpha \) and \(\beta \). Thus conformity is preferred even if \(\alpha \) is much larger than \(\beta \): this is because the minority collectively gains less than what the majority losses when the minority switches away from conformism to diversity.
An edge departing from node i towards node j without connecting j means that player i proposes a link to player j but j does not propose a link to i.
The complete network is shown as it would be in game with endogenous networks, had the complete network emerged. See the instructions in Appendix C.
Earnings for minority players are not significantly different across treatments. Similarly, for the majority earnings are not significantly different across treatments, except for endo where they earn about \(50\%\) more than in the rest. In all treatments, majority participants earn more than those in the minority.
Regarding the demographics, female participants represent \(47\%\) of all subjects in endo, and \(51\%\) in exo. All participants are undergraduate students, and the average age is 23 years old. Participants’ academic backgrounds are in law, finance, business, economics, pedagogics, tourism, and nursing.
We also analyzed the data using Wilcoxon–Mann–Whitney tests and group averages as the unit of observation. The regression’ results are consistent with those of the non-parametric tests.
We note that the treatments require a group of 15 subjects to play the same game repeatedly (20 times). In principle, therefore, we should also be considering repeated game effects. In our setting, equilibria of the repeated game will include a sequence of the static game equilibrium, and possibly other more complicated patterns of behavior. In the experiments, subjects converge fairly quickly and behave very much in line with a static equilibrium. The key finding is the contrast in outcomes between the exogenous and the endogenous linking setting. As both these treatments involve repeated interactions, repeated game effects are not central to understanding this difference.
In the Supplementary Material, we report detailed measures on the likelihood of successfully turning proposals into links, by type of player for all endogenous treatments.
Note that conformity on either action is an equilibrium as long as all agents have at least one connection and \(\alpha \le 2\beta \). These conditions are satisfied for all networks endogenously created in endo. See details in the Supplementary Material.
This design choice is justified by the minimal variations in the network structures observed in endo, which closely resemble the static nature of the fixed structures considered here. However, we realize this is not the unique option to investigate the role of endogenous linking. For example, an alternative would consist in setting the exact sequences of networks as created in endo, and ask a new group of subjects to play coordination games on those networks (which would then exogenously change over time). While such a method may offer a closer comparison with endo, we believe it would not uncover significantly more insights.
As in the design presented above, each of these new treatments consists of 6 groups of 15 subjects whose decisions are made over 20 periods, resulting in a total of 240 observations at the group level.
Treatments exosym and exoasym are aimed to resemble linking patterns as in endo, while imposing the network exogenously. Potentially, other forms of linking patterns can induce differences in outcomes and discourage conformity even more, but this was not part of our aim in this robustness check. For examples of studies exploring the effects of different linking patterns (network structures) on outcomes see Choi and Lee (2014), Antonioni et al. (2013) and Kearns et al. (2012).
In exosym, one of the remaining groups converges to conformity on the minority’s action and the remaining four groups converge to diversity. In exoasym, the remaining three groups converge to diversity.
Advani, A., & Reich, B. (2015). Melting pot or salad bowl: The formation of heterogeneous communities. Institute of Fiscal Studies, WP 15/30.
Antonioni, A., Cacault, M. P., Lalive, R., & Tomassini, M. (2013). Coordination on networks: Does topology matter? PLOS ONE.
Bernasconi, M. & Galizzi, M. (2005). Coordination in networks formation: Experimental evidence on learning and salience. Coalition Theory Network Working Papers 12159, Fondazione Eni Enrico Mattei (FEEM).
Blume, L. (1993). The statistical mechanics of strategic interaction. Games and Economic Behavior, 4, 387–424.
Bojanowski, M., & Buskens, V. (2011). Coordination in dynamic social networks under heterogeneity. Journal of Mathematical Sociology, 35, 249–286.
Camerer, C. (2003). Behavioral game theory: Experiments in strategic interaction. Princeton: Princeton University Press.
Charness, G., Feri, F., Melendez-Jimenez, M. A., & Sutter, M. (2014). Experimental games on networks: Underpinnings of behavior and equilibrium selection. Econometrica, 82, 1615–1670.
Choi, S., & Lee, J. (2014). Communication, coordination, and networks. Journal of European Economic Association, 12(1), 223–247.
Corbae, D., & Duffy, J. (2008). Experiments with network formation. Games and Economic Behaviour, 64(1), 81–120.
Crawford, V. P. (1995). Adaptive dynamics in coordination games. Econometrica, 63(1), 103–143.
Ellison, G. (1993). Learning, local interaction, and coordination. Econometrica, 61, 1047–1071.
Ellwardt, L., Hernández, P., Martínez-Canovas, G., & Muñoz-Herrera, M. (2016). Conflict and segregation in networks: An experiment on the interplay between individual preferences and social influence. Dynamic and Games, 3(2), 191–216.
Fischbacher, U. (2007). z-tree: Zurich toolbox for ready-made economic experiments. Experimental economics, 10(2), 171–178.
Goeree, J. K., Riedl, A., & Ule, A. (2009). In search of stars: Network formation among heterogeneous agents. Games and Economic Behavior, 67(2), 445–466.
Goyal, S., & Vega-Redondo, F. (2005). Network formation and social coordination. Games and Economic Behavior, 50(2), 178–207.
Isoni, A., Poulsen, A., Sugden, R., & Tsutsui, K. (2014). Efficiency, equality, and labeling: An experimental investigation of focal points in explicit bargaining. The American Economic Review, 104(10), 3256–3287.
Jackson, M. O., & Watts, A. (2002). On the formation of interaction networks in social coordination games. Games and Economic Behavior, 41(2), 265–291.
Jackson, M. O., & Wolinsky, A. (1996). A strategic model of social and economic networks. Journal of Economic Theory, 71(1), 44–74.
Kearns, M., Judd, S., Tan, J., & Wortman, J. (2009). Behavioral experiments on biased voting in networks. PNAS, 105(5), 1347–1352.
Kearns, M., Judd, S., & Vorobeychik, Y. (2012). Behavioral experiments on a network formation game. ACM EC 2012.
Lewis, D. (1969). Conventions: A philosophical study. Oxford: Harvard University Press.
Lipari, F., Stella, M., & Antonioni, A. (2019). Investigating peer and sorting effects within an adaptive multiplex network model. Games, 10.
Neary, P. R. (2012). Competing conventions. Games and Economic Behavior, 76(1), 301–328.
Riedl, A., Rohde, I. M., & Strobel, M. (2016). Efficient coordination in weakest-link games. The Review of Economic Studies, 83(2), 737–767.
Schelling, T. (1960). The strategy of conflict. Oxford: Harvard University Press.
We are grateful to the editor and two anonymous referees for comments that have significantly improved the paper. This paper has been supported by the EU through FET-Proactive Project DOLFINS (Contract No. 640772) and FET-Open Project IBSEN (Contract No. 662725), Grant FIS2015-64349-P (MINECO/FEDER, UE), Grant ECO2017-87245-R (MINECO/FEDER, UE), and Grant AICO/2019/053 (Consellera dInnovaci, Universitats, Cincia i Societat Digital, Generalitat Valenciana). We are grateful to Marina Agranov, Gary Charness, Vince Crawford, Sihua Ding, Matt Elliott, Edoardo Gallo, Joerg Kalbfuss, Jonathan Newton, Theo Offerman, Gustavo Paez, Romans Pancs, Debraj Ray, Arno Reidl, Marzena Rostek, Robert Sugden, Alan Walsh, Sevgi Yuksel, and participants at a number of seminars for helpful comments.
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Goyal, S., Hernández, P., Martínez-Cánovas, G. et al. Integration and diversity. Exp Econ 24, 387–413 (2021). https://doi.org/10.1007/s10683-020-09676-6
- Equilibrium selection
- Social coordination