Abstract
This paper studies the dynamics by which individuals with heterogeneous preferences partition themselves into groups. A novel experimental environment is developed to capture the tension between increasing returns to group size and attaining a group policy closest to an ideal point. Subjects can move freely between locations, with group policy either fixed by location or determined by member vote. A primary goal is to assess which of two stability concepts common to the group formation literature predicts which groups agents sort into. The same set of Nash stable partitions exist in each condition, with the efficient, strong Nash stable state requiring subjects to form heterogeneous groups and compromise on policy. I find that subjects who are only able to move between locations with fixed policies always over-segregate, rather than build efficient heterogeneous groups. When mobility is combined with the ability to vote on local policy, most subjects reach the efficient partition. This shows outcomes cannot be determined by considering the existence of stable states alone and that consideration must also be given to subtle aspects of the system dynamics. Further, it suggests that experiments may play an important role in understanding these group formation dynamics.
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Notes
This dichotomy originates with Tiebout (1956)’s canonical paper on free mobility and local public good provision, which seemingly refers to both types of deviations without distinction. Within these broad classifications, the definitions vary as well, both with respect to whether agents can relocate only to extant groups, or are able to establish groups of their own, and whether agents can coalitionally relocate only alongside those in their previous group or they are able to coordinate with any agents in the population.
Westhoff (1977) first formalized Tiebout’s model, while removing congestion and incorporating majority rule voting on local tax rates. He proved existence of a stable partition of agents into several communities where the median voter’s will was enacted, and that each community in this partition represented an interval of agents. Greenberg and Weber (1986) assume that agents’ preferences can be ranked by a unidimensional parameter and demonstrate existence of an equilibrium partition immune to coalition deviations by secession. Demange and Henriet (1991) incorporate a similar assumption in a market for a differentiated consumption good with free-entry and demonstrate that an optimal, stable configuration of consumers across firms exists. Demange (2005) provides a more thorough overview of this tension between increasing returns to group size and preference heterogeneity.
Cinyabuguma et al. (2005) and Maier-Rigaud et al. (2005) have found that expulsion is used frequently in public goods games. However, since the threat of expulsion also increases cooperation, subjects tend to earn more when expulsion is available. In contrast, Ahn et al. (2008) found that allowing subjects to control group size can suppress earnings in a pure public goods game. While restricted entry enables groups to sustain higher contribution rates, groups of cooperative subjects tend to earn less by being overly discerning in whom they allow to enter.
In the first three periods, subjects viewed the location features for the duration of the experiment thus far.
In the case that the group had an even-numbered population, the median policy was equal to the mean of the two middle votes.
While Nash stability requires that an outcome be immune to unilateral deviations, strong Nash stability requires that the outcome be immune to deviations by any subset of agents. Another common stability concept that considers (nested) coalitional deviations is the coalition-proof Nash equilibrium (Bernheim et al. 1987). It is less stringent than the strong Nash equilibrium, as it does not require that partitions be immune to all coalition deviations, but only those that are “self-enforcing,” i.e. from which no subset of the coalition would wish to further deviate at any step. In the experimental set-up of this paper, the unique strong Nash partition is also the unique coalition-proof partition.
While the exact payoff for each agent depends on the precise policy p, aggregate payoffs are maximized in each case when the group experiences the p at the midpoint of the range displayed. The full derivation is provided in the online appendix.
The average efficiency over the final five periods of one population of 8 subjects is taken as a single observation. The difference between conditions is significant at less than the 1 % level using a two-sample Wilcoxon rank-sum test (z = 2.664).
Taking the average group size in a population over the final five periods as a single observation, the difference in group size is significantly higher in the Voting condition at the \(p<.01\) level using a Wilcoxon rank-sum test (z = 3.30).
Subjects are in the same group as the person who shares their preference type in 85 % of observations in the Fixed Policy condition and 94 % of observations in the Voting condition (over the final five periods).
Subjects in the Voting condition are in a group with subjects of other types 61 % of the time in the Voting condition, compared to 16 % of the time in the Fixed Policy condition. The difference is significant at \(p<.01\) level using a Wilcoxon rank-sum test (z = 3.309).
In particular, subjects in the Fixed Policy condition experience policies that differ from their ideal by only .026, while the average difference in the Voting condition is .071. This difference is significant at \(p<.01\) using a Wilcoxon rank-sum test and taking the population as the unit of observation (z = 2.78).
In the Voting condition, the strong Nash stable state persists with a similar likelihood; in the Fixed Policy condition, the system does not reach the strong Nash stable state.
Recall that subjects incur a moving cost of .3 in a period in which they switch groups. Here and in the discussion to follow, a move is only considered a “best response” if the player’s payoff increases in the immediate period even after accounting for the moving cost. The presence of the moving cost does not affect the best responses in any case. In the case considered here, by switching groups this subject improves his earnings by .3 in the period that he moves and .6 in each period after.
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Acknowledgments
I thank Charles Plott, Leeat Yariv, Peter Matthews, Guillaume Fréchette, Rod Kiewiet, John Ledyard, Jean-Laurent Rosenthal, and the Harvard Decision Science Laboratory. I am also grateful to two anonymous referees and the editor, David Cooper, for their many helpful suggestions. Funding provided by the Caltech Laboratory for Experimental Economics and Political Science.
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Robbett, A. Voting with hands and feet: the requirements for optimal group formation. Exp Econ 18, 522–541 (2015). https://doi.org/10.1007/s10683-014-9418-8
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DOI: https://doi.org/10.1007/s10683-014-9418-8