Though deliberation is a critical component of democratic decision-making, existing deliberative processes do not scale to large groups of people. Motivated by this, we propose a model in which large-scale decision-making takes place through a sequence of small group interactions. Our model considers a group of participants, each having an opinion which together form a graph. We show that for median graphs, a class of graphs including grids and trees, it is possible to use a small number of three-person interactions to tightly approximate the wisdom of the crowd, defined here to be the generalized median of participant opinions, even when agents are strategic. Interestingly, we also show that this sharply contrasts with small groups of size two, for which we prove an impossibility result. Specifically, we show that it is impossible to use sequences of two-person interactions satisfying natural axioms to find a tight approximation of the generalized median, even when agents are non-strategic. Our results demonstrate the potential of small group interactions for reaching global decision-making properties.
The study of majority rule dynamics, which is an important part of deliberative decision-making, has also been a long-studied problem. A particular relevant experimental result analyzed the majority rule dynamic in groups of five and showed that the solution concept that performed best (out of 16 considered) was the majority rule equilibrium @cite_16 . Our work is a natural next step of this observation applied to the goal of scaling deliberative decision-making. Namely, given experimental evidence for the ability of small groups to come to consensus on the majority rule equilibrium, how can we use these small groups to make good decisions for larger groups?
@cite_16
2083307720 1970744965 2241340194 2151939563
This article reports the findings of a series of experiments on committee decision making under majority rule. The committee members had relatively fixed preferences, so that the process was one of making decisions rather than one of problem solving. The predictions of a variety of models drawn from Economics, Sociology, Political Science and Game Theory were compared to the experimental results. One predictive concept, the core of the noncooperative game without side payments (equivalent to the majority rule equilibrium) consistently performed best. Significantly, however, even when such an outcome did not exist, the experimental results did not display the degree of unpredictability that some theoretical work would suggest. An important subsidiary finding concerns the difference between experiments conducted under conditions of high stakes versus those conducted under conditions of much lower stakes. The findings in the two conditions differed considerably, thus calling into question the political applicability of numerous social psychological experiments in which subjects had little or no motivation. We suggest a new model for strategic voting based on local dominance, where voters consider a set of possible outcomes without assigning probabilities to them. We prove that voting equilibria under the Plurality rule exist for a broad class of local dominance relations. Furthermore, we show that local dominance-based dynamics quickly converge to an equilibrium if voters start from the truthful state, and we provide weaker convergence guarantees in more general settings. Using extensive simulations of strategic voting on generated and real profiles, we show that emerging equilibria replicate widely known patterns of human voting behavior such as Duverger's law, and that they generally improve the quality of the winner compared to non-strategic voting. We introduce a two-state opinion dynamics model where agents evolve by majority rule. In each update, a group of agents is specified whose members then all adopt the local majority state. In the mean-field limit, where a group consists of randomly selected agents, consensus is reached in a time that scales @math , where @math is the number of agents. On finite-dimensional lattices, where a group is a contiguous cluster, the consensus time fluctuates strongly between realizations and grows as a dimension-dependent power of @math . The upper critical dimension appears to be larger than 4. The final opinion always equals that of the initial majority except in one dimension. Many analyses of plurality-rule elections predict the complete coordination of strategic voting, and hence support for only two candidates. Here I suggest that stable multi-candidate support will arise in equilibrium. A group of voters must partially coordinate behind one of two challenging candidates in order to dislodge a disliked incumbent. In a departure from existing models, the popular support for each challenger is uncertain. This support must be inferred from the private observation of informative signals, such as the social communication of preferences throughout the electorate, or the imperfect observation of opinion polls. The uniquely stable voting equilibrium entails only limited strategic voting and hence incomplete coordination. This is due to the surprising presence of negative feedback: an increase in the degree of strategic voting by others reduces the incentives for an individual to vote strategically. The incentive to vote strategically is lower in relatively marginal elections, after controlling for the distance from contention of a trailing preferred challenger. A calibration of the model applied to the UK General Election of 1997 is consistent with the impact of strategic voting and the reported accuracy of voters understanding of the electoral situation. It suggests that nearly 50 seats may have been lost by the Conservative party due to strategic voting.