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 problem of scaling up decision-making is a well-studied problem in the context of voting and preference elicitation. In this context, one is typically trying to approximate or calculate the output of a social choice function while only eliciting small amounts of information from voters (e.g. through pairwise comparisons) @cite_9 @cite_23 @cite_14 @cite_30 . Our paper can be viewed as an attempt to create a thread of research mirroring preference elicitation, but for deliberative processes. Thus far, computational social choice has primarily viewed decision-making and preference elicitation from the perspective of efficiency, accuracy, and strategic issues. We propose that deliberation is a valuable new dimension to consider in social choice and provide a small step in this direction. We discuss this direction more in the concluding section.
@cite_30 @cite_9 @cite_14 @cite_23
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We present theoretical and empirical results demonstrating the usefulness of social choice functions in crowdsourcing for participatory democracies. First, we demonstrate the scalability of social choice functions by defining a natural notion of epsilon-approximation, and giving algorithms which efficiently elicit such approximations for two prominent social choice functions: the Borda rule and the Condorcet winner. This result circumvents previous prohibitive lower bounds and is surprisingly strong: even if the number of ideas is as large as the number of participants, each participant will only have to make a logarithmic number of comparisons, an exponential improvement over the linear number of comparisons previously needed. Second, we apply these ideas to Finland's recent off-road traffic law reform, an experiment on participatory democracy in real life. This allows us to verify the scaling predicted in our theory and show that the constant involved is also not large. In addition, by collecting data on the time that users take to complete rankings of varying sizes, we observe that eliciting partial rankings can further decrease elicitation time as compared to the common method of eliciting pairwise comparisons. Stated preference elicitation techniques, such as discrete choice experiments and best-worst scaling, are now widely used in health research to explore the public’s choices and preferences. In this paper, we propose an alternative stated preference elicitation technique, which we refer to as ‘trio-wise’. We explain this new technique, its relative advantages, modeling framework, and how it compares to the best-worst scaling method. To better illustrate the differences and similarities, we utilize best-worst scaling Case 2, where individuals make best and worst (most and least) choices for the attribute levels that describe a single profile. We demonstrate this new preference elicitation technique using an empirical case study that explores preferences among the general public for ways to involve them in decisions concerning the health care system. Our findings show that the best-worst scaling and trio-wise preference elicitation techniques both retrieve similar preferences. However, the capability of our trio-wise method to provide additional information on the strength of rank preferences and its ability to accommodate indifferent preferences lead us to prefer it over the standard best-worst scaling technique. Voting is a general method for aggregating the preferences of multiple agents. Each agent ranks all the possible alternatives, and based on this, an aggregate ranking of the alternatives (or at least a winning alternative) is produced. However, when there are many alternatives, it is impractical to simply ask agents to report their complete preferences. Rather, the agents' preferences, or at least the relevant parts thereof, need to be elicited. This is done by asking the agents a (hopefully small) number of simple queries about their preferences, such as comparison queries, which ask an agent to compare two of the alternatives. Prior work on preference elicitation in voting has focused on the case of unrestricted preferences. It has been shown that in this setting, it is sometimes necessary to ask each agent (almost) as many queries as would be required to determine an arbitrary ranking of the alternatives. In contrast, in this paper, we focus on single-peaked preferences. We show that such preferences can be elicited using only a linear number of comparison queries, if either the order with respect to which preferences are single-peaked is known, or at least one other agent's complete preferences are known. We show that using a sublinear number of queries does not suffice. We also consider the case of cardinally single-peaked preferences. For this case, we show that if the alternatives' cardinal positions are known, then an agent's preferences can be elicited using only a logarithmic number of queries; however, we also show that if the cardinal positions are not known, then a sublinear number of queries does not suffice. We present experimental results for all elicitation algorithms. We also consider the problem of only eliciting enough information to determine the aggregate ranking, and show that even for this more modest objective, a sublinear number of queries per agent does not suffice for known ordinal or unknown cardinal positions. Finally, we discuss whether and how these techniques can be applied when preferences are almost single-peaked. We consider a classic social choice problem in an online setting. In each round, a decision maker observes a single agent's preferences over a set of m candidates, and must choose whether to irrevocably add a candidate to a selection set of limited cardinality k. Each agent's (positional) score depends on the candidates in the set when he arrives, and the decisionmaker's goal is to maximize average (over all agents) score. We prove that no algorithm (even randomized) can achieve an approximation factor better than O(log logm logm). In contrast, if the agents arrive in random order, we present a (1-1 e-o(1))- approximate algorithm, matching a lower bound for the offline problem. We show that improved performance is possible for natural input distributions or scoring rules. Finally, if the algorithm is permitted to revoke decisions at a fixed cost, we apply regret-minimization techniques to achieve approximation 1- 1 e-o(1) even for arbitrary inputs.