voters insofar as a decision must be made, but, individually, they may still believe that ... we make deliberation models more realistic by taking into account ...
Deliberation in Networked Groups Carlo Martini and Stephan Hartmann Tilburg University How do groups reach a collective decision? One option is to apply some voting scheme and aggregate the individual judgments. By doing so, the group reaches a compromise as not everybody agreed with the compromise. A compromise is accepted by individual voters insofar as a decision must be made, but, individually, they may still believe that the judgment they originally submitted is the right one. This procedure is rather unusual for small and medium sized groups. Here, the various group members deliberate and change their judgments in the light of the judgments of others. Studying the deliberation process has attracted a lot of interest in the recent literature in philosophy and the social sciences (Aaken et. al. 2004). At the same time, in the past decade the study of social networks has been an area of extensive study in the social sciences (Jackson 2008). In this paper we aim at bringing the two strands of research together. More specifically, we make deliberation models more realistic by taking into account dependencies between the various group members. One of the main models discussed in the (philosophical) literature on deliberation and consensus is the Lehrer-Wagner model (Lehrer and Wagner 1980). According to the Lehrer-Wagner model, agents aggregate their probability judgments according to a model that makes use of converging properties of stochastic Markov chains. Agents assign (normalized) weights to all other agents in the group and probability estimates on the matter on which they are to deliberate. The set of weights is modeled into a square matrix (M ) where rows are normalized and the set of probability assignments is modeled into a vector V. The group outcome is determined by the M k × V where k tends to infinity. Lehrer and Wagner show that, under very general conditions, this process leads to a consensus and all entries in the resulting vector are equal. We see that the result of the deliberation process is not only a function of the agents’ opinions on the subject matter but also on the agents’ mutual opinion on their expertise. Several objections have been raised against the Lehrer-Wagner model (Bradley 2006). These objections typically put into doubt one or more of the assumptions the model makes. In this paper we are concerned with the assumption that the deliberating agents are independent. While a dependence of the agents can be cashed out in many ways, we focus here on the independence of agents’ mutual assignment of weights. That is to say that the weights associated with each agent are free to range between 0 and 1 independently of the assignments of the other agents. We weaken this assumption and assume that the agents are located in a social network. This network represents influence relations between the agents. In particular, we analyze certain characteristic network structures and discuss their formal properties and relevance for real life situations. The present paper looks specifically at two types of networks: the 1
Figure 1: Six-agent wheel formation wheel and the star formations. Both networks represent different types of biases in the group, the former being a case of preference bias based on distance, in a uniformly distributed neighborhood and the latter being a representation of a semi-dictatorship in which a central player leads the group. The first result is obtained by modeling agents as if they were distributed on a wheel (with no central player). See Figure 1. For any measure of distance1 between agent i and agent j, a fixed weight is assigned, and weights decrease and the distance increases. For instance, if the number of edges between i and j is 0 (the case of an agent assigning a weight to herself) the weight will be equal to 1, if there is one edge, the weight will be equal to .9, if there are two edges the weight will be equal to .7, and so forth. In this formation we obtain that the results of the iteration process exactly matches the arithmetic average of the associated vector of probabilities. The reason for this is that whenever such a formation exists, the matrix of weights produced is doubly-stochastic. It is straightforwardly shown that the fixed point vector that obtains is equal to (v1 + v2 + · · · + vn )/n where the v s are the values in the vector V and n is the number of agents. The results above are at least surprising in the intended interpretation of the LehrerWagner model. Results of the convergence process, given the wheel formation, are equal to the results from arithmetically averaging the values in the vector which is, in turn, equivalent to assuming equal weights. The dispute between advocates of the equal weight views and proponents of different weights views can find some new ground of debate here, in virtue of what we show, namely that the presence of certain network formations cancels out the role of weights in the deliberation process characteristic of the Lehrer-Wagner model. Nonetheless, one objection to the results just presented is that the schema for the assignment of weights (i.e. that the weight is a function of the distance) is too strong an assumption even in the presence of a formation in which agents are biased towards their neighbors. The idea is then to relax that assumption maintaining the idea of a bias in the deliberative group and to test the robustness of the previous analytical results. In this second step we do not assume that given a certain distance of i from j, the weight is fixed, but that it ranges within certain values. The weaker condition for assignment 1
In a real-life case distance can be interpreted in different ways, it could mean kinship, for instance, in a formation where agents are biased towards those that are genetically/culturally closer to them, or diplomatic distance, thinking of agents as countries. Several other interpretations are available but what is important is that the concept can easily be interpreted in a realistic scenario.
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Figure 2: Two data sets, the first contains the results of the iteration process and the second the arithmetic average of the vectors used. of weights is then that as the distance increases the values of the weights decreases but without being fixed at the beginning. With these weaker conditions, we systematically explore the parameter base by running simulations of the model. By using different highdimensional matrices we observe that the results of the iteration process overlap with the arithmetic average of the vectors in almost all cases up to the second or third decimal digit. Different matrices (each with a different schema for the assignment of weights) have been used in combinations with different vectors: Figure 2 shows the first 15 elements of one of the data sets obtained for a deliberative group of six agents. A third family of results can be observed with star-shaped networks. In the presence of a central player the results of the aggregation process are significantly closer to the probability estimate given by the central player himself. These results turn out to be those that we would expect in a group in which one of the agents plays the role of the leader. The group consensus is clearly biased towards the probability estimate given by the central player. In this case, like in the second case presented, the irregularity of the matrices used allows for little analytical investigation. The results from simulations with star formations, however, seem to indicate that the model is open to (typically) undesirable biases. Future scenarios for this type of investigation are vast. Ideally any possible network can be implemented in the model, as well as in other models for deliberation (for instance the bounded-confidence model (Hegselmann and Krause 2002), to test the properties of the convergence process. However, not all types of networks will be of relevance for real-life scenarios. An important class of networks still left to investigate is that of superconnected vs. minimally connected graphs; connectedness can be clearly interpreted as communication among agents. In the second place it seems worthwhile trying to study deliberation in randomly generated networks, it is expectable that different normative and descriptive results can be found by a more thorough analysis of the behavior of models for deliberation when groups of agents are networked. References 3
Aaken, A. van, C. List and C. Luetge (eds.) 2004. Deliberation and Decision: Economics, Constitutional Theory and Deliberative Democracy. London: Ashgate. Bradley, R. (2006). Taking Advantage of Difference of Opinion, Episteme 3 (3): 141-155. Hegselmann, R. and U. Krause (2002). Opinion Dynamics and Bounded Confidence: Models, Analysis and Simulation. Journal of Artificial Societies and Social Simulation vol. 5(3). Lehrer, K. and C. Wagner 1981. Rational Consensus in Science and Society. Dordrecht: Reidel. Jackson, M. 2008. Social and Economic Networks. Princeton: Princeton University Press.
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