Abstract
This chapter proposes a solution, based on the theory of social networks, to the problem of weight assignment in the Lehrer-Wagner model for consensus. The Lehrer-Wagner model of consensus is introduced, and the problem of weight assignment is outlined, together with a number of possible solutions previously suggested in the literature. The chapter argues that there is no one-size-fits-all solution to the problem of weight assignment in the Lehrer-Wagner model, and suggests an alternative solution, which is based on the idea of deriving weights from existing networks of relations in the group. This proposal, it is argued, is particularly useful for maximizing or limiting the influence of a network of relations on the consensual opinion resulting from the model.
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- 1.
In Lehrer and Wagner (1981), the unqualified expression “matrix of weights” is at times used to refer to W.
- 2.
P can contain other values, different from probabilities. Because this fact does not influence the considerations which will be made in the following sections, it will be ignored for the purposes of this chapter.
- 3.
- 4.
- 5.
Nurmi shows that the Leher-Wagner model, when weights are assigned subjectively, is manipulable (see Nurmi 1985, 15: Proposition 1).
- 6.
Details on the function and derivation of weights are left to the interested reader (see Regan et al. 2006, 172).
- 7.
- 8.
See Section 18.2, above, for the notion of reducibility. That this type of manipulation is possible can be seen from a look at the function for deriving weights in Regan et al. (2006, 172). The function is the following, \(w_{ij} = \frac{1 - | p_i - p_j|}{\sum_{j=1}^n 1 - | p_i - p_j|}\); when there are two agents in the model, giving each other weights, if i’s opinion is 1 and j’s opinion is 0, then w ij is indeterminate.
- 9.
“Confidence interval” is used here in the sense of Hegselmann and Krause (2002), not to be confused with the homonymous concept used in statistics.
- 10.
So far, I have taken both methods in Regan et al. (2006) and Hegselmann and Krause (2002) to be normative in character. Whereas the choice is not problematic for the former, it is arguable whether the latter should be taken as a normative model, at least in the authors’ intentions. In principle, however, there seems to be no reason for prohibiting that the Bounded Confidence model be taken normatively, regardless of the original authors’ intentions.
- 11.
For a recent comprehensive treatment of networks in economics and sociology see Jackson (2008).
- 12.
- 13.
An agent is a “proximate neighbor” with another agent if there is an edge that connects them without passing through any other agent.
- 14.
For an explanation of this see Hartmann et al. (2009, 120: Theorem 3)
- 15.
Indeed early versions of consensus models that use the properties of convergent Markov chains make reference to DeGroot (1974), who takes the model to be descriptive in character. In fact, if the Lehrer-Wagner model is taken as an “impossibility of disagreement” result, as Lehrer (1976) does, it is necessary to take the model to be descriptively accurate, and not only rational from the normative view point. This point cannot be developed further in the space of this chapter.
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Martini, C. (2012). Consensus Formation in Networked Groups. In: de Regt, H., Hartmann, S., Okasha, S. (eds) EPSA Philosophy of Science: Amsterdam 2009. The European Philosophy of Science Association Proceedings, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2404-4_18
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