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Probabilistic Spatial Power Indexes: The Impact of Issue Saliences and Distance Selection

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Abstract

We present a twofold generalization of probabilistic spatial voting indexes. In first instance, we introduce issue saliences in the classical application. The voting power is based on a coalition’s estimated consensus position, defined as the gravity center of the parties’ ideal positions. The issue saliences are implemented in the calculation of the gravity center as well as the Euclidean-like distance function, thus incorporating the parties’ policy priorities. The influence of the choice of distance function on voting power is the second focus of our analysis. We compare the results based on the consistent distance application for three functions: squared Euclidean, Euclidean and rectangular. All three functions can be considered either weighted or unweighted by issue saliences. This gives six possible combinations to calculate the voting power. Rather than using the gravity center, this method estimates the consensus as the point that minimizes the weighted sum of distances between itself and the parties’ bliss points. Empirical applications for Belgium illustrate the influence that allowing issue saliences to vary across parties and dimensions and using different distance functions can have on the voting power.

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Notes

  1. The reader should note that we use a somewhat different notation than the original authors as in our setting the utility function \(u\) depends on the party by which the proposition \(\theta \) is perceived.

  2. Owen and Shapley (1989) employ this definition because it means that the spatial values “are related to the so-called Copeland winner (or strong point), i.e., the policy outcome with the lowest probability to be beaten by any other alternative” (Benati and Vittucci Marzetti 2013). Because this result does not hold when preferences are defined in political spaces of three or more dimensions, the assumption is redundant in our setting and will not be continued.

  3. Plastria and Blockmans (2014) have presented alternative normalizations (e.g. product-1) but their implementation falls outside the scope of the present work.

  4. Authors such as Hinich and Munger (1997) and Gärdenfors (2004) use a different definition for a salience-weighted Euclidean-like distance. The deviation exists in the place of \(s_{ip}\) in the formula, which are squared in Debus (2008) definition but not so in the alternative approaches where they fall outside of the brackets. We have opted to square both the saliences (\(s_{ip}\)) and the disagreement \((|x^*_{ip} - \theta _i|)\) so as to give them a proportionate impact in the measurement of distances. This assures that their influence is comparable to the rectangular distance, where the absolute value of the saliences and disagreement is used.

  5. The absolute issue saliences \(s_{ip}^{abs}\) that were measured on a scale from 1 to 20 by Benoit and Laver (2006) did not comply with normalization 13. Therefore, we applied the transformation by Debus (2008), where “the relative weight of the policy dimensions for each party is measured by the absolute value of party \(p\)’s salience for dimension \(i\) (\(s_{ip}^{abs}\)) divided by the sum of absolute saliences of all policy dimensions”.

  6. In this context, weighted refers to the fact that each distance is multiplied with the party’s electoral weight \(w_p\). It has no relation to whether or not the distances are salience-weighted.

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Authors and Affiliations

  1. Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussels, Belgium

    Tom Blockmans & Marie-Anne Guerry

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  1. Tom Blockmans
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  2. Marie-Anne Guerry
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Correspondence to Tom Blockmans.

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Blockmans, T., Guerry, MA. Probabilistic Spatial Power Indexes: The Impact of Issue Saliences and Distance Selection. Group Decis Negot 24, 675–697 (2015). https://doi.org/10.1007/s10726-014-9408-4

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