Abstract
Traditional power indices are not suited to take account of explicit preferences, strategic interaction, and particular decision procedures. This chapter studies a new way to measure decision power, based on fully specified spatial preferences and strategic interaction in an explicit voting game with agenda setting. We extend the notion of inferior players to this context, and introduce a power index which—like the traditional ones—defines power as the ability to have pivotal influence on outcomes, not as the (often just lucky) occurrence of outcomes close to a player’s ideal policy. Though, at the present state, formal analysis is based on restrictive assumptions, our general approach opens an avenue for a new type of power measurement.
This article was first published in Homo Oeconomicus 19(3), 327–343, 2002
Mika Widgrén unexpectedly passed away on 16.8.2009 at the age of 44.
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Notes
- 1.
- 2.
We use this abbreviation instead of SPI to avoid confusion with the Strict Power Index, which is defined in Napel and Widgrén (2001) and abbreviated as SPI.
- 3.
Note that this is not the standard way to define a dummy player.
- 4.
We only consider proper games in which the complement of a winning coalition is losing, i. e. \(S\in \mathcal W \Rightarrow N-S\in \mathcal L \). We do not assume that the game is decisive, i.e. additionally \(N-S\in \mathcal L \Rightarrow S\in \mathcal W \), because this would preclude the analysis of qualified majority voting. If both \(S\in \mathcal L \) and \(N-S\in \mathcal L \), then status quo prevails (see definition below).
- 5.
- 6.
Note that the first part of Definition 1 implies that \(j\) belongs to all minimal winning coalitions to which \(i\) belongs and is (by definition) crucial in these. Assuming that \(i\) has a swing in a non-minimal winning coalition without \(j\) also having one leads to a contradiction once non-crucial members of that coalition—including \(j\)—are dropped. Therefore, \(i\) does never have a swing without \(j\) also having a swing in the same coalition—but \(j\) has at least one swing in a coalition without \( i\) having one.
- 7.
We do not explicitly analyse agenda-setting power here. For a corresponding extension see the general framework in Napel and Widgrén (2004).
- 8.
In our setting, one might use the more specific term \(\Uplambda \)-right -connected coalition to stress that a formed coalition necessarily includes all players to the right of a given member.
- 9.
Note that \(S=N_{\chi \succsim 0}\) is the only \((\chi ,\Uplambda )\)-IR coalition, meaning that \(\mathcal C _{i}\left( \chi ,\Uplambda \right)\) is well-defined.
- 10.
Identity of two or more players’ ideal points has zero probability for a continuous distribution of \(\Uplambda \). This case will therefore be neglected in the following.
- 11.
One may assume small costs of being rejected for agenda setter \(A\) to ensure uniqueness of \(A\)’s proposal in the last sub-case. There are, depending on \( \Uplambda \), multiple subgame perfect equilibria corresponding to the same unique equilibrium proposal by agenda setter \(A\). We focus on \((\chi ^{*},\Uplambda )\)-IR coalitions.
- 12.
Note that the ideal point \(\lambda _{\left( n-m+1\right) }\) of the pivotal player is unique. In qualified majority voting there are two potential pivotal players but agenda setting makes the equilibrium unique.
- 13.
The possible event for which \(\chi ^{*}(\cdot )\)’s derivative is not defined has zero probability and is therefore neglected.
- 14.
Equivalently, the \(m\)th player in a given order can be considered—this is just a matter of convention. A truly alternative assumption is to consider any coalition equally probable and any player in a given coalition as equally likely to leave. This leads to the Banzhaf index.
- 15.
If the \(\hat{\lambda }_{i}\) are \(U(0,1)\)-distributed, this means that \(\hat{\lambda }_{(p)}\) is Beta-distributed with parameters \((p,n-p+1)\).
- 16.
Alternatively we can think that the agenda setter is really like a voter and a proposal is made by an intelligent benevolent machine after the players have told it their ideal points.
- 17.
The values of the SSI and the SSPI are comparable as probabilities. The values of the SSPI shed some light how much difference strategic agenda setting makes to the SSI under different assumptions of the domains of preference distributions. Note, however, that the purpose of this chapter is not a beauty contest between the SSI and the SSPI. Our attempt is to assess the relationship between spatial preferences and power. As a special case we get the SSI.
- 18.
Strictly speaking we let the ratio \(\frac{\alpha }{\beta }\) vary. This ratio affects the re-scaling presented above.
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Widgrén, M., Napel, S. (2013). The Power of a Spatially Inferior Player. In: Holler, M., Nurmi, H. (eds) Power, Voting, and Voting Power: 30 Years After. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35929-3_14
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