On the Possibility of a Preference-Based Power Index: The Strategic Power Index Revisited

This index was presented, in non-normalized form, for the first time in 1996 (see Steunenberg et al. 1996). There is another attempt to develop a strategic power index labelled strict power index (Napel and Widgren 2002). As with our index, spatial preferences and strategic agenda setting are its main building blocks. However, in the framework of the strict power index, power is defined as the ability of a player “to change the current state of affair” (Napel and Widgren 2002: 4). Following the reasoning of traditional power indices power relates to the ability of being decisive or pivotal.

These ‘classical’ indices have been supplemented with more recent power measures, such as the Johnston index, the Deegan–Packel index and the Holler index. The main differences between these indices are the ways in which coalition members share the benefits of their cooperation, and the kind of coalition players chose to form (see Colomer 1999). For a comparative investigation of traditional power indices see Felsenthal and Machover (1998), Holler and Owen (2001) and Laruelle and Valenciano (2008).

For example, the Shapley–Shubik index measuring what Felsenthal and Machover call P-power, which posits an office-seeking motivation of voting behavior (see Felsenthal and Machover 1998: 171).

See Penrose (1946), Banzhaf (1965), Coleman (1971, 1986), which take a policy-seeking viewpoint focusing on the degree to which a member’s vote is able to influence the outcome of a vote. These indices reflect I-power in the sense of Felsenthal and Machover (1998: 36).

That is not to say that traditional power indices are unable to take account of voters’ preferences or spatial voting (see Straffin 1994). In probabilistic characterizations of voting power indices each voter i’s probability p _i of voting “yes” on a proposal is a random variable. Taking the p _i as an indicator of the acceptability of a proposal to voter i (see Straffin 1994: 1137), homogeneous as well as heterogeneous preferences can be modelled. If each p _i is chosen independently from the uniform distribution on [0,1] we have the Banzhaf index. The independence assumption means that the acceptability of a proposal to voter i is independent of its acceptability to any other voter j (see Straffin 1994: 1137). Note, that p _i  = 1/2, which means that voter j voting “yes” is similar to flipping a coin. Note further, that the probability characterization of the Banzhaf index is restricted to its non-normalized version. If random variable p is chosen from the uniform distribution on [0,1], and p _i  = p for all i (homogeneity assumption), we have the Shapley–Shubik index. Here the acceptability of a proposal is the same to all voters.

For a more general version of the SPI, it is only necessary that X is some metric space, i.e. a space on which a metric (distance function) is defined, which, for every two points in X, gives the distance between them as a nonnegative real number. Such a metric space must satisfy the axioms of symmetry, positive definiteness and triangle inequality. The most familiar metric space is the (one- or multidimensional) Euclidean space which we assume in this paper. The Euclidean space is translation and rotation invariant and stretching, shrinking or mirroring at the origin does not alter the SPI.

At this point we focus on a unique equilibrium outcome only for expositional convenience. The strategic power index can also be applied to games for which multiple equilibria exist. If the game does not have a unique equilibrium, but multiple equilibria, the simple Euclidean distance can be replaced by the average Euclidean distance, i.e. the sum of the Euclidean distances between each equilibrium outcome and the player’s ideal point for all equilibria in a particular state of the world, divided by the number of equilibria.

The relative power of player i can be defined as \( {\tilde{\Psi}}_i^{\pi}=\frac{\Delta_d^{\pi}-{\Delta}_i^{\pi}}{{ \sum\limits_{j=1}^n\left({\Delta}_d^{\pi}-{\Delta}_i^{\pi}\right)}} \). The relative power scores of all players add up to 1.

See Barry’s critique of the Shapley–Shubik index (Barry 1980).

This part is from Schmidtchen and Steunenberg (2002: 208–210).

This is a traditional constitutional choice problem (Buchanan 1990). On the constitutional level, society must choose the rules (choice of rules) that govern decision-making on the post-constitutional level. On the post-constitutional level, choices have to be taken within the rules decided upon on the constitutional level.

Note the difference to the strict power index approach favored by Widgren and Napel, where A is not treated as a ‘pure’ ultimatum player. Whereas A’s SPI score is 1, the strict power index is 5/7. However, according to both indices B and C are powerless.

This part is based on Schmidtchen and Steunenberg (2002: 212–214).

Of course, our approach can also be applied to games of incomplete information, which would require making assumption regarding the possible types of players.

Another difference is worth to be mentioned: Whereas in the Laruelle et al. model proposals are submitted by an external agency, the agenda setter in our model is a player, thinking strategically.

Note the difference between our definition and Barry’s definition, which has recently been given more precision by Laruelle and Valenciano (2008: 54–55, 58). In their view a player is successful ex post, i.e. once the players have voted on a given proposal, if he/she obtains an outcome—acceptance or rejection of a proposal—that he/she has been voted for. A voter has been decisive if he/she is successful and his/her vote was crucial (critical) to that outcome. Luck is simply success without decisiveness, i.e. a player’s vote is irrelevant for the outcome. Thus, Laruelle and Valenciano interpret decisiveness, success and luck as binary variables.

Our definition of terms is more general than Laruelle and Valenciano’s, first, in that it refers not only to veto-players but also includes the agenda-setter. Second, in our framework, a player is successful if his/her vote influences the content of the proposal such that the equilibrium outcome moves towards his/her ideal point (including the case in which the status quo remains). Contrary to Laruelle and Valenciano, in our framework a player can be more or less successful, since the distance between the equilibrium outcome and a player’s ideal point can vary.

Note again the difference between our approach and that proposed by Laruelle and Valenciano (2008: 58). They define the ex ante version of the three terms success, decisiveness and luck (irrelevance) using probabilities. The probability of a player being decisive is simply the difference between his/her probability of being successful minus the probability of being lucky.

Napel and Widgren present an example in which a player n always has a position “opposite” of his n − 1 colleagues (Napel and Widgren 2002: 11).

We agree with Felsenthal and Machover (2001: 94) that this is not sufficient, “because the geometric structure of the state space itself also carries some information. In particular, any asymmetry of this space implies a bias in favour of some states and against others”.

Examples are: in the discrete case the set of vertices of a regular polygon or regular polyhedron; in the continuous case a circle or the surface of a sphere of some higher dimension.

We are particularly indebted to Stefan Klößner for several illuminating suggestions.

As done by Widgren and Napel (2001) and Napel and Widgren (2004). See also the discussion in Sect. 3.3.2.

Simulations indicate that in the case of perfectly symmetric state spaces the values of the SPI match those of the Banzhaf index, but they differ considerably for asymmetric state spaces.