Excluded coalitions and the distribution of power in parliaments

ABSTRACT

In this article, we introduce a new value for cooperative games. This value is based on the Shapley (1953) value and takes into account that players exclude coalitions with other players. One example of such exclusions are the coalition statements of parliamentary parties. A case study demonstrates the application of the new value for these situations.

KEYWORDS:

• Shapley value
• EC value
• voting power
• excluded coalitions

JEL CLASSIFICATION:

• C71
• D72

Acknowledgments

The author thanks one anonymous referee for his insightful suggestions.

Notes

¹ Also, the Banzhaf-Penrose index (Penrose 1946, 1952; Banzhaf 1965) and the Holler index (Holler 1982) assume that all players cooperate. A more detailed survey of values is presented by Gambarelli (1994), Felsenthal and Machover (1998), Saari and Sieberg (2001) and Felsenthal and Machover (2005).

² They seize the suggestions by Owen (1971).

³ In a similar way, the normalized Banzhaf–Penrose index is determined (Dubey and Shapley 1979; van den Brink and van der Laan 1998).

⁴ CDU: Christian Democratic Union, FDP: Free Democratic Party, SPD: Social Democratic Party of Germany, Grüne: Green Party, and Linke: Left Party.

⁵ The set of minimum winning coalitions of $\left(N,v\right)$ is defined by$:$

$M\left(N,v\right)=\left\{S\subseteq \left(N\right|v\left(S\right)=1\mathrm{a}\mathrm{n}\mathrm{d}v\left(T\right)=0\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l}T⊊S\right\}.$

These coalitions are minimal in that any player’s defection will reduce the worth of the coalition to zero.

⁶ For the specific coalition statements in 2013, see Christian Democratic/Social Union (2013), Social Democratic Party of Germany (2013), Free Democratic Party (2013), Green Party (2013) and Left Party (2013).

⁷ In addition, the following relations between the parties’ seats are predicted by the opinion polls: $w_{CDU}>w_{SPD},$ $w_{SPD}\approx 2.5\cdot w_{Gr\ddot{u}ne},$ $w_{Gr\ddot{u}ne}\approx$ $w_{Linke}\approx 2\cdot w_{FDP}.$

⁸ We use the seat distribution directly after the election (Bundeswahlleiter 2013).

⁹ According to Aumann and Drèze (1974), components are active groups as in our understanding of government coalitions. In contrast, the Owen (1977) value interprets components as bargaining unions.