Abstract
We consider situations of majority voting, where the players are ordered linearly. This order may be based on, for example, ideology or political preferences over economic policy, ethical principles, environmental issues, and so on. Winning and losing coalitions are given by a majority voting game, while restrictions on cooperation are determined by a line graph, where only connected coalitions are feasible and can form a (winning) coalition. Various solutions for line-graph games can then be viewed as power indices measuring the ability of political parties to turn losing coalitions into winning ones, taking into account the cooperation restrictions among the parties. Here, we start by observing that a number of existing power indices either are not core stable, or do not reward intermediate veto players. Then, we take a closer look at the average hierarchical outcome, called hierarchical index in the context of this paper, and the \(\tau \)-index. These indices are core stable and, moreover, reward all veto players. Specifically, the \(\tau \)-index rewards all veto players equally, while the hierarchical index always assigns higher power to the two extreme veto players than to intermediate veto players. We axiomatically characterize the (i) hierarchical index by core stability and a weaker version of component fairness and (ii) the \(\tau \)-index by core stability and a weaker version of Myerson’s [Math Oper Res 2(3), 225–229 (1977)] fairness property.
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Notes
- 1.
For a recent survey on the Shapley value, see Algaba et al. (2019).
- 2.
Alternatively, for any game, the Shapley value distributes equally the Harsanyi dividend of coalition S over the players in S.
- 3.
Efficient normalizations of the Banzhaf value are the multiplicative and additive normalizations. The multiplicative normalization allocates v(N) proportionally to the Banzhaf values of the players [see (van den Brink and van der Laan 1998)]. The additive normalization is the least square value [see (Ruiz et al., 1998)] obtained by adding or subtracting the same amount from the Banzhaf value payoffs of the players so that an efficient payoff vector results.
- 4.
In the more general model of Myerson (1977), the players belong to a communication structure that is represented by a graph (N, A), where the player set N is the set of nodes and where \(A \subseteq \{\{i,j\}~|~i,j \in N,~i \ne j\}\), a collection of unordered pairs, is the set of edges reflecting the communication possibilities among the players. A coalition \(S \subseteq N\) can realize its worth v(S) when S is connected in graph (N, A), that is, when for any two players i and j in S, there is a subset \(\{\{i_k,i_{k+1}\} ~|~k=1,\ldots ,t\} \subseteq A\) of edges such that \(i_1 = i\), \(i_{t+1} = j\), and \(\{i_2,\ldots ,i_t\} \subseteq S\).
- 5.
For definitions on arbitrary graph games, we refer to Myerson (1977).
- 6.
See also Bilbao (1998) for the more general class of cycle-complete graphs.
- 7.
In van den Brink et al. (2007), the last line of (4) is shown to hold for \(\overline{L}\). Applying it to \((v^L)^{\overline{L}}\) yields, for \([i,j] \in {\mathcal {I}}(L)\) and \(j > i\), \(v^L[i,j] -v^L[i+1,j] - v^L[i,j-1] + v^L[i+1,j-1]\). This expression reduces to (4) since \(v^L(S)=v(S)\) and \(\Delta _S(v^L)=\Delta _S(v^{\overline{L}})\) for any connected coalition \(S \in {\mathcal {I}}(L)\).
- 8.
This follows since, if v is a superadditive game, then for any \(S, T \subseteq N\) with \(S \cap T = \varnothing \), it holds that \(v^L(S \cup T) = \sum _{H \in C_L(S \cup T)} v(H) \ge \sum _{H \in C_L(S)} v(H) + \sum _{H \in C_L(T)} v(H) = v^L(S) + v^L(T)\).
- 9.
In van den Brink et al. (2007), these four solutions are axiomatized as so-called Harsanyi solutions of the restricted game. These latter solutions allocate the Harsanyi dividends of coalitions over the corresponding players according to a fixed weight system per coalition, which implies component efficiency.
- 10.
- 11.
- 12.
In van den Nouweland and Borm (1991), this is shown for all so-called cycle-complete graphs, being those graphs such that if there is a cycle, then the subgraph on that cycle is complete. This class obviously contains all cycle-free graphs and thus all line graphs.
- 13.
In van den Brink et al. (2007), a line-graph game with an almost positive \(v^{L}\) is called linear-convex.
- 14.
- 15.
- 16.
\(v(\varnothing ) = 0\) requires that unanimous opposition implies rejection, and \(v(N) = 1\) requires that unanimous support implies acceptance. In the literature, sometimes \(v(N)=1\) is not required for a game to be a simple game.
- 17.
Note that even though we require that the grand coalition be winning in the majority game v (that is, \(v(N)=1)\), we do not require that N be winning in the restricted game \(v^{L}\). That is, \(v(N)=1\) does not imply \(v^{L}(N)=1\). For example, when \(L = \varnothing \) and \(s_i < q\) for all \(i \in N\), \(v(N)=1\) but \(v^{L}(N)=0\).
- 18.
Notice that, since \(q > w/2\), the restricted game \(v^{L}\) is a proper simple game; that is, the complement of any winning coalition, including a MWC, is a losing coalition.
- 19.
Note that a MWC in the line-graph game (v, L) need not be a MWC in the majority game v.
- 20.
This shows that the restricted game of a majority line-graph game is not almost positive.
- 21.
We remark that a different core concept for games with restricted cooperation assigns to every graph game the set of component-efficient payoff vectors such that the sum of the payoffs of all players in any connected coalition is at least equal to the worth of this coalition. For line-graph games, this gives the solution \(\overline{core}(v,L) = \{x \in \textrm{IR}^N ~|~ \sum _{i \in C} x_i = v(C)\) for all \(C \in C_L(N)\), and \(\sum _{i \in S} x_i \ge v(S)\) for all \(S \in \mathcal{I}(L)\}\). It is obvious that \(\overline{core}(v,L)=core(v^L)\) if the game v is monotonic (i.e. \(v(S) \le v(T)\) if \(S \subseteq T \subseteq N\)) and superadditive. Since majority voting games are monotonic and superadditive, for the games considered in this paper, the two core concepts boil down to the same.
- 22.
With some abuse of notation, we define [h, k] to be the empty set \(\varnothing \) if \(k < h\).
- 23.
A strategic implementation of the hierarchical outcomes and their average can be found in van den Brink et al. (2013).
- 24.
We remark that the proof follows the one in Herings et al. (2008), except that we have fewer equations from component veto fairness, but more equations from core stability.
- 25.
- 26.
Recall also the observation in Footnote 16.
- 27.
Another solution, specifically defined for line-graph games, is the spectrum value introduced in Álvarez-Mozos et al. (2013). However, as the Shapley value and the Banzhaf value, this solution is also not core stable for majority voting line-graph games.
- 28.
Note, however, that null players in v can only be intermediate, and never extreme, veto players in \(v^{L}\).
- 29.
Famously, Luxembourg was a null player in the Council at that time, which consisted of Germany, Italy, France, The Netherlands, Belgium, and Luxembourg with weights of 4, 4, 4, 2, 2, and 1, respectively, and a quota of 12 (Felsenthal & Machover, 1997). It is easy to verify that in all orders where Luxembourg ‘separates’ the big four-weight countries and the small two-weight countries (e.g. take the order Belgium, Italy, Luxembourg, France, The Netherlands, and Germany), Luxembourg is an intermediate veto player in the respective majority line-graph game.
- 30.
This also follows from the fact that the kernel of cooperative games satisfies symmetry (it rewards symmetric players equally; see [Maschler 1992, p. 621] and that veto players in line-graph games are symmetric.
- 31.
See Casajus (2021) for such a recent contribution with respect to the Shapley value for cooperative games, where what we call a player’s ‘leverage’ is called a player’s ‘second-order productivity’.
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van den Brink, R., van der Laan, G., Uzunova, M., Vasil’ev, V. (2023). Political Power on a Line Graph. In: Kurz, S., Maaser, N., Mayer, A. (eds) Advances in Collective Decision Making. Studies in Choice and Welfare. Springer, Cham. https://doi.org/10.1007/978-3-031-21696-1_16
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