Power Indices When Players can Commit to Reject Coalitions

Abstract

Power indices have been used to evaluate the allocation of power in a wide range of voting situations. While they use the language of game theory, known measures of a priori voting power are hardly more than statistical expectations assuming the players’ random behaviour. We introduce a model where players can reject certain partnerships in cooperation. For normalised power indices strategic rejection may increase power. Our notion of a strategic power index is well defined if power is measured by an index that takes only minimal winning coalitions into account.

1 Introduction

A power index measures the statistical probability of a voter being instrumental in a voting situation. They have been used to evaluate the distribution of power in currency unions (Berger and Mueller 2007), the International Monetary Fund (Leech 2002), the Bretton Woods institutions (Leech and Leech 2005), the European Union (Leech 2002; Kóczy 2012) or the US Supreme Court (Kaniovski and Leech 2009). There are more than one possible power indicies; in fact several have been proposed and studied by game theorists since Shapley and Shubik (1954) defined the Shapley–Shubik index by applying the Shapley value to simple games, despite the fact that there is nothing “game theoretical,” strategic about them.

In cooperative games it is normal that strategies are implicit, but players do maximise their payoffs. This is fairly clear when evaluating games using concepts such as the core, but the fairness represented by values assumes unconditional cooperation. There is nothing wrong with this in problems of cost sharing or risk allocation, but when we turn to voting situations, this approach seems less fit. Existing definitions of power indices seem to measure voting luck.

So what is power? Power is the “ability to act or produce an effect.” (Definition by the Merriam-Webster online dictionary.) Strategic decisions that increase a voter’s ability to make decisions are a manifestation of power. What are the strategic decisions a voter can take? Kilgour (1974, 1977) introduced the idea of quarrelling to voting games and axiomatised the Shapley value for games with a quarrelling subset of players. Quarrelling players refuse to cooperate, two such players refuse to join the same coalition so a coalition may contain at most one quarrelling player. As the paradox of quarrelling members (Brams 1975) illustrates, two players may mutually benefit from their quarrelling. Highly artificial these may seem (Laruelle and Valenciano 2005), the paradoxes of voting power actually arise in real voting situations (van Deemen and Rusinowska 2003).

Our approach is motivated by the idea of quarrelling, but we allow for individual coalitions to be rejected unilaterally.

Can we remove winning coalitions just like that? Is it possible that some winning coalitions do not form? The structure of communication (Myerson 1977, 1980; Faigle and Kern 1992) or physical or ideological position (Bilbao et al. 1998) may put restrictions on the set of feasible coalitions. Would all coalitions form in the absence of such restrictions? When a player is offered to join a winning coalition, accepting will clearly cause no harm, although a voter may turn down membership in a coalition where she gets no, or too little credit for membership expecting more elsewhere. A priori, however, she may strictly benefit from a strategic commitment to refuse to join this coalition.

We formalise this possibility by augmenting voting games with a pre-game rejection stage, where players can choose the coalitions they want to join in the voting game. This is a simple noncooperative game, where “the acquisition of power is the payoff” (Shapley 1962, p. 59). We show that all known normalised indices are affected by such strategic behaviour.

The structure of the paper is as follows. We start with a brief introduction to voting games and an overview of the known indices. We briefly explain the paradox of quarrelling members, introduce a framework for strategic indices and prove that our definition of strategic power index is well defined for a wide class power indices.

2 Power Indices

A voting situation or voting game is a pair $(N,{\mathcal {W}})$, where N is the set of voters and ${\mathcal {W}}\subseteq 2^N$ denotes the set of winning coalitions. A game is simple if

• 1. $\varnothing \notin {\mathcal {W}}$,

• 2. $N\in {\mathcal {W}}$

• 3. if $C\subset D$ and $C\in {\mathcal {W}}$, then $D\in {\mathcal {W}}$

• 4. If $C\in {\mathcal {W}}$ and $D\in {\mathcal {W}}$, then $C\cap D\ne \varnothing$.

Conditions 1 and 2 state that the empty set is not winning, while the coalition of all players is winning. Condition 3 is a monotonicity property: if a coalition is winning additional members cannot make it losing. Condition 4 requires the game to be proper, that is, no two subsets of the voters can make decisions simultaneously.

2.1 Convex Voting Games

We consider convex voting games without the assumption that the grand coalition is winning.

• 1. $\varnothing \notin {\mathcal {W}}$,

• 2. if $C\subset D\subset E$ and $C,E\in {\mathcal {W}}$, then $D\in {\mathcal {W}}$

• 3. If $C\in {\mathcal {W}}$ and $D\in {\mathcal {W}}$, then $C\cap D\ne \varnothing$.

Condition 2 is a convexity condition on the poset formed by the winning coalitions. When compared with the more common definition of simple games, Condition 2 replaces Condition 3, but the two coincide as soon as $N\in {\mathcal {W}}$. Let $\Gamma$ denote the collection of proper convex voting games satisfying the above properties. Note that our model permits the trivial game $(N,\varnothing )$.

A player i is critical in the winning coalition $C\in {\mathcal {W}}$ if $C\setminus \left\{ i\right\} \notin {\mathcal {W}}$. Let $k^C({\mathcal {W}})$ denote the number of critical players in coalition C when the set of winning coalitions is ${\mathcal {W}}$. When this does not lead to confusion, we drop the reference to ${\mathcal {W}}$.

Let ${\mathcal {M}}\subseteq {\mathcal {W}}$ denote the set of minimal winning coalitions: the set of coalitions without proper winning subsets. This implies that if $C\in {\mathcal {M}}$ and $i\in C$, then $C\setminus \left\{ i\right\} \notin {\mathcal {W}}$. Surplus coalitions are winning, but non-minimal. All members of a minimal winning coalition are critical.

Given a game $\Gamma$ a power measure $\kappa :\Gamma \longrightarrow {\mathbb{\!R}}_+^{N}$ assigns to each player i a non-negative real number $\kappa _i$, its power; a power measure $\kappa$ satisfying $\sum _{i\in N}\kappa _i=1$ is called a power index. The 0-game $(N,\varnothing )$ is an exception as here each player has a power 0.

2.2 A Characterisation

In the following we introduce a class of power indices $\kappa$, that can be expressed as follows:

$$ \kappa _i=\sum _{C\in 2^N\setminus \varnothing }a^C\mu _i^C, \quad {\text {where }}\sum _{C\in 2^N\setminus \varnothing }a^C=1, $$

(2.1)

that is, the power of a player can be expressed as a weighted average of her power or contribution $\mu _i^C$ in the (winning) coalitions $C\in 2^N\setminus \varnothing$ weighted by the (relative) impact $a^C\ge 0$ of a coalition C. Note that $a^C$ accounts for both the importance and the success of a coalition.

This class is sufficiently large to include all mainstream indices. Moreover we can only talk about contributions by members, so $\mu _i^C=0$ when $i\not \in C$ and these indices recognise the contribution of critical players only, but for these equally.

$$\begin{aligned} \mu _i^C&={\left\{ \begin{array}{ll}\frac{1}{k^C}& \quad {\text {if }}\;i\,{\text { is\,critical}}\\ \frac{1}{\left| C\right| }& \quad {\text {if\;no}}\,i\in C\,{\text {is\,critical,}}\\ 0& \quad {\text {otherwise.}}\end{array}\right. }\end{aligned}$$

(2.2)

This also means that the known indices differ only in the weights $a_C$.

Let ${\mathcal {K}}=\left\{ C\in {\mathcal {W}}|k^C>0\right\}$ denote the set of winning coalitions with at least one critical player. Clearly ${\mathcal {M}}\subseteq {\mathcal {K}}\subseteq {\mathcal {W}}$. Then we have

$$ a^C_{\phi }=\frac{(\left| C\right| -1)!k^C(n-\left| C\right| )!}{n!}. $$

(2.3)

for the Shapley–Shubik index (Shapley and Shubik 1954),

$$ a^C_{\beta }=\frac{k^C}{\sum _{C\in {\mathcal {W}}}k^C}. $$

(2.4)

for the Banzhaf index (Penrose 1946; Banzhaf 1965),

$$\begin{aligned} a^C_{\gamma }={\left\{ \begin{array}{ll}\frac{1}{\left| {\mathcal {K}}\right| }& \quad {\text {if }}\;C\in {\mathcal {K}}\\ 0 & \quad {\text {otherwise.}}\end{array}\right. } \end{aligned}$$

(2.5)

for the Johnston index $\gamma$ (Johnston 1978),

$$\begin{aligned} a^C_{\rho }={\left\{ \begin{array}{ll}\frac{1}{\left| {\mathcal {M}}\right| }& \quad {\text {if }}\;C\in {\mathcal {M}}\\ 0 & \quad {\text {otherwise.}}\end{array}\right. } \end{aligned}$$

(2.6)

for the Deegan–Packel index $\rho$ (Deegan and Packel 1978), and

$$\begin{aligned} a^C_{h}={\left\{ \begin{array}{ll}\frac{k^C}{\sum _{C\in {\mathcal {M}}}k^C}& \quad {\text {if }}\;C\in {\mathcal {M}}\\ 0 & \quad {\text {otherwise.}}\end{array}\right. } \end{aligned}$$

(2.7)

for the Holler–Packel or Public Good Index h (Holler and Packel 1983).

3 Strategic Power Indices

There may be various power indices, but they all work with an exogenously given set ${\mathcal {W}}$ of winning coalitions and it is implicitly assumed that all these coalitions will form. This seems indeed natural—why would players give up part of their power? If, for instance, two players start to “quarrel” and refuse to cooperate making any coalition they both belong to losing, their power should decrease. Not necessarily. The so-called “Paradox of Quarrelling Members” (Kilgour 1974; Brams 1975) precisely describes paradoxical situations where two players mutually benefit from refusing to cooperate with each other.

Paradoxical or not is a matter of interpretation, but players can certainly acquire a larger share of power by approving/rejecting coalitions. In this paper we extend voting games to allow for strategic considerations where player i can specify an arbitrary set of coalitions in $2^{N\setminus \left\{ i\right\} }$ they reject. We do not claim that rejected coalitions will not form, but that these should not be taken into account when calculating voting power (See also Braham and Holler (2005), on the understanding of power.). We define strategic power indices by applying power indices to this reduced set of winning coalitions.

3.1 Examples

As a motivation, we present a number of games based on weighted voting. Here N is a collection of n interest groups, or parties having $w_1, w_2, \dots , w_n$ individual representatives ($w_i\in$ ). Let $w=\sum _{i=1}^nw_i$. We assume that a quota q with $w\ge q>w/2$ is required to pass a bill. A coalition C of parties is winning if and only if $\sum _{i\in C}w_i\ge q$. Since $w\ge q$ and $w_i\ge 0$ weighted voting games are simple and proper.

Example 1

The game $G_1$ consists of four players represented by their weights (subscripts distinguish players with identical weights from each other): $3_1, 3_2, 2_1, 2_2$ and voting has a quota of 6. The set winning coalitions is ${\mathcal {W}}=\left\{ \underline{3_13_2},\underline{3_13_2}2_1,\underline{3_13_2}2_2,\underline{3_12_12_2},\underline{3_22_12_2},3_13_22_12_2\right\}$ (with critical players underlined). The Banzhaf index is $\beta =\left\{ \frac{1}{3}, \frac{1}{3}, \frac{1}{6}, \frac{1}{6}\right\}$.

Notice that in coalition $3_13_22_1$ player $2_1$ is not critical, while the two larger players are. If $2_1$ can prevent the formation of this coalition, the latter are critical in fewer coalitions, so in relative terms (in a power index) $2_1$ gains.

Given ${\mathcal {W}}'=\left\{ \underline{3_13_2},\underline{3_13_2}2_2,\underline{3_12_12_2},\underline{3_22_12_2},3_13_22_12_2\right\}$ the recalculated Banzhaf index is $\beta '=\left\{ \frac{3}{10}, \frac{3}{10}, \frac{1}{5}, \frac{1}{5}\right\}$. Player $2_1$’s rejection increased its relative power. It is therefore not in player $2_1$’s interest to join every winning coalition it is invited to. This finding is not really surprising. In coalition $3_13_22_1$ player $2_1$ assisted players $3_1$ and $3_2$ in forming a winning coalition, but without getting any credit for it.

Similarly, we find that it is not in player $2_2$’s interest to join $3_13_22_2$ and we get ${\mathcal {W}}''={\mathcal {M}}=\left\{ \underline{3_13_2},\underline{3_12_12_2},\underline{3_22_12_2},3_13_22_12_2\right\}$ and $\beta ^*=\left\{ \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}\right\}$.

Minimal winning coalitions may also be subject to rejections:

Example 2

$G_{2}$ is a 9-player game with players $5_1, 5_2, 5_3, 1_1, 1_2, 1_3, 1_4, 1_5, 1_6$ and a quota of 11. Here ${\mathcal {M}}=\left\{ 5_15_25_3, 5_i5_j1_k, 5_i1_11_21_31_41_51_6\right\}$, where $k\in \left\{ 1,2,3,4,5,6\right\}$ and $i,j\in \left\{ 1,2,3\right\}$ with $i\ne j$. Let ${\mathcal {W}}={\mathcal {M}}$. Then the Banzhaf index is given by $\beta =\left\{ \frac{7}{39},\frac{7}{39},\frac{7}{39},\frac{1}{13},\frac{1}{13},\frac{1}{13},\frac{1}{13},\frac{1}{13},\frac{1}{13}\right\}$.

Now consider ${\mathcal {W}}{'}=\left\{ 5_15_25_3, 5_i5_j1_k, 5_l1_11_21_31_41_51_6\right\}$, where $k\in \left\{ 1,2,3,4,5,6\right\}$, $i,j\in \left\{ 1,2,3\right\}$ and $l\in \left\{ 2,3\right\}$. Then $\beta '=\left\{ \frac{13}{71},\frac{14}{71},\frac{14}{71},\frac{5}{71},\frac{5}{71},\frac{5}{71},\frac{5}{71},\frac{5}{71},\frac{5}{71}\right\}$. The set ${\mathcal {W}}'$ does not contain the minimal winning coalition ${5_11_11_21_31_41_51_6}$, yet the critical player $5_1$ is better off as $\frac{13}{71}>\frac{7}{39}$.

Similarly we find that players $5_2$ and $5_3$ will respectively reject the coalitions ${5_21_11_21_31_41_51_6}$ and ${5_31_11_21_31_41_51_6}$ resulting in ${\mathcal {W}}'''=\left\{ 5_i5_j1_k\right\}$, where $k\in \left\{ 1,2,3,4,5,6\right\}$ and $i,j\in \left\{ 1,2,3\right\}$. Then $\beta {'''}=\left\{ \frac{2}{9},\frac{2}{9},\frac{2}{9},\frac{1}{18},\frac{1}{18},\frac{1}{18},\frac{1}{18},\frac{1}{18},\frac{1}{18}\right\}$.

For completeness note that it is not in the interest of any of the players to further reduce the set of winning coalitions, so $\beta ^*=\beta '''$.

In this example $5_1$ is a large player with a large share of power, so he does not benefit from participating in coalitions, where he plays a relatively minor role as one of the many members. Such large players will only benefit from joining coalitions with few members. Some players can increase their power beyond what is given by the standard power indices simply by refusing to participate in certain coalitions. Such coalitions that never form should not be included in the calculations to determine voting power.

3.2 Model

We generalise the idea of quarrelling to coalitions: a coalition Q is rejected if it is rejected by any of its members. A player i’s strategy $s_i$ therefore corresponds to quarrelling in certain coalitions that i belongs to, thus $s_i\subseteq \left\{ C\vert \ i\in C\right\}$ and its strategy space $S_i\subset 2^{\left\{ C\ni i\right\} }$.

Power comes from the possibility of influencing decisions: in a way, voters benefit from these possibilities. When a coalition is rejected it does not form, or even it if forms a player could have prevented it from forming, it could have prevented the decision so here no credits are distributed. Winning coalitions must therefore not only be able to pass decisions, but must be beneficial to all members. Note that rejection extends to supersets of coalitions, so $C\in s_i$ and $C\subseteq D$ imply that D is, in effect, rejected, too. Therefore, given $s=\left\{ s_i\right\} _{i\in N}$ the strategy profile, they are collected by

$$ {\mathcal {W}}(s)=\left\{ W\in {\mathcal {W}}\vert \ W\nsupseteq R\ \forall R\in s_i,\ \forall i\in N\right\} ={\mathcal {W}}\setminus \bigcup _{i\in N}s_i. $$

(3.1)

Observe that $(N,{\mathcal {W}}(s))$ is a voting game, thus each strategy profile s determines a voting game. In this game Conditions 1 and 3 clearly hold, since no new winning coalitions have been added. On the other hand, as the addition of new members to a rejected coalition does not make it winning, convexity, that is: Condition 2 holds, too.

Definition 1

A strategic voting game is a quadruple $(N,S,{\mathcal {W}},\kappa )$ consisting of a set of players N, a strategy space S, a collection of initial winning coalitions ${\mathcal {W}}$ and a power index $\kappa$.

As the objective of this game is to maximise power, the utility function is nothing, but $\kappa : S\longrightarrow {\mathbb{\!R}}_+^N, s\longmapsto \kappa (N,{\mathcal {W}}(s))$. Strategies are sets of coalitions, the strategy space can be derived from the player set, therefore the triple $(N,{\mathcal {W}},\kappa )$ fully defines the game.

The game consists of two stages: a first, noncooperative game of rejecting coalitions and a second, implicit, cooperative game of power allocation. We seek allocations under equilibrium rejection, and the resulting equilibrium sets of winning coalitions ${\mathcal {W}}^*$.

3.3 Equilibria with No More Rejection

Note the asymmetry in rejecting coalitions: a single player can reject a coalition, but all members must stop rejecting it to make the coalition acceptable again. Since our interest is not so much in the strategies selected by the players, but the equilibrium sets of winning coalitions ${\mathcal {W}}^*$, it is rather natural to seek equilibria in a game where only more rejection is permitted.

Proposition 1

The equilibrium sets of winning coalitions ${\mathcal {W}}^*$ under this game coincide with those of the original game.

Proof

Firstly note that the restricted game allows for less deviations. Clearly, the profitability of a particular deviation is the same in the two cases. Therefore, if for ${\mathcal {W}}$ and for all s such that ${\mathcal {W}}(s)={\mathcal {W}}$ rejecting more coalitions is profitable for one of the players, this will be true also in the original game. Therefore the set of equilibrium sets of winning coalitions in the restricted game contains the set of equilibrium sets of winning coalitions in the unrestricted case.

On the other hand for all equilibrium sets of winning coalitions ${\mathcal {W}}^*$ from the restricted game the strategy vector s where $s_i=\left\{ C\vert C\ni i, C\not \in {\mathcal {W}}^*\right\}$ is also an equilibrium in the unrestricted game. $\square$

In the following we focus on the “restricted” game, where only more rejection is possible. Once a coalition has been rejected there is no renegotiation despite incentives to make peace ex-post, and therefore only deviations with more rejection are possible. A strategy profile is a Nash equilibrium if s is a best response to itself, that is, $\kappa (s)\ge \kappa (s'_i,s_{-i})$ for all i and $s'_i\in S_i$ such that $s'_i\supseteq s_i$. The latter condition expresses the commitment to rejection, they can only reject more and not less coalitions.

Given a power index $\kappa$ the strategic $\kappa$ power index is then a vector of equilibrium payoffs, that is $\kappa (s^*)=\kappa (N,{\mathcal {W}}(s^*))$, where $s^*$ is a Nash equilibrium: for all $i\in N$ and all $s_i\subseteq s_i^*$, $s_i\in S_i$ we have $\kappa _i(s^*) \ge \kappa _i(s_i,s_{-i}^*).$

Such a strategic power index always exists (${\mathcal {W}}(s^*)=\varnothing$ is an equilibrium), but is generally not unique.

In the sequel we provide a unique refinement for indices that can be written as Eq. 2.1.

3.4 Only Minimal Winning Coalitions

Rejecting a coalition B affects a player in two ways. On the one hand for all $C\supseteq B$ the coalition’s weight (recall the definition in Sect. 2) becomes $(a^C)'=0$ and hence the player loses $\sum _{C\supseteq B}a^C\mu _i^C$, on the other hand, due to the normalisation, the weight of other coalitions increases, and hence the credit it gets from other coalitions is scaled up by

$$ \frac{\sum _{C\in 2^N\setminus \varnothing }a^C}{\sum _{C\in 2^N\setminus \varnothing }a^C-\sum _{C\supseteq B}a^C}. $$

(3.2)

Null players are unaffected and are therefore ignored in our analysis.

Proposition 2

Surplus coalitions containing critical players are rejected.

Proof

Consider a coalition B containing a surplus player i. If i is not critical in B, it is also not critical in $C\supset B$ (as, by monotonicity if $B\setminus \left\{ i\right\}$ is winning, so is $C\setminus \left\{ i\right\} \supset B\setminus \left\{ i\right\}$) and therefore $a^C\mu _i^C=0$ for all $C\supseteq B$. In sum, neither B nor $C\supset B$ yields any profit for i.

On the other hand $a^B>0$ (and possibly $a^C>0$ for some $C\supset B$), so when rejecting B the power of player i is scaled up according to Expression 3.2 making the rejection profitable. $\square$

Corollary 1

For power indices we have ${\mathcal {M}}\supseteq {\mathcal {W}}^*$.

However, even minimal winning coalitions can be rejected (see Example 2).

In the following we focus on power indices for which $a^C>0$ only if $C\in {\mathcal {M}}$. Holler and Packel (1983, p. 24.) argue that “since a non-critical member ... has no incentive to vote ... only these coalitions should be considered for measuring a priori voting power.” Thus a player cannot count on the formation of coalitions that are not due to his or her power. A similar prediction is made by aspiration solution concepts (Bennett 1983, p. 15).

3.5 Elementary Rejections

Definition 2

Given a strategy profile s, let the deviation $s_i'$ is elementary if $\left| s_i'\right| -\left| s_1\right| =1$, that is, if $s_i'$ rejects a single additional coalition.

Proposition 3

Given a strategy profile s, let $s_i'$ be $i's$ best response to $s_{-i}$. Then $s_i'$ can be reproduced by a sequence of elementary deviations.

Proof

Proof by construction. Consider the best response $s_i'$ and let $s_i\setminus s_i'=\left\{ C_1, \dots , C_k\right\}$ where, without loss of generality, $\mu _i^{C_1}\ge \cdots \ge \mu _i^{C_k}$.

We show that $\kappa _i(s')\ge \mu _1$. Consider the deviation $s_i''=s_i'\setminus C_1$. By assumption

$$ \kappa _i(s_i',s_{-i}) \ge \kappa _i(s_i'',s_{-i})$$

(3.3)

$$ \kappa _i(s_i',s_{-i}) \ge \frac{\left( \sum _{C\ne s_i'}a^C\right) \kappa _i(s_i',s_{-i})+a^{C_1}\mu ^{C_1}_i}{\sum _{C\ne s_i'}a^C+a^{C_1}} $$

(3.4)

The right hand side is a weighted average of $\kappa _i(s_i',s_{-i})$ and $\mu ^{C_1}_i$, and hence $\kappa _i(s_i',s_{-i})\ge \mu ^{C_1}_i$. $\square$

In the following we will only consider elementary deviations.

3.6 Friendly Equilibrium Selection

The strategy profile, where ${\mathcal {W}}^*=\varnothing$ is clearly a Nash-equilibrium, while this is neither the only one nor the one we want; out of the many Nash equilibria we select a focal equilibrium.

We now move on to define our refinement.

The literature of power indices has been built on the assumption that all winning coalitions form. We agree that it is reasonable to assume that, unless for good reasons, players will be friendly and not reject coalitions. Therefore also in our model we take this as the status quo; when strategic considerations do not play a major role, no coalitions are rejected so the equilibrium remains $s=\varnothing$, and a coalition is only rejected if this increases a player’s power. This last observation translates to the fact that for any acceptable equilibrium, there is a sequence of elementary deviations where each of the rejections are introduced all the way back to the status quo. Put it differently: any other equilibrium builds on the assumption that at least one coalition has been rejected irrationally.

The friendly set F, defined below collects the acceptable strategy profiles.

$$\begin{aligned} s\in F\,{\text { if }}{\left\{ \begin{array}{ll}s_i=\varnothing \ \forall i\in N, & \quad {\text {or}}\\ \exists i\in N, \exists (s'_i,s_{-i})\in F, & \quad {\text {such\,that }}\,\kappa _i(s)> \kappa _i(s'_i,s_{-i}).\end{array}\right. } \end{aligned}$$

We select friendly equilibria $s^*\in F$ that are Nash equilibria and are maximal for inclusion. The equilibrium set of winning coalitions is ${\mathcal {W}}^*={\mathcal {W}}(s^*)$ and the strategic $\kappa$ power index is defined as

$$ \kappa ^*=\kappa (s^*)=\kappa (N,{\mathcal {W}}^*). $$

Fig. 1

The rejection game for Example 1: at the top no, at the bottom all coalitions are rejected. Arrows are deviations, Nash equilibria are circled

Example 1

continued. Figure 1 shows the rejection game of Example 1. Each node of this lattice corresponds to a set of winning coalitions corresponding to a strategy profile. At the top all winning coalitions are accepted, at the bottom all are rejected. Dashed lines indicate where would the game continue if a particular coalition would be rejected by one of the players, arrows show the relations where such a rejection is profitable. Nodes where no profitable deviations are possible are circled and are the Nash-equilibria of the game.

Node ${\mathcal {W}}_1$, for instance, has already coalition $3_12_12_2$ rejected, here we have $\beta (N,{\mathcal {W}}_1)=\left( \frac{3}{9},\frac{4}{9},\frac{1}{9},\frac{1}{9}\right)$. Clearly no player in the rejected coalition benefits to get these payoffs from the original $\beta =\left( \frac{1}{3}, \frac{1}{3}, \frac{1}{6}, \frac{1}{6}\right)$. For this node there are three profitable deviations: rejecting coalition $3_22_12_2$ by $3_2$ giving $\beta (N,{\mathcal {W}}_2)=\left( \frac{1}{2},\frac{1}{2},0,0\right)$, or rejecting $3_13_22_1$ by $2_1$ or $3_13_22_2$ by $2_2$ to get $\left( \frac{2}{7},\frac{3}{7},\frac{1}{7},\frac{1}{7}\right)$ in both cases.

Note that no further profitable rejections exist at ${\mathcal {W}}_2$, that is, after $3_12_12_2$ and $3_22_12_2$ have been rejected, so ${\mathcal {W}}_2$ corresponds to a Nash equilibrium of the game.

On the other hand this equilibrium assumes that the coalition $3_12_12_2$ has also been rejected and for no particular reason. Without arbitrary rejections we would get to ${\mathcal {W}}^*$, which is also a Nash-equilibrium, but this is a friendly Nash equilibrium. As a friendly equilibrium it is unique and we will see that this is a general result.

Now observe that for minimal winning coalitions $C\ne D$ we have neither $C\subset D$ nor $D\subset C$, therefore by rejecting C a player will not reject D and vice versa, a player has the possibility to reject each minimal winning coalition separately. In sum, our model can be reduced to players picking which coalitions they do not want to form. This result makes it particularly easy to work with coalitions rather than strategies. Then an equilibrium is simply ${\mathcal {W}}^*$ instead of ${\mathcal {W}}(s^*)$ and let ${\mathcal {F}}=\left\{ {\mathcal {W}}(s)|s\in F\right\}$.

Player i profitably rejects coalition B iff

$$ \kappa _i(N,{\mathcal {W}}\setminus \left\{ B\right\} )> \kappa _i(N,{\mathcal {W}})$$

(3.5)

$$ \frac{\sum _{C\in {\mathcal {W}}}a^C}{\sum _{C\in {\mathcal {W}}}a^C-a^B}\left( \sum _{C\in {\mathcal {W}}}a^C\mu _i^C-a^B\mu _i^B\right)> \sum _{C\in {\mathcal {W}}}a^C\mu _i^C $$

(3.6)

After some rearrangements we get

$$ \frac{\sum _{C\in {\mathcal {W}}}a^C\mu _i^C}{\sum _{C\in {\mathcal {W}}}a^C}=\kappa _i(N,{\mathcal {W}})>\mu _i^B, $$

(3.7)

which gives the following result.

Lemma 1

A rejection by player i is profitable if and only if player i has less power in the rejected coalition than on average, as given by the strategic power index.

Proposition 2 can also be seen as a corollary of this lemma.

Lemma 1 also suggests a relation to the theory of aspirations (Bennett 1983), although this relation turns out to be superficial. In the theory of aspirations it is not some coalition’s payoff that is bargained over: it is the players who make their claims, and unless their claims are satisfied, certain coalitions will or will not form. Here this claim is expressed by their power index, the “credit they receive in general” and players demand the same credit in coalitions. Unfortunately the link between the two concepts does not go much beyond that. While a power index satisfies $\sum _{i\in N}\kappa ^*_i=1$ a vector of aspirations will almost always be larger. For instance, Bennett (1983, p.15.) provides an example with 5 players with weights 2, 2, 1, 1, and 1, and a quota of 5. Here the unique partnered, balanced, equal gains aspiration is (0.4, 0.4, 0.2, 0.2, 0.2), while the public good index is $h=(\frac{4}{17},\frac{4}{17},\frac{3}{17},\frac{3}{17},\frac{3}{17})$.

Now we move on to our main result.

Theorem 1

The friendly equilibrium set of winning coalitions is uniquely defined and is given by

$$ {\mathcal {W}}^*=\bigcap _{s\in F}{\mathcal {W}}(s). $$

(3.8)

In order to prove this theorem we need some additional results.

Proposition 4

Let $C_i,C_j\in {\mathcal {W}}$ be coalitions that both contain both i and j and such that i and j can profitably reject $C_i$ and $C_j$ respectively. Then either i can profitably reject $C_j$ or j can profitably reject $C_i$.

Proof

Assume that the proposition is false. This means the following. Player j rejects $C_j$, hence $\mu _j^{C_j}<\kappa _j({\mathcal {W}})$. By our assumption j does not reject $C_i$, hence $\mu _j^{C_i}\ge \kappa _j({\mathcal {W}})$. Therefore $\mu _j^{C_j}<\mu _j^{C_i}$. Similarly i rejects $C_i$, hence $\mu _i^{C_i}<\kappa _i({\mathcal {W}})$. By our assumption i does not reject $C_j$, hence $\mu _i^{C_j}\ge \kappa _i({\mathcal {W}})$. Therefore $\mu _i^{C_i}<\mu _i^{C_j}$. In sum $\mu _j^{C_j}<\mu _j^{C_i}$ and $\mu _i^{C_i}<\mu _j^{C_i}$. Since $C_i$ and $C_j$ are minimal winning coalitions $\mu _i^{C_i}=\mu _j^{C_i}=\frac{1}{\left| C_i\right| }$ and $\mu _j^{C_j}=\mu _i^{C_j}=\frac{1}{\left| C_j\right| }$. Thus $\frac{1}{\left| C_j\right| }<\frac{1}{\left| C_i\right| }$ and $\frac{1}{\left| C_i\right| }<\frac{1}{\left| C_j\right| }$. Contradiction. $\square$

Proposition 5

For all ${\mathcal {W}}_i, {\mathcal {W}}_j\in {\mathcal {F}}$ we have ${\mathcal {W}}_i\cap {\mathcal {W}}_j\in {\mathcal {F}}$.

Proof

The proof is by induction on the differences between ${\mathcal {W}}_i$ and ${\mathcal {W}}_j$.

First we deal with the elementary step. Assume ${\mathcal {W}}_i=\left\{ A, C_1,C_2, \dots C_m\right\}$, ${\mathcal {W}}_j=\left\{ B, C_1,C_2, \dots C_m\right\}$, that is, the two sets only differ in 1 element each. This ensures that their intersection is non-trivial. ${\mathcal {W}}_i$ and ${\mathcal {W}}_j$ are descendants of a common ancestor ${\mathcal {W}}_0=\left\{ A, B, C_1,C_2, \dots C_m\right\}$, but after rejecting B and A, respectively by some players i and j. The proposition merely states that either rejecting A is profitable from ${\mathcal {W}}_i$ or rejecting B is profitable from ${\mathcal {W}}_j$.

${\mathcal {W}}_i$ is the result of rejecting B by i. If $j\notin B$ then $\kappa _j({\mathcal {W}}_0)\le \kappa _j({\mathcal {W}}_i)$. We know that j rejects A at ${\mathcal {W}}_0$ and hence $\kappa _j({\mathcal {W}}_0)>\mu _j^A$. Hence $\kappa _j({\mathcal {W}}_i)>\mu _j^A$, which implies that j also rejects A at ${\mathcal {W}}_i$. Thus ${\mathcal {W}}_{ij}=\left\{ C_1,C_2, \dots C_m\right\} \in {\mathcal {F}}$.

The symmetric case gives the corresponding result for i and B at ${\mathcal {W}}_j$.

Finally we must consider the case where none of the previous two cases applied, that is where $j\in B$ and $i\in A$. As only a member can reject a coalition, we also have $j\in A$ and $i\in B$. Therefore we can apply Proposition 4 to show that i rejects at ${\mathcal {W}}_j$ or j at ${\mathcal {W}}_i$, which, as before, gives the result.

We have discussed all possible cases, which completes the first part of the proof. Now we move on to the general case. Assume that we have shown the result for all pairs of sets with differences up to $k-1$.

Now consider ${\mathcal {W}}_i=\left\{ A_1, A_2, \dots , A_k, C_1,C_2, \dots C_m\right\}$ as well as ${\mathcal {W}}_j=\left\{ B_1, B_2, \dots , B_l, C_1,C_2, \dots C_m\right\}$, where $A_1, A_2, \dots , A_k$ and $B_1, B_2, \dots , B_l$ represent the rejections that did not take place and $l\le k$. (Possibly $A_p=B_q$ for some p and q.) The question is whether this difference can be eliminated.

By definition if ${\mathcal {W}}_i\in {\mathcal {F}}$ there exists a sequence of rejections starting from ${\mathcal {W}}_0$ that lead to ${\mathcal {W}}_i$ and a similar sequence exists to ${\mathcal {W}}_j$. Let ${\mathcal {W}}_i^0$ and ${\mathcal {W}}_j^0$ be the first elements that are not common, without loss of generality, as results of rejecting $B_1$ and $A_1$ respectively. By the elementary step ${\mathcal {W}}_j^1={\mathcal {W}}_i^0\cap {\mathcal {W}}_j^0$ belongs to ${\mathcal {F}}$. (Our notation is slightly misleading given that ${\mathcal {W}}_j^1$ is not necessarily on the path to ${\mathcal {W}}_j$, but this should not lead to confusion.) Now take the next set ${\mathcal {W}}_2$ along the path to ${\mathcal {W}}_i$, ${\mathcal {W}}_i^1$. By the same argument ${\mathcal {W}}_i^1\cap {\mathcal {W}}_j^1$ also belongs to ${\mathcal {F}}$. Repeating this argument we travel parallel to the path and in the penultimate step we get ${\mathcal {W}}_j^p\in {\mathcal {F}}$. For the last time by the same argument ${\mathcal {W}}_i\cap {\mathcal {W}}_j^p=\left\{ A_2, \dots , A_k, C_1,C_2, \dots C_m\right\}$ also belongs to ${\mathcal {F}}$. If $l<k$, the inductive assumption completes the proof.

In case $l=k$ it is necessary to apply the same argument once more, but on the other side: to show that $\left\{ B_2, \dots , B_l, C_1,C_2, \dots C_m\right\} \in {\mathcal {F}}$. $\square$

Proof of Theorem

By Proposition 5 pairwise intersections of elements of ${\mathcal {F}}$ also belong to ${\mathcal {F}}$. As the number of winning coalitions is finite the result on pairwise intersections implies that ${\mathcal {W}}^*$ as defined in Equation 3.8 belongs to ${\mathcal {F}}$. Clearly ${\mathcal {W}}^*\subseteq {\mathcal {W}}^{\mathcal {F}}$ for all ${\mathcal {W}}^{\mathcal {F}}\in {\mathcal {F}}$. Therefore ${\mathcal {W}}^*$ is the smallest friendly set and it is trivially an equilibrium. $\square$

Corollary 2

The strategic power index $\kappa ^*$ is well-defined.

4 Conclusion

The power index approach sees power as the probability of being critical to a winning coalition. As such, it ignores the strategic aspects of forming winning coalitions. In our model the implicit cooperative game is preceded by a noncooperative stage where players decide which (winning) coalitions may form: only those with unanimous agreement. It turns out that there is a well-defined refinement of the Nash-equilibria of this game allowing us to define strategic power indices. Ours is not the first strategic index: The use of the word strategic in the context of power indices is not entirely new. There is a stream of literature who consider tactical voting and define power indices for such situations (Steunenberg et al. 1999; Napel and Widgrén 2010).

While the calculation of strategic power indices for larger problems is computationally difficult, strategic considerations may influence power distribution in such well-studied weighted voting situations as the voting in the Council of the European Union (formerly Council of Ministers). According to the Lisbon Treaty the support of 55 % of the countries (when acting on a proposal from the Commission or from the High Representative) and 65 % of the population is required to form a coalition. We have calculated the Public Good Index values using the software IOP-Indices of Power 2.05 (Bräuninger and König 2005) and the population data currently used in the Council (Council of the European Union 2014). We found that Germany, the largest EU member state has 5.56 % or a little over $\frac{1}{18}$ of the voting power, while it is a member of minimal winning coalitions with up to 25 members. Our model suggests that it will prefer to cooperate with smaller coalitions, ideally with less than 18 members. Besides Germany, also the other large members not included in the large coalitions (that consist of small countries, mostly) also benefit from such a move. Technical limitations inhibit us from conducting a full analysis, but it is expected that large member states will only participate in minimal winning coalitions that are also minimal in size, that is, with 16 members resulting in a higher power concentration than estimated using non-strategic power indices.

There is an interesting alternative to the underlying problem. To see this consider an initial set ${\mathcal {W}}$ of winning coalitions and a rejected coalition C. By Lemma 1 $\mu _i^C>\kappa _i(N,{\mathcal {W}})$ for the rejecting player(s). Since $\sum _{i\in C}\mu _i^C=1$ and $\sum _{i\in C}\kappa _i(N,{\mathcal {W}})< 1$ the total power of the members of the coalition decreases after the rejection. While rejecting C benefits some members, it harms others. It would be more efficient to renegotiate the $\mu _i^C$ values in the coalition. Currently power indices do not allow for that, but it would be interesting to see if there exist rejection-proof allocations of credit within coalitions. In practice, such as in the Council example above, it is very natural that less powerful voters make smaller claims than larger members of a minimal winning coalition.

Two other choices we have made was to assume that rejecting coalition C also rejects $D\supset C$, and to work with power indices defined over minimal winning coalitions only. Rejecting a single coalitions would not preserve null players who could gain power for “mediation” (turning a rejected coalition into a (non-rejected) winning one by their entry—of course this coalition would be rejected soon, too) and would allow non-minimal winning coalitions that are not surplus coalitions as they would only consist of critical players. While our original model considered a variant of this alternative, in order to avoid such odd phenomena one has to separate the notions of winning and feasibility.

At last we note that the uniqueness proof of the friendly equilibrium relies on the assumption that only minimal winning coalitions are considered. In the aforementioned model we could find counterexamples, but here we are inclined to believe the result extends to all power indices. The proof of this claim remains, however, open.