Before any results on the exhibition of the strong paradox of new players are derived, I shall answer the question: what games are extendable at all? To formulate the next lemma, I need the following definition. A simple game v is called strong if \(v\left( S\right) +v\left( c(v){\setminus } S\right) =1\) for all \(S\subseteq c(v)\). That means, it is not only required that the complement of a winning coalition [in c(v)] is losing, but also that the complement of a losing coalition [in c(v)] is winning.
Lemma 1
A simple game v is extendable if and only if it is not strong.
Proof
Let v be a strong game. Then, there are exactly \(2^{\vert c(v)\vert -1}\) winning coalitions contained in c(v). Suppose there is an extension u of v. Then, v has exactly \(2^{\vert c(u)\vert -1}\) winning coalitions that are contained in c(u). As u is an extension of v, it must have at least \(2^{\vert c(u)\vert -1}+1\) winning coalitions in c(u). Hence, u cannot be proper, i.e., it cannot be an extension of v.
On the other hand, let v not be strong. Then, there is a nonempty coalition S such that \(v\left( S\right) =v\left( c(v){\setminus } S\right) =0\). Let \(k\notin c(v)\) and u be the (unique) simple game with minimal winning coalitions \(\mathcal M\left( v\right) \cup \left\{ S\cup \left\{ k\right\} \right\} \). Then, u is a non-trivial extension of v. \(\square \)
The good news is that extendable games can easily be identified. From a theoretical perspective, these games can now easily be handled and much is already known about them. The following theorem, however, shows that if a simple game is extendable then any monotonic power index exhibits the strong paradox of new members on it.
Theorem 1
Let \(\varphi \) be a monotonic power index. Then, \(\varphi \) does not exhibit the strong paradox of new members on a simple game v if and only if v is strong.
Proof
If v is strong, it is not extendable and thus \(\varphi \) cannot exhibit the paradox. So, let v not be strong. Then, there is a nonempty coalition \(S\subseteq c(v)\) such that \(v\left( S\right) =v\left( c(v){\setminus } S\right) =0\). Let \(k\notin c(v)\) and let u be the simple game with minimal winning coalitions \(\mathcal M\left( v\right) \cup \left\{ S\cup \left\{ k\right\} \right\} \). Obviously, \(c(u)=c(v)\cup \left\{ k\right\} \). Let N be a player set containing c(u), let \(i\in S\), and let \(T\subseteq N\). Suppose that T is such that i is pivotal in T with respect to v. As \(k\notin c(v)\), this means that i is also pivotal in \(T{\setminus }\left\{ k\right\} \) with respect to v. In particular, \(u\left( T\right) \ge u\left( T{\setminus }\left\{ k\right\} \right) = v\left( T{\setminus }\left\{ k\right\} \right) =1\). Further, the only minimal winning coalition with respect to u that contains k is \(S\cup \left\{ k\right\} \), and therefore it contains i as well. This means that k cannot be pivotal (with respect to u) in any coalition that does not contain i. Hence, \(u\left( T{\setminus }\left\{ i\right\} \right) =u\left( T{\setminus }\left\{ i,k\right\} \right) =v\left( T{\setminus }\left\{ i,k\right\} \right) =0\). Therefore,
$$\begin{aligned} u\left( T\right) - u\left( T{\setminus }\left\{ i\right\} \right) \ge v\left( T\right) - v\left( T{\setminus }\left\{ i\right\} \right) \end{aligned}$$
(1)
for all \(T\subseteq N\) in which i is pivotal with respect to v. If T is such that i is not pivotal in T with respect to v, then \(v\left( T\right) - v\left( T{\setminus }\left\{ i\right\} \right) =0\) and Eq. (1) holds as well. Further,
$$\begin{aligned} u\left( S\cup \left\{ k\right\} \right) - u\left( \left( S\cup \lbrace k\rbrace \right) {\setminus }\left\{ i\right\} \right) = 1 > 0 = v\left( S\cup \left\{ k\right\} \right) - v\left( \left( S\cup \lbrace k\rbrace \right) {\setminus }\left\{ i\right\} \right) \end{aligned}$$
as \(v\left( S\cup \left\{ k\right\} \right) =0\). Hence, by monotonicity, \(\varphi _i\left( u,N\right) >\varphi _i\left( v,N\right) \). \(\square \)
The following corollary is an easy consequence of the foregoing proof. It states that for every simple game v that is not strong and for every player, whether or not he is a null player with respect to v, an extension of v can be found that makes this player better off.
Corollary 1
Let \(\varphi \) be a monotonic power index and let v be a simple game that is not strong. Then, for each \(i\in \Omega \), there is an extension u of v such that \(\varphi _i\left( u,N\right) >\varphi _i\left( v,N\right) \) for each N with \(c(u)\cup \left\{ i\right\} \subseteq N\).
Proof
Let v be a simple game that is not strong. If \(i\in c(v)\), there is \(S\subseteq c(v)\) such that \(v(S)=v(c(v){\setminus } S)=0\) and \(i\in S\). Using the construction in the proof of Theorem 1, an extension u of v can be obtained with \(\varphi _i(u,N)>\varphi _i(v,N)\) for all N with \(c(v)\subseteq N\).
If \(i\notin c(v)\), there is \(S\subseteq c(v)\) such that \(v(S)=v(c(v){\setminus } S)=0\). Let u be the simple game with minimal winning coalitions \(\lbrace S\cup \lbrace i\rbrace \rbrace \cup \mathcal M(v)\). Then, by monotonicity, \(\varphi _i(u,N)>\varphi _i(v,N)\) for each N with \(c(u)\cup \left\{ i\right\} \subseteq N\), as i is not pivotal in any coalition with respect to v but is pivotal in \(S\cup \lbrace i\rbrace \) with respect to u.
The remainder of this section will deal with weak extensions. Note that every game can be weakly extended; I have already mentioned how it can be done for weighted voting games in Sect. 2. And, again, monotonicity will imply the paradox on almost all games. A dictatorship is a simple game v with \(c(v)=\lbrace i\rbrace \) for a player \(i\in \Omega \). In such a game, a decision can be made by i only, independently of the player set that v is applied to. Note that if there is a veto player in a strong game v then v is a dictatorship.
Theorem 2
Let \(\varphi \) be a monotonic power index. Then, \(\varphi \) exhibits the weak paradox of new members on all simple games that are not dictatorships.
Proof
Let v be a game that is not a dictatorship and let \(N'\) be a player set with \(c(v)\subseteq N'\). I first show that there exist a game \(v'\) and a player \(i\in c(v')\) such that \(v'\) is not strong, \(v'(S)\le v(S)\) for all coalitions S, and \(\varphi _i(v,N')\le \varphi _i(v',N')\). If v is not strong, choose \(v'=v\). If v is strong, there is \(i\in c(v)\) that is not contained in all minimal winning coalitions (otherwise, v would be a dictatorship). Let \(S\in \mathcal M(v)\) such that \(i\notin S\) and let \(v'\) be the simple game that is defined by
Then, \(v'(T)-v'(T{\setminus }\lbrace i\rbrace )\ge v(T)-v(T{\setminus }\lbrace i\rbrace )\) for all \(T\subseteq N'\) and
$$\begin{aligned} v'(S\cup \lbrace i\rbrace )-v'(S)=v(S\cup \lbrace i\rbrace )-0=1>v(S\cup \lbrace i\rbrace )-v(S). \end{aligned}$$
Hence, \(\varphi _i(v,N')<\varphi _i(v',N')\) by monotonicity of \(\varphi \).
By Corollary 1, there is an extension u of \(v'\) such that \(\varphi _i(u,N)>\varphi _i(v',N)\ge \varphi _i(v,N)\) for all N with \(c(u)\subseteq N\). Since, in particular, u is a weak extension of v, the claim is proved. \(\square \)
The following corollary is now clear and the proof is omitted.
Corollary 2
Let \(\varphi \) be a monotonic power index and let v be a simple game which is not a dictatorship. Then, for each \(i\in \Omega \), there is a weak extension u of v such that \(\varphi _i\left( u,N\right) >\varphi _i\left( v,N\right) \) for each N with \(c(u)\subseteq N\).