Union null restricted game property (UNRGP) For any player set
\(N\subset \mathrm{I\!N}\) and
\(P\varGamma \)-game
\(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{P\varGamma }\),
\(\Gamma _\mathcal{P}=\langle \Gamma _M,\{\Gamma _k\}_{k\in M}\rangle \), if for some
\(k\in M\), for all
\(S\subseteq N_k\),
\(v_k(S)\ne 0\) implies
\(S\notin C^{\Gamma _k}(N_k)\), then it holds that for all
\( i,j\in N_k\),
\( i\ne j\),
$$\begin{aligned} \xi _i(v,\Gamma _\mathcal{P})=\xi _j(v,\Gamma _\mathcal{P}). \end{aligned}$$
The last axiom determines the distribution of the total shares obtained by internally disconnected a priori unions at the upper level bargaining between a priori unions among the components of these unions. Imagine that each a priori union
\(N_k\),
\(k\in M\), is a public institution (e.g. university, hospital, or firm) of which every component
\(C\in N_k/\Gamma _k\) is an independent unit (e.g. the faculties within a university, medical departments within a hospital, or production plants within a firm). First public institutions
\(N_k\),
\(k\in M\) compete among themselves for their annual budgets from the government. Once obtained the budget, institution
\(N_k\) has to decide how much to give to each of its independent units. At this stage the independent units of an institution compete against each other for the best possible shares from the institution’s budget. Similarly as in the competition among the public institutions, the total payoff to a unit depends on the total productivity of each of the units, but not on the productivity of the smaller collaborating teams within the units. Our last axiom requires the total payoff to a component of any a priori union to be independent of the so-called internal coalitions, each of which is a proper subcoalition of some component of one of the given a priori unions, or more precisely, a coalition
\(\emptyset \ne S\subseteq N\) is
internal if there is
\(k\in M\) such that
\(S\subset C\) for some
\(C\in N_k/\Gamma _k\). From now on, given a player set
\(N\subset \mathrm{I\!N}\), a partition
\(\mathcal{P}=\{N_1,\ldots ,N_m\}\) of
N, and a set of communication graphs
\(\{\Gamma _k\}_{k\in M}\) on a priori unions
\(N_k\),
\(k\in M\), the set of all internal coalitions we denote by
\(\mathrm{Int}(N,\mathcal{P},\{\Gamma _k\}_{k\in M})\). It is worth to remark that, of course, the worths of internal coalitions play a crucial role in the redistribution of the total payoff obtained by a component among its members, but the axiom does not concern this.
Proof
(Theorem
1) I. [
Existence]. We show that under the hypothesis of the theorem the
\(P\varGamma \)-value
\(\xi \!=\!Ka^{\langle {DL}^\mathcal{P},\{{DL}^k\}_{k\!\in \!M}\rangle }\) defined on
\(\mathcal{G}_N^{DL^\mathcal{P},\{DL^k\}_{k\in M}}\) by (
16) meets the axioms QCE, QDL, UDL, COV, UNRGP and UCPIIC. Consider an arbitrary
\(P\varGamma \)-game
\(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{DL^\mathcal{P},\{DL^k\}_{k\!\in \!M}}\).
QCE By the definition (
16) of
\(\xi \) and component efficiency of each
\(DL^k\)-value for all
\(k\in M\) it holds that
$$\begin{aligned} \sum _{i\in N_k}\xi _i(v,\Gamma _\mathcal{P})=DL^\mathcal{P}_k(v_\mathcal{P},\Gamma _M). \end{aligned}$$
(17)
Thus, for any
\(K\in M/\Gamma _M\) we have
$$\begin{aligned} \sum _{k\in K}\sum _{i\in N_k}\xi _i(v,\Gamma _\mathcal{P})=\sum _{k\in K}DL^\mathcal{P}_k(v_\mathcal{P},\Gamma _M) = v_\mathcal{P}(K), \end{aligned}$$
where the second equality follows from component efficiency of
\(DL^\mathcal{P}\)-value.
QDL From (
17) we obtain that
\(\xi ^\mathcal{P}(v,\Gamma _\mathcal{P})=DL^\mathcal{P}(v_\mathcal{P},\Gamma _M)\). Whence it follows that
$$\begin{aligned} \Psi ^{DL^\mathcal{P}}\left( \xi ^\mathcal{P}(v,\Gamma _\mathcal{P}),\Gamma _M\right) =\Psi ^{DL^\mathcal{P}}\left( DL^\mathcal{P}(v_\mathcal{P},\Gamma _M),\Gamma _M\right) =0, \end{aligned}$$
where the last equality holds true since the
\(DL^\mathcal{P}\)-value for
\(\varGamma \)-games meets
\(DL^\mathcal{P}\).
UDL We need to show that if the set of axioms \(\mathrm{DL}^k\), \(k\in M\), is restricted to F and CF, then for all \(k\in M\), \(\Psi ^{DL^k}(\xi ^k(v,\Gamma _\mathcal{P}),\Gamma _k)=0\).
Let for some
\(k\in M\),
\(\mathrm{DL}^k=\mathrm{F}\). Then since by definition for any link
\(\{i,j\}\in \Gamma _k\),
\(\Gamma _\mathcal{P}|_{-ij}=\Gamma _\mathcal{P}|^k_{-ij}=\langle \Gamma _M,\{\widehat{\Gamma }_h\}_{h\in M}\rangle \) with
\(\widehat{\Gamma }_h=\Gamma _h\) for
\(h\ne k\), and
\(\widehat{\Gamma }_k=\Gamma _k|_{-ij}\), we obtain that
$$\begin{aligned}&\Psi ^F(\xi ^k(v,\Gamma _\mathcal{P}),\Gamma _k)\\&\quad {\mathop {=}\limits ^{(4)}}\sum _{i,j\in N_k|\{i,j\}\in \Gamma _k }\! \Bigl |\Bigl (\xi ^k_i(v,\Gamma _\mathcal{P})- \xi ^k_i(v,\Gamma _\mathcal{P}|_{-ij})\Bigr )\!-\! \Bigl (\xi ^k_j(v,\Gamma _\mathcal{P})- \xi ^k_j(v,\Gamma _\mathcal{P}|_{-ij})\Bigr )\Bigr |\\&\quad {\mathop {=}\limits ^{(16)}} \sum _{i,j\in N_k|\{i,j\}\in \Gamma _k }\! \Bigl |\Bigl (F_i^k(v_k,\Gamma _k)+ \frac{DL_k^\mathcal{P}(v_\mathcal{P},\Gamma _M)- v^{\Gamma _k}_k(N_k)}{n_k}\\&\quad \quad \quad -\,F_i^k(v_k,\Gamma _k|_{-ij})+\frac{DL_k^\mathcal{P}(v_\mathcal{P},\Gamma _M)- v^{\Gamma _k|_{-ij}}_k(N_k)}{n_k}\Bigr )\\&\quad \quad \quad -\,\Bigl (F_j^k(v_k,\Gamma _k)+ \frac{DL_k^\mathcal{P}(v_\mathcal{P},\Gamma _M)- v^{\Gamma _k}_k(N_k)}{n_k}-F_j^k(v_k,\Gamma _k|_{-ij})\\&\quad \quad \quad +\, \frac{DL_k^\mathcal{P}(v_\mathcal{P},\Gamma _M)- v^{\Gamma _k|_{-ij}}_k(N_k)}{n_k}\Bigr )\Bigr |\\&\quad =\sum _{i,j\in N_k|\{i,j\}\in \Gamma _k }\! \Bigl |\Bigl (F_i(v_k,\Gamma _k)-F_i(v_k,\Gamma _k|_{-ij})\Bigr )- \Bigl (F_j(v_k,\Gamma _k)-F_j(v_k,\Gamma _k|_{-ij})\Bigr )\Bigr |\\&\quad {\mathop {=}\limits ^{(4)}}\Psi ^F(F(v_k,\Gamma _k),\Gamma _k)=0, \end{aligned}$$
where the last equality holds true since the
F-value for
\(\varGamma \)-games (the Myerson value) meets fairness F. Using the similar arguments as above we prove that for all
\(k\in M\) for which
\(\mathrm{DL}^k=\mathrm{CF}\),
\(\Psi ^{CF}(\xi ^k(v,\Gamma _\mathcal{P}),\Gamma _k)=0\).
COV Pick any
\(a\in \mathrm{I\!R}_{++}\) and
\(b\in \mathrm{I\!R}^n\). Then, for all
\(i\in N\),
$$\begin{aligned}&\xi _i(av\!+\!b,\Gamma _\mathcal{P}){{\mathop {=}\limits ^{(16),(2)}}} DL_i^{k(i)}((av+b)_{k(i)},\Gamma _{k(i)})\\&\qquad \qquad \qquad + \frac{DL_{k(i)}^\mathcal{P}((av+b)_\mathcal{P},\Gamma _M)-\! \sum \nolimits _{C\!\in \!N_{k(i)}/\Gamma _{k(i)}}\!(av+b)(C)}{n_{k(i)}}\\&\qquad \qquad \quad = aDL_i^{k(i)}(v_{k(i)},\Gamma _{k(i)})+b_i\\&\qquad \qquad \qquad \quad +\frac{aDL_{k(i)}^\mathcal{P}(v_\mathcal{P},\Gamma _M)\!+\!b(N_{k(i)})\!-\! \sum \nolimits _{C\!\in \!N_{k(i)}/\Gamma _{k(i)}}\!(av(C)\!+\!b(C))}{n_{k(i)}}\\&\qquad \qquad \quad = a\xi _i(v,\Gamma _\mathcal{P})+b_i, \end{aligned}$$
where the second equality is true because each of the considered
\(DL^k\)-values,
\(k\in M\), and
\(DL^\mathcal{{P}}\)-values meets COV on its domain, and
\((av\!+\!b)_\mathcal{P}(Q)\!=\!av_\mathcal{P}(Q)\!+\!b(\cup _{k\in Q}N_k)\) for all
\(Q\subseteq M\); and the third equality is due to the equality
\(\sum _{C\in N_{k(i)}/\Gamma _{k(i)}}b(C)=b(N_{k(i)})\), since
\(N_{k(i)}/\Gamma _{k(i)}\) forms a partition of
\(N_{k(i)}\).
UNRGP Assume that for the chosen
\(P\varGamma \)-game
\(\langle v,\Gamma _\mathcal{P}\rangle \) there exists
\(k\in M\) such that for all
\(S\subseteq N_k\),
\(v_k(S)\ne 0\) implies
\(S\notin C^{\Gamma _k}(N_k)\). Every of the considered
\(DL^k\)-values, the Myerson value and the average tree solution, are determined only by worths of connected coalitions, and therefore,
\(DL_i^k(v_k,\Gamma _k)=0\) for all
\(i\in N_k\). Moreover, as it was already mentioned earlier, the above assumption is equivalent to
\(v_k^{\Gamma _k}\equiv \mathbf{0}\), which implies that
\(v(C)=0\) for every
\(C\in N_k/\Gamma _k\). Hence, from (
16) it follows that for all
\(i\in N_k\),
$$\begin{aligned} \xi _i(v,\Gamma _\mathcal{P})=\frac{DL_k^\mathcal{P}(v_\mathcal{P},\Gamma _M)}{n_k}, \end{aligned}$$
where the right side is independent of
i, from which it follows that
\(P\varGamma \)-value
\(\xi \) meets UNRGP.
UCPIIC. Take any
\(k\in M\) and
\(C\in N_k/\Gamma _k\). The component efficiency of each of the considered
\(DL^k\)-values implies
\(DL^k(v_k,\Gamma _k)(C)=v(C)\). Then from (
16) it follows that
$$\begin{aligned} \sum \limits _{i\in C}\xi (v,\Gamma _\mathcal{P})=v(C)+ \frac{c}{n_k}\Bigl (DL_k^\mathcal{P}(v_\mathcal{P},\Gamma _M)- v^{\Gamma _k}(N_k)\Bigr ), \end{aligned}$$
where the right side is independent of worths of internal coalitions, i.e.
\(P\varGamma \)-value
\(\xi \) meets UCPIIC.
II.
[Uniqueness]. Assume that a
\((m+1)\)-tuple of deletion link axioms
\(\langle \mathrm{DL}^\mathcal{P},\{\mathrm{DL}^k\}_{k\in M}\rangle \) such that the set of axioms
\(\mathrm{DL}^k\),
\(k\in M\), is restricted to F and CF, is given. We show that there exists at most one
\(P\varGamma \)-value on
\(\mathcal{G}_N^{DL^\mathcal{P},\{DL^k\}_{k\!\in \!M}}\) that satisfies axioms QCE, QDL, UDL, COV, UNRGP, and UCPIIC. Let
\(\phi \) be such
\(P\varGamma \)-value on
\(\mathcal{G}_N^{DL^\mathcal{P},\{DL^k\}_{k\!\in \!M}}\). Take an arbitrary
\(P\varGamma \)-game
\(\langle v,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{DL^\mathcal{P},\{DL^k\}_{k\!\in \!M}}\). Fix some
\(k\in M\). We start by determining the union payoffs
\(\phi ^\mathcal{P}_k(v,\Gamma _\mathcal{P})\),
\(k\in M\), by induction on the number of links in
\(\Gamma _M\) similarly as it is done in the proof of uniqueness of the Myerson value for
\(\varGamma \)-games, cf., Myerson (
1977).
Initialization: If \(|\Gamma _M|=0\) then for all \(k \in M\) the set of neighboring unions \(\{h\in M\mid \{h,k\}\in \Gamma \}=\emptyset \), and therefore by QCE and definition of the quotient game \(v_\mathcal{P}\), \(\phi ^\mathcal{P}_k(v,\Gamma _\mathcal{P})=v_\mathcal{P}(\{k\})=v(N_k)\).
Induction hypothesis: Assume that the values \(\phi ^\mathcal{P}_k(v,\Gamma '_\mathcal{P})\), \(k\in M\), have been determined for all two-level graph structures \(\Gamma '_\mathcal{P}=\langle \Gamma ',\{\Gamma _h\}_{h\in M}\rangle \) with \(\Gamma '\) such that \(|\Gamma '|<|\Gamma _M|\).
Induction step: Let
\(Q\in M/\Gamma _M\) be a component in graph
\(\Gamma _M\) on
M. If
\(Q\subseteq M\) is a singleton, let
\(Q=\{k\}\), then from QCE it follows that
\(\phi ^\mathcal{P}_k(v,\Gamma _\mathcal{P})=v(N_k)\). If
\(q\ge 2\), then there exists a spanning tree
\(\widetilde{\Gamma }\subseteq \Gamma _M|_Q\) on
Q with the number of links
\(|\widetilde{\Gamma }|=q-1\). By QDL it holds that
$$\begin{aligned} \Psi ^{DL^\mathcal{P}}(\phi ^\mathcal{P}(v,\Gamma _\mathcal{P}),\Gamma _M)=0. \end{aligned}$$
(18)
The above equality in fact provides for each link
\(\{k,l\}\in \Gamma _M\) some equality relating values of
\(\phi ^\mathcal{P}_h(v,\Gamma _\mathcal{P})\),
\(h\in M\), with values of distinct
\(\phi ^\mathcal{P}_h(v,\Gamma _\mathcal{P}|_{-kl})\),
\(h\in M\). Since
\(|\Gamma _M|_{-kl}|=|\Gamma _M|-1\), from the induction hypothesis it follows that for all links
\(\{k,l\}\in \Gamma _M\) (
\((k,l)\in \Gamma _M\)) the payoffs
\(\phi ^\mathcal{P}_h(v,\Gamma _\mathcal{P}|_{-kl})\),
\(h\in M\), are already determined. Thus with respect to
\(q-1\) links
\(\{k,l\}\in \widetilde{\Gamma }\), (
18) yields
\(q-1\) linearly independent linear equations in the
q unknown payoffs
\(\phi ^\mathcal{P}_k(v,\Gamma _\mathcal{P})\),
\(k\in Q\). Moreover, by QCE it holds that
$$\begin{aligned} \sum _{k\in Q}\phi ^\mathcal{P}_k(v,\Gamma _\mathcal{P}) = v_\mathcal{P}(Q). \end{aligned}$$
All these
q equations are linearly independent. Whence it follows that for every
\(Q \in M / \Gamma \), all payoffs
\(\phi ^\mathcal{P}_k(v,\Gamma _\mathcal{P})\),
\(k \in Q\), are uniquely determined. Notice that in the proof of the induction step, every possible spanning tree
\(\widetilde{\Gamma }\) yields the same solution for the values
\(\phi ^\mathcal{P}_k(v,\Gamma _\mathcal{P})\),
\(k \in Q\), because otherwise a solution does not exist, which contradicts the already proved “existence” part of the proof of the theorem.
Next, we show that the individual payoffs \(\phi _i(v,\Gamma _\mathcal{P})\), \(i\in N_k\), are uniquely determined. This part of the proof is also by induction, now on the number of links in \(\Gamma _k\).
Initialization: Assume that
\(|\Gamma _k|=0\). Let
\(v_0\) be the 0-normalization of the TU game
v, i.e.,
\(v_0(S)=v(S)-\sum _{i\in S}v(\{i\})\) for all
\(S\subseteq N\), and let
\((v_0)_k=v_0|_{N_k}\). As already shown above, the union payoffs
\(\phi ^\mathcal{P}_k(w,\Gamma _\mathcal{P})\),
\(k\in M\), are uniquely determined for any
\(P\varGamma \)-game
\(\langle w,\Gamma _\mathcal{P}\rangle \in \mathcal{G}_N^{DL^\mathcal{P},\{DL^k\}_{k\!\in \!M}}\). In particular, the “union payoffs”
\(\phi ^\mathcal{P}_k(v_0,\Gamma _\mathcal{P})\),
\(k\in M\), are uniquely determined. By definition
\((v_0)_k(\{i\})=0\) for every
\(i\in N_k\), and therefore,
\((v_0)_k^{\Gamma _k}\equiv \mathbf{0}\) since
\(|\Gamma _k|=0\). Then, from UNRGP it follows that
$$\begin{aligned} \phi _i(v_0,\Gamma _\mathcal{P})=\frac{\phi ^\mathcal{P}_k(v_0,\Gamma _\mathcal{P})}{n_k},\quad \text{ for } \text{ all }i\in N_k. \end{aligned}$$
Whence by COV we obtain
$$\begin{aligned} \phi _i(v,\Gamma _\mathcal{P})=v(\{i\})+\frac{\phi ^\mathcal{P}_k(v^0,\Gamma _\mathcal{P})}{n_k},\quad \text{ for } \text{ all }i\in N_k, \end{aligned}$$
i.e., for every
\(i\in N_k\),
\(\phi _i(v,\Gamma _\mathcal{P})\) is uniquely determined.
Induction hypothesis: Let \(\Gamma '_\mathcal{P}\) denote the two-level graph structure \(\langle \Gamma _M,\{\Gamma '_h\}_{h\in M}\rangle \) with \(\Gamma '_h=\Gamma _h\) if \(h\ne k\) and \(\Gamma '_k = \Gamma '\) for some graph \(\Gamma '\) on \(N_k\). Assume that the values \(\phi _i(v,\Gamma '_\mathcal{P})\) have been determined for every \(\Gamma '\) with \(|\Gamma '|<|\Gamma _k|\).
Induction step: For every
\(S\in \mathrm{Int}(N,\mathcal{P},\{\Gamma _k\}_{k\in M})\) let
\(C_S\in N_k/\Gamma _k\) be the unique component such that
\(S\subset C_S\). Consider a game
\(w\in \mathcal{G}_N\) defined as
$$\begin{aligned} w(S)=\left\{ \begin{array}{ll} v(S),&{}\quad S\notin \mathrm{Int}(N,\mathcal{P},\{\Gamma _k\}_{k\in M}),\\ {\frac{s\,v(C_S)}{c_S}},&{}\quad S\in \mathrm{Int}(N,\mathcal{P},\{\Gamma _k\}_{k\in M}),\\ \end{array} \right. \quad \text{ for } \text{ all } S\subseteq N. \end{aligned}$$
(19)
For the 0-normalization
\(w_0\) of
w,
\((w_0)_k=(w_k)_0\). The subgame
\(w_k\) is an additive game and, therefore,
\((w_k)_0\equiv \mathbf{0}\). Then, due to UNRGP, similar as in the Initialization step, it follows that
$$\begin{aligned} \phi _i(w_0,\Gamma _\mathcal{P})=\frac{\phi ^\mathcal{P}_k(w_0,\Gamma _\mathcal{P})}{n_k},\quad \text{ for } \text{ all }i\in N_k. \end{aligned}$$
Whence by COV we obtain
$$\begin{aligned} \phi _i(w,\Gamma _\mathcal{P})=w(\{i\})+\frac{\phi ^\mathcal{P}_k(w_0,\Gamma _\mathcal{P})}{n_k},\quad \text{ for } \text{ all }i\in N_k. \end{aligned}$$
Consider a component
\(C\in N_k/\Gamma _k\). From the above equality it follows that
$$\begin{aligned} \sum _{i\in C}\phi _i(w,\Gamma _\mathcal{P})=\sum _{i\in C}w(\{i\})+ \frac{c}{n_k}\phi ^\mathcal{P}_k(w_0,\Gamma _\mathcal{P}) {\mathop {=}\limits ^{(19)}} v(C)+\frac{c}{n_k}\phi ^\mathcal{P}_k(w_0,\Gamma _\mathcal{P}). \end{aligned}$$
By the definition (
19) of
w,
\(w(S)=v(S)\) for all
\(S\subseteq N\),
\(S\notin \mathrm{Int}(N,\mathcal{P},\{\Gamma _k\}_{k\in M})\). Whence by UCPIIC it follows that
$$\begin{aligned} \sum _{i\in C}\phi _i(v,\Gamma _\mathcal{P})=v(C)+\frac{c}{n_k}\mathrm{DL}_k^\mathcal{P}(w_{0,\mathcal{P}},\Gamma _M),\quad \text {for all}C\in N_k/\Gamma _k. \end{aligned}$$
(20)
Next, if
\(c=1\), then
C is a singleton and the payoff
\(\phi _i (v,\Gamma _\mathcal{P})\) of the only player
\(i\in C\) is uniquely determined by (
20). If
\(c\ge 2\), then there exists a spanning tree
\(\widetilde{\Gamma }\subseteq \Gamma _k|_C\) on
C with the number of links
\(|\widetilde{\Gamma }|=c-1\). By UDL it holds that
$$\begin{aligned} \Psi ^{DL^k}\left( \phi ^k(v,\Gamma _\mathcal{P}),\Gamma _k\right) =0. \end{aligned}$$
(21)
The above equality in fact provides for any link
\(\{i,j\}\in \Gamma _k\) some equality relating values of
\(\phi _h(v,\Gamma _\mathcal{P})\),
\(h\in N_k\), with values of distinct
\(\phi _h(v,\Gamma _\mathcal{P}|^k_{-ij})\). Since
\(|\Gamma _k|_{-ij}|=|\Gamma _k|-1\), by the induction hypothesis it follows that for all links
\(\{i,j\}\in \Gamma _k\) the payoffs
\(\phi _h(v,\Gamma _\mathcal{P}|^k_{-ij})\),
\(h\in N_k\), are already determined. Thus with respect to
\(c-1\) links
\(\{i,j\}\in \widetilde{\Gamma }\), (
21) yields
\(c-1\) linearly independent linear equations in the
c unknown payoffs
\(\phi _i(v,\Gamma _\mathcal{P})\),
\(i\in C\). These
\(c-1\) equations together with (
20) form a system of
c linearly independent equations in the
c unknown payoffs
\(\phi _i(v,\Gamma _\mathcal{P})\),
\(i\in C\). Hence, for every
\(C \in N_k/ \Gamma _k\), all payoffs
\(\phi _i(v,\Gamma _\mathcal{P})\),
\(i \in C\), are uniquely determined.
\(\square \)