text:
\section{Introduction}

Guardiola et al. (2009) introduced Production-Inventory games (henceforth
PI-games) as a new class of totally balanced combinatorial optimization
games. That paper proposed the so-called
\textit{Owen point}  core-allocation that allows all players to operate at minimum cost at the price of not compensating  essential users by the cost reduction that they induce over the remaining players (fans). This allocation has proven to be rather appealing and in another paper, Guardiola et al. (2008) analyze its properties and propose three axiomatic
characterizations for the Owen point.
These papers also contribute to a better knowledge of the core of PI-games. Nevertheless, it was missing a deeper analysis of its complexity. Specifically speaking the two following aspects were not considered: testing core membership and the extreme points structure of the core of these games. Complexity issues in cooperative game theory raise important questions only partially answered for particular classes of games. The {core of} any convex game is the convex hull of its marginal vectors (Shapley 1971), and the same property holds true for those games satisfying the Co-Ma property which
include, among others, assignment and information games, see Hamers et al.
(2002) and Kuipers (1993) respectively. It is also well-known that the core
of assignment games coincide with the allocations induced by dual solutions
and it is a complete lattice with only two extreme points, see Sotomayor
(2003). Also, for transportation games, which constitute an extension of the
assignment games, some results about the relationship between the core and
the allocations induced by dual solutions are provided by S\'{a}nchez-Soriano et al. (2001).
Moreover, Perea et al. (2012) study cooperation situations in linear production
problems. In particular, that paper proposes a new solution concept called EOwen set as an improvement of the Owen set that contains at least one allocation that assigns a strictly positive payoff to players necessary for optimal production plans.

For minimum cost spanning tree games, flow games, linear
production games, cooperative facility location games or  min-coloring
games among others, testing whether a given allocation is in the core is an
NP-complete problem (see Faigle et al. (1997), Fang et al. (2002), Goemans and
Skutella (2004) and Deng et al. (1999), respectively). On the other hand,
there are some classes of games for which testing core membership is polynomially solvable as for instance for routing games, see Derks and Kuipers (1997), $s-t$ connectivity games, $r$-arborescence games, max matching games, min vertex cover games, min edge cover
games or max independent set games, see e.g., Deng et al. (1999). However, for many other classes of cooperative games answering that question is still open, as it is the case of PI-games.

In this paper we investigate the structure of the core of PI-games by determining its algorithmic complexity. Our contribution is to prove that testing core membership is an NP-complete problem and moreover that the number of extreme points of the core of PI-games is exponential on the number of players. Specifically, we characterize an exponential size subset of them. In addition, we look for alternative cost allocations improving the fairness properties of the Owen  point in that they recognize the role of the essential players on reducing the costs of the remaining players.

To present our results the rest of the paper is organized as follows. We start by introducing some preliminary
concepts in section \ref{preli}. In section \ref{structure} we prove that testing core membership of PI-games in an NP-complete problem, and we analyze the
core structure of PI-games. We define what we call the \textit{extreme functions}, which help us to prove that the core of a PI-game, in general, has an exponential number
of extreme points. In section \ref{structure copy 2} we introduce a new
core-allocation for PI-games, the Omega point, and provide an axiomatic characterization. Finally, in section \ref{QPQ set} we define the set of Quid Pro Quo
allocations (henceforth, QPQ allocations). Every QPQ allocation is a convex
combination of the Owen and the Omega point. We focus then on the equally
weighted QPQ allocation, the \textit{Solomonic} allocation, and we provide some
necessary conditions for the coincidence of the latter with the Shapley value
and the Nucleolus.

\section{\label{preli}Preliminaries}

A cost game with transferable utility (henceforth TU cost game) is a pair $%
(N,c)$, where $N=\left\{ 1,2,...,n\right\} $ is the finite set of players,
and the characteristic function $c:\mathcal{P}(N)\rightarrow \mathbb{R}$, is
defined over $\mathcal{P}(N)$ the set of nonempty coalitions of $N$. By
agreement, it always satisfies $c(\varnothing )=0.$ For all $S\subseteq N$,
we denote by $\left\vert S\right\vert $ the cardinal of the set $S$.

A distribution of the costs of the grand coalition, usually called
cost-sharing vector, is a vector $x\in \mathbb{R}^{N}$. For every coalition $%
S\subseteq N$ we denote by $x_{S}:=\sum_{i\in S}x_{i}$ the cost-sharing of
coalition $S$ (where $x_{\varnothing }=0).$ The core of a TU cost game
consists of those cost-sharing vectors $x$ which allocate the cost of the grand
coalition $N$ in such a way that no coalition $S$ has incentives to leave $N$ because $x(S)$ is smaller than the original cost of $S$, $c(S)$. Formally, the core of $(N,c)$
is given by $Core(N,c)=\left\{ x\in \mathbb{R}^{n}\left/ x_{N}=c(N)\text{
and }x_{S}\leq c(S)\text{ for all }S\subset N\right. \right\} .$ In the
following, core-allocations will be cost-sharing vectors belonging to the
core. A cost game $(N,c)$ is balanced if and only if has a nonempty core
(see Bondareva 1963 or Shapley 1967). Shapley and Shubik (1969) describe
totally balanced games as those games whose subgames are also balanced;
i.e., the core of every subgame is nonempty. A cost game $(N,c)$ is concave
if for all $i\in N$ and all $S,T\subseteq N$ such that $S\subseteq T\subset
N $ with $i\in S,$ then $c(S)-c(S\setminus \{i\})\geq c(T)-c(T\setminus
\{i\}). $

The Shapley value (Shapley, 1953) is a linear function on the class of all
TU games and for a cost game $(N,c)$ it is defined as $\phi (N,c)=(\phi
_{i}(N,c))_{i\in N}$ where for all $i\in N$

\begin{equation*}
\phi _{i}(N,c)=\sum\limits_{S\subseteq N\backslash \{i\}}\frac{s!(n-s-1)!}{n!%
}\cdot \left[ c(S\cup \{i\})-c(S)\right] .
\end{equation*}

The Nucleolus $\eta (N,c)$ (Schmeidler, 1969) is the allocation that
lexicographically minimizes the vector of excesses. It is well-known that the Nucleolus is a
core-allocation provided that the core is nonempty.

Let $Q$ be a bounded convex polyhedron in $\mathbb{R}^{n}$. We say that $%
x\in Q$ is an extreme point if $y,z\in Q$ and $x=\frac{1}{2}y+\frac{1}{2}z$
imply $y=z$. From now on, we denote, respectively, by $Ext\left( Q\right)$ and by $\partial
\left( Q\right)$ the set of extreme points and the boundary of the set of $Q$. Moreover, for the sake of readability, we use $e_{i}$ to refer to the $i$-th element of the canonical basis of $\mathbb{R}^{n}$ and $val(P)$ stands for the optimal value of the mathematical programming problem $P$.

It is well-known that $x\in Ext\left( Q\right) $ if
and only if $x$ satisfies as equalities at least $n$ linearly independent constraints of those defining $Q$. Since the core is a bounded
convex polyhedron, it has a finite number of extreme points. Moreover, the
core is a convex set. Therefore, characterizing the extreme core-allocations
is important to know its intrinsic structure.

From now on, and for the sake of readability, we follow the same notation as
Guardiola et al. (2009) to describe Production-Inventory situations
(henceforth: PI-situations) and PI-games. Consider first a situation with
several agents facing each one a Production-Inventory problem. Then, they
decide to cooperate to reduce costs. Here the cooperation is considered as
sharing technologies in production, inventory carrying and backlogged
demand. We mean that if a group of agents agree on cooperation then at each
period they will produce and pay inventory carrying and backlogged demand at
the cheapest costs among the members of the coalition. This situation is
called a PI-situation.

Formally, let $U$\ be an infinite set, the universe of players. A
PI-situation is a 3-tuple $(N,D,\Re )$ where $N\subset U$ is a finite set of
players $\left( \left\vert N\right\vert =n\right) $ and  $D$ an integer
matrix of demands with $D=[d^{1},\ldots ,d^{n}]^{\prime }$, $d^{i}=[d_{1}^{i},%
\ldots ,d_{T}^{i}]\geq 0$, $d_{t}^{i}$ is the demand of the player $i$ during
period $t\in T$ and $T$ is the planning horizon. In addition, $\Re =(H|B|P)$ is a cost matrix, so that $\quad H=[h^{1},\ldots ,h^{n}]^{\prime
}$, $B=[b^{1},\ldots ,b^{n}]^{\prime }$ and $P=[p^{1},\ldots
,p^{n}]^{\prime };$ where  $h^{i}=[h_{1}^{i},\ldots ,h_{T}^{i}]\geq 0$, $h_{t}^{i}$ is the unit inventory carrying costs of the player $i$ in period $t$, $%
b^{i}=[b_{1}^{i},\ldots ,b_{T}^{i}]\geq 0$, $b_{t}^{i}$ is the unit backlogging
carrying costs of the player $i$ in period $t$, and $p^{i}=[p_{1}^{i},\ldots
,p_{T}^{i}]\geq 0$, $p_{t}^{i}$ the unit production costs of the player $i$ in
period $t$, for $t=1,\ldots ,T$. The decision variables of the model, which
are required to be integer quantities, are the production during period $t$ (%
$q_{t}$), the inventory at hand at the end of period $t$ ($I_{t}$), and the
backlogged demand at the end of period $t$ ($E_{t}$). We denote by $\Upsilon $ the set of PI-situations $(N,D,\Re )$ defined over $U$, being $n\geq 1,T\geq 1$ and $D$ an integer
matrix.\medskip

Now given a PI-situation $(N,D,\Re )$, we can associate the corresponding TU
cost game $(N,c)$ with the following characteristic function $c$: $%
c(\varnothing )=0$ and for any $S\subseteq N,c(S)=val(PI(S))$, where $
PI(S) $ is given by
\begin{eqnarray*}
(PI(S))\quad &\min
&\sum_{t=1}^{T}(p_{t}^{S}q_{t}+h_{t}^{S}I_{t}+b_{t}^{S}E_{t}) \\
&\mbox{s.t.}&I_{0}=I_{T}=E_{0}=E_{T}=0, \\
&&I_{t}-E_{t}=I_{t-1}-E_{t-1}+q_{t}-d_{t}^{S},\quad t=1,\ldots ,T, \\
&&q_{t},\;I_{t},\;E_{t},\text{ non-negative, integer, }t=1,\ldots ,T;
\end{eqnarray*}%
with
\begin{equation*}
p_{t}^{S}=\min_{i\in S}\{p_{t}^{i}\},\;h_{t}^{S}=\min_{i\in
S}\{h_{t}^{i}\},\;b_{t}^{S}=\min_{i\in
S}\{b_{t}^{i}\},\;d_{t}^{S}=\sum_{i\in S}d_{t}^{i}.
\end{equation*}

Every TU cost game defined as above is called a Production-Inventory game.
Guardiola et al. (2009) points out that the problem $ PI(S)$
has integer optimal solutions provided that the demands are integer. We know
that the dual problem of $PI(S)$, for any coalition $S\subseteq N$, is the following
mathematical programming problem,
\begin{eqnarray*}
(DLPI(S))\quad &\max &\sum_{t=1}^{T}d_{t}^{S}y_{t}\mbox{   } \\
&\mbox{s.t.}&y_{t}\leq p_{t}^{S},\qquad \qquad t=1,\ldots ,T, \\
&&y_{t+1}-y_{t}\leq h_{t}^{S},\quad t=1,\ldots ,T-1, \\
&&-y_{t+1}+y_{t}\leq b_{t}^{S},\quad t=1,\ldots ,T-1.
\end{eqnarray*}

Moreover, Guardiola et al. (2009) also proves that an optimal solution of problem $%
DLPI(S)$ is $y_{t}^{\ast }(S)=\min \Big\{p_{t}^{S},\min_{k<t}%
\{p_{k}^{S}+h_{kt}^{S}\},\min_{k>t}\{p_{k}^{S}+b_{tk}^{S}\}\Big\}$, for all $%
t=1,\ldots ,T,$ with
\begin{eqnarray*}
p_{k}^{S} &=&\left\{
\begin{array}{cc}
p_{1}^{S} & \text{if }k<1, \\
p_{T}^{S} & \text{if }k>T,%
\end{array}%
\right. \\
h_{kt}^{S} &=&\sum_{r=k}^{t-1}h_{r}^{S},\quad \mbox{for any
}k<t,t=2,\ldots ,T;h_{k1}^{S}=0,k<1, \\
b_{tk}^{S} &=&\sum_{r=t}^{k-1}b_{r}^{S},\quad \mbox{ for any }%
k>t,\;t=1,\ldots ,T-1;b_{Tk}^{S}=0,k>T.
\end{eqnarray*}

It is important to note that those optimal solutions satisfy a monotonicity
property with respect to coalitions : $y_{t}^{\ast }(S)\geq y_{t}^{\ast }(R)$
for all $S\subseteq R\subseteq N$\ and all $t\in \{1,...,T\}$. Moreover, the
characteristic function of PI-games can be rewritten as follows: for any $%
\varnothing \neq S\subseteq N,c(S)=\sum_{t=1}^{T}d_{t}^{S}y_{t}^{\ast
}\left( S\right) $. \label{p:seis}

PI-games are not concave in general as shown by Example 4.4 in Guardiola et
al. (2009). The allocation $\left( \sum_{t=1}^{T}d_{t}^{i}y_{t}^{\ast
}\left( N\right) \right) _{i\in N}\hspace*{-0.3cm}=Dy^{\ast }(N)$ is called
the Owen point, and it is denoted by $Owen(N,D,\Re )$. At times, for the
sake of simplicity, we use $o$ to refer to the Owen point. That same paper also proves
that the Owen point is a core-allocation which can be reached
through a PMAS (Sprumont, 1990); hence every PI-game is a totally balanced
game. In some situations we will use $c^{(N,D,\Re )}(S)$ instead of $c(S)$,
in order to denote that the game $\left( N,c\right) $ comes from the
situation $(N,D,\Re )$.

 We say that a player $i\in N$ is essential if there exists $t\in
\{1,...,T\}$ with $d_{t}^{N\backslash \{i\}}>0$ such that $y_{t}^{\ast
}(N\backslash \{i\})>y_{t}^{\ast }(N)$. An essential player is the one for
which there exists at least one period in which he is needed by the rest of
players in order to produce a certain demand at a minimum cost. The set of
essential players is denoted by $\mathcal{E}$. Those players not being
essential are called inessential. We can easily check that for each
inessential player $i$,\ $o_{N\setminus\{i\}}=c(N\setminus \{i\})$.
Guardiola et al. (2009) showed that the core of PI-games shrinks to a single
point, the Owen point, just only when all players are inessential for the
PI-situation.

Finally, to conclude this section devoted to preliminaries, we recall the class of PS-games introduced by {Kar et al. (2009)}. A PS-game $(N,c)$ is a TU cost game satisfying that for all player $i\in N, $ there {exists} a real constant $c_{i\text{ }}$ such that $\Delta
_{i}(S)+\Delta _{i}(N\setminus \left( S\cup \{ i \}\right)=c_{i\text{ }}$ for
all $S\subseteq N\setminus \{i\},$ where $\Delta _{i}(S):=c(S\cup
\{i\})-c(S).$ The above mentioned paper proves that, for this class of games, the Shapley value and the Nucleolus coincide; i.e. $\phi (N,c)=\eta (N,c)$.

\section{\label{structure} Extreme points of the core of PI-games}

Guardiola et al. (2009) demostrated that the core of PI-games without
essential players ($\mathcal{E}=\varnothing $) shrinks to a singleton,
the Owen point. However, for those PI-games with essential players ($
\mathcal{E}\neq \varnothing $), the core is large. We focus here on those
PI-games with large cores and study the structure of its core by
analyzing its extreme points. First of all, we remark that testing core membership for PI-games cannot be done in polynomial time. One can adapt the
reduction proposed in Fang et al. (2002) to prove
that checking if an imputation belongs to the core of a PI-game is
an NP-complete decision problem. In spite of that, it is important to know the structure of the core and still very little is known about the extreme points complexity of PI-games. This is the goal of this section.
\smallskip

We begin this analysis by defining the essential player fan set.

Let $(N,D,\Re )$ be a PI-situation with $D$ being an integer matrix ($
(N,D,\Re )\in \Upsilon $), and let $i$ be an essential player. We define the
fan set of $i$\ as follows:
\begin{equation*}
F_{i}:=\{j\in N\backslash \{i\}\left\vert \exists t\in \{1,...,T\}\text{
with }d_{t}^{j}>0\text{ and }y_{t}^{\ast }(N\backslash \{i\})>y_{t}^{\ast
}(N)\}.\right.
\end{equation*}

The fan set of player $i$ consists of all players who need
him to operate at a lower cost. It is always a non-empty set.
Indeed, $F_{i}\neq \varnothing$ since taking $i\in \mathcal{E},$ there exists
$t^{\ast }\in \{1,...,T\}$ such that $y_{t^{\ast }}^{\ast }(N\backslash \{i\})>y_{t^{\ast }}^{\ast }(N)$
and $d_{t^{\ast }}^{N\backslash \{i\}}>0$. In that case, there must be, at least, a player
$j\in N\backslash \{i\}$ such that $d_{t^{\ast }}^{j}>0$ and $y_{t^{\ast }}^{\ast }(N\backslash \{i\})>y_{t^{\ast }}^{\ast }(N)$ .
\smallskip

In addition, you may notice that there is a pairwise relationship among essential players and their fans,
in the sense that the latter are interested in taking on a portion of the costs of the former.
This relationship allows us to introduce the concept of essential-fan pair.

Let $(N,D,\Re )\in \Upsilon $. The essential-fan pair set, denoted by $\mathbb{P}$, is:
\begin{equation*}
\mathbb{P}:=\{(i,j)|i\in \mathcal{E}\text{ and }j\in F_{i}\}.
\end{equation*}

We are now interested in determining the cost that can be transferred within every essential-fan pair with a cost  allocation;  i.e., the maximum portion of the essential player cost that his fan could assume while maintaining cooperation.

Given a essential-fan pair $p=\left( i,j\right) \in \mathbb{P}$ and a allocation $
x\in \mathbb{R}^{n},$ the transferred cost induced by $p$ regarding $x$ is:
\begin{equation*}
\alpha _{p}(x):=\min_{R\in \Delta _{p}}\{c(R)-x_{R}\},
\end{equation*}%
where
\begin{equation*}
\Delta _{(i,j)}:=\{R\subseteq N\backslash \{i\}\text{ such that }j\in R\}.
\end{equation*}

$\alpha _{p}(x)$ can be interpreted as the maximum portion of cost of player $i$ that can be awarded by player $j$ while maintaining the cooperation of the group.  It is worth nothing that if $x\in Core(N,c)$ then $\alpha _{p}(x)\geq 0$.
\smallskip

Next result states that there are always a positive transferred cost within every essential-fan pair with the
Owen point.

\begin{lemma}
\label{o}Let $(N,D,\Re )\in \Upsilon $ and $(N,c)$ be the corresponding
PI-game. Then $\alpha _{p}(o)>0$ for all $p\in \mathbb{P}$.
\end{lemma}

\begin{proof}
As $\mathcal{E}\neq \varnothing $, we can take $i\in \mathcal{E}$ and
therefore $F_{i}\neq \varnothing $. Let $R\subseteq N\setminus \{i{\}}$ such
that $R\cap F_{i}\neq \varnothing $ and let $j\in R\cap F_{i}$. By
definition, there exists $t^{\ast }\in \{1,...,T\}$ such that $y_{t^{\ast
}}^{\ast }(N\backslash \{i\})>y_{t^{\ast }}^{\ast }(N)$ and $d_{t^{\ast
}}^{j}>0$. Then $d_{t^{\ast }}^{R}>0$ and moreover $y_{t^{\ast }}^{\ast
}(N)<y_{t^{\ast }}^{\ast }(N\backslash \{i\})\leq y_{t^{\ast }}^{\ast }(R)$.
Thus, $o_{R}<c(R)$. Hence, $\alpha _{p}(o)=\min_{R\in \Delta
_{p}}\{c(R)-o_{R}\}>0$.
\end{proof}
\smallskip

We introduce now a function that transforms any cost allocation into a new cost allocation in which a fan player charges with the maximum cost of his essential player. That is, for each $p=(i,j)\in \mathbb{P}$, the function $f_{p}$
transforms any allocation $x$ into a new allocation $f_{p}(x),$ in which the fan player $j$ assumes as much cost as possible from his essential player $i.$ It is called the extreme function.

\begin{definition}[extreme function]
Let $(N,D,\Re )\in \Upsilon $ and $(N,c)$ be the corresponding PI-game. For
any $p=(i,j)\in \mathbb{P}$, the extreme function $f_{p}$ is defined by:
\begin{equation*}
f_{p}(x)=x+\wedge _{p}(x),
\end{equation*}%
where $x\in \mathbb{R}^{n}$ and $\wedge _{p}(x)=e_{j}\cdot \alpha
_{p}(x)-e_{i}\cdot \alpha _{p}(x).$
\end{definition}

Let us denote by $\mathbb{P}^{\left\vert \mathbb{P}\right\vert }$ the $%
\left\vert \mathbb{P}\right\vert $-fold cartesian product of the set $%
\mathbb{P}$. We consider now the composition of extreme functions.
For each $\sigma \in \mathbb{P}^{\left\vert \mathbb{P}\right\vert }$ we define
the extreme composite function, $F_{\sigma }$, as the composition of extreme
functions for all the pairs in $\sigma $, that is,
\begin{equation*}
F_{\sigma }(x):=\left( f_{\sigma _{\left\vert \mathbb{P}\right\vert }}\circ
f_{\sigma _{\left\vert \mathbb{P}\right\vert -1}}\circ ...\circ f_{\sigma
_{1}}\right) \left( x\right) .
\end{equation*}

Notice that if $\sigma =\left( p,p,...,p\right) \in \mathbb{P}^{\left\vert
\mathbb{P}\right\vert }$ then $F_{\sigma }(x)=f_{p}(x).$

\begin{example}
\label{example 1} The following table shows a PI-situation with three
periods and three players:

\begin{equation*}
\begin{tabular}{|c|c|c|c||c|c|c||c|c||c|c|}
\hline
& \multicolumn{3}{|c||}{Demand} & \multicolumn{3}{|c||}{Production} &
\multicolumn{2}{|c||}{Inventory} & \multicolumn{2}{|c|}{Backlogging} \\
\hline
P1 & 10 & 10 & 5 & 1 & 2 & 1 & 1 & 1 & 1 & 1 \\ \hline
P2 & 8 & 12 & 6 & 2 & 1 & 1 & 1 & 1 & 1 & 1 \\ \hline
P3 & 6 & 5 & 2 & 3 & 1 & 1 & 1 & 1 & 2 & 2 \\ \hline
\end{tabular}%
\end{equation*}

We can easily check that $c(S)=\displaystyle\sum_{t=1}^{3}p_{t}^{S}d_{t}^{S},$ for all $S\subseteq N$.
Hence, the characteristic function of the corresponding PI-game is given in the following table:

\begin{equation*}
\begin{array}{|c|c|c|c||c|c|c||c|c||c|c||c||}
\hline
& d_{1}^{_{S}} & d_{2}^{_{S}} & d_{3}^{_{S}} & p_{1}^{_{S}} & p_{2}^{_{S}} &
p_{3}^{_{S}} & h_{1}^{_{S}} & h_{2}^{_{S}} & b_{1}^{_{S}} & b_{2}^{_{S}} & c
\\ \hline
\{1\} & 10 & 10 & 5 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 35 \\ \hline
\{2\} & 8 & 12 & 6 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 36 \\ \hline
\{3\} & 6 & 5 & 2 & 3 & 1 & 1 & 1 & 1 & 2 & 2 & 25 \\ \hline
\{1,2\} & 18 & 22 & 11 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 51 \\ \hline
\{1,3\} & 16 & 15 & 7 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 38 \\ \hline
\{2,3\} & 14 & 17 & 8 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 53 \\ \hline
\{1,2,3\} & 24 & 27 & 13 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 64 \\ \hline
\end{array}%
\end{equation*}

In this example, $y^{\ast }(N)=\left( 1,1,1\right) $\ and $y^{\ast
}(N\setminus \{1\})=\left( 2,1,1\right) $\ are the optimal solution for $%
\left( DLPB(N)\right) $\ and $\left( DLPB(N\setminus \{1\})\right) ,$\
respectively. Then, the Owen point is $o=(25,26,13)$. Moreover, $\mathcal{E}%
=\{1\}$, $F_{1}=\{2,3\}$ and $\mathbb{P}=\{(1,2),(1,3)\}$.

The transferred cost within every essential-fan pair in $\mathbb{P}$ with the
Owen point are,
\begin{eqnarray*}
\alpha _{(1,2)}(o) &=&\min_{R\in \Delta _{(1,2)}}\{c(R)-o_{R}\}=10, \\
\alpha _{(1,3)}(o) &=&\min_{R\in \Delta _{(1,3)}}\{c(R)-o_{R}\}=12.
\end{eqnarray*}

Therefore, the extreme functions are
\begin{eqnarray*}
f_{(1,2)}(o) &=&o+\wedge _{(1,2)}(o)=(15,36,13), \\
f_{(1,3)}(o) &=&o+\wedge _{(1,3)}(o)=(13,26,25).
\end{eqnarray*}

In this case, both the extreme functions, $f_{(1,2)}(o)$, $f_{(1,3)}(o)$, and the Owen point, $o$,  are extreme points of the core.
\end{example}

The previous example shows that the Owen point is an extreme point of the core, and that the extreme functions transform it into other extreme points of the core. We wonder then if this fact occurs in general for any PI-game. First, we find a very interesting property that relates the extreme functions to the core boundary.

\begin{proposition}
Let $(N,D,\Re )\in \Upsilon $ and $(N,c)$ be the corresponding PI-game. For
all $p\in \mathbb{P}$%
\begin{equation*}
f_{p}(Core(N,c))\subseteq \partial {(Core(N,c)).}
\end{equation*}
\end{proposition}

\begin{proof}
Let $p=(i,j)\in \mathbb{P}$ and take $x\in Core(N,c).$ Then $\alpha
_{p}(x)\geq 0$. Applying the extreme function $f_{p}$ at $x$, we have:
\begin{equation*}
f_{p}(x)=(x_{1},...,x_{i-1},x_{i}-\alpha
_{p}(x),x_{i+1},...,x_{j-1},x_{j}+\alpha _{p}(x),x_{j+1},...,x_{n}).
\end{equation*}

To prove that $y:=f_{p}(x)\in Core(N,c)$ we distinguish four possibilities:

\begin{itemize}
\item $i,j\in S.$ Then $y_{S}=x_{S}+\alpha _{p}(x)-\alpha _{p}(x)=x_{S}\leq
c(S)$.

\item $i,j\notin S.$ Then $y_{S}=x_{S}\leq c(S)$.

\item $i\notin S,j\in S.$ Then $y_{S}=x_{S}+\alpha _{p}(x)\leq
x_{S}+c(S)-x_{S}=c(S)$.

\item $i\in S,j\notin S.$ Then $y_{S}=x_{S}-\alpha _{p}(x)\leq x_{S}\leq
c(S) $.
\end{itemize}

Hence, $y\in Core(N,c)$ since $y_{S}\leq c(S)$ for any coalition $S\subseteq
N$. Let us proof now that $y$ belongs to the frontier of the core.

If $\alpha _{p}(x)=0$ then there exists $R\in \Delta _{p}$ such that $%
c(R)=x_{R}$. Since $y$ belongs to the core and satisfies as equality one of
the constraints defining the core, we can conclude that $y\in \partial {%
(Core(N,c))}$.

If $\alpha _{p}(x)>0$ then for all $\lambda \in (0,1),$ $(1-\lambda
)x+\lambda y\in Core(N,c).$ Take $\lambda =1+\epsilon $ with $\epsilon >0$
to have%
\begin{equation*}
(1-\lambda )x+\lambda y=-\epsilon x+(1+\epsilon )(x+\wedge
_{p}(x))=x+(1+\epsilon )\wedge _{p}(x).
\end{equation*}

We can check that if $R^{\ast }\in \Delta _{p}$ is such that $c(R^{\ast
})-x_{R^{\ast }}=\min_{R\in \Delta _{p}}\{c(R)-x_{R}\}$ then $x_{R^{\ast
}}+\alpha _{p}(x)=c(R^{\ast })$, therefore $x+(1+\epsilon )\wedge
_{p}(x)\notin Core(N,c)$. Hence, $y$ is not an interior point.
\end{proof}
\smallskip

It follows straightforward from the above proposition,  that $F_{\sigma
}(Core(N,c))\subseteq$ $\partial {(Core\left( N,c\right) )}$ for all $\sigma
\in \mathbb{P}^{\left\vert \mathbb{P}\right\vert }$.
\smallskip

The main Theorem of this Section provides a partial answer to our previous question
about the transformation of the Owen point into extreme points of the core of PI-games.
It states that for PI-situations with a single essential player, all the different compositions of extreme functions
over the Owen point generate extreme points of the core.

\begin{theorem}
Let $(N,D,\Re )\in \Upsilon $ and $(N,c)$ be the corresponding PI-game. If $%
\mathcal{E}=\{i\}$, then $F_{\sigma }(o)\in Ext\left( Core(N,c)\right) $ for
all $\sigma \in \mathbb{P}^{\left\vert \mathbb{P}\right\vert }$.
\end{theorem}

\begin{proof}
Let $j\in F_{i}$ then the pair $p_{j}=(i,j)\in \mathbb{P}$. $f_{p_{j}}(o)$
is an extreme point if for any $y,z\in Core\left( N,c\right) $ such that

\begin{equation}
f_{p_{j}}(o)=\frac{1}{2}y+\frac{1}{2}z\mbox{ we have that }y=z.  \label{eq1}
\end{equation}

By definition, we know that%
\begin{equation*}
f_{p_{j}}(o)=(o_{1},...,o_{i-1},o_{i}-\alpha
_{p_{j}}(o),o_{i+1},...,o_{j-1},o_{j}+\alpha _{p_{j}}(o),o_{j+1},...,o_{n}).
\end{equation*}

Let us suppose that $z_{k}<o_{k}$ for any $k\neq i,j$ then $z_{N\setminus
\{k\}}>o_{N\setminus \{k\}}=c\left( N\setminus \{k\}\right) .$ However this
is not possible, therefore $y_{k},z_{k}\geq o_{k}$ for all $k\neq i,j$. Now,
apply (\ref{eq1}) to get that $y_{k}=z_{k}=o_{k}$ $\forall k\neq i,j$.
Moreover, $y_{j},z_{j}\leq o_{j}+\alpha _{p_{j}}(o)$ since $\alpha
_{p_{j}}(o)>0$ is the maximum possible increment for $o_{j}$ (see Lemma \ref%
{o}). Then by (\ref{eq1}) we have that $z=y$ and hence $f_{p_{j}}(o)\in
Ext\left( Core(N,c)\right) $.

Now, we consider $p_{l}=(i,l)\in \mathbb{P}$, and apply the corresponding
extreme function for this pair. We have that%
\begin{equation*}
f_{p_{l}}\left( f_{p_{j}}(o)\right) =(o_{1},...,o_{i}-\alpha
_{p_{j}}(o)-\alpha _{p_{l}}(f_{p_{j}}(o)),...,o_{l}+\alpha
_{p_{l}}(f_{p_{j}}(o)),...,o_{j}+\alpha _{p_{j}}(o),...,o_{n}).
\end{equation*}

We distinguish two possibilities:

\begin{enumerate}
\item $\alpha _{p_{j}}(o)$ attains its minimum in a coalition $R^{\ast }$
that contains player $l$. In this case $\alpha _{p_{l}}(f_{p_{j}}(o))=0$,
thus $f_{p_{l}}\left( f_{p_{j}}(o)\right) =f_{p_{j}}(o)$ and by the argument
above $f_{p_{l}}\left( f_{p_{j}}(o)\right) $ is an extreme point of $%
Core(N,c)$.

\item $\alpha _{p_{j}}(o)$ attains its minimum in a coalition $R^{\ast }$
that does not contain player $l$. This case implies that $\alpha
_{p_{l}}(f_{p_{j}}(o))>0$. Take $y,z\in Core(N,c)$ and assume that
\begin{equation}
f_{p_{l}}\left( f_{p_{j}}(o)\right) =\frac{1}{2}y+\frac{1}{2}z.  \label{eq2}
\end{equation}

Using the same argument as above we conclude that $y_{k}=z_{k}=o_{k}$ for
all $k\neq i,j,l.$ Consider now the $j$-th coordinate. Suppose that $%
z_{j}>o_{j}+\alpha _{p_{j}}(o)$. The coalition $R^{\ast }$ does not contain
neither $i$ nor $l$, which implies $c(R^{\ast })=o_{R^{\ast }}+\alpha
_{p_{j}}(o)<z_{R^{\ast }}.$ Since this is a contradiction, it means that $%
z_{j}\leq o_{j}+\alpha _{p_{j}}(o)$ (Notice that the same argument applies
to $y_{j}$\ and thus $y_{j}\leq o_{j}+\alpha _{p_{j}}(o)$). Therefore, by (%
\ref{eq2}) we get that $y_{j}=z_{j}=o_{j}+\alpha _{p_{j}}(o)$.

Next, consider the $l$-th coordinate. Assume that $z_{l}>o_{l}+\alpha
_{p_{l}}(f_{p_{j}}(o))$, and let $S^{\ast }$\ be the coalition where $\alpha
_{p_{l}}(f_{p_{j}}(o))$ attains its minimum, then $c(S)=\sum_{k\in S}\left(
f_{p_{j}}(o)\right) _{k}+\alpha _{p_{l}}(f_{p_{j}}(o))<z_{S}$. Again using
the same argument as in the $j$-th coordinate we conclude that $%
y_{l}=z_{l}=o_{l}+\alpha _{p_{l}}(f_{p_{j}}(o)).$

Finally, we get the same conclusion for the $i$-th coordinate since $%
f_{p_{l}}\left( f_{p_{j}}(o)\right) $ must be efficient. In conclusion $z=y.$
Hence, $f_{p_{l}}\left( f_{p_{j}}(o)\right) \in Ext\left( Core(N,c)\right) $.
Notice that $f_{p_{l}}\left( f_{p_{j}}(o)\right) $ is different from $%
f_{p_{j}}(o)$ since we have assumed that $\alpha _{p_{l}}(f_{p_{j}}(o))>0$.
\end{enumerate}

This construction can be repeated a finite number of times for each $p\in
\mathbb{P}$. Specifically, for any $\sigma \in \mathbb{P}^{\left\vert
\mathbb{P}\right\vert },$ the transformation $F_{\sigma }(o)\in Ext\left(
Core(N,c)\right) $.
\end{proof}

\begin{corollary}
Let $(N,D,\Re )\in \Upsilon $ with $\mathcal{E}=\{i\}$, and $(N,c)$ be the corresponding PI-game.
The Owen point is always an extreme point.
\end{corollary}

\begin{proof}
Take $y_{k},z_{k}\geq o_{k}$ for all $k\in N$, therefore $o=z=y$ and $o\in Ext\left( Core(N,c)\right).$
\end{proof}
\smallskip

At this point we know that PI-games with a single essential player have, at least,
$\left\vert \mathbb{P}\right\vert+1$ extreme points. Next example shows that
the core of a PI-game, in general, cannot be explicitly described
in polynomial time.

\begin{example}
\label{neco} Now we consider a PI-situation with n periods and n players:

\begin{equation*}
\begin{tabular}{|c|c|c|c|c||c|c|c|c||c|c|c|c||c|c|c|c|}
\hline
& \multicolumn{4}{|c||}{Demand} & \multicolumn{4}{|c||}{Production} &
\multicolumn{4}{|c||}{Inventory} & \multicolumn{4}{|c|}{Backlogging} \\
\hline
P1 & 1 & 1 & $\dots $ & 1 & $\frac{1}{n}$ & $\frac{1}{n}$ & $\dots $ & $%
\frac{1}{n}$ & 2 & 2 & $\dots $ & 2 & 2 & 2 & $\dots $ & 2 \\ \hline
P2 & 1 & 1 & $\dots $ & 1 & 1 & 1 & $\dots $ & 1 & 2 & 2 & $\dots $ & 2 & 2
& 2 & $\dots $ & 2 \\ \hline
$\vdots $ & $\vdots $ & $\vdots $ & $\ddots $ & $\vdots $ & $\vdots $ & $%
\vdots $ & $\ddots $ & $\vdots $ & $\vdots $ & $\vdots $ & $\ddots $ & $%
\vdots $ & $\vdots $ & $\vdots $ & $\ddots $ & $\vdots $ \\ \hline
Pn & 1 & 1 & $\dots $ & 1 & 1 & 1 & $\dots $ & 1 & 2 & 2 & $\dots $ & 2 & 2
& 2 & $\dots $ & 2 \\ \hline
\end{tabular}%
\end{equation*}

The corresponding PI-game is given by $c(S)=\displaystyle%
\sum_{t=1}^{n}p_{t}^{S}d_{t}^{S}$, for all $S\subseteq N$. Moreover, it is
easy to see that $\mathcal{E}=\{1\}$ and $F_{1}=\{2,3,...,n\}$. Then, we can
rewrite the characteristic function as follows:

\begin{equation*}
c(S)=\left\{
\begin{array}{cc}
\left\vert S\right\vert & \text{if }1\in S, \\
&  \\
n\cdot \left\vert S\right\vert & \text{if }1\notin S.%
\end{array}%
\right.
\end{equation*}

In this example, the Owen point is $o=(1,1,...,1)$. For all $i\in F_{1}$,
\begin{equation*}
\alpha _{(1,i)}(o)=\min_{R\in \Delta _{(1,i)}}\{c(R)-o_{R}\}=\min_{R\in
\Delta _{(1,i)}}\{n\cdot \left\vert R\right\vert -\left\vert R\right\vert
\}=n-1
\end{equation*}%
then%
\begin{equation*}
f_{(1,i)}(o)=(2-n,1,...,1,\underset{i}{\underbrace{n}},1,...,1).
\end{equation*}

For all $k\neq i,$ $k\in F_{1}$,
\begin{eqnarray*}
\alpha _{(1,k)}(f_{(1,i)}(o)) &=&\min_{R\in \Delta
_{(1,k)}}\{c(R)-\sum_{j\in R}\left( f_{(1,i)}(o)\right) _{j}\} \\
&=&\min_{R\in \Delta _{(1,k)}}\{\underset{\text{if }i\notin R}{\underbrace{%
n\cdot \left\vert R\right\vert -\left\vert R\right\vert }},\underset{\text{%
if }i\in R}{\underbrace{\left( n-1\right) \cdot \left\vert R\right\vert -n+1}%
}\}=n-1,
\end{eqnarray*}%
then%
\begin{equation*}
f_{(1,k)}\left( f_{(1,i)}(o)\right) =(3-2n,1,...,1,\underset{i}{\underbrace{n%
}},1,...,1,\underset{k}{\underbrace{n}},1,...,1).
\end{equation*}

Hence, we have as many extreme points as possible ways to place $"n"$ and $%
"1"$ in $n-1$ positions; i.e. in this example the core has $2^{n-1}+1$
extreme points.
\end{example}

Therefore, we can conclude that the cardinality of the extreme
points is exponential in the number of players. Hence, we cannot explicitly
describe the core of a PI-game in polynomial time.
\smallskip

We propose below an alternative core allocation to the Owen point that recognizes the role
played by essential players on reducing the cost of their fans.

\section{\label{structure copy 2} Omega point}

Guardiola et al. (2009) proposed the Owen point as a natural core allocation for PI-games that
arises when focusing on shadow prices of each period that each player must pay to meet their
demand in that period. It  makes it possible for all players in the joint venture to operate
at minimum cost.  If there is no essential player, the Owen point is the unique core
allocation. However, for those PI-situations with at least one essential player, the Owen point
reveals the altruistic character of them because of it does not take into account the role that these
essential players play in reducing the cost of their fans. As the core of the PI-games with essential
players is large, we are looking for a core allocation that motivates the essential players to continue
in the join venture obtaining a reduction in their demand costs in each period.
\smallskip

Let $(N,D,\Re )$ be a PI-situation with $D$ being an integer matrix ($
(N,D,\Re )\in \Upsilon $), and $\mathcal{E}\neq \varnothing $. Remember
that for all  $i\in\mathcal{E}$, there is a period $t^{\ast }\in \{1,...,T\}$
such that $y_{t^{\ast }}^{\ast }(N\backslash \{i\})>y_{t^{\ast }}^{\ast }(N)$
and there also exists at least one player $j\in N\backslash \{i\}$ such that
$d_{t^{\ast }}^{j}>0$. We denote by $\mathcal{E}^{t}$ and $F^{t}$ the sets of essential players and fans
for every period $t\in \{1,...,T\}$. We note in passing that $\mathcal{E}=\underset{t\in T}{\bigcup }\mathcal{E}^{t}$.
\smallskip

First, we consider the marginal contribution of the shadow prices of a player $i$ to the grand coalition $N$,
that is, $ y_{t}^{\ast }(N\backslash \{i\})-y_{t}^{\ast }(N).$  We then define the cost reduction that a player $i\in N$ can produce in another player $j\in N$ in a period $t$ as follows:
\begin{equation*}
q_{t}(i,j):=\left\{
\begin{array}{ccc}
\left( y_{t}^{\ast }(N\backslash \{i\})-y_{t}^{\ast }(N)\right) \cdot
d_{t}^{j} & if & i\neq j \\
0 & if & i=j\text{ }%
\end{array}%
\right.
\end{equation*}

The reader may notice that $q_{t}(i,j)>0$ only if $i\in \mathcal{E}^{t}$ and $j\in F^{t},$
otherwise $q_{t}(i,j)=0.$ That is to say that only essential players can reduce
their fan costs in a given period. Alternatively, the amount of the cost $q_{t}(i,j)$
can be interpreted as the maximum cost increase that a fan $j\in F^{t}$ is able to assume, in a certain period $t,$ to incentivize the essential player $i\in \mathcal{E}^{t}.$
\medskip

Next we define a new cost allocation rule, the Omega point, that considers the maximum cost increase mentioned above.

\begin{definition}[Omega point]
Let $(N,D,\Re )\in \Upsilon $ and $(N,c)$ be the corresponding PI-game.
The Omega point $\omega \in \mathbb{R}^{n}$ is defined as
$\omega _{i}=\sum_{t=1}^{T}\omega _{i}^{t}$ for all player $i\in N,$
where for each period $t=1,\ldots ,T$,
\begin{equation*}
\omega _{i}^{t}:=\left\{
\begin{array}{cc}
y_{t}^{\ast }(N)d_{t}^{i}+\overset{Q_{i}^{t}}{\overbrace{q_{t}(\mathcal{E}%
^{t},i)-\sum_{j\in F^{t}}q_{t}(i,j)}} & if\text{ \ }\left\vert \mathcal{E}%
^{t}\right\vert =1 \\
y_{t}^{\ast }(N)d_{t}^{i} & \text{otherwise}%
\end{array}%
\right. \text{ }
\end{equation*}
\end{definition}

The Omega point means that, in each of the periods with a single essential player, i.e. without competition, this essential player gets a cost reduction from his fans. The amount $Q_{i}^{t}$ represents the cost reduction or increase, depending on the sign, for player $i\in N$ in the period $t$. Notice that $Q_{i}^{t}<0,$ only if $i$ is an essential player, otherwise $Q_{i}^{t}\geq 0.$ In addition, $Q^{t}=\left( Q_{i}^{t}\right) _{i\in N}$ for all $t\in
\{1,...,T\}.$

The reader may also note that  $\omega =o+Q,$ where $Q\in \mathbb{R}^{n}$ with $Q_{i}=\sum_{t=1}^{T}Q_{i}^{t}.$  It is worth noting that $Q_{i}$ represents the marginal cost reduction or increase, of player $i$ to the rest players.  Moreover, $\sum_{i\in N}Q_{i}=0.$ In this setting, those players with $Q_{i}<0,$ would prefer the Omega point to the Owen point. On the contrary, those players with $Q_{i}>0$  would like the Owen point more.
\smallskip

The following example illustrates the cost reduction that the Omega point applies to essential players while increasing the cost of fans.

\begin{example}
In example \ref{example 1} the Owen point is $o=(25,26,13)$. Moreover, $%
\mathcal{E}^{1}=\{1\}$, $F^{1}=\{2,3\},\mathcal{E}^{2}=\mathcal{E}%
^{3}=\varnothing $ . The cost reduction for the essential player $1$ from his fans $2$ and $3$ are:

\begin{equation*}
q_{1}(1,2)=8;q_{1}(1,3)=6;
\end{equation*}

Therefore,
\begin{eqnarray*}
\omega _{1} &=&o_{1}-q_{1}(1,2)-q_{1}(1,3)=25-8-6=11 \\
\omega _{2} &=&o_{2}+q_{1}(1,2)=34 \\
\omega _{3} &=&o_{3}+q_{1}(1,3)=19
\end{eqnarray*}

In this case $\omega =(11,34,19)=(25,26,13)+Q$, with $Q=(-14,8,6)$.
It is also a core-allocation. Note that player 1 obtains a cost reduction of 14
units, while players 2 and 3 are increasing their costs by 8 and 6 units, respectively.
Here, the Omega point is a core-allocation that recognizes the essential role of player 1
through a cost reduction assumed by his fans. Next we demonstrate that this always holds
for any PI-game.
\end{example}

\begin{proposition}
\label{wcore}Let $(N,D,\Re )\in \Upsilon $ and $(N,c)$ be the corresponding
PI-game. The Omega point is a core-allocation.
\end{proposition}

\begin{proof}
Consider any period $t$ and a coalition $S\subseteq N$. If $t$ does not have
essential players or has more than one, then $\omega _{S}^{t}=\sum_{i\in
S}y_{t}^{\ast }(N)d_{t}^{i}\leq \sum_{i\in S}y_{t}^{\ast
}(S)d_{t}^{i}=y_{t}^{\ast }(S)d_{t}^{S}$.
\medskip

Otherwise, suppose that player $k$ is essential in the period $t$ ($\mathcal{%
E}^{t}=\{k\}$ ). we distinguish two possibilities:

\begin{itemize}
\item $k$ $\in S$, then%
\begin{eqnarray*}
\omega _{S}^{t} &=&\omega _{k}^{t}+\sum_{i\in S\cap F^{t}}\omega
_{i}^{t}+\sum_{i\in S\backslash F^{t}}\omega _{i}^{t}=y_{t}^{\ast
}(N)d_{t}^{k}-\sum_{j\in F^{t}}q_{t}(k,j) \\
&&+\sum_{i\in S\cap F^{t}}\left( y_{t}^{\ast }(N)d_{t}^{i}+q_{t}(k,i)\right)
+\sum_{i\in S\backslash F^{t}}y_{t}^{\ast }(N)d_{t}^{i} \\
&=&y_{t}^{\ast }(N)d_{t}^{S}+\sum_{i\in S\cap F^{t}}q_{t}(k,i)-\sum_{j\in
F^{t}}q_{t}(k,j) \\
&=&y_{t}^{\ast }(N)d_{t}^{S}-\sum_{j\in F^{t}\backslash S}q_{t}(k,j)\leq
y_{t}^{\ast }(N)d_{t}^{S}\leq y_{t}^{\ast }(S)d_{t}^{S}
\end{eqnarray*}%
\newline

\item $k$ $\notin $ $S$, then%
\begin{eqnarray*}
\omega _{S}^{t} &=&\sum_{i\in S\cap F^{t}}\omega _{i}^{t}+\sum_{i\in
S\backslash F^{t}}\omega _{i}^{t}=\sum_{i\in S\cap F^{t}}\left( y_{t}^{\ast
}(N)d_{t}^{i}+q_{t}(k,i)\right) +\sum_{i\in S\backslash F^{t}}y_{t}^{\ast
}(N)d_{t}^{i} \\
&=&y_{t}^{\ast }(N)d_{t}^{S}+\sum_{i\in S\cap F^{t}}\left( y_{t}^{\ast
}(N\backslash \{k\})-y_{t}^{\ast }(N)\right) d_{t}^{i} \\
&=&\sum_{i\in S\backslash F^{t}}y_{t}^{\ast }(N)d_{t}^{i}+\sum_{i\in S\cap
F^{t}}y_{t}^{\ast }(N\backslash \{k\})d_{t}^{i} \\
&\leq &\sum_{i\in S}y_{t}^{\ast }(N\backslash \{k\})d_{t}^{i}\leq \sum_{i\in
S}y_{t}^{\ast }(S)d_{t}^{i}=y_{t}^{\ast }(S)d_{t}^{S}
\end{eqnarray*}
\end{itemize}

Hence, $\omega _{S}^{t}\leq y_{t}^{\ast }(S)d_{t}^{S}$ for all $t\in T$.
Then, $\omega _{S}=\sum_{t=1}^{T}\sum_{i\in S}\omega
_{i}^{t}=\sum_{t=1}^{T}\omega _{S}^{t}\leq \sum_{t=1}^{T}y_{t}^{\ast
}(S)d_{t}^{S}=c(S)$ for any coalition $S\subseteq N$. Moreover, $\omega
_{N}=o_{N}+\sum_{i\in N}Q_{i}=o_{N}=c(N).$ Therefore $\omega \in Core(N,c)$.
\end{proof}

\subsection{\protect\bigskip Characterization of the Omega point}

To complete the study of the Omega point, we here propose an axiomatic characterization based on a set of desirable properties that make it unique. In order to do that, we denote by $\varphi $ a
generic allocation rule on $\Upsilon $ and consider the following properties,  some of which have been used in the literature to axiomatize alternative allocations:

\begin{itemize}
\item[(EF)] \textit{Efficiency}. For all $x\in \varphi (N,D,\Re )$ and for
any PI-game $(N,D,\Re )\in \Upsilon $, $x_{N}=c^{(N,D,\Re )}(N)$.

\item[(NE)] \textit{Nonemptiness}. For any PI-game $(N,D,\Re )\in \Upsilon $%
, $\varphi (N,D,\Re )\neq \varnothing $.

\item[(IBC)] \textit{Inessential {bounded} cost}. For any PI-game $(N,D,\Re
)\in \Upsilon $ and for all $x\in \varphi (N,D,\Re )$, if $i$ is an
inessential player for $(N,D,\Re )$, then $x_{i}\leq
\sum_{t=1}^{T}y_{t}^{\ast }(N\backslash \mathcal{E}^{t})d_{t}^{i}$

\item[(TI)] \textit{Tyranny}. For all $(N,D,\Re )\in \Upsilon $ and for all $%
x\in \varphi (N,D,\Re )$, if $k$ is a single essential player then $%
x_{N\setminus \{k\}}=c^{(N,D,\Re )}(N\setminus \{k\})$.

\item[(ACP)] \textit{{Additive} combination of periods' demands}. For all $%
(N,D,\Re )\in \Upsilon $ and for all $x\in \varphi (N,D,\Re )$, there exists
$(z_{t})_{t\in T}\in (\mathbb{R}^{N})^{N}$ such that $x=\sum_{t=1}^{T}z_{t}$
and for all $t\in T$, $z_{t}\in \varphi (N,D_{t},\Re )$ if $\left\vert
\mathcal{E}^{t}\right\vert \leq 1$ and $z_{t}=Owen(N,D_{t},\Re )$ otherwise,
where
\begin{equation}
D_{t}=\left( d^{ip}\right) _{\substack{ i=1,...,n  \\ p=1,...,T}},\text{ }%
d^{ip}=\left\{
\begin{array}{cc}
d_{p}^{i} & \text{if }t=p, \\
0 & \text{otherwise.}%
\end{array}%
\right.  \label{eq3}
\end{equation}
\end{itemize}

The first two properties were already used in Guardiola et al. (2008), among
many other papers, to characterize the Owen point solution, and they are also
important to our new characterization of the Omega point. Recall that
\textit{Efficiency} ensures that the total cost of any PI-situation is
entirely allocated among the players. Analogously, \textit{Nonemptiness}
guarantees that this allocation rule always return a feasible allocation of
the overall cost when applied to any PI-situation. \textit{Inessential {%
bounded} cost} imposes a maxim cost for every {inessential} player in
situations which an essentials players has left. \textit{Tyranny} implies
that a single essential player will assert all his power over the rest so
that they assume the maximum possible cost.
\smallskip

Finally, an allocation rule satisfies the property of \textit{Additivity
combination of periods' demands} if it is additive with respect to the
demand of the periods that has at most an essential player plus the Owen
point of those periods with more than one essential player.
We emphasize that this additivity results from the following
relationship $c^{(N,D,\Re )}=\sum_{t=1}^{T}c^{(N,D_{t},\Re )}$\textbf{\ }for
all\textbf{\ }$(N,D,\Re )\in \Upsilon .$ Thus, we are interested on allocation
rules, for PI-situations, compatible with this form of distribution of their
demands.
\medskip

First, we prove that the Omega point satisfies all the properties mentioned above.

\begin{proposition}
\label{con} The Omega point defined on the set $\Upsilon ,$ satisfies EF, NE, IBC, TI and ACP.
\end{proposition}

\begin{proof}
For any PI situation $(N,D,\Re )\in \Upsilon $ we know by proposition \ref%
{wcore} that $\omega (N,D,\Re )\in Core(N,c)$ by . Hence, the Omega point
verifies the properties of EF and NE. An inessential player satisfy IBC
since if for all $i\in N$
\begin{eqnarray*}
\omega _{i}(N,D,\Re ) &=&\sum_{t=1}^{T}\omega _{i}^{t}(N,D,\Re
)=\sum_{t=1}^{T}\omega _{i}(N,D_{t},\Re ) \\
&=&\sum_{t\in T/\left\vert \mathcal{E}^{t}\right\vert =1}\left( y_{t}^{\ast
}(N)d_{t}^{i}+q_{t}(\mathcal{E}^{t},i)\right) +\sum_{t\in T/\left\vert
\mathcal{E}^{t}\right\vert \neq 1}y_{t}^{\ast }(N)d_{t}^{i} \\
&=&\sum_{t\in T/\left\vert \mathcal{E}^{t}\right\vert =1}y_{t}^{\ast
}(N\backslash \mathcal{E}^{t})d_{t}^{i}+\sum_{t\in T/\left\vert \mathcal{E}%
^{t}\right\vert \neq 1}y_{t}^{\ast }(N)d_{t}^{i}\leq
\sum_{t=1}^{T}y_{t}^{\ast }(N\backslash \mathcal{E}^{t})d_{t}^{i}
\end{eqnarray*}

\noindent if there is only one essential player $k$, then:
\begin{eqnarray*}
\sum_{t=1}^{T}\omega _{N\setminus \{k\}}^{t} &=&\sum_{t=1}^{T}\left(
y_{t}^{\ast }(N)d_{t}^{N\setminus \{k\}}+\sum_{j\in N\setminus
\{k\}}q_{t}(k,i)\right) \\
&=&\sum_{t=1}^{T}\left( y_{t}^{\ast }(N)d_{t}^{N\setminus \{k\}}+\sum_{j\in
N\setminus \{k\}}\left( y_{t}^{\ast }(N\backslash \{k\})-y_{t}^{\ast
}(N)\right) d_{t}^{j}\right) \\
&=&\sum_{t=1}^{T}\left( y_{t}^{\ast }(N\backslash \{k\})d_{t}^{N\setminus
\{k\}}\right) =c^{(N,D,\Re )}(N\setminus \{k\}).
\end{eqnarray*}%
Then satisfy {TI}. Finally, considering $D_{t}$ as it was already defined in
(\ref{eq3}), we obtain that $\sum_{t=1}^{T}D_{t}=D$ and
\begin{eqnarray*}
\omega (N,D,\Re ) &=&\left( \sum_{t=1}^{T}\omega _{i}^{t}(N,D,\Re )\right)
_{i\in N}=\sum_{t=1}^{T}\left( \omega _{i}(N,D_{t},\Re )\right) _{i\in N} \\
&=&\sum_{t\in T/\left\vert \mathcal{E}^{t}\right\vert \leq 1}\omega
_{i}(N,D_{t},\Re )+\sum_{t\in T/\left\vert \mathcal{E}^{t}\right\vert \geq
2}Owen(N,D_{t},\Re )
\end{eqnarray*}%
Hence, the Omega point satisfies ACP.
\end{proof}

\medskip
Second, we focus on PI-situations without essential players and show that, in this setting,
the Omega point matches the Owen point, and both can be characterized by using only three of the previous properties.

\begin{proposition}
\label{con2} Let $(N,D,\Re )\in \Upsilon $ be a PI situation with $\left\vert \mathcal{E}\right\vert
=0$. Then, $\varphi (N,D,\Re )=\omega (N,D,\Re )=Owen(N,D,\Re )$ if and only
if $\varphi $ satisfies NE, EF and IBC.
\end{proposition}

\begin{proof}
(If) Immediately follows by Proposition \ref{con}. \medskip

\noindent (Only if) By NE, $\varphi (N,D,\Re )\neq \varnothing $. Take $x\in
\varphi (N,D,\Re ).$ Since all players $i\in N$ are inessential, by IBC, it
holds that $x_{i}\leq \sum_{t=1}^{T}y_{t}^{\ast }(N\backslash \mathcal{E}%
^{t})d_{t}^{i}$ $=\sum_{t=1}^{T}y_{t}^{\ast }(N)d_{t}^{i}=\omega
_{i}(N,D,\Re )$ for each $i\in N$ $.$ Therefor,e by EF, $\varphi (N,D,\Re
)=\omega (N,D,\Re )=Owen(N,D,\Re )$.\medskip
\end{proof}

The main Theorem of this section shows that the Omega point is
the unique allocation rule that satisfies the aforementioned five properties.

\begin{theorem}
\label{charac} An allocation rule on $(N,D,\Re )\in \Upsilon$\ satisfies the
properties EF, NE, IBC, TI and ACP if and only if it coincides with the
Omega point.
\end{theorem}

\begin{proof}
(If) The \textit{if} part of the proof is direct from Proposition \ref{con}.
\medskip

\noindent (Only if) Let $\varphi $ be an allocation rule. The case where the
number of essential players is zero, namely $\left\vert \mathcal{E}%
\right\vert =0$, follows from Proposition \ref{con2}. Then, it remains to
prove the case when $\left\vert \mathcal{E}\right\vert \geq 1.$In this case,
we know that $D=D_{1}+D_{2}+...+D_{T}$ where $D_{t}$ is (see (\ref{eq3}%
)):
\begin{equation*}
D_{t}=\left(
\begin{array}{ccccccc}
0 & \dots & 0 & d_{t}^{1} & 0 & \dots & 0 \\
0 & \dots & 0 & d_{t}^{2} & 0 & \dots & 0 \\
\vdots & \dots & 0 & \vdots & 0 & \dots & \vdots \\
0 & \dots & 0 & d_{t}^{n} & 0 & \dots & 0%
\end{array}%
\right).
\end{equation*}%
Then for all $t\in T$, $(N,D_{t},\Re )$ is a PI-situation with $D_{t}$ an
integer matrix. This implies that $(N,D_{t},\Re )$ belongs to $\Upsilon $.
Therefore, for any $t\in T,$ the Omega point for $(N,D_{t},\Re )$ is $%
(\omega _{i}(N,D_{t},\Re ))_{i=1,\ldots ,n}$:

By NE, $\varphi (N,D_{t},\Re )\neq \varnothing $. for each situation $%
(N,D_{t},\Re )$ we have two cases:

\begin{itemize}
\item $\left\vert \mathcal{E}^{t}\right\vert =0,$ then by Proposition \ref%
{con2} $\varphi (N,D_{t},\Re )=\omega (N,D_{t},\Re )=Owen(N,D_{t},\Re ).\ $

\item $\mathcal{E}^{t}=\{k\}.$ Take $u\in \varphi (N,D_{t},\Re )$, by IBC $%
u_{i}\leq y_{t}^{\ast }(N\backslash \{k\})d_{t}^{i}$ for all $i\in
N\backslash \{k\}$ and by TY $u_{N\backslash \{k\}}=y_{t}^{\ast
}(N\backslash \{k\})d_{t}^{N\setminus \{k\}}$. Hence for all $i\in
N\backslash \{k\}$ $u_{i}=y_{t}^{\ast }(N\backslash
\{k\})d_{t}^{i}=y_{t}^{\ast }(N)d_{t}^{i}+q_{t}(\mathcal{E}^{t},i)=\omega
_{i}(N,D_{t},\Re )$. Finally, by EF $u_{k}=c(N)-c(N\backslash
\{k\})=y_{t}^{\ast }(N)d_{t}^{N}-y_{t}^{\ast }(N\backslash
\{k\})d_{t}^{N\setminus \{k\}}=y_{t}^{\ast }(N)d_{t}^{k}-\left( y_{t}^{\ast
}(N\backslash \{k\})-y_{t}^{\ast }(N)\right) d_{t}^{N\backslash
\{k\}}=y_{t}^{\ast }(N)d_{t}^{k}-\sum_{j\in F^{t}}q_{t}(k,j)=\omega
_{k}(N,D_{t},\Re )$.
\end{itemize}

Therefore, if $x\in \varphi (N,D,\Re )$ by ACP one has that $%
x=z_{1}+...+z_{t}$ with $z_{t}\in \varphi (N,D_{t},\Re )$ for all $t\in T$,
and so
\begin{equation*}
x=\sum_{t=1}^{T}z_{t}=\sum_{t\in T/\left\vert \mathcal{E}^{t}\right\vert
\leq 1}\omega (N,D_{t},\Re )+\sum_{t\in T/\left\vert \mathcal{E}%
^{t}\right\vert \geq 2}Owen(N,D_{t},\Re )=\omega (N,D,\Re ).
\end{equation*}%
The above equation implies that $\varphi (N,D,\Re )=\omega (N,D,\Re )$.
\end{proof}
\bigskip

Finally, we prove that all the properties used in Theorem \ref{charac} are logically independent.
That is, the characterization of the Omega point is tight in the sense that no property is redundant.

\begin{example}
Let $\varphi $ be a solution rule defined on $\Upsilon $ as
\begin{equation*}
\varphi (N,D,\Re ):=\left\{
\begin{array}{cc}
\left( \frac{c^{(N,D,\Re )}(N)}{2},\frac{c^{(N,D,\Re )}(N)}{2}\right) , &
(N,D,\Re )\in \Upsilon ^{1} \\
&  \\
\omega (N,D,\Re ), & \text{otherwise,}%
\end{array}%
\right.
\end{equation*}%
\noindent where
\begin{equation*}
\Upsilon ^{1}:=\left\{ (N,D,\Re )\in \Upsilon \left/ \left\vert N\right\vert
=2,T=2,\mathcal{E}^{1}=\{1,2\},\mathcal{E}^{2}\text{ }=\varnothing \right.
\right\} .
\end{equation*}%
\noindent $\varphi (N,D,\Re )$ satisfies EF, NE, IBC and TI, but not ACP.
\end{example}

\begin{example}
Let $\varphi $ be a solution rule defined on $\Upsilon $ as
\begin{equation*}
\varphi (N,D,\Re ):=\left\{
\begin{array}{cc}
\left( c^{(N,D,\Re )}(N),0\right) , & (N,D,\Re )\in \Upsilon ^{2} \\
&  \\
\omega (N,D,\Re ), & \text{otherwise,}%
\end{array}%
\right.
\end{equation*}%
\noindent where
\begin{equation*}
\Upsilon ^{2}:=\left\{ (N,D,\Re )\in \Upsilon \left/ \left\vert N\right\vert
=2,T=1,\mathcal{E}=\varnothing \text{ }\right. \right\} .
\end{equation*}

\noindent {$\varphi $}$(N,D,\Re )${\ } satisfies EF, NE, ACP and TI, but not
IBC.
\end{example}

\begin{example}
Let $\varphi $ be a solution rule defined on $\Upsilon $ as

\begin{equation*}
\varphi (N,D,\Re ):=Owen(N,D,\Re )
\end{equation*}

\noindent {$\varphi $}$(N,D,\Re )${\ } satisfies EF, NE, IBC, and ACP, but
not TI.
\end{example}

\begin{example}
Let $\varphi $ be a solution rule defined on $\Upsilon $ as
\begin{equation*}
\varphi (N,D,\Re ):=\left\{
\begin{array}{cc}
\left( c^{(N,D,\Re )}(N\setminus \{\mathcal{E}^{1}\}),c^{(N,D,\Re
)}(N\setminus \{\mathcal{E}^{1}\})\right) , & (N,D,\Re )\in \Upsilon ^{3} \\
&  \\
\omega (N,D,\Re ), & \text{otherwise,}%
\end{array}%
\right. ,
\end{equation*}%
where%
\begin{equation*}
\Upsilon ^{3}:=\left\{ (N,D,\Re )\in \Upsilon \left/ \left\vert N\right\vert
=2,T=1,\left\vert \mathcal{E}^{1}\right\vert =1\text{ }\right. \right\} .
\end{equation*}

\noindent {$\varphi $}$(N,D,\Re )${\ } satisfies NE, IBC, TI and ACP, but
not EF.
\end{example}

\begin{example}
Let $\varphi $ be a solution rule defined on $\Upsilon $ as
\begin{equation*}
\varphi (N,D,\Re ):=\varnothing .
\end{equation*}

\noindent {$\varphi $}$(N,D,\Re )${\ satisfies }EF, ACP, IBC, and TI. but
not NE.
\end{example}

\section{\protect\bigskip \label{QPQ set}Quid Pro Quo allocations}

As we already mentioned, the Omega point can be considered the natural aspiration of
the essential players to achieve the biggest cost reduction while the Owen
point reflects their altruistic character. We combine both extreme characteristics and
define the $\lambda$-agreement $a(\lambda ):=\lambda \omega +\left( 1-\lambda \right)
o $ with $\lambda \in \left[ 0,1\right] ,$ as the convex linear combination
of the Owen point and the Omega point. The parameter $\lambda $ represents
here the weight given to individual
behavior, by those players who want to maximize their cost reduction,
compared to altruistic behavior (by $1-\lambda$), which benefits the other players.
\smallskip

The set of all the above agreements is called Quid Pro Quo allocation set.

\begin{definition}[Quid Pro Quo allocation set]
\bigskip Let $(N,D,\Re )\in \Upsilon $ and $(N,c)$ be the corresponding
PI-game. We define the Quid pro quo allocation set as follows:
\begin{equation*}
QPQ(N,c):=\left\{ a(\lambda )\ \text{such that }\lambda \in \left[ 0,1\right]
\right\} .
\end{equation*}
\end{definition}

The Quid Pro Quo allocation set, henceforth QPQ-set, is a parametric family of core-allocations.
That is, $QPQ(N,c)\subseteq Core(N,c)$.
\medskip

The following example illustrate the wealth of the QPQ set of a PI-situation with multiple  essential
players.

\begin{example}
\label{example 3} Let us consider a PI-situation with four players in four
periods with demand, and production, inventory and backlogging costs given
in the following table:

\begin{equation*}
\begin{tabular}{|c|c|c|c|c||c|c|c|c||c|c|c|c||c|c|c|c|}
\hline
& \multicolumn{4}{|c}{Demand} & \multicolumn{4}{||c||}{Production} &
\multicolumn{4}{|c||}{Inventory} & \multicolumn{4}{||c|}{Backlogging} \\
\hline
P1 & 2 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 1 & 1 & 1 &  & 2 & 2 & 2 &  \\ \hline
P2 & 2 & 2 & 1 & 2 & 2 & 1 & 2 & 2 & 1 & 1 & 1 &  & 2 & 2 & 2 &  \\ \hline
P3 & 2 & 1 & 2 & 2 & 2 & 2 & 1 & 2 & 1 & 1 & 1 &  & 2 & 2 & 2 &  \\ \hline
P4 & 2 & 1 & 1 & 2 & 2 & 2 & 2 & 1 & 1 & 1 & 1 &  & 2 & 2 & 2 &  \\ \hline
\end{tabular}%
\end{equation*}%
\newline
\newline
The above table described a cooperative game with a characteristic function
detailed in the following table:

\begin{equation*}
\begin{array}{|c|c|c|c|c||c|c|c|c||c|c|c||c|c|c||c||}
\hline
& d_{1}^{_{S}} & d_{2}^{_{S}} & d_{3}^{_{S}} & d_{4}^{_{S}} & p_{1}^{_{S}} &
p_{2}^{_{S}} & p_{3}^{_{S}} & p_{4}^{_{S}} & h_{1}^{_{S}} & h_{2}^{_{S}} &
h_{3}^{_{S}} & b_{1}^{_{S}} & b_{2}^{_{S}} & b_{3}^{_{S}} & c \\ \hline
\{1\} & 2 & 1 & 2 & 2 & 1 & 2 & 2 & 2 & 1 & 1 & 1 & 2 & 2 & 2 & 12 \\ \hline
\{2\} & 2 & 2 & 1 & 2 & 2 & 1 & 2 & 2 & 1 & 1 & 1 & 2 & 2 & 2 & 12 \\ \hline
\{3\} & 2 & 1 & 2 & 2 & 2 & 2 & 1 & 2 & 1 & 1 & 1 & 2 & 2 & 2 & 12 \\ \hline
\{4\} & 2 & 1 & 1 & 2 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 10 \\ \hline
\{1,2\} & 4 & 3 & 3 & 4 & 1 & 1 & 2 & 2 & 1 & 1 & 1 & 2 & 2 & 2 & 21 \\
\hline
\{1,3\} & 4 & 2 & 4 & 4 & 1 & 2 & 1 & 2 & 1 & 1 & 1 & 2 & 2 & 2 & 20 \\
\hline
\{1,4\} & 4 & 2 & 3 & 4 & 1 & 2 & 2 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 18 \\
\hline
\{2,3\} & 4 & 3 & 3 & 4 & 2 & 1 & 1 & 2 & 1 & 1 & 1 & 2 & 2 & 2 & 22 \\
\hline
\{2,4\} & 4 & 3 & 2 & 4 & 2 & 1 & 2 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 19 \\
\hline
\{3,4\} & 4 & 2 & 3 & 4 & 2 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 19 \\
\hline
\{1,2,3\} & 6 & 4 & 5 & 6 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 2 & 2 & 2 & 27 \\
\hline
\{1,2,4\} & 6 & 4 & 5 & 6 & 1 & 1 & 2 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 24 \\
\hline
\{1,3,4\} & 6 & 3 & 5 & 6 & 1 & 2 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 23 \\
\hline
\{2,3,4\} & 6 & 4 & 4 & 6 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 26 \\
\hline
\{1,2,3,4\} & 8 & 5 & 6 & 8 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 2 & 2 & 2 & 27 \\
\hline
\end{array}%
\end{equation*}

Here, $y^{\ast }(N)=\left( 1,1,1,1\right) $ and the Owen point
is $o=(7,7,7,6)$. Moreover, $\mathcal{E}=N,$ because of each player is essential
just in one period. For example in period 1, $\mathcal{E}^{1}=\{1\}$ and $%
F^{1}=\{2,3,4\}$. In addition, $q_{1}(1,j)=2$ for $j\in F^{1}$, $%
Q^{1}=(-6,2,2,2),Q^{2}=(1,-3,1,1),Q^{3}=(2,1,-4,1)$ and $Q^{4}=(2,2,2,-6).$
\medskip

It is easy to check that $Q=(-1,2,1,-2)$ that is, players 1 and 4 are interested in improving
the Owen point, and they would prefer the Omega point. However, players 2 and 3, still being essential, get some benefit with the Owen point's and they would prefer to keep on it.
\smallskip

On the other hand, here the omega point is $\omega =(6,9,8,4)$ and the QPQ set is given by:
\begin{equation*}
QPQ(N,c):=\left\{ (7-\lambda ,7+2\lambda ,7+\lambda ,6-2\lambda )\ \text{%
such that }\lambda \in \left[ 0,1\right] \right\}
\end{equation*}

If we consider the same weight for both individual and altruistic behaviors, we get the Shapley,
which also matches the Nucleolus. That is, for $\lambda =\frac{1}{2}$ the Shapley
value and Nucleolus coincides and both are equal to $\left( \frac{13}{2},8,%
\frac{15}{2},5\right) $.
\end{example}
\medskip

At this point we wonder whether this coincidence always holds for every PI-game. The
answer is no, in general, as example \ref{example 4} reveals.
\medskip

The main result of this section shows that, if no player can get a cost reduction
in any coalition without an essential player, then the equal agreement, $a\left( \frac{1}{2}\right) $,
coincides with the Shapley value and the Nucleolus. In some sense, it is a Solomonic
agreement between the players who demand cost reductions (individual behaviour) and those who do
not (altruistic behaviour). For that, we call $a\left( \frac{1}{2}\right) $ Solomonic allocation
and denote it $\varsigma (N,c).$

\begin{proposition}
Let $(N,D,\Re )\in \Upsilon $ and $(N,c)$ be the corresponding PI-game. Assume that
for each $t=1,\ldots ,T$ the following conditions are simultaneously fulfilled:

\begin{enumerate}
\item[(i)] $\left\vert \mathcal{E}^{t}\right\vert \leq 1$,

\item[(ii)] $y_{t}^{\ast }(\mathcal{E}^{t})=y_{t}^{\ast }(N)$ \ if $\mathcal{%
E}^{t}\neq \phi ,$

\item[(iii)] $y_{t}^{\ast }(N\backslash \mathcal{E}^{t})=y_{t}^{\ast
}(\{i\}) $ for all $i\in N\backslash \mathcal{E}^{t}.$ \medskip

\noindent Then, $\varsigma (N,c)=$ $\phi (N,c)=\eta (N,c).$
\end{enumerate}

\begin{proof}
Consider $(N,D_{t},\Re )\in \Upsilon $ and $(N,c^{t})$ be the corresponding
PI-game, with, (only period $t$ has demand)%
\begin{equation*}
D_{t}=\left( d^{ip}\right) _{\substack{ i=1,...,n  \\ p=1,...,T}},\text{ }%
d^{ip}=\left\{
\begin{array}{cc}
d_{p}^{i} & \text{if }t=p, \\
0 & \text{otherwise.}%
\end{array}%
\right.
\end{equation*}

We will denote to simplify notation $o(N,D_{t},\Re )$ and $\omega
(N,D_{t},\Re )$ as $o^{t}$ and $\omega ^{t}$, respectively. By (i) we
consider only two cases:

\begin{itemize}
\item If $\left\vert \mathcal{E}^{t}\right\vert =0$ then $\omega
^{t}=o^{t}=Core(N,c^{t})=\eta (N,c^{t})$ since the Nucleolus always belongs
to the core of a game. Moreover, because of the condition (iii) $y_{t}^{\ast
}(N)=y_{t}^{\ast }(\{i\})$ for all $i\in N,$ then $c^{t}(S)=o_{S}^{t}$ for
all $S\subseteq N,$ It is easy to verify that all players are dummy players
then $\phi (N,c^{t})=o^{t}.$

\item If $\left\vert \mathcal{E}^{t}\right\vert =1,\ (\mathcal{E}%
^{t}=\{k\}). $

Note that if $k\in S$ then $c^{t}(S\cup \{i\})-c^{t}(S)=y_{t}^{\ast
}(N)d_{t}^{i},$ otherwise $(k\notin S)$ then by condition (iii) $c^{t}(S\cup
\{i\})-c^{t}(S)=y_{t}^{\ast }(S\cup \{i\})d_{t}^{S\cup \{i\}}-y_{t}^{\ast
}(S)d_{t}^{S}=y_{t}^{\ast }(N\backslash \mathcal{E}^{t})d_{t}^{i}$ for all $%
i\in N\backslash \mathcal{E}^{t}.$

If $i\in N\backslash \mathcal{E}^{t}$ then,
\begin{eqnarray*}
\phi _{i}(N,c^{t}) &=&\sum_{S\subseteq N\diagdown \{i\}}\gamma (S)\cdot
\left[ c^{t}(S\cup \{i\})-c^{t}(S)\right] = \\
&&\sum_{S\subseteq N\diagdown \{i\}/k\in S}\gamma (S)\cdot \left[
c^{t}(S\cup \{i\})-c^{t}(S)\right] \\
&&+\sum_{S\subseteq N\diagdown \{i\}/k\notin S}\gamma (S)\cdot \left[
c^{t}(S\cup \{i\})-c^{t}(S)\right] \\
&=&\sum_{S\subseteq N\diagdown \{i\}/k\in S}\gamma (S)\cdot y_{t}^{\ast
}(N)d_{t}^{i}+\sum_{S\subseteq N\diagdown \{i\}/k\notin S}\gamma (S)\cdot
y_{t}^{\ast }(N\backslash \mathcal{E}^{t})d_{t}^{i} \\
&=&y_{t}^{\ast }(N)d_{t}^{i}\cdot \left( \sum_{S\subseteq N\diagdown
\{i\}/k\in S}\gamma (S)\right) \\
&&+y_{t}^{\ast }(N\backslash \mathcal{E}^{t})d_{t}^{i}\cdot \left(
\sum_{S\subseteq N\diagdown \{i\}/k\notin S}\gamma (S)\right) \text{ } \\
&=&\frac{1}{2}\cdot y_{t}^{\ast }(N)d_{t}^{i}+\frac{1}{2}\cdot y_{t}^{\ast
}(N\backslash \mathcal{E}^{t})d_{t}^{i} \\
&=&\frac{1}{2}\cdot y_{t}^{\ast }(N)d_{t}^{i}+\frac{1}{2}\cdot \left(
y_{t}^{\ast }(N)d_{t}^{i}+\left( y_{t}^{\ast }(N\backslash \{\mathcal{E}%
^{t}\})-y_{t}^{\ast }(N)\right) \cdot d_{t}^{i}\right) \\
&=&\frac{1}{2}\cdot o_{i}^{t}+\frac{1}{2}\cdot \omega _{i}^{t}
\end{eqnarray*}

By efficiency of Shapley value $\phi _{\mathcal{E}^{t}}(N,c^{t})=\frac{1}{2}%
\cdot o_{\mathcal{E}^{t}}^{t}+\frac{1}{2}\cdot \omega _{\mathcal{E}%
^{t}}^{t}. $ Moreover, Shapley value {satisfies additivity} property, thus
for all player $i\in N$
\begin{eqnarray*}
\phi _{i}(N,c) &=&\sum_{t=1}^{T}\phi _{i}(N,c^{t})=\sum_{t=1}^{T}\left(
\frac{1}{2}\cdot o_{i}^{t}+\frac{1}{2}\cdot \omega _{i}^{t}\right) \\
&=&\frac{1}{2}\cdot \sum_{t=1}^{T}o_{i}^{t}+\frac{1}{2}\cdot
\sum_{t=1}^{T}\omega _{i}^{t} \\
&=&\frac{1}{2}\cdot o_{i}(N,D,\Re )+\frac{1}{2}\cdot \omega _{i}(N,D,\Re )
\end{eqnarray*}

since the Owen point is additive for the demands (demonstrated in Guardiola
et al. (2008)) and $\omega _{i}(N,D_{t},\Re )=\omega _{i}^{t}(N,D,\Re )$.
Hence, $\varsigma (N,c)=\phi (N,c).$

Now, we will prove that the {Shapley} value coincides with the Nucleolus. As
we have seen previously if the properties $(i),(ii)$ and $(iii)$ are
satisfied for a period $t=1,\ldots ,T$ and for each $i\in N$ and for all $%
S\subseteq N\setminus \{i\}$
\begin{equation*}
\Delta _{i}^{t}(S):=c^{t}(S\cup \{i\})-c^{t}(S)=\left\{
\begin{array}{ccc}
y_{t}^{\ast }(N)d_{t}^{i} & if & \mathcal{E}^{t}\in S\text{ and }i\notin
\mathcal{E}^{t} \\
y_{t}^{\ast }(N\backslash \mathcal{E}^{t})d_{t}^{i} & if & \mathcal{E}%
^{t}\notin S\text{ and }i\notin \mathcal{E}^{t} \\
y_{t}^{\ast }(N)d_{t}^{i}-y_{t}^{\ast }(N\backslash \mathcal{E}^{t})d_{t}^{i}
& if & i\in \mathcal{E}^{t}%
\end{array}%
\right.
\end{equation*}

similarly we get that

\begin{equation*}
\Delta _{i}^{t}(N\setminus \left( S\cup \{i\}\right) )=\left\{
\begin{array}{ccc}
y_{t}^{\ast }(N\backslash \mathcal{E}^{t})d_{t}^{i} & if & \mathcal{E}%
^{t}\in S\text{ and }i\notin \mathcal{E}^{t} \\
y_{t}^{\ast }(N)d_{t}^{i} & if & \mathcal{E}^{t}\notin S\text{ and }i\notin
\mathcal{E}^{t} \\
y_{t}^{\ast }(N)d_{t}^{i}-y_{t}^{\ast }(N\backslash \mathcal{E}^{t})d_{t}^{i}
& if & i\in \mathcal{E}^{t}%
\end{array}%
\right.
\end{equation*}

Hence, $\Delta _{i}^{t}(S)+\Delta _{i}^{t}(N\setminus \left( S\cup
\{i\}\right) )=y_{t}^{\ast }(N)d_{t}^{i}+y_{t}^{\ast }(N\backslash \mathcal{E%
}^{t})d_{t}^{i}$ if $i\in N\backslash \mathcal{E}^{t}$ for all $S\subseteq
N\setminus \{i\}$ and $\Delta _{i}^{t}(S)+\Delta _{i}^{t}(N\setminus \left(
S\cup \{i\}\right) )=2\cdot \left( y_{t}^{\ast }(N)d_{t}^{i}-y_{t}^{\ast
}(N\backslash \mathcal{E}^{t})d_{t}^{i}\right) $ if $i\in \mathcal{E}^{t}$
for all $S\subseteq N\setminus \mathcal{E}^{t}.$ We consider $\Delta
_{i}(S):=\sum_{t=1}^{T}\Delta _{i}^{t}(S)$ for each $i\in N$ and for all $%
S\subseteq N\setminus \{i\}.$ Thus $\Delta _{i}(S)+\Delta _{i}(N\setminus
\left( S\cup \{i\}\right) )$ is a constant for all $S\subseteq N\setminus
\{i\}$ and for all $i\in N$. Then $(N,c)$ is a PS-game and $\varsigma (N,c)=$
$\phi (N,c)=\eta (N,c).$
\end{itemize}
\end{proof}
\end{proposition}

The reader may notice that for those situations in which the properties
$(i)$, $(ii)$ {and} $(iii)$ hold and, in addition, $Q=0$ (i.e., $o=\omega$),
then $QPQ(N,c)=\left\{ o(N,D,\Re )\right\} =$ $\left\{ \phi (N,c)\right\}
=\left\{ \eta (N,c)\right\} $. Otherwise, the core is larger.
\medskip

Finally, we analyze the relationships between conditions $(i)$, $(ii)$, $(iii)$
and concavity of PI-games.

\begin{proposition}
Let $(N,D,\Re )\in \Upsilon $ and $(N,c)$ be the corresponding PI-game. If
for each $t=1,\ldots ,T$ conditions (i), (ii) and (ii) are fulfilled
simultaneously the $(N,c)$ is concave.

\begin{proof}
Consider $(N,D_{t},\Re )\in \Upsilon $ and $(N,c^{t})$ be the corresponding
PI-game,

\begin{itemize}
\item[(a)] If $\left\vert \mathcal{E}^{t}\right\vert =0$ then $y_{t}^{\ast
}(\{i\})=y_{t}^{\ast }(N)$ for all $i\in N$ henceforth $c^{t}(S)-c^{t}(S%
\setminus \{i\})=y_{t}^{\ast }(N)d_{t}^{i}$ for all $i\in N$ and for all $%
S\subseteq N.$ Hence $(N,c^{t})$ is concave.
\end{itemize}

\begin{itemize}
\item[(b)] If $\left\vert \mathcal{E}^{t}\right\vert =1,\ $let say $\mathcal{%
E}^{t}=\{k\}$. Then two cases can be {distinguished}:

\item[(b1)] $k\in S$ $\subseteq T\subset N$ then $c^{t}(S)-c^{t}(S\setminus
\{i\})=y_{t}^{\ast }(N)d_{t}^{i}=c^{t}(T)-c^{t}(T\setminus \{i\})$ for all $%
i\in N\setminus \{k\}.$ Finally
\begin{eqnarray*}
c^{t}(S)-c^{t}(S\setminus \{k\}) &\geq &c^{t}(T)-c^{t}(T\setminus \{k\}); \\
y_{t}^{\ast }(N)d_{t}^{S}-y_{t}^{\ast }(N\setminus \{k\})d_{t}^{S\setminus
\{k\}} &\geq &y_{t}^{\ast }(N)d_{t}^{T}-y_{t}^{\ast }(N\setminus
\{k\})d_{t}^{T\setminus \{k\}}; \\
y_{t}^{\ast }(N\setminus \{k\})d_{t}^{T\setminus S} &\geq &y_{t}^{\ast
}(N)d_{t}^{T\setminus S}.
\end{eqnarray*}%
It is true since $y_{t}^{\ast }(S)\geq y_{t}^{\ast }(R)$ for all $S\subseteq
R\subseteq N$\ and all $t\in \{1,...,T\}.$

\item[(b2)] $k\notin S$ and $k\in T.$ By condition (iii) $%
c^{t}(S)-c^{t}(S\setminus \{i\})=y_{t}^{\ast }(N\setminus
\{k\})d_{t}^{i}\geq c^{t}(T)-c^{t}(T\setminus \{i\})$ since if $k\in T$ is {%
satisfied} $c^{t}(T)-c^{t}(T\setminus \{i\})=y_{t}^{\ast }(N)d_{t}^{i}$ and
if $k\notin T$ we have that $c^{t}(T)-c^{t}(T\setminus \{i\})=y_{t}^{\ast
}(N\setminus \{k\})d_{t}^{i}.$
\end{itemize}

Finally, by additivity property of PI-games with respect to periods (see
Guardiola et al. (2008)) $(N,c)$ is concave.
\end{proof}
\end{proposition}

Next example shows that conditions $(i)$, $(ii)$, $(iii)$, although
necessaries, are no sufficient for concavity.

\begin{example}
\label{example 4} Let us consider a PI-situation with three players in three
periods with demand, and production, inventory and backlogging costs given
in the following table:

\begin{equation*}
\begin{tabular}{|c|c|c|c||c|c|c||c|c||c|c|}
\hline
& \multicolumn{3}{|c||}{Demand} & \multicolumn{3}{|c||}{Production} &
\multicolumn{2}{|c||}{Inventory} & \multicolumn{2}{|c|}{Backlogging} \\
\hline
P1 & 10 & 10 & 10 & 1 & 2 & 3 & 1 & 2 & 1 & 1 \\ \hline
P2 & 10 & 10 & 10 & 2 & 1 & 3 & 1 & 2 & 1 & 1 \\ \hline
P3 & 10 & 10 & 10 & 3 & 3 & 1 & 1 & 2 & 2 & 2 \\ \hline
\end{tabular}%
\end{equation*}

Using those data one can obtain the cooperative
game with characteristic function described below:

\begin{equation*}
\begin{array}{|c|c|c|c||c|c|c||c|c||c|c||c||}
\hline
& d_{1}^{_{S}} & d_{2}^{_{S}} & d_{3}^{_{S}} & p_{1}^{_{S}} & p_{2}^{_{S}} &
p_{3}^{_{S}} & h_{1}^{_{S}} & h_{2}^{_{S}} & b_{1}^{_{S}} & b_{2}^{_{S}} & c
\\ \hline
\{1\} & 10 & 10 & 5 & 1 & 2 & 3 & 1 & 2 & 1 & 1 & 45 \\ \hline
\{2\} & 10 & 10 & 10 & 2 & 1 & 3 & 1 & 1 & 1 & 1 & 50 \\ \hline
\{3\} & 10 & 10 & 10 & 3 & 3 & 1 & 1 & 2 & 2 & 2 & 70 \\ \hline
\{1,2\} & 20 & 20 & 15 & 1 & 1 & 3 & 1 & 1 & 1 & 1 & 70 \\ \hline
\{1,3\} & 20 & 20 & 15 & 1 & 2 & 1 & 1 & 2 & 1 & 1 & 75 \\ \hline
\{2,3\} & 20 & 20 & 20 & 2 & 1 & 1 & 1 & 1 & 1 & 1 & 80 \\ \hline
\{1,2,3\} & 30 & 30 & 25 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 85 \\ \hline
\end{array}%
\end{equation*}

It is easy to check that the above game is concave, but condition $(iii)$ does
not hold. Indeed, for the first period, $\mathcal{E}^{1}=\left\{ 1\right\}$ but $y_{1}^{\ast }(\left\{ 2,3\right\})
=2<3= y_{1}^{\ast }(\left\{ 3\right\} ).$
\smallskip

Moreover, the Nucleolus $\eta (N,c)=\left( \frac{70}{3},
\frac{85}{3},\frac{100}{3}\right)$ is lightly different from the Shapley value,
$\phi (N,c)=\left( \frac{125}{6},\frac{155}{6}, \frac{115}{3}\right)$.
\smallskip

Finally, $o=(30,30,25)$, $\omega
=(25,30,30)$ and so the Solomonic allocation is $\varsigma (N,c)=\left( \frac{55}{2},30,\frac{55}{2}\right).$
\end{example}

\section{Concluding remarks}

This paper completes the study of the PI-games presented in Guardiola et al.
(2008, 2009). Those two papers proposed the Owen point as a natural core-allocation,
which does not pay attention to the role that essential players play in reducing the costs of their fans.
In that sense, essential players could consider the Owen point as an altruistic core-allocation.
However, the core was not studied in depth there.

Here we have analyzed carefully the core structure of PI-games,
and we have realized that the number of extreme point of its core is exponential
in the number of players. Then, we have proposed a new core-allocation, the Omega
point, that compensates the essential players for their role in reducing the
costs of their fans. Based on the Owen and Omega points we have defined the QPQ-set.
Since every QPQ allocation is a convex combination of the Owen and the Omega points,
we have paid special attention to the equally weighted QPQ allocation, the Solomonic allocation. Finally, we have provided some necessary conditions for the coincidence of the latter with the
Shapley value and the Nucleolus.

redpajama_set_name: RedPajamaArXiv


text:
Lyn McGettigan
Memoir, entertainment and performing arts, humour
Lyn, one of three children, was born in the city, raised in the bush, in Glen Innes and Baradine. Her father was in the Forestry Commission and her mother was a nurse. She was educated at St Ursula's, Kingsgrove, then graduated from Bathurst Teachers' College.
The work ethic of her parents, the rough and tumble of country life, then challenging teaching in 'difficult 'country and city High Schools, equipped her well for her life in pubs.
In 1971 Lyn met Brian the Publican at The Penshurst Hotel. Their first hotel together was the red-brick, suburban Union Hotel at North Sydney, folllowed by Sheila's- a unique, new entertainment concept, then McGettigan's, where an under-age disco brought together the teenagers of Sydney.
Tatts Tavern at Penrith was a tough pub. Never a dull moment, fondly remembered.
A tree change to The Lake Jindabyne Hotel, a huge entertainment venue showcasing Australian and International bands. Then her last hotel, The Kincumber on the Central Coast, where she was greeted by a warm caring community.
Lyn led a parallel life. After gaining her Hospitality qualifications at Ryde TAFE, 1985, and from Cornell University, she taught and trained Hospitality students throughout NSW, in clubs, pubs TAFE and UNSW. The slogan, "No More It's the Law" is the lasting result of one training session in Jindabyne.
She fulfilled a childhood dream when she graduated from UNE in 2001 with a BA in Archaeology (Aboriginal) and Palaeoanthropology and worked with the Indigenous mobs on the Central Coast.
Her three children, Danielle, Kate and Brian, lived and survived through this life and have followed her into Hospitality.
As she passed through the bar-room door for the last time, Lyn knew there were too many good stories to go untold.
She is divorced, writing, and enjoying life with her friends, family and grandchildren, and consulting in the Hospitality industry.
All books by Lyn McGettigan
Behind the Bar-Room Door: Tales of a Publican's Wife
Underworld characters, S P bookies, prostitutes, bikies, coppers, journos, footy players and salt-of-the-earth patrons. This publican's wife knew them all. Take a walk through her bar room door and read the extraordinary stories from forty years of owning and operating pubs in NSW, Australia.

redpajama_set_name: RedPajamaCommonCrawl


Computer applications technology associate's degree programs prepare students to work as computer support specialists or computer applications specialists. Learn about three U.S. schools that offer associate's degrees in computer applications technology ranked by student enrollment. Schools offering Mobile Development degrees can also be found in these popular choices.
When seeking to obtain an associate's degree in computer applications technology, students should consider the format of the program and whether the courses correspond to their academic and professional goals.
Delta College's associate program in business information technology focuses heavily on Microsoft Windows and its applications, including MS Word, Excel, PowerPoint and Access. Programming in Visual BASIC is part of the curriculum. In addition, students can learn about accounting, small business management, advertising, e-mail, Web page management, networking and information technology ethics and law. Delta College had an enrollment of 9,842 as of Fall 2014.
Pennsylvania College of Technology (PCT), an affiliate school of Pennsylvania State University, offers the Associate of Applied Science in Information Technology with a technical support technology emphasis. Degree candidates study networking, programming,computer maintenance, information security and system analysis. They also may choose to add multimedia applications, several programming languages or business transaction systems to their knowledge base or take part in a semester-long internship. PCT had a student population of 5,623 in Fall 2014.
Which Schools Offer Computer Technology Training in West Virginia?


text:
<?xml version="1.0" encoding="utf-8"?>
<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
    android:layout_width="fill_parent"
    android:layout_height="fill_parent"
    android:orientation="vertical" >

    <TextView
        android:layout_width="fill_parent"
        android:layout_height="wrap_content"
        android:text="@string/username" />

    <EditText
        android:id="@+id/txt_username"
        android:layout_width="fill_parent"
        android:layout_height="wrap_content"
        android:hint="@string/username" />

    <TextView
        android:layout_width="fill_parent"
        android:layout_height="wrap_content"
        android:text="@string/password" />

    <EditText
        android:id="@+id/txt_password"
        android:layout_width="fill_parent"
        android:layout_height="wrap_content"
        android:hint="@string/password"
        android:inputType="textPassword" />

    <Button
        android:id="@+id/btnSave"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:text="@string/save" />

    <Button
        android:id="@+id/btnTest"
        android:layout_width="fill_parent"
        android:layout_height="wrap_content"
        android:text="@string/testConnection" />

    <Button
        android:id="@+id/btnShowAllTips"
        android:layout_width="fill_parent"
        android:layout_height="wrap_content"
        android:text="@string/showAllTips" />

</LinearLayout>

redpajama_set_name: RedPajamaGithub


What does Bladder Health UK do?
Bladder Health UK gives support to people with all forms of cystitis, overactive bladder and continence issues together with their families and friends. We are the largest bladder patient support charity in the UK. We have a busy chat room/Forum , a widely used website and active Twitter and Facebook accounts for those bladder illness sufferers seeking information, help and support.
"To help people live positively with bladder illness through the provision of personalised support and care"
Commitment to our members: We will…..
Bladder Health UK (formerly The COB Foundation) has been in existence since 2003 and we have grown in to the largest bladder patient support charity in the UK. Our membership is increasing year-on-year and our telephone advice line receives a constant stream of enquiries from those in need of support and information. Our membership also includes health care professionals, who join us to learn more about our resource information and the current therapies and treatment options that are available to aid sufferers.
We work with department of health organisations and other medical bodies to improve patient care and share information.
Joining Bladder Health UK will enable you to understand your bladder illness and find ways to help manage your symptoms. Your membership is a very important part of our on-going work to raise awareness of bladder illness, not only for sufferers, but also their families, friends and employers. To find out more about the full benefits of becoming a member here.
Check out our resources section to find out what else is available to help inform you about your bladder problem. There are also our very popular message boards which enable sufferers to discuss conditions and support each other live online. In addition to providing information via this website, Bladder Health UK also offers more help to members via local groups and our magazine.
Since the beginning, the charity has produced a very informative magazine, providing up-to-date information on the latest treatments, research, resources and story features. There is a facility to write and have letters, opinions and experiences published or pose questions to our medical advisers.
We have a network of support available in some geographical areas for our members. The groups are run by volunteers who are fellow sufferers. We also operate a "phone-pals" system, which enables members to talk to each other for advice and support.
Please note: Bladder Health UK does not engage in the practice of medicine. It is not a medical authority nor does it claim to have medical knowledge. In all cases we recommend that you consult your own doctor or urologist regarding any course of treatment or medication.


text:
Grand Canyon (1991) – Review
March 19, 2013 March 20, 2013 Jason Rugaard	2 Comments	1991, Alfre Woodard, Danny Glover, Grand Canyon, Jeremy Sisto, Kevin Kline, Lawrence Kasdan, Mary McDonnell, Mary-Louise Parker, Paul Haggis, Paul Thomas Anderson, Silence of the Lambs, steve martin, The Big Chill
Lawrence Kasdan is a filmmaker that achieved mainstream success with 1983's The Big Chill, yet it is Grand Canyon that serves as the masterpiece in this underrated directors' film cannon. Influenced by the storytelling of Altman, that would later serve Paul Thomas Anderson and Paul Haggis, Kasdan and his wife Meg have crafted one of the great screenplays of modern cinema. Interweaving the lives of a dozen people living in various parts of Los Angeles over the course of a summer, Grand Canyon is a deft and sometimes mystical look at society and our relationship to one another as a civilization.
The film begins ominously as the Kevin Kline character gets lost leaving a Lakers game and ends up in a rough neighborhood. His expensive car breaks down and while waiting for a tow truck, he is confronted by a group of young men that have bad intentions on the mind. In the midst of this Danny Glover arrives playing the truck driver and he somehow gets Kline out of this dangerous situation. Leading to an unlikely friendship between the two men that has implications for both far beyond their wildest comprehension.
There are too many story threads and surprises to divulge here, the movie works best with limited knowledge of the events. I would like to point out a career highlight performance from Steve Martin as a Joel Silver-type producer, only interested in violent blockbuster escapism who experiences a brief epiphany when he is assaulted at gun point. It's a riveting portrayal and goes overlooked when Martin's dramatic turns are mentioned. Equally solid supporting work from the entire cast make round out the top-notch acting, each delivering their intelligent dialogue in a thoughtful way that never comes across as 'acting'.
My only gripe is the inclusion (or exclusion for large chunks of the film) of the Mary-Louise Parker character. Her arc feels flat and forced, something that is foreign to the entire film and the final scene in which she finds comfort in the hands of a concerned police officer is too neat. This is a minor quibble for an otherwise near flawless movie that deserves to be seen by every cinephile or open-minded audience members. Silence of the Lambs may have taken home the Oscar for Best Picture, but Grand Canyon is the Best film of 1991, and on my list of top ten for the entire decade.
Director: Lawrence Kasdan
Stars: Kevin Kline, Danny Glover, Steve Martin
← Writer's Journey: The Treatment
Snitch (2013) – Review →
The Human Shield (1991) – Review
May 24, 2014 May 24, 2014 Jason Rugaard	0
Death Race 4: Beyond Anarchy (2018) – Review
Highlander 2: The Quickening (1991) – Review
February 28, 2019 March 24, 2019 Jason Rugaard	0
2 thoughts on "Grand Canyon (1991) – Review"
I know of Big Chill but I have never even heard of this movie. Great cast, great review. Sounds like I definitely need to watch it.
Another film that reminds me of Big Chill (and may be similar to Grand Canyon) is Indian Summer. Have you seen that one?
Jason RugaardPost author
Nick, I'm a big fan of Indian Summer, though that film is a bit sweeter and has a touch of sentimentality. Grand Canyon is tougher but more abstract in a sense. I would compare it to a more successful version of Magnolia or Short Cuts. Regardless it's a must-watch, available on Netflix streaming for those days when you have exhausted your instant que options. I never took to The Big Chill but I'm in the minority on that one.


I wrote a piece after backpacking a grueling trail in the Great Smoky Mountains a few years back referring to the Swallow Fork Trail as my "daddy." The terrain was not so difficult, but the trail slowly gained elevation for four miles. This was like Chinese water torture, but when backpacking Baxter State Park, I found a trail even more difficult.
It was a ten mile day from Russell Pond to Chimney Pond. The first six miles to Davis Pond were pretty non eventful, and I made decent time. The only difficult part was a somewhat tricky river crossing of Wassataquoik Stream. The trail was full of roots and rocks and was never easy, but for the most part, the first seven miles went quickly.
This all changed on the other side of Davis Pond. I picked up the Northwest Basin Trail and started gaining elevation. Elevation is tough when hiking, but it makes it even tougher when the trail is uneven. When the Northwest Basin started to climb it went straight up the mountain. There were no switchbacks, and if not for the blue blazes, the trail would have been difficult to follow. This was because the trail was so steep that it went in places it would have been unbelievable to go. I kept thinking, that cannot be the trail as it went right up a rock face only to have a blue blaze emboldened right at the top.
After scrambling through the trees up cliff faces, I came to a pond. I could see Mt. Katahdin and the ridge above. I was not sure where the trail would go. I was hoping it plateaued for a bit before making the last push to the top.
The trail instead went straight up through the boulder field above me. If I thought the trail went in unbelievable places before this, it even got more treacherous. The boulders got so steep that I was going virtually straight up. I was grabbing the rock above me with my hand while pushing off below me with my walking stick (see the video below).
The only good thing was the boulders were lichen covered and provided good traction as it was not wet. The tread on my hiking boots griped the rocks securely. I never slipped off a rock although I did struggle at times to keep my balance. Since the danger was so great if I fell backwards, it just was not an option, so I focused on moving up. If I lost my balance, I made sure to fall forward.
I finally made it up to Mt. Hamline, which was above the treeline at 4,720. I was only 500 feet below Mt. Katahdin, but I was not going to summit today. My plan was to descend down to Chimney Pond and try to climb Katahdin the next day.
The view atop Mt. Hamline was sensational. I could see the Allagash Riverway slightly to the west. In just a couple of days I would be looking back at Mt. Katahdin from our first night's camping site while paddling the waterway. I could also look up to Katahdin as well as the rugged landscape I just climbed up from.
Finally reached the top of the ridge. Only 500 feet higher to the top of Katahdin.
The Saddleback Trail down was a replay of the Northwest Basin Trail going up. Going downhill at a steep rate like this is not as physically demanding as climbing up, but it is awkward and still difficult. I was exhausted after the climb up, so I had to be very careful not to loose my footing going down.
So close to summiting Katahdin. In fact, I did most of the difficult work. Unfortunately, did not have enough daylight to complete on this day.
I finally stumbled into Chimney Pond Campground with my headlight on in almost complete darkness. It took me another 15 minutes to find my lean-to campsite. There was a sign for every tent and lean-to except the one I was staying at. I finally figured out where it was and collapsed on the the wooden platform. I was too tired to cook, so I ate nuts, granola bars, and dried fruit for dinner and passed out.
The next morning I was still beat up from the 4.2 mile climb and descent over the boulder field ridge. A blanket of clouds covered Katahdin. I decided I was not up to climbing Katahdin today either. There would have been no view. I saw enough from Mt. Hamline the day before. The view could not be much better from another 500 feet up, especially when shrouded in fog.
I hiked in the three miles to Roaring Brook Campground and reunited with my car. I drove towards Millinocket to get some hot dogs and ice and camped from my car at Nesowadnehunk Campground to complete my Baxter State Park experience.
If you are hiking up to Katahdin or up the ridge to Mt. Hamline, do not take a tent. Make sure you get lean-to reservations and leave your tent in the car. The extra weight is a killer.
Take advantage of the $1.00 an hour canoe rentals at Russell Pond and elsewhere. It is $8.00 for the whole day.
I would hike Katahdin first from Roaring Brook when you are fresh if I were doing it over again.
For information on planning your own hike in Baxter State Park. Check out the park online.


for problem E, did you mean to write dig=1...9 (or 0...9)?
I'm sorry, this is a mistake. Hope it fixed now!
"we will look for the first point of the common part as root", how to look for the first point of the common part as root? Which is the first point of the common part as root, is the center of the origin tree?
Why do we have to keep iterating until T becomes smaller than the minimum priced sweet in Problem D. What I am thinking is when we do T=T mod C , we will be having money only for a single round of circle. Please correct me If I am wrong. Thank you.
Your assumption in second line i.e. " When we do T=Tmodc.... ...single round of circle. " is wrong.
give it another thought. it's easy you will get it.
After T%C, The number of round of circle is multiple, instead of single.
Yes! I got the error. Thank you!
Can someone please explain the logic used in problem C?
In C we apply binary search on the length of string to find whether it is possible or not to reach the desired position from the position reached by applying all the operations measured in non effective range and changing the operations in substring of given length(lets say len) and we would do that for all substrings of length len.
Eg: You have a string RRRUUL and you want to check if you could reach position (-4,2). So we apply binary search and lets say mid = 3. Then you would check for indices 0 to 2 to see if by changing the instructions in this range can I reach my desired position. Now you have to apply the rest of operations(which are not affected in the range selected, here positions 3 to 5 in the string) anyhow, so to check the validity we make our initial position to be the one that is obtained by applying the operations in non effective range(here UUL position 3 to 5 in original string). So now your initial position is (-1,2). Now from this position you check whether I can reach to (-4,2) by changing the at most 3 instructions in given range(position 0 to 2 in string). In this case you can(change RRR to LLL) and so call to ok(3) would return true. Like this the binary sear occurs and at the end you would have the minimum value of length.
In the condition of (do<=len) , should not also there be condition that (len-do)%2==0 ? Since odd difference would go wrong in some testcases as well.
I thought this was my comment lol.
PikMike In C problem why is value of l initialized with -1 and not 0 which is the minimum value if answer is possible(and it is, as we already checked it by ok(n)). Also how to determine the loop condition to be r - l > 1 and not the standard r > l?
There are two ways to write a binary search: on segment [l, r] or on interval [l, r) or (l, r]. This is the (l, r] case, so we should be sure that ok(l) = false and ok(r) = true. In case of invervals, (l, r] is a degenerate case .
Can someone explain me solution to problem E?
Not sure what the editorial does exactly, but one way to solve it with DP is two store two values.
I saw a solution with segment tree for problem D. Anyone to explain the method?
I have solved it using two segment trees so it may be similar solution. Both were for suming on interval and changing in 1 point. The first one was for suming cost of candies, the latter for suming candies. We are skipping whole circles as long as we can so ater that we will have some T < C. Then we can binary search how far can we go by checking sum of cost candies using segment tree. The we can add to result sum from the second tree on the same interval, change cost of candie to 0 and change value of candie to 0 (because we will never buy it again). Then we move our starting point just after the last fair we have binary searched for, remove fairs on which we will never buy candies, and repeat whole process again. In each loop we are decreaing number of fairs by at least 1 and cost of every loop is O(log^2 N) for segment tree and binary search so total complexity should be O(Nlog^2 N). In my solution it was handy to be only on fairs on which we can still buy candies so i kept set of indices of those.
It's so simple, it's perfect. All you do is loop over the bitmasks of sets of digits. Then, for each one, binary search on the largest string that's not greater than our boundaries. Then just subtract the total sum of all possible numbers generated by our digit set up to the right boundary from the total sum from the left boundary to find its contribution to the answer!
Of course, you have some handling with things like overcounting, leading zeroes, and such, but it turns out with this algorithm those are all gone with a wave of the hand.
BOOM! Now this is the kind of thing competitive programming is about.
Please put this into Contest materials of the Round . Thanks.
For any string "abcdef..." you can get n-1 substrings with len=2: "ab", "bc", ... If string contains at least 2 different symbols (a and b, e.g.), it contains substring "ab" or "ba", both of them are "YES" answers. So, only strings that are built from one symbol are "NO" strings.
I am solving problem C, I am stuck trying to understand why ** If there is at least one position of the beginning of the segment for which d0≤len, then we can change the segment ** anyone could explain me why, please?
Server time: Apr/26/2019 04:03:34 (e3).


Components and virtual DOM nodes can have lifecycle methods, also known as hooks, which are called at various points during the lifetime of a DOM element.
All lifecyle methods receive the vnode as their first arguments, and have their this keyword bound to vnode.state.
Lifecycle methods are only called as a side effect of a m.render() call. They are not called if the DOM is modified outside of Mithril.
A DOM element is typically created and appended to the document. It may then have attributes or child nodes updated when a UI event is triggered and data is changed; and the element may alternatively be removed from the document.
After an element is removed, it may be temporarily retained in a memory pool. The pooled element may be reused in a subsequent update (in a process called DOM recycling). Recycling an element avoids incurring the performance cost of recreating a copy of an element that existed recently.
The oninit(vnode) hook is called before a vnode is touched by the virtual DOM engine. oninit is guaranteed to run before its DOM element is attached to the document, and it is guaranteed to run on parent vnodes before their children, but it does not offer any guarantees regarding the existence of ancestor or descendant DOM elements. You should never access the vnode.dom from the oninit method.
This hook does not get called when an element is updated, but it does get called if an element is recycled.
Like in other hooks, the this keyword in the oninit callback points to vnode.state.
The oninit hook is useful for initializing component state based on arguments passed via vnode.attrs or vnode.children.
You should not modify model data synchronously from this method. Since oninit makes no guarantees regarding the status of other elements, model changes created from this method may not be reflected in all parts of the UI until the next render cycle.
The oncreate(vnode) hook is called after a DOM element is created and attached to the document. oncreate is guaranteed to run at the end of the render cycle, so it is safe to read layout values such as vnode.dom.offsetHeight and vnode.dom.getBoundingClientRect() from this method.
This hook does not get called when an element is updated.
Like in other hooks, the this keyword in the oncreate callback points to vnode.state. DOM elements whose vnodes have an oncreate hook do not get recycled.
The oncreate hook is useful for reading layout values that may trigger a repaint, starting animations and for initializing third party libraries that require a reference to the DOM element.
You should not modify model data synchronously from this method. Since oncreate is run at the end of the render cycle, model changes created from this method will not be reflected in the UI until the next render cycle.
The onupdate(vnode) hook is called after a DOM element is updated, while attached to the document. onupdate is guaranteed to run at the end of the render cycle, so it is safe to read layout values such as vnode.dom.offsetHeight and vnode.dom.getBoundingClientRect() from this method.
This hook is only called if the element existed in the previous render cycle. It is not called when an element is created or when it is recycled.
DOM elements whose vnodes have an onupdate hook do not get recycled.
The onupdate hook is useful for reading layout values that may trigger a repaint, and for dynamically updating UI-affecting state in third party libraries after model data has been changed.
The onbeforeremove(vnode) hook is called before a DOM element is detached from the document. If a Promise is returned, Mithril only detaches the DOM element after the promise completes.
This hook is only called on the DOM element that loses its parentNode, but it does not get called in its child elements.
Like in other hooks, the this keyword in the onbeforeremove callback points to vnode.state. DOM elements whose vnodes have an onbeforeremove hook do not get recycled.
The onremove(vnode) hook is called before a DOM element is removed from the document. If a onbeforeremove hook is also defined, the onremove hook runs after the promise returned from onbeforeremove is completed.
This hook is called on any element that is removed from the document, regardless of whether it was directly detached from its parent or whether it is a child of another element that was detached.
Like in other hooks, the this keyword in the onremove callback points to vnode.state. DOM elements whose vnodes have an onremove hook do not get recycled.
The onremove hook is useful for running clean up tasks.
The onbeforeupdate(vnode, old) hook is called before a vnode is diffed in a update. If this function is defined and returns false, Mithril prevents a diff from happening to the vnode, and consequently to the vnode's children.
This hook by itself does not prevent a virtual DOM subtree from being generated unless the subtree is encapsulated within a component.
Like in other hooks, the this keyword in the onbeforeupdate callback points to vnode.state.
This hook is useful to reduce lag in updates in cases where there is a overly large DOM tree.
You should only use onbeforeupdate to skip diffing as a last resort. Avoid using it unless you have a noticeable performance issue.
Typically performance problems that can be fixed via onbeforeupdate boil down to one large array of items. In this context, typically "large" means any array that contains a large number of nodes, be it in a wide spread (the infamous 5000 row table), or in a deep, dense tree.
If you do have a performance issue, first consider whether the UI presents a good user experience and change it if it doesn't. For example, it's highly unlikely that a user would ever sift through 5000 rows of raw table data, and highly likely that it would be easier for a user to use a search feature that returns only the top few most relevant items.
If a design-based solution is not feasible, and you must optimize a UI with a large number of DOM element, apply onbeforeupdate on the parent node of the largest array and re-evaluate performance. In the vast majority of cases, a single check should be sufficient. In the rare case that it is not, rinse and repeat, but you should be increasingly wary of each new onbeforeupdate declaration. Multiple onbeforeupdates are a code smell that indicates prioritization problems in the design workflow.
Avoid applying the optimization to other areas of your application "just-in-case". Remember that, generally speaking, more code incurs a higher maintenance cost than less code, and onbeforeupdate related bugs can be especially difficult to troubleshoot if you rely on object identity for its conditional checks.
Again, the onbeforeupdate hook should only be used as a last resort.


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Q: mutt regex for limit filters Does mutt support multi-line regex for its search filters? I cannot find an yes or no in the documentation. I'm assuming its not supported, but cannot tell if maybe I am improperly escaping my regex.
I'm trying to match email body that contains something along these lines:
...
...(22)

sent ...
....

My regex I'm using for the limit command is:
~b \\(22\\)\n\nsent

I assume no multiline because this works as expected:
~b \\(22\\)\n

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