Let us consider a company of three, sirs Gandalf, Frodo and Aragorn, visiting the halls of lady Galadriel in Lothlorien.
Let us consider a company of three, sirs Gandalf, Frodo and Aragorn, visiting the halls of lady Galadriel in Lothlorien.
The three of them are discussing which outfit they should wear and trying to find the best overall look their reduced fellowship can have. Worn out from the journey, their hosts show them to individual baths. Before leaving them to bathe and relax, their hosts leave them with two clean outfits sets of robes, made of either black velvet or white silk. After relaxing in the hot baths for a while, it will be up to them to decided which outfit they should wear.
The three of them are discussing which outfit they should wear and trying to find the best overall look their reduced fellowship can have. Worn out from the journey, their hosts show them to individual baths. Before leaving them to bathe and relax, their hosts leave them with two clean outfits sets of robes, made of either black velvet or white silk. After relaxing in the hot baths for a while, it will be up to them to decided which outfit they should wear.
This graph-colouring problem can be formalised as follows (see Fiure~\ref{fig:galadriel-utility}):
This graph-colouring problem can be formalised as follows (see Figure~\ref{fig:galadriel-utility}):
\begin{itemize}
\begin{itemize}
\item Set of agents, the three companions: $=(G, F, A)$
\item Set of agents, the three companions: $=(G, F, A)$
\item Set of variables, the colour of their each robe: $=(R_1, R_2, R_3)$.
\item Set of variables, the colour of their each robe: $=(R_1, R_2, R_3)$.
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@@ -13,8 +13,8 @@ This graph-colouring problem can be formalised as follows (see Fiure~\ref{fig:ga
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@@ -13,8 +13,8 @@ This graph-colouring problem can be formalised as follows (see Fiure~\ref{fig:ga
\item$\alpha$ : simply the function (here the elf who gave each robe to each man!) mapping $R_1$ to Gandalf, $R_2$ to Frodo and $R_3$ to Aragorn.
\item$\alpha$ : simply the function (here the elf who gave each robe to each man!) mapping $R_1$ to Gandalf, $R_2$ to Frodo and $R_3$ to Aragorn.
\end{itemize}
\end{itemize}
Here, the goal is to maximise the global \textbf{utility} function $F_g$ -aka charisma- so they look as dashing as they can.
Here, the goal is to maximise the global \textbf{utility} function $F_g$ -considered to be their charisma for intance- so they look as dashing as they can.
We know by computing manually the sums that there are three best option which each yield 17:
We know by computing manually the sums that there are three best option which each yield a total of 17:
\begin{itemize}
\begin{itemize}
\item G=W, F=W, A=W (see Figure~\ref{fig:galadriel-best1})
\item G=W, F=W, A=W (see Figure~\ref{fig:galadriel-best1})
\item G=W, F=B, A=B (see Figure~\ref{fig:galadriel-best2})
\item G=W, F=B, A=B (see Figure~\ref{fig:galadriel-best2})
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@@ -22,7 +22,6 @@ We know by computing manually the sums that there are three best option which ea
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@@ -22,7 +22,6 @@ We know by computing manually the sums that there are three best option which ea
