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Betrachten wir das Beispiel \(n=2\). \begin{beis}[2-Kaffee-Problem] - Sind nur 2 Personen \( p_0 \) und \( p_1 \) an der Kaffeerunde beteiligt, - lässt sich das \(2\)-Kaffee-Problem - mit Hilfe einer Zahl \( x \in \integers \) beschreiben. Hierbei gilt: + Sei \(K\) ein 2-Kaffeekränzchen und + \(k\) eine K-Kaffeekasse. + Nehmen wir an, dass bisher \(p_1\) zweimal + einen Kaffee für \(p_2\) bezahlt hat, + dann ergibt sich der Zustand \(k = (-2,2)\). + + Es lässt sich leicht erkennen, + dass stets $\Delta_1 = - \Delta_2$ gelten muss. + Somit können wir ohne Informationsverlust die + zweite Komponente der Kaffekasse vernachlässigen + und nur noch $\Delta_1$ betrachten. Hierbei gilt: \begin{align*} - x=0 &: \enspace \text{Das Verhältnis ist ausgeglichen, niemand hat Kaffeeschulden.} \\ - x > 0 &: \enspace p_0 \text{ schuldet } p_1 \text{ noch } x \text{ Kaffees.} \\ - x < 0 &: \enspace p_1 \text{ schuldet } p_0 \text{ noch } |x| \text{ Kaffees.} + \Delta_1 < 0 &: \enspace p_2 \text{ schuldet } p_1 \text{ noch } |\Delta_1| \text{ Kaffees.} \\ + \Delta_1 = 0 &: \enspace \text{Die Kaffeekasse ist ausgeglichen, niemand hat Kaffeeschulden.} \\ + \Delta_1 > 0 &: \enspace p_1 \text{ schuldet } p_2 \text{ noch } \Delta_1 \text{ Kaffees.} \end{align*} - Die Zahl \(x\) kann aufgefasst werden - als \(\Delta_1\) in der Kaffeekasse. - \(\Delta_2\) wird nicht benötigt. \end{beis} Die im Beispiel angedeutete Vereinfachungsmöglichkeit @@ -237,30 +242,47 @@ Das ist die Aussage des folgenden Satzes. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%% Bilanzierende Kaffeekassen und das Kaffee-Paradoxon -\subsection{Bilanzierende Kaffeekassen und das Kaffee-Paradoxon} +%% Explizite Kaffeekassen und das Kaffee-Paradoxon +\subsection{Explizite Kaffeekassen und das Kaffee-Paradoxon} \label{sub:kaffeeparadoxon} -\todo[inline]{Die Unterscheidung zwischen bilanzierenden und expliziten Kaffeekassen einführen. -Das Kaffeeparadoxon beschreiben und auflösen.} +Die in \autoref{def:kaffeekasse} definierte Kaffeekasse +modelliert einen gemeinsamen Kaffeepool innerhalb des Kaffeekränzchens. +Diese Kaffeekasse kann auch als \emph{bilanzierende Kaffeekasse} bezeichnet werden. +Als alternative Notation könnte auch eine explizite Kaffeekasse vorgehalten werden, +in der alle Kaffeeschulden innerhalb des Kaffeekränzchens einzeln ausgewiesen werden. + +\begin{defi}[Explizite Kaffeekasse] + Sei \(n \in \naturals_{\geq2}\) und \(K\) ein \(n\)-Kaffeekränzchen. + Eine \emph{explizite K-Kaffeekasse} ist die Menge + \[ k = \{ \delta_{ij} \in \integers | i,j \in \naturals_{\leq n} , i < j \} \] + Hierbei verhalten sich die einzelnen Kaffeedeltas $\delta_{ij}$ wie folgt: + \begin{align*} + \delta_{ij} < 0 &: \enspace p_j \text{ schuldet } p_i \text{ noch } |\delta_{ij}| \text{ Kaffees.} \\ + \delta_{ij} = 0 &: \enspace \text{Niemand hat Kaffeeschulden.} \\ + \delta_{ij} > 0 &: \enspace p_i \text{ schuldet } p_j \text{ noch } \delta_{ij} \text{ Kaffees.} + \end{align*} +\end{defi} +Hierdurch werden sogenannte \emph{transitive Kaffeeschulden} vermieden. + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Visualisierung des Kaffeeproblems \section{Visualisierung des Kaffeeproblems} \label{sec:visualisierung} -\begin{figure} - \centering - \includegraphics[scale=1]{2KaffeeProblem} - \caption{% - Visualisierung des \(2\)-Kaffee-Problems. - In diesem Fall schuldet \(p_0\) \(p_1\) - genau \(2\) Kaffees. - } - \label{fig:2_kaffee_problem} -\end{figure} +% \begin{figure} +% \centering +% \includegraphics[scale=1]{2KaffeeProblem} +% \caption{% +% Visualisierung des \(2\)-Kaffee-Problems. +% In diesem Fall schuldet \(p_0\) \(p_1\) +% genau \(2\) Kaffees. +% } +% \label{fig:2_kaffee_problem} +% \end{figure}