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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, +Als alternative Notation könnte auch eine explizite Kaffeekasse vorgehalten werden, in der alle Kaffeeschulden innerhalb des Kaffeekränzchens einzeln ausgewiesen werden. \begin{defi}[Explizite Kaffeekasse] + \label{def:explizitekaffeekasse} Sei \(n \in \naturals_{\geq2}\) und \(K\) ein \(n\)-Kaffeekränzchen. - Eine \emph{explizite K-Kaffeekasse} ist die Menge - \[ k = \{ \delta_{ij} \in \integers | i,j \in \naturals_{\leq n} , i < j \} \] + Eine \emph{explizite K-Kaffeekasse} ist die Matrix + \( \kappa \in \naturals^{n\times n} \) wobei gilt: + \begin{enumerate} + \item \( \delta_{i,i} = 0 \) für alle \( i \in \naturals_{\leq n} \) + \item \( \delta_{i,j} = -\delta_{j,i} \) für alle \( i,j \in \naturals_{\leq n} \) + \end{enumerate} 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.} \\ @@ -264,9 +271,73 @@ in der alle Kaffeeschulden innerhalb des Kaffeekränzchens einzeln ausgewiesen w \end{align*} \end{defi} +Für die Verwaltung der expliziten Kaffeekasse definieren wir die Transistion +analog zu \autoref{def:kaffeekassentransition}. +\begin{defi}[Explizite Kaffeekassentransition] + \label{def:explizitekaffeekassentransition} + Sei \(n \in \naturals_{\geq 2}\), + \(K\) ein Kaffeekränzchen über \(n\), + \(\kappa\) eine explizite \(K\)-Kaffeekasse + sowie \(i \neq j \in {[n]}\) so, + dass \(p_i\) \(p_j\) einen Kaffee ausgibt. + Die explizite Kaffeekassentransition liefert zu \(\kappa\) + eine neue explizite Kaffeekasse \( \kappa' \in \integers^{n\times n} \) mit + \[ \delta_{k,l}' = \left\{ + \begin{array}{ll} + \delta_{k,l} - 1 & \text{, falls } k=i \text{ und } l=j \\ + \delta_{k,l} + 1 & \text{, falls } k=j \text{ und } l=i \\ + \delta_{k,l} & \text{, sonst} + \end{array} \right. + \] + für \(k,l \in \naturals_{\leq n}\). +\end{defi} + +\begin{satz} + Sei \(n \in \naturals\), \(K\) ein \(n\)-Kaffeekränzchen + und \( \kappa \) eine explizite \(K\)-Kaffeekasse. Dann existiert eine entsprechende, + bilanzierende \(K\)-Kaffeekasse. +\end{satz} +\begin{proof} + Sei \(n \in \naturals\), \(K\) ein \(n\)-Kaffeekränzchen + und \(\kappa\) eine explizite \(K\)-Kaffeekasse. + Dann existieren laut \autoref{def:explizitekaffeekasse} \(\delta_{i,j} \in \integers\) + für alle \(i,j \in \naturals_{\leq n}\) als Einträge in \(\kappa\). Setze nun + \[ \Delta_i = \sum_{l=1}^{n} \delta_{i,l} . \] + Dann ergibt sich die bilanzierende Kaffeekasse als + \( k = (\Delta_1,\ldots,\Delta_n) \). +\end{proof} + +\begin{beob}[Kaffeeparadoxon] + \label{beob:kaffeparadoxon} + Betrachten wir das 3-Kaffeekränzchen \( K = {p_1, p_2, p_3} \) + und ihre explizite Kaffeekasse \(\kappa\). Bisher hat \(p_2\) + jeweils einen Kaffee für \(p_1\) und \(p_3\) bezahlt. + Weiter hat \(p_3\) \(p_1\) einen Kaffee ausgegeben. + Somit ergibt sich + \[\kappa = \left( + \begin{array}{rrr} + 0 & 1 & 1 \\ + -1 & 0 & -1 \\ + -1 & 1 & 0 + \end{array} \right). + \] + Wie aus \autoref{def:explizitekaffeekassentransition} zu erkennen ist, + werden stets zwei symmetrische Komponenten der Matrix modifiziert, wenn + ein Kaffee ausgegeben wird. Somit müssen mindestens 3 Kaffees getrunken werden, + damit die Kaffeekasse wieder im schuldenfreien Zustand ist. + + Bilden wir zu \(\kappa\) nun die bilanzierende Kaffeekasse \(k\), + so ergibt sich \(k = (2,-2,0) \). + Hierbei ist zu erkennen, das nur 2 Kaffees benötigt werden, um die Kaffeeschulden + auszugleichen. + + Dieses Phänomen, welches wir als \emph{Kaffeeparadoxon} bezeichnen, lässt sich + in dem Auftreten von \emph{transitiven Kaffeeschulden} begründen. + Die bilanzierende Kaffeekasse vermeidet derartige Kaffeeschulden direkt, + während sie bei der expliziten Kaffeekasse manuell aufgelöst werden müssen. +\end{beob} -Hierdurch werden sogenannte \emph{transitive Kaffeeschulden} vermieden. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% Visualisierung des Kaffeeproblems