From 80416f0ab893f2b80a01be4acc13bd03c7a03682 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Thu, 20 May 2021 10:09:26 +0200
Subject: add new slides

---
 vorlesungen/slides/9/parrondo/spielbmod.tex | 91 +++++++++++++++++++++++++++++
 1 file changed, 91 insertions(+)
 create mode 100644 vorlesungen/slides/9/parrondo/spielbmod.tex

(limited to 'vorlesungen/slides/9/parrondo/spielbmod.tex')

diff --git a/vorlesungen/slides/9/parrondo/spielbmod.tex b/vorlesungen/slides/9/parrondo/spielbmod.tex
new file mode 100644
index 0000000..ee1d12d
--- /dev/null
+++ b/vorlesungen/slides/9/parrondo/spielbmod.tex
@@ -0,0 +1,91 @@
+%
+% spielb.tex -- slide template
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\bgroup
+\begin{frame}[t]
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\frametitle{Modifiziertes Spiel $B$}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.48\textwidth}
+\begin{block}{Definition}
+Gewinn $\pm 1$, Wahrscheinlichkeit abhängig vom 3er-Rest des
+aktuellen Kapitals $K$:
+\begin{center}
+\begin{tikzpicture}[>=latex,thick]
+\coordinate (A0) at (90:2);
+\coordinate (A1) at (210:2);
+\coordinate (A2) at (330:2);
+
+\node at (A0) {$0$};
+\node at (A1) {$1$};
+\node at (A2) {$2$};
+
+\draw (A0) circle[radius=0.4];
+\draw (A1) circle[radius=0.4];
+\draw (A2) circle[radius=0.4];
+
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A1);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A0) -- (A2);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) -- (A2);
+
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A1) to[out=90,in=-150] (A0);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=90,in=-30] (A0);
+\draw[->,shorten >= 0.4cm,shorten <= 0.4cm] (A2) to[out=-150,in=-30] (A1);
+
+\def\R{1.9}
+\def\r{0.7}
+
+\node at (30:{0.9*\r}) {\tiny $\frac{9}{10}+\varepsilon$};
+\node at (150:{0.9*\r}) {\tiny $\frac1{10}-\varepsilon$};
+\node at (270:\r) {$\frac34-\varepsilon$};
+
+\node at (30:{1.1*\R}) {$\frac{3}{4}-\varepsilon$};
+\node at (150:{1.1*\R}) {$\frac1{4}+\varepsilon$};
+\node at (270:\R) {$\frac14+\varepsilon$};
+
+\end{tikzpicture}
+\end{center}
+\end{block}
+\end{column}
+\begin{column}{0.48\textwidth}
+\begin{block}{Markov-Kette $\tilde{Y}$}
+Übergangsmatrix
+\[
+\tilde{B}=
+B+\varepsilon F
+=
+B+\varepsilon\begin{pmatrix*}[r]
+0&1&-1\\
+-1&0&1\\
+1&-1&0
+\end{pmatrix*}
+\]
+Gewinnmatrix:
+\[
+G=\begin{pmatrix*}[r]
+0&-1&1\\
+1&0&-1\\
+-1&1&0
+\end{pmatrix*}
+\]
+\end{block}
+\begin{block}{Gewinnerwartung}
+\begin{align*}
+E(\tilde{Y})
+&=
+U^t(G\odot \tilde{B})p
+\\
+&=
+E(Y) + \varepsilon U^t(G\odot F)p
+=
+\frac1{15}+2\varepsilon
+\end{align*}
+\end{block}
+\end{column}
+\end{columns}
+\end{frame}
+\egroup
-- 
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