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
-- 
cgit v1.2.1


From d3b772e811ac42cb912cce367b8e7bee07881084 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Andreas=20M=C3=BCller?= <andreas.mueller@ost.ch>
Date: Thu, 20 May 2021 20:24:26 +0200
Subject: add slides

---
 vorlesungen/slides/9/parrondo/spielbmod.tex | 46 ++++++++++++++++++-----------
 1 file changed, 29 insertions(+), 17 deletions(-)

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

diff --git a/vorlesungen/slides/9/parrondo/spielbmod.tex b/vorlesungen/slides/9/parrondo/spielbmod.tex
index ee1d12d..66d39bc 100644
--- a/vorlesungen/slides/9/parrondo/spielbmod.tex
+++ b/vorlesungen/slides/9/parrondo/spielbmod.tex
@@ -7,7 +7,7 @@
 \begin{frame}[t]
 \setlength{\abovedisplayskip}{5pt}
 \setlength{\belowdisplayskip}{5pt}
-\frametitle{Modifiziertes Spiel $B$}
+\frametitle{Modifiziertes Spiel $\tilde{B}$}
 \vspace{-20pt}
 \begin{columns}[t,onlytextwidth]
 \begin{column}{0.48\textwidth}
@@ -39,13 +39,13 @@ aktuellen Kapitals $K$:
 \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:{0.9*\r}) {\tiny $\frac{9}{10}\uncover<2->{+\varepsilon}$};
+\node at (150:{0.9*\r}) {\tiny $\frac1{10}\uncover<2->{-\varepsilon}$};
+\node at (270:\r) {$\frac34\uncover<2->{-\varepsilon}$};
 
-\node at (30:{1.1*\R}) {$\frac{3}{4}-\varepsilon$};
-\node at (150:{1.1*\R}) {$\frac1{4}+\varepsilon$};
-\node at (270:\R) {$\frac14+\varepsilon$};
+\node at (30:{1.1*\R}) {$\frac{3}{4}\uncover<2->{-\varepsilon}$};
+\node at (150:{1.1*\R}) {$\frac1{4}\uncover<2->{+\varepsilon}$};
+\node at (270:\R) {$\frac14\uncover<2->{+\varepsilon}$};
 
 \end{tikzpicture}
 \end{center}
@@ -56,14 +56,17 @@ aktuellen Kapitals $K$:
 Übergangsmatrix
 \[
 \tilde{B}=
-B+\varepsilon F
-=
+B\uncover<2->{+\varepsilon F}
+\uncover<3->{=
 B+\varepsilon\begin{pmatrix*}[r]
 0&1&-1\\
 -1&0&1\\
 1&-1&0
-\end{pmatrix*}
+\end{pmatrix*}}
 \]
+\vspace{-12pt}
+
+\uncover<4->{%
 Gewinnmatrix:
 \[
 G=\begin{pmatrix*}[r]
@@ -71,20 +74,29 @@ G=\begin{pmatrix*}[r]
 1&0&-1\\
 -1&1&0
 \end{pmatrix*}
-\]
+\]}
 \end{block}
+\vspace{-12pt}
+\uncover<5->{%
 \begin{block}{Gewinnerwartung}
 \begin{align*}
-E(\tilde{Y})
+\uncover<6->{E(\tilde{Y})
 &=
-U^t(G\odot \tilde{B})p
+U^t(G\odot \tilde{B})p}
 \\
+&\uncover<7->{=
+E(Y) + \varepsilon U^t(G\odot F)p}
+\uncover<8->{=
+{\textstyle\frac1{15}}+2\varepsilon}
+\\
+\uncover<9->{
+\text{rep.}
 &=
-E(Y) + \varepsilon U^t(G\odot F)p
-=
-\frac1{15}+2\varepsilon
+-{\textstyle\frac{294}{169}}\varepsilon+O(\varepsilon^2)
+\quad\text{Verlustspiel}
+}
 \end{align*}
-\end{block}
+\end{block}}
 \end{column}
 \end{columns}
 \end{frame}
-- 
cgit v1.2.1

