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author | Andreas Müller <andreas.mueller@ost.ch> | 2021-06-14 07:26:10 +0200 |
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committer | GitHub <noreply@github.com> | 2021-06-14 07:26:10 +0200 |
commit | 114633b43a0f1ebedbc5dfd85f75ede9841f26fd (patch) | |
tree | 18e61c7d69883a1c9b69098b7d36856abaed5c1e /vorlesungen/slides/9/parrondo/spielbmod.tex | |
parent | Delete buch.pdf (diff) | |
parent | Fix references.bib (diff) | |
download | SeminarMatrizen-114633b43a0f1ebedbc5dfd85f75ede9841f26fd.tar.gz SeminarMatrizen-114633b43a0f1ebedbc5dfd85f75ede9841f26fd.zip |
Merge branch 'master' into master
Diffstat (limited to 'vorlesungen/slides/9/parrondo/spielbmod.tex')
-rw-r--r-- | vorlesungen/slides/9/parrondo/spielbmod.tex | 103 |
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diff --git a/vorlesungen/slides/9/parrondo/spielbmod.tex b/vorlesungen/slides/9/parrondo/spielbmod.tex new file mode 100644 index 0000000..66d39bc --- /dev/null +++ b/vorlesungen/slides/9/parrondo/spielbmod.tex @@ -0,0 +1,103 @@ +% +% 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 $\tilde{B}$} +\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 $\frac{9}{10}\uncover<2->{+\varepsilon}$}; +\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}) {$\frac{3}{4}\uncover<2->{-\varepsilon}$}; +\node at (150:{1.1*\R}) {$\frac1{4}\uncover<2->{+\varepsilon}$}; +\node at (270:\R) {$\frac14\uncover<2->{+\varepsilon}$}; + +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\begin{block}{Markov-Kette $\tilde{Y}$} +Übergangsmatrix +\[ +\tilde{B}= +B\uncover<2->{+\varepsilon F} +\uncover<3->{= +B+\varepsilon\begin{pmatrix*}[r] +0&1&-1\\ +-1&0&1\\ +1&-1&0 +\end{pmatrix*}} +\] +\vspace{-12pt} + +\uncover<4->{% +Gewinnmatrix: +\[ +G=\begin{pmatrix*}[r] +0&-1&1\\ +1&0&-1\\ +-1&1&0 +\end{pmatrix*} +\]} +\end{block} +\vspace{-12pt} +\uncover<5->{% +\begin{block}{Gewinnerwartung} +\begin{align*} +\uncover<6->{E(\tilde{Y}) +&= +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.} +&= +-{\textstyle\frac{294}{169}}\varepsilon+O(\varepsilon^2) +\quad\text{Verlustspiel} +} +\end{align*} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup |