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%
% Spring 2007 Masters Thesis Defense: Optimal Foraging Theory Revisited
% 
% (c) Copyright 2007 by Theodore P. Pavlic
%

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% --------------------------- 72 characters ---------------------------|
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%%faketop{Title Information}
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\begin{document}

% Setup title page
\title{Optimal Foraging Theory Revisited\\%
{\small Masters Thesis Defense and Doctoral Qualifier Examination%
\fn{Typeset with \LaTeX{} using \texttt{powerdot}.}}}
\author{Theodore (Ted) Pavlic}
\date{June 5, 2007}

% Setup for citations
\bibliographystyle{plainnat}
\nobibliography{optimal_foraging_theory}

% Title Slide
\maketitle

%%faketop{Main Document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{slide}[toc=,bm=]{Overview}
        \tableofcontents[content=sections]
\end{slide}

%\begin{slide}{a slide} 
%\overtrans
%        Here are two columns.
%        \twocolumn
%        {
%                \onslide{1,2}{left}\\
%                \onslide{3}{done}
%        }{
%                \onslide{2,3}{right}
%        }
%        Those were two columns.
%\deftrans
%\end{slide}

\section[slide=false]{Introduction}
\begin{slide}[toc=,bm=]{Introduction}
%
\begin{itemize}
        \item<2-> Goal\onslide*{3-}{: Discover the physics of high-level
                control}
        \item<4-> How to model? 
        \item<5-> How to measure performance?
        \item<6-> High-level actuators are behaviors
        \item<7-> Behavioral ecology replaces physics
\end{itemize}
%
\end{slide}

\section{Solitary Agent Model}

\begin{slide}[toc=\ac{OFT} Cycle]{\ac{OFT} Renewal Cycle}
        \begin{figure}[!ht]\centering
        \begin{picture}(265,42)(0,0)
                \thinlines
                \put(0,25){\rnode{oftsearchleft}{\makebox(0,0){}}}
                \put(33,25){\makebox(0,0)[l]{\rnode{oftsearch}{Search}}}
                \ncline[nodesepB=3pt]{->}{oftsearchleft}{oftsearch}
                \put(100,25){\makebox(0,0)[l]{\rnode{oftencounter}{Encounter}}}
                \ncline[nodesep=3pt]{->}{oftsearch}{oftencounter}
                %
                \put(210,37){\makebox(0,0)[l]{\rnode{oftprocess}{Process}}}
                \ncdiag[nodesep=3pt,armA=1cm,armB=0cm,angleA=0,angleB=180]{->}{oftencounter}{oftprocess}
                \put(265,37){\rnode{oftprocessright}{\makebox(0,0){}}}
                \ncline[nodesepA=3pt]{-}{oftprocess}{oftprocessright}
                %
                \put(210,13){\makebox(0,0)[l]{\rnode{oftignore}{Ignore}}}
                \ncdiag[nodesep=3pt,armA=1cm,armB=0cm,angleA=0,angleB=180]{->}{oftencounter}{oftignore}
                \put(265,13){\rnode{oftignoreright}{\makebox(0,0){}}}
                \ncline[nodesepA=3pt]{-}{oftignore}{oftignoreright}
                %
                \ncline{-}{oftprocessright}{oftignoreright}
                \ncbar[angle=-90]{-}{oftignoreright}{oftsearchleft}
                %
                \put(27,33){\circle*{5}}
        \end{picture}
        \caption{Classical \ac{OFT} Markov Renewal Process}
        \label{fig:oft_markov_process}
\end{figure}
\onslide*{2-13}{\textbf{Parameters and Decision Variables}}
\onslide*{3,4,5}{\begin{itemize}
        \item[$n$:]<3-> Number of task types.
        \item[$\lambda_i$:]<4-> {\color{blue}Poisson} encounter rate with tasks of
                type $i$ (encounters/second).
        \item[$\lambda$:]<5-> {\color{darkgreen}Merged Poisson} encounter
                rate with all tasks (encounters/second).
\end{itemize}}
%
\onslide*{6}{\vspace{\stretch{1}}
\begin{equation*}
        \boxed{{\color{darkgreen}\lambda} = \sum\limits_{i=1}^n {\color{blue}\lambda_i}}
\end{equation*}
\vspace{\stretch{1}}}
%
\onslide*{7,8}{%\vspace{\stretch{1}}
\begin{itemize}
        \item[$c^s$:]<7-> Cost rate for searching (points/second).
        \item[$\frac{c^s}{\lambda}$:]<8-> Average cycle
                {\color{red}search} cost (points/encounter).
\end{itemize}
\vspace{\stretch{1}}}
%
\onslide*{9,10}{%\vspace{\stretch{1}}
\begin{itemize}
        \item[$p_i$:]<9-> Probability of processing a task of type $i$.
        \item[$\tau_i$:]<10-> Average time processing a task of type $i$ (seconds).
\end{itemize}
\vspace{\stretch{1}}}
%
\onslide*{11,12,13}{%\vspace{\stretch{1}}
\begin{itemize}
        \settowidth{\labelwidth}{\hfil$g_i(\tau_i) - c_i \tau_i$}%
        \item[$g_i(\tau_i)$:]<11-> Average gain from processing a task of
                type $i$ (points).
        \item[$c_i$:]<12-> Average cost rate while processing a task of
                type $i$ (points/second).
        \item[$g_i(\tau_i) - c_i \tau_i$:]<13> Average net gain from
                processing a task of type $i$ (points).
\end{itemize}
\vspace{\stretch{1}}}
%
\onslide*{14-18}{\textbf{Random Variables}}
%
\onslide*{15,16,17}{%\vspace{\stretch{1}}
\begin{itemize}
        \item[$g$:]<15-> Gross {\color{red}processing} gain from $k\th$ cycle.
        \item[$c$:]<16-> {\color{red}Processing} cost from $k\th$ cycle.
        \item[$\tau$:]<17-> {\color{red}Processing} time from $k\th$ cycle.
\end{itemize}
\vspace{\stretch{1}}}
%
\onslide*{18}{%\vspace{\stretch{1}}
\begin{itemize}
        \item[$\oft{G}_k$:] {\color{darkgreen}Total} net gain from $k\th$ cycle.
        \item[$\oft{C}_k$:] {\color{darkgreen}Total} cost from $k\th$ cycle.
        \item[$\oft{T}_k$:] {\color{darkgreen}Total} time from $k\th$ cycle.
\end{itemize}
\vspace{\stretch{1}}}
%
\onslide*{19,20}{\textbf{Random Processes}}
%
\onslide*{20}{\vspace{\stretch{1}}
\begin{equation*}
        \boxed{ \oft{G}^N \triangleq \sum\limits_{k=1}^N \oft{G}_k }
        \qquad
        \boxed{ \oft{C}^N \triangleq \sum\limits_{k=1}^N \oft{C}_k }
        \qquad
        \boxed{ \oft{T}^N \triangleq \sum\limits_{k=1}^N \oft{T}_k }
\end{equation*}
\vspace{\stretch{1}}}
%
\onslide*{21-25}{\textbf{Statistics}}
%
\onslide*{22}{\vspace{\stretch{1}}
\begin{align*}
        \overline{g} \triangleq \E\left(g\right) &= \sum\limits_{i=1}^n \frac{\lambda_i}{\lambda} p_i g_i(\tau_i)\\
        \overline{c} \triangleq \E\left(c\right) &= \sum\limits_{i=1}^n \frac{\lambda_i}{\lambda} p_i c_i \tau_i\\
        \overline{\tau} \triangleq \E\left(\tau\right) &= \sum\limits_{i=1}^n \frac{\lambda_i}{\lambda} p_i \tau_i
\end{align*}
\vspace{\stretch{1}}}
%
\onslide*{23}{\vspace{\stretch{1}}
\begin{equation*}
        \var\left(g-c\right) = \sum\limits_{i=1}^n \frac{\lambda_i}{\lambda} p_i ( g_i(\tau_i) - \overline{g} )^2
\end{equation*}
\vspace{\stretch{1}}}
%
\onslide*{24}{\vspace{\stretch{1}}
\begin{align*}
        \E\left( \oft{G}^N \right) = N \E\left( \oft{G}_1 \right) &= N \left( \overline{g} - \overline{c} - \frac{c^s}{\lambda} \right)\\
        \E\left( \oft{C}^N \right) = N \E\left( \oft{C}_1 \right) &= N \left( \overline{c} + \frac{c^s}{\lambda} \right)\\
        \E\left( \oft{T}^N \right) = N \E\left( \oft{T}_1 \right) &= N \left( \overline{\tau} + \frac{1}{\lambda} \right)
\end{align*}
\vspace{\stretch{1}}}
%
\onslide*{25}{\vspace{\stretch{1}}
\begin{equation*}
        \var\left( \oft{G}^N \right) = N \var\left( \oft{G}_1 \right) = N \left( \var\left(\overline{g} - \overline{c}\right) + \left( \frac{c^s}{\lambda} \right)^2 \right)
\end{equation*}
\vspace{\stretch{1}}}
%
\onslide*{26}{\vspace{\stretch{1}}
\begin{center}\textbf{Finite Lifetime?}\end{center}
\vspace{\stretch{1}}}
\end{slide}

\begin{slide}[toc=Processing Cycle]{Processing-Only Renewal Cycle}
        \begin{figure}[!ht]\centering
        \begin{picture}(230,30)(0,0)
                \thinlines
                \put(0,20){\rnode{searchleft}{\makebox(0,0){}}}
                \put(33,20){\makebox(0,0)[l]{\rnode{search}{Find and Process}}}
                \put(230,20){\rnode{searchright}{\makebox(0,0){}}}
                \ncline[nodesepB=3pt]{->}{searchleft}{search}
                \ncline[nodesepA=3pt]{-}{search}{searchright}
                \ncbar[angle=-90,arm=.65cm]{-}{searchright}{searchleft}
                %
                \put(27,28){\circle*{5}}
        \end{picture}
        \caption{Finite Task Lifetime Markov Renewal Process}
        \label{fig:markov_process}
\end{figure}
\onslide*{2-6}{\textbf{Split Poisson Parameters}}
\onslide*{3,4,5,6}{\begin{itemize}
        \item[$\lambda^p_i$:]<3-> {\color{blue}Split} Poisson encounter
                rate with {\color{blue}processed} tasks of type $i$
                (encounters/second).
        \item[$\lambda^p$:]<5-> {\color{darkgreen}Merged}
                {\color{blue}Split} Poisson encounter rate with all
                {\color{blue}processed} tasks (encounters/second).
\end{itemize}
\showall
\begin{equation*}
        \onslide{4-}{\boxed{\lambda^p_i \triangleq p_i \lambda_i}}
        \qquad \qquad
        \onslide{6-}{\boxed{\lambda^p \triangleq \sum\limits_{i=1}^n \lambda^p_i}}
\end{equation*}}
%
\onslide*{7-9}{\textbf{Random Variables}}
%
\onslide*{8}{\begin{itemize}
        \item[$g^p$:] Gross processing gain from $k\th$
                {\color{blue}processing} cycle.
        \item[$c^p$:] Processing cost from $k\th$
                {\color{blue}processing} cycle.
        \item[$\tau^p$:] Processing time from $k\th$
                {\color{blue}processing} cycle.
\end{itemize}
\vspace{\stretch{1}}}
%
\onslide*{9}{\begin{itemize}
        \item[$G_k$:] Total net gain from $k\th$
                {\color{blue}processing} cycle.
        \item[$C_k$:] Total cost from $k\th$ {\color{blue}processing}
                cycle.
        \item[$T_k$:] Total time from $k\th$ {\color{blue}processing}
                cycle.
\end{itemize}
\vspace{\stretch{1}}}
%
\onslide*{10,11,12}{\textbf{Random Processes}}
%
\onslide*{11}{\vspace{\stretch{1}}
\begin{equation*}
        \boxed{ G^{N^p} \triangleq \sum\limits_{k=1}^{N^p} G_k }
        \qquad
        \boxed{ C^{N^p} \triangleq \sum\limits_{k=1}^{N^p} C_k }
        \qquad
        \boxed{ T^{N^p} \triangleq \sum\limits_{k=1}^{N^p} T_k }
\end{equation*}
\vspace{\stretch{1}}}
%
\onslide*{12}{\vspace{\stretch{1}}
\begin{equation*}
        \boxed{ G^{\color{red}N^p} \triangleq \sum\limits_{k=1}^{{\color{red}N^p}} G_k }
        \qquad
        \boxed{ C^{\color{darkgreen}N^p} \triangleq \sum\limits_{k=1}^{{\color{darkgreen}N^p}} C_k }
        \qquad
        \boxed{ T^{\color{blue}N^p} \triangleq \sum\limits_{k=1}^{{\color{blue}N^p}} T_k }
\end{equation*}
\vspace{\stretch{1}}}
%
\onslide*{13-17}{\textbf{Statistics}}
%
\onslide*{14}{\vspace{\stretch{1}}
\begin{align*}
        \overline{g^p} \triangleq \E\left(g^p\right) &=
        \sum\limits_{i=1}^n \frac{\lambda^p_i}{\lambda^p} g_i(\tau_i)\\
        \overline{c^p} \triangleq \E\left(c^p\right) &=
        \sum\limits_{i=1}^n \frac{\lambda^p_i}{\lambda^p} c_i \tau_i\\
        \overline{\tau^p} \triangleq \E\left(\tau^p\right) &=
        \sum\limits_{i=1}^n \frac{\lambda^p_i}{\lambda^p} \tau_i
\end{align*}
\vspace{\stretch{1}}}
%
\onslide*{15}{\vspace{\stretch{1}}
\begin{equation*}
        \var\left(g^p-c^p\right) = \sum\limits_{i=1}^n
        \frac{\lambda^p_i}{\lambda^p} ( g_i(\tau_i) - \overline{g^p} )^2
\end{equation*}
\vspace{\stretch{1}}}
%
\onslide*{16}{\vspace{\stretch{1}}
\begin{align*}
        \E\left( G^{{\color{red}N^p}} \right) = {\color{red}N^p}
        \E\left( G_1 \right) &= {\color{red}N^p} \left( \overline{g^p} -
        \overline{c^p} - \frac{c^s}{\lambda^p} \right)\\
        \E\left( C^{{\color{darkgreen}N^p}} \right) =
        {\color{darkgreen}N^p} \E\left( C_1 \right) &=
        {\color{darkgreen}N^p} \left( \overline{c^p} +
        \frac{c^s}{\lambda^p} \right)\\
        \E\left( T^{{\color{blue}N^p}} \right) = {\color{blue}N^p}
        \E\left( T_1 \right) &= {\color{blue}N^p} \left(
        \overline{\tau^p} + \frac{1}{\lambda^p} \right)
\end{align*}
\vspace{\stretch{1}}}
%
\onslide*{17}{\vspace{\stretch{1}}
\begin{equation*}
        \var\left( G^{{\color{red}N^p}} \right) = {\color{red}N^p}
        \var\left( G_1 \right) = {\color{red}N^p} \left(
        \var\left(\overline{g^p} - \overline{c^p}\right) + \left(
        \frac{c^s}{\lambda^p} \right)^2 \right)
\end{equation*}
\vspace{\stretch{1}}}
%
\end{slide}

\section{Optimization}

\begin{slide}[toc=\ac{OFT} Rate]{Importance of Rate in Classical \ac{OFT}}
%
\onslide*{2,3,4,5,6,7,8,9,10,11}{
\begin{itemize}
        \item<9-> \onslide*{9}{{\color{darkgreen}Time maximizer}}\onslide*{10-}{Time maximizer} when net gain is negative
        \item<2-> \onslide*{2-5,7-}{Time minimizer}\onslide*{6}{{\color{red}Time minimizer}}\onslide*{9-}{ when net gain is positive}
        \item<3-> \onslide*{2-6,8-}{Net gain maximizer}\onslide*{7}{{\color{darkgreen}Net gain maximizer}}
        \item<4-> \onslide*{2-7,9-}{Lifetime}\onslide*{8}{{\color{blue}Lifetime}} pressure
\end{itemize}
\vspace{\stretch{1}}
\showall
\onslide{5,6,7,8,9,10,11}{\begin{equation*}
        \onslide*{5-7,9-}{\aslim\limits_{N \to \infty}} 
        \onslide*{8}{\aslim\limits_{N \to {\color{blue}\infty}}}
        \onslide*{5,8,10-}{\frac{ \oft{G}^N }{ \oft{T}^N }}
        \onslide*{6}{\frac{ \oft{G}^N }{\color{red}\oft{T}^N}}
        \onslide*{7}{\frac{\color{darkgreen}\oft{G}^N}{ \oft{T}^N }}
        \onslide*{9}{\frac{ \oft{G}^N }{\color{darkgreen}\oft{T}^N}}
        \onslide{10,11}{= \ovalnode[linecolor=blue]{oftrate}{\onslide*{5,8,10}{\frac{ \E\Bigl(\oft{G}_1\Bigr) }{ \E\Bigl(\oft{T}_1\Bigr) }}\onslide*{11}{\frac{ \E\Bigl({\color{magenta}G_1}\Bigr) }{\E\Bigl({\color{magenta}T_1}\Bigr) }}
        \onslide*{6}{\frac{ \E\Bigl(\oft{G}_1\Bigr) }{\color{red}\E\Bigl(\oft{T}_1\Bigr)}}
        \onslide*{7}{\frac{\color{darkgreen}\E\Bigl(\oft{G}_1\Bigr)}{ \E\Bigl(\oft{T}_1\Bigr) }}
        \onslide*{9}{\frac{ \E\Bigl(\oft{G}_1\Bigr) }{\color{darkgreen}\E\Bigl(\oft{T}_1\Bigr)}}} =}
        \onslide*{5-7,9-}{\lim\limits_{N \to \infty}} 
        \onslide*{8}{\lim\limits_{N \to {\color{blue}\infty}}}
        \onslide*{5,8,10-}{\E\left( \frac{ \oft{G}^N }{ \oft{T}^N } \right)}
        \onslide*{6}{\E\left( \frac{ \oft{G}^N }{\color{red}\oft{T}^N} \right)}
        \onslide*{7}{\E\left( \frac{\color{darkgreen}\oft{G}^N}{ \oft{T}^N } \right)}
        \onslide*{9}{\E\left( \frac{ \oft{G}^N }{\color{darkgreen}\oft{T}^N} \right)}
\end{equation*}}
\vspace{\stretch{1}}}
%
\end{slide}

\begin{wideslide}[toc=\ac{OFT} Optimization]{Rate Maximization in Classical \ac{OFT}}
\vspace{\stretch{1}}
        \begin{figure}[!ht]\centering
        \begin{picture}(100,109)(0,0)
                \onslide{2-}%
                {%
                \put(50, 54.5){%
                \makebox(0,0){\scalebox{0.9}{\framebox(87.5,95){
                \shortstack[c]%
                {%
                \onslide{4-}%
                  {$\E(\oft{G}_1) = g_1(\tau_1)$}\\%
                \onslide{4-}%
                  {$\E(\oft{T}_1) = \tau_1 + \frac{1}{\lambda}$}\\%
                \onslide*{2-4}%
                  {$\oft{\gamma} \triangleq \frac{\E(\oft{G}_1)}%
                   {\E(\oft{T}_1)}$\\}%
                \onslide*{5-}%
                  {$\oft{\gamma} \triangleq \frac{g_1(\tau_1)}%
                   {\tau_1 + \frac{1}{\lambda}}$\\}%
                \onslide{3-}%
                  {$\lambda = \lambda_1$, $p_1 = 1$\\}%
                \onslide{10-}%
                  {$\lozenge^* \triangleq
                   \max\left\{\lozenge\right\}$}}}}}}%
                }%
        \end{picture}
        \onslide+{6-}{%
        \begin{picture}(220,111)(-72,-22)
                %\put(-72,-22){\framebox(220,111){}}
                % Horizontal Axis
                \thicklines
                \onslide*{5-}%
                  {\put(-70,0){\vector(1,0){184}}}
                % Horizontal axis has 1/lambda for these
                \onslide*{5-11}%
                  {\put(115,0){\makebox(0,0)[l]{$\tau_1+\frac{1}{\lambda}$}}}
                % Horizontal axis has no 1/lambda for these
                \onslide*{12-}%
                  {\put(115,0){\makebox(0,0)[l]{$\tau_1\phantom{{}+\frac{1}{\lambda}}$}}}
                  %{\put(115,0){\makebox(0,0)[l]{$\tau_1$}}}
                % Vertical Axis
                \thicklines
                % Vertical axis starts on the left 
                \onslide*{5-11}%
                  {\put(-24,-16){\vector(0,1){90}} 
                   \put(-24,75){\makebox(0,0)[b]{$g_1(\tau_1)$}}}
                % Vertical axis shifts to the right
                \onslide*{12-}%
                  {\put(0,-16){\vector(0,1){90}} 
                   \put(0,75){\makebox(0,0)[b]{$g_1(\tau_1)$}}}
                % Everything else - thinlines
                \thinlines
                %% These slides occur before shift in y axis
                %% (i.e., shows location of 1/lambda)
                \onslide*{6-11}
                  {\put(0,-3){\line(0,1){6}}
                   \put(0,-6){\makebox(0, 0)[t]{$\frac{1}{\lambda}$}}}
                %% This slide shows tau-vs-g curve
                \onslide*{6-}%
                  {\qbezier(0,0)(0,5)(5,10)
                   \qbezier(5,10)(14,19)(20,22)
                   \qbezier(20,22)(36,30)(51,35)
                   \qbezier(51,35)(66,40)(78,40)
                   \qbezier(78,40)(81.752,40)(86.2,38)
                   \qbezier(86.2,38)(108.44,28)(114,10)}
                %% This slide shows the direction of increasing tau
                \onslide*{7}%
                  {\put(0,0){\rnode{oftcurvestart}{\makebox(0,0){}}}%
                   \put(5,10){\rnode{oftcurvealong}{\makebox(0,0){}}}%
                   \nccurve[linecolor=blue,linewidth=\unitlength,angleA=90,angleB=225]{->}{oftcurvestart}{oftcurvealong}}
                %% These slides show a generic rate (as slope and line) at generic taus
                \onslide*{8,9}%
                  {\put(-24,0){\line(29,10){138}}         % Line  
                   \put(34,20){\line(1,0){47.125}}        % Slope
                   \put(81.125,20){\line(0,1){16.25}}
                   \put(79.125,22){\makebox(0,0)[rb]{$\oft{\gamma}$}}}
                %% This slide shows generic rate at small tau
                \onslide*{8}%
                  {\put(-27,10){\line(1,0){6}}%
                   \put(5,-3){\line(0,1){6}}%
                   \linethickness{3\unitlength}%
                   \put(-24,0){\rnode{oftgline}{\color{darkgreen}\line(0,1){10}}}%
                   \put(-40,-10){\rnode{oftgnote}{\makebox(0,0)[r]{\color{darkgreen}$g_1({\color{red}\tau_1})$}}}%
                   \nccurve[linecolor=darkgreen,angleA=35,angleB=195]{<-}{oftgnote}{oftgline}%
                   \put(0,0){\rnode{ofttline}{\color{red}\line(1,0){5}}}%
                   \put(60,-10){\rnode{ofttnote}{\makebox(0,0)[l]{\color{red}$\accentset{ }{\tau}_1$}}}%
                   \nccurve[linecolor=red,angleA=190,angleB=-90]{<-}{ofttnote}{ofttline}%
                   \thinlines%
                   \put(5,10){\circle*{3}}}
                %% This slide shows same generic rate at large tau
                \onslide*{9}%
                  {\put(-27,38){\line(1,0){6}}%
                   \put(86.2,-3){\line(0,1){6}}%
                   \linethickness{3\unitlength}%
                   \put(-24,0){\rnode{oftgline}{\color{darkgreen}\line(0,1){38}}}%
                   \put(-40,-10){\rnode{oftgnote}{\makebox(0,0)[r]{\color{darkgreen}$g_1({\color{red}\tau_1})$}}}%
                   \nccurve[linecolor=darkgreen,angleA=35,angleB=195]{<-}{oftgnote}{oftgline}%
                   \put(0,0){\rnode{ofttline}{\color{red}\line(1,0){86.2}}}%
                   \put(60,-10){\rnode{ofttnote}{\makebox(0,0)[l]{\color{red}$\accentset{ }{\tau}_1$}}}%
                   \nccurve[linecolor=red,angleA=190,angleB=-90]{<-}{ofttnote}{ofttline}%
                   \thinlines%
                   \put(86.2,38){\circle*{3}}}
                %% These slides shows g and tau at optimum point
                \onslide*{10,11}%
                  {\put(-27,22){\line(1,0){6}}
                   \linethickness{3\unitlength}%
                   \put(-24,0){\rnode{oftghatline}{\color{darkgreen}\line(0,1){22}}}%
                   \put(-40,-10){\rnode{oftghatnote}{\makebox(0,0)[r]{\color{darkgreen}$g_1({\color{red}\hat{\tau}_1})$}}}%
                   \nccurve[linecolor=darkgreen,angleA=35,angleB=195]{<-}{oftghatnote}{oftghatline}%
                   \put(0,0){\rnode{oftthatline}{\color{red}\line(1,0){20}}}%
                   \put(60,-10){\rnode{oftthatnote}{\makebox(0,0)[l]{\color{red}$\hat{\tau}_1$}}}%
                  \nccurve[linecolor=red,angleA=190,angleB=-90]{<-}{oftthatnote}{oftthatline}%
                   \thinlines}
                %% Marks position of maximum tau
                \onslide*{10-13,17,19-}%
                  {\put(20,-3){\rnode{ofttau}{\line(0,1){6}}}}
                %% Shows slope that is optimal rate
                \onslide*{10-13,15}%
                  {\put(-24, 0){\line(2,1){138}}        % Optimal Line
                   \put(74,67){\line(1,0){36}}          % Optimal Slope
                   \put(74,49){\line(0,1){18}}}
                %% Shows optimal rate as slope label
                %% Also shows optimal point
                \onslide*{10-12,15}%
                  {\put(20,22){\circle*{3}}                             % Optimal point
                   \put(76,65.5){\makebox(0,0)[lt]{$\oft{\gamma}^*$}}}  % Optimal slope label
                %% Shows the location of maximum of g
                \onslide*{11}%
                  {\put(-27,40){\line(1,0){6}}
                   \put(-29,40){\makebox(0,0)[r]{$g_1^*$}}
                   \put(-24,40){\dashbox{3}(102,0){}}}
                %% These slides occur just after shift in y axis
                \onslide*{12}%
                  {\put(-5,22){\makebox(0,0)[r]{$g_1(\hat{\tau}_1)$}}
                   \put(20,-6){\makebox(0,0)[t]{$\hat{\tau}_1$}}}
                %% Labels -1/lambda
                \onslide*{12,13}
                  {\put(-24,-6){\makebox(0, 0)[t]{$\frac{-1}{\lambda}$}}}
                %% Marks position of optimal gain at lambda 
                %% (lambda is also marked)
                \onslide*{12-13,17,19-}%
                  {\put(-3,22){\rnode{oftg}{\line(1,0){6}}}
                   \put(-24,-3){\rnode{oftlambda}{\line(0,1){6}}}}
                \onslide*{15,16}%
                  {\put(-24,-3){\rnode{oftlambda}{\line(0,1){6}}}}
                %% MVT illustration slide
                \onslide*{13}%
                  {\put(76,65.5){\makebox(0,0)[lt]{\color{blue}$\oft{\gamma}^*$}}%
                   \put(-5,22){\makebox(0,0)[r]{$g_1({\color{red}\hat{\tau}_1})$}}%
                   \put(20,-6){\makebox(0,0)[t]{\color{red}$\hat{\tau}_1$}}%
                   \put(20,22){\color{purple}\circle*{3}}%
                   \put(-70,60){\makebox(70,0)[t]{\shortstack{{\color{purple}\ac{MVT}}\\%
                        ${\color{blue}g_1'(}{\color{red}\hat{\tau}_1}{\color{blue})}={\color{blue}\oft{\gamma}^*}$}}}}
                %% Motivate question of how picture changes with
                %% increasing lambda
                \onslide*{14}
                  {\put(-70,52){\makebox(70,0)[t]{$\phantom{\lambda^- < {}} \lambda < \lambda^+$}}%
                   \put(0,4.5){\makebox(114,0)[b]{$\operatorname*{\gtreqless}\limits^\text{\ac{MVT}?}$}}}
                %% Shows shift in optimization from increasing lambda
                \onslide*{15-17}
                  {% Label for this slide
                   \put(-70,52){\makebox(70,0)[t]{$\phantom{\lambda^- < {}} \lambda < {\color{purple}\lambda^+}$}}%
                   % Shifted lambda (old lambda at (-24,0))
                   \put(-5,-3){\rnode{oftlambdaplus}{\line(0,1){6}}}
                   \put(-14.5,-6){\rnode{oftlambdaplusbox}{\makebox(0,0){}}}
                   \nccurve[linecolor=purple,angleA=-45,angleB=180]{-}{oftlambda}{oftlambdaplusbox}%
                   \nccurve[linecolor=purple,angleA=0,angleB=-135]{->}{oftlambdaplusbox}{oftlambdaplus}%
                   \put(-14.5,-7){\makebox(0,0)[t]{\scalebox{0.75}{$\frac{-1}{\color{purple}\lambda^+}$}}}
                   \onslide*{16,17}
                     {% New optimal slope/rate
                      \put(-5,0){\line(1,1){69}}   % New rate line
                      % Position of new optimal point
                      \put(5,10){\circle*{3}}}
                   \onslide*{16}
                     {% Optimal slope and optimal rate
                      \put(32,37){\line(0,1){30}}  % Optimal slope
                      \put(32,67){\line(1,0){30}}
                      \put(34,65){\makebox(0,0)[lt]{$\oft{\gamma}^*_+$}}} % Optimal rate label
                   \onslide*{17}
                     {% Shifted tau (old tau at (20,0))
                      \put(5,-3){\rnode{ofttauplus}{\line(0,1){6}}}
                      \put(12.5,-6){\rnode{ofttauplusbox}{\makebox(0,0){}}}
                      \nccurve[linecolor=red,angleA=-135,angleB=0]{-}{ofttau}{ofttauplusbox}%
                      \nccurve[linecolor=red,angleA=180,angleB=-45]{->}{ofttauplusbox}{ofttauplus}%
                      \put(12.5,-7){\makebox(0,0)[t]{\scalebox{0.75}{$\hat{\tau}^+_1$}}}
                      % Shifted g (old g at (0,22))
                      \put(-3,10){\rnode{oftgplus}{\line(1,0){6}}}
                      \put(-6,16){\rnode{oftgplusbox}{\makebox(0,0){}}}
                      \nccurve[linecolor=green,angleA=-145,angleB=90]{-}{oftg}{oftgplusbox}%
                      \nccurve[linecolor=green,angleA=-90,angleB=145]{->}{oftgplusbox}{oftgplus}%
                      \put(-7,16){\makebox(0,0)[r]{\scalebox{0.75}{$g_1(\hat{\tau}^+_1)$}}}
                      % Optimal slope and optimal rate
                      \put(32,37){\line(0,1){30}}  % Optimal slope
                      \put(32,67){\line(1,0){30}}
                      \put(34,65){\makebox(0,0)[lt]{\color{blue}$\oft{\gamma}^*_+$}} % Optimal rate label
                      % Label change in rate
                      \put(0,8){\makebox(114,0)[b]{$\phantom{\oft{\gamma}^*_- < {}} \oft{\gamma} < {\color{blue}\oft{\gamma}^*_+}$}}}}
                %% Motivate question of how picture changes with
                %% decreasing lambda
                \onslide*{18}
                  {\put(-70,52){\makebox(70,0)[t]{$\lambda^- < \lambda \phantom{{} < \lambda^+}$}}%
                   \put(0,4.5){\makebox(114,0)[b]{$\operatorname*{\gtreqless}\limits^\text{\ac{MVT}?}$}}}
                %% Shows shift in optimization from increasing lambda
                \onslide*{19}
                  {% Shifted lambda (old lambda at (-24,0))
                   \put(-54,-3){\rnode{oftlambdaminus}{\line(0,1){6}}}
                   \put(-39,-6){\rnode{oftlambdaminusbox}{\makebox(0,0){}}}
                   \nccurve[linecolor=purple,angleA=-135,angleB=0]{-}{oftlambda}{oftlambdaminusbox}%
                   \nccurve[linecolor=purple,angleA=180,angleB=-45]{->}{oftlambdaminusbox}{oftlambdaminus}%
                   \put(-39,-7){\makebox(0,0)[t]{\scalebox{0.75}{$\frac{-1}{\color{purple}\lambda^-}$}}}
                   % Shifted tau (old tau at (20,0))
                   \put(51,-3){\rnode{ofttauminus}{\line(0,1){6}}}
                   \put(35.5,-6){\rnode{ofttauminusbox}{\makebox(0,0){}}}
                   \nccurve[linecolor=red,angleA=-45,angleB=180]{-}{ofttau}{ofttauminusbox}%
                   \nccurve[linecolor=red,angleA=0,angleB=-135]{->}{ofttauminusbox}{ofttauminus}%
                   \put(35.5,-7){\makebox(0,0)[t]{\scalebox{0.75}{$\hat{\tau}^-_1$}}}
                   % Shifted g (old g at (0,22))
                   \put(-3,35){\rnode{oftgminus}{\line(1,0){6}}}
                   \put(-6,28.5){\rnode{oftgminusbox}{\makebox(0,0){}}}
                   \nccurve[linecolor=green,angleA=145,angleB=-90]{-}{oftg}{oftgminusbox}%
                   \nccurve[linecolor=green,angleA=90,angleB=-145]{->}{oftgminusbox}{oftgminus}%
                   \put(-7,28.5){\makebox(0,0)[r]{\scalebox{0.75}{$g_1(\hat{\tau}^-_1)$}}}
                   % Position of new optimal point
                   \put(51,35){\circle*{3}}
                   % New optimal slope/rate
                   \put(-54,0){\line(3,1){168}}   % New rate line
                   \put(57,37){\line(0,1){18}}    % Optimal slope
                   \put(57,55){\line(1,0){54}}
                   \put(59,53){\makebox(0,0)[lt]{\color{blue}$\oft{\gamma}^*_-$}}  % Optimal slope label
                   % Label for this slide
                   \put(-70,52){\makebox(70,0)[t]{${\color{purple}\lambda^-} < \lambda \phantom{{} < \lambda^+}$}}%
                   \put(0,8){\makebox(114,0)[b]{${\color{blue}\oft{\gamma}^*_-} < \oft{\gamma} \phantom{{} < \oft{\gamma}^*_+}$}}}
        \end{picture}}
        \caption{Visualization of Classical OFT Rate Maximization}
        \label{fig:oft_rate_maximization}
        \end{figure}
\vspace{\stretch{1}}
\end{wideslide}

\begin{wideslide}[toc=\ac{OFT} \& Pareto]{Efficient \ac{OFT} Rate Maximization}
\vspace{\stretch{1}}
\onslide*{2-}{\begin{equation*}
        \overline{\tau}
        \triangleq 
        \onslide*{2,3}
          {\sum\limits_{i=1}^n \frac{\lambda_i}{\lambda} p_i \tau_i}
        \onslide*{4}
          {\sum\limits_{i=1}^n \rnode{lambdailambda1}{\color{red}\frac{\lambda_i}{\lambda}} p_i \tau_i}
        \onslide*{5}
          {\sum\limits_{i=1}^n \rnode{lambdailambda4}{\color{darkgreen}\Lambda_i} p_i \tau_i}
        \onslide*{6-}
          {\sum\limits_{i=1}^n \Lambda_i p_i \tau_i}
        \qquad
        \overline{g}
        \triangleq
        \onslide*{2,3}
          {\sum\limits_{i=1}^n \frac{\lambda_i}{\lambda} p_i g_i(\tau_i)}
        \onslide*{4}
          {\sum\limits_{i=1}^n \rnode{lambdailambda2}{\color{red}\frac{\lambda_i}{\lambda}} p_i g_i(\tau_i)}
        \onslide*{5}
          {\sum\limits_{i=1}^n \rnode{lambdailambda5}{\color{darkgreen}\Lambda_i} p_i g_i(\tau_i)}
        \onslide*{6-}
          {\sum\limits_{i=1}^n \Lambda_i p_i g_i(\tau_i)}
        \qquad
        \overline{c}
        \triangleq 
        \onslide*{2,3}
          {\sum\limits_{i=1}^n \frac{\lambda_i}{\lambda} p_i c_i \tau_i}
        \onslide*{4}
          {\sum\limits_{i=1}^n \rnode{lambdailambda3}{\color{red}\frac{\lambda_i}{\lambda}} p_i c_i \tau_i}
        \onslide*{5}
          {\sum\limits_{i=1}^n \rnode{lambdailambda6}{\color{darkgreen}\Lambda_i} p_i c_i \tau_i}
        \onslide*{6-}
          {\sum\limits_{i=1}^n \Lambda_i p_i c_i \tau_i}
        \onslide*{4}{\ncbar[linecolor=red,angle=90]{lambdailambda1}{lambdailambda2}\ncbar[linecolor=red,angle=90]{lambdailambda2}{lambdailambda3}}
        \onslide*{5}{\ncbar[linecolor=darkgreen,angle=90,armA=.5cm,armB=.5cm]{<->}{lambdailambda4}{lambdailambda5}\ncbar[linecolor=darkgreen,angle=90,armA=.5cm,armB=.5cm]{->}{lambdailambda5}{lambdailambda6}}
\end{equation*}
\vspace{\stretch{1}}}
        \begin{figure}[!ht]\centering
        \begin{picture}(100,109)(0,0)
                \put(50, 54.5){%
                \makebox(0,0){\scalebox{0.9}{\framebox(103,95){%
                \shortstack[c]%
                {%
                \onslide{3-}%
                {$\E(\oft{G}_1) = \overline{g}-\overline{c}-\frac{c^s}{\lambda}$}\\%
                \onslide{3-}%
                  {$\E(\oft{T}_1) = \overline{\tau} +  \frac{1}{\lambda}$}\\%
                \onslide*{1,2}%
                  {$\oft{\gamma} \triangleq \frac{\E(\oft{G}_1)}%
                   {\E(\oft{T}_1)}$\\}%
                \onslide*{3-}%
                  {$\oft{\gamma} \triangleq \frac{\overline{g}-\overline{c}-\frac{c^s}{\lambda}}%
                  {\overline{\tau} + \frac{1}{\lambda}}$\\}%
                \onslide{14-}%
                  {$\lozenge^* \triangleq
                   \max\left\{\lozenge\right\}$}}}}}}%
        \end{picture}
        \onslide+{7-}{%
        \begin{picture}(220,121)(-72,-22)
                %\put(-72,-22){\framebox(220,111){}}
                % Horizontal Axis
                \thicklines
                \onslide*{6-12}%
                  {\put(-70,10){\vector(1,0){184}}}
                \onslide*{13-}%
                  {\put(-70,0){\vector(1,0){184}}}
                % Horizontal axis has 1/lambda for these
                \onslide*{6-12}%
                  {\put(115,10){\makebox(0,0)[l]{$\overline{\tau}+\frac{1}{\lambda}$}}}
                % Horizontal axis has no 1/lambda for these
                \onslide*{13-}%
                  {\put(115,0){\makebox(0,0)[l]{$\overline{\tau}\phantom{{}+\frac{1}{\lambda}}$}}}
                  %{\put(115,0){\makebox(0,0)[l]{$\tau_1$}}}
                % Vertical Axis
                \thicklines
                % Vertical axis starts on the left 
                \onslide*{6-12}%
                  {\put(-10,-16){\vector(0,1){100}} 
                   \put(-10,85){\makebox(0,0)[b]{$\overline{g}-\overline{c}-\frac{c^s}{\lambda}$}}}
                % Vertical axis shifts to the right
                \onslide*{13-}%
                  {\put(0,-16){\vector(0,1){100}} 
                   \put(0,85){\makebox(0,0)[b]{$\overline{g}-\overline{c}$}}}
                % Everything else - thinlines
                \thinlines
                \onslide*{7-11}
                {
                  \onslide*{7-}
                    {\put(15,30){\circle*{2}}}
                  \onslide*{8-}
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                  \onslide*{9-}
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                  \onslide*{10-}
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                     \put(35.04,7.81){\circle*{2}}
                     \put(41.04,52.42){\circle*{2}}
                     \put(62.7,13.4){\circle*{2}}
                     \put(54.86,8.83){\circle*{2}}
                     \put(38.94,50.67){\circle*{2}}
                     \put(33.34,46.23){\circle*{2}}
                     \put(51.98,14.64){\circle*{2}}
                     \put(58.03,4.75){\circle*{2}}
                     \put(52.87,17.94){\circle*{2}}
                     \put(49.49,21.93){\circle*{2}}
                     \put(111.51,16.44){\circle*{2}}
                     \put(43.84,39.79){\circle*{2}}
                     \put(37.85,36.26){\circle*{2}}
                     \put(82.95,52.81){\circle*{2}}
                     \put(45.37,9.18){\circle*{2}}
                     \put(48.44,28.82){\circle*{2}}
                     \put(37.45,60.79){\circle*{2}}
                     \put(71.29,21.6){\circle*{2}}
                     \put(86.14,30.35){\circle*{2}}
                     \put(46.45,17.11){\circle*{2}}
                     \put(111.32,16.8){\circle*{2}}
                     \put(71.11,56.42){\circle*{2}}
                     \put(67.48,53.19){\circle*{2}}
                     \put(32.4,41.25){\circle*{2}}
                     \put(70.81,27.1){\circle*{2}}
                     \put(66.51,32.31){\circle*{2}}
                     \put(75.89,39.91){\circle*{2}}
                     \put(40.44,62.73){\circle*{2}}
                     \put(99.53,43.03){\circle*{2}}
                     \put(37.86,55.92){\circle*{2}}
                     \put(44.86,3.95){\circle*{2}}
                     \put(41.72,7.35){\circle*{2}}
                     \put(53.09,15.18){\circle*{2}}
                     \put(114,5){\circle*{2}}}}
                %% These slides occur before shift in y axis
                %% (i.e., shows location of 1/lambda)
                \onslide*{11-12}
                  {\put(0,7){\line(0,1){6}}
                   \put(1,14){\makebox(0,0)[rb]{$\frac{1}{\lambda}$}}
                   \put(-13,0){\line(1,0){6}}
                   \put(-14,0){\makebox(0,0)[r]{$\frac{-c^s}{\lambda}$}}
                   }
                %% This slide shows tau-vs-g curve
                \onslide*{11-}%
                  {\qbezier(0,0)(0,22.5)(30,60)
                   \qbezier(30,60)(38,70)(50,70)
                   \qbezier(50,70)(70,70)(114,38)}
                %% Labels -1/lambda and c^s/lambda
                \onslide*{14}
                  {\put(-10,-3){\line(0,1){6}}
                   \put(-10,-6){\makebox(0, 0)[t]{$\frac{-1}{\lambda}$}}
                   \put(-3,10){\line(1,0){6}}
                   \put(6,10){\makebox(0, 0)[l]{$\frac{c^s}{\lambda}$}}
                %% Shows slope that is optimal rate
                   \put(0,0){\rnode{oftcssource}{\makebox(0,0){}}}
                   \put(-60,60){\rnode{oftcstarget}{\makebox(0,0){}}}
                   \ncline[linestyle=dotted]{->}{oftcssource}{oftcstarget}
                   \put(-32,32){\line(0,-1){17}}           % c^s Dotted Line Slope
                   \put(-32,15){\line(1,0){17}}
                   \put(-31,16){\makebox(0,0)[lb]{\scalebox{0.5}{$-c^s$}}}
                   \put(-10,10){\circle*{2}}
                   \put(-10,10){\line(8,10){51.2}}      % Optimal Line
                   \put(6,30){\line(0,1){25}}           % Optimal Slope
                   \put(6,55){\line(1,0){20}}
                %% Shows optimal rate as slope label
                %% Also shows optimal point
                   \put(30,60){\circle*{3}}                             % Optimal point
                   \put(8,53){\makebox(0,0)[lt]{$\oft{\gamma}^*$}}}  % Optimal slope label
                %% Labels -1/lambda and c^s/lambda
                \onslide*{15,16}
                  {\put(-10,-3){\line(0,1){6}}
                   \put(-10,-6){\makebox(0, 0)[t]{$\frac{-1}{\lambda}$}}
                   \put(-3,70){\line(1,0){6}}
                   \put(6,69){\makebox(0, 0)[lt]{$\frac{c^s}{\lambda}$}}
                %% Shows slope that is optimal rate
                   \put(0,0){\rnode{oftcssourcenew}{\makebox(0,0){}}}
                   \put(-12,84){\rnode{oftcstargetnew}{\makebox(0,0){}}}
                   \ncline[linestyle=dotted]{->}{oftcssourcenew}{oftcstargetnew}
                   \onslide*{15}
                     {\put(0,0){\rnode{oftcssourceold}{\makebox(0,0){}}}
                      \put(-60,60){\rnode{oftcstargetold}{\makebox(0,0){}}}
                      \ncline[linestyle=dotted]{->}{oftcssourceold}{oftcstargetold}
                      \nccurve[linecolor=blue,angleA=45,angleB=187]{->}{oftcstargetold}{oftcstargetnew}
                      \put(-9,11){\makebox(0,0)[lb]{\scalebox{0.35}{\color{blue}$-c^s$}}}}
                   \put(-9.5,66.5){\line(0,-1){56}}           % c^s Dotted Line Slope
                   \put(-9.5,10.5){\line(1,0){8}}
                   \onslide*{16}
                     {\put(-9,11){\makebox(0,0)[lb]{\scalebox{0.35}{$-c^s$}}}
                      \put(-10,70){\circle*{2}}
                      \put(-10,70){\line(1,0){124}}        % Optimal Line
                %% Shows optimal rate as slope label
                %% Also shows optimal point
                      \put(50,70){\circle*{3}}                              % Optimal point
                      \put(50,73){\makebox(0,0)[b]{$\oft{\gamma}^* = 0$}}}} % Optimal slope label
        \end{picture}}
        \caption{Visualization of Generalized OFT Rate Maximization}
        \label{fig:efficient_rate_maximization}
        \end{figure}
\vspace{\stretch{1}}
\end{wideslide}

\begin{slide}[toc=Finite Approach]{Finite-Lifetime Approach}
        \begin{itemize}
                \item<2-> Lifetime ends after $N^p$ {\onslide*{2}{\color{red}}processed} tasks
                \item<3-> Lifetime success threshold of $G^T$ points
                \item<4-> Average cycle success threshold of $\frac{G^T}{N^p}$ points
                \item<5-> Blends rate maximization and risk sensitivity
        \end{itemize}
\end{slide}

\begin{wideslide}[toc=Finite Rate]{Finite-Lifetime Rate Maximization}
\vspace{\stretch{1}}
\begin{figure}[!ht]\centering
        \begin{picture}(155,155)(0,0)
                \put(77.5,77.5){\makebox(0,0){\scalebox{0.8}{\framebox(155,85){
                \shortstack[c]%
                {$\mu \triangleq \E(G_1) = %
                \overline{g^p} - \overline{c^p} - \frac{c^s}{\lambda^p}%
                $\\%
                $\E(T_1) = \overline{\tau^p} + \frac{1}{\lambda^p}$\\%
                $\gamma \triangleq \frac{\mu - %
                \frac{G^T}{N^p}}{\overline{\tau^p}%
                + \frac{1}{\lambda^p}}$\\%
                $\lozenge^* \triangleq \max\left\{\lozenge\right\}$}}}}}
        \end{picture}
        \begin{picture}(182,155)(-54,-20)
                % Horizontal Axis
                \thicklines
                \put(-30, 0){\vector(1, 0){144}}
                \put(115, 0){\makebox(0, 0)[l]{$\overline{\tau^p}$}}
                % Vertical Axis
                \thicklines
                \put(0, -16){\vector(0, 1){136}}
                \put(0, 121){\makebox(0, 0)[b]%
                {$\overline{g^p}-\overline{c^p}$}}
                % Rate Line
                \thinlines
                \put(-12, 92){\line(2,-1){132}}
                \put(84, 44){\circle*{5}}
                % Maximal Rate (Slope)
                \thinlines
                \put(20,76){\line(1,0){34}}
                \put(54,76){\line(0,-1){17}}
                \put(53,75){\makebox(0, 0)[rt]{$\gamma^*$}}
                % Threshold Value
                \thicklines
                \put(-4, 80){\line(1,0){8}}
                \put(-5, 80){\makebox(0, 0)[rt]%
                {\scalebox{0.8}{$\frac{G^T}{N^p}$}}}
                \put(-4, 92){\line(1,0){8}}
                \put(5, 90){\makebox(0, 0)[lb]%
                {$\frac{G^T}{N^p}+\frac{c^s}{\lambda^p}$}}
                \put(0, 80){\circle*{2}}
                \put(-2, 82){\circle*{1}}
                \put(-4, 84){\circle*{1}}
                \put(-6, 86){\circle*{1}}
                \put(-8, 88){\circle*{1}}
                \put(-10, 90){\circle*{1}}
                \put(-12, 92){\circle*{3}}
                \put(-14, 94){\circle*{1}}
                \put(-16, 96){\circle*{1}}
                \put(-18, 98){\circle*{1}}
                \put(-20, 100){\circle*{1}}
                \put(-22, 102){\circle*{1}}
                \put(-24, 104){\circle*{1}}
                \put(-26, 106){\circle*{1}}
                \put(-28, 108){\circle*{1}}
                \put(-30, 110){\circle*{1}}
                \put(-32, 112){\circle*{1}}
                \put(-34, 114){\circle*{1}}
                \put(-36, 116){\circle*{1}}
                \put(-38, 118){\vector(-1,1){1}}
                \put(-28,108){\line(0,-1){6}}
                \put(-28,102){\line(1,0){6}}
                \put(-29,105){\makebox(0, 0)[r]
                {\scalebox{0.8}{$-c^s$}}}
                % Curve, starting origin and moving CW
                \thinlines
                \qbezier(0,0)(10,55)(40,55)
                \qbezier(40,55)(58.5,55)(84,44)
                %\qbezier(84,44)(92,41)(114,21)
                \qbezier(84,44)(92,41)(100,21)
                % Max and mins
                \thicklines
                \put(-4,55){\line(1,0){8}}
                \put(100,-4){\line(0,1){8}}
                \thinlines
                \put(-5,54){\makebox(0, 0)[r]%
                {$\left(\overline{g^p}-\overline{c^p}\right)^*$}}
                \put(0,55){\dashbox{3}(40,0){}}
                \put(100,-5){\makebox(0, 0)[t]%
                {$\left(\overline{\tau^p}\right)^*$}}
                \put(100,0){\dashbox{3}(0,21){}}
                % Origin strategy
                \thicklines
                \put(-12,-4){\line(0,1){8}}
                %\put(-4,12){\line(1,0){8}}
                \thinlines
                \put(-12,-5){\makebox(0, 0)[t]%
                {$\frac{-1}{\lambda^p}$}}
                %\put(-5,12){\makebox(0, 0)[r]%
                %{$-\frac{c^s}{\lambda^p}$}}
        \end{picture}
        \caption{Excess Rate Maximization ($N^p$ cycles and $G^T$
        threshold)}
        \label{fig:rate_maximization_cycle_visualization}
\end{figure}
\vspace{\stretch{1}}
\end{wideslide}

\begin{wideslide}[toc=Finite R-to-V]{Reward-to-Variability Maximization}
\vspace{\stretch{1}}
\begin{figure}[!ht]\centering
        \begin{picture}(150,42)(0,0)
                \put(75,21){\makebox(0,0){\scalebox{0.75}{\framebox(363,55){
                \shortstack[c]%
                {$\mu \triangleq \E(G_1) = %
                \overline{g^p} - \overline{c^p} - \frac{c^s}{\lambda^p}%
                $ \qquad %
                $\sigma \triangleq \std(G_1) = \sqrt{%
                \var\left(g^p - c^p\right) %
                + \left(\frac{c^s}{\lambda^p}\right)^2%
                }$\\%
                $\rho \triangleq \frac{\mu - %
                \frac{G^T}{N^p}}{\sigma}$ \qquad \qquad%
                $\lozenge^* \triangleq \max\left\{\lozenge\right\}$}}}}}
        \end{picture}\\
        \vspace{\stretch{4}}
        \begin{picture}(236,149)(-52,-22)
                \thicklines
                % Horizontal Axis
                \put(-10, 0){\vector(1, 0){155}}
                \put(147, 0){\makebox(0, 0)[l]%
                {$\sigma-\frac{|c^s|}{\lambda^p}$}}
                % Vertical Axis
                \put(24, -15){\vector(0, 1){125}}
                \put(24, 111){\makebox(0, 0)[b]%
                %{$\mu + \frac{c^s}{\lambda}$}}
                {$\overline{g^p} - \overline{c^p}$}}
                % Reward-to-Variability Line
                \thinlines
                \put(12, 85.2){\line(5,-2){133}}
                \put(110, 46){\circle*{5}}
                % Maximal Reward-to-Variability (Slope)
                \thinlines
                \put(31,77.6){\line(1,0){45}}
                \put(76,77.6){\line(0,-1){18}}
                \put(75,76.6){\makebox(0, 0)[rt]{$\rho^*$}}
                % Threshold Value
                %\thinlines
                %\put(0,85.2){\dashbox{3}(12,0){}}
                \thicklines
                \put(20,73.2){\line(1,0){8}}
                \put(12,-4){\line(0,1){8}}
                \thinlines
                %\put(-3,95.2){\makebox(0,0)[r]{$-\frac{c^s}{|c^s|}$}}
                \put(-3,95.2){\makebox(0,0)[r]{$-\sgn(c^s)$}}
                \put(-2,99.2){\line(0,-1){8}}
                \put(-2,91.2){\line(1,0){8}}
                \put(24,73.2){\circle*{1}}
                \put(22,75.2){\circle*{1}}
                \put(20,77.2){\circle*{1}}
                \put(18,79.2){\circle*{1}}
                \put(16,81.2){\circle*{1}}
                \put(14,83.2){\circle*{1}}
                \put(12,85.2){\circle*{3}}
                \put(10,87.2){\circle*{1}}
                \put(8,89.2){\circle*{1}}
                \put(6,91.2){\circle*{1}}
                \put(4,93.2){\circle*{1}}
                \put(2,95.2){\circle*{1}}
                \put(0,97.2){\circle*{1}}
                \put(-2,99.2){\circle*{1}}
                \put(-4,101.2){\circle*{1}}
                \put(-6,103.2){\circle*{1}}
                \put(-8,105.2){\circle*{1}}
                \put(-10,107.2){\circle*{1}}
                \put(-12,109.2){\vector(-1, 1){1}}
                \put(17,73.2){\makebox(0,0)[r]{$\frac{G^T}{N^p}$}}
                \put(12,-5){\makebox(0,0)[t]%
                {$-\frac{|c^s|}{\lambda^p}$}}
                % Curve, starting from upper left and moving CW
                \thinlines
                \qbezier(35,30)(35,55)(51,55)
                \qbezier(51,55)(80,55)(110,46)
                \qbezier(110,46)(120,40)(120,33)
                \qbezier(120,33)(120,10)(24,0)
                % Max and mins
                \thicklines
                \put(120,-4){\line(0,1){8}}
                \put(20,55){\line(1,0){8}}
                \thinlines
                \put(120,-5){\makebox(0, 0)[t]%
                {$\sigma^*-\frac{|c^s|}{\lambda^p}$}}
                \put(120,0){\dashbox{3}(0,33){}}
                \put(129,33){\circle*{1}}
                \put(126,33){\circle*{1}}
                \put(123,33){\circle*{1}}
                \put(120,33){\circle*{1}}
                \put(117,33){\circle*{1}}
                \put(114,33){\circle*{1}}
                \put(111,33){\circle*{1}}
                \put(108,33){\circle*{1}}
                \put(105,33){\circle*{1}}
                \put(102,33){\vector(-1, 0){1}}
                \put(19,55){\makebox(0, 0)[r]%
                {$\left( \overline{g^p} - \overline{c^p} \right)^*$}}
                \put(24,55){\dashbox{3}(27,0){}}
        \end{picture}
        \caption{Sharpe Ratio Maximization ($N^p$ cycles and $G^T$ threshold)}
        \label{fig:risk_sensitivity_sharpe_cycle_visualization}
\end{figure}
\vspace{\stretch{1}}
\end{wideslide}

\begin{slide}[toc=Pareto Discounts]{Optimization by Discounting}
%
\onslide*{2,3,4,5,6,7,8,9}{\begin{itemize}
        \item<2-> Objectives
                \onslide*{2-5}{$A$}\onslide*{6-}{{\color{darkgreen}$A$}}
                and \onslide*{2-6}{$B$}\onslide*{7-}{{\color{red}$B$}}
        \item<3-> Discount \onslide*{2-8}{$\onslide*{2-7}{w}\onslide*{8-}{{\color{blue}w}} \in \R_{\geq0}$}\onslide*{9}{${\color{magenta}\oft{\gamma}^* \in \R}$}
        \item<4-> Maximization of
                $\onslide*{2-5}{A}\onslide*{6-}{{\color{darkgreen}A}} -
                \onslide*{2-7}{w}\onslide*{8}{{\color{blue}w}}\onslide*{9-}{{\color{magenta}\oft{\gamma}^*}}
                \onslide*{2-6}{B}\onslide*{7-}{{\color{red}B}}$ is Pareto optimal
        \item<9-> \onslide{9}{\rnode{oppcost}{\color{magenta}Opportunity cost}}
\end{itemize}}
%
\showall
%
\onslide*{5,6,7,8,9}{\vspace{\stretch{1}}
\begin{equation*}
        \boxed{%
        \onslide{6-}{{\color{darkgreen}\E\left(\oft{G}_1\right)}}
        \onslide{7-}{-}
        \onslide*{5,6,7}{\phantom{w}}\onslide*{8}{{\color{blue}w}}\onslide*{9}{\rnode{ocgamma}{\color{magenta}\oft{\gamma}^*}}
        \onslide{7-}{{\color{red}\E\left(\oft{T}_1\right)}}%
        }
\end{equation*}
\onslide*{9}{\nccurve[nodesep=1pt,linecolor=magenta,angleA=-90,angleB=90]{<-}{oppcost}{ocgamma}}
\vspace{\stretch{1}}}
%
\onslide*{10}{\textbf{Other Possibilities}
\vspace{\stretch{1}}
\twocolumn%
{
\quad\\
\quad\\
\begin{gather*}
        \E\left(\oft{G}_1\right) - \oft{w}_0 \E\left(\oft{T}_1\right)\\
        \qquad\\
        \E\left(\oft{G}_1\right) - \oft{w}_1 \std\left(\oft{G}_1\right)\\
        \qquad\\
        \E\left(\oft{G}_1 + \oft{C}_1\right) - \oft{w}_2 \E\left(\oft{C}_1\right)
\end{gather*}
}{
\quad\\
\quad\\
\begin{gather*}
        \phantom{\Bigl(}\E\left(G_1\right) - w_0 \E\left(T_1\right)\phantom{\Bigr)}\\
        \qquad\\
        \phantom{\Bigl(}\E\left(G_1\right) - w_1 \std\left(G_1\right)\phantom{\Bigr)}\\
        \qquad\\
        \phantom{\Bigl(}\E\left(G_1+C_1\right) - w_2 \E\left(C_1\right)\phantom{\Bigr)}
\end{gather*}
}%
\vspace{\stretch{1}}}
%
\end{slide}

\section{Results}

\begin{slide}[toc=A-to-D Functions]{Advantage-to-Disadvantage Ratio}
%
\onslide*{2,3,4}{\vspace{\stretch{2}}
\begin{align*}
        \onslide{2-}{{\onslide*{4}{\color{darkgreen}}A} &\triangleq a + \sum\limits_{i=1}^n p_i a_i(\tau_i)}
        \\ 
        \onslide{3-}{{\onslide*{4}{\color{red}}D} &\triangleq d + \sum\limits_{i=1}^n p_i d_i(\tau_i)}
\end{align*}
%
\vspace{\stretch{3}}
%
\begin{equation*}
        \onslide{4-}{\ovalnode[linecolor=blue]{JAD}{ J \triangleq \frac{\color{darkgreen}A}{\color{red}D} }}
\end{equation*}
\vspace{\stretch{2}}}
%
\onslide*{5}{\textbf{Examples: Gain and Time Objectives}
\vspace{\stretch{2}}
%
\begin{equation*}
        \frac{ \E(G_1) - \frac{G^T}{N^p} }{ \E(T_1) }
        =
        \frac%
        { {\color{darkgreen}-c^s} + \sum\limits_{i=1}^n p_i {\color{darkgreen}\lambda_i \left( g_i(\tau_i) - c_i \tau_i - \frac{G^T}{N^p} \right)} }%
        { {\color{red}1} + \sum\limits_{i=1}^n p_i {\color{red}\lambda_i \tau_i} }%
\end{equation*}
\vspace{\stretch{1}}
\begin{equation*}
        \E(G_1) - w \E(T_1)
        =
        \frac%
        { {\color{darkgreen}-(c^s+w)} + \sum\limits_{i=1}^n p_i {\color{darkgreen}\lambda_i \left( g_i(\tau_i) - c_i \tau_i - w_i \tau_i \right)} }%
        { \sum\limits_{i=1}^n p_i {\color{red}\lambda_i} }%
\end{equation*}\vspace{\stretch{2}}}
%
\onslide*{6}{\textbf{Examples: Efficiency-Type Objectives}
\vspace{\stretch{2}}
\begin{equation*}
        \frac{ \E(G_1) + \E(C_1) - \frac{G_g^T}{N^p} }{ \E(C_1) }
        =
        \frac%
        { \sum\limits_{i=1}^n p_i {\color{darkgreen}\lambda_i \left( g_i(\tau_i) - \frac{G_g^T}{N^p} \right)} }%
        { {\color{red}c^s} + \sum\limits_{i=1}^n p_i {\color{red}\lambda_i c_i \tau_i} }%
\end{equation*}
\vspace{\stretch{1}}
\begin{equation*}
        \E(G_1) + \E(C_1) - w \E(C_1)
        =
        \frac%
        { {\color{darkgreen}-w c^s} + \sum\limits_{i=1}^n p_i {\color{darkgreen}\lambda_i \left( g_i(\tau_i) - w_i c_i \tau_i \right)} }%
        { \sum\limits_{i=1}^n p_i {\color{red}\lambda_i} }%
\end{equation*}
\vspace{\stretch{2}}}
%
\end{slide}

\begin{slide}[toc=Lumped Tasks]{Constant Disadvantage Case}
        \textbf{Assume that}
        \begin{itemize}
                \item $d_j$ is constant for all $\tau_j$
                \item $\frac{a_j}{d_j}$ achieves its maximum at $\tau_j^*$
        \end{itemize}
        %
        \pause
        \textbf{Index so that}
        %
        \begin{equation*}
                \frac{ a_1(\tau_1^*) }{ d_1(\tau_1^*) } >
                \frac{ a_2(\tau_2^*) }{ d_2(\tau_2^*) } > 
                \cdots >
                \frac{ a_{n-1}(\tau_{n-1}^*) }{ d_{n-1}(\tau_{n-1}^*) } > 
                \frac{ a_n(\tau_n^*) }{ d_n(\tau_n^*) }
        \end{equation*}
        %
        \pause
        \textbf{Exclude when}
        %
        \begin{equation*}
                \frac{ a + \sum\limits_{i=1}^k a_i(\tau_i^*) }{ d + \sum\limits_{i=1}^k d_i(\tau_i^*) }
                >
                \frac{ a_{k+1}(\tau_{k+1}^*) }{ d_{k+1}(\tau_{k+1}^*) }
        \end{equation*}
        %
\end{slide}

\begin{slide}[toc=Variable Time]{Generalized \ac{MVT}}
\onslide*{1-4}{%
        \textbf{Assume that}
        \begin{equation*}
                \left( \frac{a_j}{d_j} \right)' < 0 \text{ for all } \tau_j 
                \pause 
                \quad
                \text{ (maximum at } \tau_j^* = 0 \text{)}
        \end{equation*}
        %
        \pause
        \textbf{Index so that}
        %
        \begin{equation*}
                \frac{ a_1(0) }{ d_1(0) } >
                \frac{ a_2(0) }{ d_2(0) } > 
                \cdots >
                \frac{ a_{n-1}(0) }{ d_{n-1}(0) } > 
                \frac{ a_n(0) }{ d_n(0) }
        \end{equation*}
        %
        \pause
        \textbf{Exclude when}
        %
        \begin{equation*}
                \frac{ a + \sum\limits_{i=1}^k a_i(\tau_i^k) }{ d + \sum\limits_{i=1}^k d_i(\tau_i^k) }
                >
                \frac{ a_{k+1}(0) }{ d_{k+1}(0) }
        \end{equation*}%
}
%
\onslide*{5,6}{\vspace{\stretch{1}}
        \textbf{where $\tau_j^k$ is found with}
        %
        \begin{equation*}
                \frac{ a'_k(\tau_j^k) }{ d'_k(\tau_j^k) }
                =
                \frac{ a + \sum\limits_{i=1}^k a_i(\tau_i^k) }{ d + \sum\limits_{i=1}^k d_i(\tau_i^k) }
        \end{equation*}
        %
        \onslide{6}{This is a generalized version of the marginal value theorem.}
        %
        \vspace{\stretch{1}}%
}
        %
\end{slide}

\section[slide=false]{Remarks}
\begin{slide}[toc=,bm=]{Concluding Remarks}
%
\begin{itemize}
        \item<2-> Simple stochastic model of a solitary agent 
        \item<3-> Provided analytical tools for finding optimal
                agent behavior
        \item<4-> Model can be expanded (\eg, recognition cost,
                behavior-dependent rates, nonlinear fuel costs)
        \item<5-> Optimization approaches can be enhanced using methods
                from post-modern portfolio theory
        \item<6-> Behaviors should be implemented on real agents
\end{itemize}
%
\end{slide}

\section[slide=false]{Questions?}
\begin{slide}[toc=,bm=]{Questions?}
        \textbf{Major Influences}
        \begin{itemize}
                \item \bibentry{Cha76}.
                \item \bibentry{Cha73}.
                \item \bibentry{SK86}.
        \end{itemize}
\end{slide}
\section[tocsection=hidden]{Follow-Up}

\begin{slide}{Applications}
        \onslide*{2,3}{\begin{itemize}
                \item<2-> Military
                        \begin{itemize}
                                \item Surveillance 
                                \item Offensive action
                        \end{itemize}
                \item<3-> {\color{blue}Robotic Exploration} (\eg, \emph{\texttt{DEPTHX}})
                        \begin{itemize}
                                \item Deep Sea
                                \item Space
                        \end{itemize}
        \end{itemize}}
        \onslide*{4,5,6,7}{\begin{itemize}
                \item<4-> Limited resources
                        \begin{itemize}
                                \item<5-> Retrieval space
                                \item<5-> Objects to deploy 
                        \end{itemize}
                \item<6-> Prioritization during real-time search
                \item<7-> Failure thresholds (\eg, $G^T$)
        \end{itemize}}
        \onslide*{8,9,10,11}{\begin{itemize}
                \item Information foraging of human beings (\ie,
                        analysis)
                        \begin{itemize}
                                \item<9-> Web behavior consistent with
                                        foraging predictions
                                \item<9-> Design technology based on
                                        behavioral analysis
                                \item<10-> Content delivery based on
                                        encounter characteristics of
                                        search
                                \item<11-> Advertising
                        \end{itemize}
        \end{itemize}}
\end{slide}

\begin{slide}{Future Directions}
        \onslide*{2,3,4,5,6,7,8,9,10}{\begin{itemize}
                \item<2-> Model can be expanded
                        \begin{itemize}
                                \item<3-> Behavior-dependent encounter
                                        rates
                                \item<4-> Recognition costs
                                \item<5-> Nonlinear cost functions 
                                \item<6-> Parameter uncertainty and
                                        estimation
                                \item<7-> Relax Poisson assumption
                        \end{itemize}
                \item<8-> New stochastic optimization criteria
                        \begin{itemize}
                                \item<9-> Skew-sensitive objectives
                                \item<10-> Use of lower-partial moments 
                        \end{itemize}
        \end{itemize}}
        \onslide*{11,12,13,14,15,16,17,18}{\begin{itemize}
                \item<11-> \ac{PMPT}: Stochastic dominance
                        \begin{itemize}
                                \item<12-> {\color{red}Static} approach
                                \item<13-> Embraces risk sensitivity and a
                                        wider range of return
                                        distributions
                        \end{itemize}
                \item<14-> State-based {\color{darkgreen}dynamic} programming methods
                        \begin{itemize}
                                \item<15-> Modern approach
                                \item<16-> {\color{darkgreen}Improves performance}
                                \item<17-> {\color{red}Increases behavioral
                                        complexity}
                                \item<18-> Better machines and
                                        human-in-loop systems prevent
                                        complexity problems
                                \item<18-> Parameter estimation can be
                                        built-in
                        \end{itemize}
        \end{itemize}}
\end{slide}

\begin{slide}{Questions?}
        \textbf{Influences for Future Work}
        %
        \twocolumn{\begin{itemize}
                \item {\tiny\bibentry{Bawa75}.}
                \item {\tiny\bibentry{Bawa82}.}
                \item {\tiny\bibentry{Bawa78}.}
                \item {\tiny\bibentry{BL77}.}
        \end{itemize}}{\begin{itemize}
                \item {\tiny\bibentry{Caraco80}.}
                \item {\tiny\bibentry{HouMc99}.}
                \item {\tiny\bibentry{MnglClk88}.}
        \end{itemize}}
        %
\end{slide}

\end{document}