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\title[Prospect Theory for Intertemporal Choice]{Prospect Theory for Intertemporal Choice}
\author[Lampe \& Weber]{Immanuel Lampe \and Matthias Weber}

\institute[]{ \\ University of St.Gallen}
\date{} 


\begin{document}
	
	
\begin{frame}
\titlepage
\end{frame}

\linespread{1.1}



\section*{Outline}

\begin{frame}
\frametitle{Outline}
\tableofcontents
\end{frame}


\section{Introduction}

\subsection{Motivation}

\begin{frame}{Motivation}
\begin{itemize}
	\item Prospect Theory (PT) in general describes decisions under risk better than Expected Utility Theory (EUT)
	\item Daniel Kahneman won the Nobel Prize for his work on PT
	\item PT can explain several phenomena that EUT cannot explain \vspace{0.75cm}
	\begin{itemize}
		\item In atemporal settings  (= outcomes materialize at one point in time)
		\item PT in atemporal settings is well understood 
	\end{itemize}

\end{itemize}
\end{frame}


\begin{frame}{Motivation}
\begin{itemize}
	\item However, many (most?) important decisions in economics and finance involve a risk and a time dimension:
	\begin{itemize}
		\item Saving and consumption (retirement savings)
		\item Asset allocation
		\item Buying a house vs.\ renting
		\item Insurance
		\item Etc.
	\end{itemize}
	\item Still unclear how to apply prospect theory when outcomes materialize at different points in time (intertemporal contexts)	
	\begin{itemize}
		\item In particular, two potential application methods mentioned in the literature
	\end{itemize}
\end{itemize}
\end{frame}



\begin{frame}{Motivation}
What we do, in a nutshell:
\begin{itemize}
	\item Conduct an experiment on a representative sample
	\item Subjects evaluate intertemporal lotteries
	\item Find out which application method describes risky choices best (out-of-sample prediction performance)
	\item Deliver a calibration for intertemporal PT 
\end{itemize}
\end{frame}

\begin{frame}{Notes}
\begin{itemize}
	\item This is about how probability weighting should be applied, not about reference points
	\begin{itemize}
		\item Reference points of zero by design
	\end{itemize}
	\item We use PT with neutral propbabilities (lotteries)
	\begin{itemize}
		\item We provide the benchmark: adaptations to specific environments are possible
		\item Paper is also relevant for other theories with non-objective probabilities
	\end{itemize}
\end{itemize}
\end{frame}




\subsection{Background: (Atemporal) Prospect Theory}


\begin{frame}{Prospect Theory}
\begin{itemize}
	\item A prospect/lottery consists of outcomes arising with given probabilities, $(x_1:p_1;...;x_n:p_n)$, e.g., $(100:0.2;\ 50:0.15;\ 0:0.65)$ 
	\begin{figure}[H]
	\includegraphics[width=0.18 \textwidth]{Figures/atemporal_3_payouts.jpg}
\end{figure}
	\item Value of the prospect under EUT, in utility terms: \vspace{-0.4cm}
	$$ (x_1:p_1;...;x_n:p_n) \overset{EUT}{\rightarrow} \sum_{i=1}^{n}{p_i u(x_i)} $$ \vspace{-0.4cm}
	\item Two differences in PT
	\begin{itemize}
	\item Different utility/value function (incl.\ reference dependence)
	\item Probability weighting
	\end{itemize}
\end{itemize}
\end{frame}


\begin{frame}{PT Value Function}
\begin{itemize}
	\item Gains and losses with respect to a reference point $R=0$
	\item Kink around the reference point (loss aversion)
	\item Often slightly concave for gains, slightly convex for losses
\end{itemize}
\begin{figure}[H]
	\includegraphics[width=0.4 \textwidth]{Figures/value_function.png}
\end{figure}
\end{frame}


\begin{frame}{Probability Weighting}
\begin{itemize}
	\item Value of the prospect under PT, in utility/value terms: 
	$$ (x_1:p_1;...;x_n:p_n) \overset{PT}{\rightarrow} \sum_{i=1}^{n}{\textcolor{red}{\pi_i} v(x_i)} $$
	\item How does this probability weighting work?
		\begin{itemize}
	\item There is a weighting function $w$
	\item The weighting is not $\pi_i = w(p_i)$
	\end{itemize}
\end{itemize}
\end{frame}

\begin{frame}
\begin{figure}[H]
	\includegraphics[width=1.15 \textwidth, trim= 0cm 2cm 0cm 1.5cm, clip=false]{Figures/prop_weighting_example.png}
\end{figure} \pause
\begin{itemize}
	\item Done separately for gains and losses\end{itemize}
\end{frame}






\subsection{Applying PT to Intertemporal Prospects}


\begin{frame}[label=intertemporal_lottery]{Intertemporal Prospects}
	\begin{itemize}
	\item An intertemporal prospect yields (uncertain) payouts at different points in time.
\end{itemize}

\begin{figure}[H]
	\includegraphics[width=0.4 \textwidth]{Figures/coin_toss_4_v2.jpg}
\end{figure}\vfill
	\hfill \hyperlink{example_lottery}{\beamergotobutton{Additional example}}
\end{frame}





\begin{frame}{Time-Separation Method}
\vspace{0.2cm}
\begin{tabular}{cl}  
	\begin{tabular}{l}
		\includegraphics[width=0.33 \textwidth]{Figures/coin_toss_time_sep_v2.jpg}
	\end{tabular}  
	\begin{tabular}{l}
		\parbox{0.50\linewidth}{\footnotesize{ 
				$PT_1=w(0.5)v(60)+(1-w(0.5))v(0) \\ $ 
				$PT_2=w(0.65)v(60)+(1-w(0.65))v(0) \\ $ 
	  } 
		}
	\end{tabular}
\end{tabular}
\vspace{0.2cm}
\footnotesize{
		$  \text{\hspace{0.6cm}}$ $PT = \delta(1)PT_1 $ $\text{\hspace{0.05cm}}$ $+$ $\text{\hspace{0.05cm}}$ $ \delta(2)PT_2$  }
\end{frame}




\begin{frame}{Risk-Separation Method}
\vspace{0.2cm}
\begin{tabular}{cl}  
	\begin{tabular}{l}
		\includegraphics[width=0.33 \textwidth]{Figures/coin_toss_present_val_v2_nocolor.jpg}
	\end{tabular}  
	\begin{tabular}{l}
		\parbox{0.50\linewidth}{\footnotesize{ 
				$PV_1= \delta(1)v(60)+\delta(2)v(60) \\ $ 
				$PV_2= \delta(1)v(60)+\delta(2)v(0) \\ $
				$PV_3= \delta(1)v(0)+\delta(2)v(60) $} 	
		}
	\end{tabular}
\end{tabular}
\vspace{-0.1cm}
\footnotesize{\begin{align*}
		 & PT  = \pi_1 PV_1 + \pi_2 PV_2 + \pi_3 PV_3  
		\end{align*}}
\end{frame}


\begin{frame}{Application Methods}
\begin{itemize}
	\item These two methods have been proposed in the literature
		\begin{itemize}
		\item Time-separation method: e.g., \cite{andreoni2012risk}, \cite{krause2020hyperopic} 
		\item Risk-separation method: e.g., \cite{halevy2008strotz}, \cite{epper2015risk}
	\end{itemize}
		\item Note: without (= with linear) probability weighting, both methods give the same results \vspace{0.4cm}
\end{itemize}
\end{frame}




\subsection{Research Questions, Preview of the Results}



\begin{frame}{Research Questions}
\begin{itemize}
	\item[1] Which way of applying PT describes risky choices best?
	\begin{itemize}
		\item Papers with similar scope are scarce and very different (\citealp{andreoni2017stream}; \citealp{rohdecorrelation}), with inconclusive results
	\end{itemize}
	\vspace{0.2cm}	
	\item[2] What are good calibrations to apply prospect theory to intertemporal contexts?
	\begin{itemize}
	\item Intertemporal applications usually use parametric specifications from atemporal contexts
	\item  Good reasons to assume that calibrations should be different in intertemporal contexts (e.g., \citealp{abdellaoui2013there})
	\end{itemize}
\end{itemize}
\end{frame}



\begin{frame}{Preview of the Results}
	\begin{itemize}
	\item Risk-separation method performs much better than time-separation method
	\item Calibration:
	\begin{itemize}
	\item almost linear value functions (in loss and gain domains)
	\item loss aversion parameter close to one
	\item inverse-s shaped probability weighting functions (as in atemporal PT)
	\item moderate discounting of all quarters (exponential) or distinction between now and future (quasi-hyperbolic; both versions predict equally well)
	\end{itemize}
	\end{itemize}
	\end{frame}








\section{Experiment}

\subsection{Procedures and Decision Tasks}

\begin{frame}
\tableofcontents[currentsection]
\end{frame}


\begin{frame}[label=experiment_slide]{Experiment}
\begin{itemize}
	\item Experiment on a sample representative for the Dutch population
	\begin{itemize}
		\item LISS panel (Centerdata, Tilburg University)
	\end{itemize}
	\item Study was pre-registered (data analysis follows pre-analysis plan)
	\end{itemize} \vfill
	\hfill \hyperlink{exclusion_slide}{\beamergotobutton{Details}}
\end{frame}


\begin{frame}{Experiment}

\begin{itemize}
	\item A total of 48 decision tasks. 
	\item In each task subjects see a lottery ("risky option") with three uncertain payouts (today, in three months, in six months). 
	\item We elicit the switching point from the lottery to a safe option that yields three certain and identical payouts (multi-period certainty equivalent; CE). 
	\item For 75\% of subjects hypothetical choices (T1), for the rest (T2) part of the choices incentivized
	\begin{itemize}
	\item T1: 15 EUR for participation
	\item T2: 15 EUR for participation, on average 84 EUR in addition 
	\item Incentivized and hypothetical choices do not differ [no strategic interaction, social image, self image]
	\end{itemize}
\end{itemize}
\end{frame}




\begin{frame}{Decision Screen}
 \begin{figure}[htb]
  \includegraphics[width=0.6  \textwidth]{Figures/Slider_1.jpg}
\visible<2-3>{
	\includegraphics[width=0.75  \textwidth]{Figures/Slider_2.jpg}
} \end{figure}\vspace{-0.7cm}
\visible<3>{\hspace{4cm}  $\uparrow$ \colorbox{white!30}{CE: 7-8}}

\end{frame}



\subsection{Lottery Design}


\begin{frame}{Lotteries}
\begin{itemize}
	\item $6$ sets of $8$ lotteries each 
\end{itemize}
\begin{small}
\begin{table}[ht]
	\begin{tabular}{cccc}  \cline{1-4} 
		& Gains & Losses & Mixed    \\
		\cline{1-4} 
		Small Stakes &  Set 1  &  Set 2 & Set 3    \\
		Large Stakes & Set 4 & Set 5 &  Set 6    \\ 	\cline{1-4}	
	\end{tabular}
\end{table}
\end{small}
\begin{table}[]
	\begin{tabular}[t]{>{\columncolor[gray]{0.9}}ll@{}}
\hspace{4.34cm} 48 Lotteries \hspace{4.37cm} \\
\end{tabular} \\
\resizebox{0.95\textwidth}{!}{
	\begin{tabular}[t]{>{\columncolor[gray]{0.9}}ll@{}}
		\\ 36 Calibration Lotteries  \\ 
		For any combination of the value, weighting \\ and discount function TS = RS  \\
				$\Rightarrow$ Estimation leads to same parameters \\ 
				for both methods \\ 	\vspace{0.1cm}
	\end{tabular} \hspace{0.2cm}
	\begin{tabular}[t]{>{\columncolor[gray]{0.9}}ll@{}}
		 \\ 12 Test Lotteries (2 from each set) \\
		 1-6: TS $>$ Linear Probability Weighting $>$ RS
		 \\ 7-12: TS $<$ Linear Probability Weighting $<$ RS \\
		 \ \\ \ \\ \vspace{0.1cm}
	\end{tabular}}
\end{table}
\end{frame}





\section{Estimation Procedure}

\begin{frame}
\tableofcontents[currentsection]
\end{frame}


\begin{frame}[label=functions_slide]{Parametric Specification}
\begin{itemize}
 \item For each application method, we use 12 combinations of value, probability weighting, and time-discount functions  
\end{itemize}
\begin{small}
\begin{table}[htbp]
	\label{tab:model_combinations}
		\resizebox{\textwidth}{!}{
		\begin{tabular}{llcccccccccccc}  \hline	
   & & C1 & C2 & C3 & C4 & C5 & C6 & C7 & C8 & C9 & C10 & C11 & C12 \\ \hline
\multirow{2}{*}{Value} & Power & x & x & x & x & x & x & & & & \\
 & Exponential & & & & & & & x & x & x & x & x & x \\ \hline
\multirow{3}{*}{Weighting}   & T+K (1992) & x & x & & & & & x & x & & & & \\
    & Prelec (1998) & & & x & x & & & & & x & x & & \\
     & G+E (1987) & & & & & x & x & & & & & x & x \\ \hline
  \multirow{2}{*}{Discount}    & Exponential & x & & x & & x &  & x &  & x &  & x & \\
    & Quasi hyp. & & x & & x & & x & & x & & x & & x \\ \hline
	\end{tabular}}
\end{table}
\end{small} \vfill
\hfill \hyperlink{function_details}{\beamergotobutton{Details}}
\end{frame}








\begin{frame}[label=MLE_slide]{Maximum Likelihood Estimation (Calibration Set)}
Assumptions (standard) for the log-likelihood function: 
\begin{itemize}
	\item Stated certainty equivalents are affected by noise $\epsilon_{i,j}$, with $\epsilon_{i,j} \sim \mathcal{N}(0,\sigma_{i,j}^2)$.
	\item Standard deviation $\sigma_{i,j}=\epsilon_i w_j$
	\begin{itemize} 
		\item subject-specific $\epsilon_i$
		\item proportional to lotteries payout range $w_j$
	\end{itemize}
\end{itemize} \vfill
\hfill \hyperlink{MLE_details}{\beamergotobutton{Details}}
\end{frame}


\begin{frame}{Measurement of Prediction Performance (Test Set)}
\begin{itemize}
	\item \textbf{Main outcome variable:} $MSE= \frac{1}{n} \sum_{i=1}^{n}{ MSE_i}.$
	\item $MSE_i= \frac{1}{12} {\sum_{j=1}^{12}{  \left( \frac{1}{w_j} \left( CE_{i,j} - \widehat{ce_{j}} \right) \right)^2 }}$ \vspace{0.2cm}
		\begin{itemize}
		\item  $\widehat{ce_{j}}$: predicted certainty equivalent for lottery $j$ by given model
		\item $CE_{i,j}$: certainty equivalent reported by player $i$ for lottery $j$
		\item $w_j$: payout range of lottery $j$
	\end{itemize} \vspace{0.3cm}
	\vspace{0.3cm}
	\item Standard errors are bootstrapped (at the participant level)
	\item Tests are paired bootstrap tests
\end{itemize}

\end{frame}











\section{Results}

\begin{frame}
\tableofcontents[currentsection]
\end{frame}


\subsection{Comparison of the Application Methods}

\begin{frame}[label=Main_result_slide]{Main Result}
	\only<1>{\begin{figure}[H]
	\includegraphics[width=0.70  \textwidth]{Figures/Figure7_part1.pdf}
\end{figure}}
	\only<2>{\begin{figure}[H]
		\includegraphics[width=0.70  \textwidth]{Figures/Figure7.jpg}
\end{figure}}

\only<1-2>{\begin{itemize}
	\item For any combination (C1,C2,...,C12), the risk-separation method predicts decisions better than the time-separation method [even holds for all lotteries!]
\end{itemize}}
\hfill \hyperlink{deviation_lotteries}{\beamergotobutton{Illustration I}}
\hfill \hyperlink{participant_classification}{\beamergotobutton{Illustration II}}
\end{frame}




\begin{frame}{Statistical Tests}
{\begin{figure}[H]
		\includegraphics[width=0.7  \textwidth]{Figures/Figure8.jpg}
\end{figure}}

\end{frame}


\begin{frame}{Intuition of the Main Result}
\begin{itemize}
	\item It seems more natural for people to think along outcome streams than separating time periods
	\begin{itemize}
		\item Different outcome streams are disjoint probabilistic events, while different time periods are not 
		\item Separating time periods (where outcomes depend on other time periods) may be not natural
	\end{itemize}
	\item Note that it is not necessary that people make calculations to determine their certainty equivalents (in general, they do not)
\end{itemize}
\end{frame}




\subsection{Calibration}


\begin{frame}{Example Calibration}
\begin{itemize}
	\item C6: i) power value function, ii) G+E (1987) probability weighting functions, iii) quasi-hyperbolic discounting)

	\begin{tabular}{cl}  
		\begin{tabular}{c}
			\parbox{0.4\linewidth}{
				\begin{figure}[H]
					\includegraphics[width=0.35 \textwidth]{Figures/example_value.jpg}
				\end{figure}
			}
		\end{tabular} \hspace{0.5cm}
		& 	\begin{tabular}{c}
			\parbox{0.4\linewidth}{
					\begin{figure}[H]
						\includegraphics[width=0.4 \textwidth]{Figures/example_weighting.jpg}
				\end{figure}
			}
		\end{tabular}
\end{tabular}

\item quasi-hyperbolic disc.: $\delta(t) = k   \exp(-rt)$, $k =0.885$, $r=0.001$
\end{itemize} \hfill \hyperlink{MPV_slide}{\beamergotobutton{MPV}}
\end{frame}



\subsection{Additional Analyses}


\begin{frame}{PT Components and Comparison to EDU}
\only<1>{\begin{figure}[H]
		\includegraphics[width=0.9  \textwidth]{Figures/weighted_MSE_components_1.jpg}
\end{figure}}
\only<2>{\begin{figure}[H]
		\includegraphics[width=0.9  \textwidth]{Figures/weighted_MSE_components_all.jpg}
\end{figure}}

\begin{itemize}
		\item It's all about probability weighting
\end{itemize}
\end{frame}




\begin{frame}[label=MPV_slide]{Risk-Separation Present-Value Method}
\vspace{0.2cm}
\begin{tabular}{cl}  
	\begin{tabular}{l}
		\includegraphics[width=0.33 \textwidth]{Figures/coin_toss_present_val_v2_nocolor.jpg}
	\end{tabular}  
	\begin{tabular}{l}
		\parbox{0.50\linewidth}{\footnotesize{ 
				$MPV_1= \delta(1)60+\delta(2)60 \\ $ 
				$MPV_2= \delta(1)60+\delta(2)0 \\ $
				$MPV_3= \delta(1)0+\delta(2)60 $} 	
		}
	\end{tabular}
\end{tabular}
\vspace{-0.1cm}
\footnotesize{\begin{align*}
		 & PT  = \pi_1 v(MPV_1) + \pi_2 v(MPV_2) + \pi_3 v(MPV_3)  
		\end{align*}}
\end{frame}



\begin{frame}[label=MPV_slide_2]{Risk-Separation Present-Value Method}
{\begin{figure}[H]
		\includegraphics[width=0.7  \textwidth]{Figures/Figure11.jpg}
\end{figure}} 
\end{frame}









\section{Application Example (Sketch)}

\begin{frame}
\tableofcontents[currentsection]
\end{frame}


\begin{frame}{Sketch of an Application Example (Insurance)}
Does it matter for real-world applications which method is used? \vspace{0.2cm}
\begin{itemize}
\item Think of (private) long-term care insurance
\item Small probabiliy that an insurable event appears in any single time period
\begin{itemize}
\item Overweighting by time-separation method (overinsurance)
\end{itemize}
\item Relatively large probability that an event appears at some point
\begin{itemize}
\item Underweighting by risk-separation method (underinsurance)
\end{itemize}
\item Underinsurance is in line with empirical evidence (e.g., \citealp{brown2011insuring})
\end{itemize}
\end{frame}


\section{Conclusion}

\begin{frame}
\tableofcontents[currentsection]
\end{frame}


\begin{frame}[label=conclusion_slide]{Conclusion}
\begin{itemize}
	\item Risk-separation method performs much better than time-separation method
	\item Risk-separation present-value method as good as regular risk-separation method
	\item Calibration:
	\begin{itemize}
	\item almost linear value functions (in loss and gain domains)
	\item loss aversion parameter close to one
	\item inverse-s shaped probability weighting functions (as in atemporal PT)
%	\item moderate discounting of all quarters (exponential) or distinction between now and future (quasi-hyperbolic; both versions predict equally well)
	\end{itemize}
	\end{itemize}
\end{frame}



\begin{frame}[label=conclusion_slide2]{Conclusion}
\begin{itemize}
	\item Does this matter in applications? 
	\begin{itemize}
		\item Yes, e.g.\ explaining long-term care insurance choices
	\end{itemize}
	\item Does this matter for other theories with non-objective probabilities? 
	\begin{itemize}
		\item Yes, with both time and risk, there is always the matter time-first vs.\ risk-first
	\end{itemize}
\end{itemize}
\end{frame}


\begin{frame}[noframenumbering]
\vspace{2cm}
\begin{center}
\Large{Thank you for your attention!}
\end{center}
\vspace{2cm}
\end{frame}









\section*{Appendix}

\begin{frame}[noframenumbering, label= example_lottery, standout]{Example Lottery (Same Evaluation)}
\begin{figure}[H]
	\includegraphics[width=0.7 \textwidth]{Figures/Option8.jpg}
\end{figure} \vfill
	\hfill \hyperlink{intertemporal_lottery}{\beamerreturnbutton{Return}}
\end{frame}



\begin{frame}[noframenumbering, label=exclusion_slide]{Number of Subjects}
\begin{itemize}
	\item 378 subjects completed the experiment 
	\item Data exclusion is strict (ensures that results are not driven by carelessness or misunderstandings) and follows pre-registration:
	\begin{itemize}
	\item subjects stated comprehension difficulties or low attention in at least one post-experimental question
	\item short median decision times
	\end{itemize}
	\item Left with 100 subjects
	\item Main result (comparison of the methods) identical when conducted with all subjects
	\item Demographic variables between excluded and general subject pool very similar
	\end{itemize} \vfill
	\hfill \hyperlink{experiment_slide}{\beamerreturnbutton{Return}}
\end{frame}



\begin{frame}[noframenumbering, label=function_details]{Function Specifications}
\begin{tiny}
\begin{table}[htb]
	\label{tab:used_parameters}
	\begin{tabular}{ll}
		\toprule
		Specification & Parameters   \\ 
		\midrule 
		Utility functions&  \\
		\midrule
		\begin{tabular}[c]{@{}l@{}} \hspace{10pt} Power utility: \\ \hspace{10pt} $v(x) =  \mathbbm{1}_{x\geq 0} x^ {\alpha}  - \mathbbm{1}_{x < 0} \lambda$ $(-x)^ {\alpha} $\end{tabular} & $\alpha, \lambda$   \\
		\begin{tabular}[c]{@{}l@{}} \hspace{10pt} Exponential utility:\\  \hspace{10pt} $v(x) =  \mathbbm{1}_{x\geq 0} \frac{1- \exp(- \alpha x)}{\alpha} - \mathbbm{1}_{x < 0} \lambda  \frac{1- \exp( \beta x)}{\beta} $\end{tabular} & $\alpha, \beta, \lambda$  \vspace{5pt}
		\\ 
		\midrule 
		Probability weighting functions (gains and losses)&   \\ 
		\midrule
		\begin{tabular}[c]{@{}l@{}}  \hspace{10pt} Tversky and Kahneman (1992):\\ \hspace{10pt} $w(p) = \frac{p^{\gamma}}{ \left( p^{\gamma} + (1-p)^{\gamma}\right) ^{1 / \gamma}} $\end{tabular} &  $\gamma^+,\gamma^-$  \vspace{3pt} \\ 
		\begin{tabular}[c]{@{}l@{}} \hspace{10pt} Prelec (1998): \\ 
			\hspace{10pt} $w(p) =  \exp(-\nu (-ln(p))^ {\gamma})$\end{tabular} &   $\gamma^+,\gamma^-, \nu^+,\nu^-$ \\
		\begin{tabular}[c]{@{}l@{}} \hspace{10pt} Goldstein Einhorn (1987): \\ 
			\hspace{10pt} $w(p) = \frac{\nu p^{\gamma}}{\nu p^{\gamma} +\left(1-p \right)^{\gamma}} $ \end{tabular} &  $\gamma^+,\gamma^-, \nu^+,\nu^-$
		\vspace{5pt}
		\\  	\midrule
		Time-discount functions &   \\ 
		\midrule
		\begin{tabular}[c]{@{}l@{}}  \hspace{10pt} Exponential discounting:\\ \hspace{10pt} $\delta(t) = \exp(-rt) $\end{tabular} &  $r$  \vspace{2pt} \\ 
		\begin{tabular}[c]{@{}l@{}} \hspace{10pt} Quasi-Hyperbolic discounting: \\ 
			\hspace{10pt} $\delta(t) =  \mathbbm{1}_{t = 0} + \mathbbm{1}_{t > 0}   k   \exp(-rt) $ \end{tabular} &  $k, r$  \vspace{2pt}  \\
		\bottomrule \\[-1.8ex] 
	\end{tabular}
\end{table}
\end{tiny}% \vfill
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\end{frame}


\begin{frame}[noframenumbering, label=MLE_details]{Maximum Likelihood Procedure (Details)}
\begin{itemize}
	\item $ce_{j}$ = the certainty equivalent of lottery $j$ resulting from an evaluation under one model specification. 
	\item $CE_{i,j}$ = certainty equivalent player j reports for lottery $i$. 
	\item Assumptions (as \citealp{{hey2009noise, bruhin2010risk}})
	\begin{itemize}
		\item Noise: $CE_{i,j} = ce_{j} + \epsilon_{i,j}$, with $\sim \mathcal{N}(0,\sigma_{i,j}^2)$.
		\item SD subject specific and payout range dependent $\sigma_{i,j}=\epsilon_i w_j$. 
	\end{itemize}
	\item contribution of participant $i$:
	\footnotesize{$$ f( \theta, \epsilon_i | CE_i) = \prod_{j=1}^{36}  \frac{1}{\sigma_{i,j}} \phi \left(\frac{CE_{i,j} - ce_{j}(\theta)}{\sigma_{i,j}}\right)$$}
	\item All $n$ participants
	\footnotesize{$$  log \ L (\theta, \epsilon | CE) =  \sum_{i=1}^{n}{ log \ f( \theta, \epsilon_i | CE_i)} =  \sum_{i=1}^{n}{ \sum_{j=1}^{36}{ log \ \left[ \frac{1}{\sigma_{i,j}} \phi \left( \frac{CE_{i,j} - ce_{j}(\theta)}{\sigma_{i,j}}\right) \right] }}$$}
\end{itemize} 
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\end{frame}




\begin{frame}[noframenumbering, label=deviation_lotteries]{Mean Absolute Prediction Error (By Lottery)}
\begin{table}[htbp] \centering 
	\label{Error_by_Lottery} 
	\resizebox{\textwidth}{!}{
		\begin{tabular}{lcccccccccccc} \toprule
			&   \multicolumn{6}{c}{Low-stake lotteries} & \multicolumn{6}{c}{High-stake lotteries} \\
			&  L7 & L8 & L15 & L16 & L23 & L24 & L31 & L32 & L39 & L40 & L47 & L48 \\ 
			\midrule
			Payout range & $23$ & $20$ & $23$ & $20$ & $30$ & $50$ & $467$ & $400$ & $467$ & $400$ & $600$ & $1000$ \\  
			Mean error TS & $6$ & $8.1$ & $7.3$ & $5.6$ & $7.9$ & $16.6$ & $144.7$ & $148.6$ & $146.4$ & $124.6$ & $180.3$ & $319.5$
			\\  
			Mean error RS &$5.2$ & $5.6$ & $6.1$ & $5.6$ & $7.2$ & $14.6$ & $108.7$ & $118.1$ & $114.4$ & $115.1$ & $150.7$ & $276.7$  \\
\bottomrule \\[-1.8ex] 
\multicolumn{13}{p{490pt}}{Notes: The mean absolute prediction error of Lottery $j$ is calculated as mean$(|CE_{i,j} - \widehat{ce_{j}}|)$, with $CE_{i,j}$ denoting the certainty equivalent subject $j$ reported for lottery $j$ and $\widehat{ce_{j}}$ denoting the predicted certainty equivalent resulting from the parameters estimated on the calibration set.}
		\end{tabular}}
\end{table}  \vfill
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\begin{frame}[noframenumbering, label=participant_classification]{Participant Types}
\begin{table}[htbp]
	\label{tab:classification_by_typ}
		\resizebox{\textwidth}{!}{
	\begin{tabular}{lcccccccccccc}  \toprule
		&  C1 & C2 & C3 & C4 & C5 & C6 & C7 & C8 & C9 & C10 & C11 & C12 \\ \midrule
		Time-separation types   & 10 & 5 & 1 & 1 & 1  & 1  & 5 & 5 & 2 & 1 & 2  & 2 \\ 
		Risk-separation types   & 81 & 85 & 84 & 81 & 89 & 88 & 87 & 87  & 86 & 89 & 88 & 88 \\ 
		Unclassified   & 9 & 10 & 15 & 18 & 10 & 11 & 8 & 8  & 12 & 10 & 10 & 10 \\ 
		\bottomrule \\[-1.8ex] 
		\multicolumn{13}{p{450pt}}{Notes: Subject $i$ is classified as time-separation type if $MSE^{RS}_i - SE(\triangle MSE) >MSE^{TS}_i$ or as risk-separation type if $MSE^{TS}_i - SE(\triangle MSE) >MSE^{RS}_i$. $SE(\triangle MSE)$ denotes the standard error of the difference in MSE between the two methods.}
	\end{tabular}}
\end{table} \vfill
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\end{frame}











\begin{frame}[noframenumbering,allowframebreaks] 
\frametitle{References}
\bibliographystyle{apalike}
\bibliography{references}
\end{frame}








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