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\title{Well-established Effects in Behavioural Economics Depend on Access to No Feedback, Partial Feedback, or Full Feedback}
\author[1]{Ola Andersson}
\affil[1]{Dept. of Economics, Uppsala University}  
\author[2]{Mattias Forsgren}
\affil[2]{Dept. of Psychology, Uppsala University}  
\author[1]{Gustav Karreskog}
\date{\today}




\begin{document}
\begin{comment}

20241204

Effekter att titta på:
- Prospect theory
- Common consequence
- Common ratio
- Salience

Betingelser:
Full feedback, partial feedback & no feedback

20230523 
- Tydlig feedback effekt både för spel med och utan ambiguity, med ambiguity mycket större PT effekter generellt. 
- Implementera en salience modell och jämför
- I experimentet kan det vara instressant att jämföra stor och liten variation mellan olika trials, exempelvis alltid samma sannolikheter men olika monetära värden, och att titta på learning curves. 
- Hur får Peterson et al sin ambiguity? 


09052023
- Prospect theory: titta på "kinken" (loss/gain). Replikerar just dessa resultat? Feedback i tidigare experiment?
- Rank-dependent utility: droppa.

- Common consequence, common ratio: mycket gjort.

Köra alla effekter i ett enda stort experiment. Variera endast feedbacktyp.

Gustav tittar på Peterson-datan.

Experiment 

- Vi måste veta att allt replikerar 
- Prospect-theory (den behöver vi inte kolla på) (ska sammanfattas av Mattias) 
- Ambiguity (Ola) 
- Common consequence, common ratio, osv (see slides from behavioral econ course) (Gustav) 
- Salience (Ola)
- Rank-dependent utility (Mattias) 
- Pro-social preferences (kanske)
- Fyra paradigm (sampling, full feedback, partial feedback, description).
- Bayesianskt också, inte minst för Bayesfaktorer så vi kan kolla power
- Använd två bingo blowers, ibland med utskrivna sannolikheter


Väg framåt: 

- Vi sammanfattar våra olika områden i dokumentet
    1. Replikerar dem?
    2. Har någon gjort vad vi vill göra? 
    3. Hur skiljer man på feedback och inte? 
(Gammal)
1. Identifiera en större grupp möjliga effekter
2. Kolla så att de gjorts tillräckligt 
3. Går det att göra med feedback? 
4. Replikerar det? 

25102023
1. Mellangrupps: full feedback, partial feedback, description.
2. Konstanthåll presentationsformat: siffror (tabeller).
3. Slumpar stimuli tailored för effekter vi undersöker.
4. Hur många


\end{comment}


\maketitle

\begin{abstract}
Your abstract.
\end{abstract}

\section{Introduction}

In description \citep{Hertwig2004, Hertwig2009} paradigms, the information which participants may base their decisions on is stated explicitly. They then make a hypothetical choice between the options or courses of action. Much like a one-shot game can be played several times, several trials of such hypothetical choices can be included back-to-back in a single session. When doing so in behavioural economic experiments, choices are typically (REFS WITH EXAMPLES) only realised after the session has concluded, if at all. (Often, only a single randomly selected choice is realised e.g. REFS). Psychologists further distinguish between partial feedback and full feedback paradigms \citep{Camilleri2011}. In partial feedback paradigms, each choice participants make is realised then and there: immediately after locking in their answer, they are informed of the outcome. Full feedback paradigms are like partial feedback paradigms except that participants are also informed of the foregone outcome(s): the opportunity cost(s) of their decision.

We will here focus on four well-established empirical phenomena:

\begin{itemize}
    \item Fourfold pattern of risk attitudes (REF)
    \item Common consequence effect
    \item Common ratio effect
    \item Salience effect
\end{itemize}

If these phenomena disclose deeply held tastes \citep{Bentham1780-BENITT} of participants, they should be robust to arbitrary design choices \citep[cf.][]{Oprea2024} such as provision of No, Partial, or Full feedback. If the choices are instead only quite indirectly related to tastes \citep{McGranaghan2024}, rather emerging from the mechanics of human information processing \citep{Simon1955}, Partial and Full feedback paradigms should allow participants to use cognitive mechanisms \citep{Fodor1983} that are unavailable under No feedback. Most conspicuously, the feedback may disclose contingencies between outcomes and features of the options \citep{Juslin2003, Brunswik1955} that participants may rely on. If prominent phenomena are not robust to provision of No, Partial and Full feedback, theories should not treat them as universals of economic behaviour \citep[e.g.][]{Kahneman1979, Tversky1992}. Rather, they should be treated as interesting and perhaps provocative \citep[e.g.][]{Allais1953} special cases. Theory should be able to accommodate them \citep{Rabin2013} without overgeneralising them \citep{Morgenstern1979}.


\section{Related literature}

\subsection{Prospect theory and rank-dependent utility}

In a standard expected utility model, the agents initial state is typically not explicitly modelled as affecting their behaviour. Rather, initial states (e.g. existent wealth) are incorporated into kitchen-sink concepts (e.g. "preference for money"). This could be thought of as related to the philosophical tradition of thought experiments in that decision making is abstracted and decontextualised: a choice problem is presented as if occurring in a universe of its own to facilitate closer inspection \citep[cf. "credible worlds"][]{Sugden2000}. Prospect theory \cite{Kahneman1979, Tversky1992} may be regarded as a reaction to this in that it considers the agent as looking out from some perceived reference state while also connecting it to the notion from heuristics-and-biases programme that humans tend to simplify complex decision problems \citep[see e.g. "attribute substitution",][]{Kahneman2002}.

The original prospect theory \citep{Kahneman1979} consists of one "editing" stage and one "evaluation" stage. In the editing stage, the decision problem - a choice between monetary gambles in that article - is subjected to interpretative, simplifying operations (e.g. cancellation of branches of options). In the evaluation stage, risky options are evaluated as per expected utility theory but with two major differences. Firstly, subjective weights are imposed on the objective, stated probabilities. Although not clearly stated, these weights seem to represent actual risk attitudes (in terms of deeply held convictions) rather than imperfect mapping between the stated probability and an internal number line \citep[see e.g.][]{Communicatingrisk}. Secondly, stated outcomes are also filtered through subjective weights but these differ depending on whether the outcome is a gain or a loss relative to some reference point. This reference point is determined by intentional aspects: a loss of some sum may still be a gain if the individual had reconciled themselves with an even greater expected loss. The same end state can thus be valued differently depending on whether one arrives there through a (perceived) windfall and then a (perceived) loss (e.g. receive £1000, then lose £100) or just a gain (e.g. receive £900). For options involving both certain and risky outcomes, people are claimed to consider the \emph{value difference} between the two under a subjective probability weight.

However, prospect theory implies violations of first stochastic dominance - a dominated option may be selected over the dominating one. This may be avoided by assuming that dominated options are cancelled but this instead introduces intransitivity. As a response to this challenge, \cite{Quiggin1981} introduced the rank-dependent utility model (RDU). RDU not only concedes that people operate on subjectively weighted probabilities. Its perhaps more radical proposal is that the perceived attractiveness of an option is not determined merely by its outcome, but by how favourably it compares to the outcomes of the other options \cite{Abdellaoui2009}. In a nutshell: what do I gain by selecting the option \emph{over and above what I could have gained from the alternative}?

This feature was introduced into the successor of prospect theory, Cumulative prospect theory \citep[CPT,][]{Tversky1992}. In addition, the model was extended to apply to uncertain \citep{Knight1921-KNIRUA} outcomes and constraints were placed on the subjective probability weights to bring them more in line with normative theory. Also, the weighting functions for gains and losses were more clearly stated to be ''S shaped'' - concave near the reference point and convex far away from it for gains and convex near the reference point and concave far away from it for losses (in \cite{Kahneman1979}, the authors appeared more agnostic as to the graph of the weighting functions).

CPT has been falsified \citep{Birnbaum2008} and is not well-supported even in the original data \citep{Regenwetter2022} from \cite{Tversky1992}. The experiment effects that inspired it, however, ... ... ...


\citet{list2006using} finds evidence of prospect theory among inexperienced consumers, but that experienced consumers look more like rational behavior. 

A lot of data on binary gambles \citet{Peterson2021}. 

For common consequence, common ratio effect: 
\citet{Cubitt1998} seems like a relevant experiment
\citet{Blavatskyy2022} does kind of a meta-study on common consequences, and finds that it is sensitive to specification. 




\section{Ambiguity attitudes}
From \citet{Trautmann_amb_summary}.
\subsection{Elicitation Methods}
\subsubsection{Ellsberg urns (2 or 3 color)}
The two- and three-color Ellsberg tasks are methods used to elicit ambiguity attitudes. In the two-color task, an urn contains 30 balls, 10 of which are red and 20 of which are either yellow or black in an unknown proportion. In the three-color task, an urn contains nine balls, three of which are red and six of which are either yellow or black in an unknown proportion. Participants are asked to choose between two bets: one that pays off if a red ball is drawn from the urn and one that pays off if a non-red ball is drawn. The first bet has a known probability of success (1/3 in the three-color task and 1/4 in the two-color task), while the second bet has an unknown probability of success due to the ambiguous nature of the urn contents. The choices made by participants can reveal their attitudes towards ambiguity.

In the Ellsberg tasks, researchers can control for beliefs by using a baseline probability that is known to all participants. For example, in the two-color task, the probability of drawing a red ball from the urn is always 1/3, while in the three-color task, the probability is always 1/4. By holding this baseline probability constant across all participants, researchers can ensure that any differences in choices between risky and ambiguous bets are due to ambiguity aversion rather than differences in beliefs about the underlying probabilities.

\subsubsection{Second order probabilites}
In this method, participants are presented with a series of compound lotteries, each consisting of two stages. In the first stage, a probability distribution is presented that describes the likelihood of different outcomes in the second stage. In the second stage, a set of outcomes and their associated probabilities are presented. The probabilities in the second stage are drawn from the distribution presented in the first stage, but participants do not know what that distribution is. Participants are asked to choose between two compound lotteries, one with known probabilities and one with unknown probabilities due to ambiguity about the underlying distribution. The choices made by participants can reveal their attitudes towards ambiguity.

This method allows for easy variation of the degree of ambiguity while controlling for beliefs. However, it is not clear how far compound lotteries capture ambiguity in the sense of Ellsberg's unknown probabilities.

\subsubsection{Source method}
The source method is another way to elicit ambiguity attitudes. In this method, participants are presented with a series of bets that involve different sources of uncertainty. For example, participants might be asked to choose between a bet on the temperature in Beijing and a bet on the temperature in Tokyo, both of which are unknown to the participant. The bets are designed so that they have equal expected values but differ in the degree of ambiguity.

To control for beliefs, participants are asked to provide estimates of the probabilities associated with each source of uncertainty before making their choices. This allows researchers to determine whether differences in choices between risky and ambiguous bets are due to ambiguity aversion or differences in beliefs about the underlying probabilities.

The source method has been used successfully to elicit ambiguity attitudes in various contexts, including financial decision-making and medical decision-making. However, it can be challenging to design bets that differ only in their degree of ambiguity while holding other factors constant. Additionally, there may be individual differences in how people interpret and respond to different sources of uncertainty, which could affect the reliability and validity of results. 

\subsection{Findings}

The findings summarized in the meta study by \citep{Trautmann_amb_summary} suggests that ambiguity attitudes is a robust phenomenon across various elicitation methods, but the degree to which people exhibit ambiguity aversion can vary depending on how uncertainty is presented. For example, people may be more likely to exhibit stronger ambiguity aversion when uncertainty is presented using natural sources (such as asset markets or weather conditions) rather than artificial sources (such as balls and urns). Moreover, elicitation method seems to matter, where choice tasks (between e.g. risky and ambiguous bets) elicit lower ambiguity aversion than corresponding valuation tasks (WTP). Ambiguity attitudes seems to show a pattern remenicient of that of prospect theory. Indeed, subjects are ambiguity averse for moderate percieved likelihoods but shows ambiguity seeking behavior for unlikely events. Same is true over the payoff domain: Ambiguity averse for gain- and mixed domain  but ambiguity seeking in the loss domain \citep{kocher2018ambiguity}. Interestingly,  in a three-color urn experiment about 30 \% are inconsistant across identical tasks.  

\section{Focusing/salience}
A theoretical framework proposed by Kőszegi and Szeidl (2013), Bordalo et al. (2014) to explore how people make choices. According to this framework, decision makers evaluate consumption options from a set of options referred to as the consideration set. The authors test the behavioral predictions of this model in the context of intertemporal choice, which involves making decisions about how to allocate resources over time. 

One key assumption of the model is that decision makers place more weight on attributes where options differ more. In other words, they are more likely to focus on differences between options rather than similarities. The authors conduct experiments to test whether this assumption holds up in practice, and find evidence that supports it.

\subsection{Model \citet{bordalo2012salience}}

\begin{itemize}
\item \textbf{Value Function:} The decision maker uses a value function \(v\) to evaluate lottery payoffs relative to the reference point of zero. The expected utility maximizer evaluates \(Li\) as:
\[
V(Li) = \sum_{s \in S} \pi_s v(x_i^s)
\]
where \(\pi_s\) is the probability of state \(s\) and \(x_i^s\) is the payoff of lottery \(Li\) in state \(s\).

The Local thinker instead evaluates

\(Li\) as:
\[
V(Li) = \sum_{s \in S} \pi_s \omega^i_s v(x_i^s)
\]
where $\omega$ is a weight that is increasing in the salience of a state $s$.

\item \textbf{Salience Function:} The local thinker departs from the standard model by overweighting the lottery’s most salient states in \(S\). The salience of state \(s\) for lottery \(Li\), \(i = 1, 2\), is a continuous and bounded function \(\sigma(x_i^s, x_{-i}^s)\) that satisfies three conditions:
  \begin{enumerate}
  \item \textbf{Ordering:} If for states \(s, \tilde{s} \in S\) we have that \([x_{min}^s, x_{max}^s] \subseteq [x_{min}^{\tilde{s}}, x_{max}^{\tilde{s}}]\), then \(\sigma(x_i^s, x_{-i}^s) \leq \sigma(x_i^{\tilde{s}}, x_{-i}^{\tilde{s}})\).
  \item \textbf{Diminishing Sensitivity:} If \(x_j^s \geq 0\) for \(j = 1, 2\), then for any \(\delta > 0\), \(\sigma(x_i^s + \delta, x_{-i}^s + \delta) \leq \sigma(x_i^s, x_{-i}^s)\).
  \item \textbf{Reflection:} For any two states \(s, \tilde{s} \in S\) such that \(x_j^s, x_j^{\tilde{s}} \geq 0\) for \(j = 1, 2\), we have \(\sigma(x_i^s, x_{-i}^s) = \sigma(x_i^{\tilde{s}}, x_{-i}^{\tilde{s}})\) if and only if \(\sigma(-x_i^s, -x_{-i}^s) = \sigma(-x_i^{\tilde{s}}, -x_{-i}^{\tilde{s}})\).
  \end{enumerate}
\end{itemize}

The local thinker evaluates a lottery by inflating the relative weights attached to the lottery's most salient states. Specifically, \(s\) is overweighted if and only if it is more salient than average.

\subsection{Model \citet{kHoszegi2013model}}
\begin{enumerate}
\item{Choice Set} $C \in R^K$ \\
\item{Attributes} $K$ could be interpreted as the states of the world, with corresponding probabilities $\pi_k$ and  $\sum_{k=1}^{K} \pi_k$ 
\item{Expected Utility} $U(c) = \sum_{k=1}^{K} \pi_k u_k(c_k)$ \\
\item{Focus-Weighted Expected Utility}  $\sum_{k=1}^{K} g_k \pi_k u_k(c_k)$ \\
\item{Focus weight} $g_k = g(\Delta_k(C))$ \\
\item{Range}  $\Delta_k(C) = \max_{c \in C} u_k(c_k) - \min_{c \in C} u_k(c_k)$ \\
\item{Focus weight increasing Function}: $g'(\Delta)>0$

\end{enumerate}
Model easier to handle but more narrow that \citet{bordalo2012salience} and essentially only considers the order effect in that model.

\subsection{Elicitation methods}
\subsubsection{Indirect method}
Method in Andersson et al. (20XX) is indirect. For two given target options alternate a third decoy option to make different aspects of target options more salient. Study conducted a series of experiments to elicit preferences related to intertemporal choice, but in principle they could concern risky bets. In these experiments, participants were presented with a series of hypothetical choices involving different options for receiving rewards at different points in time. 

To test the idea that differences between options can attract decision makers attention, the authors manipulated the attributes of these options in various ways. For example, they added options that increased the maximal difference in the original option's strongest attribute. They also varied the number of options available and the order in which they were presented.

Participants were then asked to choose between these different options, and their choices were recorded and analyzed. The authors used statistical methods to determine whether there was evidence to support their hypothesis that people are more likely to choose an option when it has a stronger difference in its strongest attribute. Evidence from several experiments support this hypothesis. 

We estimated salience for risky bets also but it never made it to a WP. We wanted to look at eye-tracking but when we introduced that the salience effect disappeared. 

\subsubsection{Direct method}
A more direct method is used in \citet{dertwinkel2020salience}. They compare two risky gambles with same mean and variance but with different skewness (i.e. Mao pairs). By changing the correlation between the lotteries, they can increase/decrease focus on either lottery. They find evidence supporting this prediction.  



\section{Methods}

\subsection{Gamble parameters}

All the empirical phenomena under investigation here are less than universal (see e.g. REFs). We endeavoured to select parameter values for the gambles such that we maximised the chances of obtaining the phenomena in the no-feedback condition; if we cannot demonstrate the phenomena to begin with, it makes little sense to ask if they are robust to feedback. We therefore reviewed the literature to identify parameter values that would provide us with a good chance of observing the effects.

\subsection{Fourfold pattern of risk attitudes}

Se mejl från Ola.

\subsubsection{Common ratio effect}

We reviewed the 143 experiments reported in Table 1 in \cite{Blavatskyy2023} to identify the one with the highest proportion of participants who exhibited choice patterns consistent with the common ratio effect. The front runners were Experiment 1 \& 3 in \cite{DeKay2016} (60.9\%) and what \cite{Blavatskyy2023} called the ``1m p0.05'' condition in \cite{MacCrimmon1979} (70.6\%). However, \cite{MacCrimmon1979} only included 17 participants in that experiment why there is a good deal of uncertainty in the estimated prevalence. We therefore opted to follow \cite{DeKay2016}.

We used the 22 gambles from Experiment 1 \& 3 in \cite{DeKay2016}.

\begin{table}[h!]
\centering
\begin{tabular}{c c|cc|cc}
\hline
\textbf{Exp.} & \textbf{Choice} & \multicolumn{2}{c|}{\textbf{Option A}} & \multicolumn{2}{c}{\textbf{Option B}} \\
 &  & Probability & Payoff & Probability & Payoff \\
\hline

\multicolumn{6}{c}{\textbf{Experiment 1}} \\
\hline
1 & 1  & 30\% / 70\% & \$110 / \$0 & 40\% / 60\% & \$70 / \$0 \\
1 & 2  & 25\% / 75\% & \$60 / \$0  & 20\% / 80\% & \$100 / \$0 \\
1 & 3  & 15\% / 85\% & \$140 / \$0 & 30\% / 70\% & \$80 / \$0 \\
1 & 4  & 2\% / 98\%  & \$50 / \$0  & 1\% / 99\%  & \$120 / \$0 \\
1 & 5  & 50\% / 50\% & \$60 / \$0  & 25\% / 75\% & \$90 / \$0 \\
1 & 6  & 40\% / 60\% & \$80 / \$0  & 30\% / 70\% & \$70 / \$0 \\
1 & 7  & 65\% / 35\% & \$150 / \$0 & 85\% / 15\% & \$120 / \$0 \\
1 & 8  & 80\% / 20\% & \$100 / \$0 & 100\% / 0\% & \$60 / \$0 \\
1 & 9  & 2\% / 98\%  & \$130 / \$0 & 4\% / 96\%  & \$70 / \$0 \\
1 & 10 & 45\% / 55\% & \$120 / \$0 & 90\% / 10\% & \$50 / \$0 \\
1 & 11 & 10\% / 90\% & \$150 / \$0 & 25\% / 75\% & \$70 / \$0 \\

\hline
\multicolumn{6}{c}{\textbf{Experiment 3}} \\
\hline
3 & 1  & 30\% / 70\% & \$110 / \$0 & 40\% / 60\% & \$70 / \$0 \\
3 & 2  & 25\% / 75\% & \$60 / \$0  & 20\% / 80\% & \$100 / \$0 \\
3 & 3  & 15\% / 85\% & \$140 / \$0 & 30\% / 70\% & \$80 / \$0 \\
3 & 4  & 2\% / 98\%  & \$50 / \$0  & 1\% / 99\%  & \$120 / \$0 \\
3 & 5  & 50\% / 50\% & \$60 / \$0  & 25\% / 75\% & \$90 / \$0 \\
3 & 6  & 40\% / 60\% & \$80 / \$0  & 30\% / 70\% & \$70 / \$0 \\
3 & 7  & 65\% / 35\% & \$150 / \$0 & 85\% / 15\% & \$120 / \$0 \\
3 & 8  & 80\% / 20\% & \$100 / \$0 & 100\% / 0\% & \$60 / \$0 \\
3 & 9  & 2\% / 98\%  & \$130 / \$0 & 4\% / 96\%  & \$70 / \$0 \\
3 & 10 & 45\% / 55\% & \$120 / \$0 & 90\% / 10\% & \$50 / \$0 \\
3 & 11 & 10\% / 90\% & \$150 / \$0 & 25\% / 75\% & \$70 / \$0 \\

\hline
\end{tabular}
\caption{Choice sets for Experiments 1 and 3}
\end{table}

%Experiment 1 included the following 11 options:

%Option A:
%30\% chance that you get \$110
%70\% chance that you get no money
%Option B:
%40\% chance that you get \$70
%60\% chance that you get no money

%Option A:
%25\% chance that you get \$60
%75\% chance that you get no money
%Option B:
%20\% chance that you get \$100
%80\% chance that you get no money

%Option A:
%15\% chance that you get \$140
%85\% chance that you get no money
%Option B:
%30\% chance that you get \$80
%70\% chance that you get no money

%Option A:
%2\% chance that you get \$50
%98\% chance that you get no money
%Option B:
%1\% chance that you get \$120
%99\% chance that you get no money

%Option A:
%50\% chance that you get \$60
%50\% chance that you get no money
%Option B:
%25\% chance that you get \$90
%75\% chance that you get no money

%Option A:
%40\% chance that you get \$80
%60 \% chance that you get no money
%Option B:
%30\% chance that you get \$70
%70\% chance that you get no money

%Option A:
%65\% chance that you get \$150
%35\% chance that you get no money
%Option B:
%85\% chance that you get \$120
%15\% chance that you get no money

%Option A:
%80\% change that you get \$100
%20\% chance that you get no money
%Option B:
%100\% chance that you get \$60

%Option A:
%2\% chance that you get \$130
%98\% chance that you get no money
%0ption B:
%4\% chance that you get \$70
%96\% chance that you get no money

%Option A:
%45\% chance that you get \$120
%55\% chance that you get no money
%Option B:
%90\% chance that you get \$50
%10\% chance that you get no money

%Option A:
%10\% chance that you get \$150
%90\% chance that you get no money
%Option B:
%25\% chance that you get \$70
%75\% chance that you get no money


%Experiment 3 included the following 11 options:

%Option A:
%30\% chance that you get \$110
%70\% chance that you get no money
%Option B:
%40\% chance that you get \$70
%60\% chance that you get no money

%Option A:
%25\% chance that you get \$60
%75\% chance that you get no money
%Option B:
%20\% chance that you get \$100
%80\% chance that you get no money

%Option A:
%15\% chance that you get \$140
%85\% chance that you get no money
%Option B:
%30\% chance that you get \$80
%70\% chance that you get no money

%Option A:
%2\% chance that you get \$50
%98\% chance that you get no money
%Option B:
%1\% chance that you get \$120
%99\% chance that you get no money

%Option A:
%50\% chance that you get \$60
%50\% chance that you get no money
%Option B:
%25\% chance that you get \$90
%75\% chance that you get no money

%Option A:
%40\% chance that you get \$80
%60 \% chance that you get no money
%Option B:
%30\% chance that you get \$70
%70\% chance that you get no money

%Option A:
%65\% chance that you get \$150
%35\% chance that you get no money
%Option B:
%85\% chance that you get \$120
%15\% chance that you get no money

%Option A:
%80\% chance that you get \$100
%20\% chance that you get no money
%Option B:
%100\% chance that you get \$60

%Option A:
%2\% chance that you get \$130
%98\% chance that you get no money
%0ption B:
%4\% chance that you get \$70
%96\% chance that you get no money

%Option A:
%45\% chance that you get \$120
%55\% chance that you get no money
%Option B:
%90\% chance that you get \$50
%10\% chance that you get no money

%Option A:
%10\% chance that you get \$150
%90\% chance that you get no money
%Option B:
%25\% chance that you get \$70
%75\% chance that you get no money



\subsection{Common consequence effect}

We reviewed the 81 sets of experiments in Table 1 in \cite{Blavatskyy2022} to identify the one with the highest Conlisk z-statistic \citep{Conlisk_z_stat}, indicating statistical evidence of the common consequence effect, which was \cite{Sopher1993}.

We use the gambles from both experiments in that article:

\begin{table}[htbp]
\centering
\caption{Lotteries in Experiment I}
\begin{tabular}{llcccc}
\hline
\textbf{Treatment} & \textbf{Question \#} & \textbf{Lottery} & \textbf{Prob [\$0]} & \textbf{Prob [\$1M]} & \textbf{Prob [\$5M]} \\
\hline
\multirow{10}{*}{\parbox{2.5cm}{on-border\\and framing}}
& 1 & A & 0.00 & 1.00 & 0.00 \\
&   & B & 0.01 & 0.89 & 0.10 \\
& 2 & C & 0.89 & 0.11 & 0.00 \\
&   & D & 0.90 & 0.10 & 0.00 \\
& 3 & E & 0.00 & 0.11 & 0.89 \\
&   & F & 0.01 & 0.00 & 0.99 \\
& 4 & G & 0.79 & 0.11 & 0.10 \\
&   & H & 0.80 & 0.00 & 0.20 \\
& 5 & J & 0.01 & 0.89 & 0.10 \\
&   & K & 0.02 & 0.78 & 0.20 \\
\hline
\multirow{10}{*}{off-border}
& 1 & A$^*$ & 0.01 & 0.98 & 0.01 \\
&   & B$^*$ & 0.02 & 0.87 & 0.11 \\
& 2 & C$^*$ & 0.80 & 0.19 & 0.01 \\
&   & D$^*$ & 0.81 & 0.08 & 0.11 \\
& 3 & E$^*$ & 0.01 & 0.19 & 0.80 \\
&   & F$^*$ & 0.02 & 0.08 & 0.90 \\
& 4 & G$^*$ & 0.70 & 0.19 & 0.11 \\
&   & H$^*$ & 0.71 & 0.08 & 0.21 \\
& 5 & J$^*$ & 0.02 & 0.87 & 0.11 \\
&   & K$^*$ & 0.03 & 0.76 & 0.21 \\
\hline
\end{tabular}
\end{table}


\begin{table}[htbp]
\centering
\caption{Lotteries in Experiment II}
\begin{tabular}{llcccc}
\hline
\textbf{Treatment} & \textbf{Question \#} & \textbf{Lottery} & \textbf{Prob [\$0]} & \textbf{Prob [\$1M]} & \textbf{Prob [\$5M]} \\
\hline
\multirow{10}{*}{on-border}
& 1 & A & 0.000 & 1.000 & 0.000 \\
&   & B & 0.100 & 0.000 & 0.900 \\
& 2 & C & 0.700 & 0.300 & 0.000 \\
&   & D & 0.730 & 0.000 & 0.270 \\
& 3 & E & 0.000 & 0.300 & 0.700 \\
&   & F & 0.030 & 0.000 & 0.970 \\
& 4 & G & 0.380 & 0.620 & 0.000 \\
&   & H & 0.442 & 0.000 & 0.558 \\
& 5 & J & 0.000 & 0.620 & 0.380 \\
&   & K & 0.062 & 0.000 & 0.938 \\
\hline
\multirow{10}{*}{off-border}
& 1 & A$^*$ & 0.010 & 0.900 & 0.090 \\
&   & B$^*$ & 0.090 & 0.100 & 0.810 \\
& 2 & C$^*$ & 0.710 & 0.200 & 0.090 \\
&   & D$^*$ & 0.720 & 0.100 & 0.180 \\
& 3 & E$^*$ & 0.010 & 0.200 & 0.790 \\
&   & F$^*$ & 0.020 & 0.100 & 0.880 \\
& 4 & G$^*$ & 0.390 & 0.520 & 0.090 \\
&   & H$^*$ & 0.432 & 0.100 & 0.468 \\
& 5 & J$^*$ & 0.010 & 0.520 & 0.470 \\
&   & K$^*$ & 0.052 & 0.100 & 0.848 \\
\hline
\end{tabular}
\end{table}


\subsection{Salience effect}

We generate gambles intended to test the salience effect by constructing so-called Mao pairs \citep[see for example][]{DertwinkelKalt2019}. These have identical expected values and variances, but one option has skewness of $S$ while the other has skewness of $-S$. That is, one option is left skewed while the other is right skewed. We set the range of expected values and variances of the generated gambles to be equal to the range used in Experiment 2 in \cite{DertwinkelKalt2019}.

\begin{table}[htbp]
\centering
\caption{Mao pairs used in Experiment 2 in \cite{dertwinkel2020salience}}
\begin{tabular}{l l c c cc}
\hline
\textbf{Left-skewed lottery} & \textbf{Right-skewed lottery} & \textbf{Variance} & \textbf{Abs. skewness} & \multicolumn{2}{c}{\textbf{Rel. skewness}} \\
 & & & & $\eta = 0$ & $\eta = 1$ \\
\hline
(120, 90\%; 0, 10\%)   & (96, 90\%; 216, 10\%)  & 1296 & $\pm 2.7$ & $-2.7$ & $-1.5$ \\
(135, 64\%; 60, 36\%)  & (81, 64\%; 156, 36\%)  & 1296 & $\pm 0.6$ & $-0.6$ & $1.0$ \\
(40, 90\%; 0, 10\%)    & (32, 90\%; 72, 10\%)   & 144  & $\pm 2.7$ & $-2.7$ & $-1.5$ \\
(45, 64\%; 20, 36\%)   & (27, 64\%; 52, 36\%)   & 144  & $\pm 0.6$ & $-0.6$ & $1.0$ \\
(80, 90\%; 0, 10\%)    & (64, 90\%; 144, 10\%)  & 576  & $\pm 2.7$ & $-2.7$ & $-1.5$ \\
(90, 64\%; 40, 36\%)   & (54, 64\%; 104, 36\%)  & 576  & $\pm 0.6$ & $-0.6$ & $1.0$ \\
\hline
\end{tabular}
\end{table}


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