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\title{Better is worse, worse is better: Reexamination of violations of dominance in intertemporal choice}

\author{
Cheng-Ming Jiang\thanks{Institute of Neural Management Sciences, College of Economics and
Management, Zhejiang University of Technology}
\and
Hong-Mei Sun\thanks{College of Management, Tianjin University of Traditional Chinese
Medicine}
\and
Long-Fei Zhu\thanks{Institute of Neural Management Sciences, College of Economics and
Management, Zhejiang University of Technology}
\and
Lei Zhao\footnotemark[3]
\and
Hong-Zhi Liu\thanks{CAS Key Laboratory of Behavioral Sciences, Institute of Psychology,
Chinese Academy of Sciences}
\and
Hong-Yue Sun\thanks{Corresponding author. Department of Psychology,
  College of Education, Shanghai Normal University, 100 Guilin Road, 
  Xuhui District, Shanghai (200234), P.R. China. E-mail: sunhy @ shnu.edu.cn.}
}

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\newcommand{\jdate}{May 2017}
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{\small \href{\jref}{\jhead}, \jdate, pp.\ 253--259}
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\maketitle

%\begin{latexonly}
\vspace{-3mm}
\begin{onecolabstract}
%\end{latexonly}
  Recently, Scholten and Read (2014) found new violations of dominance
  in intertemporal choice. Although adding a small receipt before a
  delayed payment or adding a small delayed receipt after an immediate
  receipt makes the prospect objectively better, it decreases the
  preference for that prospect (better is worse). Conversely, although
  adding a small payment before a delayed receipt or adding a small
  delayed payment after an immediate payment makes the prospect
  objectively worse, it increases the preference for that prospect
  (worse is better). Scholten and Read explained these violations in
  terms of a preference for improvement. However, to produce
  violations such as these, we find that the temporal sequences need
  not be constructed as Scholten and Read suggested. In this study,
  adding a small receipt before a dated receipt (thus constructed as
  improving) or adding a receipt after a dated payment (thus
  constructed as improving) decreases preferences for those
  prospects. Conversely, adding a small payment after a dated receipt
  (thus constructed as deteriorating) or adding a small payment before
  a delayed payment (thus constructed as deteriorating) increases
  preferences for those prospects.

\smallskip
\noindent
Keywords: sequences, intertemporal choice, dominance, a preference for
improvement, salience
%\begin{latexonly}
\end{onecolabstract}\bigskip
]
%\end{latexonly}

{\renewcommand{\thefootnote}{}

\footnotetext{

  This research was supported by the National Natural Science
  Foundation of China (No.\ 71571164, 71601121), Zhejiang Provincial
  Natural Science Foundation of China (No. LY15C090005, LQ15C090002),
  and the Humanity and Social Science Research Youth foundation of
  Ministry of Education of China (No.\ 14YJC190009), the key research
  base of Philosophy and Social Sciences in Zhejiang Province–Research
  Center of technology innovation and enterprise
  internationalization. The authors would like to thank Jonathan Baron
  and the referees for their valuable comments and suggestions on the
  draft of this paper.

Copyright: \copyright\ 2017.
The authors license this article under the terms of the
\href{http://creativecommons.org/licenses/by/3.0/}{Creative Commons
  Attribution 3.0 License.}
}}

\saythanks

\section{Introduction}
Intertemporal choices are decisions with outcomes that occur over time;
for example, to buy a consumable now or save money for future
education, to prepare a proposal soon or delay it until the last
minute. Researchers have shown that intertemporal choices are often
inconsistent with the economic principles of rationality such as
dominance, which is a cornerstone of any normative theory. Relying both
on the assumption of monotonicity (i.e., that more money is better) and
on the assumption of positive time preferences (i.e., that earlier
rewards are better than later ones and later losses are better than
earlier ones) (Fisher, 1930), in intertemporal choice with money as
outcomes, dominance means that if the cumulative money obtained from
one option in successive time periods is never less than that obtained
from another option, and more is obtained in at least one time period,
then the first option is a dominant option and should therefore be
preferred (Scholten \& Read, 2014).

Scholten and Read have reported new violations of dominance in which
“an intertemporal prospect fares better by making it worse, and fares
worse by making it better.” They found that compared with the choice
between “A. Receive £75 today vs.\ B. Receive £100 in 1 year,” the
preference for the later and larger (LL) options rather than the
sooner and smaller (SS) options was increased given the choices of
“C. Receive £75 today and receive £5 in 1 year vs.\ B. Receive £100 in
1 year” and “A. Receive £75 today vs.\ D.  Pay £5 today and receive
£100 in 1 year.” Adding a small delayed reward to an immediate reward
(C option) makes the prospect objectively better but decreases the
likelihood that the prospect will be chosen; adding an immediate small
loss to a delayed reward (D option) makes the prospect objectively
worse but increases the attractiveness of the prospect. In addition,
when the focal temporal options were changed to losses — i.e.,
“A’. Pay £75 today vs.\ B’. Pay £100 in 1 year,” “A’. Pay £75 today
vs.\ D’. Receive £5 today and Pay £100 in 1 year” and “C’. Pay £75
today and pay £5 in 1 year vs.\ B’. Pay £100 in 1 year” — there was
also a preference shift and a violation of the dominance principle.
Scholten and Read explained these violations in terms of the account
of a preference for improvement\footnote{Actually, Scholten and Read
  (2014) explained these violations in terms of the sequences model.
  However, the sequences model (Loewenstein \& Prelec, 1993) allows
  for both preferences for improvement and spreading and the opposite
  preferences. Therefore, it is misleading to state that improving
  sequences should be preferred according to the sequences model and
  we substitute “a preference for improvement” for “the sequences
  model” in the current paper.}, which states a preference for
improvement over deterioration (the C and D’ options are deteriorating
sequences, whereas the D and C’ options are improving sequences).

Nevertheless, we doubt that a preference for improvement can account for
these violations of dominance. The doubt arises from two avenues of
research. One avenue of research is concerned with whether the account
of a preference for improvement can apply to sequences of unlabeled
money, meaning money that is not attributed to a specific origin, such
as wage payments or rental income (e.g., Loewenstein \& Sicherman,
1991). The evidence for a preference for improvement is primarily
derived from research on sequences of nonmonetary, qualitative
outcomes. For example, in Loewenstein and Prelec’s Example 2, most
people preferred visiting an abrasive aunt on one weekend and then
visiting friends on the next weekend than the reverse. However, there
is no convincing evidence of preferences for improvement in choices
between elementary sequences involving money as outcomes (Read \&
Scholten, 2012). Even in Frederick and Loewenstein’s (2008) Study 2a, a
preference for deterioration emerged using a matching task involving
monetary two-outcome sequences:
\begin{quote}
G. Receive \$2,000 now for signing up as a participant in a 1-year study
and receive another \$1,000 when the study is complete.

H. Receive \$1,000 now for signing up as a participant in a 1-year study
and receive another \$\_\_\_\_\_ when the study is complete.
\end{quote}
Almost no one gave a response below \$2,000, and the mean response was
above \$2,000.

Given Frederick and Loewenstein’s Study 2a using a match other than a
choice task, we conducted a small-scale survey in a classroom with a
choice task asking students to choose between ``L. Receive CNY 200 in
26 weeks and receive CNY 300 in 52 weeks'' and ``M. Receive CNY 300 in
26 weeks and receive CNY 200 in 52 weeks,'' finding that an
overwhelming majority preferred a deteriorated sequence (i.e., option
M) to an improved one (i.e., option L).\footnote{ Forty-five students
  in a classroom chose between ``L. receive CNY 200 in 26 weeks and
  receive CNY 300 in 52 weeks vs.\ M. receive CNY 300 in 26 weeks and
  receive CNY 200 in 52 weeks,'' which was presented in a paper
  questionnaire. The order of options L and M was counterbalanced
  among students. In other words, approximately half of the students
  were offered the choice with the L option presented first (95.5\%
  chose M), whereas the other half of the students were offered the
  choice with the M option presented first (82.6\% chose M). Because
  the order did not affect the students' choices (\textit{p} = 0.17),
  we combined all students' data and found that an overwhelming
  majority (88.9\%) preferred option M (\textit{p} {\textless}
  0.001).}

The other source of doubt arises from our own recent research (Jiang,
Hu \& Zhu, 2014; Jiang, Sun, Zheng, Wang \& Qin, 2016; Sun \& Jiang,
2015). We found some effects in intertemporal choice that might be
related to the violations in Scholten and Read (2014). For example, in
Sun and Jiang's Experiment 3, compared with a choice of ``P. Gaining
CNY 220 in three weeks vs.\ Q. Gaining CNY 270 in five weeks,'' more
people preferred the larger and later (LL) option in the choice of
``P'.  Gaining CNY 220 in three weeks and losing CNY 21 in six weeks
vs.\ Q'. Gaining CNY 270 in five weeks and losing CNY 27 in six
weeks.'' Given both that options P' and Q' were constructed by adding
later losses to options P and Q and that the loss added to option Q
was a little larger than that added to option P, the comparison of the
Q' option with the P' option was objectively worse than the comparison
of the Q option with the P option, and therefore the proportion of
preference for the Q' option should be less than the proportion of
preference for the Q option. However, the real choice data revealed
the reverse trend. The explanation of the preference for improvement
cannot account for this effect because options P' and Q' were both
designed as decreasing sequences and the precipitation speed is even
greater in option Q' (from CNY 270 to --27) than in option P' (from
CNY 220 to --21). We supposed that the effect could be explained by
the salience account; that is, introducing extra dated money to simple
(one dated-money) options could render the money dimension more
salient and cause people to focus and weight money more heavily,
therefore, increasing their preference for the LL options in
intertemporal rewards.

Moreover, we found that, regardless of whether the extra dated-money
was a loss or a gain, inserted before the SS option, inserted between
the SS and LL options, or inserted later than the LL options, its
introduction reduced intertemporal discounting (Sun \& Jiang,
2015). And, the extra dated-money effect can also be extended to
intertemporal choice when the focal intertemporal options are losses
(Jiang et al., 2016). Therefore, we suspect that the salience account
can also explain Scholten and Read's violation of dominance because in
a choice between one dated-money and a sequence of two dated-money,
the sequence is a focal dated-money added with extra dated-money,
which could render the money dimension more salient. Thus, we
considered that, to produce the violations like Scholten and Read's,
it's not necessary to construct sequences as they suggested. To extend
Scholten and Read's violations of dominance and examine whether a
preference for improvement can account for the violations of
dominance, we designed two experiments — Experiment A and Experiment B
— using rewards and payments as focal intertemporal option stimuli. In
the experiments, the two-outcome money sequences are constructed as
deteriorating and objectively worse or improving and objectively
better compared with single-dated money options. If the preference for
an improving (and objectively better) sequence or the distaste for a
deteriorating (and objectively worse) sequence is not obtained, this
result will contradict that explanation of a preference for
improvement.

\subsection{Experiment A}
\subsection{Method}
In Experiment A, 300 students from Zhejiang University of Technology
were approached in the library and randomly assigned to one of three
conditions — the R (reference) condition, the S$^+$ (improving
sequence) and the S$^-$ (deteriorating sequence) conditions, which are
as follows:

\medskip
\noindent
\textbf{R condition:}

U. Receive CNY 200 in 26 weeks 

V. Receive CNY 300 in 52 weeks

\medskip
\noindent
\textbf{S$^+$ condition:}

 W. \parbox[t]{3in}{Receive CNY 5 in 25 weeks and receive CNY 200 in 26 weeks}\smallskip

 V. Receive CNY 300 in 52 weeks

\medskip
\noindent
\textbf{S$^-$ condition:}

 U. Receive CNY 200 in 26 weeks 

 X. \parbox[t]{3in}{Receive CNY 300 in 52 weeks and pay CNY 5 in 53 weeks}
\medskip

The sequence (i.e., option W) in the S$^+$ condition was
thus constructed as an improving sequence and the sequence (i.e.,
option X) in the S$^-$ condition was constructed as a
deteriorating sequence. The choices were presented in a questionnaire
on paper containing other unrelated studies and were listed first to
avoid potential contamination by other studies. After completion, each
participant received a small gift.

\begin{figure}[t]
\includegraphics[width=\columnwidth]{fig1.eps} 
\caption*{Figure 1. Proportion of responses (\%) in the S$^+$, R and
  S$^-$ conditions in Experiment A. Numbers with dashes are 95\%
  confidence intervals.}
\end{figure}

\subsection{Results and Discussion}
One questionnaire for condition S$^+$ was missed during
data collection, leaving 299 participants (143 males,
\textit{M}\textsubscript{age} = 21.69, \textit{SD} = 2.56) for the
final analysis. 

As seen in Figure 1, compared with the choices in R condition, the
preference for the SS option in the S$^+$ condition was decreased
(from 82\% to 48.5\%), ${\chi^2}$(1) = 24.67, \textit{p }{\textless}
0.001, \textit{${\varphi}$} = 0.35, although introducing a small
reward (i.e., receive CNY 5 in 25 weeks) to option U (therefore
becomes option W) renders the prospect objectively better. In the
S$^-$ condition, the preference for the LL option was increased
compared with the choices in R condition (from 18\% to 30\%),
$\chi^2$(1) = 3.95, \textit{p }= 0.047, \textit{${\varphi}$} = 0.14,
although introducing a small payment (i.e., pay CNY 5 in 53 weeks) to
option V (therefore becomes option X) makes the prospect objectively
worse. These preference shifts directly contradict the account of a
preference for improvement. The sequence in S$^+$ condition was
constructed as improving and therefore should be preferred according
to a preference for improvement; whereas the sequence in the S$^-$
condition was constructed as a deteriorating sequence and thus should
be disfavored.

\section[Experiment B]{Experiment B}
In Experiment B, we substituted rewards of focal intertemporal options
with payments.

\subsection{Method}
Another 300 students from Zhejiang University of Technology participated
in this experiment. The data were collected in the
library's study room. Each subject was randomly
assigned to one of three conditions—the R (reference) condition, the
S\textsuperscript{+ }(improving sequence) and the
S$^-$ (deteriorating sequence) conditions, which are
as follows:

\medskip
\noindent
\textbf{R condition:}

 U’. Pay CNY 200 in 26 weeks 

 V’. Pay CNY 300 in 52 weeks

\medskip
\noindent
\textbf{S$^+$ condition:}

 U’. Pay CNY 200 in 26 weeks 

 W’. \parbox[t]{2.8in}{Pay CNY 300 in 52 weeks and receive CNY 5 in 53 weeks}

\medskip
\noindent
\textbf{S$^-$ condition:}

 X’. Pay CNY 5 in 25weeks and pay CNY 200 in 26 weeks  

 V’. Pay CNY 300 in 52 weeks
\medskip

\renewcommand{\baselinestretch}{1.03} \large\normalsize

The sequence (i.e., option W’) in the S$^+$ condition was
thus constructed as an improving sequence and the sequence (i.e.,
option X’) in the S$^-$ condition was constructed as a
deteriorating sequence. The choices were presented in a questionnaire
on paper with other unrelated studies and were listed as the first
problems. After completing the questionnaire, each participant received
a small gift.

\begin{figure}[t]
\includegraphics[width=\columnwidth]{fig2.eps} 
\caption*{Figure 2. Proportion of responses (\%) in the S$^+$, R and
  S$^-$ conditions in Experiment B. Numbers with dashes are 95\%
  confidence intervals.}
\end{figure}

\subsection{Results and Discussion}
Two questionnaires for condition R were missed during data collection,
and one participant in condition S$^-$ did not respond
to the intertemporal choice, leaving 297 participants (142 males,
\textit{M}\textsubscript{age} = 20.88, \textit{SD} = 2.1) for the final
analysis.

As demonstrated in Figure 2, compared with the choices in R condition,
the preference for the LL option was decreased (from 43.9\% to 32\%)
in the S$^+$ condition, ${\chi^2}$(1) = 2.97, \textit{p }= 0.085,
\textit{${\varphi}$} = 0.12, although introducing a small reward
(i.e., receive CNY 5 in 53 weeks) to option V’ (therefore becomes
option W’) makes the prospect objectively better. In the S$^-$
condition, the preference for the SS option was increased compared
with the choices in R condition (from 56.1\% to 70.7\%), ${\chi^2}$(1)
= 4.52, \textit{p }= 0.034, \textit{${\varphi}$} = 0.15, although
introducing a small payment (i.e., pay CNY 5 in 25 weeks) to option U’
(therefore becomes option X’) makes the prospect objectively
worse. These preference shifts are inconsistent with the predictions
of a preference for improvement. The sequence in S\textsuperscript{+
}condition was constructed as improving and therefore should be
preferred, and the sequence in the S$^-$ condition was constructed as
a deteriorating sequence and thus should be disfavored.

\section{General Discussion}
In this study, we reexamined the violations of dominance discovered by
Scholten and Read (2014) and found that the intertemporal options
designed to produce these violations do not necessarily conform to the
principles of a preference for improvement (Loewenstein \& Prelec,
1993). The sequences in this study that were constructed to be
objectively better and to improve (``Receive CNY 5 in 25 weeks and
receive CNY 200 in 26 weeks'' in Experiment A and ``Pay CNY 300 in 52
weeks and receive CNY 5 in 53 weeks'' in Experiment B) had reduced
preferences. The sequences that were constructed to be objectively
worse and to deteriorate (``Receive CNY 300 in 52 weeks and pay CNY 5
in 53 weeks'' in Experiment A and ``Pay CNY 5 in 25 weeks and pay CNY
200 in 26 weeks'' in Experiment B) had increased preferences.

Earlier, in the introduction, we suggested that the salience account
may explain Scholten and Read’s violations of dominance, which was one
of the reasons that we designed the current study. Although the
results reported here are consistent with the salience account’s
predictions that extra-dated money makes the money dimension more
salient and more heavily weighted, thus reducing intertemporal
discounting (Jiang et al., 2014; Jiang et al., 2016; Sun \& Jiang,
2015), we did not examine this account directly. The main purpose of
this paper was to extend Scholten and Read’s violations and to provide
possible evidence against the explanation of a preference for
improvement. An alternative hypothesis to the salience account that
the money dimension weights differently for the sequences conditions
vs.\ the one dated-outcome conditions is that utilities
change. Recently, multiple studies have suggested that, when making
intertemporal choices, people adopt attribute-based approaches such as
comparing difference between outcomes of pairwise options with that of
delays (Dai \& Busemeyer, 2014; Ericson, White, Laibson \& Cohen,
2015; Jiang, Liu, Cai \& Li, 2016; Scholten \& Read, 2010). For
example, when choosing between ``Receive CNY 200 in 26 weeks'' and
``Receive CNY 300 in 52 weeks,'' people compare the difference between
``Receive CNY 200'' and ``Receive CNY 300'' with the difference
between ``26 weeks Delayed'' and ``52 weeks Delayed.'' The
utility-changing hypothesis argues that the comparison between
outcomes seems larger when a small outcome is introduced; for example,
the presence of CNY 5 makes the comparison of ``Receive CNY 200'' and
``Receive CNY 300'' seem larger. This hypothesis is similar to the
evaluability hypothesis proposed by Bateman, Dent, Peters, Slovic and
Starmer (2007) to explain why the attractiveness of playing a simple
game (7/36 to win \$9; otherwise win nothing) is greatly increased by
introducing a small loss (7/36 win \$9; otherwise lose 5¢). Bateman et
al.\ proposed that in the second game, comparison with the small loss
makes \$9 more valued, thus increasing the gamble’s
attractiveness. However, the utility changing hypothesis is confined
to the findings reported here and cannot accommodate the findings of
Jiang et al.\ (2014), Sun and Jiang (2015) and Jiang et al.\ (2016),
because some of the extra-dated money introduced in their studies is
rather large; for example, a common ``Gain CNY 4250 now'' was added to
both of the pairwise options of ``Gain CNY 4800 in a year'' and ``Gain
CNY 8000 in 4 years'' (in Experiment 1A, Jiang et al., 2014).

Recently, several sequence effects, which are when single-dated
outcomes are expanded into sequences of two outcomes, choice results
are contradictory to the predictions of the exponentially-discounted
model\footnote{ The exponentially-discounted model assumes that
  outcomes in future times are discounted by a constant personal
  interest. This is the normative model considered by
  psychologists. }, have been discovered by researchers (Jiang, et al,
2014; Jiang et al.  2016; Magen, Dweck \& Gross, 2008; Rao \& Li,
2011; Read, Olivola \& Hardisty, in press; Read \& Scholten, 2012;
Scholten \& Read, 2014; Sun \& Jiang, 2015; Urminsky \& Kivetz, 2011;
Wu \& He, 2012). The salience account was proposed to explain the
decreased discounting rate when a common money amount was added to
both single-outcome SS and LL options (Jiang et al., 2014; Jiang et
al., 2016; Sun \& Jiang, 2015), and was also suggested to explain the
effects reported here. We should point out that, in these studies, the
amount added was smaller than outcomes of SS options. When the amount
is larger than outcomes of SS options or is zero, other factors may
kick in and make the salience effect trivial (e.g., Read et al., in
press; Read \& Scholten, 2012). The Appendix summarizes whether
different sequence effects can be accommodated by alternative models;
it extends a similar summary by Scholten, Read \& Sanborn (2016).

It can be noted that the timing of SS options delivered in our study
(in 26 weeks) is somewhat different from that in Scholten and Read's
study, in which SS options were delivered immediately (today). The
reason that the timing was designed in this manner, instead of in the
manner of Scholten and Read, is that to construct choice stimuli in
this study, such as ``Receive CNY 5 in 25 weeks and receive CNY 200 in
26 weeks'' (which should be objectively better and improving), the SS
option (``Receive CNY 200 in 26 weeks'') must be delayed instead of
delivered now. However, the sequence structure was retained
(Loewenstein \& Prelec, 1993). Furthermore, in the current study we
cannot describe the proportions of participants who were affected by
the manipulation and exhibited a violation, for we did not use the
same subjects across conditions. Further studies identifying the
source of the individual differences would be more helpful to
understand the nature of this violation of dominance.

\section{Conclusions}
We found the extensions of Scholten and Read (2014)'s violations of
dominance. That is, sequences that were constructed to be objectively
better and to improve returned reduced preferences, whereas sequences
that were constructed to be objectively worse and to deteriorate
returned increased preferences. These violations contradict the
predictions of a preference for improvement originally proposed by
Scholten and Read to explain their violations.

\renewcommand{\baselinestretch}{1.00} \large\normalsize

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%\break
\end{hangparas}

\onecolumn
\noindent
\textbf{Appendix: Whether Sequence Effects Can Be Accommodated by
  the Sequences Model (SM), the Trade off Model (TM), or the Salience
  Account (SA)?}\smallskip

{\small
\noindent
\begin{tabular}{@{}L{2.2in}L{1.25in}L{1.9in}@{}C{.5in}ccc@{}}\toprule
 & \multicolumn{2}{c}{Comparison} &  & \multicolumn{3}{c}{Model}\\\cmidrule{2-3}\cmidrule{5-7}
Sequence effect & Control & Experimental & Effect  & SM & TM & SA\\\midrule
Mere-token effect (Urminsky \& Kivetz, 2011) & $SS$: ($x_S$,$t_S$)\newline
$LL$: ($x_L$, $t_L$) & $SS_M$: ($x_M$, $t_M$; $x_S$, $t_S$)\newline
$LL_M$: ($x_M$, $t_M$; $x_L$, $t_L$) & Less discounting & Yes & Yes & Yes\\\cmidrule{1-7}
Amplification effect (Read \& Scholten, 2012) & $SS_M$: ($x_M$, $t_M$;$x_S$, $t_S$)\newline
$LL_M$: ($x_M$, $t_M$;$x_L$, $t_L$) & $SS_M$: ($x_A$ + $x_M$, $t_M$;$x_S$, $t_S$)\newline
$LL_M$: ($x_A$ + $x_M$, $t_M$;$x_L$, $t_L$) & Less discounting & Yes & Yes & No\\\cmidrule{1-7}
Common-consequence effect (Read \& Scholten, 2012) & $SS$: ($x_S$, $t_S$)\newline
$LL$: ($x_L$, $t_L$) & $SS_C$: ($x_S$, $t_S$; $x_C$, $t_C$)\newline
$LL_C$: ($x_C$, $t_C$; $x_L$, $t_L$) & Less discounting & Yes & Yes & Yes\\\cmidrule{1-7}
Extra-dated money effect$^a$ (Jiang et al., 2014; Jiang et al., 2016; Sun \& Jiang, 2015) & $SS$: ($x_S$, $t_S$)\newline
$LL$: ($x_L$, $t_L$) & Unmixed $SS_E$: ($x_S$, $t_S$) + ($x_E$, $t_E$)\newline
Sequence$^b$ $LL_E$: ($x_L$, $t_L$) + ($x_E$, $t_E$) & Less discounting & No & Yes & Yes\\
 &  & Mixed $SS_E$: ($x_S$, $t_S$) + ($x_E$, $t_E$)\newline
Sequence$^c$ $LL_E$: ($x_L$, $t_L$) + ($x_E$, $t_E$) & Less discounting & No & No & Yes\\\cmidrule{1-7}
Front-end amount effect (Rao \& Li, 2011; Read \& Scholten, 2012) & $SS$: ($x_S$, $t_S$)\newline
$LL$: ($x_L$, $t_L$) & $SS_F$: ($x_F$ + $x_S$, $t_S$)\newline
$LL_F$: ($x_F$, $t_S$; $x_L$, $t_L$) & More discounting & Yes & Yes & No\\\cmidrule{1-7}
Reverse front-end amount effect (Read \& Scholten, 2012) & $SS_A$: ($x_A$ + $x_S$, $t_S$)\newline
$LL_A$: ($x_A$ + $x_L$, $t_L$) & $SS_{FA}$: ($x_F$ + $x_A$ + $x_S$, $t_S$)\newline
$LL_{FA}$: ($x_F$, $t_S$; $x_A$ + $x_L$, $t_L$) & Less discounting & Yes & Yes & Yes\\\cmidrule{1-7}
Relocation effect (Read \& Scholten, 2012) & $SS_F$: ($x_R$ + $x_S$, $t_S$)\newline
$LL_F$: ($x_R$, $t_S$; $x_L$, $t_L$) & $SS_B$: ($x_S$, $t_S$;$x_R$, $t_L$)\newline
$LL_B$: ($x_R$ + $x_L$, $t_L$) & More discounting & Yes & Yes & No\\\cmidrule{1-7}
Violations of dominance (Scholten \& Read , 2014) & $SS$: ($x_S$, $t_S$)\newline
$LL$: ($x_L$, $t_L$) & Unmixed $SS_T$: ($x_S$, $t_S$;$x_T$, $t_T$)\newline
Sequence$^d$ $LL_T$: ($x_L$, $t_L$) & Less discounting & Yes & Yes & Yes\\
 &  & Mixed $SS_T$: ($x_S$, $t_S$)\newline
Sequence$^e$ $LL_T$: ($x_T$, $t_T$;$x_L$, $t_L$)  & Less discounting & Yes & No & Yes\\\cmidrule{1-7}
Extension of violations of dominance (this article) & $SS$: ($x_S$, $t_S$)\newline
$LL$: ($x_L$, $t_L$) & Unmixed $SS_V$: ($x_V$, $t_V$;$x_S$, $t_S$)\newline
Sequence$^f$ $LL_V$: ($x_L$, $t_L$) & Less discounting & No & No & Yes\\
 &  & Mixed $SS_V$: ($x_S$, $t_S$)\newline
Sequence$^g$ $LL_V$: ($x_L$, $t_L$;$x_V$, $t_V$)  & Less discounting & No & No & Yes\\\cmidrule{1-7}
Hidden-zero effect (Magen, Dweck, \& Gross, 2008) & $SS$: ($x_S$, $t_S$)\newline
$LL$: ($x_L$, $t_L$) & $SS_0$: ($x_S$, $t_S$; 0, $t_L$)\newline
$LL_0$: (0, $t_S$; $x_L$, $t_L$) & Less discounting & Yes & Yes & Yes\\\cmidrule{1-7}
Asymmetric hidden-zero effect (Read et al., in press; Wu \& He, 2012) & $SS$: ($x_S$, $t_S$)\newline
$LL$: ($x_L$, $t_L$) & Sooner $SS$: ($x_S$, $t_S$)\newline
zero $LL_0$: (0, $t_S$; $x_L$, $t_L$) & No effect & No & Yes & No\\
 &  & Later $SS_0$: ($x_S$, $t_S$; 0, $t_L$)\newline
zero $LL$: ($x_L$, $t_L$) & Less discounting & Yes & Yes & Yes\\\bottomrule
\end{tabular}

$^a$ $t_E$ can be before $t_S$, between $t_S$ and $t_L$, or after $t_L$. When
$t_E$ is between $t_S$ and $t_L$, extra-dated money effect becomes
common-consequence effect.
$^b$ $x_E$ has a same sign with $x_S$ and $x_L$.
$^c$ $x_E$ has a different sign with $x_S$ and $x_L$.  
$^d$ $x_T$ has a same sign with $x_S$ and $x_L$.
$^e$ $x_T$ has a different sign with $x_S$ and $x_L$.
$^f$ $x_V$ has a same sign with $x_S$ and $x_L$.
$^g$ $x_V$ has a different sign with $x_S$ and $x_L$.
}

\medskip

{\footnotesize
Note 1. Utility functions for SM and TM are assumed to be concave over gains
and convex over losses, which is common in our field (Kahneman \& Tversky, 1979; Loewenstein \& Prelec, 1992).

Note 2. One model that holds the instantaneous utilities of the outcomes in a
sequence are discounted as a function of  their delays and summed as a
discounted utility of the sequence (Scholten et al., 2016) is not
included in the table, because it cannot accommodate most of the sequence
effects listed here, except for the reverse front-end amount effect and
the relocation effect.

Note 3. SM allows for both preferences for improvement and spreading
and the opposite preferences. If we allow SM to accommodate some
sequence effects, we should assume preferences for improvement and
spreading, however, on the other way, if we allow it to accommodate
some other sequence effects, we should assume the opposite
preferences. Therefore, we hold the preferences for improvement and
spreading as priors for SM, because the empirical results in
Loewenstein and Prelec (1993) suggested so.

Note 4. SA is a qualitative rather than a quantitative model at present, which generally assumes that outcomes in sequences, relative to delays, are more weighted compares with single dated outcomes relative to
delays, therefore less discounting is in a choice of sequences than in
a choice of single dated outcomes. If more assumptions, such as that a
utility function is concave over gains, is allowed into SA, more
sequence effects can be accommodated.
\vspace*{\fill}

}
\end{document}
