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{\title Decision Making on the Internet: Comparison of Laboratory Research with Data from Financially-Motivated Participants on the Wo}{\author Michael H. Birnbaum}{\operator Michael H. Birnbaum}{\creatim\yr1999\mo2\dy22\hr13\min29}
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Decision Making on the Internet }{\field{\*\fldinst {\cs21 PAGE }}{\fldrslt {\cs21\lang1024 30}}}{
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Testing Critical Properties of Decision Making on the Internet}{\b
\par }{Michael H. Birnbaum
\par California State University, Fullerton and
\par }\pard \qc\ri-540\sl480\slmult0\widctlpar\outlinelevel0\adjustright {Decision Research Center
\par }\pard \ri-540\widctlpar\outlinelevel0\adjustright {
\par }\pard \qc\ri-540\widctlpar\adjustright {
\par }\pard \ri-540\sl480\slmult1\widctlpar\adjustright {\b File}{: DM on the Internet-8\tab Date: 02-10-99
\par
\par }\pard \ri-540\widctlpar\adjustright {\b Address}{:\tab Michael H. Birnbaum
\par \tab \tab Department of Psychology H-830M
\par \tab \tab California State University, Fullerton
\par \tab \tab P. O. Box 6846
\par \tab \tab Fullerton, CA 92834-6846
\par }\pard \ri-540\sl360\slmult1\widctlpar\adjustright {
\par }{\b Phones}{: \tab (714)-278-2102\tab (714)-278-3514 (Psychology Dept.)
\par }{\b fax}{:\tab \tab (714) 278-7134
\par }{\b e-mail:}{ mbirnbaum@fullerton.edu
\par }\pard \ri-540\widctlpar\outlinelevel0\adjustright {\b Running Head:}{ Testing Decision Theories on the WWW
\par }{\b Word Counts:}{ Abstract 146; Paper 3838 Appendix 838 (total = 4676); 28 References
\par }\pard \ri-540\sl480\slmult0\widctlpar\adjustright {\b Footnotes
\par }{\tab Author\rquote s address: Department of Psychology, C.S.U.F., P.O. Box 6846, Fullerton, CA 92834-6846. Email address: mbirnbaum@fullerton.edu. Web URL
\par }{\field\flddirty{\*\fldinst { HYPERLINK http://psych.fullerton.edu/mbirnbaum/home.htm }{\fs20 {\*\datafield
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\par \tab Support was received from National Science Foundation Grant, SBR-9410572. Thanks are due Jamie N. Patton, Melissa Lott, Juan Navarrete, and Dana Storm for assistance in collecting the laboratory sample.
\par \tab 1. To compute predictions for CPT and the configural weight, TAX models, use Netscape to load the on-line calculator at URL
\par }{\field\flddirty{\*\fldinst { HYPERLINK http://psych.fullerton.edu/mbirnbaum/taxcalculator.htm }{\fs20 {\*\datafield
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\par Additional information on model fitting (including source listings of computer programs) is available from this web site.
\par }\pard \qc\ri-540\sl480\slmult1\widctlpar\adjustright {\page Abstract
\par }\pard \ri-540\sl480\slmult1\widctlpar\adjustright {This paper reviews recent findings that violate a broad class of descriptive theories of decision
making. A new study compares 1224 participants tested via the Internet and 124 undergraduates tested in the laboratory. Both samples confirm systematic violations of stochastic dominance and cumulative independence; new tests also find violations of coa
l
escing. The Internet sample was older, more highly educated, more likely male, and also more demographically diverse than the lab sample. Internet participants were more likely than undergraduates to choose the gamble with higher expected value, but no
o
ne conformed exactly to expected value. Violations of stochastic dominance decrease as education increases, but violations of stochastic dominance and coalescing are still substantial in persons with doctoral degrees who have read a scientific work on de
cision making. In their implications, Internet research and lab findings agree: descriptive decision theories cannot assume that identical consequences can be coalesced.
\par \page \tab Some say that psychological science is based on research with rats, the mentally dis
turbed, and college students. We study rats because they can be controlled, the disturbed because they need help, and college students because they are available. The Internet now makes available a worldwide population. This new medium not only provide
s new research opportunities, but it also raises new questions about sampling and experimental control (Krantz, Ballard, & Scher, 1997; Krantz & Dalal, 1999). How do results from the Internet compare with those obtained in the laboratory?
\par \tab This study explo
res this question with new tests that refute descriptive theories of decision making. Reviews of modern theories, including rank-dependent utility (RDU), rank-and sign-dependent utility (RSDU) and cumulative prospect theory (CPT), can be found in Quiggin
(1993), Luce (1990; 1998), Luce and Fishburn (1991; 1995), Stevenson, Busemeyer, and Naylor (1991), Tversky and Kahneman (1992), Weber (1994), and Wu and Gonzalez (1998). These modern theories account for phenomena that were not explained by earlier theo
ries of Edwards (1954), Karmarkar (1979), and Kahneman & Tversky (1979). However, even these modern theories are now challenged by evidence with newly devised tests.
\par }{\ul Stochastic Dominance
\par }\pard \fi720\ri-540\sl480\slmult1\widctlpar\adjustright {Stochastic dominance is not only considered rational, it is also impli
ed by many descriptive theories, including RSDU, RDU, CPT, and others (Becker & Sarin, 1987; Machina, 1982). A test of stochastic dominance is illustrated in Choice 5 of Table 1 (first row). Birnbaum (1997) proposed this choice as a test between theories
that satisfy stochastic dominance and models that violate it. Configural weight, RAM and TAX models (with parameters estimated in previous studies) imply violation of stochastic dominance in this choice (Birnbaum, 1997; 1998; Appendix).}{\super 1}{
Birnbaum and Navarrete (1998) found that 70% of undergraduates violated stochastic dominance by choosing }{\ul J}{ = }{\ul G}{\endash over }{\ul I}{ = }{\ul G}{+, even though }{\ul G}{+ dominates }{\ul G}{\endash .
\par }\pard \ri-540\sl480\slmult1\widctlpar\adjustright {\ul Coalescing/Event Splitting
\par }{\tab Coalescing also distinguishes decision theories (Birnbaum & Navarrete, 1998; Luce, 1998). C
oalescing (Table 2) assumes that branches with identical consequences can be combined (adding their probabilities), without affecting preference. Coalescing was assumed as an editing principle of prospect theory (Kahneman & Tversky, 1979), but it also fo
llows from RSDU, RDU, CPT, and other theories (Luce, 1998). }{\chbrdr\brdrs\brdrw10 Insert Tables 1 and 2 about here.}{
\par }\pard \fi720\ri-540\sl480\slmult1\widctlpar\adjustright {Because stochastic dominance can be deduced from transitivity, consequence monotonicity, and coalescing (Table 2), Birnbaum and Navarrete (1998) and Bi
rnbaum, et al. (1999) argued that violations of coalescing might cause violations of stochastic dominance, but their studies did not test coalescing directly. Starmer and Sugden (1993) and Humphrey (1995) reported event-splitting effects (violations of co
alescing combined with transitivity); however, Luce (1998) described that evidence as "decidedly weak." This study will use strong, within-subjects tests to determine if event splitting can reverse violations of stochastic dominance.
\par According to coalescing, Choice 11 is the same as Choice 5, since }{\ul GS}{+ (}{\ul U}{ in Table 1) is simply the split version of }{\ul G}{+ (}{\ul I}{) and }{\ul GS}{\endash (}{\ul V}{) is the split version of }{\ul G}{\endash (}{\ul J}{
). Any theory assuming coalescing and transitivity implies }{\ul G}{+ }{\f20 \u-3994\'66}{ }{\ul G}{\endash iff }{\ul GS}{+ }{\f20 \u-3994\'66}{ }{\ul GS}{\endash , where }{\f20 \u-3994\'66}{ represents preference
. The configural weight, RAM and TAX models with previously estimated parameters predict that people should prefer }{\ul G}{\endash }{\f20 \u-3994\'66\u-4064\'20}{\ul G}{+ in Choice 5 and }{\ul GS}{+ }{\f20 \u-3994\'66\u-4064\'20}{\ul GS}{\endash
in Choice 11.}{\super 1}{
\par }\pard \ri-540\sl480\slmult1\widctlpar\adjustright {\ul Lower and Upper Cumulative Independence
\par }{\tab Birnbaum (1997) also devised two cumulative indepe
ndence conditions that test modern theories. Any theory that assumes comonotonic branch independence, consequence monotonicity, transitivity, and coalescing implies both lower and upper cumulative independence (Table 2). Whereas RDU, RSDU, and CPT must s
atisfy these properties, RAM and TAX models violate them.
\par }\pard \fi720\ri-540\sl480\slmult1\widctlpar\adjustright {Choices 6 and 8 of Table 1 test lower cumulative independence. Suppose }{\ul S}{ }{\f20 \u-3994\'66}{ }{\ul R}{
in Choice 6. By comonotonic independence, we can increase the common branch in both gambles from $2 to $10, implying ($10, .8; $40, .1; $44, .1) }{\f20 \u-3994\'66}{ ($10, .8; $10, .1; $98, .1). Increasing $40 to $44 should make }{\ul S}{
even better. Therefore, ($10, .8; $44, .1; $44, .1) }{\f20 \u-3994\'66}{ ($10, .8; $10, .1; $98, .1); by coalescing, ($10, .8; $44, .2) }{\f20 \u-3994\'66}{ ($10, .9; $98, .1); i.e., }{\ul S}{'' }{\f20 \u-3994\'66}{ }{\ul R}{'' in Choice 8.
\par Upper cumulative independence is illustrated with Choices 10 and 9 of Table 1. Suppose }{\ul S}{' = ($40, .1; $44, .1, $110, .8) }{\f20 \u-3984\'70}{ }{\ul R}{
' = ($10, .1; $98, .1; $110, .8) in Choice 10. Reduce the (common) prize from $110 to $98 in both gambles. Reducing $44 to $40 makes }{\ul S}{' even worse, so }{\ul S}{''' }{\f20 \u-3984\'70}{ }{\ul R}{''' in Choice 9, by coalescing.
\par }\pard \ri-540\sl480\slmult1\widctlpar\adjustright {\ul Branch Independence
\par }\pard \fi720\ri-540\sl480\slmult1\widctlpar\adjustright {Branch independence postulates that if two
gambles have a common branch (the same prize with the same, known probability), then that common consequence can be changed without changing the preference order of the gambles. For example, }{\ul S}{ and }{\ul R}{
in Choice 6 have a common branch of .8 probability to win $2. Branch independence assumes that $2 can be changed to $110 in both gambles, so the preference between }{\ul S}{' and R' in Choice 10 should match that in Choice 6, i.e., }{\ul S}{ }{\f20
\u-3994\'66}{ }{\ul R}{ if and only if }{\ul S}{' }{\f20 \u-3994\'66}{ }{\ul R}{'.
\par The theory that decision-makers cancel common branches prior to choice (Kahneman & Tversky, 1979) implies branch independence; however, without this assumption, RDU, RSDU, and CPT violate it. The inverse-S weighting function an
d CPT model of Tversky and Kahneman (1992) predicts that R }{\f20 \u-3994\'66}{ S and S' }{\f20 \u-3994\'66}{ R', opposite of what has been observed (Birnbaum & McIntosh, 1996; Birnbaum & Chavez, 1997; Birnbaum, et al., 1999).
\par }\pard \ri-540\sl480\slmult1\widctlpar\adjustright {\tab Because violations of stochastic dominance and cumulative independ
ence potentially refute a large class of descriptive theories, it is vital to know if laboratory studies hold up outside the lab with people other than college students who make choices with real monetary consequences. If event splitting can reverse the v
iolations, it pinpoints coalescing as the property that must be revised in descriptive theory.
\par }\pard \qc\ri-540\sl480\slmult1\widctlpar\adjustright {Method
\par }\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\tx0\tx720\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24 \tab
Participants completed the experiment on-line by visiting the web site (now retired) at URL }{\field\fldedit{\*\fldinst {\f4\fs24 HYPERLINK http://psych.fullerton.edu/mbirnbaum/exp2.htm }{\f4 {\*\datafield
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ce118c8200aa004ba90b5c00000068007400740070003a002f002f00700073007900630068002e00660075006c006c006500720074006f006e002e006500640075002f006d006200690072006e006200610075006d002f0065007800700032002e00680074006d0000000000}}}{\fldrslt {\cs18\f4\fs24\ul\cf2 ht
tp://psych.fullerton.edu/mbirnbaum/exp2a.htm}}}{\f4\fs24 .
\par }\pard\plain \ri-540\sl480\slmult1\widctlpar\adjustright \f4\cgrid {\tab Instructions in the page included the following: "Would you rather play:
\par }\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24 A: fifty-fifty chance of winning either $100 or $0 (nothing),
\par OR
\par B: fifty-fifty chance to win either $25 or $35.
\par }{\i\f4\fs24 "... (If a person chooses A)}{\f4\fs24
... half the time they might win $0 and half the time $100. But in this study, you only get to play a gamble once, so the prize will be either $0 or $100. Gamble B's bag has 100 tickets also, but 50 of them say $25 and 50 of them
say $35. Bag B thus guarantees at least $25, but the most you can win is $35...
\par "For each choice below, click the button beside the gamble you would rather play... after people have finished their choices... (}{\i\f4\fs24 1% of participants}{\f4\fs24 ) will be selected randomly to play one gamble for real money... (}{\i\f4\fs24
If you are selected}{\f4\fs24 ), you will get to play the gamble you chose on the trial selected. ... Any one of the 20 choices might be the one you get to play, so choose carefully." (Games were played as promised, and 11 par
ticipants won $90 or more; 8 won smaller prizes).
\par }{\f4\fs24\ul Stimuli
\par }{\f4\fs24 \tab Gambles were displayed as in the following example:
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1. Which do you choose?
\par }{\field\flddirty{\*\fldinst {MACROBUTTON "HTMLDirect"}{\field{\*\fldinst {PRIVATE ""}}{\fldrslt }}{{\pict{\*\picprop\shplid1026{\sp{\sn shapeType}{\sv 75}}{\sp{\sn fFlipH}{\sv 0}}
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\picw12\pich12\picwgoal240\pichgoal240\macpict\bliptag-451223088\blipupi72{\*\blipuid e51ae1d088caac829fbb1b69c396a098}019e00000000000c000c001102ff0c00ffffffff0000000000000000000c0000000c000000000000001e0001000a800180017fff7fff009a000000ff80180000
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032108000000002108fd7fff17037fff7fff0000294afe739c045ef7252900007fff7fff17047fff00006318739c7bdefe7fff03739c4a5200007fff190b7fff
294a739c7bde7fff7fff7bde7bde6f7b631821087fff190b2108739c7bde7fff7fff7bde7bde739c739c5ef74a522108190b0000739c7fff7fff7bde7bde739c
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\par .50 probability to win $100
\par OR
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\par .50 probability to win $35
\par }{\f4\fs24\ul Design
\par }\pard\plain \ri-720\sl480\slmult1\widctlpar\adjustright \f4\cgrid {\tab The 20 choices are listed in Tables 1 and 3. There were two tests each of stochastic dominance, event-splitting, lower cumulative independence, upper cumulative independence, a
nd branch independence, with position (first or second gamble) counterbalanced. }{\chbrdr\brdrs\brdrw10 Insert Table 3 here. }{
\par }\pard \ri-540\sl480\slmult1\widctlpar\adjustright {\ul Questionnaire
\par }{\tab The form requested the participant's email address (to contact prize-winners), her or his country, age, gender, and education. Subjects we
re also asked the question (Yes or No), "Have you ever read a scientific paper (i.e., a journal article or book) on the theory of decision making or the psychology of decision making?" Participants were invited to write comments in a box.
\par }\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24\ul Recruitment of Participants}{\f4\fs24
\par }\pard\plain \ri-540\sl480\slmult1\widctlpar\adjustright \f4\cgrid {\tab }{\ul Lab Sample}{
. 124 undergraduates from the subject pool served as one option toward an assignment in introductory psychology. In the lab, the experimental web page was already displayed on several computers. Experimenters checked that participan
ts could use the mouse to click and to scroll through the page. After completing the form by clicking the "submit" button, each lab participant repeated the same task on a fresh page.
\par \tab }{\ul Reliability of Lab Data}{. The mean number of agreements between the fi
rst and second repetitions was 16.4 (82% agreement). The median was 17, and 80% made 15 or more identical choices (75% agreement). The average within-person correlation between choices, averaged over participants, was .63, which is higher than .24, the
average correlation between different people.
\par \tab }{\ul Internet Sample}{. The Internet sample consisted of 1224 participants from 44 nations who completed the experiment on-line. The web site was announced by email sent to members of the Societies for Judgment and
Decision Making and for Mathematical Psychology. It was suggested to major search engines and to web sites that list contests and games with prizes.
\par }{\ul Demographic Characteristics of the Samples
\par }{\tab The lab sample ranged from 18 to 28 years; 91% were 22 and und
er. The Internet sample ranged from 18 to 86, with 77% over 22, 50% over 28, and 20% above 40. Of the lab sample, 91% had three years of college or less (none had degrees). For the Internet, 60% were college graduates, including 333 with postgraduate s
t
udies, of whom 134 had doctoral degrees. Only 1% of the Internet sample had less than 12 years of education. Females represented 73% of lab subjects and 56% of the Internet sample. Of the lab sample, 13% indicated having read a scientific work on decis
ion making, compared to 31% of the Internet sample.
\par \tab All lab subjects were residents of the U.S., whereas the Internet sample represented 44 different nations. Countries with 8 or more people were Australia (23), Canada (66), Germany (42), Netherlands (6
2), Norway (12), Spain (8), United Kingdom (36), and U.S. (896).
\par }\pard\plain \s15\qc\ri-540\sl480\slmult1\nowidctlpar\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24 Results}{\f4\fs24\ul
\par }\pard \s15\ri-540\sl480\slmult1\nowidctlpar\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright {\f4\fs24\ul Comparison of Choice Percentages
\par }\pard\plain \ri-540\sl480\slmult1\widctlpar\adjustright \f4\cgrid {\tab Tables 1 and 3 show the percentage of preferences for the gamble shown on the right in the tables. Internet and lab samples give similar choice perc
entages (correlation = .94). Six choices show differences of 10% or more: Choices 1, 5, 6, 7, 9, and 17; in these cases, the Internet sample is more likely to choose the gamble with higher expected value (EV). For example, in Choice 1, 58% of the lab sa
mple (against 48% of Internet sample) chose the "safe" gamble (to win $25 or $35) rather than the "risky" gamble to win either $0 or $100, even though the risky gamble has the higher EV ($50 versus $30).
\par \tab }{\ul Risk aversion}{ refers to preference for a sure gain
over a risky gamble with the same or higher EV. Consistent with previous findings (e.g., Tversky & Kahneman, 1992), majorities of both samples are risk averse for gambles with medium or high probabilities to win (Choices 2, 18, and 19). Also consistent
with previous findings, majorities of both samples are }{\ul risk seeking}{ when the probability to win is small (Choice 15).
\par }\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24\ul Consequence Monotonicity
\par }\pard\plain \ri-540\sl480\slmult1\widctlpar\adjustright \f4\cgrid {\tab If two gambles are identical except for the value(s) of one (or more) consequence(s), then the gamble with the hi
gher consequence(s) should be preferred. There were four direct tests of consequence monotonicity (Choices 3, 4, 11, and 13). There were also six choices that indirectly tested monotonicity. Consider Choices 1 and 2 of Table 3. If a person prefers }{
\ul B}{ }{\f20 \u-3994\'66}{ }{\ul A}{ in Choice 1, then that person should also prefer }{\ul D}{ }{\f20 \u-3994\'66}{\ul C}{ in Choice 2 because }{\ul A}{ is the same as }{\ul C}{, and }{\ul D}{ dominates }{\ul B}{
. The term "indirect" is used because the test involves transitivity as well as monotonicity, not to mention memory.
\par \tab For the Internet sample, only 7 of 1224 violated indirect monotonicity on Choices 1 and 2; for the lab sample, 5 choices of 248 were violations. For Choices 15 and 16, there were 12 for the Internet sample and 6 in the lab sample who chose }{\ul c}{
}{\f20 \u-3994\'66}{ }{\ul d}{ and }{\ul f}{ }{\f20 \u-3994\'66\u-4064\'20}{\ul e}{. For Choices 18 and 19, there were 38 from the Internet and 10 in the lab who chose }{\ul i}{ over }{\ul j}{ and }{\ul l}{ over }{\ul k}{
. The mean rates of violation of indirect monotonicity are 1.6% and 2.8% for Internet and lab samples, respectively.
\par \tab For Choices 3, 4, 11, and 13 (direct tests of conseq
uence monotonicity) there were 71, 42, 54, and 127 violations in the Internet and 21, 14, 36, and 21 in the lab samples, respectively. The average rates of violation in direct tests of monotonicity were therefore 6.0% and 9.3%. Interestingly, indirect t
e
sts, which were on adjacent trials, showed fewer violations than direct tests. When the data are analyzed separately for first and second replications in the lab sample, there were found slightly fewer violations in the second replicate. If consequence
monotonicity were considered an index of the quality of the data, then the Internet data would be judged higher in quality than the lab data.
\par }\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24\ul Stochastic Dominance and Event-Splitting}{\f4\fs24
\par }\pard\plain \ri-540\sl480\slmult1\widctlpar\adjustright \f4\cgrid {\tab Table 4 tests stochastic dominance and event splitting for Internet and l
aboratory samples. Entries are percentages of each combination of preferences in Choices 5 and 11, and in 7 and 13 of Table 1. If everyone satisfied stochastic dominance, then 100% would have chosen }{\ul G}{+ and }{\ul GS}{
+. Instead, half or more of choice combinations of Choices 5 and 11 are }{\ul G}{\endash }{\f20 \u-3994\'66}{ }{\ul G}{\endash and }{\ul GS}{+ }{\f20 \u-3994\'66}{ }{\ul GS}{\endash , (labeled }{\ul G}{\endash }{\ul GS}{
+ in Table 4 and shown in bold type). }{\chbrdr\brdrs\brdrw10 Insert Table 4 about here. }{
\par \tab To compare probabilities of choosing }{\ul G}{+ }{\f20 \u-3994\'66}{ }{\ul G}{\endash against }{\ul GS}{+ }{\f20 \u-3994\'66}{ }{\ul GS}{\endash , one can use the test of correlated proportions. This binomial sign test compares t
he frequencies of reversal combinations, i.e., }{\ul G}{\endash }{\ul GS}{+ against }{\ul G}{+}{\ul GS}{\endash . For example, 632 of 1224 in the Internet sample violated stochastic dominance by choosing }{\ul G}{\endash }{\f20 \u-3994\'66}{ }{\ul G}{
+ on Choice 5 }{\ul and}{ switched preferences by choosing }{\ul GS}{+ }{\f20 \u-3994\'66}{\ul GS}{\endash on Choice 11, compared to only 57 who showed the opposite combination of preferences. In this case, the binomial has }{\f3 \u-3987\'6d}{
= 344.5 [(632 + 57)/2], with }{\f3 \u-3981\'73}{ = 13.12; therefore, }{\ul z}{ = 21.91*. The critical value of }{\ul z}{ with }{\f3 \u-3999\'61}{ = .05 is 1.96, so the result is significant. Asterisks in text or tabl
es (*) denote statistical significance throughout this paper.
\par \tab One can also test separately the (conservative) hypothesis that 50% of people violated stochastic dominance on Choice 5, with the binomial sign test on the split of the 704 (57.5%) who chose }{\ul G}{\endash over }{\ul G}{
+ against the 519 who chose }{\ul G}{+ over }{\ul G}{\endash , }{\ul z}{ = 5.29*. "Only" 46% of the Internet sample violated stochastic dominance in Choice 7, but significantly more had the }{\ul G}{\endash }{\ul GS}{
+ combination of preferences than the opposite switch, }{\ul z}{ = 21.49*.
\par \tab Significantly mo
re than half of the lab sample violated stochastic dominance in both tests, averaging 68.3% violations. The lab sample also showed significant event-splitting effects. The lab sample has higher rates of violation of stochastic dominance than the Interne
t sample, a difference explored in terms of demographic correlates in a later section.
\par }\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24\ul Cumulative Independence
\par }\pard\plain \ri-540\sl480\slmult1\widctlpar\tx540\adjustright \f4\cgrid {\tab Table 5 tests lower cumulative independence. Recall that a shift from }{\ul S}{ }{\f20 \u-3984\'70}{ }{\ul R}{ to }{\ul S}{'' }{\f20 \u-3994\'66}{ }{\ul R}{
'' would be consistent with the property, but the opposite shift is a violation: }{\ul S}{ }{\f20 \u-3994\'66}{ }{\ul R}{ and }{\ul S}{'' }{\f20 \u-3984\'70}{ }{\ul R}{
''. Table 5 shows that violations (bold type) significantly exceed shifts that are consistent with the property in all four tests. }{\chbrdr\brdrs\brdrw10
Insert Tables 5 and 6 about here. }{
\par \tab Upper cumulative independence is tested in Table 6. If }{\ul S}{' }{\f20 \u-3984\'70}{ }{\ul R}{' then }{\ul S}{''' }{\f20 \u-3984\'70}{ }{\ul R}{'''. It is a violation to prefer }{\ul R}{' and }{\ul S}{
'''. Table 6 shows significantly more violations (bold) than switches that are consistent in all four tests.
\par }{\ul Branch Independence
\par }{\tab Table 7 tests branch independence. Asymmetry of the preference shifts indicates systematic deviations. Consistent with previous tests (Birnbaum & Beeghley, 1997; Birnbaum
& McIntosh, 1996; Birnbaum & Chavez, 1997; Birnbaum & Navarrete, 1998; Birnbaum, et al., 1999), there are significantly more violations of the }{\ul SR}{' type (bold) than of the opposite (}{\ul RS'}{
) in three of four tests (the fourth is not significant). }{\chbrdr\brdrs\brdrw10 Insert Table 7 about here.}{
\par }{\ul Demographic Correlates
\par }\pard \ri-540\sl480\slmult1\widctlpar\adjustright {\tab Because the Internet sample is large and diverse, it is possible to subdivide the sample by gender, education level, experience reading a scientific work on decision making, and
nationality. The data were then analyzed as in Tables 4\endash
7 within each of these subdivisions. These separate tests led to essentially the same conclusions (i.e., there were significant violations of stochastic dominance, event splitting effects, and violations of upper and lower cumulative independence).
\par \tab However, the incidence of violations of stochastic dominance correlates with education and gender. Of the 686 females, 414 (60.3%) violated stochastic dominance on Choice 5, and 378 (55.1%) violated s
tochastic dominance on Choice 7. Of the 526 males, 281 (53.4%) and 182 (34.6%) violated stochastic dominance on Choices 5 and 7, respectively.
\par }\pard \fi720\ri-540\sl480\slmult1\widctlpar\adjustright {Table 8 shows the relationship between violations of stochastic dominance (Choice 5), consequence monotonicity
(Choice 11), education, and gender. Violations of stochastic dominance are less frequent among the more highly educated. Females with bachelor's degrees have 65% violations, and equally educated males have 52.3% violations. For the 54 females with doc
torates, there were 40.7% violations; for doctorate males, the rate is 48.8%. Similar results (not shown) were observed for Choices 7 and 13. }{\chbrdr\brdrs\brdrw10 Insert Table 8 about here. }{
\par }\pard \ri-540\sl480\slmult1\widctlpar\adjustright {\tab Violations of stochastic dominance were also
less frequent among those who had read a scientific work on decision making. Of the 837 people who had not read about decision making, 59.9% violated stochastic dominance on Choice 5; among the 382 who had read such a work, Choice 5 had 52.6% violations
[}{\f3 \u-3997\'63}{(1) = 5.41*].
\par }\pard \fi720\ri-540\sl480\slmult1\widctlpar\adjustright {Among those who had read about decision making were 95 who also held doctorates, including many members of Society for Judgment and Decision Making. This }{\ul expert}{
group had 50% violations of stochastic dominance on Choice 5. This group included 46* with the preference combination }{\ul G}{\endash }{\ul GS}{+ against only 7 with the combination, }{\ul G}{+}{\ul GS}{\endash , }{\ul z}{
= 5.36*. Only 32 of these 95 violated stochastic dominance on Choice 7, but 30* switched preferences from }{\ul G}{\endash in Choice 7 to }{\ul GS}{+ in Choice 13, compared to 0 (none
) who had the opposite switch, indicating significant event-splitting effects in this expert group. The expert group has "only" 41.6% violations, averaged over the two tests, compared to 68.3% for the lab sample of undergraduates.
\par }\pard \ri-540\sl480\slmult1\widctlpar\adjustright {\tab Violations of cumulativ
e independence did not appear to be systematically related to gender, education, or other demographic characteristics. For example, among the 95 in the expert group, 31* had the preference order }{\ul SR}{
'' for Choices 17 and 20, violating the property, against 8 who had the opposite pattern, }{\ul RS}{''. For Choices 12 and 14, these 95 showed 25* preference combinations }{\ul R'S}{
''', in violation of upper cumulative independence, against only 5 of the pattern }{\ul S'R}{'''. These are similar to the overall rates in Tables 5 and 6.
\par \tab
There were 328 subjects from nations outside the U.S. They were more highly educated on average than the U.S. sample (e.g., 62 had doctorates). Their data were similar to those from the U.S., once education is considered. Correlations with gender and
education (Table 8) were also observed. For example, of the 59 foreign females with bachelor's degrees, 66% violated stochastic dominance on Choice 5, compared to 44% of the 64 foreign males with the same degrees. There were 41 with doctoral degrees wh
o had read on decision making; these had an average of 44% violations of stochastic dominance against 8.5% violations of consequence monotonicity.
\par }\pard \qc\ri-540\sl480\slmult1\widctlpar\adjustright {Discussion
\par }\pard \ri-540\sl480\slmult1\widctlpar\adjustright {\tab Results show systematic violations of stochastic dominance, lower cumulative independence and uppe
r cumulative independence. These violations contradict the implications of a class of decision models including RDU, RSDU, and CPT, but they are consistent with configural weight RAM and TAX models. Although there are differences between Internet and la
b results, both lead to the same conclusions concerning these properties.
\par }\pard \fi720\ri-540\sl480\slmult1\widctlpar\adjustright {Violations of stochastic dominance are largely eliminated by event-splitting, suggesting that coalescing is the key to violations of stochastic dominance, cumulative independence, a
nd ordinal independence observed here and in previous research (Birnbaum & Navarrete, 1998; Wu, 1994). Event splitting also violates this class of RDU, RSDU, and CPT models. The Appendix presents a comparison of fit, which shows that a TAX model of conf
igural weighting predicts more choices correctly than a CPT model with the same number of estimated parameters.
\par Internet and lab samples yield similar conclusions, indicating that the findings are not unique to lab studies of college students. Internet re
search has two potential problems, sampling and control. In the lab, we can ensure that subjects do not use calculators to compute expected value, for example, or we could require them to do so. With an Internet study, we have less control over the cond
i
tions. We can ask people to follow instructions, and we can ask them if they did, but we can't know for certain. One might hope that variations of conditions would simply introduce random error that would be overcome in large samples. Ultimately, we mu
st rely on honesty, indirect checks, or the hope that deviations of protocol don't matter to the case at hand.
\par }\pard \ri-540\sl480\slmult1\widctlpar\adjustright {\tab With respect to sampling, the Internet is not really a single population, but many. The demographics of Internet users are changing, and variat
ions in recruitment could potentially have powerful effects. This study used methods intended to reach a highly educated population, knowledgeable in decision making. The fact that 95 with doctorates were recruited who have studied decision making sugge
sts that the recruitment succeeded.
\par \tab Although one can use methods intended to reach certain groups, Internet experimenters do not have complete control of recruitment. Another well-meaning person can place a link to a study in a place that recruits people
the original investigator did not intend to sample.
\par \tab If demographic or individual difference variables affect behavior, then one can measure these and study their correlations with the results. In the present study, rates of violation of stochastic domina
nce were found to correlate with gender, education, and experience reading a scholarly work on decision making. The Internet sample was less likely to violate stochastic dominance than the lab sample, but the Internet sample also had a lower percentage o
f females, more education, greater age, and were more likely to have read a paper on decision making. Thus, differences in results fit those expected from demographic differences between samples.
\par \tab Education, which correlates with violations of stochastic d
ominance, probably correlates with unmeasured variables that might be the actual causal agents. Those with more education are probably also higher in intelligence and wealth. Therefore, fewer violations of stochastic dominance among the highly educated
might be due to higher intelligence, for example, rather than to the causal effects of education. Experiments with differential education will test if specific training teaches people to satisfy stochastic dominance.
\par \tab In summary, Internet data clarify and
reinforce results from the lab. The data indicate that descriptive theory should not assume coalescing. Dropping coalescing allows violations of rationality (e.g., stochastic dominance), but evidence shows that people violate implications of coalescing,
e
ven when there are monetary consequences. Thus, these data and others (Birnbaum & Navarrete, 1998; Birnbaum, et al., 1999) call for a major revision in theories of decision making. Results can be approximated by the configural weight, TAX model, which im
plies violations of stochastic dominance, event-splitting effects, and violations of cumulative independence.
\par }\pard\plain \s17\qc\fi-259\li259\ri-540\sl480\slmult1\widctlpar\adjustright \f4\cgrid {Appendix: Comparison of Models
\par }\pard\plain \ri-540\sl480\slmult1\widctlpar\adjustright \f4\cgrid {\tab The Configurally Weighted Utility (CWU) of a gamble can be written as follows:
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CPT, RAM, TAX, EU, and EV) are special cases of Equation 1, with different assumptions about the weights. Nonlinear utility models in which weights depend on the utilities of the consequences are also special cases of Equation 1.
\par \tab For positive outcomes, RSDU reduces to RDU, which assumes that weights are as follows,
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. Equation 2 implies stochastic dominance (Quiggin, 1993), coalescing (Luce, 1998), and cumulative independence (Birnbaum, 1997).
\par }\pard \fi720\ri-540\sl480\slmult1\widctlpar\adjustright {The model of CPT (Tversky & Wakker, 1995) further assumes that }{\i W(P)}{ in Equation 2 is given by the following,
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\par }\pard \ri-540\sl480\slmult1\widctlpar\adjustright {where }{\i c}{ and }{\i\f3 \u-3993\'67}{ have been estimated to be < 1, giving }{\i W(P)}{ an inverse-S shape.
\par \tab The transfer of attention exchange (TAX) model assumes that weights are transferred among branches (distinct probability-conseque
nce pairs) according to ranks of consequences and the weight each branch has to lose. When lower outcomes have greater configural weight, lower valued branches "tax" weight from higher valued ones; with }{\f3 \u-3982\'72}{
< 0, relative weights are as follows,
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. Although EV and EU have been rejected in previous studies (e.g., Kahneman & Tversky, 1979), they provide benchmarks for assessing the accuracy of more complex models.
\par \tab TAX, CPT, EU, and EV models were fit to compare the relative accuracy of the
models in describing individual data. Each person's data were fit (minimizing the negative log likelihood of the choices given each model by methods of Birnbaum & Chavez, 1997). After a model was fit to each person, utility differences were computed to s
ee if the model correctly predicted each choice. The percentages of subjects for whom the models predicted 15 or more choices (75% accuracy or better) are given in Table 9. }{\chbrdr\brdrs\brdrw10
Insert Table 9 about here. }{
\par \tab The TAX model can account for violations of stochastic dominance, event-splitting effects, and violations of lower and upper cumulative independence. The model was fit assuming }{\i u(x) = x}{. In the Internet sample, median estimates of }{\f3
\u-3993\'67}{ and }{\f3 \u-3996\'64}{ for the TAX model are .791 and \endash
.333, respectively. The mean number of choices correctly predicted was 15.53 choices (78%), including perfect scores of 20 for 66 people. For the lab sample, mean correct predictions were 14.75 and 15.36; median estimates are }{\i\f3 \u-3993\'67}{
= .982 and .681; and }{\i\f3 \u-3996\'64}{ = \endash .463 and \endash .574, in first and second replicates, respectively. Correlations between replicates are .31, .726, and .543, for }{\i\f3 \u-3993\'67}{, }{\i\f3 \u-3996\'64}{
, and number correct, respectively, all significant. The TAX model is not quite as accurate on average in the lab (mean = 15.05, or 75%) as other choices by the same subject (16.4, or 82%).
\par }\pard \fi720\ri-540\sl480\slmult1\widctlpar\adjustright {For the CPT model in the lab, means of correct predictions were 14.22 and 13.99; median estimates are }{\i\f3 \u-3993\'67}{ = .759 and .749, }{\i c}{
= .481 and .438, respectively. Correlations between replicates were .63 and .56 for }{\i\f3 \u-3993\'67}{ and }{\i c}{, respectively. For the Internet sample, medians }{\i\f3 \u-3993\'67}{ = .743 and }{\i c}{
=.597, respectively. For Internet data, CPT had an average of 14.91 correct (75%), significantly worse than for the TAX model, }{\i\ul t}{(1223) = \endash 8.05.* Within -per
son, the TAX model predicted more choices correctly than CPT for 614* people; 414 had more correct predictions by CPT, and 196 were tied.
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. For Internet, first and second replicates of lab samples, median estimates of }{\i\f3 \u-3998\'62}{
= .611, .587, and .585, respectively (correlation between lab replicates = .89). EU correctly predicted averages of 13.55, 12.93, and 12.43 choices correctly (68%, 65%, and 62%) in the Internet and two lab replicates, respectively.
\par No one in either samp
le was perfectly consistent with EV. This seemed a bit surprising because a number of people (including several with doctorates) sent comments that they simply chose by EV. However, no one wrote that they actually computed EV, and apparently no one did.
For the Internet and lab replicates, EV correctly predicts a mean of 12.4 choices (62%), 10.44 and 10.15 (51%), respectively; none in the lab had more than 17 consistent with EV. In summary, TAX, CPT, EU, and EV models predicted 75% or more of individua
l choices in 65%, 47%, 25% and 8% of the fits, respectively.
\par }\pard \qc\ri-540\sl480\slmult1\widctlpar\adjustright {\page References
\par }\pard\plain \s17\fi-259\li259\ri-540\sl480\slmult1\widctlpar\adjustright \f4\cgrid {Becker, J., & Sarin, R. (1987). Lottery dependent utility. }{\ul Management Science}{, }{\ul 33}{, 1367-1382.
\par Birnbaum, M. H. (1997). Violations of monotonicity in judgment and decision making. In A. A. J. Marley (Eds.), }{\ul Choice, decision, and measurement: Essays in honor of R. Duncan Luce}{ (pp. 73-100). Mahwah, NJ: Erlbaum.
\par Birnbaum, M. H. (1998). Paradoxes of Allais, stochastic dominance, and decision weights. In J. C. Shanteau, B. A. Mellers, & D. Schum (Eds.), }{\ul Decision science and technology: Reflections on the contributions of Ward Edwards}{
. Norwell, MA: Kluwer Academic Publishers.
\par Birnbaum, M. H., & Beeghley, D. (1997). Violations of branch independence in judgments of the value of gambles. }{\ul Psychological Science}{, }{\ul 8}{, 87-94.
\par Birnbaum, M. H., & Chavez, A. (1997). Tests of theories of decision making: Violations of branch independence and distribution Independence. }{\ul Organizational Behavior and Human Decision Processes}{, }{\ul 71}{(2), 161-194.
\par Birnbaum, M. H., & McIntosh, W. R. (1996). Violations of branch independence in choices between gambles. }{\ul Organizational Behavior and Human Decision Processes}{, }{\ul 67}{, 91-110.
\par Birnbaum, M. H., & Navarrete, J. B. (1998). Testing descriptive utility theories: Violations of stochastic dominance and cumulative independence. }{\ul Journal of Risk and Uncertainty}{, }{\ul 17}{, 49-78.
\par Birnbaum, M. H., Patton, J. N., & Lott, M. K. (1999). Evidence against rank-dependent utility theories: Violations of cumulative independence, interval independence, stochastic dominance, and transitivity. }{\ul
Organizational Behavior and Human Decision Processes}{, }{\ul 77}{, 44-83.
\par }\pard \s17\fi-259\li259\ri-720\sl480\slmult1\widctlpar\adjustright {Edwards, W. (1954). The theory of decision making. }{\ul Psychological Bulletin}{, }{\ul 51}{, 380-417.
\par Humphrey, S. J. (1995). Regret aversion or event-splitting effects? More evidence under risk and uncertainty. }{\ul Journal of risk and uncertainty}{, }{\ul 11}{, 263-274.
\par Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk. }{\ul Econometrica}{, }{\ul 47}{, 263-291.
\par Karmarkar, U. S. (1979). Subjectively weighted utility and the Allais paradox. }{\ul Organizational Behavior and Human Performance}{, }{\ul 24}{, 67-72.
\par }\pard \s17\fi-259\li259\ri-540\sl480\slmult1\widctlpar\adjustright {Krantz, J. H., Ballard, J., & Scher, J. (1997). Comparing the results of laboratory and world-wide web samples on the determinants of female attractiveness. }{\ul Behavior Resear
ch Methods, Instruments, & Computers}{, }{\ul 29}{, 264-269.
\par Krantz, J. H., & Dalal, R. (1999). Validity of web-based psychological research. In M. H. Birnbaum (Ed.), }{\ul Psychological Experiments on the Internet}{ (manuscript).
\par Luce, R. D. (1990). Rational versus plausible accounting equivalences in preference judgments. }{\ul Psychological Science}{, }{\ul 1}{, 225-234.
\par }\pard \s17\fi-259\li259\ri-720\sl480\slmult1\widctlpar\adjustright {Luce, R. D. (1998). Coalescing, event commutativity, and theories of utility. }{\ul Journal of Risk and Uncertainty}{, }{\ul 16}{, 87-113.
\par }\pard \s17\fi-259\li259\ri-540\sl480\slmult1\widctlpar\adjustright {Luce, R. D., & Fishburn, P. C. (1991). Rank- and sign-dependent linear utility models for finite first order gambles. }{\ul Journal of Risk and Uncertainty}{, }{\ul 4}{, 29-59.
\par Luce, R. D., & Fishburn, P. C. (1995). A note on deriving rank-dependent utility using additive joint receipts. }{\ul Journal of Risk and Uncertainty}{, }{\ul 11}{, 5-16.
\par Machina, M. J. (1982). Expected utility analysis without the independence axiom. }{\ul Econometrica}{, }{\ul 50}{, 277-323.
\par }\pard \s17\fi-259\li259\ri-720\sl480\slmult1\widctlpar\adjustright {Quiggin, J. (1993). }{\ul Generalized expected utility theory: The rank-dependent model}{. Boston: Kluwer.
\par }\pard \s17\fi-259\li259\ri-540\sl480\slmult1\widctlpar\adjustright {Starmer, C., & Sugden, R. (1993). Testing for juxtaposition and event-splitting effects. }{\ul Journal of Risk and Uncertainty}{, }{\ul 6}{, 235-254.
\par Stevenson, M. K., Busemeyer, J. R., & Naylor, J. C. (1991). Judgment and decision-making theory. In M. Dunnette & L. M. Hough (Eds.), }{\ul New handbook of industrial-organizational psychology}{ (pp. 283-374). Palo Alto, CA: Consulting Psychologist Press.
\par Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of uncertainty. }{\ul Journal of Risk and Uncertainty}{, }{\ul 5}{, 297-323.
\par Tversky, A., & Wakker, P. (1995). Risk attitudes and decision weights. }{\ul Econometrica}{, }{\ul 63}{, 1255-1280.
\par Wakker, P., & Tversky, A. (1993). An axiomatization of cumulative prospect theory. }{\ul Journal of Risk and Uncertainty}{, }{\ul 7}{, 147-176.
\par Weber, E. U. (1994). From subjective probabilities to decision weights: The effects of asymmetric loss functions on the evaluation of uncertain outcomes and events. }{\ul Psychological Bulletin}{, }{\ul 114}{, 228-242.
\par }\pard \s17\fi-259\li259\ri-720\sl480\slmult1\widctlpar\adjustright {Wu, G. (1994). An empirical test of ordinal independence. }{\ul Journal of Risk and Uncertainty}{, }{\ul 9}{, 39-60.
\par }\pard \s17\fi-259\li259\ri-540\sl480\slmult1\widctlpar\adjustright {Wu, G., & Gonzalez, R. (1998). Common consequence conditions in decision making under risk. }{\ul Journal of Risk and Uncertainty}{, }{\ul 16}{, 115-139.
\par }\pard\plain \ri-540\sl480\slmult1\widctlpar\adjustright \f4\cgrid {\page Table 1. Choices used to test models of risky decision making.
\par }\trowd \trgaph108\trleft-180\trkeep \clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx1620\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx8370\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\fs20 \cell \cell }\pard \widctlpar\intbl\adjustright {
\fs20 \row }\trowd \trgaph108\trleft-180\trkeep \clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx1620\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx6840\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx8370\pard \ri-540\sl480\slmult1
\widctlpar\intbl\adjustright {\f6\fs18 Choice Choice
\par No. Type\cell }\pard \qc\ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 Choice
\par \cell }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 % Choice
\par Internet Lab\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\trowd \trgaph108\trleft-180\trkeep \clvertalt\cltxlrtb \cellx450\clvertalt\cltxlrtb \cellx1620\clvertalt\cltxlrtb \cellx6840\clvertalt\cltxlrtb \cellx7560\clvertalt\cltxlrtb \cellx8370
\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 5\cell }{\f6\fs18\ul G}{\f6\fs18 + }{\f6\fs18\ul G}{\fs18 \endash \cell }\pard \ri-540\widctlpar\intbl\adjustright {\f6\fs18\ul I}{\f6\fs18 : .05 to win $12\tab \tab }{\f6\fs18\ul J}{
\f6\fs18 : .10 to win $12
\par .05 to win $14\tab \tab .05 to win $90
\par .90 to win $96\tab \tab .85 to win $96
\par \cell }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 58\cell 73\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 6\cell }{\f6\fs18\ul S}{\f6\fs18 }{\f6\fs18\ul
R}{\fs18 \cell }\pard \ri-540\widctlpar\intbl\adjustright {\f6\fs18\ul K}{\f6\fs18 : .80 to win $2\tab \tab }{\f6\fs18\ul L}{\f6\fs18 : .80 to win $2
\par .10 to win $40\tab \tab .10 to win $10
\par .10 to win $44\tab \tab .10 to win $98
\par \cell }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 69\cell 58\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 7\cell }{\f6\fs18\ul G}{\fs18 \endash }{\f6\fs18
}{\f6\fs18\ul G}{\f6\fs18 +\cell }\pard \ri-540\widctlpar\intbl\adjustright {\f6\fs18\ul M}{\f6\fs18 : .06 to win $6\tab \tab }{\f6\fs18\ul N}{\f6\fs18 : .03 to win $6
\par .03 to win $96\tab \tab .03 to win $8
\par .91 to win $99\tab \tab .94 to win $99
\par \cell }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 54\cell 36\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 8\cell }{\f6\fs18\ul S}{\f6\fs18 '' }{\f6\fs18\ul
R}{\f6\fs18 ''\cell }\pard \ri-540\widctlpar\intbl\adjustright {\f6\fs18\ul O}{\f6\fs18 : .80 to win $10\tab \tab }{\f6\fs18\ul P}{\f6\fs18 : .90 to win $10
\par .20 to win $44\tab \tab .10 to win $98
\par \cell }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 75\cell 69\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 9\cell }{\f6\fs18\ul S}{\f6\fs16 '''}{\f6\fs18 }{
\f6\fs18\ul R}{\f6\fs16 '''}{\fs18 \cell }\pard \ri-540\widctlpar\intbl\adjustright {\f6\fs18\ul Q}{\f6\fs18 . .20 to win $40\tab \tab }{\f6\fs18\ul R}{\f6\fs18 : .10 to win $10
\par .80 to win $98\tab \tab .90 to win $98
\par \cell }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 47\cell 34\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 10\cell }{\f6\fs18\ul S}{\f6\fs18 ' }{
\f6\fs18\ul R}{\f6\fs18 '\cell }\pard \ri-540\widctlpar\intbl\adjustright {\f6\fs18\ul S}{\f6\fs18 : .10 to win $40\tab \tab }{\f6\fs18\ul T}{\f6\fs18 : .10 to win $10
\par .10 to win $44\tab \tab .10 to win $98
\par .80 to win $110\tab \tab .80 to win $110
\par \cell }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 73\cell 72\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 11\cell }{\f6\fs18\ul GS}{\f6\fs16 + }{
\f6\fs18\ul GS}{\fs16 \endash }{\fs18 \cell }\pard \ri-540\widctlpar\intbl\adjustright {\f6\fs18\ul U}{\f6\fs18 : .05 to win $12\tab \tab }{\f6\fs18\ul V}{\f6\fs18 : .05 to win $12
\par .05 to win $14\tab \tab .05 to win $12
\par .05 to win $96\tab \tab .05 to win $90
\par .85 to win $96\tab \tab .85 to win $96
\par \cell }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 10\cell 15\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 12\cell }{\f6\fs18\ul R}{\f6\fs18 ' }{
\f6\fs18\ul S}{\f6\fs18 '\cell }\pard \ri-540\widctlpar\intbl\adjustright {\f6\fs18\ul W}{\f6\fs18 : .05 to win $12\tab \tab }{\f6\fs18\ul X}{\f6\fs18 : .05 to win $48
\par .05 to win $96\tab \tab .05 to win $52
\par .90 to win $106\tab \tab .90 to win $106
\par \cell }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 50\cell 49\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 13\cell }{\f6\fs18\ul GS}{\fs16 \endash }{\f6\fs16
}{\f6\fs18\ul GS}{\f6\fs16 +}{\fs18 \cell }\pard\plain \s17\ri-540\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18\ul Y}{\f6\fs18 : .03 to win $6\tab \tab }{\f6\fs18\ul Z}{\f6\fs18 : .03 to win $6
\par }\pard \s17\fi-259\li259\ri-540\widctlpar\intbl\adjustright {\f6\fs18 .03 to win $6\tab \tab .03 to win $8
\par .03 to win $96\tab \tab .03 to win $99
\par .91 to win $99\tab \tab .91 to win $99
\par }\pard\plain \ri-540\widctlpar\intbl\adjustright \f4\cgrid {\fs18 \cell }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 95\cell 92\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard \ri-540\sl480\slmult1
\widctlpar\intbl\adjustright {\f6\fs18 14\cell }{\f6\fs18\ul R}{\f6\fs16 '''}{\f6\fs18 }{\f6\fs18\ul S}{\f6\fs16 '''}{\fs18 \cell }\pard\plain \s17\fi-259\li259\ri-540\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18\ul a}{\f6\fs18 : .05 to win $12\tab
\tab }{\f6\fs18\ul b}{\f6\fs18 : .10 to win $48
\par }\pard\plain \ri-540\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18 .95 to win $96\tab \tab .90 to win $96
\par \cell }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 74\cell 81\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 17\cell }{\f6\fs18\ul R}{\f6\fs18 }{
\f6\fs18\ul S}{\fs18 \cell }\pard \ri-540\widctlpar\intbl\adjustright {\f6\fs18\ul g}{\f6\fs18 : .90 to win $3\tab \tab }{\f6\fs18\ul h}{\f6\fs18 : .90 to win $3
\par .05 to win $12\tab \tab .05 to win $48
\par .05 to win $96\tab \tab .05 to win $52
\par }\pard\plain \s17\fi-259\li259\ri-540\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18 \cell }\pard\plain \ri-540\sl480\slmult1\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18 49\cell 61\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\trowd
\trgaph108\trleft-180\trkeep \clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx450\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx1620\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx6840\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx7560\clvertalt
\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx8370\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 20\cell }{\f6\fs18\ul R}{\f6\fs18 '' }{\f6\fs18\ul S}{\f6\fs18 ''\cell }\pard\plain \s17\fi-259\li259\ri-540\widctlpar\intbl\adjustright
\f4\cgrid {\f6\fs18\ul m}{\f6\fs18 : .95 to win $12\tab \tab }{\f6\fs18\ul n}{\f6\fs18 : .90 to win $12
\par }\pard\plain \ri-540\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18 .05 to win $96\tab \tab .10 to win $52
\par \cell }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 28\cell 31\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar
\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24 Note: % Choice indicates the percentage who chose the gamble shown on the right in the table.}{\b\f4\fs24
\par }\pard\plain \ri-540\sl480\slmult1\widctlpar\adjustright \f4\cgrid {\page Table 2. Testable Properties of Decision Making Theories
\par }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr
\brdrs\brdrw10 \cltxlrtb \cellx3330\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb \cellx9270\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {Property Name\cell Illustration \cell
}\pard \widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl
\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb \cellx3330\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb \cellx9270\pard \ri-540\sl480\slmult1
\widctlpar\intbl\adjustright {Transitivity\cell }{\ul\lang2 A}{ }{\f20 \u-3994\'66}{ }{\ul\lang2 B}{ and }{\ul\lang2 B}{ }{\f20 \u-3994\'66}{ }{\ul\lang2 C}{ }{{\field{\*\fldinst SYMBOL 222 \\f "Symbol" \\s 12}{\fldrslt\f3\fs24}}}{ }{\ul\lang2 A}{ }{\f20
\u-3994\'66}{ }{\ul\lang2 C}{\lang2 .}{\ul\lang2 }{\cell }\pard \widctlpar\intbl\adjustright {\row }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {Consequence monotonicity\cell (}{\ul\lang2 x}{\super +}{, }{\ul\lang2 p}{; }{\ul\lang2 y}{, }{
\ul\lang2 p}{; }{\ul\lang2 z}{, }{\ul\lang2 r}{) }{\f20 \u-3994\'66}{ (}{\ul\lang2 x}{, }{\ul\lang2 p}{; }{\ul\lang2 y}{, }{\ul\lang2 p}{; }{\ul\lang2 z}{, }{\ul\lang2 r}{) }{{\field{\*\fldinst SYMBOL 219 \\f "Symbol" \\s 12}{\fldrslt\f3\fs24}}}{ }{
\ul\lang2 x}{\super +}{ }{\f20 \u-3994\'66}{ }{\ul\lang2 x}{. \cell }\pard \widctlpar\intbl\adjustright {\row }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {Coalescing\cell (}{\ul\lang2 x}{, }{\ul\lang2 p}{; }{\ul\lang2 x}{, }{\ul\lang2 q}{; }{
\ul\lang2 z}{, }{\ul\lang2 r}{) ~ (}{\ul\lang2 x}{, }{\ul\lang2 p}{ + }{\ul\lang2 q}{; }{\ul\lang2 z}{, }{\ul\lang2 r}{). \cell }\pard \widctlpar\intbl\adjustright {\row }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {Stochastic Dominance \cell
}{\ul\lang2 P}{(}{\ul\lang2 x}{ > }{\ul\lang2 t}{ | }{\ul\lang2 A}{) \u8805\'b3 }{\ul\lang2 P}{(}{\ul\lang2 x}{ > }{\ul\lang2 t}{ | }{\ul\lang2 B}{) }{{\field{\*\fldinst SYMBOL 34 \\f "Symbol" \\s 12}{\fldrslt\f3\fs24}}}{\ul\lang2 t}{ }{
{\field{\*\fldinst SYMBOL 222 \\f "Symbol" \\s 12}{\fldrslt\f3\fs24}}}{ }{\ul\lang2 A}{ }{\f20 \u-3994\'66}{ }{\ul\lang2 B}{ or }{\ul\lang2 A}{ ~ }{\ul\lang2 B}{.\cell }\pard \widctlpar\intbl\adjustright {\row }\pard \ri-540\sl480\slmult1
\widctlpar\intbl\adjustright {Restricted
\par Branch Independence\cell }{\ul S}{ = (}{\ul\lang2 x}{, }{\ul\lang2 p}{; }{\ul\lang2 y}{, }{\ul\lang2 q}{; }{\ul\lang2 z}{, }{\ul\lang2 r}{) }{\f20 \u-3994\'66}{ }{\ul R}{ = (}{\ul\lang2 x}{', }{\ul\lang2 p}{; }{\ul\lang2 y}{', }{\ul\lang2 q}{; }{
\ul\lang2 z}{, }{\ul\lang2 r}{) }{{\field{\*\fldinst SYMBOL 219 \\f "Symbol" \\s 12}{\fldrslt\f3\fs24}}}{
\par }{\ul S}{' = (}{\ul\lang2 x}{, }{\ul\lang2 p}{; }{\ul\lang2 y}{, }{\ul\lang2 q}{; }{\ul\lang2 z}{', }{\ul\lang2 r}{) }{\f20 \u-3994\'66}{ }{\ul R}{' = (}{\ul\lang2 x}{', }{\ul\lang2 p}{; }{\ul\lang2 y}{', }{\ul\lang2 q}{; }{\ul\lang2 z}{', }{\ul\lang2 r}{
). \cell }\pard \widctlpar\intbl\adjustright {\row }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {Lower
\par Cumulative Independence\cell }{\ul S}{ = (}{\ul z}{, }{\ul\cf1 r}{; }{\ul x}{, }{\ul\cf1 p}{; }{\ul y}{, }{\ul\cf1 q}{) }{\f20 \u-3994\'66}{ }{\ul R}{ = (}{\ul z}{, }{\ul\cf1 r}{; }{\ul x}{', }{\ul\cf1 p}{; }{\ul y}{', }{\ul\cf1 q}{) }{
{\field{\*\fldinst SYMBOL 222 \\f "Symbol" \\s 12}{\fldrslt\f3\fs24}}}{
\par }{\ul S}{'' = (}{\ul x}{', }{\ul\cf1 r}{ ; }{\ul y}{, }{\ul\cf1 p}{ + }{\ul\cf1 q}{) }{\f20 \u-3994\'66}{ }{\ul R}{'' = (}{\ul x}{', }{\ul\cf1 r}{ + }{\ul\cf1 p}{; }{\ul y}{', }{\ul\cf1 q}{).\cell }\pard \widctlpar\intbl\adjustright {\row }\trowd
\trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr
\brdrs\brdrw10 \cltxlrtb \cellx3330\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb \cellx9270\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {Upper
\par Cumulative Independence\cell }{\ul S}{' = (}{\ul x}{, }{\ul\cf1 p}{; }{\ul y}{, }{\ul\cf1 q}{; }{\ul z}{', }{\ul\cf1 r}{) }{\f20 \u-3984\'70}{ }{\ul R}{' = (}{\ul x}{', }{\ul\cf1 p}{; }{\ul y}{', }{\ul\cf1 q}{; }{\ul z}{', }{\ul\cf1 r}{) }{
{\field{\*\fldinst SYMBOL 222 \\f "Symbol" \\s 12}{\fldrslt\f3\fs24}}}{
\par }{\ul S}{''' = (}{\ul x}{, }{\ul\cf1 p}{ + }{\ul\cf1 q}{; }{\ul y}{', }{\ul\cf1 r}{) }{\f20 \u-3984\'70}{ }{\ul R}{''' = (}{\ul x}{', }{\ul\cf1 p}{; }{\ul y}{', }{\ul\cf1 q}{ + }{\ul\cf1 r}{).\cell }\pard \widctlpar\intbl\adjustright {\row }\pard\plain
\s15\ri-540\sl480\slmult1\nowidctlpar\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24 Notation: Let }{\f4\fs24\ul A}{\f4\fs24 = (}{\f4\fs24\ul x}{\f4\fs24 , }{\f4\fs24\ul p}{\f4\fs24 ; }{
\f4\fs24\ul y}{\f4\fs24 , }{\f4\fs24\ul q}{\f4\fs24 ; }{\f4\fs24\ul z}{\f4\fs24 , }{\f4\fs24\ul r}{\f4\fs24 ) represent a gamble to win }{\f4\fs24\ul x}{\f4\fs24 with probability }{\f4\fs24\ul p}{\f4\fs24 , }{\f4\fs24\ul y}{\f4\fs24 with probability }{
\f4\fs24\ul q}{\f4\fs24 , and }{\f4\fs24\ul z}{\f4\fs24 with probability }{\f4\fs24\ul r}{\f4\fs24 , where }{\f4\fs24\ul\cf1 p}{\f4\fs24 + }{\f4\fs24\ul\cf1 q}{\f4\fs24 + }{\f4\fs24\ul\cf1 r}{\f4\fs24 = 1. }{\f4\fs24\ul A}{\f4\fs24 }{\f20\fs24
\u-3994\'66}{\f4\fs24 }{\f4\fs24\ul B}{\f4\fs24 means }{\f4\fs24\ul A}{\f4\fs24 is preferred to }{\f4\fs24\ul B}{\f4\fs24 , and ~ represents indifference. }{\f4\fs24\ul P}{\f4\fs24 (}{\f4\fs24\ul x}{\f4\fs24 > }{\f4\fs24\ul t}{\f4\fs24 | }{
\f4\fs24\ul A}{\f4\fs24 ) represents the probability of winning more than }{\f4\fs24\ul t}{\f4\fs24 given }{\f4\fs24\ul A}{\f4\fs24 . Branch independence is }{\i\f4\fs24 restricted}{\f4\fs24
when the number of distinct branches and their probabilities are fixed. }{\i\f4\fs24 Comonotonic}{\f4\fs24 branch independence is the special case where consequences
maintain the same ranks. In tests of (noncomonotonic) branch independence and cumulative independence, 0 < }{\f4\fs24\ul z}{\f4\fs24 < }{\f4\fs24\ul x}{\f4\fs24 ' < }{\f4\fs24\ul x}{\f4\fs24 < }{\f4\fs24\ul y}{\f4\fs24 < }{\f4\fs24\ul y}{\f4\fs24 ' <
}{\f4\fs24\ul z}{\f4\fs24 '.
\par }\pard\plain \ri-540\sl480\slmult1\widctlpar\adjustright \f4\cgrid {\page Table 3. Choices used to assess risk attitudes and consequence monotonicity.
\par }\trowd \trgaph108\trleft-180\trkeep \clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx1530\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx8370\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\fs20 \cell \cell }\pard \widctlpar\intbl\adjustright {
\fs20 \row }\trowd \trgaph108\trleft-180\trkeep \clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx1530\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx6840\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx8370\pard \ri-540\sl480\slmult1
\widctlpar\intbl\adjustright {\f6\fs18 Choice
\par No. \cell }\pard \qc\ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 Choice
\par \cell }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 % Choice
\par Internet Lab\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\trowd \trgaph108\trleft-180\trkeep \clvertalt\cltxlrtb \cellx450\clvertalt\cltxlrtb \cellx1530\clvertalt\cltxlrtb \cellx6840\clvertalt\cltxlrtb \cellx7560\clvertalt\cltxlrtb \cellx8370
\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 1\cell \cell }\pard \ri-540\widctlpar\intbl\adjustright {\f6\fs18\ul A}{\f6\fs18 : .50 to win $0\tab \tab }{\f6\fs18\ul B}{\f6\fs18 : .50 to win $25
\par .50 to win $100\tab \tab .50 to win $35
\par \cell }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 48\cell 58\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 2\cell \cell }\pard
\ri-540\widctlpar\intbl\adjustright {\f6\fs18\ul C}{\f6\fs18 : .50 to win $0\tab \tab }{\f6\fs18\ul D}{\f6\fs18 : .50 to win $45
\par .50 to win $100\tab \tab .50 to win $50
\par \cell }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 60\cell 69\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 3\cell \cell }\pard
\ri-540\widctlpar\intbl\adjustright {\f6\fs18\ul E}{\f6\fs18 : .50 to win $4\tab \tab }{\f6\fs18\ul F}{\f6\fs18 : .50 to win $4
\par .30 to win $96\tab \tab .30 to win $12
\par .20 to win $100\tab \tab .20 to win $100
\par \cell }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 6\cell 8\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 4\cell \cell }\pard
\ri-540\widctlpar\intbl\adjustright {\f6\fs18\ul G}{\f6\fs18 : .40 to win $2\tab \tab }{\f6\fs18\ul H}{\f6\fs18 : .40 to win $2
\par .50 to win $12\tab \tab .50 to win $96
\par .10 to win $108\tab \tab .10 to win $108
\par \cell }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 96\cell 94\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard \ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 15\cell \cell }\pard\plain
\s17\fi-259\li259\ri-540\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18\ul c}{\f6\fs18 : $1 for sure \tab \tab }{\f6\fs18\ul d}{\f6\fs18 : .99 to win $0
\par }\pard\plain \ri-540\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18 \tab .01 to win $100
\par }\pard\plain \s17\fi-259\li259\ri-540\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18 \cell }\pard\plain \ri-540\sl480\slmult1\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18 58\cell 55\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard
\ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 16\cell \cell }\pard\plain \s17\fi-259\li259\ri-540\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18\ul e}{\f6\fs18 : $3 for sure \tab \tab }{\f6\fs18\ul f}{\f6\fs18 : .99 to win $0
\par }\pard\plain \ri-540\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18 \tab \tab .01 to win $100
\par }\pard\plain \s17\fi-259\li259\ri-540\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18 \cell }\pard\plain \ri-540\sl480\slmult1\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18 43\cell 50\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard
\ri-540\sl480\slmult1\widctlpar\intbl\adjustright {\f6\fs18 18\cell \cell }\pard\plain \s17\fi-259\li259\ri-540\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18\ul i}{\f6\fs18 : $90 for sure\tab \tab }{\f6\fs18\ul j}{\f6\fs18 : .01 to win $0
\par \tab \tab .99 to win $100
\par \cell }\pard\plain \ri-540\sl480\slmult1\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18 33\cell 38\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\trowd \trgaph108\trleft-180\trkeep \clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx450\clvertalt
\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx1530\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx6840\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx7560\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx8370\pard \ri-540\sl480\slmult1
\widctlpar\intbl\adjustright {\f6\fs18 19\cell \cell }\pard\plain \s17\fi-259\li259\ri-540\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18\ul k}{\f6\fs18 : $96 for sure\tab \tab }{\f6\fs18\ul l}{\f6\fs18 : .01 to win $0
\par \tab \tab .99 to win $100
\par \cell }\pard\plain \ri-540\sl480\slmult1\widctlpar\intbl\adjustright \f4\cgrid {\f6\fs18 26\cell 32\cell }\pard \widctlpar\intbl\adjustright {\fs18 \row }\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar
\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24 Note: % Choice indicates the percentage who chose the gamble shown on the right in the table.
\par
\par }\pard\plain \sl480\slmult1\widctlpar\adjustright \f4\cgrid {\page
\par Table 4. Percentages of each choice combination in tests of stochastic dominance and event-splitting
\par }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx1663\clvertalt\clbrdrt\brdrs\brdrw10
\cltxlrtb \cellx3434\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx8747\pard \sl480\slmult1\widctlpar\intbl\adjustright {Choices 5 & 11\cell }{\ul
G+GS+\cell G+GS\endash }{\cell }{\ul G\endash GS+}{\cell }{\ul G\endash GS\endash }{\cell }\pard \widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10
\trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\cltxlrtb \cellx1663\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx3434\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt
\brdrs\brdrw10 \cltxlrtb \cellx8747\pard \sl480\slmult1\widctlpar\intbl\adjustright { Internet\cell 37.4\cell 4.7\cell }{\b 51.6*\cell }{5.8\cell }\pard \widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl
\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\cltxlrtb \cellx1663\clvertalt\cltxlrtb \cellx3434\clvertalt\cltxlrtb \cellx5205\clvertalt\cltxlrtb \cellx6976\clvertalt\cltxlrtb
\cellx8747\pard \sl480\slmult1\widctlpar\intbl\adjustright { Lab
\par \cell 21.8\cell 4.8\cell }{\b 62.5*\cell }{9.7\cell }\pard \widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv
\brdrs\brdrw10 \clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx1663\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx3434\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt
\brdrs\brdrw10 \cltxlrtb \cellx8747\pard \sl480\slmult1\widctlpar\intbl\adjustright {Choices 7 & 13\cell }{\ul G+GS+}{\cell }{\ul G+GS\endash }{\cell }{\ul G\endash GS+}{\cell }{\ul G\endash GS\endash }{\cell }\pard \widctlpar\intbl\adjustright {\row
}\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\cltxlrtb \cellx1663\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx3434
\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx8747\pard \sl480\slmult1\widctlpar\intbl\adjustright { Internet\cell 51.3\cell 2.1\cell }{\b
43.7*\cell }{2.3\cell }\pard \widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\clbrdrb
\brdrs\brdrw10 \cltxlrtb \cellx1663\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx3434\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx8747\pard
\sl480\slmult1\widctlpar\intbl\adjustright { Lab\cell 33.1\cell 2.8\cell }{\b 58.5*\cell }{5.6\cell }\pard \widctlpar\intbl\adjustright {\row }\pard \sl480\slmult1\widctlpar\adjustright {
Notes: Percentages based on n = 1224 for Internet and n = 248 (two replications by 124 judges); these do not always sum to 100, due to occasional nonresponse.
\par
\par \page
\par Table 5. Percentages of choice combinations in tests of lower cumulative independence
\par }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx1800\clvertalt\clbrdrt\brdrs\brdrw10
\cltxlrtb \cellx3510\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx8747\pard \sl480\slmult1\widctlpar\intbl\adjustright {Choices 6 & 8\cell }{\ul
SS''\cell SR''}{\cell }{\ul RS''}{\cell }{\ul RR''}{\cell }\pard \widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv
\brdrs\brdrw10 \clvertalt\cltxlrtb \cellx1800\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx3510\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb
\cellx8747\pard \sl480\slmult1\widctlpar\intbl\adjustright { Internet\cell 12.7\cell }{\b 18.0*\cell }{12.0\cell 56.5\cell }\pard \widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb
\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\cltxlrtb \cellx1800\clvertalt\cltxlrtb \cellx3510\clvertalt\cltxlrtb \cellx5205\clvertalt\cltxlrtb \cellx6976\clvertalt\cltxlrtb \cellx8747\pard \sl480\slmult1
\widctlpar\intbl\adjustright { Lab
\par \cell 18.5\cell }{\b 23.0*\cell }{11.7\cell 46.0\cell }\pard \widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv
\brdrs\brdrw10 \clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx1800\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx3510\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt
\brdrs\brdrw10 \cltxlrtb \cellx8747\pard \sl480\slmult1\widctlpar\intbl\adjustright {Choices 17 & 20\cell }{\ul SS''\cell SR''}{\cell }{\ul RS''}{\cell }{\ul RR''}{\cell }\pard \widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt
\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\cltxlrtb \cellx1800\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx3510\clvertalt\clbrdrt\brdrs\brdrw10
\cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx8747\pard \sl480\slmult1\widctlpar\intbl\adjustright { Internet\cell 18.5\cell }{\b 30.3*\cell }{9.6\cell 40.4\cell }\pard
\widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx1800
\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx3510\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx8747\pard \sl480\slmult1
\widctlpar\intbl\adjustright { Lab\cell 21.0\cell }{\b 39.5*\cell }{10.1\cell 29.0\cell }\pard \widctlpar\intbl\adjustright {\row }\pard \sl480\slmult1\widctlpar\adjustright {
Notes: Percentages based on n = 1224 for Internet and n = 248 (two replications by 124 judges); these do not always sum to 100, due to occasional nonresponse.
\par
\par \page Table 6. Percentages of choice combinations in tests of upper cumulative independence
\par }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx1800\clvertalt\clbrdrt\brdrs\brdrw10
\cltxlrtb \cellx3510\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx8747\pard \sl480\slmult1\widctlpar\intbl\adjustright {Choices 10 & 9\cell }{\ul
S'S'''\cell S'R'''}{\cell }{\ul R'S'''}{\cell }{\ul R'R'''}{\cell }\pard \widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10
\trbrdrv\brdrs\brdrw10 \clvertalt\cltxlrtb \cellx1800\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx3510\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb
\cellx8747\pard \sl480\slmult1\widctlpar\intbl\adjustright { Internet\cell 19.2\cell 6.9\cell }{\b 33.3*\cell }{39.7\cell }\pard \widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb
\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\cltxlrtb \cellx1800\clvertalt\cltxlrtb \cellx3510\clvertalt\cltxlrtb \cellx5205\clvertalt\cltxlrtb \cellx6976\clvertalt\cltxlrtb \cellx8747\pard \sl480\slmult1
\widctlpar\intbl\adjustright { Lab
\par \cell 23.0\cell 3.6\cell }{\b 42.7*\cell }{29.4\cell }\pard \widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv
\brdrs\brdrw10 \clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx1800\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx3510\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt
\brdrs\brdrw10 \cltxlrtb \cellx8747\pard \sl480\slmult1\widctlpar\intbl\adjustright {Choices 12 & 14\cell }{\ul S'S'''\cell S'R'''}{\cell }{\ul R'S'''}{\cell }{\ul R'R'''}{\cell }\pard \widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108
\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\cltxlrtb \cellx1800\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx3510\clvertalt\clbrdrt\brdrs\brdrw10
\cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx8747\pard \sl480\slmult1\widctlpar\intbl\adjustright { Internet\cell 42.2\cell 7.8\cell }{\b 31.8*\cell }{17.5\cell }\pard
\widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx1800
\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx3510\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx8747\pard \sl480\slmult1
\widctlpar\intbl\adjustright { Lab\cell 42.3\cell 6.5\cell }{\b 38.3*\cell }{12.9\cell }\pard \widctlpar\intbl\adjustright {\row }\pard \sl480\slmult1\widctlpar\adjustright {
Notes: Percentages based on n = 1224 for Internet and n = 248 (two replications by 124 judges); these do not always sum to 100, due to occasional nonresponse.
\par
\par \page Table 7. Percentages of choice combinations in tests of branch independence
\par }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx1800\clvertalt\clbrdrt\brdrs\brdrw10
\cltxlrtb \cellx3510\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx8747\pard \sl480\slmult1\widctlpar\intbl\adjustright {Choices 6 & 10\cell }{\ul
SS'\cell SR'}{\cell }{\ul RS'}{\cell }{\ul RR'}{\cell }\pard \widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv
\brdrs\brdrw10 \clvertalt\cltxlrtb \cellx1800\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx3510\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb
\cellx8747\pard \sl480\slmult1\widctlpar\intbl\adjustright { Internet\cell 12.0\cell }{\b 18.7*\cell }{14.1\cell 54.3\cell }\pard \widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb
\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\cltxlrtb \cellx1800\clvertalt\cltxlrtb \cellx3510\clvertalt\cltxlrtb \cellx5205\clvertalt\cltxlrtb \cellx6976\clvertalt\cltxlrtb \cellx8747\pard \sl480\slmult1
\widctlpar\intbl\adjustright { Lab
\par \cell 18.5\cell }{\b 22.2*\cell }{7.7\cell 50.0\cell }\pard \widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv
\brdrs\brdrw10 \clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx1800\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx3510\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt
\brdrs\brdrw10 \cltxlrtb \cellx8747\pard \sl480\slmult1\widctlpar\intbl\adjustright {Choices 17 & 12\cell }{\ul SS'\cell SR'}{\cell }{\ul RS'}{\cell }{\ul RR'}{\cell }\pard \widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt
\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\cltxlrtb \cellx1800\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx3510\clvertalt\clbrdrt\brdrs\brdrw10
\cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt\brdrs\brdrw10 \cltxlrtb \cellx8747\pard \sl480\slmult1\widctlpar\intbl\adjustright { Internet\cell 32.0\cell 17.1\cell 17.8\cell 32.4\cell }\pard
\widctlpar\intbl\adjustright {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx1800
\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx3510\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx8747\pard \sl480\slmult1
\widctlpar\intbl\adjustright { Lab\cell 37.5\cell }{\b 23.4*\cell }{11.3\cell 27.8\cell }\pard \widctlpar\intbl\adjustright {\row }\pard \sl480\slmult1\widctlpar\adjustright {Notes: Percentages based on n = 1224 for Internet and n =
248 (two replications by 124 judges); these do not always sum to 100, due to occasional nonresponse.
\par
\par }\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24 \page
Table 8. Violations of stochastic dominance and monotonicity related to gender and education.
\par }\trowd \trgaph108\trleft-108 \clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx1663\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx3434\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \cltxlrtb
\cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx7110\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx9090\pard \s15\ri-540\sl480\slmult1\nowidctlpar\intbl
\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright {\f4\fs24 Gender\cell Education
\par \cell Stochastic Dominance (%)
\par }{\f4\fs24\ul G}{\f4\fs24 \endash }{\f20\fs24 \u-3994\'66}{\f4\fs24 }{\f4\fs24\ul G}{\f4\fs24 +\cell Monotonicity (%)
\par }{\f4\fs24\ul GS}{\f4\fs24 \endash }{\f20\fs24 \u-3994\'66}{\f4\fs24 }{\f4\fs24\ul GS}{\f4\fs24 +\cell Number of
\par Subjects\cell }\pard\plain \widctlpar\intbl\adjustright \f4\cgrid {\row }\trowd \trgaph108\trleft-108 \clvertalt\cltxlrtb \cellx1663\clvertalt\cltxlrtb \cellx3434\clvertalt\cltxlrtb \cellx5205\clvertalt\cltxlrtb \cellx7110\clvertalt\cltxlrtb \cellx8910
\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\intbl\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24 F\cell < 16\cell 62.3 (75.3)\cell 12.6 (14.3)\cell 318 (91)\cell }\pard\plain
\widctlpar\intbl\adjustright \f4\cgrid {\row }\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\intbl\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24 F\cell 16\cell 65.0\cell 13.1\cell 206\cell
}\pard\plain \widctlpar\intbl\adjustright \f4\cgrid {\row }\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\intbl\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24 F\cell 17-19\cell 55.6\cell
12.0\cell 108\cell }\pard\plain \widctlpar\intbl\adjustright \f4\cgrid {\row }\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\intbl\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24 F\cell 20
\cell 40.7\cell 1.9\cell 54\cell }\pard\plain \widctlpar\intbl\adjustright \f4\cgrid {\row }\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\intbl\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {
\f4\fs24 M\cell < 16\cell 60.1 (65.2)\cell 9.8 (15.2)\cell 163 (33)\cell }\pard\plain \widctlpar\intbl\adjustright \f4\cgrid {\row }\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\intbl
\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24 M\cell 16\cell 52.3\cell 10.3\cell 195\cell }\pard\plain \widctlpar\intbl\adjustright \f4\cgrid {\row }\pard\plain \s15\ri-540\sl480\slmult1
\nowidctlpar\intbl\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24 M\cell 17-19\cell 47.7\cell 2.3\cell 88\cell }\pard\plain \widctlpar\intbl\adjustright \f4\cgrid {\row }\trowd
\trgaph108\trleft-108 \clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx1663\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx3434\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrb\brdrs\brdrw10 \cltxlrtb \cellx7110\clvertalt\clbrdrb
\brdrs\brdrw10 \cltxlrtb \cellx8910\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\intbl\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24 M\cell 20\cell 48.8\cell 11.2\cell 80\cell
}\pard\plain \widctlpar\intbl\adjustright \f4\cgrid {\row }\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24 Notes: Educati
on < 16 indicates less than bachelor's degree; 16= Bachelor's degree; 17-19 = Postgraduate studies; 20 = doctorate. Lab sample results shown in parentheses. Percentages for Choices 5 and 11 indicate percentage of violations of stochastic dominance and c
onsequence monotonicity.
\par \page Table 9. Percentage of cases for whom 15 or more choices (of 20) are correctly predicted
\par }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr
\brdrs\brdrw10 \cltxlrtb \cellx1710\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb \cellx3510\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr
\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr
\brdrs\brdrw10 \cltxlrtb \cellx8747\pard \s15\ri-540\sl480\slmult1\nowidctlpar\intbl\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright {\f4\fs24 Sample\cell TAX\cell CPT\cell EU\cell EV\cell }\pard\plain
\widctlpar\intbl\adjustright \f4\cgrid {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl
\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb \cellx1710\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb \cellx3510\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl
\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl
\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb \cellx8747\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\intbl\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24
Internet (n=1224)\cell 65\cell 46\cell 23\cell 7\cell }\pard\plain \widctlpar\intbl\adjustright \f4\cgrid {\row }\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\intbl\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright
\f6\fs20\lang2\cgrid {\f4\fs24 Lab 1 (n=124)\cell 67\cell 58\cell 37\cell 17\cell }\pard\plain \widctlpar\intbl\adjustright \f4\cgrid {\row }\trowd \trgaph108\trleft-108\trbrdrt\brdrs\brdrw10 \trbrdrl\brdrs\brdrw10 \trbrdrb\brdrs\brdrw10 \trbrdrr
\brdrs\brdrw10 \trbrdrh\brdrs\brdrw10 \trbrdrv\brdrs\brdrw10 \clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb \cellx1710\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb
\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb \cellx3510\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb \cellx5205\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb
\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb \cellx6976\clvertalt\clbrdrt\brdrs\brdrw10 \clbrdrl\brdrs\brdrw10 \clbrdrb\brdrs\brdrw10 \clbrdrr\brdrs\brdrw10 \cltxlrtb \cellx8747\pard\plain \s15\ri-540\sl480\slmult1\nowidctlpar\intbl
\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24 Lab 2 (n=124)\cell 60\cell 43\cell 28\cell 9\cell }\pard\plain \widctlpar\intbl\adjustright \f4\cgrid {\row }\pard\plain
\s15\ri-540\sl480\slmult1\nowidctlpar\tx0\tx959\tx1918\tx2877\tx3836\tx4795\tx5754\tx6713\tx7672\tx8631\adjustright \f6\fs20\lang2\cgrid {\f4\fs24 Note: Lab 1 choice correctly predicts 15 or more choices in Lab 2 in 80% of cases.
\par }}