Tversky Amos 1969 â€å“intransitivity of Preferencesã¢â‚¬â Psychological Review 76 31ã¢â‚¬â€œ48
Judgment and Decision Making, Vol. 13, No. 3, May 2018, pp. 217-236
Predictably intransitive preferences
David J. Butler* Ganna Pogrebna# $ $
The transitivity precept is common to nearly all descriptive and normative utility theories of choice under take chances. Contrary to both intuition and common assumption, the piffling-known 'Steinhaus-Trybula paradox' shows the relation 'stochastically greater than' will not always be transitive, in contradiction of Weak Stochastic Transitivity. Nosotros bespoke-design pairs of lotteries inspired by the paradox, over which individual preferences might cycle. We run an experiment to look for prove of cycles, and violations of expansion/contraction consistency between option sets. Fifty-fifty afterward considering possible stochastic but transitive explanations, we evidence that cycles can be the modal preference pattern over these simple lotteries, and we discover systematic violations of expansion/contraction consistency.
Keywords: intransitivity, cycles, lotteries, experiment, expansion consistency
1 Introduction
Researchers take questioned the adequacy of Expected Utility Theory (EUT) every bit an account of choice nether risk since Allais (1953) presented his famous 'paradox' examples. Economists question one precept of EUT less than most: transitivity. Bar-Hillel & Margalit (1988) quote Luce & Raiffa'due south (1957) definition of transitivity as "if A is preferred in the paired comparing (A, B) and B is preferred in the paired comparing (B. C), then A is preferred in the paired comparison (A, C)" (Luce & Raiffa, 1975, p. xvi). Notice that the binary comparing A, C is therefore superfluous: a rational chooser can rely on transitivity to deliver the best choice. Choice cycles cannot occur unless this chooser is exactly indifferent between A, B and C, or makes a fault. In brusk, economists regard transitivity as a defining characteristic of rational pick.
In lite of this consensus, Butler & Blavatskyy (2018) propose the following scenario. A fund manager offers a reward to the broker who selects one portfolio that outperforms the others over the following year. The conclusion maker's (DM) preference and then is to maximise the probability of earning the greater sum. Suppose there are iii (statistically contained) portfolios; portfolio A yields $4m with probability 2/3 and $1m with probability ane/3; portfolio B yields $3m for sure; and portfolio C yields $5m with probability 1/iii and $2m with probability ii/3.
Suppose the fund director begins by comparing {A,B}; she will choose portfolio A because A yields a higher result than B with probability two/3. Next she compares {B,C}; she chooses portfolio B considering B will yield a higher result than C besides with probability 2/3. Then as a rational determination maker relying on transitivity for choice gear up {A,C}, she selects portfolio A over C. Nonetheless, her faith in transitivity is disadvantageous. Had she not relied upon transitivity, her revealed preference in {A,C} would accept been for C, considering C yields a higher outcome than A with probability v/9, or 55.v%. The advantageous preference ordering beyond the ready of pairwise choices is the bicycle A≻ B≻ C≻ A, contradicting the transitivity axiom.ane While some may say this makes probable winner preferences whether induced or elicited unreasonable (e.g., Pratt, 1972), this paper takes a dissimilar view (e.g., Blyth, 1972; Bar-Hillel & Margalit, 1988). Offset, businesses do employ these kinds of incentives, which for pairwise decisions tin lead to the preference (and choice) cycle in our example. Second, no utility theories with a transitivity precept currently come up with a alarm that they cannot business relationship for probable winner or related preferences, in which case it is non clear why we should deem their preferences unworthy of maximization.
This cycle in our example is an illustration of a paradox first described past Steinhaus and Trybula (Steinhaus & Trybula, 1959). As a mathematical puzzle their paradox, which we denote STP for curt, has inspired a small literature in applied statistics (due east.g., from Usiskin 1964 to Conrey et al., 2016). Nosotros may country it as follows: let choice objects A, B, C exist independent random variables and let Pr(A≻ B) denote the probability of choosing A over B. Information technology is possible for Pr(A≻ B), Pr(B ≻ C) and Pr(C≻ A) to all exceed fifty%, given a preference for the winner, contrary to Weak Stochastic Transitivity (WST).
Steinhaus & Trybula proved that, for iii choice objects, each with three equiprobable attributes, the theoretical maximum 'minimum' (max-min) winning probability is (√five−ane)/2 or 61.viii%. It is because this value exceeds 50% that preference cycles may arise. In our earlier example, the smallest of the three 'winning' probabilities is 55.5%. While nosotros focus in the residual of this newspaper on preferences over simple lotteries, we should not forget the relevance of these objects to real economic decisions for option under take chances. Steinhaus & Trybula gave an application to testing the relative strength of randomly selected steel bars A, B, C, for which successive comparisons could showroom a wheel: A stochastically stronger than B, B than C but with C stochastically stronger than A. Other examples are not difficult to imagine.
Given the STP relies on a preference for the well-nigh likely winner, every bit incentivised by the fund manager in our before example, does it have much relevance for decision theory and individual preferences more generally? This paper suggests that the answer is aye, even though the STP has passed mostly unnoticed in the decision theory literature (exceptions include Butler & Hey, 1987; Anand, 1993; Blavatskyy, 2006; Rubinstein & Segal, 2012). Even if very few individual preferences over lottery pairs are simply for the probable winner, the STP withal serves equally an important demonstration that imposing transitivity on an unrestricted domain of preference profiles will sometimes result in an inferior selection. We theorize the STP tin can also serve every bit a heuristic in amalgam new lotteries, over which a broader range of preferences may wheel, a merits we render to in section on Inferences for Lottery Design and an Experiment.
Kahneman (2012) reminds us that "The errors of a theory are rarely found in what it asserts explicitly; they hide in what information technology ignores or tacitly assumes". Transitivity must hold either if a value attaches to each pick without reference to other alternatives (choice-set independence), or if an equivalent value results after comparing and contrasting the attributes of the bachelor choice options. The latter process points us towards the flaw in how transitivity is applied to multi-attribute option. This process volition produce an equivalent value only if utility is sufficiently 'linear in the differences' betwixt the options' attribute values; see Tversky (1969); Fishburn (1982); or Loomes & Sugden, (1982) for details. The STP relies on an extreme case of a non-linear additive departure choice dominion, for which a larger difference in an attribute's magnitude carries no extra weight.
However, individuals often form valuations of options in a comparative, context-dependent way rather than attracting a context-independent value (Russo & Dosher, 1983; Arieli et al, 2009; Noguchi & Stewart, 2014). Prove from centre-tracking experiments shows clear empirical bear witness against option-ready independence, at to the lowest degree when expected utilities are sufficiently 'close' to prompt a DM to compare the attributes of the alternatives.two Non-linearity sufficient to produce a preference bike (dependent on the relative size of the attributes) may then occur.
The rest of the paper is organised as follows. In the next Section, we talk over inferences for lottery design and describe our experiment. After that, we present the experimental results and conclude with a general discussion
2 Inferences for Lottery Design and Experiment
ii.ane General Implications
At present we transition to using the STP objects to design our ain lotteries without either inducing or bold probable winner preferences. Let us consider decision making nether chance when choice alternatives are lotteries — i.due east. probability distributions over a nonempty finite fix of outcomes. A decision maker faces a fix of choice alternatives that contains at least ii distinct elements. Next, the DM chooses the choice alternative that yields a strictly greater, context independent, expected utility. Often, no such choice culling is present, then she compares the attributes of the available options to recognize where her preference lies. This step is required to avoid arbitrary choice; a growing torso of evidence shows preferences are often known imperfectly (inter alia, Butler & Loomes, 2007).
Descriptively, the consensus is that true intransitive preference cycles are vanishingly exceptional. Show once taken to indicate systematic intransitivity (Tversky, 1969; Loomes, Starmer & Sugden, 1991) has since been either reinterpreted as not reflecting fundamental intransitivity (inter alia: Starmer & Sugden, 1993) or establish past newer statistical methods to exist compatible with noisy but transitive responses (Baillon et al, 2015; Birnbaum & Diecidue, 2015). In his highly influential 1969 commodity 'Intransitivity of Preference', Amos Tversky lamented "...in the absence of a model that guides the construction of the alternatives, i is unlikely to discover consistent violations of weak stochastic transitivity (WST)".
Figure ane: Experimental Flow.
One reason why experiments to date have simply rarely found testify of intransitive behaviour is lack of guidance from theory to select suitable lottery parameters. This lack of guidance is probably a effect of the supposition any utility part must apply to all lottery pairs. Even so, this presumes there is no 'black hole' in parameter space from which question sets can trigger preferences of a different kind. Drawing on the STP as a heuristic, nosotros may address the trouble Tversky faced and bespoke design candidate lotteries.
Let each consequence represent a sum of money, in £, a very familiar, directly comparable upshot for which magnitudes are easily interpretable by our subjects. For simplicity and comparability, the probability of each outcome in our design is e'er 1/3, 2/iii or 1. We imposed some filters to guide our option of triples. Since expected value differences between each choice object, inside a given triple, reach a maximum of £4 ii −vmu/ −iiimu 3 ± £1 2 −5mu/ −3mu 3 ± 35.7%, our first filter is to focus on sets with larger EV differences (see Figure 1 for the distribution). We avoid constructing triples informed by sets with equal or nigh equal EV's to avoid tipping the residue towards a cycle just through noise.
Next, the 2d filter reduces cognitive load by requiring each of the three STP objects to reuse integers such that there are no more than two different amounts as consequences (e.g., 5, two and 2); we therefore exclude any triple with 3 unlike money consequences on any lottery. It is also important to keep the presentation of the number of attributes in each object equal rather than coalescing identical outcomes. This is because past experiments accept found the dissimilarity between coalesced and non-coalesced outcomes (also known as event-splitting; Starmer and Sugden, 1993) can be confused every bit show for intransitivity (Birnbaum & Schmidt, 2008; Baillon et al, 2015). To command for this we display each result fifty-fifty when all iii lead to the aforementioned sum. We then made a number of modifications that increased the prizes on offer and then allowed for take chances disfavor in our experiment (run a risk-disfavor plays no role for probable winner preferences). We brand no claim this step in the parameter selection process involves more than a mix of informed guesswork and personal judgment.
Finally, equally a tertiary filter, to mimic the preference reversal (PR) miracle problems (Lichtenstein & Slovic, 1971) we decided to focus on the STP triples which accept expected values strictly in the post-obit guild: $ ≻ P ≻ CE. Hither, $ is a dollar bet (lottery which yields a big issue with low probability), P is a probability bet (lottery which gives a small consequence with large probability) and CE is certainty. If nosotros accept 3 lotteries: X, Y and Z, we will assume that Z stands for the dollar bet, X - for the probability bet, and Y for certainty (degenerate lottery).
Table 1: Binary choices and intransitivity.
Triple
Option
p1=1/3
p2=1/3
p3=i/three
EV
N intransitivities in Repetition 1 (frequency)
N intransitivities in Repetition 2 (frequency)
N consistently intransitive with the same blueprint (frequency)
Northward consistently intransitive including different patterns (frequency)
X
12
12
3
nine.00
Y
8
8
8
8.00
1
Z
24
four
iv
10.67
0.22
0.21
0.10
0.11
10
11
xi
1
seven.67
Y
5
5
5
5.00
two
Z
twenty
2
2
8.00
0.eleven
0.25
0.05
0.05
X
12
12
2
viii.67
Y
eight
8
8
8.00
3
Z
20
4
iv
9.33
0.29
0.26
0.12
0.15
10
15
fifteen
iii
11.00
Y
10
x
10
10.00
four
Z
27
five
5
12.33
0.59
0.sixty
0.26
0.34
X
15
15
6
12.00
Y
11
11
11
11.00
5
Z
28
6
6
xiii.33
0.23
0.21
0.09
0.09
10
17
17
five
xiii.00
Y
12
12
12
12.00
half dozen
Z
xxx
half dozen
6
xiv.00
0.19
0.16
0.09
0.09
X
nine
ix
3
7.00
Y
half dozen
6
6
6.00
7
Z
xvi
four
iv
eight.00
0.35
0.24
0.12
0.13
Ten
15
15
5
11.67
Y
10
ten
10
10.00
viii
Z
30
3
3
12.00
0.27
0.15
0.10
0.11
X
12
12
0
8.00
Y
7
7
seven
7.00
9
Z
28
0
0
9.33
0.08
0.04
0.01
0.01
X
14
14
2
10.00
Y
8
eight
eight
8.00
10
Z
21
half-dozen
6
xi.00
0.23
0.30
0.17
0.17
10
14
xiv
two
10
Y
24
five
5
eleven.33
11
Z
8
eight
9
8.33
0.23
0.xx
0.07
0.09
An important implication of our blueprint choices is that the management of cycles may be subject to ii opposing forces, in aggregate, considering the direction of cycles for standard PR lotteries is opposite to the 'probable winner' cycle. Nevertheless, this should not terminate systematic cycles appearing at the private level, if some people showroom one tendency more strongly. This in mind, we can now put frontward our first testable hypothesis. Birnbaum & Schmidt (2008) succinctly country the currently dominant view regarding the prove for intransitive preferences: "...nosotros think the burden of proof should shift to those who argue that intransitive models are descriptive of more five percentage of the population".
Hypothesis 1: Cartoon on the STP ingredients, we tin design sets of lotteries for which cycles volition occur with significantly greater frequency than 5%.
We tin see the lotteries nosotros designed in the left hand columns (columns 1–6) of Tabular array 1.
In the spirit of Allais' famous example, consider 1 such 'bespoke' lottery set up, and assume your preferred lottery is incentivized. The three option objects are statistically independent lotteries: each result is a monetary amount with a one-3rd probability attached. For each of the binary choice sets {X,Y}; {Y,Z}; {Z,Ten}, viewed separately, we ask the reader to consider her preference, ideally looking only at each decision in isolation. In combination, there are eight possible binary preference patterns, of which just two are intransitive. Consider Ten versus Y, where X provides £fifteen with probability one/3, £fifteen with probability 1/three, or £iii with probability 1/3; and Y yields £ten for sure - i.e., £ten with probability 1/iii, £10 with probability 1/three, or £10 with probability 1/3 (see Table ane). Suppose Y ≻ X. Now compare Y which gives £ten for certain and Z which provides £27 with probability one/3, £5 with probability 1/3, or £5 with probability 1/3. Perhaps, here Z ≻ Y. Finally, compare Z which yields £27 with probability 1/3, £5 with probability 1/3, or £five with probability i/three and X which provides £fifteen with probability 1/3, £15 with probability 1/three, or £3 with probability 1/three.3
In this case, maybe you found X ≻ Z. If you prefer Y ≻ Ten, Z ≻ Y and X ≻ Z, y'all take exhibited the preference cycle X ≻ Z ≻ Y ≻ X, the modal preference pattern for our subjects. The opposite cycle here is X ≻ Y ≻ Z ≻ 10; we found these 2 intransitive patterns together exceeded, by a minor bulk, the half dozen transitive patterns combined. Referring back to our opening instance, suppose the consequences on each of X, Y and Z refer to investment returns on iii portfolios and the probabilities are the historical frequencies. A consumer'due south binary preferences over those risk-return combinations might potentially cycle likewise with implications for the construction of portfolios in finance.
two.2 Design Implications from Models of Probabilistic Choice
Although pick is often stochastic, and an intransitive cycle may arise from transitive latent preference due to noise, distinguishing structurally intransitive latent preferences from stochastic transitivity in experiments is not straightforward. How frequently can intransitive cycles ascend for individuals with transitive preferences, but who choose probabilistically? For example, individuals may have transitive core preferences but choice probabilities are determined by embedding these preferences into a model of random errors (e.g., Blavatskyy, 2014). Such a modelling approach can generate a statistically significant disproportion betwixt the two possible wheel directions, just it cannot generate a proportion of intransitive cycles above 25% of all observed selection patterns, for whatsoever triple.
A more promising model of probabilistic choice for rationalizing intransitive cycles is the random preference approach (e.g., Loomes & Sugden, 1995). As the extreme instance, let u.s.a. consider an private who has three transitive preference orderings X ≻ Y ≻ Z, Z ≻ X ≻ Y and Y ≻ Z ≻ X with each ordering equally probable to be drawn when a choice is to be made. Information technology is straightforward to see that in a direct binary choice between X and Y, this private chooses 10 with probability two/3. Likewise, in a directly binary choice between Y and Z, this private chooses Y with probability two/3. Finally, in a direct binary choice between X and Z, this individual chooses Z with probability 2/iii, thereby violating weak stochastic transitivity. Thus, a model of random transitive preferences generates a maximum of (ii3)/(iiiiii) = 8/27 (29.six%) intransitive choice cycles. This limit involves a strong asymmetry between the two possible intransitive patterns; the maximum frequency of a particular bike given random sampling is 1/4; see Rubinstein & Segal (2012) for proofs of these propositions.
However, a model of random transitive preferences has another testable implication so far overlooked by a literature focused on binary selection sets. When comparing binary choice data with the choice data from a ternary set, we derive a new set of constraints that any stochastic but exclusively transitive preferences must meet. In such models of stochastic choice, the probability of choosing Ten from the ternary set up {X,Y,Z} is given by the probability that a decision maker draws a preference order in which 10 is preferred to Y and X is preferred to Z. In contrast, for a direct binary choice between 10 and Y, this conclusion maker chooses X with a probability that is equal to the probability that he or she draws a preference order in which 10 is preferred to Y (but 10 may or may not exist preferred to Z). Similarly, for a straight binary choice between Ten and Z, this decision maker chooses X with a probability that is equal to the probability that he or she draws a preference society in which X is preferred to Z (but X may or may non be preferred to Y). Hence, any model of random transitive preferences must make the following testable hypotheses. If whatever one of the 3 hypotheses fails to hold, no model of stochastic transitive preferences tin be consequent with the information.
Hypothesis 2: The probability of choosing 10 from the ternary set {Ten,Y,Z} cannot exceed
|
Hypothesis 3: The probability of choosing Y from the ternary set {10,Y,Z} cannot exceed
|
Hypothesis iv: The probability of choosing Z from the ternary set {X,Y,Z} cannot exceed
|
Since, by definition, the probabilities of choosing X, Y and Z from the ternary set {Ten,Y,Z} must sum up to ane, we have the post-obit implication of any model of stochastic but transitive preferences:
|
A decision maker who violates weak stochastic transitivity, such that P(X,Y)>0.5, P(Y,Z)>0.5 and P(Z,Ten)>0.5, must withal satisfy the inequality
|
which can exist simplified as a triangle inequality
|
The triangle inequalities (v) and (vi), it is unremarkably argued, produce a stronger test than WST to separate genuine intransitivity from stochastic transitivity. However, Birnbaum (2011) showed that the triangle inequalities could exist satisfied even by underlying intransitive preferences. Furthermore, recent work past Müller-Trede et al. (2015) demonstrates how these inequalities may be violated even when underlying preferences are 100% transitive. Their experiment also shows articulate violations of these inequalities.
Figure 2: Binary selection display.
In other words, the triangle inequalities for stochastic transitive preferences tin be satisfied when preferences are intransitive and violated when preferences are transitive, raising a business organization that they are not as useful for identifying true intransitive preference cycles every bit generally believed, though see Cavagnaro & Davis-Stober (2014) for an alternative view. For these reasons, among others, our experiment was non designed specifically to test the triangle inequalities, which ideally would require multiple repetitions of the same lottery pairs for every person. Nevertheless, we can examination H2-H4 below, for each triple violating WST. We also follow Birnbaum & Diecidue (2015) and repeat each ready of choices once after a distractor task, which facilitates additional methods of separating noise from true preferences.
Consider a decision maker who: a) makes a directly binary choice between choice alternatives X and Y; and b) ranks choice alternatives X and Y as part of a ternary choice prepare in terms of their desirability. Ignoring the ranking of Z, this decision maker tin reveal iv different preference patterns:
- X ≻ Y and X is ranked more desirable than Y (revealed preferences i);
- Y ≻ X and Y is ranked more desirable than 10 (revealed preferences 2);
- Ten ≻ Y and Y is ranked more desirable than X (revealed preferences iii);
- Y ≻ X and X is ranked more desirable than Y (revealed preferences iv).
Revealed preferences i and ii are both consequent with the independence of irrelevant alternatives axiom. Revealed preferences iii and iv are inconsistent with this axiom. Thus, if preferences of a conclusion maker satisfy contraction and expansion consistency, nosotros should observe patterns i and ii and non patterns iii and iv. Our experiment investigates.
two.3 Experimental Design
Figure 3: Ternary Choice Brandish.
In total 100 subjects (all undergraduate students at the Academy of Warwick) were invited to take part in the experiment. We programmed the experiment using the Qualtrics software and consisted of 100 questions divided into v parts (run into Figure i).
Earlier tests for preference cycles primarily used country-contingent consequences in matrix-style displays. Those displays facilitate between-act comparisons and raise the possibility of, for case, anticipated regret when consequences are state-contingent and thus the potential for cycles. Our blueprint maintains statistical independence between the choice objects such that any observed preference cycles are more probable to be rooted in clarification-invariant, intransitive latent preferences. Finally, we include a 'standard PR' control set (Triple 9) to compare to our 'new PR' gambles that is the focus of our experiment.
In Part one, nosotros broke upwards the 11 triples into binary choices betwixt individual lotteries and asked subjects to answer 33 questions (3 binary choice questions per each triple). Table i provides a detailed listing of all triples. We present each binary choice in the format shown in Figure 2 with 2 options — Left and Correct. Each option shows a lottery with three equiprobable outcomes.
All binary pick questions gave subjects 4 unlike options on a slider. The initial starting point for each slider was "No preference". However, subjects were non able to go along by leaving the slider in the original position (i.e., the selection of "No preference" was not allowed). Subjects were able to motion the slider to the right and opt for "Slightly prefer Right" or "Strongly prefer Right" or, alternatively, to move the slider to the left and cull "Slightly prefer Left" or "Strongly prefer Left". Irrespective of whether a subject indicated slight or strong preference, we used simply revealed preferences for "Left" or "Right" in the payoff calculations and we do not written report the strength of preference results hither. We randomized all 33 questions for each individual separately. In Part three, we repeated all 33 binary questions over again only presented them in a different random gild to each subject.
Tabular array ii: Frequency of intransitive cycles obtained from binary choices, in per centum.
Triple Profile 121 123 131 133 221 223 231 233 INTR TR R1 ix 2 5 12 29 eight 20 15 22 78 -two*1 R2 xiii 3 6 10 27 ix eighteen fourteen 21 79 R1 30 9 25 17 7 viii 2 2 eleven 89 -2*ii R2 25 17 23 5 2 10 8 10 25 75 R1 16 eleven 5 0 31 12 18 vii 29 71 -ii*3 R2 14 x v eight 25 xiii sixteen 9 26 74 R1 11 0 6 6 9 5 59 4 59 41 -two*iv R2 eight 18 v 3 6 14 42 4 threescore 40 R1 8 3 five 17 25 seven twenty 15 23 77 -ii*5 R2 xiii 5 6 12 31 8 sixteen nine 21 79 R1 15 4 8 7 32 9 15 10 nineteen 81 -2*vi R2 15 four 5 11 33 12 12 viii sixteen 84 R1 three vii 15 16 11 nine 28 xi 35 65 -ii*7 R2 7 1 17 19 16 ten 23 7 24 76 R1 12 14 5 8 xl 3 13 5 27 73 -2*8 R2 13 seven vii 9 47 7 8 2 15 85 R1 17 4 3 i 57 10 four 4 8 92 -two*9 R2 15 3 ii ane 57 19 ane two 4 96 R1 4 0 sixteen 16 8 7 23 26 23 77 -2*10 R2 5 one 10 13 12 ten 29 20 thirty seventy R1 8 22 xiii xiii 15 21 i 7 23 77 -2*11 R2 10 10 9 11 18 fourteen ten 18 xx 80 Notes: R1 — Repetition ane; R2 — Repetition 2; INRT - intransitive preferences; TR - transitive preferences.
In Office 2, we asked subjects to make 2 choices in each ternary fix, for the virtually and next most preferred object for each of the 11 triples (see Figure 3).
We sought to maximize the similarity to the binary choice task and and then we incentivized the choice of best and next best in each ternary set to obtain the total ordering in the triple. Nosotros explained that we would describe two of the 3 lotteries at random and they would play out whichever of the two they had positioned college, if the ternary set is selected for payment. Nosotros utilise the standard 'random lottery incentive system' for the binary choices. Our aim was to keep to a minimum any inapplicable 'option versus ranking' disparities, to allow as clean a exam equally possible of the expansion consistency property and the transitive random preference restrictions, H2–iv.
We randomized the guild in which the ternary sets appeared, too as the order in which the lotteries appeared on each screen. To avoid lazy credence of the default ordering, information technology was not possible to accept the default ranking. If the default was by risk preferred, they first had to move away from the default then movement back to it by deliberate pick.
In Part iii, subjects were offered a Distractor Task in order to create a interruption between the two repetitions of binary and ternary choice tasks. The distractor job consisted of 12 risky choices using a unlike display, the results of which are not reported hither. In Role 4, all 11 ternary option gear up problems were repeated in a different random order.
We asked subjects to consummate all tasks in the experiment online in a 5-day window. All 100 subjects received an invitation to the laboratory to play out their decisions for real money. Each subject drew a question number (between 1 and 100) at random and received payment based on his/her choice in that question. In each question, we looked at the lottery option called past the subject and played out that lottery according to the description on the experimental brandish (e.one thousand., Figure 2 and Effigy iii). Immediately after the describe, subjects received their payoff in cash. All 100 subjects turned up to play out the lottery and receive their winnings. Finally, all subjects completed a detailed online survey covering questions such as domain-specific risk attitudes and a variety of demographic variables not reported on here.
Figure 4: Histogram of Wheel Frequency by Private.
3 Results
3.1 Descriptive Statistics
Table 2 reports the frequency of intransitive cycles for each of the xi triples obtained from binary choices. Other than the PR control (Triple 9) nosotros discover the proportion of preference cycles averaged across both repetitions ranges from a depression of 18% (Triples 2 and 6) to a loftier of 59% (Triple 4).4
In contrast the PR 'control' triple institute just half-dozen% intransitive patterns. Other than the command triple, the proportion of cycles for each triple is strongly significantly greater than v%, using Fisher'south exact test, supporting H1. The average proportion intransitive in the showtime cake was 27.1% followed by 25.eight% for the second block, giving an overall intransitive proportion of 26.5%. If learning occurred between blocks, it did not reduce the occurrence of cycles noticeably, giving our first clue that error may not be the primary cause of cycles. Of the eight possible preference orderings (two intransitive, vi transitive), the mode was intransitive for three of the ten triples and an intransitive blueprint was runner-upward in a farther 5 triples.
At the individual level, betwixt 30% and 85% of subjects cycled in each of the ten triples either once or on both repetitions. For instance, in triple 4 alone, 85 of the 100 individuals cycled at least one time and 34 cycled on both occasions. It appears that a significant minority, a plurality, even an occasional majority, exhibit intransitive choice cycles, for these statistically independent pairs of simple, incentivized lotteries. Across the 10 triples, every single 1 of the 100 subjects cycled at to the lowest degree once. A Spearman correlation of the number of cycles by individual between repetitions was +0.93, suggesting once again that the cycles nosotros observe are not simply random errors but latent intransitive preferences. Figure 4 shows the histogram of cycles by private.
The predominant management of cycles for our lotteries is consistent with 'contrary' cycles, rather than the 'likely winner' cycles.v The 26% average figure breaks down nineteen:7 in favour of the 'contrary' direction. For the 8 triples where an intransitive blueprint is either the mode or runner-up, six follow the 'reverse' direction and two the 'probable winner' management. This likely reflects our selection of expected value rankings for X, Y and Z, as nosotros noted previously. Unlike the STP, the random lottery incentive arrangement does not induce a particular preference design; it elicits preferences rather than prescribes them.
Figure 5: Individual Inconsistency versus Frequency of Cycles.
3.ii Noisy Transitivity or Noisy Intransitivity?
A reasonable approximation to a true, 'error free' proportion for each preference pattern is to place those subjects making the same three binary choices within a triple on both repetitions. To do so means they avoid vi possible pick errors, for each 'true' preference blueprint. Across the ten sets of triples (excluding the control), the diagonals for each triple in Table 2 reveal this occurs on 366 occasions out of thou. Of these, 117 were of i of the two intransitive orderings and 249 were for one of the six transitive orderings. Thus, the share of revealed consistently intransitive preference patterns among all revealed consequent preference patterns was 32% (117 out of 366). To go a sense of how striking this finding is, a recent and unusually conscientious and thorough investigation of intransitive selection patterns was able to conclude: "...very few people repeat the same intransitive pattern on two replications of the same test. In other words, nearly violations that have been observed can exist attributed to fault rather than to truthful intransitivity" (Birnbaum & Diecidue, 2015). Yet we find that on boilerplate a typical intransitive pattern is 41% (117/2 vs 249/6) more likely to replicate than a typical transitive pattern.
Delving deeper, we see from the middle panels in Tabular array 2 that the intransitive proportion of consistently revealed patterns across the triples varies from 17.viii% (triple 8) to 83.9% (triple iv). The modal consistently revealed patterns are intransitive for triples four, 7, 10 and 11 and runner-upwardly for Triples 1, 3, 5 and 6; that is, eight of the 10 triples have an intransitive modal or second modal consistently revealed preference design. Therefore, consistently intransitive preferences appear to be revealed relatively more frequently than intransitive preferences that may or may not replicate. This result suggests that noise diminishes (rather than increases) intransitive preferences in revealed choice patterns. In other words, as the dissonance washes out, cyclical selection patterns increase their share of the total, a finding replicated at the private level, as we next show. Equally a comparing, in our command, triple nine, we notice the contrary: just 1 consistently intransitive person but 51 consistently transitive people, a 98% transitive share, in line with the consensus view on the rarity of cycles.
Another fashion to check whether cardinal intransitivity or racket is driving the information is to split the subjects into 2 equal-size groups by rate of pick switching between repetitions. On inspection, nosotros discover a threshold of ten or fewer inconsistencies separates 51 individuals with fewer and 49 individuals with more inconsistencies (or stochastic preferences). In Effigy v, we plot the number of cycles exhibited by each individual confronting the number of his or her option switches between repetitions.
The graph shows a broad tendency for more than cycles amongst the more consistent individuals, with cycles decreasing as the rate of choice switching increases. The most consequent group of 51 subjects committed an boilerplate 6.sixteen cycles (out of a maximum possible xx), which is 33% more than the group of 49 noisier individuals who averaged 4.63 cycles. This determination is not dependent on the threshold of x, every bit is clear from Figure 5, offer further prove that true intransitivity rather than dissonance is responsible for most of the observed cycles.vi
Weak stochastic transitivity (WST), requires Pr (C ≻ A) to be at least as large as the minimum of Pr (B ≻ A) and Pr (C ≻ B), a requirement diametrically at odds with the STP. Every bit noted earlier, violations of WST can also result if subjects take random preferences over exclusively transitive preference orders. Withal, we also showed an overlooked implication of this claim is the set of constraints we identified in H2-H4. The frequency with which each lottery is called from the respective ternary choice fix must satisfy each of H2-H4. In other words, any random preference model over transitive orderings capable of violating WST must encounter H2-H4. If it does not, latent intransitive preferences are presumably the only remaining explanation. Our data shows that WST is violated for Triples i, 3, 4, 5, and 7 (run across Table 3).vii
Table 3: Intransitive cycles across binary and ternary choices.
Total Triple Selection p1=1/three p2=one/3 p3=1/3 X ≻ Y Y ≻ 10 Y ≻ Z Z ≻ Y X ≻ Z Z ≻ X i X 12 12 3 60 140 100 100 127 73 Y 8 8 8 43 81 76 53 95 41 Z 24 4 4 72% 58% 76% 53% 75% 56% 2 X xi 11 1 152 49 108 92 122 78 Y 5 5 5 104 27 49 59 87 48 Z 20 2 2 69% 55% 45% 64% 71% 62% three X 12 12 2 69 131 132 68 130 seventy Y 8 viii viii 35 84 83 27 107 52 Z 20 4 4 51% 64% 63% 40% 82% 74% 4 X 15 15 iii 57 143 71 129 146 54 Y 10 10 10 37 97 49 32 106 20 Z 27 v 5 65% 68% 69% 25% 73% 37% v 10 xv 15 half-dozen 69 131 100 100 124 76 Y xi 11 11 33 91 72 37 105 51 Z 28 half-dozen 6 48% 69% 72% 37% 85% 67% 6 X 17 17 5 69 131 124 76 135 65 Y 12 12 12 38 71 80 27 103 36 Z 30 half-dozen six 55% 54% 65% 36% 76% 55% 7 X ix 9 3 85 115 64 136 120 lxxx Y 6 6 6 50 55 37 87 47 45 Z 16 4 4 59% 48% 58% 64% 39% 56% eight X xv 15 5 75 125 143 57 145 55 Y 10 10 x 38 58 61 29 93 29 Z 30 3 3 51% 46% 43% 51% 64% 53% 9 Ten 12 12 0 46 154 182 18 156 44 Y 7 7 vii 27 86 97 9 76 24 Z 28 0 0 59% 56% 53% 50% 49% 55% 10 X xiv 14 2 65 135 47 153 107 93 Y 8 8 8 34 76 31 86 76 67 Z 21 vi 6 52% 56% 66% 56% 71% 72% 11 X 14 14 2 96 104 118 82 84 116 Y eight 8 ix 44 59 32 32 33 73 Z 24 5 5 46% 57% 27% 39% 39% 63% Notes: In each Triple in Columns 6-11: row 1 shows information for binary option task; row 2 shows data for ternary selection task; row 3 shows (data from ternary/data from binary) in %.
Tabular array four: Testing Hypotheses 2–4.
Triples Hypothesis 2 Hypothesis 3 Hypothesis 4 ane WST H2: Pr(X) <0.36
Information: Pr(Ten) = 0.32H3: Pr(Y) <0.3
Data: Pr(Y) = 0.39H4: Pr(Z) <0.5
Information: Pr(Z) = 0.293 WST H2: Pr(10) <0.35
Data: Pr(X) = 0.37H3: Pr(Y) <0.34
Information: Pr(Y) = 0.43H4: Pr(Z) <0.35
Data: Pr(Z) = 0.29four WST H2: Pr(X) <0.27
Data: Pr(X) = 0.34H3: Pr(Y) <0.285
Data: Pr(Y) = 0.485H4: Pr(Z) <0.355
Data: Pr(Z) = 0.1755 WST H2: Pr(X) <0.38
Data: Pr(X) = 0.23H3: Pr(Y) <0.34
Data: Pr(Y) = 0.54H4: Pr(Z) <0. 5
Information: Pr(Z) = 0.237 WST H2: Pr(10) <0.4
Information: Pr(Ten) = 0.23H3: Pr(Y) <0.42
Information: Pr(Y) = 0.315H4: Pr(Z) <0.32
Data: Pr(Z) = 0.455Notes: Shaded cells highlight cases when a hypothesis is rejected by the information.
Triple 4 exhibits the strongest violation followed closely by Triple 3. Averaged across both repetitions, in triple iv we found: Pr (Y ≻ 10)= 71.5%; Pr (Z ≻ Y) = 64.5% and also that Pr (Z ≻ Ten) = 27%. At 37.5 per centum points, this is a strikingly stiff violation of WST. It is besides a violation of 'elementary scalability' (Tversky & Russo, 1968). Table iv shows the results of the hypothesis tests.
In summary, for each triple where WST did not agree for the binary selection sets, any transitive random preference model must satisfy H2-H4 in the ternary sets. This is essential if the WST violations in the binary sets were a result of stochastic merely transitive latent preferences. Taken as a whole, the tests reported to a higher place to dissever noisy only transitive latent preferences from underlying intransitivity lean heavily in favor of the latter suggestion.
Finally, triple 8 was one of ii triples where intransitive patterns were relatively infrequent. The lotteries comprising triple 8 were designed to be a examination of one intriguing 'ingredient' in the STP recipe: a higher minimum consequence for Z than for 10. A modest airplane pilot experiment had previously identified triple 4 as particularly prone to exhibit cycles (19 of 27 subjects cycled). We decided to make as few changes as possible to the lottery pairs of triple 4 when swapping the lowest payoff in X with that in Z. This change then required an increase in the maximum payoff in Z to keep the expected value above that for X.
The combined effect of these ii changes is to drastically reduce the number of observed cycles, from 119 of 200 in triple 4 down to 42 of 200 in triple eight (triple 8 resembles, only does not strictly satisfy, the STP). Even more striking is the reduction in the intransitive share of consistently revealed patterns, from 83.9% in triple 4 to 17.viii% in triple 8. Nosotros conjecture that if researchers prefer this new PR design, they may find even stronger reversals than those that have comprised the PR paradox to date.
3.three Testing Expansion Consistency and IIA
The second purpose for eliciting preferences in the respective ternary choice sets is to test for expansion consistency. Table iii presents the results for all decisions, contrasting the binary and ternary choices, by triple. Table five reports separately the most preferred elements in all the ternary sets.
Table 5: Average ternary top preferences past triple.
Triple Selection p1=1/three p2=i/3 p3=1/3 Ten in a 1st choice Y in a 1st pick Z in a 1st choice 1 X 12 12 iii 32% 39% 29% Y viii 8 viii Z 24 4 four 2 X 11 11 1 49% 14.v% 36.5% Y 5 5 5 Z 20 2 2 three X 12 12 ii 43% 28% 29% Y 8 8 8 Z xx four 4 four X xv xv iii 34% 48.5% 17.v% Y 10 10 10 Z 27 5 5 v X 15 15 6 23% 54% 23% Y xi xi 11 Z 28 6 6 vi Ten 17 17 v 33% 41.five% 25.five% Y 12 12 12 Z 30 6 6 7 X 9 9 3 23% 31.five% 45.5% Y 6 half-dozen 6 Z xvi 4 four 8 10 xv 15 5 33.five% 38% 28.five% Y 10 x x Z thirty 3 3 nine 10 12 12 0 44% 42% 14% Y 7 7 7 Z 28 0 0 10 X fourteen 14 2 34% xxx% 36% Y viii eight 8 Z 21 half-dozen 6 11 Ten 14 14 2 25.5% 31% 43.v% Y 8 eight nine Z 24 five 5
The rightmost panels in Table 3 show the following. The binary choices in repetition 1 were consistent with binary choices in repetition 2 for 2043/3000 or 68.i% of decisions (excluding the control), which we accept as the benchmark. Aggregating across these ten sets of triples, for those choosing 10 ≻ Z in the binary ready, 852 of 1240, or 68.seven% maintain the ranking in the ternary set. The binary/ternary comparison for X ≻ Z is the aforementioned every bit the benchmark consistency supporting (stochastic) expansion consistency when Y is included. Even so, this is just one of half-dozen binary preference ranks tested and is the only 1 to match the binary/binary consistency.
For the binary preference Y ≻ X, 699 of 1204 decisions, maintain this ranking in the ternary fix, or but 58.1%. This is statistically significantly lower than 68.1%. Finally at that place are 993 binary choices of Z ≻ Y; of these merely 469, or 47.2%, maintain that rank in the respective ternary comparisons so expansion consistency is rejected. This is the everyman consistency rate of the six binary ranks. It seems that preference for the riskiest selection over the certainty reverses when the intermediate option is included in the gear up.
The other 4 comparisons are all around 58% consistent. Hence, most binary/ternary comparisons show clear evidence, beyond dissonance, of ready-dependent preferences, inconsistent with expansion to, or contraction from, the ternary set. While some of the violations reflect the well-known compromise consequence, the largest inconsistency to a higher place cannot exist explained past whatever of the three known effects: attraction, similarity or compromise. We can as well test whether preferences revealed from a binary selection are consistent with preferences revealed from rankings within a ternary set at a more fine-grained level using the Conlisk (1989) test. This test compares the relative frequency of two inconsistent choice patterns. The first pattern is when a conclusion maker chooses X over Y in a binary choice simply ranks Y over X in a ternary ready. The second pattern is when a conclusion maker chooses Y over X in a binary choice but ranks 10 over Y in a ternary set. If these two option patterns are due to indifferences, random errors, imprecision, noise or indecisiveness, they should occur, a priori, with equal or similar frequencies. In dissimilarity, a determination maker who systematically reveals i pick pattern significantly more ofttimes than the other (for instance, by following the most likely winner) is unlikely to just reflect indifference or noise. The Conlisk's (1989) test formalizes this idea.
Table 6 presents the results of Conlisk (1989) exam comparison the consistency of binary choices with ternary rankings.
Table half-dozen: Conlisk z (p-value) comparing binary choices X vs Y, Y vs Z and X vs Z with the corresponding rankings from a ternary {X,Y,Z}.
Triple
Y vs Z
Conlisk z
(p-value)X vs Y
Conlisk z
(p-value)X vs Z
Conlisk z
(p-value)1
−ii.7749
(0.0028)−v.1114
(0.0000)0.0000
(0.5000)2
two.7550
(0.0029)iii.0725
(0.0011)0.6192
(0.2679)3
0.8427
(0.1997)−i.4484
(0.0738)0.7801
(0.2177)4
−vii.8478
(0.0000)−3.2774
(0.0005)0.6966
(0.2430)5
−three.7896
(0.0001)−0.4579
(0.3235)−0.9041
(0.1830)6
−0.5175
(0.3024)−iii.1050
(0.0010)0.3833
(0.3508)7
−two.5583
(0.0053)−2.6017
(0.0046)iii.7758
(0.0001)eight
5.5142
(0.0000)−3.0000
(0.0013)three.0023
(0.0013)9
9.3944
(0.0000)−v.6441
(0.0000)6.6093
(0.0000)10
−vi.0806
(0.0000)−3.0104
(0.0013)0.6613
(0.2542)11
3.1553
(0.0008)0.7099
(0.2389)0.8245
(0.2048)
A significant positive or negative z value indicates that inconsistencies between binary choices and ternary rankings are non due to indifferences, random errors, imprecision or noise. In triples 7-10 binary option is e'er statistically significantly different (at 5% significance level) from ternary rankings for all three comparisons (10 vs Y, Y vs Z and 10 vs Z). For triple 3 binary choices are non statistically significantly different from ternary rankings (for all three comparisons). For other triples, there are significant differences for some just not all combinations (typically 10 vs Y and Y vs Z are significantly dissimilar but 10 vs Z is non). Thus, for all triples but triple 3 in that location appear to be significant inconsistencies between binary choice and ternary rankings that go beyond imprecision or noise.
More specifically, at that place is an disproportion when Z is involved; when the binary choice favors Z (over X or over Y) the ranking is more than likely to be overturned in the ternary set than if Z is disfavored in the binary set up (see Figure 6). These results appear to confirm a preponderance of set-dependent preferences, in violation of expansion-consistency, as predicted earlier.
Effigy six: Ready-Dependent Preferences.
The consequence captured in Effigy 6 suggests that when there is a directly selection between our modified $-bet and certainty, the bulk of subjects (129 of 200 people) prefer the new $-bet. Of those 129, 97 reversed their preference when new P-bet was included in the choice set. The set-dependent anchoring and adjustment event (Slovic & Lichtenstein, 1983) may assistance explicate Figure half-dozen. Specifically, when people consider the new $-bet versus certainty, they tend to focus on the pale rather than probability and concentrate on the nigh extreme result.8 However, when we add the new P-bet to the choice ready, our results suggest that subjects tend to concentrate on probabilities rather than stakes. They pay attention to the most extreme probability of i (certainty) and, hence, tend to choose certainty over the new $-bet.
four Determination: Predictably Intransitive?
Where do our arguments leave the most pop utility theories of choice nether risk (which all include transitivity)? The Steinhaus-Trybula Paradox shows that for multi-attribute risky choice objects, which we evaluate in binary and ternary choice sets, relying on transitivity tin neglect to select the most advantageous lotteries. The domain of application for transitive theories excludes choice-rules that are common and cognitively plausible and can violate transitivity where utility differences between options are not also big. It is likely that ternary choice sets using smaller differences in expected values than we used would show even greater rates of intransitive binary comparisons.
An innovative characteristic of our experiment was eliciting preferences in the ternary choice sets likewise as the constituent binary sets. This allows u.s.a. to investigate the predicted set up-dependence of preferences and test transitive just random preference theory as a possible caption of cycles. Results back up our conjectures that the cycles reverberate latent intransitive preference rather than noisy implementation of transitive preferences. We saw that although very little solid evidence for true intransitive preferences existed prior to our experiment, this 'absence of evidence' should not exist mistaken for 'prove of the absence' of preference cycles. This paper has identified limits to the descriptive invocation of transitivity. Our findings also point to a deeper underlying process at work in option under risk, office of the growing testify (Louie, Glimcher & Webb, 2015) that choices often bear the postage of other options in the choice prepare, too every bit latent preferences (Noguchi & Stewart, 2014).
Ane implication of our arguments and our experimental results is that rather than modelling individuals equally possessing a core utility office (transitive or intransitive), many typically transitive individuals are the same people who violate transitivity in the circumstances we identify. This suggests neither a transitive nor intransitive 'core' utility function tin can accurately describe preferences over all lotteries a person may encounter. Stewart, Reimers and Harris (2015) recently concluded, "The shape of the revealed utility ... function is, at least in function, a property of the question set and not the individual", in line with a constructed-preference paradigm. Our results indicate towards the synthetic-preference image every bit the more promising way forrad.
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Appendix
Figure A1: Experimental Instructions: Screenshot ane.
Effigy A2: Experimental Instructions: Screenshot 2.
Figure A3: Experimental Instructions: Screenshot 3.
Figure A4: Experimental Instructions: Screenshot 4.
Figure A5: Experimental Instructions: Screenshot 5.
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Department of Accounting, Finance and Economic science, Griffith Concern School, Griffith Academy, Gold Coast, Queensland
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Corresponding author: Section of Economics, Birmingham Business School, University of Birmingham, JG Smith Building, Birmingham, B15 2TT, email: gpogrebna@gmail.com
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Alan Turing Establish, 96 Euston Road, Kings Cross, London, NW1 2DB
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We would like to thank seminar participants at Warwick Business concern Schoolhouse, Academy of East Anglia, University of California, Irvine, Cal State Fullerton, University of Arizona, Curtin University, Murdoch University, Sydney Academy, Griffith University, University of Queensland and Queensland University of Technology for comments on before versions of this manuscript. David Butler acknowledges the back up of the Australian Inquiry Council (grant: DP1095681). Ganna Pogrebna acknowledges financial support from RCUK/EPSRC grants EP/N028422/1 and EP/P011896/one.
Copyright: © 2018. The authors license this article under the terms of the Creative Eatables Attribution 3.0 License.
- i
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We discuss her preference club over the ternary choice set up {A,B,C} in the next department.
- ii
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Evidence from response fourth dimension's shows fast decisions when i choice is clearly better. Response times lengthen, equally the DM needs to accumulate more evidence to trigger a choice.
- 3
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In order to brand sure that subjects in our experiment understand probabilities, we use display with 3 differently colored marbles for each lottery option. We provide screenshots of this brandish in afterward sections of this paper.
- 4
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Table A in the Online Supplementary Fabric provides detailed summary statistics including the frequency of intransitive cycles for each of the xi triples.
- v
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The 'likely winner' wheel refers to a case when CE ≻ $−bet ≻ P ≻ CE and the 'reverse' refers to a case when $−bet ≻ CE ≻ P−bet ≻ $−bet.
- 6
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The caveat is there may be a modest uptick of cycles for the nigh inconsistent of all; but inspection of the graph shows just two out of 100 individuals drive the uptick, and so it may non be reliable.
- 7
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Kim, Seligman & Kable (2012) show in an eye-tracking study that when either $-bet or P-bet is compared to certainty; people tend to pay more attention to stakes rather than probabilities which is consequent with our findings. Kim et al. (2012) also find that when $-bet is compared with P-bet in a straight binary choice, people tend to focus on probabilities rather than stakes.
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