Testing transitivity of preferences using linked designs

Abstract

Three experiments tested if individuals show violations of transitivity in choices between risky gambles in linked designs. The binary gambles varied in the probability to win the higher (better) prize, the value of the higher prize, and the value of the lower prize. Each design varied two factors, with the third fixed. Designs are linked by using the same values in different designs. Linked designs allow one to determine if a lexicographic semiorder model can describe violations of transitivity in more than one design using the same parameters. In addition, two experiments tested interactive independence, a critical property implied by all lexicographic semiorder models. Very few people showed systematic violations of transitivity; only one person out of 136 showed violations of transitivity in two designs that could be linked by a lexicographic semiorder. However, that person violated interactive independence, as did the majority of other participants. Most individuals showed systematic violations of the assumptions of stochastic independence and stationarity of choice responses. That means that investigators should evaluate models with respect to response patterns (response combinations) rather than focusing entirely on choice proportions.

1 Introduction

Descriptive theories of risky decision making can be divided into two groups: those that satisfy transitivity of preference and those that do not. Transitivity of preference is the assumption that, if a person prefers A to B and prefers B to C, then that person should prefer A to C, apart from random error. We use the symbol, ≻, to denote preference, so the property can be denoted as follows: A ≻ B and B ≻ C ⇒ A ≻ C.

Theories that represent each gamble by a single number automatically imply transitivity. These theories assume that A ≻ B ⇔ U(A) > U(B), where U(A) and U(B) are the numerical values or utilities of the two gambles. Expected utility theory (EU), cumulative prospect theory (CPT), and the transfer of attention exchange model (TAX), as well as many other theories, fall in this class of theories that satisfy transitivity (Birnbaum, 2008b; Tversky & Kahneman, 1992; Luce, 2000; Wakker, 2011).

Theories that represent choice in terms of contrasts between the components of the alternatives, however, need not satisfy transitivity of preference. Theories that violate transitivity include the family of lexicographic semiorder (LS) models, the priority heuristic, regret theory (RT), the stochastic difference model (SDM) and others (Birnbaum, 2010; Birnbaum & Gutierrez, 2007; Birnbaum & Schmidt, 2008; Brandstätter, Hertwig, & Gigerenzer, 2006; González-Vallejo, 2002; Loomes, Starmer, & Sugden, 1991; Luce, 1956; 2000; Myung, Karabatsos, & Iverson, 2005; Regenwetter, Dana, and Davis-Stober, 2010, 2011; Rieskamp, Busemeyer, & Mellers, 2006; Tversky, 1969).

An example of a lexicographic semiorder (LS) is presented next to illustrate how such a model can account for intransitive preferences.

1.1 Lexicographic semiorders

Let G = (x, p; y) represent a two-branch gamble in which prize x is received with probability p and otherwise y is received, where x > y ≥ 0. In such two-branch gambles, there are three variables that can be manipulated experimentally: y = Lowest (L) consequence; x = Highest (H) consequence; and p = Probability (P) to win the higher prize.

We use the notation LPH LS to refer to the lexicographic semiorder (LS) model in which the person is assumed to compare the attributes in the order L, then P, then H. The three attributes might be examined by a participant in any of five other possible orders: LHP, HPL, HLP, PLH, and PHL.

In the LPH LS model, a person is assumed to compare two such gambles, G = (x, p; y) and F = (x ^′, q; y ^′) by contrasting attributes with thresholds (Δ_L and Δ_P) as follows:

1. 1. First compare L: If , choose the gamble with the higher lowest consequence;

2. 2. Else, compare P: if , choose the gamble with the higher probability to win the better prize;

3. 3. Else, check H: if , chose the gamble with the higher best prize;

4. 4. Else, choose randomly.

The priority heuristic of Brandstätter, et al. (2006) is a variant of this LPH LS in which it is assumed that Δ_L equals one tenth of the highest consequence in either gamble, rounded to the nearest prominent number, where prominent numbers are integer powers of 10 plus one-half and twice those values; i.e., 1, 2, 5, 10, 20, 50, 100, etc. If the highest prize always rounds to $100 (as in the experiments of this article), then Δ_L = $10. Further, the priority heuristic assumes that Δ_P = 0.10, presumably, due to the base 10 number system. Therefore, in these studies the priority heuristic is a special case of the LPH LS model. Brandstätter, et al. (2006) showed that with these selected parameters, this model approximates the results of several previously published papers; in addition, they claimed that the priority heuristic is more accurate than other models for these selected studies.

To illustrate how this LPH LS model can violate transitivity, consider the following five gambles: K = ($100, .50; $0), L = ($96, .54; $0), M = ($92, .58; $0), N = ($88, .62; $0), and O = ($84, .66; $0). According to the priority heuristic, Δ_L = $10 and Δ_P = 0.10, so people should prefer K ≻ L, L ≻ M, M ≻ N, and N ≻ O, because the differences in probability are only 0.04; these are too small to be decisive (less than Δ_P = 0.10), so preferences are determined by the highest consequences. However, O ≻ K, because the difference in probability is 0.16, which exceeds the threshold of Δ_P = 0.10. As long as 0.16 ≥ Δ_P > 0.04, the LPH LS implies: K ≻ L, L ≻ M, M ≻ N, and N ≻ O, but O ≻ K, violating transitivity. When Δ_P = 0.10, as in the priority heuristic, then two other violations are also predicted, O ≻ L and N ≻ K. If Δ_P ≤ 0.04, the LPH LS model implies the transitive order ONMLK, and if Δ_P > 0.16, it predicts the transitive order KLMNO.

Now consider a second design with choices among the following gambles: A = ($84, 0.5; $24), B = ($88, 0.5; $20), C = ($92, 0.5; $16), D = ($96, 0.5; $12), and E = ($100, 0.5; $8). According to the priority heuristic, E ≻ D, D ≻ C, C ≻ B, and B ≻ A, because in each of these choices, the lowest consequences differ by less than $10, and probabilities are equal, so these choices are determined by the highest consequences. However, in the choice between A and E, the lowest consequences differ by $16, which exceeds Δ_L= $10, so A ≻ E, violating transitivity. As long as $16 ≥ Δ_L > $4, the LPH LS implies there should be at least one intransitivity in this design, with A ≻ E. If Δ_L = $10, as in the priority heuristic, then two other violations are also predicted, A ≻ D and B ≻ E. If Δ_L ≤ $4, the LPH LS predicts the transitive order, ABCDE, and if Δ_L > $16, it predicts the transitive order, EDCBA.¹

1.2 Intransitive preferences in linked designs

The LS models, including the priority heuristic, imply that choices in linked designs will be related. This study uses the two designs described above, and also a third design with choices among the following: F = ($100, 0.5; $24), G = ($100, 0.54; $20), H = ($100, 0.58; $16), I = ($100, 0.62; $12), and J = ($100, 0.66; $8). Note that the levels of lowest consequence match those in the design with A, B, C, D, and E and that probability values match those in the first design, with K, L, M, N, and O. Those two designs are linked in turn by the levels of the highest consequence. These designs with linked levels should show predictable patterns of transitivity or intransitivity, if a person used the same LS model in all three designs.

The stimuli used in these studies with linked levels are listed in Table 1. The designs are named after the variables manipulated: the LH design varies the lowest consequence (L) and the highest consequence (H), the LP design varies the lowest consequence and probability (P), and the PH design varied probability and highest consequence.

Table 1: Gambles used in linked tests of transitivity.

For example, suppose that a person conformed to the LPH LS model. If that person showed data consistent with the transitive order ONMLK in the PH design, it means that Δ_P ≤ 0.04; and suppose that the same person showed intransitive choices in the LH design consistent with Δ_L = $10. In that case, the model implies intransitive data in the LP design such that J ≻ I, I ≻ H, H ≻ G, G ≻ F and yet F ≻ J, F ≻ I, and G ≻ J. So, if results conformed to this prediction, they would represent successful confirmations of new predictions, and if not, the model(s) that predicted them would be disconfirmed. See Appendix A for all possible linked patterns in the LPH LS model.

1.3 TAX Model

A transitive model that has been fairly successful in describing violations of EU and CPT is Birnbaum’s (1999; 2008b) special Transfer of Attention Exchange (TAX) model. This model represents the utility of a gamble as a weighted average of the utilities of the consequences, but weight in this model depends on the probabilities of the branch consequences and ranks of the consequences in the gamble. This model can be written for gambles of the form G = (x, p; y) where x > y ≥ 0 as follows:

(1)

where a = t(p) – ωt(p); b = t(q) + ωt(p), where q = 1 – p and when ω > 0. In this case (ω > 0), there is a transfer of attention from the branch leading to the best consequence to the branch leading to the worst consequence. In the case where ω < 0, weight is transferred from lower-valued branches to higher ones; in that case, a = t(p) – ωt(q); b = t(q) + ωt(q). The configural parameter, ω, can produce risk aversion (ω > 0) or risk-seeking (ω < 0), even when u(x) = x. When ω = 0 and t(p) = p, this model reduces to expected utility (EU). Expected Value (EV) is a special case of EU in which u(x) = x.

When fitting individual data in a suitable experiment, parameters can be estimated from the data. However, for the purpose of making predictions before conducting new studies, a simple version of the special TAX model has been used (e.g., Birnbaum, 2008b): u(x) = x for 0 < x < $150; t(p) = p ^0.7, and ω = 1/3, where ω = δ/(n + 1), δ = 1, and n = 2 is the number of branches. These have been called “prior” parameters, because they have been used in previous studies to design new experiments and predict modal results with similar participants, contexts, and procedures. Although these are not optimal, they have had reasonable success predicting aggregate results of new studies with American undergraduates who choose among gambles with small gains (e.g., Birnbaum, 2004, 2005, 2008b, 2010).

The TAX model with these prior parameters implies the transitive orders, ABCDE, FGHIJ, and ONMLK. Although the TAX model successfully predicted new results that violated CPT, in the tests of transitivity of Table 1, TAX with these parameters makes virtually the same predictions as CPT with the parameters of Tversky and Kahneman (1992). With other parameters, TAX, CPT, and EU could account for other transitive orders, but these models always imply transitivity. Experiments 2 and 3 also include tests between TAX and CPT.

Transitivity can therefore be considered a critical property of TAX, CPT and EU because these models (with any parameters) cannot account for systematic violations of transitivity. The family of LS models could handle either transitive or intransitive data, so finding transitive preferences would not refute LS models. For example, the LPH LS model with Δ_P ≤ 0.04 and Δ_L ≤ $4 makes the same transitive predictions for this study as TAX model with its prior parameters. There are critical properties of LS models, however, that can lead to refutation of those models (Birnbaum, 2008a, 2010; Birnbaum & LaCroix, 2008), described next.

1.4 Critical properties of LS models

Birnbaum (2010) considered a general family of LS models in which each person might have a different priority order in which to compare the features; each person might have a different monotonic utility function for monetary prizes and a different subjective function for probability; and each person might have different thresholds for determining if a given subjective difference is decisive. Birnbaum (2010) showed that this general family of LS models implies properties of priority dominance, integrative independence, and interactive independence. In Experiments 2 and 3, we test interactive independence, which can be written:

In these two choice problems, note that F and G share a common probability to win (p), and F ^′ and G ^′ also share a common probability (q). According to this most general family of LS models, a person should either prefer F to G and F ^′ to G ^′ or prefer G to F and G ^′ to F ^′, or be indifferent in both cases, but a person should not shift from F to G ^′ or from G to F ^′ as the common probability is changed, except by random error. To test such a property with real data requires a theory to separate random error from systematic violation.

1.5 Testing algebraic properties with probabilistic data

Testing properties such as transitivity or interactive independence is complicated by the fact that people are not completely consistent in their responses. Different people can make different responses when asked the same question, so we must allow for individual differences. Furthermore, the same person might make different responses when the same choice problem is presented on a later trial, following other intervening trials. It is possible that the person has changed her or his “true” preferences, that responses contain “error”, or both. Exactly how to analyze data containing variability has been the topic of debate (Loomes & Sugden, 1995; Birnbaum & Bahra, 2012; Regenwetter, et al., 2011).

Morrison (1963) reviewed two properties that are implied by certain stochastic choice models such as Luce’s (1959) choice model: Weak Stochastic Transitivity (WST) and the Triangle Inequality (TI). These properties were also viewed as methods for analyzing transitive models with variable data. The TI can be written:

where p(A, B) is the probability to choose A over B. WST can be written:

and

Morrison (1963) advised that both of these properties should be tested. Tversky (1969) cited Morrison but reported only tests of WST. Tversky’s statistical tests were challenged by Iverson and Falmagne (1985), who noted that Tversky’s tests did not properly allow for individual differences in preference orders (cf., Myung, et al., 2005).

Regenwetter, et al. (2010, 2011) also criticized Tversky’s failure to test the TI, and proposed statistical tests of these properties based on the assumptions that repeated responses to the same choice problem are independent and identically distributed (iid). They argued that, if each person’s responses can be modeled as an iid sample from a mixture of different transitive preferences (so that choice proportions satisfy the linear order polytope, which includes TI), there would be no reason to argue for LS models.

Birnbaum (2011, 2012) questioned Regenwetter, et al. (2011) for not testing the crucial iid assumptions; when iid assumptions are false, neither WST nor TI (nor any analysis of the linear order polytope defined on choice proportions for individual choice items) can be regarded as unambiguous tests of transitivity.

There are two problems: WST can be violated even when a person has a mixture of strictly transitive orders, and the TI (and the linear order polytope) can be satisfied even when a person has a mixture that includes intransitive patterns (Birnbaum, 2011, 2012). These two properties can be more informative when they agree, but Birnbaum (2011) argued that we should also examine response patterns in order to ensure that choice proportions also reflect individual behavior that might in fact change systematically during a study.

A debate between Birnbaum (2011, 2012) and Regenwetter, Dana, Davis-Stober, and Guo (2011) has arisen concerning methods for analyzing variability of choice responses. The approach of Regenwetter, et al. (2010, 2011) analyzes only binary choice proportions based on the assumption of iid, whereas the “true and error” (TE) models analyze relative frequencies of response patterns, based on the assumption that errors are independent.

The true and error (TE) model, as applied by Birnbaum and Gutierrez (2007) and Birnbaum and Schmidt (2008) assumes that different people may have a different patterns of true preferences and that different choice problems may have different error rates. Different individuals might also have different levels of “noise” in their data. This type of model has not been the subject of much debate, because these models assume that behaviors of people tested separately are independent (i.e., that people do not influence each other via ESP).

More controversial is the proposal that TE model be applied to individual data with the assumption that a person might have different “true” preferences in different blocks of trials during the course of a long study (Birnbaum, 2011; Birnbaum & Bahra, 2012). This approach uses the variability of response by the same person to the same item within the same block of trials in order to separate variability due to “error” from “true” intention. This model contradicts the iid assumptions of Regenwetter, et al. (2011), which until recently have been assumed but not tested empirically.

This second type of TE model, called individual true and error theory (iTET), allows that, in a long experiment, a person might have different “true” preferences at the end of the study from those at the beginning. If a person has only one “true” pattern of preferences in all blocks, then iTET model implies that responses will satisfy the assumptions of iid. However, when a person has more than one true pattern (changing systematically during the study), the assumptions of iid will not in general be satisfied in this model (Birnbaum, 2011). The present studies tested these iid properties.

1.6 Overview of tests, results and implications

Experiment 1 found overwhelming evidence against iid. The violations of iid suggest that many if not most participants systematically changed their “true” preferences during the course of the study. Violations of iid mean that we cannot properly restrict our analysis to choice proportions, but we should examine response patterns in order to test properties such as transitivity.

Nevertheless, choice proportions are analyzed for comparison with related theories, such as the priority heuristic, that make predictions at the level of average choice proportions (summarized in appendices). The averaged choice proportions did not agree with this heuristic and not one person had data consistent with it.

If a LS model holds, there can be linked patterns of intransitivity in linked designs. This was the key idea that led to Experiment 1, but as shown below, very few participants showed evidence of intransitive preferences in any of the three experiments and only one person showed intransitivity in two linked designs that might be compatible with a LS model. A further analysis of response patterns in every individual block of data found little evidence that many, if any, people held intransitive patterns as portions of a mixture of strategies.

The property of interactive independence should be satisfied according to all LS models; however, most participants in Experiments 2 and 3 violated this property systematically (as predicted by interactive models such as EU, CPT, and TAX) including even the one participant who showed linked violations of transitivity. The results led to the surprisingly strong conclusion that LS models can be rejected for nearly every participant.

2 Method

Each participant made choices between gambles, knowing that 10 participants would play one of their chosen gambles for real cash prizes. Each gamble was described as an urn containing 100 otherwise identical tickets, which differed only in the prize values printed on them. A ticket would be drawn randomly from the chosen urn to determine the cash prize. Participants were told that any of the choice problems might be selected for play, so they should choose carefully. At the conclusion of the study, randomly selected participants were awarded prizes, as promised.

2.1 Stimuli and designs

Each choice was displayed as in the following example:

• First Gamble:

• 50 tickets to win $100

• 50 tickets to win $0

OR

• Second Gamble:

• 50 tickets to win $35

• 50 tickets to win $25

Participants viewed the choices via computer and indicated their decisions by clicking one of two buttons to identify the gamble they would rather play in each choice.

Three linked sub-designs were used to test transitivity (Table 1). The LH design used 5 binary gambles in which probability was 0.5 and in which the Lowest (L) and Highest (H) consequences were varied. In the LP design, the highest prize was fixed to $100 and both probability (P) and lowest consequence (L) were varied. In the PH design, the lowest consequence was fixed to $0 and both probability (P) and highest consequence (H) were varied.

The five gambles within each of the LH, LP, or PH designs could appear as either First or Second gamble, making 5 × 5 = 25 possible choice trials; however, a gamble was not presented with itself, leaving 20 trials in each of these three sub-design. Note that each of 10 distinct choice problems was presented in each of two counterbalanced arrangements in each block.

There were 5 other “filler” sub-designs containing 6 to 48 choices each. These other sub-designs included trials in which a person was asked to choose between gambles with up to five branches (including choices listed in Table 11 of Birnbaum, 2008b), or to choose between gambles and cash prizes to be received for certain. For the purpose of this article, trials in these other designs served as “fillers” that separated blocks of trials. Complete instructions and materials, including the filler tasks, can be viewed at the following URL: http://psych.fullerton.edu/mbirnbaum/Birnbaum\_Bahra\_archive.htm.

2.2 Procedures and participants of Experiment 1

Trials in the three main subdesigns (LH, LP, and PH) were blocked in sets of 25 to 26 choices each. Each block included all 20 trials from one sub-design, intermixed with 5 or 6 fillers, and put in restricted random order. This means that each of the 10 choices was presented twice within each block of trials, with position (first or second gamble) counterbalanced. A block of trials including any of the LH, LP, or PH designs was not presented again until at least 98 intervening trials and at most 175 intervening trials with choices from other designs had been presented.

Participants of Experiment 1 were 51 undergraduates enrolled in lower division psychology at California State University, Fullerton. Participants were tested in a lab via computers. Each participant served in two sessions of 1.5 hours each, separated by one week.

Each person worked alone, viewing instructions and materials via computer, and worked at his or her own pace for the time allotted. Therefore, some participants completed more repetitions than others. In Experiment 1, the limit was 20 blocks of trials, meaning each of the choice problems testing transitivity was judged up to 40 times by each person. For additional detail, see Bahra (2012).

2.3 LS design in Experiments 2 and 3

Experiments 2 and 3 included LH, LP, and PH designs plus additional trials that tested interactive independence, the priority heuristic, and CPT. The LS design consisted of 16 choices. Five choices testing interactive independence were of the form, R = ($95, p; $5, 1 – p) versus S = ($55, p; $20, 1 – p), where p = 0.95, 0.9, 0.5, 0.1, or 0.05. Six others were formed by presenting each of three choices: S = ($99, p; $1, 1 – p) versus R = ($40, p; $35, 1 – p), where p = 0.9, 0.5, or 0.1, with either S or R presented first. There were five additional trials, as follows: R = ($90, 0.05; $88, 0.05; $2, 0.9) versus S = ($45, 0.2; $4, 0.2; $2, 0.6), R+ = ($90, 0.1; $3, 0.7; $2, 0.2) versus S– = ($45, 0.1; $44, 0.1; $2, 0.8), S2 = ($40, 0.4; $5, 0.1; $4, 0.5) versus R2 = ($80, 0.1; $78, 0.1; $3, 0.8), S3– = ($40, 0.2; $39, 0.2; $3, 0.5) versus R3+ = ($80, 0.2; $4, 0.7; $3, 0.1), and G4 = ($99, 0.30; $15, 0.65; $14, 0.05) versus F4 = ($88, 0.12; $86, 0.70; $3, 0.18). These five trials test implications of the priority heuristic and CPT (see Birnbaum, 2008c).

2.4 Procedures and participants in Experiment 2

Experiment 2 was conducted as a replication of Experiment 1 with new participants, except with different filler designs between blocks and the addition of the LS design, which allowed us to test if those people showing signs of intransitive preferences also satisfied a critical property of LS models. In Experiment 2, blocks containing LH, LP, and PH designs (each with 5 or 6 intermixed trials from the LS design) were separated by at least 76 intervening trials, which included different intervening choices from those used in Experiment 1. There were 43 different undergraduates from the same “participant pool” tested with this procedure. Further description of the “filler” tasks, which tested restricted branch independence and stochastic dominance, are described in Birnbaum and Bahra (2012, Study 1).

2.5 Procedures and participants in Experiment 3

Experiment 3 was conducted to investigate two conjectures: First, it was conjectured that if all three transitivity designs were intermixed to make larger and more heterogeneous blocks, it might be more “confusing” to subjects, and this might induce more intransitivity. In the first two experiments, where a block of trials contained 25 trials, it was argued, people might remember preferring A to B and B to C when they were comparing A versus C, so they might obey transitivity because of an experimental demand for consistency. The idea was that by intermixing trials and spreading them out over larger blocks, memory would be overburdened, so intransitive data might be observed. Therefore, trials of all three transitivity designs were intermixed in Experiment 3.

Second, it was argued (Regenwetter, et al., 2011) that if trials from one design are separated by 3 intervening “filler” trials, responses might satisfy iid, presumably also due to the burdens of memory. Therefore, order was constrained in Experiment 3 such that any two trials from the LH, LP, or PH designs were separated by at least 3 trials from other designs. Perhaps these procedures would “help” iid to be satisfied.

Each block of Experiment 3 consisted of 107 trials (including 20 trials of the LH design, 20 trials of the PH design, 20 trials of the LP design, 16 trials of the LS design, and 31 other trials consisting of choices problems like those in Birnbaum, 2008b). Following a warmup of four “filler” trials, each pair of trials from any of the LH, LP, or PH designs was separated by at least 3 intervening trials from other designs. Each block of 107 mixed trials was separated by a separate “filler” task with 57 trials involving choices between three-branch gambles, with equally likely consequences. Experiment 3 used 42 undergraduates from the same pool, who served in two, 1.5-hr sessions, one week apart, and who completed at least 10 blocks. Materials from Experiment 3, including, the filler task, can be viewed from the following URL: http://ati-birnbaum-2009.netfirms.com/Spr\_2010/thanks3.htm.

3 Results

3.1 Data reliability and consistency

Each of the ten basic choices in each of the LH, LP, and PH designs was presented twice in each block of trials, with positions counterbalanced. We define within-block consistency as the number of consistent choices out of 10 in each block, which required the participant to push opposite buttons for the two versions of each choice. If a person “went to sleep” and clicked the same button throughout a block, within-block consistency would be 0; if a person clicked buttons randomly, expected consistency would be 5, and if a person made perfectly consistent preferences (appropriately clicking opposite buttons), this index would be 10 (100%).

Within-block consistency was apparently high in the first two experiments; mean consistencies were 86% and 91% in Exps 1 and 2, respectively). The least consistent individuals in Exps 1 and 2 had self-agreements of 67% and 73%, respectively. Similar figures were found within each subdesign; mean within block consistency was 86%, 85%, and 87% in LH, LP, and PH designs of Exp 1, and 91%, 84%, and 86% in Exp 2, respectively. In Exp 3, where each block had 107 mixed choices, within-block consistency was 77%.

We defined between-block consistency as the mean number of consistent responses out of 20 choices between every pair of trial blocks. If a participant completed 20 blocks of trials, for example, this person judged each of the 10 choices 40 times, and there are 190 pairs of blocks (20*19/2) for which the number of agreements can be counted. Comparing two blocks of trials, if a person made the same decisions in all 20 choice problems, the score would be 20; if a person randomly pushed buttons; the expected score would be 10 (50%), and the lowest score possible is 0, if a person made exactly opposite choices on all 20 trials. Note that if a person used the same button on all 20 responses in two blocks (which would produce within-block consistency of 0), between block consistency would be 20 (100%). If a person had a response bias, for example, clicking button 2 when unsure or indifferent, such a bias would increase between-block consistency and decrease within-block consistency.

We found that mean between-block consistency was significantly lower than within-block consistency: 80%, 84%, and 75% in Exps 1, 2, and 3 respectively, t(50) = 6.19, t(42) = 7.13, and t(41) = 4.26.

3.2 Violations of stochastic independence and stationarity

Examining individual data of Experiment 1, we found a surprising result: Some individuals had exactly opposite responses on 20 out of 20 choices between two trial blocks and had perfect within-block consistency on both blocks. The probability of observing this result, assuming iid is as probable as predicting the exact sequence of 20 tosses of a fair coin: less than 1 in a million! Yet 10 people out of 51 (#101, 106, 109, 113, 124, 130, 134, 141, 145, 149) showed such patterns in Exp 1, and most of these showed multiple instances of perfect reversals in their data. In addition, three other people showed reversals of 19 out of 20 choices, which has a probability less than 1 in 50,000. Such results mean that the assumptions of iid are seriously and systematically violated.

Table 2 shows raw data for Participant #134, who showed complete reversals in all three designs. Responses to the 20 choice problems are listed in the LH design in the order: AB, AC, AD, AE, BA, BC, BD, BE, CA, CB, CD, CE, DA, DB, DC, DE, EA, EB, EC, ED. The integers 1 and 2 indicate responses indicating preference for first or second gamble, respectively. Entries under the columns labeled “order” indicate cases where all 20 responses in a block are perfectly consistent with a transitive order. This person started the experiment with three trial blocks showing inconsistency, but on the seventh block, this person finished the first day with all 60 responses perfectly consistent with the transitive orders, ABCDE, JIHGF, and ONMLK.

Table 2: Raw data from Case #134 in the LH, LP, and PH Designs. Day indicates the day on which the participant completed each block, denoted “blk”. “Order” indicates where all 20 responses in a block were perfectly consistent with a transitive order. Note that all 60 responses are opposite between Block 7 and Block 15.

Participant #134 began the second session (day 8) with the same behavior in the LH and PH designs, but the LP data were different; by the 10^th block of trials, all 60 responses were now perfectly consistent with the orders, ABCDE, FGHIJ, and ONMLK. During the 11^th block, data were not perfectly consistent in any of the designs, but on the 12^th and 15^th blocks, data were now perfectly consistent with EDCBA, FGHIJ, and KLMNO, exactly the opposite of that shown at the end of the first day. The odds of a single such perfect reversal, assuming iid, of 60 trials is less than one in a million cubed!

Such dramatic and surprising results from Exp 1 led us to conduct Exp 2 as a replication with different “filler” tasks between blocks and with new participants. In Exp 2 there were 7 of 43 participants who showed at least one such complete reversal of 20 responses (#201, 221, 235, 212, 222, 230, and 232); an additional 4 showed 19 reversals between at least two blocks.

In Experiment 3, where all three designs were intermixed in blocks of 107 trials, one person (#334) showed a complete reversal in the LP design. Summed over the three studies, we observed 410 instances of perfect reversals of 20 responses between blocks. So many “1 chance in a million” outcomes cannot be reconciled with the assumptions of iid.

These cases with perfect reversals involve data that are so clean and response patterns that are so different that it is easy to detect obvious and systematic changes in preferences between blocks. Such findings suggest that there might be subtler cases where people change between similar patterns of preference or where responses contain enough variability that one could not spot violations of iid without a statistical tool to detect them.

Birnbaum (2012) devised two statistical tests of iid based on the Monte Carlo procedure suggested by Smith and Batchelder (2008): One test uses the variance of preference reversals between pairs of trial blocks and the other uses the correlation between preference reversals and the separation between blocks (which is correlated with the intervening time between blocks). These tests, summarized in Appendix B, show that most participants in all three studies had significant violations of iid. In Exps 1 and 2 there were six tests of iid for each person (variance and correlation methods in LH, LP, and PH designs). We found that only 3 out of 51 and only 4 of 43 participants did not have at least one violation of iid significant at the .01 level of significance in Exps 1 and 2, respectively.

Only 4 of 42 in Exp 3 did not have at least one of two tests significant at the .01 level. These results show that the assumptions of iid must be rejected as empirical descriptions. Violations of iid suggest that people are changing their “true” preferences during a study, in which case it could easily be misleading to analyze only marginal choice proportions. Instead, we should examine response patterns.

3.3 Analysis of modal response patterns

According to the family of LS models, it is possible to prefer A ≻ B, B ≻ C, C ≻ D, D ≻ E, and yet prefer E ≻ A. This pattern is denoted 11112, where 1 (or 2) represents a preference response for the alphabetically higher (or lower) labeled gamble, in Choices AB, BC, CD, DE, and AE, respectively. The opposite pattern, 22221, is also intransitive. All other patterns for these five choice problems are compatible with transitivity.

In each block of trials testing transitivity, there are two such tests, each based on 5 choice problems, where the positions of the gambles are reversed in the two tests. For each person, we determined the most frequent response patterns for these five choice problems separately for each presentation order. Out of 408 possible cases (136 participants by 3 designs), 333 cases were consistent; that is, the same modal pattern was observed in both presentation arrangements.

The number of participants who showed each of the modal response patterns are shown in Table 3, for consistent cases in the LH, LP, and PH designs, respectively. There were only 7 matrices (involving just 6 participants) with intransitive, consistent modal patterns; 7 cases out of 333 represent only about 2%. Cases were numbered starting with #101, 201, and 301 in the three experiments, respectively, and case numbers for those people showing consistent modal violations of transitivity are listed in Table 3. Only one person (#214) showed intransitivity in two linked designs: 22221 and 22221 in LH and LP designs. This case will be reexamined later.

Table 3: The frequency of consistent, modal response patterns in LH, LP, and PH designs. To be consistent, the participant had to have the same modal response pattern, over repetition blocks, in both ways of presenting the choices. Patterns 11112 and 22221 are intransitive. There were 51, 43, and 42 participants in Experiments 1, 2, and 3 with three designs each; only 7 cases out of 333 consistent modal patterns were intransitive.

3.4 Analysis of the Priority Heuristic

The priority heuristic implies the patterns, 22221 and 11112 in Table 3 for LH and PH designs, respectively. No consistent case showed these modal patterns, so no one obeyed the predictions of the priority heuristic.

The priority heuristic was proposed as a theory to describe the process that (most) people use when making choices (Brandstätter, et al., 2006). It is supposed to fit modal preferences, averaged over participants. Averaging our data (see Appendix C), the median response proportions are perfectly consistent with both WST and the TI. The averaged data agree with the transitive orders implied by the TAX model with its prior parameters: ABCDE, FGHIJ, and ONMLK; i.e., 11111, 11111, and 22222, in designs LH, LP, and PH, respectively.

As shown in Appendix C, the priority heuristic correctly predicted the most often chosen gamble in the averaged data in only three out of ten choice problems in each sub-design. By predicting only 9 of 30 modal choice proportions correctly, this model performed significantly worse than a random coin toss, which would have a binomial probability of .98 of scoring 10 out of 30 or higher. Therefore, the priority heuristic not only failed to fit the data of any individual, it also failed to describe the averaged data in this study.

3.5 Individual response patterns

The finding in Table 3 that 98% of individual modal response patterns are transitive does not rule out the possibility that some individuals might have intransitive patterns of preferences as “true” patterns in a mixture of response patterns. In order to explore this possibility, we tabulated all response patterns (see Appendix D). The proportions are shown in Table 4, which shows that intransitive response patterns (11112 and 22221) amounted to 5% or less of all individual response patterns in all three designs of all three experiments. Because some of these responses could occur by random error, this low rate of intransitivity of individual response patterns provides little support for the notion that more than a small number have “true” intransitive patterns as part of a mixture of response patterns. A more detailed analysis is presented in Appendix D, which describes the search for individuals who might have intransitive patterns as secondary or tertiary patterns in a mixture. Appendix H further analyzes these response patterns with respect to iid and TE models.

Table 4: Percentages of all response patterns in LH, LP, and PH Designs. Column sums may differ from 100, due to rounding.

3.6 WST and TI

Whereas Tables 3 and 4 analyze response patterns for five of the choice problems, WST and TI can be examined for all 10 choice problems, which might detect other violations of transitivity besides those implied by LS models.

Violations of iid (Appendix B) cast doubt on any analysis that focuses strictly on marginal choice proportions, including tests of WST and TI. Nevertheless, we examined WST and TI as a third tactic to search for evidence of intransitivity. There were 136 participants with 3 matrices per person, making 408 data matrices; of these, only 18 matrices (4.4%) violated (were not perfectly consistent with) both WST and TI (In LH, #120, 126, 137, 140, 151, 214, and 311; in LP, #102, 122, 125, 214, and 239; in PH, #137, 147, 202, 239, 309, and 338). If one were to apply statistical tests (a dubious procedure given the violations of iid), some of these 18 cases (out of 408) might be declared “nonsignificant;” but in the big picture, it matters little whether the rate of violation of WST and TI is 4.4% or, say, 2%. Further details are in Appendix E.

3.7 Intransitive individuals

Because violations of transitivity (by any of the definitions) are so rare in these data, it might be tempting to conclude that no one is ever intransitive. However, we think that such a conclusion is too strong for two reasons: First, it argues from failure to reject the null hypothesis to its “truth”. Second, some intransitive cases appear to be systematic. Table 5 shows response patterns (for the five choice problems of Table 3) for Participants #125, 214, and 309 in each block. Responses are shown for each block in each of two sessions (days), with responses reflected so that identical numbers represent consistent choices. Participant #125 shows evidence of intransitivity in the LP design, showing the exact, intransitive response pattern 22221 in 20 of 30 presentations, and showing the same intransitive pattern 7 times in both versions within a block.

Table 5: Analysis of Participants #125, #214, and #309. LH and LH2 show the response patterns for choice problems AB, BC, CD, DE, and AE when the alphabetically higher gamble was presented first or second. The patterns, 22221 and 11112 are intransitive (bold font).

One might argue that #125 was transitive in the LP design with a “true” pattern of 22222, but responded randomly on the last choice, which produced the modal pattern 22221. But we need to explain why, 27 out of 30 times, this person chose F over J (a response of 1 for the last choice listed); the binomial probability of 27 or more out of 30 with p = ½ is less than .00001. This binomial calculation assumes a single “true” pattern (22222) and independence of responses both within and between blocks.

A more complex analysis by means of the TE model (see Appendix F) allows all 30 transitive “true” patterns (but no intransitive ones). Fitting this more general model, the probability to observe the pattern of 22221 both times in a block is estimated to be only 0.15, which under the weaker independence assumptions of the TE model yields a probability of finding 7 or more repeated patterns of 22221 (out of 15 blocks) of only p = .003). So even with this mixture model, these data are unlikely to arise from such a transitive model.

This response pattern for #125 in the LP design, 22221, would be consistent with LPH, LHP, or HLP LS models, if $16 ≥ Δ_L > $4. According to these respective LS models, however, #125 should have shown a pattern of 22221, 22221, or 22222 in the LH design, respectively. Instead, #125 had the modal response pattern of 11111 in the LH design (also 20 of 30 times, with 8 repeats), which requires Δ_L ≤ $4 under any of these three models, contradicting the behavior in the LP design. Therefore, we cannot use any of the LS models to connect the modal response patterns of Case #125 in these two linked designs. So even if we conclude that Case #125 was truly intransitive in the LP design, the results in the LH design contradict the compatible linkages of the LS models.

Participant #214 is the only case in which a person had intransitive modal data in two linked designs that might be consistent with a LS model. This person showed the modal pattern 22221 in both the LH and LP designs and the transitive pattern, 22222, in the PH design. These modal patterns of behavior (22221, 22221, and 22222 in LH, LP, and PH Designs) are consistent with the LPH LS model with $16 ≥ Δ_L > $4 and Δ_P ≤ 0.04.

If we argue that #214 was truly transitive with “true” patterns of 22222 in both LH and LP conditions, we need to explain why this person repeated the exact 22221 pattern on eleven blocks out of 22 opportunities and why the last choice is “1” 19 times out of 22 in the LH condition, and 19 of 22 in the LP task. Suppose the probability of choosing “1” in the last choice is 0.5; if so, the binomial probability to show 19 or more out of 22 is 0.0004, so it is unlikely that two sets of such data arose from true patterns of 22222, combined with independent random responding on the last choice. Case #214 is the only case of intransitive behavior in more than one design consistent with a single LS model. However, this same person violated a critical property of LS models, as shown in the next section, as did most of the other participants.

3.8 Testing interactive independence

The LS design of Experiments 2 and 3 provides critical tests of the family of LS models. Four tests of interactive independence (each consisting of two choices) per block of trials were constructed from the LS design. For example, consider these two choice problems: R = ($95, .95; $5) versus S = ($55, .95; $20) and R ^′ = ($95, .10; $5) versus S ^′ = ($55, .1; $20).²

For each of four tests, there are four possible response patterns, SS ^′, SR ^′, RS ^′, and RR ^′ in each block. According to any LS model, a person should prefer either S and S ^′ or R and R ^′; that is, SS ^′ and RR ^′. With any mixture of LS models, a person might show a mixture of these two response patterns, but should not switch systematically from R in the first choice to S ^′ in the second choice, denoted the RS ^′ pattern. TAX with its prior parameters, implies this RS ^′ reversal.

Consider again Case #214, whose modal data conformed to the LPH LS model in two linked designs. This person completed 11 blocks of trials with 4 tests of interactive independence each, making 44 possible tests. Out of 44 tests (two choice problems per test), this person had the exact response pattern of RS ^′ in 43 of 44 tests. Therefore, the data of Case #214 cannot be represented by any LS model or mixture of LS models.

In Experiments 2 and 3, there were just four cases that showed consistent evidence of intransitivity: #214, 239, 309, and 311. For these cases, the scores were 43 to 0, 27 to 0, 11 to 0, and 48 to 0, comparing RS ^′ reversals (predicted by interactive models) versus opposite reversals, SR ^′, respectively. Three other cases (from Exps 2 and 3) were identified with partial indicators of intransitivity: #202, 218, and 338. The scores for these cases on the tests of interactive independence are 37 to 0, 33 to 0, and 8 to 0, respectively. So even for those cases that seem most promising for evidence of intransitivity, the data refute interactive independence, which is implied by any LS model or mixture of LS models.

Most individuals, including those whose data appear compatible with transitivity, showed evidence of interaction: Out of the 85 participants in Experiments 2 and 3, there were 79 (93%) who had more response patterns of RS ^′ against only 2 who had more of the opposite reversal and only 4 who had equal numbers or no reversals. Summed over participants and blocks, there were 1807 blocks with the RS ^′ pattern compared to only 98 with the SR ^′ pattern. Interaction rules out all LS models.

Another test from the LS design rules out a sub-class of LS models including the priority heuristic. Any person who uses a LS, starting with the four variables of the priority heuristic (lowest consequence, probability of the lowest consequence, highest consequence, probability of highest consequence), considered in any order, should prefer G4 = ($99, 0.3; $15, 0.65; $14, 0.05) over F4 = ($88, 0.12; $86, 0.7; $3, 0.18), if Δ_L and Δ_H ≤ $11 and Δ_P ≤ 0.13). Instead, 84% of participants chose F4 over G4 more than half the time. If we retain a LS starting with any of these four attributes, we must conclude that Δ_L > $11, Δ_H > $11 and Δ_P > 0.13, contrary to published parameters needed to account for previous data.

In order for LPH, LHP, and HLP LS models to mimic the transitive predictions of the prior TAX model (which are the most commonly observed patterns in the data), these LS models all require Δ_L ≤ $4. These are the only LS models that mimic the TAX model this way, but we must reject them for those people who systematically prefer F4 over G4, since that requires Δ_L> $11, which contradicts the assumption (Δ_L ≤ $4) needed in order to mimic that transitive model’s predictions.

3.9 Tests of Cumulative Prospect Theory and the Priority Heuristic

Also included in the LS design were direct tests of CPT that also test the priority heuristic. For any monotonic utility function and any probability weighting function, CPT implies that if R = ($90, 0.05; $88, 0.05; $2, 0.9) ≻ S = ($45, 0.2; $4, 0.2; $2, 0.6) ⇒ R+ = ($90, 0.1; $3, 0.7; $2, 0.2) ≻ S– = ($45, 0.1; $44, 0.1; $2, 0.8). Note that R+ stochastically dominates R, and that S stochastically dominates S–. CPT therefore allows the response pattern SR+ but not the opposite, RS–, which is implied by TAX with its prior parameters (proofs in Birnbaum, 2008c).

The LPH LS and the priority heuristic imply that a person should choose S ≻ R and R+ ≻ S–, as long as Δ_P ≤ 0.30, so the priority heuristic implies the SR+ pattern that is also compatible with CPT. The PHL and PLH LS models with Δ_P ≤ 0.30 also imply the same pattern. The HLP, HPL, and LHP LS models imply the pattern RR+ when Δ_H ≤ $45. These parameter ranges are extremely large and include by a wide margin plausible values.

There were two tests of this type per block in Experiments 2 and 3. In the two tests, 38 of 43 and 35 of 42 participants in Experiments 2 and 3, respectively, showed more response patterns of RS– than of the opposite, against only 4 and 4 who showed more of the SR+ pattern compatible with CPT and the priority heuristic. These findings rule out CPT and the priority heuristic as well as the other LS models (with wide parameter ranges) for those participants who systematically show the SR+ pattern. Additional results in the LS design, including individual results, are presented in Appendix G.

4 Discussion

Our first experiment was initially designed to test whether those participants who showed evidence of intransitive behavior consistent with use of a LS model in one design would show evidence of linked intransitivity between designs. However, we were surprised by two results from that first study: First, few participants showed plausible evidence of intransitivity in even one design, and no one in that study had consistent evidence of linked intransitivity in two designs.

Second, several individuals completely reversed their preferences between blocks of trials, which refutes the assumption of iid that is required for meaningful analysis of marginal choice proportions, averaged over response patterns. These findings led to a second experiment with new participants and new “filler” tasks between blocks, which also included tests of critical properties of LS models.

The second experiment confirmed that some people completely and perfectly reversed preferences between blocks. Because this behavior has less than one chance in a million under the assumptions of iid, we must reject that assumption. Evidence of intransitive behavior was again quite minimal.

The third experiment was an attempt to alter our procedures more drastically in an attempt to “confuse” participants by intermixing many different types of trials within blocks and by including multiple “fillers” between related items in order to put a greater burden on memory, which was conjectured as the reason that people behaved transitively. It was suggested that these changes in procedure might also produce better satisfaction of iid. Although these procedures increased “error” and reduced the incidence of perfect reversals, they did not prevent them, nor did these changes in procedure increase the incidence of violations of transitivity. Targeted statistical tests indicated that iid was violated strongly in all three studies by all but a very small number of participants.

When iid can be assumed, it means that an investigator can simplify data analysis by examining only choice proportions. But when iid is dubious, it means that we need to also examine response patterns because choice proportions could easily misrepresent individual data. When testing transitivity, it means that choice proportions can appear transitive when the person’s data are perfectly intransitive and it means that choice proportions can appear intransitive when every single response pattern by the person was transitive.

In a search for intransitive patterns of the type consistent with LS models, only a few cases gave credible evidence of intransitivity. However, these cases also showed evidence of violation of critical properties of LS models, including systematic violation of interactive independence. Other tests led to contradictions in the value of difference thresholds required by LS models to handle the data.

Only one person showed intransitive behavior in two designs that could be linked by a LS model. Case #214 showed data consistent with the LPH LS model with $16 ≥ Δ_L > $4 and Δ_P ≤ 0.04. However, this same person chose F4 over G4 100% of the time, which means that Δ_P > 0.13, contradicting the LS model that links these two designs. And this person also systematically violated the critical property of interactive independence 43 times in 44 tests, which means that no LS model can account for this person’s data.

Had we tested only transitivity in separate designs, we would have concluded that cases of intransitive preference are rare. Such findings might modify our assessment of the incidence of this behavior. Some studies claimed evidence of systematic violations (e.g., Tversky, 1969; Myung, et al., 2005) and others claimed that “significant” violations of transitivity might be due to chance (Regenwetter, et al., 2010). Based on these new data, the estimated incidence of violation of transitivity in these designs is below 5%, which is compatible with recent studies with PH designs.

Because LS models can handle transitive response patterns as well as intransitive ones, and because studies done to date have examined only a tiny region of the space of all possible sets of choice problems, evidence concerning the incidence of violations of intransitivity says very little about the empirical standing of the class of LS models and it says little about the general validity of transitivity in the infinite space of all choice problems. The failure to find predicted intransitivity might mean only that the researchers did not yet do the right study.

4.1 Refutation of LS models

However, by using linked designs and by including critical tests of the LS models, we can reach much stronger conclusions regarding the LS family; namely, these models can be rejected as descriptive for most people, including even those who appeared to show indications of intransitivity as well as for those whose data appear to be transitive. These findings agree with other tests of critical properties of LS models (Birnbaum, 2010; Birnbaum & Gutierrez, 2007; Birnbaum & LaCroix, 2008)

If those few cases of systematic intransitivity we observed are “real” (and not due to statistical coincidence), then some other origin must be sought to account for them besides the family of LS models. One possibility that has been suggested is that people use an interactive, integrative model but have a tendency to “round off” via editing (i.e., they assimilate subjective values of attributes that are similar) in a choice problem before applying an interactive, integrative model (Kahneman & Tversky, 1979). Such a model could produce intransitive choices and also violate the critical property of interactive independence (Birnbaum & Gutierrez, 2007).

This “rounding” or “editing” model should also cause linked violations as long as the rules for rounding stay the same in all designs; if so, it might describe the data of Case #214, who was the only case showing evidence of linked intransitivity in two designs, but not Case #125, whose intransitive data in one design contradicted the modal pattern in another design under any such interpretation. Although the “editing” operations were originally proposed as general descriptions, only one case (out of 136) appears to call for this “rounding” rule. Systematic violations of the editing rules of cancellation and combination have been observed in other studies (Birnbaum, 2008b), so the empirical status of the editing rules remains doubtful.

4.2 Refutation of the Priority Heuristic

These experiments were designed to produce robust, linked violations of transitivity if a person used the priority heuristic. The priority heuristic is a variant of LPH LS, with specified parameters that are part of the theory. As Birnbaum (2008a) noted, the optimal parameters for this model to describe certain published data are indeed close to the values postulated by Brandstätter, et al., (2006). However, neither averaged choice proportions nor individual data of any person agreed with the predictions of the priority heuristic. Furthermore, violations of interactive independence refute the priority heuristic along with the other LS models.

These failures of priority heuristic to predict new data are consistent with findings of other recent studies that tested other implications of this heuristic (Birnbaum, 2008a; 2010; Birnbaum & Bahra, 2012; Birnbaum and LaCroix, 2008; Fiedler, 2010; Gloeckner & Betsch, 2008; Gloeckner & Herbold, 2011; Hilbig, 2008; Rieskamp, 2008).

In response to previous failures of the priority heuristic, Brandstätter, Gigerenzer & Hertwig (2008) proposed that the priority heuristic is preceded by use of an “adaptive toolbox” of other procedures for comparing risky gambles. These other processes were proposed to handle cases where the priority heuristic was refuted by experiments designed to test its implications. For example, they suggested that the model does not apply when there is a large discrepancy in expected value (a ratio exceeding 2), when there is a no conflict resolution, when there are branches that might be cancelled, when two or more branches lead to the same consequence, etc. However, this study contains none of the “triggering conditions” yet postulated to invoke the other heuristics. The gambles compared in each set (Table 1) are very close in expected value, choices between them do not have “no conflict” resolutions, and all gambles have exactly two branches. None of the excuses yet published provide a reason for the model to fail in this study.

4.3 Refutation of CPT

The LS design of Experiments 2 and 3 also included individual tests of CPT and the priority heuristic. These tests extend previous findings (Birnbaum, 2008c; Birnbaum, 2010) and show that, analyzed at the level of individuals, the majority show systematic violations of CPT as well as of the priority heuristic. Because the property tested does not assume any specific functional form of the value function on money and it does not assume any particular probability weighting function, the refutation of CPT holds for all parametric versions of that model. This evidence against CPT is consistent with other studies that found systematic violations of that model (Birnbaum, 1999, 2004, 2008b, 2008c; Birnbaum & Bahra, 2012).

4.4 Refutation of iid assumptions

The present data show extremely strong evidence against iid. These violations were found even in Experiment 3, where multiple fillers separated related trials within blocks and more than 50 intervening trials separated blocks. Therefore, the assumption of iid should be considered as a dubious basis for determining whether or not formal properties in choice data satisfy structural properties such as transitivity.

The refutation of iid creates difficulties for the analysis of individual choice proportions, rather than response patterns. Regenwetter, et al. (2010, 2011) proposed that marginal choice proportions (averaged over response patterns) could be used in a general method for analyzing algebraic models with probabilistic data. However, the overwhelming and extreme violations of iid indicate that this method might, in principle, lead to wrong theoretical conclusions as well as erroneous statistical results.

There are two general forms of the TE model that require weaker assumptions than iid. These models typically violate iid, except in special cases. In iTET, a person is assumed to have the same “true” preferences within a block of trials, but it is allowed that a person might change “true” preferences from block to block. This model implies that iid can be violated within a person, if that person changes systematically during the study.

In a sense, the debate between the methods of Regenwetter, et al. (2011) and of Birnbaum (2011) is a debate about how often a person might change “true” preferences. The method of Regenwetter, et al. (2011) assumes that responses are resampled independently (“true” preferences can change) between every pair of trials, as long as there are several intervening filler trials, whereas the TE approach of Birnbaum (2011) assumes that a person’s true preferences last longer and are theorized to be constant within a block of trials. They are allowed to change between blocks.

In the TE models, iid occurs only in special cases, such as when a person maintains only one “true” preference pattern throughout a study. Because the TE model can allow iid to be satisfied or violated, we think the Regenwetter, et al. iid assumptions are stronger than those required by or implied by the TE model. Trials within the same block and between blocks are assumed to be independent in the iid approach, whereas the TE model assumes only that errors within and between blocks are independent, allowing choices within blocks to be dependent.

When the TE model holds, it can provide more information than is available when iid holds: one can estimate the distribution of “true” preferences in a mixture, whereas in the method of Regenwetter, et al. (2011) one can test mixture models but one cannot discover the distribution of preferences in a person’s mixture. A practical difficulty of the TE model is that more data are required in order to fit this model to the data.

The correlations between behavior in different blocks represent a problem for both iid models and for a sub-class of TE models in which “true” choice patterns are re-sampled independently between blocks. Instead, the present data appear more consistent with the idea that a person follows a model in which parameters change gradually but systematically throughout a study. For example, parameters of the TAX model might change gradually via a random walk from trial to trial and block to block.

It would be useful to produce a statistical method that detects when a person changes “true” preferences; such a test might provide a more accurate model and also allow one to compare the assumptions of iid and TE as special cases of a more general model. According to the iid models, this test should identify that people change “true” preference between every trial (or that they never change), and according to the TE models, this test should indicate that people change true preferences only between blocks.

The finding that iid assumptions are violated agrees with Birnbaum’s (2012) reanalysis of data from Regenwetter, et al. (2011), which concluded there were significant deviations from iid in those data as well. By using two repetitions of each choice problem within each block, and by separating blocks by more than 50 trials, the present studies provide stronger tests of iid than were possible in that study, where each choice problem was presented only once per block and only three trials separated blocks.

4.5 Does transitivity hold everywhere?

Because so few “significant” violations of transitivity have been found in recent studies, including this one, some might argue that they should be dismissed as due to chance. If one uses a 5% level of significance, one expects 5 cases per hundred to be “significant” by chance, so finding a small number of “significant” violations of transitivity does not refute the null hypothesis that transitivity holds for everyone.

But even if one can retain the null hypothesis, it does not mean that the null hypothesis has been proved to be true, and there may indeed be some people who truly violate transitivity. Those particular individuals who significantly violated transitivity might continue to do so if they were tested again. A new study might discover stimuli in which violations would be more apparent. Another procedure might be found in which transitivity is easier to detect. Therefore, despite the weakness of evidence against transitivity in the literature, we think the case for transitivity is still open.

4.6 Conclusions

In contrast, the cases against the priority heuristic, the family of LS models, CPT, and assumptions of iid are quite strong: This study found significant and systematic violation of all four of these theoretical ideas. We found that data systematically violated the predictions of the priority heuristic and no one satisfied them. Only one person showed intransitivity in two designs that could be linked via a LS model; and that person (along with the majority of others) also showed systematic violations interactive independence, a critical property of LS models. Tests of CPT with any monotonic value and probability weighting functions led to systematic violations by most individuals. Finally, tests of iid showed extremely strong violations indicating that people are likely changing their true preferences during a long study.