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Testing Lexicographic Semiorders as Models of Decision Making:
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Priority Dominance, Integration, Interaction, and Transitivity
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Michael H. Birnbaum*
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California State University, Fullerton
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Short Title: Testing Lexicographic Semiorder Models
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Filename: Testing_heuristic_38a.doc
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Contact Info: Mailing address:
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Prof. Michael H. Birnbaum,
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Department of Psychology, CSUF H-830M,
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P.O. Box 6846,
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Fullerton, CA 92834-6846
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Email address: [email protected]
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Phone: 714-278-2102 Fax: 714-278-7134
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* This research was supported by grants from the National Science Foundation, BCS-0129453, and
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SES-0202448. Thanks are due to Jeffrey P. Bahra, Roman Gutierrez, Adam R. LaCroix, Cari
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Welsh, Christopher Barrett, and Phu Luu, who assisted with pilot studies and data collection for this
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research. Thanks are due to Eduard Brandstätter for providing a preprint of his paper in 2005 and
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for discussions of these issues and to Jerome Busemeyer, Anthony J. Bishara, and Jörg
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Rieskamp for comments on an earlier draft.
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Testing Heuristic Models 2
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Abstract
Three new properties are devised to test a family of lexicographic semiorder models of
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choice. Lexicographic semiorders imply priority dominance, the principle that when an attribute
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with priority determines a choice, no variation of other attributes can overcome that preference.
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Attribute integration tests whether differences on two attributes that are too small, individually, to
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be decisive can combine to reverse a preference. Attribute interaction tests whether preference due
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to a given contrast can be reversed by changing an attribute that is the same in both alternatives.
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Additive contrast models as well as lexicographic semiorder models assume that attributes do not
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interact, but additive contrast models assume integration of attributes. Additive contrast and
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lexicographic semiorder models also imply systematic violations of transitivity. Four new studies
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show that priority dominance is systematically violated, that most people integrate attributes, and
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that most people show interactions between probability and consequences. In addition, very few
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people show the pattern of intransitivity predicted by the priority heuristic, which is a variant of the
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lexicographic semiorder model with threshold parameters chosen to reproduce certain previous
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data. When individual differences are analyzed in these three new tests, it is found that few people
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exhibit data compatible with any of the lexicographic semiorder models. The most frequent pattern
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of individual data is that predicted by the TAX model with parameters used in previous research.
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Testing Heuristic Models 3
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The most popular theories of choice between risky or uncertain alternatives are models that
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assign a computed value (utility) to each alternative and assume that people prefer (or at least tend
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to prefer) the alternative with the higher utility (Luce, 2000; Starmer, 2000; Wu, Zhang, &
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Gonzalez, 2004). Let A
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Transitivity is the assumption that, A
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preferred to C, then A is preferred to C.
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B represent systematic preference for gamble A over gamble B.
B& B C  A C . If A is preferred to B and B is
The class of transitive utility models includes Bernoulli’s (1738/1954) expected utility (EU),
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Edwards’ (1954) subjectively weighted utility (SWU), Quiggin’s (1993) rank-dependent utility
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(RDU), Luce and Fishburn’s (1991; 1995) rank-and sign-dependent utility (RSDU), Tversky and
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Kahneman’s (1992) cumulative prospect theory (CPT), Birnbaum’s (1997) rank affected
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multiplicative weights (RAM), Birnbaum’s (1999; Birnbaum & Chavez, 1997) transfer of attention
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exchange (TAX), Lopes and Oden’s (1999) Security Potential Aspiration Level Theory, Marley and
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Luce’s (2001; 2005) gains decomposition utility (GDU), Busemeyer and Townsend’s (1993)
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decision field theory (DFT), and others. These models can be compared by testing classic and new
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paradoxes, which are properties that must be satisfied by different subsets of the models (Marley &
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Luce, 2005; Birnbaum, 2004a; 2004b; in press-b).
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Four Classes of Decision Models
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Although the transitive utility models can be tested against each other, they all have in
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common that there is a single, integrated value or utility for each gamble, that these utilities are
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compared, and people tend to choose the gamble with the higher utility. Let U(A) represent the
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utility of gamble A. This class of models assumes,
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A
B U(A)  U(B) ,
(1)
where  denotes the preference relation. Aside from “random error,” these models imply
Testing Heuristic Models 4
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transitivity of preference, because A
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U(A)  U(C)  A C .
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B  U(A)  U(B) and B C  U(B)  U(C)
Let A  (x, p; y,1  p) represent a ranked, two-branch gamble with probability p to win x,
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and otherwise receive y, where x  y  0 . This binary gamble can also be written as A  (x, p; y) .
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The two branches are probability-consequence pairs that are distinct in the gamble’s presentation:
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(x, p) and (y,1  p) .
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In EU theory, the utility of gamble A  (x, p; y,1  p) is given as follows:
EU(A)  pu(x) (1 p)u(y) .
(2)
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where u(x) and u(y) are the utilities of consequences x and y . In expected utility, the weights of
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the consequences are simply the probabilities of receiving those consequences.
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Another transitive model that has been shown to be a more accurate description of risky
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decision making than EU or CPT is the special transfer of attention exchange (TAX) model. This
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model also represents the utility of a gamble as a weighted average of the utilities of the
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consequences, but weight in this model depends on the probabilities of the consequences and their
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ranks in the gamble. In this model, people assign attention to each consequence as a function of its
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probability, but weight is transferred from branch to branch, according to the participant’s point of
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view. A person who is risk-averse may transfer weight from the highest consequence to the lowest
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consequence. This model can be written for two branch gambles (when  > 0) as follows:
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TAX(A) 
au(x)  bu(y)
ab
(3)
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where a  t(p)   t( p) , b  t(q)   t( p) , and q  1  p. Intuitively, when the parameter  > 0
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there is a transfer of attention from the branch leading to the best consequence to the branch leading
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to the worst consequence. This parameter can produce risk aversion even when u(x)  x . In the
Testing Heuristic Models 5
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case where   0 , weight is transferred from lower-valued branches to higher ones; in this case,
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a  t(p)   t(q) and b  t(q)   t(q) . The formulas for three-branch gambles include weight
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transfers among all pairs of branches (Birnbaum & Navarrete, 1998; Birnbaum, 2004a)
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The special TAX model was to data of individuals by Birnbaum and Navarrete (1998), with
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

the assumptions that t(p)  p and u(x)  x . The median best-fit parameters were   0.41 ,
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  0.79 , and   0.32 .
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For the purpose of making “prior” predictions for TAX in this article, a still simpler version
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0.7
of the model will be used. Let u(x)  x , 0  x  $150 ; t(p)  p , and   1/ 3 . These functions
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and parameters, which approximate certain data for gambles with positive consequences less than
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$150, are called the “prior” parameters, because they have been used in previous studies to predict
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new data with similar participants and procedures. They have had some success predicting results
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with undergraduates who choose among gambles with small prizes (e.g., Birnbaum, 2004a). Use of
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these parameters to predict modal patterns of data does not assume, however, that everyone is
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assumed to have the same parameters. The main use of the prior parameters is for the purpose of
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designing studies that are likely to find violations of rival models.
Several papers have shown that the TAX model is more accurate in predicting choices
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among risky gambles than CPT or RDU (Birnbaum, 1999; 2004a; 2004b; 2005a; 2005b; 2006;
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2007; in press; Birnbaum & Chavez, 1997; Birnbaum & McIntosh, 1996; Birnbaum & Navarrete,
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1998; Weber, 2007). This model made correct predictions for critical properties that distinguish
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these models. Critical properties are theorems that can be deduced from at least one theory and
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which are false according to at least one other theory. There are a number of such properties
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involving three-branch gambles that distinguish TAX and RDU models (including CPT). For
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example, CPT and RDU models imply first order stochastic dominance, but the TAX model does
Testing Heuristic Models 6
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not. Experiments designed to test stochastic dominance have found violations where they are
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predicted to occur based on the TAX model with its prior parameters (e.g., Birnbaum, 2004a).
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Similarly, TAX correctly predicted violations of lower and upper cumulative independence, upper
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tail independence, 3-lower distribution independence, 3-upper distribution independence, dissection
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of the Allais paradox, and other properties (Birnbaum, in press-b).
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For the purpose of this paper, however, TAX, CPT, RDU, GDU, EU, DFT and other
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theories in this family are all in the same category and their predictions will be virtually identical.
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These models are all transitive, they imply that attributes are integrated, and they all imply
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interactions between probability and consequences. The experiments in this article will test
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properties that these integrative models share in common against predictions of other families of
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models that disagree. Therefore, it should be kept in mind that what is said about TAX in this paper
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applies as well to the other transitive, integrative models in its class.
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A second class includes non-integrative but transitive models. For example, if people
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compared gambles by just one attribute (for example, by comparing their worst consequences), they
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would satisfy transitivity, but no change in the best consequence or probability of receiving that
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consequence could overcome a difference due to the lowest consequence.
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A third class of theories violates transitivity but assumes that people integrate contrasts
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(e.g., differences) between alternatives in terms of contrasts along attributes of the choices
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(Tversky, 1969; González-Vallejo, 2002). For the purpose of this paper, this class of theories will
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be described as integrative contrast models, since they involve the calculation of contrasts and
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integration of the values of these contrasts.
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An example of such a model is the additive contrasts model:
n
n
i 1
i 1
D(G, F)   i ( piG , piF )    i (xiG , xiF )
(4)
Testing Heuristic Models 7
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where the i are functions of branch probabilities in the two gambles and  i are functions of
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consequences on corresponding branches of the two gambles. These functions are assumed to be
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strictly increasing in their first arguments, strictly decreasing in their second arguments; they are
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assumed to be zero when the two components are equal in any given contrast; further, let
i ( piG , piF )   i (piF , piG ) and  i (x iG ,x iF )   i (xiF , xiG ) .
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This type of model can violate transitivity but it assumes integration of information from different
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attributes. For example, consider three-branch gambles with equally likely outcomes,
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G  (x1, 13; x 2, 13; x3, 13) . Because the probabilities are all equal, we assume i ( piG , piF )  0 in this
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case. Now suppose
1 , x iG  x iF  0

 i (x iG ,x iF )  0, x iG  x iF  0

1, x iG  x iF  0
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(5)
F  D(G, F)  0 . Let A  ($80; $40; $30) represent a gamble that is equally likely to
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where G
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win $80, $40, or $30. It will be preferred to B  ($70; $60; $20) because D(A, B)  1  0 (A has two
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consequences higher than corresponding consequences of B). Similarly, B will be preferred to
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C  ($90; $50; $10) for the same reason; however, C will be preferred to A, violating transitivity.
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Rubinstein (1988) showed that such models could account for the Allais paradox, and
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Leland (1994) noted that such models might account for other “anomalies” of choice, empirical
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findings that violate expected utility theory. González-Vallejo (2002) incorporated a normally
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distributed error component in an additive contrast model called the stochastic difference model,
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and showed that, with specified  i and  i functions, her model could fit several empirical results in
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risky decision making. These functions incorporated the difference in probability or in
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consequence divided by the maximum probability or consequence in the choice, respectively.
Testing Heuristic Models 8
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Integrative contrast models can also violate stochastic dominance. For example, suppose
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i ( piG , piF )  piG  piF , and  i (x iG ,x iF )  xiG  x iF , for all branches. It follows that
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G  ($100,0.7;$90,0.1;$0, 0.2) will be preferred to F  ($100,0.8;$10,0.1;$0,0.1) , even though F
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dominates G . In this case, D(G, F)  $80. To avoid such implications, some have assumed that
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people first check for stochastic dominance before employing such models (e.g., Leland, 1994).
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Another type of contrast model includes products of terms representing probabilities with
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terms representing contrasts in consequence. A member of this class of models is regret theory
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(Loomes, Starmer, & Sugden, 1991). See the discussion in Leland (1998). Another example is
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called the most probable winner model (Blavatskyy, 2003), sometimes called majority rule (Zhang,
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Hsee, & Xiao, 2006):
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D(G, F)   i ( piG ) i (xiG , xiF )
(6)
i 1
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In this case, gambles are defined on mutually exclusive states of the world (indexed with i),
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which have subjective probabilities, i , and the  i are functions that represent, for example, “regret”
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for having chosen gamble G instead of F given their consequences under that state of the world.
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This model can also violate transitivity (as in the example above); but unlike the additive case
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(Equation 4), this model allows interactions between probability and consequences.
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A fourth class, explored in this paper, includes non-integrative intransitive models.
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Examples are the lexicographic semiorder (Tversky, 1969) and the priority heuristic (PH) of
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Brandstätter, Gigerenzer, & Hertwig (2006) in restricted domains. In this class of models, people
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move in a particular order from attribute to attribute, and consider attributes with lower priority
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only if differences on the more important attributes are too small to be decisive. Once a decisive
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contrast is found on an attribute, a decision is made without influence of less important attributes.
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A semiorder is a structure in which the indifference relation need not be transitive (Luce,
Testing Heuristic Models 9
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1956). Suppose two stimuli are indifferent if and only if their difference in utility is less than a
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critical threshold. Let ~ represent the indifference relation. If A ~ B and B ~ C it need not follow
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that A ~ C .
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A lexicographic order is illustrated by the task of alphabetizing a list of words. Two words
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can be ordered if they differ in their first letter. However, if the first letter is the same, one needs to
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examine the second letters in the two words. At each stage, letters following the one on which they
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differ have no effect on the ordering of the two words.
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In a lexicographic semiorder, people compare the first attribute, and if the difference is less
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than a threshold, they go on to examine the second attribute; if that attribute differs by less than a
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threshold, they go on to the third, and so on.
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Lexicographic Semiorder for Risky Decision Making
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For analysis of new tests that follow, it is useful to state a lexicographic semiorder for the
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special case of two-branch gambles with strictly non-negative consequences, A  (x, p; y,1  p) and
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B  ( x, p ;y,1  p) , where x  y  0 and x y 0 . As in Brandstätter, et al. (2006), suppose
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people consider first the lowest consequences of the two gambles, next the probabilities of the two
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gambles, and finally, the higher consequences of the two gambles, as follows:
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If y  y L , choose A.
(7a)
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If y y  L , choose B.
(7b)
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In these expressions, people choose a gamble if the difference in lowest outcomes exceeds the
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threshold,  L . Only if neither condition (7a or 7b) holds, does the decider examine the difference
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between probabilities. In this stage, the decision is made as follows:
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Else if p  p P , choose A.
(8a)
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Else if p  p  P , choose B.
(8b)
Testing Heuristic Models 10
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Only if none of the above four conditions hold, does the decider compare the largest consequences.
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In the case of two-branch gambles, the decision would then be based on that reason alone:
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Else if x  x  choose A
(9a)
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Else if x  x  choose B.
(9b)
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If none of the above six conditions holds, then the decider has no preference.
In two-branch gambles, there are six possible lexicographic semiorders (LS), which will be
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denoted: LPH, LHP, PHL, PLH, HPL, and HLP. For example, the LPH semiorder is the one
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described in Expressions 7, 8, and 9 in which the lowest consequence, probability, and highest
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consequence are examined in that order. There are thus four “parameters” in the LS family for two-
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branch gambles: the first attribute examined, the threshold for deciding on that attribute; the second
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attribute examined, and the threshold for deciding on that attribute (Once the first and second
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attributes have been chosen, the third is determined for two-branch gambles).
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Priority Heuristic
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The priority heuristic (PH) is similar to a lexicographic semiorder, but it has some
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additional features. According to the priority heuristic, people act as if they compare risky gambles
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with positive consequences as follows: First, they compute one tenth of the largest consequence in
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either gamble, and round this to the nearest prominent number. Prominent numbers include integer
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powers of 10 plus one-half and twice their values (e.g, 1, 2, 5, 10, 20, 50, 100, …). This figure
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yields difference threshold for comparing lowest consequences:  L  H  R[max( x, x)] , where R
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represents the rounded value of one-tenth. For example, if the largest prize of either gamble were
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$90, 10% of this would be $9, which would be rounded to  L  $10 .
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Second, people examine the lowest consequences of the two gambles. If the difference
between these two consequences equals or exceeds  L , the decision is made based on that factor
Testing Heuristic Models 11
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alone, as in Expressions 7a and 7b.
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Third, if this difference does not reach this threshold, the decider next compares the
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probabilities to receive the worst consequence. If the probabilities differ by  P = 0.1 or more, they
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decide on this factor alone.
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Fourth, if this difference is too small, they examine the best consequences of the two
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gambles. In the case of two-branch gambles, if there is any difference on the best consequence,
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people would decide on this factor alone, as in Expressions 9a and 9b. In the case of gambles with
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three branches or more, however, if the difference in the highest consequences exceeds the
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aspiration level,  H , they decide on that factor. Only if this difference is smaller than the aspiration
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level, the probability to win the best consequence is considered. Fifth, if there is any difference on
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this attribute, the final decision is based on this factor alone.
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Sixth, if none of these reasons is decisive, the priority heuristic holds that people choose
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randomly between the gambles. This heuristic assumes that people never use more than the above
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four reasons, no matter how many branches there are in the gambles. Except for the computation of
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threshold as a function of the largest prize, the priority heuristic is a lexicographic semiorder.
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Although Brandstätter, et al. (2006) indicate in their title that their model has no “trade-
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offs,” the comparison of a difference to the largest consequence does involve a tradeoff, by the
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usual definition of this term. A given difference of $10 would be sufficient to be decisive in the
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case of gambles with a maximal prize of $100, but this difference would not be decisive in a choice
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with a highest prize of $1000. So there is a tradeoff between the difference in the lowest
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consequences and the highest consequence. However, if the largest prize is fixed in a given
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experiment (e.g., when the largest prize is always $100), we can treat the priority heuristic as a
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lexicographic semiorder within that restricted experimental domain.
Testing Heuristic Models 12
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Comparing Models
This article will assess models by two methods. First, the priority heuristic with its
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threshold parameters is compared against the simplified version of the TAX model with its prior
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parameters. To compare the models this way, one simply counts the number of participants who
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show the pattern of behavior predicted by each model. This method puts both models on equal
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footing since no parameters are from the data. It has the drawback, however, that individual
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differences or poor choices of parameters can confound conclusions from such comparisons.
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Second, critical properties that are implied by one class of models (with any parameters)
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that are not implied by another class of models are tested. Testing critical properties has the
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advantage that it allows rejection of an entire class of models without assuming that all people are
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the same or that their parameters are known in advance of the experiment. When there are
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systematic violations, the model whose properties are violated can be rejected, and the model
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whose predicted violations are observed can be retained. The negative argument (against the
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disproved class of models) is, of course, stronger than the positive argument in favor of the model
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that predicted the violations because other models might have also predicted the same violations.
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In order to test whether violations of critical properties are “real” rather than produced by
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“random error,” a true and error model is used to specify the frequencies of violation of critical
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properties that might be expected due to the unreliability of choice. This model allows each person
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to have a different “true” pattern of preferences, and it allows each choice to have a different rate of
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“error.”
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Generalizing the Models
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In this paper, I provide tests of lexicographic semiorder models that have been generalized
in several ways. For example, different people might have different priority orders for the different
Testing Heuristic Models 13
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factors. In addition, within any order, different people might have different threshold parameters
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for comparing the lowest consequences, highest consequences, and probabilities,  L ,  P , and  H .
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Brandstätter, et al. (2006) emphasize that their priority heuristic has only one order, LPH for
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two branch gambles, and LQHP for gambles with three or more branches. In addition, they assume
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that there is only one threshold parameter; indeed, they are skeptical about allowing parameters to
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be estimated from data, and they argue that their threshold parameters are determined by the base-
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10 number system. Presumably, all people who have the same number system should have the
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same parameters.
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In this paper, however, I propose tests that allow us to refute the entire family of
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lexicographic semiorders, even allowing each person to have a different priority order and different
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threshold parameters.
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Three new diagnostic tests allow us to test and compare generalized lexicographic
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semiorder models with a class of rival models. Whereas the lexicographic semiorders have been
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used in previous studies to describe violations from implications of the family of expected utility
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models, in this paper, three implications of lexicographic semiorder models will be tested.
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New Diagnostic Tests
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The three new properties are priority dominance, attribute integration, and attribute
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interaction. These tests allow one to assess if different individuals are using different lexicographic
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semiorder models. Tests of attribute integration and interaction also allow evaluation of a mixture
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model: in this model, each person may be using a mixture of lexicographic semiorders, switching
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priority orders and threshold parameters randomly from trial to trial.
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Relations among theories and the properties to be tested in four new studies are listed in
Table 1. To my knowledge, the first three properties of lexicographic semiorder models (aside from
Testing Heuristic Models 14
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transitivity) have not been previously proposed or empirically tested.
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Insert Table 1 about here.
The properties will be stated in terms of three attributes, L, P, and H. These might represent
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any three factors that can be manipulated, but for the tests presented here, L represents the lowest
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consequence in a two-branch gamble, P the probability to win the highest consequence, and H the
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highest (best) consequence in the gamble. Let A  (xA , pA ; yA ) represent the gamble to win x A with
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probability p A and otherwise win yA , where x A > yA  0 . In a choice between gambles A and B, let
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x A and x B represent values of H, the highest consequences in gambles A and B, respectively, yA
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and yB represent the respective values of the lower consequence, and p A and pB represent levels of
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probability to win the better consequence. The expression, A
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Priority Dominance
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B, denotes A is preferred to B.
Priority Dominance holds that attributes with lower priority cannot over-rule a decision
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based on an attribute with higher priority. Once a critical threshold is reached on an attribute with
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priority, variation in all other attributes, no matter how great, should have absolutely no effect.
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I use the term, “dominance” here by analogy to a dominant allele in Mendelian genetics. A
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heterozygous genotype shows the observed characteristic (phenotype) of the dominant allele. Here,
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priority dominance refers to the implication that if an attribute has priority and is the reason for one
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choice, if that same reason is present in a second choice and pitted against manipulation of
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attributes with lower priority, the attribute with higher priority should determine the choice. Hence,
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no change in “recessive” attributes can overcome a decision based on a dominant attribute.
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Priority dominance is the assumption that one and only one of the three attributes has first
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(highest) priority and that all three of the following conditions hold:
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If L has first priority, then for all yA  yB  0; p B  pA  0; xB  xA  0; p
B  p 
A  0; x
B  x
A  0,
Testing Heuristic Models 15
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317
A  (xA , pA; yA)
B  (xB, pB; yB )  A (xA, p
A ; yA )
B ( x
; yB) .
B , pB
(10a)
If P has first priority, then for all p A  pB  0; xB  xA  0; yB  yA  0; xB  xA  0; yB  yA  0,
A  (xA , pA; yA)
B  (xB, pB; yB )  A (xA, pA; y
A)
B ( x
B , pB ; y
B) .
(10b)
If H has first priority, then for all xA  xB  0; pB  pA  0; yB  yA  0; p
B  p 
A  0; y
B  y
A 0,
318
A  (xA , pA; yA)
B  (xB, pB; yB )  A (xA , p
A ; y
A)
B (xB, pB; y
B) .
319
Note that in Expression 10a, there is only one reason to prefer A
(10c)
B; namely, A has a
320
better lowest consequence. Both of the other attributes favor B . If L has first priority in a
321
lexicographic semiorder, then we know that the difference in lowest consequence was enough to
322
meet or exceed the threshold; otherwise, the person would have chosen B . But if L has first
323
priority, then there should be no effect of any changes in P and H. Therefore, because the same
324
contrast in L is present in A and B, changing the levels of P and H cannot reverse the preference.
325
Similarly, if A B, then if L has first priority, then A
326
B by the same argument.
The priority heuristic implies a particular pattern of priority dominance; namely, the first
327
attribute considered is lowest consequence (L), then probability, followed by highest consequence
328
(i.e., LPH). If L has first priority, and if a $20 difference is enough to prefer A to B, then no
329
variation in the other two attributes should reverse this preference. For example,
330
A  ($98,0.10;$20)
331
Suppose, however, that different participants might have different priorities. In that case, we must
332
test all three propositions that each of the three attributes might be most important. If each person
333
uses a lexicographic semiorder but different people have different priorities, then each person
334
should show priority dominance in one and only one of the three tests, aside from random error. To
335
reject priority dominance, it is necessary to show that for each person, no attribute satisfies priority
336
dominance.
B  ($100,0.11;$0)  A  ($22,0.1;$20)
B ($100,0.99;$0) .
Testing Heuristic Models 16
337
338
Attribute Integration
Integrative independence is the assumption that small differences on two factors, if not large
339
enough by themselves to reverse a decision, cannot combine to reverse a decision. The term
340
attribute integration will be used to indicate systematic violations that cause us to reject integrative
341
independence. This independence property will be tested by factorial manipulations in which two
342
small differences are independently varied. We examine whether the combined effect of two
343
attributes produces a different decision from that produced by each one independently. For
344
example, integrative independence of L and H can be expressed, for all pB  pA  0 ; x A  xB ;
345
yA  yB  0; x
  y
A  x 
B  0 ; and yA
B  0 , as follows:
346
If A  (xA , pA; yA)
B  (xB, pB; yB ) ,
347
And if A  ( x
A , pA ;yA )
B (xB, pB; yB ) ,
(11b)
348
And if A (xA, pA; yA)
B (xB, pB;yB) ,
(11c)
349
Then A ( x
A , pA ; y
A)
B ( x
B , pB ; y
B)
(11d)
350
In this case, note that changing the highest consequences results in the same choice between
(11a)
351
Aand B as in the choice between A and B (11a and 11b). Similarly, changing the lowest
352
consequences (11a and 11c) does not reverse the choice between A and B (it is the same as that
353
between A and B). If people do not integrate information between these attributes, then the
354
combination of both changes (in 11d) should not reverse preference in the choice between Aand
355
B. The four choices represent a 2 X 2 factorial design of contrasts in the lower consequences,
356
(yA ,yB ) or ( yA, y
B ) by contrasts in the higher consequences, (x A , x B ) or ( x 
A , x 
B ) . The key to finding
357
violations (evidence of integration) is to devise choices such that the first three choices (11a, 11b,
358
and 11c) yield one decision and the fourth (11d, combining the two factors) reverses that decision.
359
If LH integrative independence is systematically violated, we say that attributes L and H
Testing Heuristic Models 17
360
show evidence of attribute integration.
361
With three attributes (e.g., binary gambles), there are three possible pair-wise tests of
362
attribute integration. See Appendix A.1 for a proof that integrative independence in all three
363
pairwise tests is implied by lexicographic semiorders. The proof consists of enumerating all
364
possible combinations of assumptions concerning priority orders and threshold parameters to show
365
that the choice pattern, BBBA is not consistent with any of these LS models.
366
Note that if priority dominance held, 11a is true if and only if 11d is true, assuming P is
367
highest in priority. Priority dominance must be violated to observe evidence of integration for two
368
nondominant attributes. However, violation of priority dominance does not rule out integrative
369
independence.
370
Attribute Interaction
371
Lexicographic semiorders and the priority heuristic posit no interactions between
372
probabilities and consequences. Thus, in a choice between two binary gambles, if all four
373
consequences are held fixed, it should not be possible to reverse the preference by changing the
374
probability to win the larger consequence, as long as that probability stays the same in both gambles
375
in each choice. Indeed, if people consider one factor at a time, any factor that is the same in both
376
gambles of a choice should have no effect on the decision. If people are unaffected by any factor
377
that is the same in both alternatives, they will show no attribute interaction. Interactive
378
independence for probability and consequences can be written as follows:
379
A  (x A , p;y A )
B  (x B , p; yB )  A (x A , p; y A )
B  (x B , p ; yB )
(12)
380
In this case, the probability has been changed but it is the same in both gambles within each choice.
381
In order to find violations, we choose xB  xA  0 , yA  yB  0, and p  p  0 . For example,
382
A  ($55,0.1;$20)
B  ($95;0.1;$5) if and only if A  ($55,0.9;$20)
B ($95,0.9;$5) can be
Testing Heuristic Models 18
383
refuted. According to any lexicographic semiorder, people should choose either A and A or B
384
and B, but they should not switch from A to B. Violations of interactive independence are
385
evidence of attribute interaction. With three attributes, there are three possible pairwise tests of
386
interaction.
387
This test of interaction should not be confused with the test of sign dependence; Expression
388
12 does not require mixed consequences. It has some similarities to but is not the same as
389
Birnbaum’s (1974) “scale-free” test of interaction. Violations of Expression 12 are stronger than
390
Birnbaum’s scale-free test because they represent reversals of preference rather than just
391
inequalities of strengths of preference.
392
393
Predictions
The predictions of families of models are listed in Table 1. The family of compensatory,
394
integrative, interactive, transitive utility models (including EU, CPT, TAX, and many others)
395
violates priority dominance, violates integrative independence (displays attribute integration),
396
violates interactive independence (shows attribute interaction), and satisfies transitivity.
397
One-attribute heuristics (OAH) are transitive and show priority dominance, but they show
398
integrative independence and interactive independence. If a person used only one attribute in
399
making a choice, he or she would not show interactions or integration among attributes, even if that
400
person used a different randomly selected attribute on each trial.
401
The majority rule model, most probable winner model, and regret theory (Loomes, Starmer,
402
& Sugden, 1991; Loomes and Sugden, 1982) imply no priority dominance; they show integration
403
and interaction, but they violate transitivity.
404
405
Additive difference models, including stochastic difference model (González-Vallejo,
2002), similarity models (Rubinstein, 1988; Leland, 1994; 1998) and regret theory violate priority
Testing Heuristic Models 19
406
407
dominance, exhibit attribute integration, imply no attribute interactions and violate transitivity.
The family of lexicographic semiorders shows priority dominance, integrative
408
independence, interactive independence, and it violates transitivity. As long as 10% of the largest
409
consequence rounds to the same prominent number, the priority heuristic is a particular
410
lexicographic semiorder. When the largest consequence can be varied, there is an integrative
411
tradeoff between the lowest consequences and that largest consequence in either gamble.
412
Error Model
413
With real data, it is not expected that properties such as those tested here will hold perfectly.
414
Indeed, when the same choice is presented to the same person, that person does not always make
415
the same decision. For that reason, we need a model of error to construct the statistical null
416
hypothesis that the property holds, except for random “error.”
417
For example, suppose we conduct a test of integrative independence. According to the
A, B A,
418
property, we should not observe response patterns where a person chooses B
419
B A and yet A B. Some violations might arise because a person has the true pattern
420
BBBB, which is compatible with the property, but makes an “error” on the fourth choice. How
421
do we decide that a given number of violations is enough to refute the property? A “true and error”
422
model allows us to distinguish whether violations of properties such as those listed in Table 1 are
423
systematic or can instead be attributed to unreliability of choice. The error model used here is like
424
that of Sopher and Gigliotti (1993) and Birnbaum (2004), except with Birnbaum’s (2005)
425
improvement using replications to estimate the error rates.
426
When there are replications, error rates can estimated from cases where the same person
427
makes different decisions when presented the same choice problems. Consider a choice between a
428
“safe” and a “risky” gamble, denoted S and R, respectively. Let p represent the “true” probability
Testing Heuristic Models 20
429
of preferring the safe gamble, and let e represent the error rate. If this choice is presented twice,
430
there are four possible response patterns, SS, SR, RS, and RR, where RR denotes choice of the
431
“risky” gamble on both presentations. The theoretical probability that a person would choose the
432
risky gamble on both replicates, P(RR), is then given by the following expression:
P(RR)  pe  (1  p)(1 e)
2
433
2
(13)
434
In other words, this observed pattern can come about in two mutually exclusive ways: people who
435
truly preferred the safe gamble and made two errors or people who truly preferred the first gamble
436
and twice expressed their preferences correctly. Similarly, the predicted probability of switching
437
from R in the first choice to S in the second choice is given by the following:
P(RS)  pe(1 e)  (1  p)e(1  e)  e(1  e)
438
(14)
439
The probability of making the opposite reversal of preferences P(SR), is also predicted to be
440
e(1 e) , same as P(RS). The probability of two choices of the “safe” gamble is
441
P(SS)  p(1 e)  (1  p)e .
2
442
2
This model has been extended to test such properties as gain loss separability (Birnbaum &
443
Bahra, 2007) and transitivity (Birnbaum & Gutierrez, 2007), and will be extended below to evaluate
444
the properties in Table 1.
445
Study 1: A Test of Priority Dominance
446
According to the class of lexicographic semiorder models, once a decisive reason is found,
447
attributes with lower priority are ignored. Therefore, there should be no effect of altering the values
448
of attributes with lower priority.
449
Priority dominance can be tested in a pair of choices such as the following:
450
Choice 1
R:
10 tickets to win $100
S:
10 tickets to win $98
Testing Heuristic Models 21
90 tickets to win $0
451
90 tickets to win $20
Choice 2
R: 99 tickets to win $100
S :
01 ticket to win $0
10 tickets to win $22
90 tickets to win $20
452
Each row represents a different choice. In both choices, the risky gamble ( R or Ron the left) has a
453
lowest consequence of $0 and the safe gamble (on the right) has a lowest consequence of $20
454
(shown in bold font). According to the LPH lexicographic semiorder with  L = $10, people should
455
choose the safe gamble in both cases ( S and S , because the lowest consequence has highest
456
priority, the lowest consequence is better by $20 in both cases, $20 exceeds 10% of the biggest
457
prize in either lottery; i.e., 10% of $100).
458
According to integrative models, however, people may switch from the safe gamble, S , in
459
the first choice to the risky one, R in the second choice as the other attributes (probability and
460
highest consequence) are improved in the risky gamble in the second choice relative to the “safe”
461
one. So, if people systematically switch, it is evidence against the priority heuristic (i.e., the LPH
462
LS).
463
Now suppose that people used a lexicographic semiorder but that highest consequence had
464
highest priority. If so, people following that priority order might also switch from S to R because
465
the contrast in the highest consequences is too small in the first choice to be decisive (just $2), but
466
large enough in the second choice to be decisive. In order to test the hypothesis that highest
467
consequence has priority against the theory that people are integrating information, we need a test
468
in which the highest contrast is fixed and the other two attributes altered. By testing all three
469
hypotheses as to the attribute with highest priority, we can test the whole family of lexicographic
470
semiorder models, allowing the possibility that different people have different priorities and
471
different threshold parameters.
Testing Heuristic Models 22
472
Table 2 displays three such tests of priority dominance and a fourth test that allow us to
473
evaluate ten variations of lexicographic semiorders with different priority orders and different
474
ranges for their threshold parameters. The first two choices are described above, which test the
475
hypothesis that the lowest consequence has priority. This test used Choices 5 and 9, which were
476
replicated in Choices 13 and 17, respectively).
477
The next two choices in Table 2 (Choices 10, and 6, replicated in 18 and 14, respectively)
478
test the hypothesis that probability (shown in bold font) has the highest priority, and the third pair
479
of choices (Choices 7, 19, replicated in 11, 15) tests the hypothesis that the highest consequence
480
(bold font) has highest priority. By combining these three replicated tests with the same
481
participants, we can test the theory that different people might place highest priority on different
482
attributes.
483
The fourth test (Choices 8, 20, replicated in 12, 16) is a test of interaction. Here all four
484
consequences are the same in both choices. Probabilities are the same in within each choice, but
485
differ between choices. According to all lexicographic semiorder models, there should be no effect
486
of changing probability, as long as probability is the same in both gambles in a choice. The entries
487
in bold font show the attributes that have fixed values in each test.
488
The predictions of the prior TAX model (TAX with simplified parameters used in previous
489
research) are shown in the column labeled “TAX.” With suitable parameters, other integrative
490
models, like EU and CPT, can also make the same predictions as TAX. Entries of R and S indicate
491
predicted preference for the “risky” and “safe” gambles, respectively. Each test was devised so that
492
TAX with its prior parameters predicts a reversal of preference in contrast to each dominance
493
prediction. Thus, TAX implies a preference pattern of SRSRRSSR. None of the lexicographic
494
semiorders predicts this pattern that is predicted by prior TAX.
Testing Heuristic Models 23
495
The next ten columns of Table 2 show predictions of six lexicographic semiorder models
496
with different assumptions about the priority order and threshold parameters; these produce 10 sets
497
of predictions, shown in the next ten columns. (LPH indicates that lowest consequence has highest
498
priority, followed by probability and highest consequence, respectively; HLP indicates that highest
499
consequence has highest priority, followed by lowest consequence and probability). The model
500
labeled LPH-1 agrees with the priority heuristic of Brandstätter, et al. (2006). According to that
501
model, the threshold parameter is one tenth of the largest consequence in either gamble, which is
502
$100 in all eight choices; therefore  L  $10 . Note that LPH-1 predicts choice of the safe gamble
503
(S) in the first two choices. In contrast, HLP-1,4 predicts a switch from the safe gamble to the risky
504
one (R) in the first two choices. This HLP model assumes that highest consequence has first
505
priority, that $10  H  $40 , and that 2  L  $20 .
506
Insert Table 2 about here.
507
508
Method of Study 1
Participants viewed the materials on computers in the lab connected to the Internet. They
509
made 20 choices between gambles, clicking a button beside the gamble in each pair that they would
510
rather play. Gambles were represented as urns containing 100 identical tickets with different prize
511
values printed on them. The prize of each gamble was the value printed on a randomly drawn ticket
512
from the chosen urn. Gambles were displayed as in the following example:
513
A: 50 tickets to win $100
514
50 tickets to win $0
515
OR
516
B: 50 tickets to win $35
517
50 tickets to win $25
518
They were told that one person per 100 would be selected randomly, one choice would be selected
Testing Heuristic Models 24
519
randomly, and winners would receive the prize of their chosen gamble on that choice. This study
520
was included as one among several similar studies on judgment and decision making. Instructions
521
and the trials can be viewed from the following URLs:
522
http://psych.fullerton.edu/mbirnbaum/decisions/prior_dom.htm
523
Participants were 238 undergraduates enrolled in lower division psychology courses, 62%
524
were female and 82% were between 18 and 20 years of age.
525
Results of Study 1
526
The sums in the last three rows of Table 2 show the numbers of people who conformed to
527
each pattern of eight responses in the first replication, second replication, and the number who did
528
so in both replicates, respectively. There are 2 8  256 possible response patterns in each replicate.
529
The 119 under the first replicate for TAX, for example, indicates that 119 of the 238 participants
530
(50%) had the exact pattern of eight responses predicted by TAX model (with prior parameters) in
531
the first replicate. This was the most frequent pattern in the data. The same number showed this
532
pattern in the second replicate. The number in the last row indicates that 96 people showed this
533
exact pattern on all sixteen choices (in both replicates). For the priority heuristic model of
534
Brandstätter, et al. (2006), whose predictions are the same as LPH-1 in Table 2, there were only 6,
535
10, and 4 people whose data were consistent with that model in the first, second, and both
536
replicates, respectively. None of the other lexicographic semiorder models (with different priority
537
orders and threshold parameters) had more than 3 people who showed its predicted pattern on both
538
replicates.
539
We can use preference reversals between replicates to estimate rates of “error” in the eight
540
choices. Table 3 shows how many people made each combination of choices on the two
541
replications of each choice. Insert Table 3 about here
Testing Heuristic Models 25
542
Table 3 shows results only for cases in which people were exactly compatible with one of
543
the models on at least one replicate (people who were not consistent are not depicted). Table 4
544
represents all of the data in a format that allows analysis of error rates. The parameters, p and e, are
545
2
estimated by minimizing the  (1) between predicted and obtained frequencies for the “true and
546
error” model.
547
2
The estimated parameters and  (1) are displayed in three columns of Table 3. The true
548
and error model appears to give a reasonable approximation to the replication data; indeed, none of
549
2
the  (1) are significant. The tests of independence of replications (that the probability of a
550
2
repeated choice is the product of two probabilities), in contrast, are all significant; those  (1) are
551
shown in the last column of Table 3. Both of these tests are based on the same four frequencies,
552
and both estimate two parameters from the data, so these results indicate that the “true and error”
553
model provides a much better description of the data than any theory that implies independence.
554
Expressions 13 and 14 imply independence only when everyone has the same true preferences (i.e.,
555
when p = 0 or p = 1).
556
The estimated true choice probabilities show large deviations from the predictions of the
557
priority heuristic models and smaller deviations from the assumption that every subject agrees with
558
the TAX model and its prior parameters. For example, if everyone were perfectly consistent with
559
the TAX model, the true probabilities of choosing the safe gamble in the first two rows (Choices 5
560
and 9) should be 1 and 0. Instead, the estimated probabilities are 0.93 and 0.10, corrected for the
561
“attenuation” of the error, which are estimated to be e  0.06 and 0.10 for these two choices,
562
respectively.
563
564
The idea of priority dominance is that a person makes a decision based on one attribute at a
time, and once a decision is reached, it is unaffected by other attributes. If a person makes trade-
Testing Heuristic Models 26
565
offs, however, then it is possible for less important attributes to combine to outweigh the advantage
566
based on one attribute. Tables 2 and 3 show large violations of priority dominance.
567
Brandstätter, et al. (2006) criticized two integrative assumptions common to the utility
568
models: first, that people summate over values of consequences of mutually exclusive branches in a
569
gamble (integration); second, they questioned if people combine probabilities and consequences by
570
a multiplicative (interactive) relation. The next two sections present tests of attribute integration and
571
attribute interaction that permit direct tests of these two independence assumptions of lexicographic
572
semiorders.
573
Study 2: Attribute Integration
574
Table 4 illustrates a test of attribute integration in which the two attributes manipulated are
575
the lower consequences of the gambles and their probabilities. The highest consequences in each
576
choice are fixed to $100 and $51 in “risky” and “safe” gambles, respectively. In the first row
577
(Choice 7), the lowest consequences are the same, but the “risky” gamble has the more favorable
578
probability; therefore the priority heuristic predicts that people should choose R. In the second
579
choice (#18), both gambles have the same lower consequence ($50), but probability decides in
580
favor of the “safe” gamble. Similarly, the priority heuristic implies that the majority should choose
581
the “safe” gamble in the last two choices (#8 and 11) because the lowest consequence of that
582
gamble is $10 higher than that in the “risky” gamble. This model predicts the pattern RSSS for
583
these four choices, which are constructed from a factorial design of difference in the lowest
584
consequences ($50 versus $50 or $10 versus $0) by probability to win in the risky gamble (0.9 or
585
0.1). Predictions of lexicographic semiorder models are also displayed in Table 4. None of these
586
match the predictions of the TAX model with its prior parameters, which predicts the pattern,
587
RRRS.
Testing Heuristic Models 27
588
Insert Table 4 about here.
589
To understand the logic of the factorial design, it helps to start with Choice 7 in the first row
590
of Table 4. In this choice, there is no reason to prefer the safe gamble. Next, examine Choice 18 in
591
the second row. The change is that the probability to win the highest prize in the risky gamble has
592
been reduced from 0.90 to only 0.10, which should improve the tendency to choose S. But this
593
change is not enough by itself to switch most people to choosing the safe gamble; the majority still
594
chooses the risky gamble. Next, compare Choice 11 (third row of Table 4) with Choice 7; here,
595
the lowest consequences in both gambles have been reduced (from $50 to $0 in risky and from $50
596
to $10 in the safe gamble). This manipulation improves the safe gamble, but is not enough by itself
597
to cause the majority to choose the safe gamble. Choice 8 combines two changes, each of which
598
failed to reverse the majority preference by itself. If people do not integrate information, we expect
599
that the combination of two manipulations that failed to produce a preference for the “safe” gamble
600
should be no better than the best of them alone. Instead, we now have a strong majority (79%) who
601
choose the safe gamble. Two small changes tip the scales in favor of the safe gamble.
602
Note that the pattern RSSS, predicted by LPH and the priority heuristic, is not evidence of
603
integration since it represents a decision to choose “safe” if either lowest consequence or
604
probability favors that gamble; however, the pattern RRRS indicates that only when both of these
605
factors favor S does the person choose it. In other words, the pattern RSSS is consistent with the
606
definition of integrative independence (Expressions 11a-d) whereas RRRS violates it.
607
608
Method of Study 2
There were 242 undergraduates from the same subject pool who viewed the materials in the
609
lab via computers, clicking a button next to the gamble in each choice that they would rather play.
610
Sixteen choices were composed of four, 4-choice tests of attribute integration (Tables 4, 5, 6, and
Testing Heuristic Models 28
611
7). There were also three choices containing a test of transitivity (which is described under Study
612
4). Experimental materials can be viewed at the following URL:
613
http://psych.fullerton.edu/mbirnbaum/decisions/dim_integ_trans2.htm
614
615
Results of Study 2
Examining the percentages in Table 4, the modal choices agree with prior TAX, RRRS,
616
since 79% is significantly greater than 50% and 22%, 33% and 11% are all significantly less than
617
50%. The most frequent pattern of individual data was that predicted by prior TAX, which was
618
exhibited by 99 of the 242 participants. Only 8 people showed the pattern predicted by the priority
619
heuristic, RSSS. This comparison estimates no parameters from either model, but it does not test
620
the more general families of models that allow each person to have different parameters.
621
To compare the family of lexicographic semiorder models against the TAX model with free
622
parameters, we can examine the predictions of all possible lexicographic semiorders, as in Table
623
A.2. These 4 predicted orders are RRRR, RRSS, RSRS, and RSSS. The TAX model with free
624
parameters can accommodate these 4 patterns but it also predicts the pattern RRRS, when
625
( ,  ,  )  (1, 0.7, 1), which are the “prior” parameters. In addition, TAX predicts the patterns
626
RSSS, RSRS, RRSS and RRRR when ( ,  ,  )  (0.2; 2.5, 1), (1.4, 2.2, 1), (0.5, .1, 1) and (1.8, 0.7,
627
1), respectively. Thus, in this study, the family of lexicographic semiorders is restricted to only
628
four of five patterns that are compatible with integrative models like TAX, CPT, and the other
629
models in their class. There are sixteen possible response patterns, RRRR, RRRS, RRSR, RRSS,
630
RSRR, RSRS, RSSR, RSSS, SRRR, SRRS, SRSR, SRSS, SSRR, SSRS, SSSR, SSSS. The number of
631
people who showed each of these 16 patterns is 27, 99, 5, 21, 8, 44, 4, 8, 0, 5, 3, 1, 3, 2, 2, and 10,
632
respectively. (Patterns and frequencies set in bold type show cases consistent with TAX).
633
The “true and error” model of this four-choice property has 16 equations, each with 16
Testing Heuristic Models 29
634
terms that describe the probability of showing each observed pattern given each true pattern and
635
combination of errors. For example, the probability of showing the observed pattern, RRRS and
636
having the true pattern of RRRR is as follows:
637
638
P(RRRS  RRRR )  p(RRRR )(1 e1)(1  e2 )(1  e3)e4
That is, a person whose true pattern is RRRR will show the RRRS pattern by correctly
639
reporting the first three preferences and making an error on the fourth choice. There are fifteen
640
other ways to show this response pattern, corresponding to the other fifteen possible true patterns of
641
preference. The sum of these sixteen terms is the predicted probability of showing the pattern
642
RRRS. The TAX model is a special case of this true and error model in which five terms have non-
643
zero probabilities (RRRS, RSSS, RSRS , RRSS and RRRR) and the LS family is a still smaller special
644
case in which the probability of RRRS is fixed to zero.
645
Because the 16 observed frequencies sum to the number of participants, the data have 15
646
degrees of freedom. For the TAX model, there are five response patterns, so there are five
647
probabilities of the predicted patterns to estimate (which sum to 1, so four degrees of freedom are
648
used) and four error rates to estimate (one for each choice). That leaves 15 – 4 – 4 = 7 degrees of
649
freedom. The family of LS models is a special case of this model in which p(RRRS) is assumed to
650
2
be zero. The difference in  provides a test if the rate of RRRS is “significant.” Fitting this model
651
2
to minimize the  (7) yielded estimates of (eˆ1 , eˆ2 , eˆ3 , eˆ4 )  (0.09, 0.10, 0.11, and 0.19); pˆ (RSSS) =
652
0.00, pˆ (RSRS) = 0.44, pˆ (RRSS) = 0.00, pˆ (RRRS) = 0.53, pˆ (RRRR) = 0.03. When p(RRRS) is
653
2
2
fixed to zero (testing the family of LS models), the  increased by  (1) = 11.81. The estimated
654
error rate for the third choice in Table 4 (Choice 11) in this model was very large, eˆ3 = 0.5, which
655
also indicates an unsatisfactory solution. In sum, the observation that 99 people showed the RRRS
656
pattern, which is incompatible with any of the LS models, is greater than one can reconcile with the
Testing Heuristic Models 30
657
658
family of LS semiorders and the true and error model.
Because many of the cells in this example had low expected frequencies, statistical tests
659
were also repeated, pooling cells whose predicted frequencies were less than 4. Although values of
660
 2 and degrees of freedom are altered in these analyses, the conclusions were the same. In this
661
analysis of Table 4, (eˆ1 , eˆ2 , eˆ3 , eˆ4 )  (0.04, 0.24, 0.11, 0.20), pˆ (RRRS )  0.77, pˆ (RRSS)  0.06,
662
pˆ (RSRS)  0.09, and pˆ (SSSS)  0.08. All other parameters were set to zero. There were 7 cells
663
2
that had predicted frequencies less than 4, which were pooled, leaving 10 cells with 9 df; the  (2)
664
= 0.87. The predicted frequencies from this model are 24.9, 100.8, 5.1, 20.7, 10.4, 41.9, 2.0, 8.2,
665
1.3, 5.1, 1.0, 4.0, 0.8, 3.2, 2.5, and 10.1, respectively. When pˆ (RRRS ) was fixed to zero, the fit
666
2
was significantly worse,  (1) = 43.2.
667
A Mixture Model
668
Suppose people use lexicographic semiorder (LS) models, but they randomly switch from
669
one LS model to another (also switching threshold parameters) from trial to trial. In contrast with
670
the true and error model (which assumes that each person has a true response pattern), this mixture
671
model would not necessarily show individuals who fit the predicted patterns of individual LS
672
models, because it allows the same person to change priority order and threshold parameters from
673
trial to trial.
674
We can test whether such an LS mixture model is compatible with the overall choice
675
proportions. Let a = the probability of using the LPH LS with 0  L  $10 , b = probability of
676
using the LHP LS with 0  L  $10 ), c = probability of using one or more of PHL, PLH or LPH
677
with  L  $10 . Predictions of this mixture model are shown in the column labeled “LS Mixture”
678
in Table 4. Adding the second two choice percentages (22 + 33 = 55), we have 2a  b  c = 55%,
679
which should exceed the fourth choice percentage ( a  b  c  79%), if a > 0. Even if we assume
Testing Heuristic Models 31
680
that no one ever used the priority heuristic (a = 0), we still cannot reconcile the observed choice
681
proportions with the theory that people are using a mixture of LS semiorders because 79% is
682
substantially greater than 55%.
683
Additional Tests
684
Three other tests of attribute integration are presented in Tables 5, 6, and 7. In Table 5, the
685
contrast in the lowest consequences was fixed to a larger difference than in Table 4 ($20 versus $0),
686
and the probability to win the higher consequence (in both gambles) and the value of the higher
687
consequence in the risky gamble were varied in a factorial design. According to the priority
688
heuristic, the majority should choose the safe gamble in all four rows. Instead, the majority choices
689
agree with the TAX model’s predictions, which was also the most frequent pattern of individual
690
responses (shown by 115 people). Only 16 exhibited the pattern predicted by the priority heuristic
691
with its prior parameters. Insert Tables 5 and 6 about here.
692
The same analyses as described above for Table 4 were applied to the test in Table 5. In
693
this case, the TAX model with its prior parameters predicts the pattern SSSR, a pattern that is not
694
compatible with any of the LS models, but exhibited by 115 people. When its parameters are free,
695
the TAX model can also accommodate the patterns SSSS, SSRR, SRSR, and RRRR, when
696
( ,  ,  )  (0.5; 0.1, 1), (2; 2.5, 0), (2; 0.1, 0), and (14; 1.2, 0), respectively. The family of LS
697
models again allows only a subset of these patterns, in this case, including only SSSS, SRSR, and
698
RRRR. The observed frequencies of the 16 possible patterns from RRRR, RRRS, RRSR, RRSS,
699
RSRR, RSRS, RSSR, RSSS, SRRR, SRRS, SRSR, SRSS, SSRR, SSRS, SSSR, SSSS, are 6, 3, 5, 3, 2,
700
0, 3, 2, 12, 4, 25, 2, 39, 5, 115, and 16, respectively. Fitting the true and error model to these
701
frequencies yielded the following solution: pˆ (RRRR) = 0.07, pˆ (SRSR) = 0.19, pˆ (SSRR) = 0.0,
702
pˆ (SSSR) = 0.74, and pˆ (SSSS) = 0.0, with all others set to zero. When the model was restricted so
Testing Heuristic Models 32
703
that pˆ (SSSR) was also fixed to 0, the fit was significantly worse, as one might expect from the fact
704
that 115 people showed this pattern.
705
The data allow rejection of the LS mixture model. In this case, the mixture model implies
706
that the choice probabilities in the second and fourth choices in Table 5 (Choices 17 and 4) should
707
be equal. Instead, these are 75% (significantly greater than 50%) and 14% (significantly less than
708
50%).
709
In Table 6, an even greater difference in the lowest consequence was used ($0 versus $49).
710
In this case, the entire “safe” gamble was fixed, and the probability and value of the higher
711
consequence in the risky gamble were both varied. According to the priority heuristic, the majority
712
should choose the safe gamble in every choice. Instead, 91%, 83%, and 81% (all significantly more
713
than half) chose the safe gamble in the first three choices and 65% (significantly more than half)
714
chose the risky gamble in the fourth choice (Choice 10). Here 104 people showed the response
715
pattern predicted by the TAX model with its prior parameters (SSSR); this pattern violates all of the
716
LS models. For comparison, 62 showed the response pattern predicted by the priority heuristic
717
(SSSS). The integrative family (TAX with free parameters) can also predict the patterns SSSS,
718
SSRR, and SRSR [when ( ,  ,  )  (0.6; 0.7, 1), (2; 2.5, 0), and (2.5; 0.5, 0.5), respectively]. The
719
family of LS can handle three of these patterns, SSRR, SRSR, and SRRR, but not SSSR. The
720
observed frequencies of the 16 response patterns from RRRR, RRRS, to SSSS are 2, 7, 1, 3, 1, 2, 3,
721
2, 6, 2, 15, 5, 25, 2, 104, and 62, respectively. Fitting the true and error model, the estimated true
722
probabilities are pˆ (SRRR) = 0.06, pˆ (SRSR) = 0.04, pˆ (SSRR) = 0.12, pˆ (SSSR) = 0.69, and pˆ (SSSS)
723
= 0.09, with all others zero. The assumption that p(SSSR) = 0 can be rejected, which shows that the
724
family of LS cannot handle these data.
725
The LS mixture model also fails to fit the data of Table 6. According to the choice
Testing Heuristic Models 33
726
percentages, aˆ  35; cˆ  81 – 35 = 46; bˆ  83 – 35 = 48; therefore, dˆ  91 – 35 – 46 – 48 = – 38,
727
which is negative, in contradiction to the LS mixture model.
728
In Table 7, the “safe” gamble was again fixed, all probabilities were fixed to 0.5, and the
729
lower and upper consequences of the risky gamble were manipulated. In this case, the priority
730
heuristic implies that the majority should choose the safe gamble in all four choices. Instead, 81%
731
chose the risky gamble in the last choice, consistent with the prior TAX model.
732
The most frequent response pattern by individuals was SSSR, which is the pattern predicted
733
by the TAX model with prior parameters and which is not consistent with any of the LS models. In
734
this test, the LS models can produce the patterns, SSSS, SSRR, SRSR, and RRRR. The TAX model
735
can handle the patterns SSSR, SSSS, and SRSR [with ( ,  ,  )  (1, 0.7, 1), (0.3, 0.7, 1), and (1.5,
736
0.7, 1), respectively]. The observed frequencies of individual response patterns from RRRR to SSSS
737
were 2, 1, 4, 4, 1, 0, 2, 1, 4, 0, 80, 12, 11, 3, 92, and 25. The true and error model yielded the
738
following estimates: pˆ (RRRR) = 0.02, pˆ (SRSR) = 0.37, pˆ (SSRR) = 0.06, pˆ (SSSR) = 0.45, and
739
pˆ (SSSS) = 0.10, with all others set to zero. A simpler model with pˆ (SRSR) = 0.42, pˆ (SSSR) =
740
0.49, pˆ (SSSS) = 0.09 and all others set to zero also provided an acceptable fit; however, the LS
741
models can be rejected because they cannot account for the pattern, SSSR.
742
According to the LS mixture model, aˆ  19; cˆ = 91 – 19 = 72; bˆ  56 – 19 = 37, so
743
aˆ  bˆ  cˆ  19 + 72 + 37 = 128, which far exceeds the first choice percentage, 94%. Thus, these
744
data do not fit the LS Mixture model.
745
In all four tests (Tables 4, 5, 6, and 7), the data significantly violated the family of LS
746
models, including the LS mixture model. In all four tests, the most frequent response pattern was
747
that predicted by the prior TAX model, which is the pattern violating integration independence.
748
Insert Table 7 about here.
Testing Heuristic Models 34
749
Although the prior TAX model is the most accurate of the models with fixed parameters,
750
including Table 7, Table 7 is the least convincing of the four tests. To better test the models and
751
estimate the error terms, it is useful to employ a design with replications. Birnbaum and LaCroix (in
752
press) used a replicated test with an improved choice of attribute levels to check the integration of
753
lower and upper consequences (as in Table 7). They found more convincing evidence of
754
integration of these attributes.
755
A replicated test of attribute integration of probability and value was included in Study 4
756
(Table 8), which also tested transitivity (discussed below). Series A and B in this test used
757
consequences that were so similar that they can be treated as replications. Note that the safe
758
gamble is again fixed, as are the lower consequences of the two gambles. In this case, the priority
759
heuristic again predicts that the majority should choose the safe gamble in every choice. Instead,
760
73% and 72% (significantly more than half) chose the risky gamble in the two replications of
761
Choice 9, consistent with the prior TAX model [with ( ,  ,  )  (1, 0.7, 1)].
762
TAX (and the family of integrative models with free parameters) predicts the response
763
patterns, SSSS, SSSR, SSRR, SRSR, SRRR, and RRRR. Parameters that predict these patterns in
764
TAX, with ( ,  ,  )  (0.5, .7, 1), (1, 0.7, 1), (1, 1.5, 0), (2, 0.2, 0), (2, 0.5, –0.5), and (4, 0.2, –1),
765
respectively. The family of LS models can handle all of these patterns except SSSR. This pattern
766
was shown by 93 and 99 participants in Series A and B, and by 52 people in both series. This
767
pattern, which is consistent with the prior TAX model and refutes the LS models was the most
768
frequent pattern shown by individuals. Appendix B presents the analysis of this replicated test in
769
greater detail.
770
771
Insert Table 8 about here.
The five tests (Tables 4 – 8) agreed in that each pair of attributes tested showed significant
Testing Heuristic Models 35
772
integration. Put another way, each test allows us to reject the family of lexicographic semiorders,
773
including the LS mixture model, in favor of the assumption that people integrated each pair of
774
attributes tested.
775
Study 3: Attribute Interaction
776
The property of attribute interaction is implied by models such as EU, CPT, and TAX,
777
which include products of functions of probability and value. Tests of attribute interaction are
778
shown in Table 9. All four consequences in the choices are the same in all five choices (rows in
779
Table 9). The probabilities change from row to row, but they are always the same within any
780
choice. According to any LS model, including the priority heuristic, there should therefore be no
781
change in preference between any two rows in Table 9. Any theory that assumes that people make
782
choices by looking strictly at contrasts between the prizes and between probabilities implies no
783
change in preference between the rows.
784
Insert Table 9 about here.
The stochastic difference model (González-Vallejo, 2002) and other additive difference
785
models also assume that people choose by integrating comparisons between values of attributes as
786
well. Although these models are integrative, they assume no attribute interactions. The TAX
787
model and other interactive utility models, in contrast, imply that the difference between risky and
788
safe gambles is amplified by the probability to win the higher consequence in both gambles.
789
Predictions of the TAX model with previous parameters show that preference should reverse
790
between second and third rows in Table 9.
791
792
Method of Study 3
Gambles were described either as urns containing tickets, as in earlier studies, or as urns
793
containing exactly 100 marbles of different colors. Trials in the marble format appeared as in the
794
following example:
Testing Heuristic Models 36
795
Which do you choose?
E: 99 red marbles to win $95
796
01 white marbles to win $5
797
798
799
800
801
802
OR
F: 99 blue marbles to win $55
01 green marbles to win $20
Testing Attribute Interaction
In each repetition of the study, there were 21 choices between gambles, with two series of 5
803
choices testing attribute interaction. Series A consisted of five choices of the form,
804
($95, p; $5,1  p) vs. ($50, p; $15,1 p) , in which the five levels of p were 0.01, 0.10, 0.50, 0.90,
805
and 0.99. Series B consisted of five choices of the form, ($95, p; $5,1  p) vs. ($55, p; $20,1 p) ,
806
with the same 5 levels of p. The other 11 trials were warm-ups and fillers. Fillers were used so that
807
no two trials from the main design (Series A or B) would appear on successive choices. The two
808
conditions with different formats and different fillers gave virtually the same results, so results are
809
combined in presentation that follows.
810
Each participant completed the 21 choices twice, separated by 5 other tasks that required
811
about 25 minutes. Materials can be found at the following URL:
812
http://psych.fullerton.edu/mbirnbaum/decisions/dim_x_mhb2.htm
813
Participants were 153 undergraduates who served as one option toward an assignment in lower
814
division psychology. Of the 153, 78 (51%) were female and 86% were 18 to 20 years of age.
815
Choice-Based Certainty Equivalents
816
To further investigate the interaction between probability and prize, and to examine the
817
prominent number rounding assumption in the priority heuristic, a follow-up study was conducted
818
in which participants chose between binary gambles and sure cash. The binary gambles were of the
Testing Heuristic Models 37
819
form, G  (x, p;$0,1 p). There were 48 choices, constructed from a 4 X 3 X 4, Prize by
820
Probability by Certain Cash factorial design in which Prize, x = $75, $100, $140, or $200;
821
Probability, p  0.1, 0.5, or 0.9; and Certain Cash, c  $10, $20, $30, or $50.
822
Participants in this follow-up were 231 undergraduates from the same pool. As in earlier
823
studies, gambles were described as containers with 100 equally likely tickets, from which a ticket
824
would be drawn at random to determine the prize. Participants viewed the materials via the Internet
825
and clicked one of two buttons to indicate preference for the gamble or the cash.
826
Results of Study 3
827
The data in Table 9 show a systematic decrease from 83% choosing the “safe” gamble
828
(significantly greater than 50%) in the first row to only 17% who do so (significantly less than
829
50%) in the last row. This systematic decrease violates all lexicographic semiorder models, the
830
priority heuristic, the stochastic difference model (González-Vallejo, 2002), and other additive
831
difference models, as well as any random mixture of those models. TAX, with prior parameters,
832
correctly predicted the pattern of majority preferences.
833
Table 10 shows the frequencies of response patterns in both series. There are 32 possible
834
response patterns for each series, but only these six were repeated by at least one person in at least
835
one series. According to the priority heuristic, people should consistently choose the S gamble in
836
every choice. Only 3 people did this in both replicates of Series A; 4 did so in Series B, and only 2
837
of these did so in both A and B. Instead, most individuals switched from S to R in both series as the
838
probability to win the higher consequence was increased, consistent with the interactive models.
839
840
841
Insert Table 10 about here.
These same six patterns SSSSS, SSSSR, SSSRR, SSRRR, SRRRR, and RRRRR are all
compatible with the family of interactive models, including TAX, CPT, and EU, among others.
Testing Heuristic Models 38
842
843
Choice-Based Certainty Equivalents
According to the priority heuristic, the cash certainty value in choices between c (sure cash)
844
and (x, p; y,1  p) depends only on the highest prize in the gamble (x); that is, the choice between
845
gamble and cash depends entirely on x and is independent of p. If the sure cash amount exceeds the
846
rounded value of one tenth of x, the majority should prefer the cash, and if it falls short of this
847
amount, the majority should prefer the gamble. Because 10% of $75, $100, and $140 all round to
848
$10 (the nearest prominent number), there should be no difference between these cash values.
849
People should prefer $10 or more to any gamble with those highest prizes. When the highest prize
850
is $200, people should prefer any cash amount greater than or equal to $20 to the gamble,
851
independent of probability.
852
Of the 48 choice percentages, the priority heuristic correctly predicted the majority choice in
853
only 20 cases (only 42% of the choices). For example, 91%, 41%, and 11% preferred $30 over
854
($100, 0.1; $0, 0.9), ($100, 0.5; $0, 0.5), and ($100, 0.9; $0, 0.1), respectively. According to the
855
priority heuristic, all three percentages should have exceeded 50% since the decision should be
856
based on the difference in lowest prizes ($30), which exceeds 10% of the highest prize. Similarly,
857
93%, 41%, and 15% preferred $50 to ($200, 0.1; $0, 0.9), ($200, 0.5; $0, 0.5), and ($200, 0.9; $0,
858
0.1), respectively. Such reversals of the majority preference related to probability occurred for all
859
sixteen combinations of highest prize and certain cash, contrary to the priority heuristic.
860
In order to solve for the cash equivalent value of each gamble (the value of c that would be judged
861
higher than the gamble 50% of the time), choice proportions were fit to the following model:
862
Pij k 
1
1  exp[aij (gij  c k )]
(15)
863
where Pij k is the predicted probability of choosing cash value ck over gamble (xi , pj ;$0,1  p j ) ; the
864
values of ck were set to their objective cash values ($10, $20, $30, $50), and the logistic spread
Testing Heuristic Models 39
865
parameter aij was estimated separately for each gamble. The values of gij are the certainty
866
equivalents of the gambles; that is, the interpolated cash values that would be judged better than the
867
gamble 50% of the time. These were estimated from the data to minimize the sum of squared
868
differences between the predicted probabilities and observed choice proportions.
869
The certainty equivalents of the gambles are shown in Figure 1 as a function of the largest
870
prize value (x), with a separate curve for each level of probability to win, p. According to the
871
priority heuristic, all three curves should coincide. That is, the cash-equivalent value should depend
872
only on x in these choices and should be equal to  L . Instead, the curves never coincide.
873
874
Insert Figure 1 about here.
According to the priority heuristic, gambles with highest prizes of $75, $100, and $140
875
should all show the same cash equivalent values, since people should have rounded 10% of these
876
values to $10, the nearest prominent number. This means that the curves in Figure 1 should have
877
been horizontal until $140 and show a positive slope between $140 and $200. This might be the
878
case when probability is 0.1 (lowest curve), but the curves show positive slope when the probability
879
to win is either 0.5 or 0.9. Instead of showing the patterns predicted by the priority heuristic, the
880
curves diverge to the right, consistent with a multiplicative interaction between prize and
881
probability, such as is assumed in utility models like EU, CPT, TAX, and many others.
882
The results for tests of priority dominance, attribute integration, and attribute interaction all
883
show that the family of lexicographic semiorder models can be rejected because data systematically
884
violate predictions of those models. In those tests, the LS models had to defend the null hypothesis
885
against integrative and interactive models that predicted violations. It should be clear that the tests
886
of integration and interaction can lead to only three outcomes; they can retain both families of
887
models, they could reject LS and retain the integrative, interactive family, or they might reject both
Testing Heuristic Models 40
888
families. There is no way for the LS models to be retained and the integrative, interactive family
889
rejected in these tests because the LS models can handle only a subset of the response patterns
890
consistent with the integrative models. However, it is possible to refute the family of transitive
891
models (including TAX, CPT, EU, etc.) and retain the family of LS models in tests of transitivity
892
(Study 4). Testing transitivity, therefore, allows the priority heuristic to go on offense, putting the
893
family of transitive utility models in danger of refutation.
894
895
Study 4: Transitivity of Preference
The first three choices in Table 11 provide a new test of transitivity in which the priority
896
heuristic predicts violations. Similarly, the second set of three choices (Series B) is a near replicate
897
in which the priority heuristic also implies violation of transitivity, except that positions of the
898
gambles are counterbalanced.
899
Insert Table 11 about here.
Methods matched those of the earlier studies. Participants were 266 undergraduates, tested
900
in the lab, who completed the experiment at the following URL:
901
http://psych.fullerton.edu/mbirnbaum/decisions/dim_int_trans4.htm
902
There were 18 choices between gambles consisting of 4 warm-ups, 8 trials forming a
903
replicated test of attribute integration, and 6 trials composing two tests of transitivity. (The tests of
904
attribute integration were presented in Table 8).
905
Results of Study 4
906
The priority heuristic correctly predicted that the majority would choose the second gamble
907
in Choices 6 and 18; the lowest prizes are both $0 and the difference in probability is .08 or less, so
908
people should choose by the highest consequence. However, in Choice 14, priority heuristic
909
predicts that people should choose the first gamble, which has a lower probability to receive $0
910
(0.15 rather than 0.30, a difference exceeding 0.10). Instead, 84% chose the gamble with the higher
Testing Heuristic Models 41
911
prize. Similar results were observed in the second series, with positions reversed. So, the majority
912
data fail to show the predicted pattern of intransitivity predicted by the priority heuristic.
913
Table 12 shows the number of people who showed each response pattern in the two tests.
914
The most frequent response pattern (001) is shown by 184 and 177 of 266 participants in Series C
915
and D (reflected), respectively, and by 149 people (56%) in both tests. This pattern matches the
916
predictions of the TAX model with its prior parameters. (This pattern is also consistent with other
917
transitive utility models). The intransitive pattern predicted by the priority heuristic was shown by
918
18 and 9 people in Series A and B, respectively, but no one agreed with the priority heuristic on
919
both series.
920
Insert Table 12 about here.
The “true and error” model allows a person who is truly intransitive to show a transitive
921
pattern. For example, they might correctly report preference for the first gamble (F) on Choices 6
922
and 18 and make an “error” on trial 14 by choosing S. The probability that this FFS response
923
pattern would be observed, given that the person’s true pattern is FFF is given as follows:
924
P(observed  FFS true  FFF) (1  e1 )(1 e2 )e3
(16)
925
where e1, e2 , and e3 are the error rates for Choices 6, 18, and 14, respectively. Assuming Choices
926
15, 12, and 8 are replicates of 6, 18, and 14, respectively, then we can estimate the error terms and
927
also calculate the probability that a person would show each combination of preferences on two
928
tests, given each theoretical “true” response pattern. Indeed a person who truly has the pattern
929
predicted by TAX (FFS) might show an observed intransitive pattern, FFF, by making one error.
930
The theoretical probability of each response pattern is the sum of eight terms, representing the eight
931
mutually exclusive and exhaustive conjunctions of observed and “true” patterns.
932
933
The data are partitioned into the frequencies of showing each pattern on both replicates and
the average frequency of showing each response pattern on one replicate or the other but not both.
Testing Heuristic Models 42
934
2
The model is fit to these 16 frequencies to minimize the  between predicted and obtained
935
frequencies. The true and error model is neutral with respect to the issue of transitivity, since it
936
allows each subject to have a different true pattern, which need not be transitive. The models
937
estimates the probability of each of 8 possible response patterns, two of which are intransitive (FFF
938
and SSS).
939
When this model was fit to the data, the best-fit solution yielded the following estimates for
940
the “true” probabilities of the 8 possible response patterns: 0, 0.87, 0.03, 0, 0.01, 0, 0.07, 0.03 for
941
FFF, FFS, FSF, FSS, SFF, SFS, SSF, and SSS, respectively, where F = choice of the First gamble
942
and S = choice of the Second gamble in Table 11. . Thus, this model estimates that 87% of the
943
participants had FFS as their “true” preference pattern, that no one had the pattern of the priority
944
heuristic (FFF) as his or her “true” pattern, that 3% had the opposite intransitive pattern as their
945
“true” pattern, and the rest had other transitive orders. The estimated error rates were 0.11, 0.09,
946
and 0.08, for the first, second, and third choices in the table, respectively.
947
Study 3 (n = 242) also included a test of transitivity in binary gambles in which the lowest
948
consequence was $0. The choices are shown in Table 13. The majority preferences were transitive.
949
The modal pattern was to prefer A over B, B over C and A over C. However, 41 people (17%)
950
showed the intransitive pattern predicted by the priority heuristic. Fitting the “true and error”
951
model to the response frequencies, an acceptable fit was obtained for a solution with only three
952
“true” response patterns (all transitive): 001, 010, and 110, with estimated “true” probabilities of
953
0.86, 0.05, and 0.09, respectively, with error terms of 0.10, 0.02, and 0.24, respectively. Deviations
954
2
from this purely transitive model were not significant,  (1) = 1.75.
955
956
Insert Table 13 about here.
If people had been systematically intransitive in the manner predicted by the priority
Testing Heuristic Models 43
957
heuristic, then all of the transitive utility models such as TAX, RAM, CPT, GDU, EU, and others
958
would have been rejected or would have required revision to account for the intransitive
959
preferences. However, Studies 3 and 4 found no systematic evidence of intransitivity of the type
960
predicted by the priority heuristic.
961
Discussion
962
All four studies yield clear results that either violate or fail to confirm predictions of
963
lexicographic semiorders and the priority heuristic. When we compare models with prior
964
parameters, very few participants show the data patterns predicted by the priority heuristic, whereas
965
the TAX model with prior parameters did a good job of predicting the most common data pattern in
966
these studies. When we compare classes of models by agreement with critical properties, we find
967
that evidence against the family of lexicographic semiorders including the generalized priority
968
heuristic is very strong and evidence against the transitive models is very weak.
969
970
971
Tests of priority dominance in Study 1 show that no attribute has priority over the other two
for more than a few participants.
Tests of attribute integration show that each pair of attributes studied shows evidence of
972
integration for a large and significant number of participants. Small changes that do not reverse a
973
decision by themselves can combine to reverse a preference. The theory that people use a mixture
974
of lexicographic semiorders with different priority orders and different threshold parameters,
975
(switching from trial to trial) does not account for the choice proportions in any of five tests. There
976
were four tests in Study 3 and a replicated test in Study 4. Nor did the most promising case for a
977
lexicographic semiorder in Study 3 (Table 7 with 50-50 gambles) fit the data in the improved and
978
replicated tests of 50-50 gambles in Birnbaum and LaCroix (in press).
979
Tests of attribute interaction in Study 3 show clear evidence of interaction between
Testing Heuristic Models 44
980
probability to win and amount to win. Whereas lexicographic semiorders and additive difference
981
models imply no interactions between probability and value, Study 4 shows that preferences can be
982
reversed by changing the value of probability that is the same in both gambles in a choice.
983
Lexicographic semiorders and additive difference models require that any factor that is the same in
984
both choices cannot affect the choice. Instead, the data show very clear violations of this
985
independence property in favor of systematic interaction. Brandstätter et al. (2006) expressed
986
doubts that humans have multiplication in their adaptive toolbox. The evidence of interaction in
987
Study 3 is joined by the certainty equivalents in the follow up to show that people behave as if they
988
indeed have multiplication between functions of probability and of prize value.
989
When individual differences are analyzed in these three new tests, it is found that few
990
participants produce response patterns compatible with any of the lexicographic semiorder models,
991
including the priority heuristic.
992
993
994
The most common pattern of individual data is that predicted by the TAX model with its
prior parameters, not that of any of the lexicographic semiorder models.
Tests of transitivity do not show evidence that many, if any, participants exhibit systematic
995
violations of transitivity of the type predicted by the priority heuristic. Failure to find intransitivities
996
does not prove there are none to be found, of course. Design of a study of transitivity will be
997
sensitive to assumed threshold parameters and the priority order of the LS model used to devise the
998
test. However, the present findings fit with other recent failed attempts to find violations of
999
transitivity predicted by specific intransitive models.
1000
Although some studies have reported systematic violations of transitivity based on the
1001
Tversky (1969) gambles (see review in González-Vallejo, 2002; Iverson & Falmagne, 1985;
1002
Iverson & Myung, submitted), others who have reviewed this literature concluded that the evidence
Testing Heuristic Models 45
1003
does not warrant abandonment of transitive theories (Luce, 2000; Stevenson, Busemeyer, & Naylor,
1004
1991; Regenwetter, et al., 2006; Rieskamp, Busemeyer, & Mellers, 2006). Interestingly, Tversky
1005
went on to publish transitive theories (Kahneman & Tversky, 1979; Tversky & Kahneman, 1992),
1006
in which intransitivities might be produced by editing.
1007
Birnbaum and Gutierrez (2007) tested transitivity using the same gambles as in Tversky
1008
(1969) and a variation with prizes 100 times as large. The priority heuristic of Brandstätter, et al.
1009
(2006) predicts that the majority should show intransitivity with these gambles. For example, most
1010
should prefer A  ($500, 0.29;$0) over C  ($450,0.38;$0) , C over E  ($400,0.46;$0) , and the
1011
majority should violate transitivity by choosing E over A . The highest estimated rates of
1012
intransitivity observed by Birnbaum and Gutierrez occurred in a condition of their second
1013
experiment in which they used pie charts to display probability and no numerical information on
1014
probability was provided to participants. In this condition, with either 160 undergraduates tested in
1015
the lab or 201 Web recruits tested via the WWW, the estimated percentage who were consistent
1016
with the intransitive strategy, was 5.8% in the true and error model. Out of 361 participants, only
1017
14 (3.9%) displayed the intransitive pattern predicted by the priority heuristic on both repetitions
1018
(A
1019
incidence of 5.8% agreement with the predicted data pattern seems too low for a descriptive theory.
1020
Birnbaum and LaCroix (in press) also found transitivity to be largely satisfied in a design in
B; B C; C
A ). Although greater than zero, an empirical incidence of 3.9% and theoretical
1021
which they used 50-50 gambles: A  ($100, 0.5;$20) , B  ($60,0.5;$27) , and C  ($45,0.5;$34) .
1022
If  L = $10, people should choose A
1023
(submitted) found that most people satisfied transitivity in two designs in which regret theory
1024
predicted violations.
1025
Evaluation of the Priority Heuristic
B and B C , but C
A . Birnbaum and Schmidt
Testing Heuristic Models 46
1026
What can one make of the seemingly “good” fit of the priority heuristic to previously
1027
published data, as claimed by Brandstatter, et al? Birnbaum (in press) made three points
1028
concerning their contests of fit. First, they did not review previously published data in which their
1029
heuristic was not accurate. The priority heuristic does not account for satisfaction of stochastic
1030
dominance in cases where most people satisfy it, nor does it account for cases where most people
1031
violate stochastic dominance (Birnbaum, 1999; Birnbaum & Navarrete, 1998). Birnbaum (in press)
1032
noted that it does not account for previously published violations of restricted branch independence,
1033
lower or upper cumulative independence, nor does it account for violations of distribution
1034
independence (Birnbaum, 2005b).
1035
Second, Brandstaetter, et al used a non-diagnostic analysis of the Tversky and Kahneman
1036
(1992) data; when these are re-analyzed with diagnostic techniques, both TAX and CPT are far
1037
more accurate in fitting those data than the priority heuristic.
1038
Third, their analysis relied on a global index of fit, percentage of correct predictions of the
1039
majority choice and did not estimate parameters from the data. Such analyses can lead to wrong
1040
conclusions when parameters are not properly estimated. When parameters are estimated from the
1041
data, CPT, TAX, and the priority heuristic all fit the Kahneman and Tversky (1979) perfectly.
1042
Thus, the Kahneman and Tversky (1979) data, like other studies analyzed by Brandstätter, et al.
1043
(2006), are not suitable for comparing these theories. When parameters were estimated from the
1044
data, TAX and CPT fit the data of Erev et al (2002) and of Mellers et al. (1992) better than the best-
1045
fitting solution of the priority heuristic with the same number of free parameters. Rather than
1046
compare theories by an index of fit, it is better to test critical properties that must be satisfied by the
1047
theory with any parameters.
1048
In this paper, three new critical properties that are implied by lexicographic semiorder
Testing Heuristic Models 47
1049
models have been proposed and tested. These critical tests should hold under any lexicographic
1050
semiorder, with any threshold parameters. They should also hold for the priority heuristic when the
1051
largest consequence in either gamble is fixed. If any member of this family of models is true, we
1052
should observe evidence of priority dominance in one of the three tests for each person, there
1053
should be no evidence of systematic integration, and there should be no evidence of systematic
1054
interaction. Instead, we find very strong evidence contradicting the family of lexicographic
1055
semiorders, including the priority heuristic. Very few people satisfy these critical properties of the
1056
lexicographic semiorder.
1057
According to lexicographic semiorders and additive difference models, there should be
1058
systematic violations of transitivity, which would require that TAX, CPT, EU and other models in
1059
their class to be revised or rejected. But substantial violations of transitivity did not appear where
1060
they were predicted by the priority heuristic. Here, conclusions must be more tentative, since we
1061
cannot accept a null hypothesis based on a failure to find evidence of intransitivity. Nevertheless,
1062
in several tests, the data show few systematic violations where predicted by the priority heuristic.
1063
These negative findings shift the burden of proof to supporters of intransitive models: show us
1064
where to find evidence of intransitivity.
1065
Expected Value Exclusion
1066
Brandstätter, et al. (2006, p. 426) noted that the priority heuristic was not accurate in cases
1067
where expected value (EV) differed by a ratio greater than 2. Suppose people first compute EV of
1068
each gamble; take their ratio, choose the gamble with the higher EV if the ratio exceeds 2, and use
1069
the rest of the PH otherwise. The EV model implies attribute integration and interaction, violates
1070
priority dominance, and satisfies transitivity as long as the threshold for using EV is small enough.
1071
This EV modified priority heuristic model could handle some but not all of the previous and present
Testing Heuristic Models 48
1072
results.
1073
For example, violations of stochastic dominance reported by Birnbaum (1999; 2004a;
1074
2004b; 2005a) are violations of expected value as well as the priority heuristic. Violations of
1075
restricted branch independence and cumulative independence (Birnbaum & Navarrete, 1998) also
1076
include cases where people violate both expected value and the priority heuristic. For
1077
example, R  ($97,0.1;$11,0.1;$2,0.8) has a higher EV (12.4) than S  ($49,0.1;$45,0.1;$2,0.8) ,
1078
whose EV is 11; however, most people choose S, contrary to both EV and the priority heuristic. So,
1079
even the addition of EV to the list of reasons in the priority heuristic does not account for previous
1080
results. Brandstätter et al.(in press) replicated a portion of the choices used by Birnbaum and
1081
Navarrete (1998) and also found that the priority heuristic correctly predicts fewer than half of the
1082
modal choices in their replication.
1083
Nor does this incorporation of EV into the priority heuristic handle all of the results
1084
presented here. For example, in Table 9, the priority heuristic predicts that people should choose
1085
the “Safe” gamble in all choices, including Choices 17 and 7. In both of these choices, the EV ratio
1086
is less than 2), as it is for the last four choices in Table 9). But we see a clear effect of the common
1087
probability in these choices, contrary to the priority heuristic.
1088
Similarly, the priority heuristic predicts that people should prefer the “safe” gamble in
1089
Choice 18 of Table 4 (it has a lower probability of the lowest consequence). Instead, 67% choose
1090
the “risky” gamble, ($100,0.1;$50,0.9) over the “safe” gamble, ($51,0.5;$50,0.5) , despite the fact
1091
that the EV ratio is only 1.09. Other choices in this paper also have expected value ratios in the
1092
acceptable region and yet violate the priority heuristic.
1093
Concluding Comments
1094
Reviewing the findings with respect to Table 1, we see that one attribute heuristics can be
Testing Heuristic Models 49
1095
rejected by the strong evidence evidence showing attribute integration and attribute interaction.
1096
Additive difference models, including the stochastic difference model are compatible with attribute
1097
integration and violate priority dominance, but they assume no interaction, and they are thus not
1098
consistent with the results of Study 3. Regret theory and majority rule allow an interaction in Study
1099
3, but these violate transitivity. The tests of transitivity in this article were designed on the basis of
1100
the LS models; transitivity violations predicted by regret theory and majority rule have not been
1101
tested here. Most compatible with all of the results are the integrative, transitive utility models like
1102
EU, CPT, and TAX because the data do not require us to reject transitivity of preference.
1103
The weakest part of this argument is the conclusion that transitivity is acceptable. Finding
1104
conformity to a principle does not show that there are no violations. Therefore, the focus for
1105
empirical investigations for defenders of models like the regret model, majority rule, priority
1106
heuristic, or stochastic difference model would seem to be to find where consistent and systematic
1107
violations of transitivity can be observed.
1108
These studies were designed to investigate cases where LS models make predictions that are
1109
systematically different from predictions made by transitive, integrative models. Had the priority
1110
heuristic, for example, correctly predicted the results for priority dominance, attribute integration,
1111
attribute interaction and transitivity, such results would have required rejection or revision of rival
1112
models. Despite the opportunity in these studies for LS models to outshine transitive, integrative
1113
models, the LS models and priority heuristic failed to provide accurate descriptions of the empirical
1114
data.
1115
Appendix A
1116
Suppose three attributes can vary in each alternative, A, B, and C. The ABC lexicographic
1117
semiorder assumes that people examine these attributes in the order, A, then B, then C, using the
Testing Heuristic Models 50
1118
following rule to decide between two alternatives, G  (aG ,bG ,cG ) and F  (aF ,bF ,cF ) :
1119
If u(aG )  u(aF )   A choose G
1120
Else if u(a F )  u(aG )   A choose F
1121
Else if v(bG )  v(bF )  B choose G
1122
Else if v(bF )  v(bG )  B choose F
1123
Else if t(cG )  t(cF )  0 choose G
1124
Else if t(c F )  t(cG )  0 choose F
1125
Else choose randomly.
1126
Where  A ,  B , and  C are difference threshold parameters for comparison of the attributes A, B,
1127
and C, respectively, and u(a) , v(b) , and t(c) are strictly monotonic, subjective scales of the
1128
attribute values. Lexicographic semiorders generated by other priority orders, ACB, BAC, BCA,
1129
CAB, and CBA are similarly defined.
1130
A test of integration of attributes A and B (with levels of C fixed) is illustrated in Table 14.
1131
The four choices in the test have levels chosen such that cG  cF , aF  aG , aF  aG, bF  bG , and
1132
bF  bG.
1133
Integrative independence of A and B is the property that
1134
If G  (aG ,bG ,cG )
1135
And G (aG,bG ,cG )
F  (aF,bF ,c F )
1136
And G (aG , bG,cG )
F (aF , bF,cF )
1137
Then G (a
,cG )
G , bG
1138
However, if people integrate attributes A and B, then integrative independence can be
1139
F  (a F ,bF ,c F )
F (a
F , bF,cF )
violated with the pattern GGGF
 (i.e., G
F , G F, G F  and F G). It can be
Testing Heuristic Models 51
1140
shown that none of the lexicographic semiorders imply this response pattern. The proof amounts to
1141
enumeration of all possible combinations of priority orders with all possible assumptions
1142
concerning whether the contrast in each attribute reaches threshold. These combinations are
1143
worked out in Table 15, which shows that none of the combinations of assumptions leads to
1144
GGGF
 .
1145
Insert Tables 14 and 15 about here.
1146
The columns of Table 15 represent the six possible priority orders for attributes A, B, and C.
1147
The rows indicate 2 X 2 X 2 = 8 combinations of assumptions concerning whether each contrast in
1148
attributes is large enough to be decisive or not. The contrasts are either decisive or not; i.e.,
1149
u(a 
)  B , and t(cG )  t(cF )   C are either true (“Yes”) or not (“No”).
F )  u(a
G )   A , v(b
F )  v( bG
1150
Note that the levels in the test are devised so that the only attribute favoring G in the first choice is
1151
C and the other two attributes favor F.
1152
There are only five response patterns consistent with the 48 lexicographic semiorders
1153
(combinations of assumptions) in Table 15: FFFF, GFFF
 , GFGF
 , GGFF
 , or
1154
GGGG
 . The pattern, GFFF
 , should not be confused as evidence of integration; this pattern
1155
merely indicates that if either A or B favors F, the response is F. The pattern, GGGF
 , is not
1156
compatible with any of the lexicographic semiorders analyzed in Table 15; this pattern indicates
1157
that when both A and B favor F, the response is F.
1158
Because the labeling of the attributes (A, B, and C) is arbitrary, it should be clear that the
1159
analysis in Table 15 also shows that attributes B and C (with levels of A fixed) should show
1160
independence, as should attributes A and C (with levels of B fixed). That is, each pair of attributes
1161
should show integrative independence according to the family of lexicographic semiorders. In
1162
addition, we can exchange the roles of F and G in the test by selecting c F  c G , aF  aG , aF  aG,
Testing Heuristic Models 52
1163
bF  bG , and bF  bG. In this case, the pattern FFFGis not compatible with any of the
1164
lexicographic semiorders, whereas the patterns GGGG
 , FGGG
 , FGF G, FF GG,
1165
and FFFF would be consistent with the property.
1166
1167
Apppendix B: Replicated Test of Attribute Integration
A test with replication has the advantage that the error terms can be estimated in a manner
1168
that is independent of the substantive theories to be tested. Study 4 included such a replicated test,
1169
described in Table 8. When each choice is replicated, error terms can be estimated from preference
1170
reversals between replicates. As shown in Equation 14, reversals between repeated presentations of
1171
the same choice provide an estimate of the error rate, P(RS)  P(SR)  2e(1  e) . Using this
1172
method, the error terms for Choices 16, 13, 5, and 9 of Table 8 are estimated to be 0.05, 0.09, 0.21,
1173
and 0.15 respectively.
1174
The observed frequencies of showing each response pattern on first and second replicates
1175
and on both replicates are shown in Table 16. According to the true and error model, each person
1176
may have one of sixteen “true” preference patterns. The probability of a person showing a given
1177
pattern of data is then the sum of sixteen terms, each representing the probability of showing a
1178
given response pattern and having the pattern of errors and correct responses required to produce
1179
that response pattern. There are sixteen equations, each of which has sixteen terms. For example,
1180
the following term gives the probability of having the true pattern of SSSS and showing the
1181
observed pattern of SSSR on one replicate:
1182
p(SSSS)(1  e1 )(1  e2 )(1 e3 )e4 .
1183
The probability of showing the observed pattern SSSR would be the sum of sixteen terms, one for
1184
each of the “true” data patterns. Thus, even a person who truly satisfies the LS and has the true
1185
pattern SSSS might show the pattern predicted by TAX by making an error on the fourth choice. In
Testing Heuristic Models 53
1186
order to show this pattern on both replicates, the terms representing correct choices and errors
1187
would be squared.
1188
The true and error model was fit to the 16 frequencies of repeating a response pattern on two
1189
replications and the 16 average frequencies of showing these response patterns on either the first or
1190
second replication but not both. These 32 frequencies are mutually exclusive and sum to the
1191
number of participants, which means there are 31 degrees of freedom in the data. The substantive
1192
models to be compared are both special cases of this “true and error” model, in which many of the
1193
possible response patterns are assumed to have zero probability.
1194
In particular, the integrative models (TAX with free parameters) can handle 6 response
1195
patterns: SSSS, SSSR, SSRR, SRSR, SRRR, and RRRR. The probabilities of the other 10 patterns are
1196
set to zero. From the data, 4 error terms are estimated, and 6 “true” probabilities of response
1197
patterns are estimated. These 6 true probabilities sum to 1, so they use 5 degrees of freedom,
1198
2
leaving 31 – 4 – 5 = 22 degrees of freedom to test the model. Fitting Table 16,  (22) = 25.0, an
1199
acceptable fit. The estimated “true” probabilities are shown in the last column of Table 16. One of
1200
the patterns compatible with TAX, pˆ (SRRR) was estimated to be 0, but the diagnostic pattern,
1201
pˆ (SSSR) = 0.48. This latter probability indicates that almost half of the sample has the response
1202
pattern that is predicted by TAX with its prior parameters and which is incompatible with the
1203
family of LS models.
1204
1205
Insert Table 16 about here.
The family of LS models is compatible with five of the six patterns consistent with TAX
1206
with free parameters, but not SSSR. When the probability of this pattern, SSSR, is set to zero, the
1207
2
2
best-fit solution yields  (21) = 533.1, an increase of  (1) = 508.1. In sum, whereas these data
1208
are compatible with the family of integrative models, they are not compatible with the family of LS
Testing Heuristic Models 54
1209
1210
models, which cannot account for the pattern SSSR.
Testing Heuristic Models 55
1211
1212
1213
1214
References
Bateson, M., Healy, S. D., & Hurly, A. (2002). Irrational choices in hummingbird foraging
behaviour. Animal Behaviour, 63, 587-596.
Bernoulli, D. (1738). Specimen theoriae novae de mensura sortis. Commentarii Academiae
1215
Scientiarum Imperialis Petropoliannae, 5, 175-192. Translation by L.S. appears as
1216
Bernoulli, D. (1954). Exposition of a new theory on the measurement of risk. Econometrica,
1217
22, 23-36.
1218
1219
1220
1221
1222
1223
1224
Birnbaum, M. H. (1973). The Devil rides again: Correlation as an index of fit. Psychological
Bulletin, 79, 239-242.
Birnbaum, M. H. (1974a). The nonadditivity of personality impressions. Journal of Experimental
Psychology, 102, 543-561.
Birnbaum, M. H. (1974b). Reply to the Devil's advocates: Don't confound model testing and
measurement. Psychological Bulletin, 81, 854-859.
Birnbaum, M. H. (1982). Controversies in psychological measurement. In B. Wegener (Eds.),
1225
Social attitudes and psychophysical measurement (pp. 401-485). Hillsdale, N. J.: Erlbaum.
1226
Birnbaum, M. H. (1997). Violations of monotonicity in judgment and decision making. In A. A. J.
1227
Marley (Eds.), Choice, decision, and measurement: Essays in honor of R. Duncan Luce (pp.
1228
73-100). Mahwah, NJ: Erlbaum.
1229
1230
1231
1232
1233
Birnbaum, M. H. (1999). Testing critical properties of decision making on the Internet.
Psychological Science, 10, 399-407.
Birnbaum, M. H. (2004a). Causes of Allais common consequence paradoxes: An experimental
dissection. Journal of Mathematical Psychology, 48(2), 87-106.
Birnbaum, M. H. (2004b). Tests of rank-dependent utility and cumulative prospect theory in
Testing Heuristic Models 56
1234
gambles represented by natural frequencies: Effects of format, event framing, and branch
1235
splitting. Organizational Behavior and Human Decision Processes, 95, 40-65.
1236
Birnbaum, M. H. (2005a). A comparison of five models that predict violations of first-order
1237
stochastic dominance in risky decision making. Journal of Risk and Uncertainty, 31, 263-
1238
287.
1239
1240
1241
Birnbaum, M. H. (2005b). Three new tests of independence that differentiate models of risky
decision making. Management Science, 51, 1346-1358.
Birnbaum, M. H. (in press-a). Evaluation of the priority heuristic as a descriptive model of risky
1242
decision making: Comment on Brandstätter, Gigerenzer, and Hertwig (2006). Psychological
1243
Review, in press.
1244
1245
1246
1247
1248
Birnbaum, M. H. (in press-b). New paradoxes of risky decision making. Psychological Review, in
press.
Birnbaum, M. H., & Bahra, J. (2007). Gain-loss separability and coalescing in risky decision
making. Management Science, 53, 1016-1028.
Birnbaum, M. H., & Chavez, A. (1997). Tests of theories of decision making: Violations of branch
1249
independence and distribution independence. Organizational Behavior and Human Decision
1250
Processes, 71(2), 161-194.
1251
Birnbaum, M. H., & Gutierrez, R. J. (2007). Testing for intransitivity of preferences predicted by a
1252
lexicographic semiorder. Organizational Behavior and Human Decision Processes, 104, 97-
1253
112.
1254
1255
1256
Birnbaum, M. H., & Jou, J. W. (1990). A theory of comparative response times and "difference"
judgments. Cognitive Psychology, 22, 184-210.
Birnbaum, M. H., & LaCroix, A. R. (in press). Dimension integration: Testing models without
Testing Heuristic Models 57
1257
1258
1259
1260
trade-offs. Organizational Behavior and Human Decision Processes, xx, xx-xx.
Birnbaum, M. H., & McIntosh, W. R. (1996). Violations of branch independence in choices
between gambles. Organizational Behavior and Human Decision Processes, 67, 91-110.
Birnbaum, M. H., & Navarrete, J. B. (1998). Testing descriptive utility theories: Violations of
1261
stochastic dominance and cumulative independence. Journal of Risk and Uncertainty, 17,
1262
49-78.
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
Birnbaum, M. H., & Schmidt, U. (submitted). Empirical tests of intransitivity predicted by models
of risky choice. Submitted for Publication.
Blavatskyy, P. R. (2005). Axiomatization of a preference for most probable winner. Institute for
Empirical Research in Economics, University of Zurich, No. 230.
Bleichrodt, H., & Schmidt, U. (2002). A context-dependent model of the gambling effect.
Management Science, 48, 802-812.
Bleichrodt, H., & Schmidt, U. (2005). Context- and reference-dependent utility. Working paper,
available from U. Schmidt.
Brandstätter, E., Gigerenzer, G., & Hertwig, R. (2006). The priority heuristic: Choices without
tradeoffs. Psychological Review, 113, 409-432.
Busemeyer, J. R., & Townsend, J. T. (1993). Decision Field Theory: A dynamic cognition approach
to decision making. Psychological Review, 100, 432-459.
1275
Edwards, W. (1954). The theory of decision making. Psychological Bulletin, 51, 380-417.
1276
Gigerenzer, G., & Goldstein, D. G. (1996). Reasoning the fast and frugal way: Models of bounded
1277
1278
1279
rationality. Psychological Review, 103, 650-669.
Gigerenzer, G., & Selten, R. (Ed.). (2001). Bounded rationality: The adaptive toolbox. Cambridge,
MA: MIT Press.
Testing Heuristic Models 58
1280
1281
1282
1283
1284
1285
1286
Gigerenzer, G., Todd, P. M., & the ABC Group (1999). Simple heuristics that make us smart. New
York: Oxford University Press.
González-Vallejo, C. (2002). Making trade-offs: A probabilistic and context-sensitive model of
choice behavior. Psychological Review, 109, 137-155.
Iverson, G., & Falmagne, J.-C. (1985). Statistical issues in measurement. Mathematical Social
Sciences, 10, 131-153.
Iverson, G. J., & Myung, J. I. (submitted). Intransitivity of preference: Revisited. Submitted for
1287
publication, Available from Geoffrey J. Iverson, Institute for Mathematical Behavioral
1288
Sciences, UCI, Irvine, CA 92697.
1289
1290
1291
Johnson, J. G., & Busemeyer, J. R. (2005). A dynamic, computational model of preference reversal
phenomena. Psychological Review, 112, 841-861.
Kagel, J. H., MacDonald, D. N., & Battalio, R. C. (1990). Tests of "fanning out" of indifference
1292
curves: Results from animal and human experiments. American Economic Review, 80, 912-
1293
921.
1294
1295
1296
1297
1298
1299
1300
1301
1302
Kahneman, D., & Tversky, A. (1979). Prospect theory: An analysis of decision under risk.
Econometrica, 47, 263-291.
Leland, J., W. (1994). Generalized similarity judgments: An alternative explanation for choice
anomalies. Journal of Risk and Uncertainty, 9, 151-172.
Leland, J. W. (1998). Similarity judgments in choice under uncertainty: A re-interpretation of the
predictions of regret theory. Management Science, 44, 659-672.
Link, S. W. (1992). The wave theory of difference and similarity. Hillsdale, NJ: Lawrence Erlbaum
Associates.
Lopes, L. L., & Oden, G. C. (1999). The role of aspiration level in risky choice: A comparison of
Testing Heuristic Models 59
1303
cumulative prospect theory and SP/A theory. Journal of Mathematical Psychology, 43, 286-
1304
313.
1305
1306
1307
1308
Loomes, G., Starmer, C., & Sugden, R. (1991). Observing violations of transitivity by experimental
methods. Econometrica, 59, 425-440.
Loomes, G., & Sugden, R. (1982). Regret theory: an alternative theory of rational choice under
uncertainty. Economic Journal, 92, 805-24.
1309
Luce, R. D. (1956). Semiorders and a theory of utility discrimination. Econometrica, 24, 178-191.
1310
Luce, R. D. (2000). Utility of gains and losses: Measurement-theoretical and experimental
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
approaches. Mahwah, NJ: Lawrence Erlbaum Associates.
Luce, R. D., & Fishburn, P. C. (1991). Rank- and sign-dependent linear utility models for finite first
order gambles. Journal of Risk and Uncertainty, 4, 29-59.
Luce, R. D., & Fishburn, P. C. (1995). A note on deriving rank-dependent utility using additive
joint receipts. Journal of Risk and Uncertainty, 11, 5-16.
Marley, A. A. J., & Luce, R. D. (2001). Rank-weighted utilities and qualitative convolution. Journal
of Risk and Uncertainty, 23(2), 135-163.
Marley, A. A. J., & Luce, R. D. (2005). Independence properties vis-à-vis several utility
representations. Theory and Decision, 58, 77-143.
May, K. O. (1954). Intransitivity, utility, and the aggregation of preference patterns. Econometrica,
22, 1-13.
Navarro, A. D., & Fantino, E. (2005). The sunk cost effect in pigeons and humans. Journal of the
Experimental Analysis of Behavior, 83, 1-13.
Quiggin, J. (1993). Generalized expected utility theory: The rank-dependent model. Boston:
Kluwer.
Testing Heuristic Models 60
1326
1327
1328
1329
1330
1331
1332
1333
1334
Regenwetter, M., Stober, C. D., Dana, J., & Kim, A. (2006). Tversky's intransitivity of preference
revisited. Paper presented to Society for Mathematical Psychology, Vancouver, July, 2006.
Rieskamp, J. (2007). The probabilistic nature of preferential choice. Working Paper, Max Planck
Institue for Human Development.
Rieskamp, J., Busemeyer, J. R., & Mellers, B. A. (2006). Extending the bounds of rationality: A
review of research on preferential choice. Journal of Economic Literature, 35, 631-661.
Rubinstein, A. (1988). Similarity and decision-making under risk (Is there a utility theory resolution
to the Allais paradox?). Journal of Economic Theory, 46, 145-153.
Sopher, B., & Gigliotti, G. (1993). Intransitive cycles: Rational Choice or random error? An answer
1335
based on estimation of error rates with experimental data. Theory and Decision, 35, 311-
1336
336.
1337
1338
1339
Starmer, C. (2000). Developments in non-expected utility theory: The hunt for a descriptive theory
of choice under risk. Journal of Economic Literature, 38, 332-382.
Stevenson, M. K., Busemeyer, J. R., & Naylor, J. C. (1991). Judgment and decision-making theory.
1340
In M. Dunnette & L. M. Hough (Eds.), New handbook of industrial-organizational
1341
psychology (pp. 283-374). Palo Alto, CA: Consulting Psychologist Press.
1342
Tversky, A. (1969). Intransitivity of preferences. Psychological Review, 76, 31-48.
1343
Tversky, A., & Kahneman, D. (1992). Advances in prospect theory: Cumulative representation of
1344
1345
1346
1347
1348
uncertainty. Journal of Risk and Uncertainty, 5, 297-323.
Weber, B. J. (2007). The effects of losses and event splitting on the Allais paradox. Judgment and
Decision Making, 2, 115-125.
Wu, G., Zhang, J., & Gonzalez, R. (2004). Decision under risk. In D. Koehler & N. Harvey (Eds.),
Blackwell Handbook of Judgment and Decision Making (pp. 399-423). Oxford: Oxford
Testing Heuristic Models 61
1349
1350
1351
1352
University Press.
Zhang, J., Hsee, C. K., & Xiao, Z. (2006). The majority rule in individual decision making.
Organizational Behavior and Human Decision Processes, 99, 102-111.
Testing Heuristic Models 63
1353
Table 1. Classification of models with respect to testable critical properties.
Testable Property
Model
Priority
Attribute
Attribute
Dominance
Integration
Interaction
No
Yes
Yes
Yes
ODH
Yes
No
No
Yes
MPW, MR, RT
No
Yes
Yes
No
AD, SDM
No
Yes
No
No
PH, LS
Yes
No*
No*
No
EU, CPT, TAX,
Transitivity
GDU
1354
Notes: EU = Expected utility theory; CPT = cumulative prospect theory, TAX = transfer of attention exchange model; GDU = gains
1355
decomposition utility; ODH = one attribute heuristic; AD = Additive differences model; SDM = Stochastic Difference Model; MPW = most
1356
probable winner, MR = majority rule; RT = regret theory; PH = priority heuristic; LS = lexicographic semiorder. *The PH allows integration
1357
and interaction between lowest and highest consequences when the maximal consequence is varied such that it rounds to different prominent
1358
numbers; however, with the highest consequence fixed, PH agrees with the LS model.
Testing Heuristic Models 64
1359
Table 2. Tests of priority dominance (Study 1). TAX = predicted choices using prior parameters; “priority order” show predicted preferences
1360
under LS models; S = “safe” gamble; R = “risky” gamble. Last three rows show numbers of people (out of 238) showing each pattern.
Choice
No.
Choice
LPH-1
LHP-1,4
LPH-2
LHP2,4
LHP-1,3
LHP-1,5
LHP-2,3
LHP-2,5
PLH-1
PLH-2
PHL-3
PHL-4
PHL-5
HPL-3
HLP-2,3
HPL-4
HLP-1,4
HLP-1,5
HPL5
HLP-2,4
S
S
R
S
R*
S
R
S
R
S
S*
R
S
R
S
R
R
R
R
R
R
R
S
S
S
R
R
S
S
S
R
S
R
R
R
R
R
R
S
S
S
R
R
R
R
R
R
R
R
R
R
R
R
R
R
S
S
S
S
S
S
S
S
R
R
R
S
S
R
S
R
S
R
R
R
R
R
R
S
R
S
R
S
R
R
R
R
R
Number who showed pattern on First
Replicate
119
6
2
5
0
0
0
2
1
0
5
Number who showed pattern on Second
Replicate
119
10
1
5
2
0
1
0
1
1
3
96
4
0
3
0
0
0
0
0
0
1
5, 13
9, 17
10, 18
6, 14
7, 19
11, 15
8, 20
12, 16
Risky Gamble (R)
Safe Gamble (S)
10 to win $100
10 to win $98
90 to win $0
90 to win $20
99 to win $100
10 to win $22
01 to win $0
90 to win $20
10 to win $100
90 to win $90
90 to win $0
10 to win $0
10 to win $100
90 to win $60
90 to win $55
10 to win $0
90 to win $100
90 to win $60
10 to win $20
10 to win $22
01 to win $100
99 to win $60
99 to win $0
01 to win $55
01 to win $100
01 to win $22
99 to win $0
99 to win $20
99 to win $100
99 to win $22
01 to win $0
01 to win $20
Number who showed pattern on both
replicates.
1361
1362
1363
Priority Order and Threshold Parameters
TAX
1 $2  L  $20 ; 2 $20  L  $55 ; 3 $2  H  $10 ; 4 $10  H  $40 ; 5 $2  H  $10 ; all models assume 0  P  0.8. * HLP-2,4
model is same as HPL-5, except indecisive on first choice. * LHP-2,5 is same as LHP 2, 3, except it is indecisive on first choice.
Testing Heuristic Models 65
1364
Table 3. Estimation of “true” and “error” probabilities from replications in tests of priority dominance. TAX shows predictions based on
1365
prior parameters. RR = risky gamble chosen on both replicates; RS = risky gamble (R) preferred on the first replicate and safe (S) on the
1366
second, etc. Data Patterns show number of people (out of 238) who had each data pattern. Model fit shows estimated true probability of
1367
2
2
choosing the “safe” gamble ( pˆ ), error rate ( eˆ ), and test of fit (  (1) ); last column shows the  (1) test of independence of first and second
1368
replicates.
Choice
No.
5, 13
9, 17
10, 18
6, 14
7, 19
11, 15
8, 20
12, 16
1369
Risky Gamble (R)
Data Pattern
Safe Gamble (S)
10 to win $100
10 to win $98
90 to win $0
90 to win $20
99 to win $100
10 to win $22
01 to win $0
90 to win $20
10 to win $100
90 to win $90
90 to win $0
10 to win $0
10 to win $100
90 to win $60
90 to win $55
10 to win $0
90 to win $100
90 to win $60
10 to win $20
10 to win $22
01 to win $100
99 to win $60
99 to win $0
01 to win $55
01 to win $100
01 to win $22
99 to win $0
99 to win $20
99 to win $100
99 to win $22
01 to win $0
01 to win $20
TAX
RR
RS
SR
 2 (1)
Model Fit
SS
pˆ
eˆ
 2 (1)
indep
S
16
11
16
195
0.93
0.06
0.92
54.93
R
175
18
24
21
0.10
0.10
0.85
37.13
S
30
24
18
166
0.86
0.10
0.85
54.33
R
176
27
21
14
0.06
0.11
0.75
14.92
R
195
15
17
11
0.05
0.07
0.12
26.23
S
14
16
10
198
0.94
0.06
1.37
50.67
S
29
18
22
169
0.86
0.09
0.40
56.42
R
183
17
13
25
0.12
0.07
0.53
72.12
Testing Heuristic Models 66
1370
Table 4. Test of Attribute Integration (Study 2, n = 242). Lowest consequences and probability are manipulated. Nine variants of
1371
Lexicographic Semiorder model make four different patterns of predictions. None of these match the pattern predicted by the TAX model
1372
with prior parameters, RRRS.
Choice
No.
Risky (R)
Choice Models
Safe (S)
%S
TAX
LPH
a
7
18
11
8
90 to win $100
50 to win $51
10 to win $50
50 to win $50
10 to win $100
50 to win $51
90 to win $50
50 to win $50
90 to win $100
50 to win $51
10 to win $0
50 to win $10
10 to win $100
50 to win $51
90 to win $0
50 to win $10
Number of participants who showed the
LHP
PHL,
HLP,
LS Mixture
PLH
HPL
(Prob choosing
b
LPH1, c
LHP1
S)
11
R
R
R
R
R
0
33
R
S
R
S
R
ac
22
R
S
S
R
R
ab
79
S
S
S
S
R
abc
99
8
21
44
27
pattern
1373
Notes: LPH refers to the priority heuristic which considers first the lowest consequence (L), then the probability (P), then highest
1374
consequence (H). In this case, it agrees with the LPH lexicographic semiorder in which 0  L  $10 , $0  H  $49 , and 0  P  0.4 .
1375
LPH1 and LHP1 are the same lexicographic semiorders as LPH and LHP, respectively, except with  L  $10 . For the LS mixture model, a
1376
= the probability of LPH, b = probability of LHP, c = probability of PHL, LPH1, or PLH. Ten people showed the pattern SSSS; no other
1377
pattern had more than 8.
1378
Testing Heuristic Models 67
1379
Table 5. Test of Attribute Integration/Interaction, manipulating the probability to win higher consequence and value of the highest
1380
consequence. The contrast in lowest consequences is fixed. (Study 2, n = 242)
Choice
No.
Risky (R)
% Safe
TAX
Priority Order Model
LPH,
Safe (S)
HPL, HLP,
LHP, PLH
PHL
17
9
4
10 to win $26
10 to win $25
90 to win $0
90 to win $20
10 to win $100
10 to win $25
90 to win $0
90 to win $20
99 to win $26
99 to win $25
01 to win $0
01 to win $20
99 to win $100
99 to win $25
01 to win $0
01 to win $20
LS
HPL1, HLP1
Mixture
Model
a
13
PHL1, LPH1,
c
90
S
S
S
R
ac
75
S
S
R
R
a
71
S
S
S
R
ac
14
R
S
R
R
a
115
16
25
6
Number of Participants who showed the predicted
patterns
1381
Notes: LPH, LHP, and PLH assume that 0  L  $20 ; HPL, HLP and PHL assume $1  H  $75; PHL1, HLP1, HPL1 assume
1382
0  H  $1 ; LPH1 assumes 0  H  $1 and  L  $20 , LPH2 and LHP2 assume  L  $20 and $1  H  $75; these last two are
1383
indecisive on the first and third choices. According to the mixture model, we can estimate c in two ways: the difference between the first
1384
two or last two choice percentages. cˆ  90 – 75 = 15, and cˆ  71 – 14 = 57. These figures are significantly different, in contradiction of the
1385
model. Note as well that the choice percentage in Choice 17 is 75%, significantly greater than 50% and the corresponding percentage in
1386
Choice 4 should be equal to it, but is only 14%, significantly lower than 50%.
Testing Heuristic Models 68
1387
Table 6. Test of Attribute Integration, manipulating highest consequence and probability to win in Risky gamble. (n = 242)
Choice
No.
5
14
3
10
Risky (R)
Models
Safe (S)
10 to win $50
50 to win $50
90 to win $0
50 to win $49
10 to win $100
50 to win $50
90 to win $0
50 to win $49
90 to win $50
50 to win $50
10 to win $0
50 to win $49
90 to win $100
50 to win $50
10 to win $0
50 to win $49
%S
TAX
LPH
LPH1
HLP
HPL
LHP
PHL
LPH2
PLH
PHL2
a
b
c
d
LS Mixture
91
S
S
S
S
S
a b cd
83
S
S
S
R
R
ab
81
S
S
R
S
R
ac
35
R
S
R
R
R
a
104
62
25
15
6
Number of participants who showed each pattern
1388
Notes: LPH and LHP assume 0  L  $49 ; PHL, PLH assume 0  P  0.4 ; LPH1 assumes  L  $49 and 0  P  0.4 ; HPL assumes
1389
0  H  $50 and 0  P  0.4 ; LPH2 assumes  L  $49 and  P  0.4; PHL2 assumes  P  0.4 and 0  H  $50 ; HLP assumes
1390
0  H  $50 and 0  L  $49 .
1391
Testing Heuristic Models 69
1392
Table 7. Test of Attribute Integration, manipulating lowest and highest consequences in “risky” gamble. Probability is fixed, as is the “safe”
1393
gamble. (n = 242)
Choice
Risky (R)
Lexicographic Semiorders
Safe (S)
% S
TAX
LPH
LPH1
HLP
HLP1
LHP
LHP1
HPL
HPL1
PLH
PLH1
PHL
PHL1
b
c
a
15
1
6
12
50 to win $51
50 to win $50
50 to win $0
50 to win $40
50 to win $100
50 to win $50
50 to win $0
50 to win $40
50 to win $51
50 to win $50
50 to win $25
50 to win $40
50 to win $100
50 to win $50
50 to win $25
50 to win $40
LS
Mixture
Model
94
S
S
S
S
R
abc
56
S
S
S
R
R
ab
91
S
S
R
S
R
ac
19
R
S
R
R
R
a
92
25
11
80
2
Number of participants who showed each data pattern
1394
LPH, LHP and PLH assume 0  L  $15 ; LPH1, PLH1. LHP1 assume $15   L  $40 and 0  H  $1 ; HLP assumes $1  H  $50 and
1395
0  L  $15 ; PHL and HPL assume $1  H  $50; HLP1, HPL1, and PHL1 assume 0  H  $1 . According to the LS mixture model,
1396
aˆ  19; cˆ = 91 – 19 = 72; bˆ  56 – 19 = 37, so aˆ  bˆ  cˆ  19 + 72 + 37 = 128, which exceeds 100% and far exceeds the first choice
1397
percentage, 94%. Thus, these data also do not fit the LS Mixture model.
Testing Heuristic Models 70
1398
Table 8. Test of Attribute Integration in Study 4 (n = 266). Probability by Prize manipulated in Risky Gamble. The “safe” gamble is always
1399
the same.
Choice Percentages
Choice
No.
Choice
Risky (R)
16
13
5
9
Safe (S)
10 to win $45
60 to win $40
90 to win $0
40 to win $38
10 to win $100
60 to win $40
90 to win $0
40 to win $38
90 to win $45
60 to win $40
10 to win $0
40 to win $38
90 to win $100
60 to win $40
10 to win $0
40 to win $38
Series
A
Series B
(reflected)
%S
%S
89
93
77
81
72
69
27
28
Models
TAX
LPH
LHP
a
PLH
PHL
LPH1
b
S
S
S
HLP
HPL
HLP1
HPL1
c
d
S
S
S
R
S
S
R
R
R
S
S
R
S
R
R
R
S
R
R
R
R
Number of participants showing predicted pattern in Series A
93
47
48
26
7
7
Number of participants showing predicted pattern in Series B
99
45
52
25
2
5
Number of participants showing predicted pattern in both A and B
52
26
18
14
0
1
Mixture
Model
a b cd
ab
ac
a
1400
Series B (Choices 17, 10, 7, and 11) was the same as Series A, except $100, $45, $40, $38, and $0 were replaced by $98, $43, $42, $40, and
1401
$0, respectively, and the positions (first or second) of “risky” and “safe” gambles were reversed. According to the mixture model, aˆ  27,
1402
bˆ  77 – 27 = 50, cˆ  72 – 27 = 45; these estimates yield dˆ  89 – 27 – 50 – 45 = -33, which is negative and therefore contradicts the
1403
mixture of lexicographic semiorders model. LPH and LHP assume 0  L  $38 ; PLH and PHL assume 0  P  0.3, LPH1 assumes
1404
 L  $38 and 0  P  0.3; HLP assumes $5   H  $60 and 0  L  $38 , HPL assumes $5   H  $60 and 0  P  0.3; HLP1 and
1405
HPL1 assume  H  $5 . Priority heuristic implies that all choice percentages should exceed 50%.
1406
Testing Heuristic Models 71
1407
Table 9. Tests of attribute interaction in Series B of Study 3 (n = 153). TAX shows predicted certainty equivalents of risky and safe
1408
gambles based on prior parameters. These predict the pattern SSRRR.
Choice
Choice
TAX
No.
Risky (R)
11
17
21
7
3
Safe (S)
01 to win $95
01 to win $55
99 to win $5
99 to win $20
10 to win $95
10 to win $55
90 to win $5
90 to win $20
50 to win $95
50 to win $55
50 to win $5
50 to win $20
90 to win $95
90 to win $55
10 to win $5
10 to win $20
99 to win $95
99 to win $55
01 to win $5
01 to win $20
%S
R
LPH, LHP,
LPH1, LHP1,
PLH, PHL,
PLH1, PHL1,
HPL, HLP,
HPL1, HLP1
S
83 7.3
20.9
S
R
71 15.6
24.1
S
R
49 35
31.7
S
R
22 54.4
39.2
S
R
17 62.7
42.4
S
R
1409
In LPH, LHP, PLH, 0  L  $15 ; In LPH1, LHP1, PHL1,  L  $15 . In PHL, HPL, HLP,  H  $40 ; in PHL1, HPL1, and HLP1,
1410
0  H  $40 .EV(R) #17 = 14, EV(S) = 23.5 EV(R) = 86, EV(S) = 51.5
Testing Heuristic Models 72
1411
Table 10. Frequency of response patterns (Study 3, n = 153). Within each repetition, there are 32 possible patterns (1024 counting
1412
replications). The six patterns listed are consistent with interactive utility models; in fact, no one repeated any other pattern. Only SSSSS and
1413
RRRRR are consistent, with the class of priority heuristic models; only SSSSS is consistent with the LPH model of Brandstätter, et al. (2006).
1414
Totals sum to 150 because 3 of 153 participants skipped at least one of these items.
Choice
Pattern
1415
1416
Series A
Rep 1
Rep 2
Series B
Both
Rep 1
Rep 2
Both
SSSSS
10
10
3
14
12
4
SSSSR
3
6
0
7
14
2
SSSRR
11
12
2
39
38
21
SSRRR
50
50
31
35
30
17
SRRRR
18
16
5
14
16
5
RRRRR
22
27
15
15
13
9
Others
36
29
0
26
27
0
Total
150
150
56
150
150
58
Testing Heuristic Models 73
1417
Table 11. Test of transitivity in Study 4 (n = 266). According to the priority heuristic, majority choices should violate transitivity.
Choices in Series C
Choice
First (F)
Models
Second (S)
No.
6
18
14
%
TAX
LPH
Second
70 to win $100
78 to win $50
30 to win $0
22 to win $0
78 to win $50
85 to win $25
22 to win $0
15 to win $0
85 to win $25
70 to win $100
15 to win $0
30 to win $0
16
F
F
14
F
F
84
S
F
1418
Priority heuristic assumes  P  0.1. Series D, Choices 15, 12, and 8 were the same as Series C (#6, 18, and 14, respectively), except $100,
1419
$50, and $25 were replaced with $98, $52, and $26, respectively, and the positions of the gambles (first or second) were counterbalanced.
1420
The corresponding (reflected) choice percentages are 20, 19, and 86%, respectively.
1421
Testing Heuristic Models 74
1422
Table 12. Analysis of response patterns in test of transitivity in Study 4.
1423
Choice Pattern
Frequency of Choice Patterns
#6
#18
#14
15
12
8
F
F
F
18
9
0
F
F
S
184
177
149
F
S
F
10
11
1
F
S
S
12
17
0
S
F
F
8
8
1
S
F
S
18
22
2
S
S
F
7
9
3
S
S
S
9
13
2
266
266
158
Column Totals
1424
1425
Series C
Series D
Both
(reflected)
Testing Heuristic Models 75
1426
Table 13. Test of Transitivity (n = 242) included in Study 2.
1427
Choice
First (F)
Second (S)
TAX
No.
2
A: 60 to win $60
B: 67 to win $40
33 to win $0
19
C: 73 to win $20
27 to win $0
1428
1429
% S
heuristic
40 to win $0
16
Priority
B: 67 to win $40
F
F
17
F
F
16
S
F
70
33 to win $0
C: 73 to win $20
27 to win $0
A: 60 to win $60
40 to win $0
Testing Heuristic Models 76
Table 14. Test of Integration. Levels are selected such that cG  cF , aF  aG , aF  aG, bF  bG , and
bS bR. To devise a test, one selects levels intended to make G preferred in the first choice and
  violates integrative independence,
Fpreferred in the fourth choice. The choice pattern, GGGF
which refutes lexicographic semiorder models, as shown in Table 15.
Choice
Gamble, G
Gamble, F
1
G  (aG , bG , cG )
F  (aF , bF , cF )
2
G (aG,bG ,cG )
F ( a
F , bF , c F )
3
G (aG , b
G , cG )
F (aF, b
F , cF )
4
G (a
, cG )
G , bG
F  (aF, bF, cF)
Testing Heuristic Models 77
Table 15. Response patterns implied by lexicographic semiorders under different priority orders and with
different assumptions concerning spacing of levels relative to the thresholds for the choices in Table 14. It
is assumed that levels have been selected such that u(a F )  u(aG )   A and v(bF )  v(bG )  B . The term
“Yes” in the first column indicates that u(a 
F )  u(a
G )   A ; “No” indicates that it is not; similarly, “Yes”
in the second column means v(b
)   B ; “Yes” in the third column indicates that t(cG )  t(cF ) 
F )  v( bG
 , GFGF
 ,
 C . Only five patterns are consistent with these assumptions: FFFF, GFFF
GGFF
 , or GGGG
 . The pattern GGGF
 is not compatible with any of these 48 lexicographic
semiorders.
Parameter
Priority Order
assumptions
A
B
C
ABC
ACB
No
No
No
GGGG
 
 
FFFF GGGG
FFFF FFFF FFFF
No
No
Yes
GGGG
 
GGGG
 
GGGG
 
No
Yes
No
GGFF
 
  FFFF FFFF FFFF
FFFF GGFF
No
Yes
Yes
GGFF
  GGGG
 
Yes
No
No
GFGF
 
Yes
No
Yes
GFGF
  GFGF
  GFGF
  GGGG
 
Yes
Yes
No
GFFF
 
Yes
Yes
Yes
GFFF
  GFGF
  GGFF
  GFFF
  GGGG
  GGGG
 
BAC
GGGG
 
BCA
CAB
GGGG
 
GGFF
  GGFF
  GGGG
 
CBA
GGGG
 
GGGG
 
  FFFF FFFF FFFF
FFFF GFGF
 
FFFF GFFF
GGGG
 
GGGG
 
FFFF FFFF FFFF
Testing Heuristic Models 78
Table 16. True and Error Model Analysis of Attribute Integration in a replicated design (n = 266).
Replicate 1 used Choices 16, 13, 9, and 5; Replicate 2 used Choices 11, 7, 10, and 17, which reversed
positions of risky and safe gambles. Entries shown in bold are consistent with integrative models such as
TAX. Estimated error terms are (eˆ1 , eˆ2 , eˆ3 , eˆ4 )  (0.05, 0.09, 0.21, 0.15), respectively. Pattern SSSR is
inconsistent with family of LS models.
Response Pattern
RRRR
RRRS
RRSR
RRSS
RSRR
RSRS
RSSR
RSSS
SRRR
SRRS
SRSR
SRSS
SSRR
SSRS
SSSR
SSSS
Number who show each pattern
Estimated
Replicate 1
probability
Replicate 2
Both replicates
7
5
1
1
1
0
5
4
0
4
1
0
3
1
0
1
2
0
5
3
0
2
1
0
7
2
0
2
7
0
26
25
14
9
5
1
48
52
18
6
13
1
93
99
52
47
45
26
0.05
0
0
0
0
0
0
0
0.00
0
0.12
0
0.14
0
0.48
0.21
Testing Heuristic Models 79
Figure 1. Cash equivalent values of binary gambles, win x with probability p, otherwise $0, plotted as a
function of x with a separate curve for each p. According to the priority heuristic, the three curves should
coincide, because probability should have no effect. In addition, the (single) curve should be horizontal
for the first three cash values, since they all round to the same prominent number, $100.