Testing lexicographic semiorders as models of decision making: Priority dominance, integration, interaction, and transitivity

Three new properties are devised to test a family of lexicographic semiorder models of risky decision making. Lexicographic semiorder models imply priority dominance, the principle that when an attribute with priority determines a choice, no variation of other attributes can overcome that preference. Attribute integration tests whether two changes in attributes that are too small, individually, to be decisive can combine to reverse a preference. Attribute interaction tests whether preference due to a given contrast can be reversed by changing an attribute that is the same in both alternatives. These three properties, combined with the property of transitivity, allow us to compare four classes of models. Four new studies show that priority dominance is systematically violated, that most people integrate attributes, and that most people show interactions between probability and consequences. In addition, very few people show the pattern of intransitivity predicted by the priority heuristic, which is a variant of a lexicographic semiorder model with additional features chosen to reproduce certain previous data. When individual data are analyzed in these three new tests, it is found that few people exhibit data compatible with any of the lexicographic semiorder models. The most frequent patterns of individual data are those implied by Birnbaum’s transfer of attentionmodel with parameters used in previous research. These results show that the family of lexicographic semiorders is not a good description of how people make risky decisions. © 2010 Elsevier Inc. All rights reserved. Risky decision making involves making choices between gambles such as the following: Would you rather have gamble A with 0.5 probability to win $40 and 0.5 probability to win $30, or would you prefer gamble B, in which you win $100 with probability 0.5 and win nothing with probability 0.5? Some people prefer A, which guarantees at least $30; whereas other people prefer B, which has the higher expected value. Models of risky decision making attempt to describe and to predict such decisions. The features that differ between gambles such as the probability to win the highest prize, the value of the highest prize, and the value of the lowest prize are called ‘‘attributes’’ of a gamble. We denote gambles A and B in terms of their attributes as follows: A = ($40, 0.5; $30, 0.5) and B = ($100, 0.5; $0, 0.5). The purpose of this paper is to present empirical tests of critical properties that can be used to testmodels of risky decisionmaking. Critical properties are theorems that can be deduced from one theory but which are systematically violated by at least one rival theory. Three new critical tests are proposed here and evaluated empirically in order to compare a class of lexicographic semiorder models against alternative models. The property of transitivity E-mail address:[email protected]. of preference, which also distinguishes classes of models, is also tested. Three new properties are described that are implied by lexicographic semiordermodels that are violated by other decision making models. Priority dominance is the assumption that if a person makes a decision based on a dimension with priority, then no variation of other attributes of the gambles should reverse that decision. Integrative independence is the assumption that two changes in attributes that are not strong enough separately to reverse a decision cannot combine to reverse a decision. Interactive independence is the assumption that any attribute that is the same in both gambles of a choice can be changed (to another common value) without changing the preference between the gambles. If lexicographic semiorder models are descriptive of how people make decisions, we should not expect violations of these properties, except due to random error. Lexicographic semiorder models can violate transitivity, which is implied by many other models. Let A ! B represent systematic preference for gamble A over gamble B. Transitivity is the assumption that if A ! B and B ! C , then A ! C . The rest of this paper is organized as follows. The next section describes four classes of decision making models that can be evaluated by testing these four properties. In addition, two specific models are described including parameters chosen from previous research that will be used to make predictions to the new 0022-2496/$ – see front matter© 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jmp.2010.03.002 Author's personal copy 364 M.H. Birnbaum / Journal of Mathematical Psychology 54 (2010) 363–386 experiments. The section titled ‘‘New Diagnostic Tests’’ presents the properties to be tested in greater detail. The ‘‘Predictions’’ section presents a summary of the predicted outcomes of the tests according to the classes of models. This section also includes a description of the model of random error that is used to evaluate whether or not a given pattern of observed violations is systematic or can be attributed instead to random variability. The next four sections present experimental tests of priority dominance, attribute integration, attribute interaction, and transitivity, respectively. The results show systematic violations of the three properties implied by lexicographic semiorders that were predicted in advance using a model and parameters taken from previous research. The fourth study searched for violations of transitivity predicted by a specific lexicographic semiorder model; the data failed to confirm predictions of that model. The discussion summarizes the case against lexicographic semiorders as descriptive models of risky decision making, including related findings in the literature. Additional technical details concerning the test of integrative independence, proofs, true and error model, and supplementary statistical results are presented in Appendices. 1. Four classes of decision models Themost popular theories of choice between risky or uncertain alternatives are models that assign a computed value (utility) to each alternative and assume that people prefer (or at least tend to prefer) the alternative with the higher utility (Luce, 2000; Starmer, 2000;Wu, Zhang, & Gonzalez, 2004). These models imply transitivity of preference. The class of transitive utility models includes Bernoulli’s (1738/1954) expected utility (EU), Edwards’ (1954) subjectively weighted utility (SWU), Quiggin’s (1993) rank-dependent utility (RDU), Luce and Fishburn’s (1991; 1995) rank-and signdependent utility (RSDU), Tversky and Kahneman’s (1992) cumulative prospect theory (CPT), Birnbaum’s (1997) rank affected multiplicative weights (RAM), Birnbaum’s (1999; Birnbaum& Chavez, 1997) transfer of attention exchange (TAX)model, Marley and Luce’s (2001; 2005) gains decomposition utility (GDU), Busemeyer and Townsend’s (1993) decision field theory (DFT), and others. These models can be compared to each other by testing ‘‘new paradoxes’’, which are critical properties that must be satisfied by proper subsets of the models (Birnbaum, 1999, 2004a,b, 2008b; Marley & Luce, 2005). Although these transitive utility models can be tested against each other, they all have in common that there is a single, integrated value or utility for each gamble, that these utilities are compared, and people tend to choose the gamble with the higher utility. For the purpose of this paper, these models are all in the same class. Let U(A) represent the utility of gamble A. All members of this class of models assume, A ! B⇔ U(A) > U(B), (1) where ! denotes the preference relation. Aside from ‘‘random error,’’ these models imply transitivity of preference, because A ! B ⇔ U(A) > U(B) and B ! C ⇔ U(B) > U(C) ⇒ U(A) > U(C) ⇔ A ! C . That is, because these models represent utilities of gambles with numbers and because numbers are transitive, it follows that preferences are transitive. Let A = (x, p; y, 1− p) represent a ranked, two-branch gamble with probability p to win x, and otherwise receive y, where x > y ≥ 0. This binary gamble can also be written as A = (x, p; y). The two branches are probability-consequence pairs that are distinct in the gamble’s presentation: (x, p) and (y, 1− p). 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) where u(x) and u(y) are the utilities of consequences x and y. In expected utility, the weights of the consequences are simply the probabilities of receiving those consequences. A transitive model that has been shown to be a more accurate description of risky decision making than EU or CPT is Birnbaum’s (1999) special transfer of attention exchange (TAX) model. Birnbaum (2008b) has shown that this model correctly predicts data that refute CPT. This model also represents the utility of a gamble as a weighted average of the utilities of the consequences, butweight in thismodel depends on the probabilities of the branch consequences and ranks of branch consequences in the gamble. In this model, people assign attention to each branch as a function of its probability, but weight is transferred among ranked branches, according to the participant’s point of view. A person who is riskaverse may transfer weight from the branch leading to the highest consequence to the branch with the lowest consequence, whereas one who is risk-seeking may transfer attention (and weight) to higher-valued branches. This model can be written for two branch gambles (when ω > 0) as follows: TAX(A) = au(x) + bu(y) a + b (3) where a = t(p)−ωt(p), b = t(q)+ωt(p), and q = 1−p. Intuitively, when the parameterω > 0 there is a transfer of attention from the branch leading to the best consequence to the branch leading to the worst consequence. This parameter can produce risk aversion evenwhen u(x) = x. In the casewhereω < 0,weight is transferred from lower-valued branches to higher ones; in this case, a = t(p)− ωt(q) and b = t(q)+ωt(q). The formulas for three-branch gambles include weight transfers among all pairs of branches (Birnbaum, 2004a, 2008b; Birnbaum & Navarrete, 1998). The special TAX model was fit to data of individuals with the assumptions that t(p) = p and u(x) = x ; for example, Birnbaum and Navarrete (1998) reported median best-fit parameters as follows: β = 0.41, γ = 0.79, and ω = 0.32. For the purpose of making ‘‘prior’’ predictions for TAX in this article, a still simpler version of the TAX model is used. Let u(x) = x for 0 < x < $150; t(p) = p0.7, and ω = 1/3. These functions and parameters, which approximate certain group data for gambles with small positive consequences, are called the ‘‘prior’’ parameters, because they have been used in previous studies to predict new datawith similar participants, contexts, and procedures. They have had some success predicting results with American undergraduates who choose among gambles with small prizes (e.g. Birnbaum, 2004a, 2008b). Use of such parameters to predict modal patterns of data should not be taken to mean that everyone is assumed to have the same parameters. Themain use of these prior parameters is for the purpose of designing new studies that are likely to find violations of rival models. Several papers have shown that the TAXmodel ismore accurate in predicting choices among risky gambles than CPT or RDU (Birnbaum, 1999, 2004a,b, 2005a,b; Birnbaum, 2006; Birnbaum, 2007; Birnbaum, 2008b; Birnbaum & Chavez, 1997; Birnbaum & McIntosh, 1996; Birnbaum & Navarrete, 1998; Weber, 2007). For example, CPT and RDU models imply first order stochastic dominance, but the TAX model does not. Experiments designed to test stochastic dominance have found violations where they are predicted to occur based on the TAX model with its prior parameters (e.g. Birnbaum, 2004a, 2005a). Similarly, TAX correctly predicted other violations of this class of rank dependent models (Birnbaum, 2004a,b, 2008b). This model has also been extended to make predictions for choice response times as well as choice Author's personal copy M.H. Birnbaum / Journal of Mathematical Psychology 54 (2010) 363–386 365 probabilities and judgments (Birnbaum & Jou, 1990; Johnson & Busemeyer, 2005). For the purpose of this paper, however, TAX, CPT, RDU, GDU, EU, DFT and other theories in this family are all in the same category and (given selected parameters) can be virtually identical. These models are all transitive, all imply that attributes are integrated, and they all imply interactions between probability and consequences. The experiments in this article will test properties that these integrativemodels share in common against predictions of other families of models that disagree. Therefore, it should be kept in mind that what is said about TAX in this paper applies as well to other models in its class. A second class includes non-integrative but transitive models. For example, suppose people compared gambles by just one attribute (for example, by comparing their worst consequences). If so, they would satisfy transitivity, but no change in the other attributes could overcome a difference due to the one attribute people use to choose. A third class of theories assumes that people integrate contrasts (differences between the alternatives) but can violate transitivity (González-Vallejo, 2002). For the purpose of this paper, this class of theories will be described as integrative contrast models, since they involve contrasts and aggregation of the values of these contrasts. An example of such a model is the additive contrastsmodel: