Sequentially Rationalizable Choice By Paola Manzini and Marco Mariotti* A sequentially rationalizable choice function is a choice function that can be retrieved by applying sequentially to each choice problem the same fixed set of asymmetric binary relations (rationales) to remove inferior alternatives. These con- cepts translate into economic language some human choice heuristics studied in psychology and explain cyclical patterns of choice observed in experiments. We study some properties of sequential rationalizability and provide a full character- ization of choice functions rationalizable by two and three rationales. (JEL D0). Cyclical choice is persistently observed in a maximizer for it. In this paper, we propose experimental evidence. It typically occurs and study a family of boundedly rational choice in simple decision problems (involving only procedures that can account for these observed binary comparisons and few alternatives) and anomalies. in significant proportions, sometimes nearing In line with some prominent psychology and or even exceeding 50 percent. This is obviously marketing studies (see below), in our model we incompatible with the classical model of ratio­ assume that the decision maker uses sequen­ nal choice, in which choice is constructed as the tially two rationales to discriminate among maximizer of a single preference relation (which the available alternatives. These rationales are we call a rationale), or of a utility function. If applied in a fixed order, independently of the a decision maker exhibits cycles of choice over choice set, to remove inferior alternatives. This some set of alternatives, for any candidate procedure “sequentially rationalizes” a choice “best” alternative there is always another one in function if, for any feasible set, the process the set that is judged better still: it is not possible identifies the unique alternative specified by to express a decision maker’s preferences by a the choice function. In this case, we say that a utility function, since it is not possible to find choice function is a Rational Shortlist Method (RSM). Intuitively, the first rationale identifies * Manzini: Department of Economics, Queen Mary, Uni­ a shortlist of candidate alternatives from which versity of London, Mile End Road, London E 4NS, United the second rationale selects. The special case in Kingdom (e­mail: [email protected]); Mariotti: Depart­ ment of Economics, Queen Mary, University of London, Mile End Road, London E 4NS, United Kingdom (e­mail: [email protected]). This paper was previously cir­ (2000) find that the majority (52 percent) of choices exhib­ culated under the title “Rationalizing Boundedly Rational ited binary cycles in a universal choice set of four alterna­ Choice: Sequential Rationalizability and Rational Shortlist tives. In the experiment carried out in Loomes, Starmer, Methods.” Most of this work was carried out while we were and Sugden (99), between 4 percent and 29 percent of visiting Bocconi University in Milan. Their generous hos­ choices made by all subjects were cyclical, and a staggering pitality and financial support through fellowships are grate­ 64 percent of subjects exhibited at least one binary cycle in fully acknowledged. We thank the editor, two anonymous a universal choice set of just three alternatives. More recent referees, Geir Asheim, Sophie Bade, Walter Bossert, Robin results in this same line are in Pavlo Blavatskyy (2003), Cubitt, Eddie Dekel, Marc Fleurbay, Ozgur Kibris, Michele who finds that 55 percent of his experimental subjects Lombardi, Vincenzo Manzini, Efe Ok, Ariel Rubinstein, violate transitivity of choice. Humans seem to fare better Alejandro Saporiti, Rani Spiegler, Yves Sprumont, Bob than nonhuman animals: for instance, in an experiment of Sugden, Koichi Tadenuma, and seminar participants at choice behavior of gray jays, Thomas A. Waite (200) finds FUR XII, LSE, McGill University, and the Universities that all the birds preferred choices a to b and b to c, but none of Oslo, Osnabrueck, Pais Vasco in Bilbao, Sevilla, and preferred a over c, where all alternatives 1n, l2 consisted in Trento for useful comments. The responsibility for any going and getting n raisins at the end of a lcm long tube, error is our own. with a 5 1 raisin, 28 cm2, b 5 12 raisins, 42 cm2, and See, e.g., Amos Tversky (969), Graham Loomes, c 5 13 raisins, 56 cm2. Thus, none of the birds exhibited Chris Starmer, and Robert Sugden (99), and Peter H. M. transitive choice; moreover, 25 percent of them exhibited P. Roelofsma and Daniel Read (2000). Roelofsma and Read consistently intransitive choice. 1824 VOL. 97 NO. 5 ManZINI anD MarIottI: SEQUentIallY RatIonalIZable CHOIce 1825 which the first rationale always yields a unique by c using the Pareto criterion, and, second, c maximal element corresponds to the standard is eliminated by b using the fairness criterion. model of rationality. On the other hand, g 15a, b62 5 a, given that the A notable aspect of these procedures is that Pareto criterion has no bite, and the arbitrator they are testable based on a “revealed prefer­ would select on the basis of fairness. Similarly, ence” type of analysis that, despite the highly g 15b, c62 5 b, whereas g 15a, c62 5 c by Pareto. nonstandard choices to be explained, is not more This seems an entirely reasonable way for the demanding than the standard one.2 In other arbitrator to come to a decision. In fact, this pro­ words, we ask the following question: when are cedure has been proposed in a social choice set­ observed choices compatible with the use of ting by Koichi Tadenuma (2002). Yet it produces our boundedly rational choice procedure? The a violation of WARP and pairwise cyclical pat­ answer is: if and only if the choice data satisfy tern of choice. two testable conditions. Of these conditions, one One can think of a wide array of other practi­ is a standard Expansion axiom, and the other is cal situations where RSMs may apply. A cau­ a modification of Samuelson’s Weak Axiom of tious investor comparing alternative portfolios Revealed Preference (WARP).3 The simplicity first eliminates those that are too risky relative of our tests stands in contrast to the indirect esti­ to others available, and then ranks the surviving mation algorithms normally used (notably in the ones on the basis of expected returns. A recruit­ marketing literature) to infer boundedly rational ing selector first excludes candidates with lower procedures.4 levels of some desired skills than other appli­ Typically, RSMs will lack standard menu­inde­ cants he is considering, and then selects based pendence properties, so that it may be possible on merit from the remaining ones. The notion of for an alternative to be revealed as preferable to RSM is relevant also in other fields in the social another alternative in some choice set, but for that sciences. For instance, psychologists have often preference to be reversed in a different choice set insisted on sequential “noncompensatory”5 heu­ (thus violating WARP). Because of this feature, ristics, as opposed to one single rationale, to RSMs can exhibit cyclical patterns of choice; explain choices (though axiomatic character­ however, they still rule out other types of irratio­ izations of such boundedly rational procedures nal choice. In this sense, an RSM is a nonvacuous are lacking). Notable in this respect are the notion and this gives it empirical content: it can “Elimination by Aspects” procedure of Tversky be tested by observable choice data. (972) and the idea of “fast and frugal heuris­ For a simple example of how an RSM works, tics” of Gerd Gigerenzer, Peter M. Todd, and suppose that an arbitrator has to pick one from the ABC Research Group (999). Similarly, this the available allocations a, b, or c. Suppose that type of model is widely used and documented c Pareto dominates a, while no other Pareto in the management/marketing literature. Yee et comparisons are possible. Assume further that al. (forthcoming) provide recent and compelling the arbitrator deems a fairer than b and b fairer evidence of the use by consumers of “two­stage than c. The arbitrator decides first on the basis consideration and choice” decision­making pro­ of the Pareto criterion, invoking the fairness cri­ cedures, and also refer to firms taking account terion only when Pareto is not decisive. Then, of this fact in product development. the arbitrator’s choice function g would be such In summary, RSMs are simple boundedly that g 15a, b, c62 5 b, since, first,a is eliminated rational procedures that are introspectively plausible and can explain empirically rel­ evant “anomalies” of choice patterns. Above 2 See Hal R. Varian (2005) for a recent survey on stan­ all, whether the choice pattern of a decision dard revealed preference theory. 3 Recall that WARP, in its general form, states that if maker can be explained by an RSM is a testable an alternative a is chosen from some menu of alternatives hypothesis. Last but not least, RSMs provide where some other alternative b is present (i.e., a is directly rigorous formal underpinnings to the heuristics revealed preferred to b), then it can never be the case that alternative b is selected from any other menu including both a and b. 4 For recent examples, see, e.g., Michael Yee et al. (forth­ 5 That is, in which the several “criteria” used for choice coming) and Rajeev Kohli (forthcoming). cannot be traded off against each other. 1826 THE AMERICAN ECONOMIC REVIEW DecEMBER 2007 approach central to much psychology and mar­ outcome of maximizing behavior. Formally, a keting literature. choice function g is rationalizable if there exists In addition to providing a characterization of an acyclic binary relation P, such that RSMs, we consider a natural extension whereby the decision maker applies sequentially more 5g 1S26 5 max 1S; P2 for all S [ P 1X2.