Rationality and the speed of decision-making Michael Mandler Department of Economics Royal Holloway College, University of London Egham, Surrey TW20 0EX United Kingdom Abstract turns out to be unambiguous: the rational agents can on some domain always use a strictly shorter check- We consider agents who choose by proceeding list of criteria than the irrational agents can and on through an ordered list of criteria and give no domain do they have to use a longer list. Ratio- the lower bound on the number of criteria nal agents can in fact make preference discriminations that are needed for an agent to make deci- with great e¢ ciency; the number of discriminations sions that obey a given set of preference rank- they can make is an exponential function of the num- ings. Agents with rational preferences can ber of criteria at their disposal. And the speed advan- always use lists with the lower-bound num- tage of rationality holds even when compared to agents ber of criteria while any agent with nonra- whose sole source of irrationality is incompleteness. tional preferences must on some domains use Our conclusions depart from Herbert Simon’s view strictly more criteria. We preview some of that the assumption of utility maximization places ‘a the results in Mandler (2009) and explain in heavy (often unbearable) computational burden on the more detail the order-theoretic link between decision maker’(Simon (1990)). For checklist users, in rationality and rapid decision-making. contrast, it is the rational agents who have the easier computational task and therefore make decisions more quickly. The di¢ culty with Simon’sand kindred views 1 INTRODUCTION is that they take utility to be a set of external facts that an agent has to uncover. If instead we take an One way to choose between alternatives is to divide agent’s decision-making procedure as primitive, then the domain of possible alternatives into various cate- the computational burden of decision-making becomes gories and then decide on an ordering of those cate- a function of the procedure alone. We may then ask gories. If this step does not discriminate su¢ ciently if the procedures that lead to rational decisions carry one can then move on to a second categorization, and a heavier or lighter burden than the irrational proce- so on. For any pair of alternatives, an agent can pro- dures. ceed in this fashion through a sequence of orderings, choosing the alternative that is recommended by the The greater decision-making e¢ ciency of rational pref- …rst ordering that actually ranks the pair. While this erences derives from a fundamental fact about linear choice procedure, which we call a checklist of criteria, orders. On domains that consist of at least four ele- does not look like classical economic behavior it does ments, linear orders are the only binary relations such describe one of the rough-and-ready methods that peo- that any two subsets with the same number of elements ple use to make decisions. are order-isomorphic. In our setting, a rational pref- erence relation induces a linear ordering over its indif- A checklist user’s decisions implicitly de…ne a prefer- ference classes and the order isomorphism fact implies ence relation: for any pair of alternatives that are fed that no matter how we constrain the criteria that an into the choice procedure, we can label the chosen al- agent can use, the agent will be able to divide up his ternative to be strictly preferred, and if no criterion in indi¤erence classes into order-isomorphic subsets that a checklist discriminates between the alternatives we a single criterion can then order simultaneously. As can label the pair unranked. Which types of checklist we will see, this is the source of the speed advantage users can make decisions the most rapidly, the rational of rational agents. agents whose preferences are complete and transitive or the irrational agents? The answer to this question Checklists of criteria bear a close resemblance to Chip- man’s (1960) lexicographic theory of utility, which section 3. was designed to provide representations for preferences For example, X might be a large set of cars, crite- that do not have classical utility functions (such as lex- rion 1 could divide cars into cheap, mid-priced, and icographic preferences). In Chipman, the preference expensive categories, criterion 2 could divide cars into between two items is determined by the …rst utility in sedans, convertibles, SUV’s,and all others, and …nally a sequence of real-valued utility functions that ranks criterion 3 might divide cars by their country of ori- the items. In our model a sequence of criteria replaces gin. After the categorization step, the agent must the sequence of utilities and, to gauge decision-making …gure out how to order each set of categories. Then, e¢ ciency, we will use an accounting system that mea- to decide between a pair of cars, the agent proceeds in sures the discriminatory capacity of criteria. It turns sequence through the criteria: if one car has the bet- out that even the least discriminating criteria, when ter price ranking, then the agent buys that car; if not, strung together in a sequence, can generate a vast set the agent goes on to the next categorization, and so of preference discriminations. The absence of a mea- on until the agent comes to a categorization that does sure of discriminatory capacity in Chipman has hid rank the pair. If the pair is ranked by none of the from view the power of his representation concept. criteria, the agent declares either car to be acceptable. More concretely, the present work follows in the foot- We can de…ne a preference relation from a checklist steps of the checklists studied in Mandler, Manzini, by seeing for any pair in X which element of the pair and Mariotti (2008). The MMM checklists are con- is chosen. Formally, a preference is another asym- strained to be rational, however, which precludes the metric binary on X. Given a preference , we say rational versus irrational horse race that we pur- that C = (C ; :::; C ) is a checklist for if and only sue here. Kalai, Rubinstein, and Spiegler (2002), 1 L if, for all x; y X, Apesteguia and Ballester (2005, 2008), and Manzini 2 and Mariotti (2007) also consider choice behavior x y i with 1 i L such that x Ci y and that can be decomposed into ‘criteria.’ KRS and , 9 Apesteguia and Ballester (2005) have an agenda sim- not y Cj x for all j < i: ilar to the present paper but they seek out the most Notice that it is only the …rst criterion that ranks a concise set of rationales for decisions, not the most e¢ - pair that determines preference; if the …rst criterion cient use of discriminatory capacity. The introduction Ci that ranks and x and y has x Ci y then some later of an accounting system for discriminatory capacity of criterion Ck can have y Ck x and still x y. We will criteria marks out a distinct agenda. also say that a preference has the checklist C if C is In addition to previewing some of the results in Man- a checklist for . A checklist for a preference amounts dler (2009), we will use this forum to explain in more to a mild extension of Chipman’s (1960) de…nition of detail the order-theoretic connection between rational- a lexicographic utility. ity and the speed of decision-making. Checklists of properties. Some seemingly crude se- quential decision procedures may not at …rst appear 2 CHECKLISTS OF CRITERIA to qualify as checklists, for example, the model in Mandler, Manzini, and Mariotti (2008). Suppose an Fix some domain of alternatives X. We consider agent chooses between two items x and y by proceed- agents who make decisions between pairs of alterna- ing through a list of ‘properties,’P1;P2;P3; ::: , where tives x; y X by proceeding through a checklist of each Pi is a subset of X. The agent …rst checks if x 2 criteria C1; :::; CL, where each Ci is an asymmetric bi- and y have property 1 (i.e., if x P1 and y P1). If nary relation on X, and choosing the alternative that either or x or y has property 1 but2 the other2 does not is ranked superior by the …rst Ci that actually orders then the agent chooses the item that does. If both x x and y. One should think of each criterion as di- and y have property 1 or neither does then the agent viding X into a relatively small number of categories proceeds to property 2, and so on. A list of properties or equivalence classes. Outside of asymmetry, crite- quali…es as a checklist of criteria since the agent could ria are unrestricted; they could be rational orderings, make the same decisions by using a C1 that ranks any they could cycle, or they could leave some alternatives item in P1 as strictly superior to any item in X P1, n unranked. a C2 that ranks item in P2 as strictly superior to any item in X P2, and so on. An agent might construct a criterion Ci in two steps, n …rst a division of X into categories that partition X Checklists can be seen as a decision procedure for two- and then a ranking (though not necessarily a complete element choice sets, but they can also be used to decide ranking) of these categories. Formally, the categories from a larger choice set A by letting the Ci sequentially are the equivalence classes of Ci, which we de…ne in eliminate Ci-dominated items from A.