Salience Effects in a Model of Satisficing Behavior Christopher J. Tyson∗ September 22, 2011

Abstract A two-stage model of decision making is proposed that allows for both satisficing behavior and salience (i.e., attention) effects. In the first stage the agent maximizes preferences that are perceived only coarsely, with the degree of coarseness depending on the complexity of the choice problem. In the second stage, the resulting “pseudoindifference” between the remaining alternatives can be broken when one option is more salient than another. With a relatively unrestricted choice domain, this model is characterized behaviorally for both subjective and objective salience. The formal analysis follows Richter’s (1966) classical characterization of preference-maximizing behavior by means of the Congruence axiom. J.E.L. classification codes: D01, D03.

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Introduction

1.1

Salience and satisficing

In a famous remark, William James [11] wrote that “everyone knows what attention is. It is the taking possession by the mind, in a clear and vivid form, of one out of what seem several simultaneously possible objects or trains of thought.” Attention is a core topic in cognitive psychology (e.g., Anderson [1, pp. 72–105]), which since James’s day has found a number of systems (including vision, hearing, motor control, and central cognition) to be subject to constraints on parallel processing, and has also documented the fact that attentional resources are often allocated unconsciously. The success of a stimulus in attracting attention — the property of standing out from the rest — is known as salience. Neuroscientists have posited a mental “salience map” charting [34, p. 428] “stimulus strength and behavioral relevance” across sensory space, and of course advertisers have long sought to win customers by positioning their products advantageously in this terrain. But despite the abundant work on this topic in the nearby fields of psychology and marketing, and despite the growth of “behavioral economics” in ∗

School of Economics and Finance; Queen Mary, University of London; Mile End Road, London E1 4NS, U.K. Email: [[email protected]].

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general, decision-theoretic treatments of salience oriented towards economic applications have only recently begun to emerge. There are no doubt numerous ways that salience could influence decision making, and different strategies for incorporating attention into economic theory that could potentially be useful. One approach has been to develop formal models of the “consideration set” originally proposed in the marketing literature; that is, the subset of alternatives actively investigated by the decision maker.1 In this formulation attention intervenes at the outset of the choice process, and the agent applies his or her preferences in the second stage only to those options that are considered. Moreover, advertising here will play an informational role — seeking to force the advertised product into the consideration set and perhaps also to inhibit the investigation of competing options.2 In this paper we take a different approach. We envisage attention influencing behavior at the end of the choice process, only after the agent’s preferences have been applied to the alternatives. For an otherwise standard decision maker, relative salience could then only break indifference between multiple preference-maximal options. But our agent will engage in a form of satisficing rather than preference maximization in the first stage, and this will create room for salience to have a larger impact on both behavior and welfare.3 Indeed, under this approach there is a clear rationale for non-informative advertising, which can sway the agent’s choice among a variety of alternatives deemed good if not necessarily the best.4 Combining salience and satisficing effects in a single decision-theoretic model is natural in that both are responses to the same problem: Human cognitive capabilities are limited, while the environment in which choices are made is often highly complex — assailing our sensory and information-processing systems with a barrage of stimuli ranging from the important to the irrelevant. Satisficing deals with this problem by allowing a margin of error in the attempt to find an optimal alternative. Salience, meanwhile, focuses our cognitive resources on aspects of the environment that we are predisposed (whether for evolutionary, experiential, or other reasons) to find interesting, enticing, or alarming. How these two coping mechanisms interact is open to debate, and this paper explores one model of their relationship.

1.2

Two illustrative examples

We demonstrate the main features of our model first in the context of choice among the cells in the array shown in Figure 1. Here the payoff from each cell equals the base amount displayed in the top panel plus the bonus indicated in the bottom panel.5 An 1

Several contributions in this vein are discussed in §4.1. In his comprehensive survey of the economics of advertising, Bagwell [4] contrasts the “informative” view of marketing activities with the “persuasive” and “complementary” views. 3 Reber et al. [19, p. 701] define satisficing as “accept[ing] a choice or judgement as one that is good enough, one that satisfies.” The concept is due to Herbert Simon (e.g., [31]). 4 Such advertising is by no means uncommon: Resnik and Stern [20] review 378 commercials broadcast on American network television in 1975, and report the “gloomy” conclusion that “less than half of the sample’s advertisements met the liberal criteri[on] of possessing [any] useful informational cues.” 5 Cognitively-constrained choice has been operationalized in a similar way in innovative experimental work by Caplin et al. [7]. 2

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1

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A

$20

$10

$30

$50

$0

B

$40

$30

$10

$20

$50

$40 $0

C

$50

$20

$40

$0

$10

$30

D

$10

$50

$0

$30

$40

$20

E

$0

$40

$20

$10

$30

$50

F

$30

$0

$50

$40

$20

$10

Bonuses. A4, C1, C6, D5: +$0. A1, B3, E3: +$1. A6, B2, B4, E5: +$2. A5, C5, E6: +$3. B1, D3, F4: +$4. D1, D4, E1, F3: +$5. B5, C4, D2, F1: +$6. A2, C3, D6: +$7. A3, C2, F2, F6: +$8. B6, E2, E4, F5: +$9.

Figure 1: Satisficing with salience effects — a concrete example. The payoff from each of the thirty-six cells equals the base amount displayed in the top panel plus the bonus indicated in the bottom panel; e.g., D4 yields $30 + $5 = $35. A payoff-maximizer will choose B5 or D2. A satisficer who ignores the bonuses will choose A4, B5, C1, D2, E6, or F3. And of the latter options, font-size salience effects favor the selection of C1 or E6. agent who is free to consider the options at leisure will compute the necessary sums and proceed to choose B5 or D2 (yielding $56). On the other hand, a decision maker subject to cognitive constraints — one who, say, has only ten seconds to evaluate the alternatives and make a selection — might well ignore the bonuses and be willing to choose any of A4, B5, C1, D2, E6, or F3 (yielding at least $50). And if we take the intrinsic appeal of a cell to be the size of the font in which its base payoff is printed, then it is reasonable to imagine that from these six options the agent is most likely to choose C1 or E6 (yielding $50 or $53, respectively). Our theory will describe satisficing behavior subject to salience effects, like that of a decision maker faced with Figure 1 who ignores bonuses and is influenced by font sizes. In this case satisficing appears as the requirement that the utility (i.e., money value) of any choosable cell exceed a certain threshold (here $50) while not necessarily being maximal within the feasible set. Among the cells that meet this criterion, four are then ruled out by salience (i.e., font size) effects, leaving two that might be chosen. Note that if we restrict the set of options to those in row A, then our model does not permit A1 or A6 to be chosen instead of A4. On the contrary, salience will never trump utility as long as the relevant utility differentials are perceived. But in many settings determining one’s preferences is not as simple as adding integers and comparing money values. In these environments it is natural to suppose (as in [35]) that the resolving power

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C(wxyz) = x xIwxyz yIwxyz zPwxyz w; xTyEz θ(wxyz) = 1 C(wxy) = x xIwxy yPwxy w; xTy θ(wxy) = 1 C(wx) = x xPwx w θ(wx) = 1

C(wxz) = z zPwxz xPwxz w θ(wxz) = 3

C(wy) = y yPwy w θ(wy) = 2

C(wyz) = yz yIwyz zPwyz w; yEz θ(wyz) = 2

C(wz) = z zPwz w θ(wz) = 3

C(xy) = y yPxy x θ(xy) = 2

C(xz) = z zPxz x θ(xz) = 3

C(xyz) = z zPxyz yPxyz x θ(xyz) = 3 C(yz) = z zPyz y θ(yz) = 3

Figure 2: Satisficing with salience effects — an abstract example. Strict preference is denoted by P, “pseudo-indifference” by I, strictly higher salience by T, and equal salience by E. The decision maker’s preferences are given by zPyPxPw and represented by the utility assignments f (w) = 0, f (x) = 1, f (y) = 2, and f (z) = 3. The relative salience of the alternatives is given by xTwTyEz, with representation g(w) = 1, g(x) = 2, and g(y) = g(z) = 0. In each cell is shown the subset of choosable options (e.g., C(wxy) = x), the underlying perceived preference and relative salience relationships (e.g., xIwxy yPwxy w and xTy in problem wxy), and the applicable utility threshold (e.g., θ(wxy) = 1). of the preference-perception apparatus will be imperfect. And as noted above, satisficing at the first stage leaves room for salience to have both behavioral and welfare significance at the second stage.6 Before developing our decision-theoretic model and characterization results in detail, it is useful to consider a more abstract example that illustrates the various moving parts of the theory. Such an example appears in Figure 2, where each cell is linked to a choice problem drawn from the catalog of options wxyz.7 Denoting strict preference by P, the decision maker’s ranking of these alternatives is assumed to be zPyPxPw, which we can also represent via the utility assignments f (w) = 0, f (x) = 1, f (y) = 2, and f (z) = 3. Similarly, denoting strictly higher salience by T and equal salience by E, we assume that xTwTyEz, and can represent these rankings via the assignments g(w) = 1, g(x) = 2, and g(y) = g(z) = 0. The simplest choice problems are those involving only two alternatives. We assume that in these “binary” problems the decision maker behaves as a standard preference maximizer; indeed, it is from choices in such problems that we infer what the agent’s true preferences are. For example, in problem wx the only relevant strict preference is xPw, and to indicate that this preference is perceived we write xPwx w in the corresponding cell. It follows that only x is choosable in this problem, expressed in choice-function notation as C(wx) = x, and moreover the appropriate utility threshold is θ(wx) = f (x) = 1. 6

Rubinstein and Salant [25, p. 118] declare that they “cannot find any a priori reason to assume that an individual’s behavioral preferences, which describe his choices, fully represent or convey his mental preferences,” and go on to advocate [pp. 120–122] the exploration of models allowing “conclusions other than that the chosen element is always mentally preferred to the other elements in the [available] set.” As illustrated in the example of Figure 1, our theory is consistent with this research program. 7 Note the multiplicative notation for enumerated sets, which we shall use whenever convenient.

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In more complex problems our model allows the agent’s preferences to be perceived with varying degrees of coarseness. For example, in problem wxy the strict preferences xPw and yPw are perceived, while yPx is not detected and is replaced by the “pseudoindifference” assertion xIwxy y. Likewise, in problem wyz the undetected preference zPy is replaced by yIwyz z. In both cases salience rankings then come into play: The pseudoindifference between x and y in problem wxy is broken by xTy, which rules out y and yields C(wxy) = x. In contrast, the (equal-salience) assessment yEz leaves undisturbed the pseudo-indifference between y and z in problem wyz, yielding C(wyz) = yz. And setting θ(wxy) = f (x) = 1 and θ(wyz) = f (y) = 2 ensures that in each case an alternative is choosable if and only if it is maximally salient among the options with utilities above the relevant threshold. Incidentally, note that the choice (of x) from wxy is necessarily suboptimal, while that (of y or z) from wyz may or may not be optimal. In developing a theory of the mode of decision making illustrated in Figure 2, we shall not give an account of where particular preferences come from, why they are or are not perceived, or what makes alternatives more or less salient, since these are questions best addressed in the context of specific applications.8 Adopting the viewpoint of axiomatic choice theory, we shall instead take preferences, cognition, and salience as given and subject only to regularity assumptions; both ordering assumptions such as transitivity and “inter-menu consistency” assumptions linking the perception of preferences across different choice problems. Our primary goal will be to characterize the proposed model by identifying its behavioral implications. As usual this will establish that the model is falsifiable, show what sort of evidence could be adduced for or against it, and help to clarify its relationship to other models formulated in the same axiomatic framework. The difficulty of characterizing our model behaviorally can be appreciated by imagining that we had only the choice-function data from Figure 2 (i.e., the top entry in each cell) and wished to reconstruct the information relating to preferences, cognition, and salience. If an alternative is not choosable in a given problem, how can we know whether it was ruled out by a perceived preference in stage one of the model, or by insufficient salience in stage two? How can we separate out these two influences on behavior, at the same time making certain that the inferred preference and salience relationships exhibit all desired consistency properties? To see how mental constructs can be elicited from the choices in the figure, consider again the datum C(wxy) = x. Here y is not choosable, so it must be that either xTy or both xPy (an assertion about preference) and xPwxy y (an assertion about cognition). If we adopt the inter-menu “nestedness” hypothesis (see [35, pp. 54–56]) that preferences perceived in larger choice problems are also perceived in smaller ones, then xPwxy y would imply xPxy y and hence y ∈ / C(xy). But in fact y = C(xy), so we can conclude that xTy, yPx, yPxy x, and ¬[yPwxy x]. Our characterization results will require, in essence, that the information about tastes and cognition revealed in this way be free from contradictions. Figure 2 illustrates our theory’s three main components. The mathematical primitive of our analysis is the choice function encoding behavior in different decision problems. Formally, our results will be about these functions, their properties, and whether or not 8

In particular, we do not seek to embed the perception of preferences in an optimization model with information-gathering or contemplation costs. On this point, see [35, pp. 64–65].

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they possess certain underlying structure. On the other hand, our conceptual primitive is the preference and salience apparatus sketched in this section and defined more precisely below. It is to these structures — and not to the choice function — that we should turn when considering whether the model is plausible introspectively. Indeed, the purpose of choice-theoretic analysis is precisely to draw logical connections between hypotheses about an agent’s mental world and what we can observe about his or her behavior.9 The third component of our theory is a pair of real-valued functions f and g that represent the decision maker’s ordinal preference and salience rankings, together with a mapping θ that assigns a utility threshold to each menu in the domain of the choice function. As in the above examples, an agent facing feasible set A will be willing to choose those and only those alternatives that solve the constrained optimization problem max g(x) subject to f (x) ≥ θ(A). x∈A

(1)

It is essential to understand, however, that we do not envision this problem being solved consciously or intentionally. Of course, if f were known to the agent then he or she would maximize this function directly and would have no interest in g or θ. These functions are artifacts that may be useful to the modeler, but are not thought to correspond directly to anything in the decision maker’s experience.10

1.3

Road map

The remainder of the paper is organized as follows. In §2 we introduce the concepts needed to express our model formally (§§2.1–2.2) before developing the main characterization result (§2.3). We then adapt this result to situations in which relative salience is objective in the sense of being observable by the modeler (§2.4).11 Material secondary to the results just mentioned can be found in §3. We first briefly review the classical characterization of preference-maximizing behavior (§3.1). We then turn to the one-stage satisficing model (§3.2), improving the characterization of this case in [35, pp. 56–59] to create a version on which our two-stage theory can be based. Next we consider appending a salience-maximization stage to the standard, no-satisficing model, again treating both the subjective and objective cases (§3.3). And finally, the issue of choice data being consistent with multiple mental states, and the related question of the reliability of welfare analysis under our theory, are discussed at some length (§3.4). 9

The point that theories of decision making cannot be based solely on “internal consistency” of choice functions has been made by Sen [30] and more recently by Rubinstein and Salant [25, pp. 118–120]. 10 In the maximization problem in (1), the threshold θ(A) can be interpreted as the lowest utility of any alternative in the highest pseudo-indifference class formed when the cognitively-constrained agent chooses an option from A. The constraint f (x) ≥ θ(A) requires that the chosen option be in this class. And the maximization of g then ensures that no other alternative meeting the constraint is more salient than the chosen one. 11 The question of whether salience should be considered objective or subjective has not been addressed thus far. Both possibilities seem plausible: If attention is attracted by television advertising, then the relative salience of different products will vary according to which channels the consumers watch. On the other hand, if the marketing consists of billboard campaigns to which all consumers are equally exposed, then it would seem reasonable to treat salience as objective. Lacking a definite answer to this question, we characterize both cases.

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In §4 we conclude by discussing related work (§4.1) as well as the question of whether relative salience can be defined independently of context (§4.2). All proofs, plus some specialized definitions and results, appear in Appendix A.

2 2.1

Main results Preliminaries

Fix a finite set X of alternatives and a domain D ⊆ {A ⊆ X : A 6= ∅} =: F.12 A mapping C : D → F is a choice function if ∀A ∈ D we have C(A) ⊆ A. Here each A ∈ D is a menu and C(A) the corresponding choice set containing the alternatives that might be chosen from A given some cognitive hypothesis. We shall assume that {xy : x, y ∈ X} ⊆ D, so that C associates a choice set with each unary or binary menu, and also for convenience that X ∈ D. But D will be otherwise unrestricted and need not be the full domain F.13 A (binary) relation R is a subset of X × X, with hx, yi ∈ R more commonly written as xRy. Such a relation is a complete preorder if it is both complete (¬[xRy] only if yRx) and transitive (xRyRz only if xRz), and a complete order if it is complete, transitive, and antisymmetric (xRyRx only if x = y). Notation 2.1. We write R⇑(A) := {x ∈ A : [∀y ∈ A] xRy}. Thus R⇑(A) contains those alternatives in A ∈ D that are greatest with respect to the relation R. Recall that classical choice theory (see, e.g., Samuelson [27] and Arrow [3]) imagines the decision maker’s behavior to be determined entirely by his or her preferences among alternatives. Writing xRy when x is considered at least as good as y (“weak preference”), xPy when xRy and ¬[yRx] (“strict preference”), and xIy when xRyRx (“indifference”), this hypothesis becomes the requirement that ∀A ∈ D we have C(A) = R⇑(A), which we can express more compactly as C = R⇑. And assuming as usual that R is a complete preorder, C then simply selects from each menu the highest indifference (i.e., I-equivalence) class of alternatives according to P.

2.2

Relation systems and threshold structures

The model of decision making studied in this paper differs from the classical model in two respects. Firstly, preference maximization is imperfect, and may become increasingly so as the menu presented becomes more complex. And secondly, the initial satisficing stage is followed by a process that eliminates from the set of remaining alternatives those that 12

Finiteness of X is needed only for results relating to numerical representations of behavior; e.g., the equivalence of (iii) in Theorems 2.11 and 2.14. All other results hold for any X. 13 The inclusion of all binary menus in D is essential for our results. However, this restriction is weak by the standards of axiomatic choice theory, where D = F is commonly assumed (despite exceptions such as, e.g., Bossert et al. [6]). Our success in characterizing the present model given data from a relatively unrestricted set of choice problems can be attributed to the use of Richterian congruence conditions, an approach that is extended further in [36].

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are relatively less salient. We proceed now to formalize tools for modeling the first stage, before turning (in §§2.3–2.4) to the characterization of the full composite model. In our formulation, satisficing is represented cognitively as menu-dependence of the “perceived preferences” guiding choice behavior. That is to say, for each menu A we have a separate binary relation RA (with associated PA and IA ) that incorporates both the decision maker’s true preferences over A and the resolution at which these preferences are perceived. Assembling the various menu-specific relations into a vector then yields the agent’s overall “preference system,” a type of object that we now define more formally. Definition 2.2. A. A relation system R = hRA iA∈D is a vector of binary relations on the menus in D. B. A system of complete preorders is a relation system each component of which is a complete preorder. To require that the preference system be made up of complete preorders is to assume that while its components may be incomplete in the sense of reflecting the decision maker’s true preferences only coarsely, each component must be both complete and transitive in the relation-theoretic sense.14 Each menu A is thus partitioned into well-defined “pseudoindifference” (i.e., IA -equivalence) classes of alternatives, and maximization of perceived preferences then amounts to selecting the highest such class according to PA . Notation 2.3. We write R⇑(A) := RA ⇑(A) = {x ∈ A : [∀y ∈ A] xRA y}. In addition to the intramenu ordering requirement in Definition 2.2B, we shall impose a pair of intermenu consistency properties on our decision maker’s preference system. Definition 2.4. A. A relation system R is nested if ∀x, y ∈ A, B ∈ D such that A ⊆ B we have xPB y only if xPA y. B. A relation system R is binary transitive if ∀x, y, z ∈ X we have xRxy yRyz z only if xRxz z. Nestedness embodies an assumption that the decision maker can discriminate among alternatives no less finely when the menu on which they appear is smaller.15 Since the default relationship between any two alternatives in the context of any choice problem is pseudo-indifference, it is the agent’s strict preferences that his or her cognitive faculties must seek to discover. Given x, y ∈ A ⊆ B, we posit that a strict preference for x over y that is perceived in the context of problem B, written xPB y, will also be perceived in the context of the relatively simpler problem A, written xPA y.16 (This is equivalent to yRA x implying yRB x, but does not guarantee that xRB y implies xRA y.) 14

Perceived preferences exhibiting less stringent ordering properties are considered in [35, pp. 60–61]. The intuition for such an assumption can be understood in terms of an analogy (suggested by Robert Wilson) to either mapmaking or telescopic vision. The larger is the area one wishes to depict on one’s map or view through one’s telescope, the lower will be the resolution of the image. Zooming in on a particular region — analogous to removing alternatives from a menu — will improve the level of detail but at the cost of narrower scope. 16 The implicit assumption that complexity is aligned with set inclusion is discussed at length in [35, pp. 54–56]. 15

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The second intermenu consistency property to be imposed on the preference system concerns choice problems containing no more than two alternatives. (Recall that all such problems are taken to be included in D.) Intuitively, our assumption is that the agent perceives his or her true preferences when choosing from these particularly simple menus; that is, any strict preference between alternatives x and y that exists in principle must be perceived in problem xy. Of course, we also desire that these true preferences constitute a complete preorder, and indeed their completeness is immediate (since ∀x, y ∈ X the relation Rxy is assumed to be complete). But transitivity of the preferences perceived in two-element choice problems is guaranteed neither by the intramenu complete preorder requirement nor by nestedness, and it is the role of binary transitivity to ensure this. In summary, the first stage of our composite model describes a decision maker with “true” preferences of the classical sort; who perceives these preferences perfectly in binary problems but (perhaps) imperfectly in larger ones; whose perceived preferences partition each menu into well-defined pseudo-indifference classes of alternatives; and who perceives a strict preference in a given problem only if he or she also perceives it in each smaller problem (as measured by set inclusion) in which it is relevant. Clearly this first-stage model cannot be represented as utility maximization. Instead, the cognitive apparatus outlined above admits the following type of representation, which makes its satisficing interpretation more explicit. Definition 2.5. A. A threshold structure hf, θi is a pair of functions f : X → R and θ : D → R such that ∀x, y ∈ X we have θ(xy) = max f [xy]. B. A threshold structure hf, θi is said to be expansive if ∀A, B ∈ D such that A ⊆ B and max f [A] ≥ θ(B) we have θ(A) ≥ θ(B). Here f is a utility function representing the decision maker’s true preferences, while θ associates with each menu A a utility threshold θ(A) for viability as a potential choice. The options that are not eliminated by the agent’s perceived preferences are those whose utilities fall on the interval [θ(A), max f [A]], which is simply another way to describe the alternatives in the highest pseudo-indifference class according to PA . And the requirement that θ(xy) = max f [xy] then enforces our assumption that the agent perceives his or her true preferences in all binary choice problems. Nestedness of the preference system translates into the requirement that the corresponding threshold structure be “expansive.”17 To understand this property, let A ⊆ B and suppose contrary to the definition that max f [A] ≥ θ(B) > θ(A). Taking x ∈ A such that f (x) = max f [A] and assuming (without loss of generality) that ∃y ∈ A such that f (y) = θ(A), we then have f (x) ≥ θ(B) > θ(A) = f (y). But in this case the agent’s strict preference for x over y is perceived in the larger problem B but not in the smaller problem A, violating nestedness. Expansiveness thus requires that θ be in a sense conditionally decreasing: Larger menus must be assigned lower thresholds, but only if at least one alternative on the smaller menu achieves the threshold for the larger one. 17 This terminology originated in [35, p. 59], where the requirement was linked to the Strong Expansion condition on the choice function (see §3.2).

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2.3

Satisficing with subjective salience effects

We turn now to the task of characterizing our full two-stage model behaviorally. To do this, we shall need methods of deducing from raw choice data sufficient information about both perceived preferences between and the relative salience of the various alternatives. The first type of information is contained in the following pair of relation systems, which for each menu B search the domain D “locally” for evidence that in this problem one alternative is considered no worse than another. Definition 2.6. Given x, y ∈ B ∈ D: A. We write xR`B y and say that x is revealed pseudo-preferred to y at B if ∃A ∈ D such that both y ∈ A ⊆ B and x ∈ C(A). B. We write x[R`B ]∗ y and say that x is revealed serially pseudo-preferred to y at B if ∃n ≥ 2 and z1 , z2 , . . . , zn ∈ B such that x = z1 R`B z2 R`B · · · R`B zn = y. The logic of these definitions is not hard to appreciate. First of all, xR`B y means that x is choosable in the presence of y in at least one problem A ⊆ B. Writing R for the actual (unobserved) preference system, this implies that xRA y — since otherwise x would have been eliminated — and hence xRB y by nestedness. But then x = z1 R`B z2 R`B · · · R`B zn = y only if x = z1 RB z2 RB · · · RB zn = y, and it follows that xRB y since RB is transitive. Thus xR`B y is evidence that either xPB y (a perceived strict preference for x over y at B) or xIB y (pseudo-indifference between x and y at B). And it bears noting that this conclusion does not depend in any way on what occurs at stage two of the decision-making process. Next we deduce salience comparisons from the data contained in the choice function, collecting this information in ordinary binary relations (since salience is assumed to be menu-independent) and conducting a “global” search of the domain. Definition 2.7. Given x, y ∈ X: A. We write xSg y and say that x is revealed more salient than y if ∃A ∈ D such that both y is revealed serially pseudo-preferred to x at A and x ∈ C(A). B. We write x[Sg ]∗ y and say that x is revealed serially more salient than y if ∃n ≥ 2 and z1 , z2 , . . . , zn ∈ X such that x = z1 Sg z2 Sg · · · Sg zn = y. Writing R and S for the actual (unobserved) preference system and salience relation, we know from above that if y is revealed serially pseudo-preferred to x at A then yRA x. If in addition x ∈ C(A), then both alternatives must be maximal in A with respect to RA and so x must be at least as salient as y. We conclude that the assertion xSg y constitutes reliable evidence for the indicated salience comparison, and provided S is transitive the revealed serial relationship x[Sg ]∗ y will do so as well. (Here the conclusion of course does depend on the first stage having been modeled correctly.) Now given x, y ∈ A, let x ∈ C(A). If y is revealed serially pseudo-preferred to x at A, then we know that y is RA -maximal in this choice problem. If in addition y is revealed serially more salient than x, then y must survive both stages of the decision-making process and so it should be that y ∈ C(A). The following condition on C is therefore necessary for our two-stage model.

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Condition 2.8 (Weak Congruence). Given x, y ∈ A ∈ D, if x ∈ C(A), y is revealed serially pseudo-preferred to x at A, and y is revealed serially more salient than x, then y ∈ C(A). Here the name of the condition refers to the Congruence axiom characterizing ordinary preference maximization (see §3.1). A second necessary condition for our model imposes transitivity on the “base relation” encoding the decision maker’s behavior in binary choice problems. Condition 2.9 (Base Transitivity). Given x, y, z ∈ X, if both x ∈ C(xy) and y ∈ C(yz) then x ∈ C(xz). In binary problems the agent applies first his or her true preferences (i.e., Rxy in problem xy) and then the salience relation (i.e., S). And since a lexicographic composition of two complete preorders is itself a complete preorder, the result will be choices in such problems that satisfy the above transitivity condition. (Note that the base relation is inherently complete since each C(xy) 6= ∅.) The conjunction of Weak Congruence and Base Transitivity is not only necessary for our model, it is also sufficient. We state this finding formally as a three-way result linking the conditions both to the postulated cognitive apparatus (viz., the preference system R and salience relation S) and to the corresponding numerical representation. Notation 2.10. We write Ξ(A|f ≥ θ) := {x ∈ A : f (x) ≥ θ(A)}. Theorem 2.11. The following statements are equivalent: i. Weak Congruence and Base Transitivity hold. ii. There exist a nested, binary transitive system of complete preorders R and a complete preorder S such that C = S⇑ ◦ R⇑. iii. There exist an expansive threshold structure hf, θi and a function g : X → R such that ∀A ∈ D we have C(A) = Arg maxx∈Ξ(A|f ≥θ) g(x). As usual, moving from the representation to the cognitive model to the axioms is the more straightforward exercise. Assuming (iii), we can use f to define the decision maker’s true preferences and g to define salience comparisons, after which showing (ii) amounts to constructing a system R that agrees with the thresholds returned by θ. Moreover, from (ii) we have already seen rough arguments for the necessity of the conditions in (i). Starting from (i), the heart of the proof of Theorem 2.11 lies in demonstrating that the revealed serial pseudo-preference system [R` ]∗ and serial salience relation [Sg ]∗ constitute adequate approximations of the unobserved objects R and S.18 Then, with (ii) in hand, we can let the utility function f represent the preferences perceived in binary problems, let the salience function g represent S, and let the threshold mapping θ return the minimum utility achievable in the highest pseudo-indifference class corresponding to each menu. That is to say, under Weak Congruence we have C = [Sg ]∗ ⇑ ◦ [R` ]∗ ⇑. Here we encounter the obstacle that [Sg ]∗ is not necessarily complete, but using Szpilrajn’s Theorem [33] we can extend this relation to a complete preorder that together with [R` ]∗ yields the same behavior. 18

11

C(wxyz) = y ySg x; ySg z

` w; xI`wxyz yI`wxyz zPwxyz

C(wxy) = x xSg y

` w; xI`wxy yPwxy

C(wx) = x ` w xPwx

C(wyz) = yz zSg y

C(wxz) = z ` ` w xPwxz zPwxz

C(wy) = y ` yPwy w

` w; yI`wyz zPwyz

C(xy) = y ` yPxy x

C(wz) = z ` w zPwz

C(xz) = z ` x zPxz

C(xyz) = z ` ` x yPxyz zPxyz C(yz) = z ` zPyz y

Figure 3: A choice function that violates Weak Congruence. Since y ∈ C(xy) we have yR`wxy x, and since x ∈ C(wxy) it follows that both xSg y and xR`wxyz y. But y ∈ C(wxyz) and x ∈ / C(wxyz) together then violate the condition. Note that the function exhibited here is identical to that in Figure 2 (which satisfies Weak Congruence) except at wxyz. An example of a choice function that satisfies the conditions in Theorem 2.11 appears in Figure 2 above. (Verifying this directly is left as an exercise.) The function in Figure 3 differs only in the behavior specified for problem wxyz, but nevertheless violates Weak Congruence (so that the two choice functions can be described as a “minimal pair” for this axiom). To confirm the violation, observe that since y ∈ C(xy) and x ∈ C(wxy) we know that y is revealed pseudo-preferred to x at wxy, x is revealed more salient than y, and x is revealed pseudo-preferred to y at wxyz. But since y ∈ C(wxyz), Weak Congruence demands that also x ∈ C(wxyz), contrary to fact. Hence by Theorem 2.11 the choice function in Figure 3 is inconsistent with our model.

2.4

Satisficing with objective salience effects

When the relative salience of the various alternatives is known in advance to the modeler, information about the relation S need no longer be inferred from choice behavior. In this situation what is required is a test for the consistency of C and S, since these two observable objects may contradict each other in the context of our theory. Our consistency test will take the form of two conditions, the first of which is simply the objective analog of Weak Congruence. Condition 2.12 (Weak Consistency). Given x, y ∈ A ∈ D, if x ∈ C(A), y is revealed serially pseudo-preferred to x at A, and ySx, then y ∈ C(A). Here the assumption on C that y is revealed serially more salient than x is replaced with the assumption on S that y is observed to be more salient than x. Moreover, the rationale for the necessity of this axiom is identical to that for Weak Congruence: If both x ∈ C(A) and y is revealed serially pseudo-preferred to x at A, then y must be RA -maximal in this choice problem. And if y is also objectively more salient than x then it will survive the second stage as well, and so we should have that y ∈ C(A). The second half of our test for consistency requires that when two alternatives are choosable “together” (i.e., in the same problem), each must be as salient as the other. 12

Condition 2.13 (Togetherness). Given x, y ∈ A ∈ D, if x, y ∈ C(A) then xSy. This condition is clearly necessary for the objective two-stage model, since the two choosable alternatives can only have survived the second stage if neither is strictly more salient than the other. Observe also that the subjective analog of Togetherness is tautological: If x, y ∈ C(A) then by definition y is revealed pseudo-preferred to x at A and hence by definition xSg y, as the subjective axiom would require. Theorem 2.11 features no counterpart to Togetherness because when salience relationships are not exogenously given we are free to specify them in whatever way is most advantageous, and doing so rules out some inconsistencies that we must be concerned with in the objective case. Our characterization of objective salience now appears as follows. Theorem 2.14. Given a complete preorder S represented by g : X → R, the following statements are equivalent: i. Weak Consistency, Togetherness, and Base Transitivity hold. ii. There exists a nested, binary transitive system of complete preorders R such that C = S⇑ ◦ R⇑. iii. There exists an expansive threshold structure hf, θi such that ∀A ∈ D we have C(A) = Arg maxx∈Ξ(A|f ≥θ) g(x). Apart from the substitution of Weak Consistency and Togetherness for Weak Congruence, this result differs from the subjective characterization only in the quantification of S and its representation g. Here these constructs are taken to be fixed and known, whereas in Theorem 2.11 their existence is asserted in (ii)–(iii) and must be demonstrated. Theorems 2.11 and 2.14 are our main results characterizing satisficing behavior with (subjective or objective) salience effects. As will be clear by this point, the model describes maximization of menu-dependent perceived preferences lexicographically composed with maximization of salience. It would be misleading, however, to interpret the second stage of this procedure too literally. The decision maker has no particular interest in choosing more over less salient alternatives, and does nothing consciously to achieve this goal. Rather, some alternatives simply secure better access to his attention-allocation system than others. While salience is indeed maximized at the second stage, the agent does not “maximize salience” in the intentional sense ordinarily understood.

3 3.1

Additional characterizations Preference maximization

For the sake of comparison with our theory it is worth stating the original Congruence condition (first formulated and used by Richter [21, pp. 637]) together with the associated characterization of preference-maximizing choice behavior. Condition 3.1 (Congruence). Given x, y ∈ A ∈ D, if x ∈ C(A) and y is revealed serially pseudo-preferred to x at X then y ∈ C(A).

13

Here yR`X x means that ∃A ∈ D such that both x ∈ A and y ∈ C(A), which is the classical notion of revealed preference. And in the presence of Congruence the classical revealed serial preference assertion y[R`X ]∗ x then makes y choosable in all problems where it is available and x is in the choice set. Theorem 3.2 (Richter). The following statements are equivalent: i. Congruence holds. ii. There exists a complete preorder R such that C = R⇑. iii. There exists a function f : X → R such that ∀A ∈ D we have C(A) = Arg maxx∈A f (x). One notable aspect of this result is that most of its content lies in the equivalence of (i) and (ii). The same is true of all of the other three-way results in this paper, including Theorems 2.11 and 2.14: In each case the equivalence of the cognitive apparatus in (ii) and its numerical representation in (iii) is relatively straightforward, while determining precisely the model’s behavioral implications in (i) is considerably more challenging.

3.2

Satisficing without salience effects

To bridge the gap from Richter’s result and as a step towards proving Theorems 2.11 and 2.14, we now characterize the form of satisficing behavior that constitutes the first stage of our model. Condition 3.3 (Local Congruence). Given x, y ∈ A ∈ D, if x ∈ C(A) and y is revealed serially pseudo-preferred to x at A then y ∈ C(A). Theorem 3.4. The following statements are equivalent: i. Local Congruence and Base Transitivity hold. ii. There exists a nested, binary transitive system of complete preorders R such that C = R⇑. iii. There exists an expansive threshold structure hf, θi such that ∀A ∈ D we have C(A) = Ξ(A|f ≥ θ) = {x ∈ A : f (x) ≥ θ(A)}. Observe that Congruence, Local Congruence, and Weak Congruence all have the same structure: Given x, y ∈ A, if x ∈ C(A) and one or more ancillary conditions hold, then y ∈ C(A). In the case of Congruence the ancillary condition is that y be revealed serially pseudo-preferred to x at X (i.e., classically revealed serially preferred), a relatively weak hypothesis that leads to a relatively powerful axiom. Local Congruence strengthens the ancillary condition (weakening the axiom) to require that y be revealed serially pseudopreferred to x at A, a smaller set within which to find evidence. Weak Congruence then adds the requirement that y be revealed serially more salient than x, creating the strongest set of ancillary conditions and thus constituting the weakest of the three axioms. Theorem 3.4 differs from the characterization of satisficing in [35, pp. 56–59] in that Local Congruence and Base Transitivity replace, respectively, the following two axioms.

14

Condition 3.5 (Strong Expansion). Given x, y ∈ A, B ∈ D, if A ⊆ B, x ∈ C(A), and y ∈ C(B), then x ∈ C(B). Condition 3.6 (Base Acyclicity). Given n ≥ 3 and x1 , x2 , . . . , xn ∈ X, if for 1 ≤ k < n we have both C(xk xk+1 ) = xk and xk 6= xk+1 , then x1 6= xn . Here the first difference is merely cosmetic: Local Congruence and Strong Expansion are in fact logically equivalent.19 Base Transitivity is logically stronger than Base Acyclicity, however, and this strengthening of the characterization is reflected both in the binary transitivity restriction on R and in the requirement that θ(xy) = max f [xy].20 These modifications to the satisficing model are important in two ways. Conceptually, they are needed if we are to interpret the preferences perceived in binary problems as the agent’s “true” assessments, which as such should be transitive. And practically, they are a prerequisite to introducing salience effects at the second stage, which proves intractable with the original characterization as our starting point. The reasons for this will not be examined here, but it is worth noting that both the conceptual and practical difficulties with the earlier satisficing model relate to its toleration of “phantom” preferences that are not perceived even in the relevant binary problem. When f (x) > f (y) ≥ θ(xy), for example, x is superior to y but this preference may be impossible to infer from behavior. Incidentally, while Strong Expansion is not an implication of our two-stage model, a slight weakening of this condition creates an axiom that is necessary and is much simpler to state and test than Weak Congruence. Condition 3.7 (Simple Expansion). Given x, y ∈ A, B ∈ D, if A ⊆ B, x, y ∈ C(A), and y ∈ C(B), then x ∈ C(B).21

3.3

Salience effects without satisficing

In this section we consider the simplified model in which preference maximization replaces satisficing at the first stage. Here the agent successfully maximizes the preference relation R, after which indifferences are broken according to the salience relation S. When salience is subjective, the decision maker’s behavior can be described as maximizing a lexicographic composition of two unobservable complete preorders. Defining the composite relation by xLy if and only if either xPy or yIxSy, we have that C = L⇑ with L itself an unobservable complete preorder. But such a choice function falls under the characterization in Theorem 3.2, and hence we may conclude the following. Proposition 3.8. If there exist complete preorders R and S such that C = S⇑ ◦ R⇑, then Congruence holds. 19

Establishing this equivalence is left as an exercise. The merit of the Local Congruence formulation is that, as we have seen, its structure mirrors that of other useful congruence conditions. On the other hand, the advantage of the Strong Expansion formulation is that it makes clear the link to “expansiveness” of the threshold structure representation. (See [35, p. 59] for discussion of this point.) 20 Instead of binary transitivity, the earlier characterization requires that R admit a complete preorder R such that ∀A ∈ D we have R ∩ [A × A] ⊆ RA (see [35, p. 57]). In the presence of nestedness this amounts to the property of binary acyclicity, defined in the obvious manner. 21 Simple and Strong Expansion are Sen’s “β” [28, p. 384] and “β(+)” [29, p. 66], respectively.

15

This result shows that subjective salience without satisficing effects is observationally indistinguishable from the classical model. When the Congruence axiom holds, the most parsimonious explanation of the behavior will be that the decision maker is an ordinary preference maximizer. And if we nevertheless postulate a two-stage model we shall find that preference and relative salience are impossible to disentangle, since the components of the unobserved lexicographic composition L cannot be identified using choice data. When salience is objective, the simplified model will have new implications for choice behavior. While Togetherness is once again necessary for the same reason as in §2.4, Weak Consistency must be modified to disallow satisficing at the first stage. And as might be expected, the required modification mimics the strengthening of Local Congruence to Congruence, which also has the effect of disallowing satisficing. Condition 3.9 (Consistency). Given x, y ∈ A ∈ D, if x ∈ C(A), y is revealed serially pseudo-preferred to x at X, and ySx, then y ∈ C(A). We can then characterize behavior arising from preference maximization with objective salience effects in a result analogous to Theorem 2.14. Notation 3.10. We write Ξ(A|f max) := Arg maxx∈A f (x). Theorem 3.11. Given a complete preorder S represented by g : X → R, the following statements are equivalent: i. Consistency and Togetherness hold. ii. There exists a complete preorder R such that C = S⇑ ◦ R⇑. iii. There exists a function f : X → R such that ∀A ∈ D we have C(A) = Arg maxx∈Ξ(A|f max) g(x). The two-stage model without satisficing is thus indistinguishable or distinguishable from the classical model accordingly as salience is treated as subjective or objective. In neither case, however, can salience effects impact the decision maker’s welfare, since each alternative that survives the first stage is by construction preference-maximal: Under the representation in (iii), each x ∈ C(A) manifestly satisfies f (x) = max f [A] and so cannot be improved upon.

3.4

Multiplicity and welfare analysis

Continuing to write xSy when x is at least as salient as y, let xTy when xSy and ¬[ySx] (“strictly higher salience”) and xEy when xSySx (“equal salience”), as in §1.2. Suppose now that X = xyz and D = F, and consider a decision maker with cognition vector hR, Si given by yPxy x, zPxz x, yPyz z, xIxyz yIxyz z, and zTxEy. This structure results in the behavior C(xy) = y, C(xz) = z, C(yz) = y, and C(xyz) = z, a choice function satisfying both Weak Congruence and Base Transitivity. If we observe only the above choice function, what can we deduce about the cognition vector? Since C(yz) = y we know that yRyz z, and since also C(xyz) = z it follows that yPyz z, yIxyz z, and zTy. Moreover, C(xz) = z implies that zRxz x and hence yPxy x by the binary transitivity of R. But we cannot be sure that zPxz x, xIxyz z, zTx, or xEy. Indeed, 16

the function C could have resulted from a vector hR0 , S0 i identical to hR, Si except for 0 zPxyz x and xS0 zT0 y, or from the vector hR00 , S00 i identical to hR, Si except for xI00xz z and zT00 yT00 x. This example shows that a choice function satisfying the conditions in Theorem 2.11 can admit multiple combinations of preference system and salience relation. As a consequence, neither our revealed preference algorithm nor any substitute can reconstruct every cognition vector from the resulting choice data with perfect accuracy. And it is important, therefore, to consider both the extent of this multiplicity and its implications for welfare analysis under our model. Observe that, according to C above, x is never choosable in the presence of z, a fact we can express as zPX` x. One possible explanation of this is that a strict preference for z over x is perceived in all problems where both are available (as under R0 ). Another is that the decision maker is indifferent between these two alternatives but z is strictly more salient than x (as under R00 ). And since we cannot rule out either of these hypotheses using indirect evidence, we remain uncertain about both the preference between and the relative salience of x and z.22 In contrast, when each of two alternatives is choosable from some menu that contains the other, we can always draw definite conclusions about both preference and salience. This is true of y and z in the example, where from C(yz) = y and z ∈ C(A) 63 y ∈ A we can deduce that yPyz z and zTy. If instead C(yz) = y and yz ⊆ C(A) then we could conclude that yPyz z and yEz. And if C(yz) = yz then we would have yIyz z and yEz. (This exhausts all possibilities, since C(yz) 6= ∅.) Within our model, uncertainty about either the true preference between or the relative salience of any two alternatives must therefore be due to one bearing the classical revealed strict preference relation PX` to the other. Such uncertainty can be nonexistent, as when C(xy) = xy, C(xz) = x, C(yz) = y, and C(xyz) = z; behavior that — it is easy to show — admits a unique cognition vector. Or it can be widespread, as when C(xy) = x, C(xz) = x, C(yz) = y, and C(xyz) = x; behavior consistent with various combinations of indifference and strict preference and with any salience relation. Note that the former choice function conspicuously violates Congruence, while the latter satisfies this condition and returns the behavior of a classical agent with preferences xPyPz. Thus we see again (as in connection with Proposition 3.8) that preference and salience effects can be fully disambiguated only to the extent that the decision maker engages in satisficing. The fewer the violations of Congruence exhibited by C, the less we require the additional degree of freedom provided by the second stage of our model, and the more uncertainty will remain about the causes of behavior. Little can be done about our inability to (always) reconstruct completely the salience relation S. Even if the entire preference system R were somehow observable, attempts to find revealed salience comparisons would depend for their success on the effective domain of the search (namely, {RA ⇑(A) : A ∈ D}) containing suitable target sets. The same issue arises in the classical theory when binary menus may be absent from D.23 And 22

Note that while x is also never choosable in the presence of y, we can show indirectly that these two alternatives are not indifferent. 23 When the menus x, y, xy ∈ / D, for example, it is possible that ∀A ∈ D we have x, y ∈ / C(A). Here Congruence remains necessary and sufficient for preference maximization, but even when this condition

17

nothing about our model makes it any easier to guess cognitive states of which there is no trace in behavior. In any event, the more pressing issue is that of uncertainty about the true preferences perceived in binary choice problems, which we encode in the utility function f and use to evaluate welfare. The discussion above makes clear that such uncertainty can apply to alternatives x and y only if one of the two, say y, is never choosable in the presence of the other (i.e., xPX` y). In this case it is natural to assume that x is strictly preferred (rather than indifferent) to y, and indeed our revealed preference algorithm makes such an assumption. Thus the only type of error we could potentially make in welfare evaluation is to guess that xPxy y and thus set f (x) > f (y), when in fact yIxy xTy and we should set f (x) = f (y).24 To obtain a model in which no welfare evaluation errors of any sort are possible, it suffices to require that the decision maker’s true preferences are a complete order — a common simplifying assumption when (as in our case) the set of alternatives is finite. Definition 3.12. A relation system R is binary antisymmetric if ∀x, y ∈ X we have xRxy yRxy x only if x = y. The observable consequence of this hypothesis will be that menus listing two alternatives will possess choice sets containing only one. Condition 3.13 (Base Univalence). For each x, y ∈ X we have |C(xy)| = 1. We can then modify the characterization of Theorem 2.11 to incorporate the complete ordering assumption. Proposition 3.14. Weak Congruence, Base Transitivity, and Base Univalence hold if and only if there exist a nested, binary transitive, and binary antisymmetric system of complete preorders R and a complete preorder S such that C = S⇑ ◦ R⇑. In this model the agent’s true preferences hRxy ix,y∈X are uniquely determined by C, as is their one-to-one representation f up to an increasing transformation.25 And here xPX` y implies for certain that xPxy y, since xIxy y is ruled out by binary antisymmetry. Incidentally, for some purposes we may desire to strengthen Base Univalence to a comprehensive single-valuedness axiom; for example, to relate the theory to models that generate unique choices. Condition 3.15 (Univalence). For each A ∈ D we have |C(A)| = 1. holds there is no basis for supposing that either xRy or yRx. 24 In particular, we can never mistakenly guess that xPxy y when in fact yPxy x. Excluding errors of this sort is one reason for the prohibition of phantom preferences and the associated strengthening of Base Acyclicity to Base Transitivity (see §3.2). 25 Under none of our characterizations is the full preference system R in general uniquely determined. Even in the case of satisficing without salience effects (§3.2), we may remain uncertain about the perception or nonperception of particular preferences. The reason for this, briefly, is that the choice set gives us information only about the highest pseudo-indifference class of alternatives on a given menu, and our ability to infer the remainder of the partition using the nestedness assumption is imperfect.

18

As our final characterization establishes, this has the incremental effect of imposing the complete ordering requirement also on the salience relation. Proposition 3.16. Weak Congruence, Base Transitivity, and Univalence hold if and only if there exist a nested, binary transitive, and binary antisymmetric system of complete preorders R and a complete order S such that C = S⇑ ◦ R⇑. Both the utility function f and the salience function g will in this case be one-to-one, though still only f will be ordinally unique.26

4 4.1

Discussion Related work

This paper belongs to a substantial literature that studies the behavioral implications of nonstandard models of decision making. The most closely related work is of two sorts, which we now discuss in turn. The first area of related research, mentioned already in §1.1, is that concerned with the “consideration set.” Recall that this is the subset of alternatives actively investigated by the agent, a notion originally proposed in the marketing literature.27 This idea leads naturally to a two-stage model in which each menu A is first narrowed down — either consciously or unconsciously — to the consideration set σ(A) ⊆ A, and then the decision maker applies his or her preferences to form the choice set C(A) = R⇑ ◦ σ(A). Versions of the latter model have been studied axiomatically by Cherepanov et al. [8], Eliaz et al. [9], Lleras et al. [12], Manzini and Mariotti [13] (specifically, their “rational shortlist methods”), Masatlioglu et al. [14], and Spears [32]. Eliaz and Spiegler [10] have used a related framework to examine the strategic interaction among firms that try to manipulate the consideration sets of their customers by means of “door opener” products and other costly marketing schemes. And Armstrong et al. [2] have applied similar ideas to consumer search behavior, analyzing a model in which one seller is more “prominent” than its rivals and is therefore sampled first by potential buyers. One hypothesis about the nature of consideration sets posits that they are influenced by status-quo or other reference-point effects, and several contributions explore this idea. Masatlioglu and Ok [15, 16] and Bossert and Sprumont [5] develop models in which the status-quo alternative is exogenous and observable to the modeler, while Ok et al. [17] consider the endogenous case. Ortoleva [18] and Riella and Teper [22] likewise formulate reference-point models in the context of choice under uncertainty. Regardless of how consideration sets are constructed, this approach differs from ours in that it allows attention or some other cognitive phenomenon to impact decision making before preferences are applied, whereas in our model salience effects follow maximization of (perceived) preferences. The motivation for introducing the first of these mechanisms is well expressed in the cited papers. It is not clear, however, that consideration sets and 26

Note also that adding Univalence to the conditions in Theorem 2.11 does not reduce our model to preference maximization, as occurs in the model of satisficing without salience effects (see [35, pp. 61–62]). 27 See Roberts and Lattin [23] for a survey.

19

the associated “early” attention effects provide an adequate justification for the enormous amounts of non-informative advertising that can be observed.28 A billboard stating only “Coca-Cola” will not bring the ubiquitous carbonated drink into the consideration set of many consumers previously innocent of its existence (though an advertisement stating the name of a newly-released film could have such a result). What the billboard might instead do is position Coke favorably on the salience maps of numerous buyers who have no perceivable preference between various brands which taste and cost much the same; a possibility allowed for by our complementary model of “late” attention effects. Expressing the general consideration-set procedure as C(A) = R⇑ ◦ σ(A) raises the question of whether our model can be reinterpreted in these terms. Simply put, can we set σ(A) = R⇑(A) and use the “salience” relation S to instead encode preferences?29 The answer is that while reinterpretation of S in terms of preferences is unproblematic, the initial stages of the two models do not correspond. The properties that one would wish to impose on a consideration-set operator σ (see, e.g., [12, 14]) are quite different from those that result from maximization of our nested system R of complete preorders. And of course without substantial first-stage structure the consideration-set model will have no empirical content.30 The second area of related research studies the impact of “frames” on decision making. In the broadest sense, a frame is any aspect of a choice problem other than the available alternatives and their payoff-relevant characteristics that may affect behavior. Examples include the order in which the options are presented or the point in time when the choice is made. Salant and Rubinstein [26] propose a theory of framing effects in which the choice function is conditioned on the new, payoff-irrelevant information: Denoting S the frame by f ∈ F , the menu A is mapped to the choice set c(A, f ) ⊆ A, and C(A) = f ∈F c(A, f ) then contains those and only those alternatives that are choosable from A in at least one frame. Assuming that |c(A, f )| = 1 for each hA, f i, the authors examine the relationship between the “extended choice function” c and the induced C, and show in particular when the latter will be consistent with maximization of a binary relation. In our model the payoff-irrelevant factor that affects behavior is precisely the salience relation S. Writing F for the collection of all complete orders on X and taking S ∈ F , we can express our decision-making procedure as c(A, S) = S⇑ ◦ R⇑(A). Moreover, since in   S this case C(A) = S∈F S⇑ ◦ R⇑(A) = R⇑(A), the characterizations of Proposition 3.16 and Theorem 3.4 are linked as follows: If for a fixed preference system and any given salience relation the decision maker’s choices are described by our model (and so satisfy Weak Congruence, Base Transitivity, and Univalence), then the first-stage behavior (sat28

Non-decision-theoretic rationales for such “persuasive” advertising are reviewed in Bagwell [4]. Similarly, models cited above in the discussion of consideration sets have multiple potential interpretations. Manzini and Mariotti’s characterization of rational shortlist methods, for example, is in essence a formal result concerning the generation of certain choice functions by a binary-relation structure, with the relations involved having no fixed interpretation. 30 An abstract analysis of two-stage choice procedures, general enough to encompass both the present model and much of the consideration-set literature, is provided in [36]. One corollary of the latter paper’s main “meta-characterization” result is a weakened version of our Theorem 2.11 without Base Transitivity in (i), binary transitivity in (ii), or any sort of equivalence to the structure in (iii). 29

20

isfying Local Congruence and Base Transitivity) will appear in aggregate as the salience relation varies across the class of complete orders. Extended choice functions could also be used to give our theory additional structure. Suppose, for example, that we were able to observe some measure ρ ∈ R of the cognitive resources available to the decision maker. Conditional on this measure, our present results would then characterize choices of the form c(A, ρ) = S⇑ ◦ R(ρ)⇑(A). The argument for nestedness would then apply not only to changes in the menu but also to changes in the resource allocation: Given x, y ∈ A, it should be that xPA (ρ1 )y and ρ1 ≤ ρ2 only if also xPA (ρ2 )y. This opens the door to revealed-preference deductions across different values of ρ and to comparative statics with respect to the resource endowment. One illustration of framing offered by Salant and Rubinstein [p. 1289] is a satisficing procedure in the setting of choice from lists (see also [24]).31 Here a frame is a complete order  on the set X of alternatives, and the decision maker “has in mind a value function [f : X → R] and an aspiration threshold [θ ∈ R].” For any menu, if at least one available alternative has a value above θ then the first such option according to  is chosen, and otherwise the last option is chosen. Taking any map g : X → R that is strictly decreasing with respect to , the output of this procedure looks much like that of our model: A welfare-irrelevant variable g (viz., list order or salience) is maximized subject to a bound on a welfare-relevant variable f (viz., value or utility). Formally, the main difference is that here the threshold is fixed and may not always be achievable, whereas our thresholds are menu-dependent and can always be reached. But there are also significant differences of interpretation. The above satisficing model follows Herbert Simon in imagining that the options arrive sequentially and the agent consciously targets an aspiration level. Our theory, in contradistinction, is one in which the menu arrives as a unit, the agent tries (but may not succeed) to maximize his or her preferences, and constrained optimization is simply a convenient device for representing the behavior with no special status in terms of cognition.

4.2

Relative salience and context dependence

One possible objection to the present theory concerns our modeling of “relative salience” using a binary relation. A black swan may be more salient than a white one if all other swans are white, but the white swan more salient if all other swans are black. So can we reasonably speak of the relative salience of a black and a white swan without reference to the other swans nearby? Is salience not inherently a context-dependent property? Admittedly this critique is a cogent one: The background of other alternatives can undoubtedly affect salience comparisons in general. What is less certain, however, is that context effects will play a central role in the decision-making environments of interest. If salience is determined by advertising, as will be natural to assume in many applications, is it sensible to suppose that a firm can make its product more salient than the heavilyadvertised products of its competitors by undertaking no advertising at all? At least in this sort of environment, treating salience as context-independent seems appropriate. 31

Relevant too is their discussion of “limited attention,” which uses extended choice functions to model the consideration set.

21

Even in a setting where context effects were known to be important, all might not be lost. If context-dependent salience were objective — that is, if we knew in advance which alternatives in a given set were the most salient — then a generalization of the approach in §2.4 seems likely to yield a successful characterization. But subjective salience combined with unrestricted context effects would lead to dire problems of unfalsifiability.

A

Proofs

Given a relation R, its transitive closure R∗ is defined by xR∗ y if and only if ∃n ≥ 2 and hzk ink=1 such that x = z1 Rz2 R · · · Rzn = y, its symmetric part R• by xR• y if and only if xRyRx, and its asymmetric part R◦ by xR◦ y if and only if both xRy and ¬[yRx]. A relation R is a strict partial order if it is both irreflexive (xRy only if x 6= y) and transitive, and a linear order if it is irreflexive, transitive, and weakly complete (x 6= y only if xRy or yRx). A relation E is an equivalence if it is reflexive (x = y only if xEy), symmetric (xEy only if yEx), and transitive. An equivalence E is termed a congruence with respect to R if we have wExRyEz only if wRz. We first define formally the base relation encoding binary choices. Definition A.1. Given x, y ∈ X, we write xRb y if x ∈ C(xy). We shall need also the following fact about binary relations, adapted from Richter’s [21, pp. 639–640] proof of Theorem 3.2 for an arbitrary domain D. Lemma A.2. Any reflexive relation R admits a complete preorder Q ⊇ R∗ such that xQy only if xR∗ y or ¬[yR∗ x]. Proof of Lemma A.2. Since R is reflexive, [R∗ ]◦ is a strict partial order and [R∗ ]• is a congruence with respect to [R∗ ]◦ . Write φ(x) for the [R∗ ]• -equivalence class containing a given x ∈ X, and define a strict partial order  on Φ = {φ(x) : x ∈ X} by φ(x)  φ(y) if and only if x[R∗ ]◦ y. Szpilrajn’s Theorem [33] then allows us to embed  in a linear order ≫ on Φ, and we can proceed to define the complete preorder Q by xQy if and only if ¬[φ(y) ≫ φ(x)]. It follows that xR∗ y only if either φ(x)  φ(y) or φ(x) = φ(y). But then φ(x) ≫ φ(y) or φ(x) = φ(y), and in either case ¬[φ(y) ≫ φ(x)] and xQy. Hence R∗ ⊆ Q. Furthermore, if xQy then ¬[φ(y)  φ(x)] and hence ¬[y[R∗ ]◦ x], which means that xR∗ y or ¬[yR∗ x]. Definition A.3. Given x, y ∈ X, we write xSs y if ∀A ∈ D such that x[R`A ]∗ y and y ∈ C(A) we have x ∈ C(A). Lemma A.4. A. Sg is reflexive. B. C ⊆ [Sg ]∗ ⇑ ◦ [R` ]∗ ⇑. C. If C ⊆ S⇑ ◦ R⇑ for some complete preorder S and nested system of complete preorders R, then [Sg ]∗ ⊆ S. D. If C = S⇑ ◦ R⇑ for some complete preorder S and nested system of complete preorders R, then S ⊆ Ss . Proof of Lemma A.4. A. Given x ∈ X, we have {x} ∈ D and C(x) = x, and so Sg is reflexive. B. Given x ∈ A ∈ D, let x ∈ / [Sg ]∗ ⇑ ◦ [R` ]∗ ⇑(A). If x ∈ / [R` ]∗ ⇑(A), then we have ` ∗ ` ∗ x∈ / C(A) by Lemma A.7B. If x ∈ [R ] ⇑(A) then ∃y ∈ [R ] ⇑(A) such that ¬[x[Sg ]∗ y] 22

and hence ¬[xSg y], and since y[R`A ]∗ x this implies once again that x ∈ / C(A). C. Let C ⊆ S⇑ ◦ R⇑ for some complete preorder S and nested system of complete preorders R. Given x, y ∈ X, the assertion ySg x means that ∃A ∈ D such that x[R`A ]∗ y and y ∈ C(A) ⊆ S⇑ ◦ R⇑(A), and we then have xRA y by Lemma A.7C. Since y ∈ R⇑(A) and R is a system of complete preorders, we have x ∈ R⇑(A) and hence ySx. Thus Sg ⊆ S, and it follows that [Sg ]∗ ⊆ S∗ ⊆ S since S is a complete preorder. D. Let C = S⇑ ◦ R⇑ for some complete preorder S and nested system of complete preorders R. Given x, y ∈ X, the assertion ¬[ySs x] means that ∃A ∈ D such that y[R`A ]∗ x, x ∈ C(A) = S⇑ ◦ R⇑(A), and y ∈ / C(A), and we then have yRA x by Lemma A.7C. Since x ∈ R⇑(A) and R is a system of complete preorders, we have y ∈ R⇑(A) and hence ¬[ySx] since S is a complete preorder. Thus S ⊆ Ss by contraposition. Lemma A.5. Base Transitivity implies that [R` ]∗ is binary transitive. Proof of Lemma A.5. Given x, y, z ∈ X, if x[R`xy ]∗ y[R`yz ]∗ z then xR`xy yR`yz z, and therefore xRb yRb z. But then xRb z by Base Transitivity, in which case xR`xz z and x[R`xz ]∗ z. Proof of Theorem 2.11. Let (i) hold and suppose that for given x ∈ A ∈ D we have x ∈ / C(A). Then ∃y ∈ C(A), and we have y ∈ [Sg ]∗ ⇑ ◦ [R` ]∗ ⇑(A) by Lemma A.4B. If x ∈ [R` ]∗ ⇑(A) then y[Sg ]∗ x, and we have also ¬[xSs y] since x[R`A ]∗ y. In this case ¬[x[Sg ]∗ y] by Weak Congruence (which is equivalent to [Sg ]∗ ⊆ Ss ). Defining Q by wQz if and only if w[Sg ]∗ z or ¬[z[Sg ]∗ w], it follows that ¬[xQy] and hence x ∈ / Q⇑ ◦ [R` ]∗ ⇑(A). ` ∗ g But then Q⇑ ◦ [R ] ⇑ ⊆ C by contraposition. Moreover, S is reflexive by Lemma A.4A, and so by Lemma A.2 there exists a complete preorder S such that [Sg ]∗ ⊆ S ⊆ Q. We then have C ⊆ [Sg ]∗ ⇑ ◦ [R` ]∗ ⇑ ⊆ S⇑ ◦ [R` ]∗ ⇑ ⊆ Q⇑ ◦ [R` ]∗ ⇑ ⊆ C (using Lemma A.4B), implying that C = S⇑ ◦ [R` ]∗ ⇑. And since by Lemmas A.5 and A.7A we have that [R` ]∗ is a nested, binary transitive system of complete preorders, (ii) is satisfied. Conversely, if (ii) holds then [Sg ]∗ ⊆ S ⊆ Ss by Lemma A.4C–D. Since the inclusion [Sg ]∗ ⊆ Ss is equivalent to Weak Congruence, the latter condition must hold. Moreover, given x, y, z ∈ X, if xRb yRb z then xRxy yRyz z since C ⊆ R⇑, and so xRxz z since R is binary transitive. We have also xIxy y only if xSy, and similarly yIyz z only if ySz; from which it follows that xIxz z only if xSySz and hence xSz since S is a complete preorder. Thus xRb z since S⇑ ◦ R⇑ ⊆ C, so that Base Transitivity holds and (i) is satisfied. The equivalence of (ii) and (iii) follows easily from Theorem 3.4. Proof of Theorem 2.14. Let (i) hold and suppose that for given x ∈ A ∈ D we have x∈ / C(A). Then ∃y ∈ C(A) ⊆ [R` ]∗ ⇑(A) by Lemma A.7B, and if x ∈ [R` ]∗ ⇑(A) then we have x[R`A ]∗ y and hence ¬[xSy] by Weak Consistency. It follows that x ∈ / S⇑ ◦ [R` ]∗ ⇑(A), and so S⇑ ◦ [R` ]∗ ⇑ ⊆ C by contraposition. Given x ∈ A ∈ D, now suppose that x ∈ C(A) ⊆ [R` ]∗ ⇑(A). If ∃y ∈ [R` ]∗ ⇑(A) such that ¬[xSy], then we have y[R`A ]∗ x and ySx (since S is a complete preorder), and it follows that y ∈ C(A) by Weak Consistency. But in this case Togetherness implies that xSy, a contradiction. Thus C = S⇑ ◦ [R` ]∗ ⇑. And since by Lemmas A.5 and A.7A we have that [R` ]∗ is a nested, binary transitive system of complete preorders, (ii) is satisfied. Conversely, if (ii) holds then for x, y ∈ A ∈ D such that x ∈ C(A) = S⇑ ◦ R⇑(A), y[R`A ]∗ x, and ySx we have yRA x by Lemma A.7C, y ∈ R⇑(A) since R is a system of 23

complete preorders, and y ∈ S⇑ ◦ R⇑(A) = C(A) since S is a complete preorder, and thus Weak Consistency holds. Moreover, given x, y ∈ A ∈ D, if x, y ∈ C(A) = S⇑ ◦ R⇑(A) then xSy is immediate, and so Togetherness holds. Finally, since Base Transitivity holds by Theorem 2.11, (i) is satisfied. The equivalence of (ii) and (iii) follows easily from Theorem 3.4. Definition A.6. Given x, y ∈ A ∈ D, we write xRuA y if ∀B ∈ D such that A ⊆ B and y ∈ C(B) we have x ∈ C(B). Lemma A.7. A. [R` ]∗ is a nested system of complete preorders. B. C ⊆ [R` ]∗ ⇑. C. If C ⊆ R⇑ for some nested system of complete preorders R, then [R` ]∗ ⊆ R. D. If C = R⇑ for some nested system of complete preorders R, then R ⊆ Ru . Proof of Lemma A.7. A. Given x, y ∈ A ∈ D, we have both A ⊇ xy ∈ D and C(xy) 6= ∅, and so R`A is complete. Hence [R` ]∗ is a system of complete preorders, and its nestedness follows from that of the system R` . B. Given x ∈ A ∈ D, let x ∈ / [R` ]∗ ⇑(A). Then ∃y ∈ A such that ¬[x[R`A ]∗ y], and hence ¬[xR`A y] and x ∈ / C(A). C. Let C ⊆ R⇑ for some nested system of complete preorders R. Given x, y ∈ B ∈ D, the assertion yR`B x means that ∃A ∈ D such that both x ∈ A ⊆ B and y ∈ C(A) ⊆ R⇑(A), and we then have yRA x and hence yRB x since R is nested. Thus R` ⊆ R, and it follows that [R` ]∗ ⊆ R∗ ⊆ R since R is a system of complete preorders. D. Let C = R⇑ for some nested system of complete preorders R. Given x, y ∈ A ∈ D, the assertion ¬[yRuA x] means that ∃B ∈ D such that A ⊆ B, x ∈ C(B) = R⇑(B), and y ∈ / C(B), and since R is a nested system of complete preorders we then have both ¬[yRB x] and ¬[yRA x]. Thus R ⊆ Ru by contraposition. Proof of Theorem 3.4. Let (i) hold and suppose that for given x ∈ A ∈ D we have x ∈ / C(A). Then ∃y ∈ C(A), so that ¬[x[R`A ]∗ y] by Local Congruence and hence x ∈ / [R` ]∗ ⇑(A). We then have [R` ]∗ ⇑ ⊆ C by contraposition and so C = [R` ]∗ ⇑ by Lemma A.7B. And since by Lemmas A.5 and A.7A we have that [R` ]∗ is a nested, binary transitive system of complete preorders, (ii) is satisfied. Conversely, if (ii) holds then we have [R` ]∗ ⊆ R ⊆ Ru by Lemma A.7C–D. Since the inclusion [R` ]∗ ⊆ Ru is equivalent to Local Congruence, the latter condition must hold. Moreover, given x, y, z ∈ X, if xRb yRb z then xRxy yRyz z since C ⊆ R⇑ and so xRxz z since R is binary transitive. But then xRb z since R⇑ ⊆ C, so that Base Transitivity holds and (i) is satisfied. Now let (ii) hold, so that Base Transitivity is satisfied. The relation Rb is then a complete preorder and admits a representation f : X → R. For each A ∈ D let θ(A) = min f [C(A)], in which case it is immediate that C(A) ⊆ Ξ(A|f ≥ θ). Given any x ∈ A ∈ D for which x ∈ / C(A) = R⇑(A), choose y ∈ C(A) such that f (y) = min f [C(A)]. We then have ¬[xRA y] since R is a system of complete preorders, ¬[xRxy y] since R is nested, x ∈ / C(xy) and hence ¬[xRb y] since C = R⇑, and f (x) < f (y) = θ(A) since f represents Rb . But then by contraposition Ξ(A|f ≥ θ) ⊆ C(A) and so C(A) = Ξ(A|f ≥ θ). To confirm that hf, θi is a threshold structure, let x, y ∈ X be such that f (x) ≥ f (y), in which case f (x) ≥ θ(xy). If f (x) > θ(xy) = f (y) then ¬[yRb x], y ∈ / C(xy), and thus f (y) < θ(xy), a contradiction. Therefore θ(xy) = f (x) = max f [xy]. To confirm that hf, θi is expansive, let A, B ∈ D be such that A ⊆ B and max f [A] ≥ θ(B). In

24

this case ∃y ∈ A ⊆ B such that f (y) ≥ θ(B), and so y ∈ C(B) = R⇑(B). For each x ∈ C(A) = R⇑(A) we have xRA y and thus xRB y since R is nested. Since y ∈ R⇑(B) and R is a system of complete preorders, this implies that x ∈ R⇑(B) = C(B). But then C(A) ⊆ C(B) and θ(A) = min f [C(A)] ≥ min f [C(B)] = θ(B). Hence (iii) is satisfied. Conversely, if (iii) holds then define a relation system R as follows: For each A ∈ D, let xRA y if and only if ∀B ∈ D such that A ⊆ B and f (y) ≥ θ(B) we have f (x) ≥ θ(B). The system R is then nested and complete by construction, and it is apparent that R⇑ ⊆ C. Given x, y, z ∈ A ∈ D such that xRA yRA z, for any B ∈ D such that A ⊆ B and f (z) ≥ θ(B) we have f (y) ≥ θ(B) since yRA z and in turn f (x) ≥ θ(B) since xRA y. It follows that xRA z, so R is transitive and hence a system of complete preorders. Given x ∈ A ∈ D, if x ∈ / R⇑(A) then ∃y ∈ A such that ¬[xRA y]. It follows that ∃B ∈ D such that A ⊆ B and f (y) ≥ θ(B) > f (x), in which case since hf, θi is expansive we have θ(A) ≥ θ(B) > f (x) and x ∈ / C(A). But then C ⊆ R⇑ by contraposition, and so C = R⇑. Finally, given x, y, z ∈ X such that xRxy yRyz z, we have f (y) ≥ θ(xy) only if f (x) ≥ θ(xy) and similarly f (z) ≥ θ(yz) only if f (y) ≥ θ(yz). Since also θ(xy) = max f [xy] and similarly θ(yz) = max f [yz], it follows that f (x) ≥ f (y) ≥ f (z). But in this case xRxz z, so R is binary transitive and (ii) is satisfied. Lemma A.8. A. C ⊆ [R`X ]∗ ⇑. B. If C ⊆ R⇑ for some complete preorder R, then [R`X ]∗ ⊆ R. Proof of Lemma A.8. A. Given x ∈ A ∈ D, suppose x ∈ / [R`X ]∗ ⇑(A). Then ∃y ∈ A / C(A). B. Let C ⊆ R⇑ for some complete such that ¬[x[R`X ]∗ y], so that ¬[xR`X y] and x ∈ preorder R. Given x, y ∈ X, the assertion yR`X x means that ∃A ∈ D such that x ∈ A and y ∈ C(A) ⊆ R⇑(A), and we then have yRx. Thus R`X ⊆ R, and hence [R`X ]∗ ⊆ R∗ ⊆ R since R is a complete preorder. Proof of Theorem 3.11. Let (i) hold and suppose that for given x ∈ A ∈ D we have x∈ / C(A). Then ∃y ∈ C(A) ⊆ [R`X ]∗ ⇑(A) by Lemma A.8A, and if x ∈ [R`X ]∗ ⇑(A) then we have x[R`X ]∗ y and hence ¬[xSy] by Consistency. It follows that x ∈ / S⇑◦[R`X ]∗ ⇑(A), and so ` ∗ S⇑ ◦ [RX ] ⇑ ⊆ C by contraposition. Given x ∈ A ∈ D, now let x ∈ C(A) ⊆ [R`X ]∗ ⇑(A). If ∃y ∈ [R`X ]∗ ⇑(A) such that ¬[xSy], then y[R`X ]∗ x and ySx (since S is a complete preorder), and therefore y ∈ C(A) by Consistency. But then Togetherness implies that xSy, a contradiction. Hence C ⊆ S⇑ ◦ [R`X ]∗ ⇑, and from Lemma A.7A it follows that [R`X ]∗ is a complete preorder such that C = S⇑ ◦ [R`X ]∗ ⇑. Thus (ii) is satisfied. Conversely, if (ii) holds then for x, y ∈ A ∈ D such that x ∈ C(A) = S⇑ ◦ R⇑(A), y[R`X ]∗ x, and ySx we have yRx by Lemma A.8B, y ∈ R⇑(A) since R is a complete preorder, and y ∈ S⇑ ◦ R⇑(A) = C(A) since S is a complete preorder, and therefore Consistency holds. Moreover, given x, y ∈ A ∈ D, if x, y ∈ C(A) = S⇑ ◦ R⇑(A) then xSy is immediate, and so Togetherness holds. Hence (i) is satisfied. The equivalence of (ii) and (iii) is immediate. Lemma A.9. Base Univalence implies that [R` ]∗ is binary antisymmetric. Proof of Lemma A.9. Given x, y ∈ X, if x[R`xy ]∗ y[R`xy ]∗ x then xR`xy yR`xy x, and therefore xRb yRb x. Hence x, y ∈ C(xy), in which case x = y by Base Univalence.

25

Proof of Proposition 3.14. The necessity of Base Univalence is immediate, while that of Weak Congruence and Base Transitivity follows from Theorem 2.11. Moreover, in proving the latter result we have shown that under Weak Congruence and Base Transitivity there exists a complete preorder S such that C = S⇑ ◦ [R` ]∗ ⇑, with [R` ]∗ a nested, binary transitive system of complete preorders. Thus it remains only to show that the addition of Base Univalence makes [R` ]∗ binary antisymmetric, and this is established by Lemma A.9. Proof of Proposition 3.16. The necessity of Univalence is immediate, while that of Weak Congruence and Base Transitivity follows from Theorem 2.11. Moreover, under Base Transitivity and Univalence (which implies Base Univalence) we have that [R` ]∗ is a nested, binary transitive, and binary antisymmetric system of complete preorders by Lemmas A.5, A.7A, and A.9. Thus it remains only to show that the addition of Weak Congruence guarantees the existence of a complete order S such that C = S⇑ ◦ [R` ]∗ ⇑. Define a relation Q by wQz if and only if both wSg z and w 6= z. If ∃x ∈ X such that ∗ xQ x then there must exist a y ∈ X such that y 6= x and xQyQ∗ x. Hence xSg y[Sg ]∗ x, and it follows that ∃A ∈ D such that x ∈ C(A) and y[R`A ]∗ x. But then y ∈ C(A) by Weak Congruence and y = x by Univalence, contradicting y 6= x, so in fact the transitive relation Q∗ is irreflexive and hence a strict partial order. Using Szpilrajn’s Theorem to embed Q∗ in a linear order L, define a complete order S by wSz if and only if either wLz or w = z. We then have [Sg ]∗ ⊆ S, and thus C ⊆ [Sg ]∗ ⇑ ◦ [R` ]∗ ⇑ ⊆ S⇑ ◦ [R` ]∗ ⇑ by Lemma A.4B. Given x ∈ A ∈ D, now suppose that x ∈ / C(A). Then ∃y ∈ C(A) ⊆ ` ∗ ` ∗ [R ] ⇑(A) by Lemma A.7B, and clearly y 6= x. If x ∈ [R ] ⇑(A) then we have x[R`A ]∗ y, and it follows that ySg x. But then yQx, yQ∗ x, yLx, and ySx, whereupon ¬[xSy] since x 6= y and S is a complete order. Therefore x ∈ / S⇑ ◦ [R` ]∗ ⇑(A), so we can conclude that ` ∗ S⇑ ◦ [R ] ⇑ ⊆ C by contraposition and C = S⇑ ◦ [R` ]∗ ⇑.

References [1] John R. Anderson. Cognitive Psychology and Its Implications. Worth, New York, 2005. [2] Mark Armstrong, John Vickers, and Jidong Zhou. Prominence and consumer search. RAND Journal of Economics, 40(2):209–233, Summer 2009. [3] Kenneth J. Arrow. Rational choice functions and orderings. Economica, New Series, 26(102):121–127, May 1959. [4] Kyle Bagwell. The economic analysis of advertising. In Mark Armstrong and Robert H. Porter, editors, Handbook of Industrial Organization, volume 3. Elsevier, New York, 2007. [5] Walter Bossert and Yves Sprumont. 76(302):337–363, April 2009.

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Economica,

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