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Rational with limited attention: a complete domain
A complete observation means that we observe the agent’s choice for all possible lists: for all (S; S) 2 M, we observe the chosen alternative C(S; S). In other words, the domain of the choice function from lists is complete.
Constant threshold of attention
k-Limited Independence of Irrelevant Alternatives The rationalization of choice behavior is based on a single axiom: Axiom 5. k-Limited Independence of Irrelevant Alternatives (k-Limited-IIA) Let k 2 f1; : : : ; ng. C satisfies k-Limited-IIA if for all (S; S); (R; R) 2 M, [C(S; S) 2 R; R fx 2 Sj S(x) kg] implies [C(S; S) = C(R; R)].
This property is a restriction of the standard Independence of Irrelevant Alternatives (IIA) on the consideration set. The disappearance of irrelevant alternatives (i.e. unchosen) in the consideration set should not affect the agent’s choice. Formally, if the chosen alternative in the consideration set of (S; S) belongs to a sublist reduced from this consideration set, it should also be chosen. The following example illustrates this axiom for a choice behavior with a constant threshold of attention k = 2: In the first list hx; y; zi, y is chosen. If the choice function of this imaginary decision maker satisfies k-Limited-IIA, then in hx; yi, he must choose y (list (2)) and not x (list (3)). Indeed, x was already available, but unchosen in hx; y; zi. Lastly, note that there is no requirement on the order of the alternatives in the reduced list. Since k is fixed, this reduced list is fully considered. In the illustration, this means that, assuming C satisfies k-Limited-IIA, C(hx; y; zi) = y also implies C(hy; xi) = y (list (4)).
In this part of the model, the selected options of all lists are known. This complete observation allows us to elicit the constant threshold of attention for a rational agent: it is the maximal rank of his chosen alternatives. Indeed, the decision maker is assumed to be rational with a constant limited attention. That is, there exists k such that he picks his preferred alternative in a subset of the first k alternatives of each list. We know that the agent has at least scanned until the rank of the chosen option. So, since we search for a unique k, this threshold of attention is the maximal rank attained. This constant threshold of a P -rational agent is denoted by k . Then “constant limited attention” becomes “k -limited attention”: Lemma 3. Let C be a choice function from lists defined on M. Let P 2 L(X) be a linear order and k 2 f1; : : : ; ng.
Variable thresholds of attention
This subsection presents the characterization of a rational choice procedure with variable limited attention. The observation is still complete, meaning the choices from all lists are observed. Now, however, the thresholds of attention may vary for each list. Consequently, the identification of this threshold cannot be generalizable as in the previous specific case with the constant limited attention. For each list, k can be different and based on an unknown rule. However, an interval for this threshold can be intuitively elicited. The decision maker views until at least the rank of the chosen alternative. He stops, at the furthest, one rank before the first unchosen alternative which is preferred to the chosen one. In terms of rationalization, the result relies on a new concept called considered cycle which is defined and explained below.
Rational with limited attention: an incomplete do-main
An incomplete observation means that we do not observe the choices of an agent for every possible list, but only on a subset of lists. In this case, the rationalization of the choice procedures and the identification of the individual parameters (linear order P and threshold(s) of attention k) is more complex due to the lack of information. The objective of this section is to provide operational results to identify individual parameters in some situations. It is assumed that the decision maker behaves rationally.
For simplicity, we focus first on a constant limited attention. Because of the restric-tion of the observed choices, the agent’s threshold of attention may not be precisely identified (contrary to the case in complete observation). However, we can elicit an interval which contains k.
Table of contents :
Note de présentation synthétique en français
General Introduction
1 Choice with Incomparable Alternatives
1.1 Introduction
1.2 Notation and basic definitions
1.3 Common Domination Implies Equivalence
1.4 General representation theorem
1.5 Specific representation theorems: categorization
1.6 Concluding remarks
1.7 Appendix
2 Choice from Lists with Limited Attention
2.1 Introduction
2.2 Notation and basic definitions
2.3 Rational with limited attention: a complete domain
2.4 Rational with limited attention: an incomplete domain
2.5 Conclusion
2.6 Appendix
3 Retention of New Customers with a Loyalty Program
3.1 Introduction
3.2 The data
3.3 Conceptual framework
3.4 Results
3.5 Focus on the loyalty program
3.6 Conclusion
3.7 Appendix
Bibliography
