In the previous section, we used the following strategy to solve the identification problem. We en- riched the choice dataset by using a frame-dependent choice function, thereby increasing the number of rationalisation conditions. This increases the number of equality constraints on the unknowns and, as a consequence, the number of equations is no longer less less than the number of unknowns. It should be apparent, however, that, for some models, this solution strategy may fail. Suppose that we have a system of equations in which the number of unknowns exceeds the number of equations, and, as a consequence, there are multiple solutions. Even if we add further equality constraints – so that the number of constraints is no longer less than the number of unknowns – this does not, of course, guarantee that we can obtain a unique solution. For example, the new equality constraints may fail to provide any additional information about the unknowns. An analogy can be made to the case of linear equations. If a system of linear equations has multiple solutions, the addition of another equality constraint that is linearly dependent on the original equations does not add any additional information about the unknowns. In some cases, therefore, the addition of the new equality constraint may not allow us to solve the identification problem.
In Appendix 3, we show that, for a model with a default option and an independent attention assignment, the use of a frame-dependent choice function does not solve the identification problem. Moreover, we present a diagnosis of the source of the identification failure. When we enrich the
choice dataset by using a frame-dependent choice function, most of the new equality constraints do not add any additional information about the unknowns. The situation is analogous, therefore, to the case of adding to a system of linear equations an equality constraint that is linearly dependent on the original equations.
We suggest, however, that the identification failure that arises with an independent attention assign- ment is something of a special case. In Appendix 4, we present a model with a default option and a “size-effect attention assignment”. A size-effect attention assignment is a simple generalization of an independent attention assignment, which accommodates the possibility that the probability of consideration sets of certain sizes may deviate systematically from the probabilities implied by an independent attention assignment. The model is fully identified for a range of frame-dependent choice functions. This model illustrates two points. First, we can interpret the identification failure for the model with an independent attention assignment as, in some sense, a special case. Sec- ond, in the size-effect model, there is not a closed-form solution for the attention parameters; they are obtained from the rationalisation conditions using numerical methods. This model illustrates that, our strategy for solving the identification problem is potentially applicable in the absence of a closed-form solution.
In the remainder of this section, we present another model with a default option that is fully identified – it is a model with a minimum attention level. For this model, we will obtain a series of identification and characterisation results akin to those in Theorems 5, 6 and 7.
The definition of rationalisation in Section 2.4 can be adapted to the case of a frame-dependent choice function with a default option.
Definition 20(Rationalisation of a default-option frame-dependent choice function)
A default-option frame-dependent choice function (¯c,¯c0) is a pair of default-option choice functions. Consider any ranking assignment on ¯X, ¯π, and any pair of attention assignments (α, α0). A default- option frame-dependent choice function (¯c,¯c0) is said to be rationalised by ¯π and (α, α0) if ¯c is rationalised by ¯π and α and ¯c0 is rationalised by ¯π and α0.
eration set is 1. As discussed in Section 2.3, for such a case, a natural assumption is that the DM’s attention assignment is conditional. We will refer to this model as the “default-option” model, or DO model. In order to develop models with higher minimum attention levels, we could use, instead, the restriction on attention assignments suggested in Appendix 2.
The rationalisation conditions for the DO model imply that ¯c(xi, xix) = ¯¯ π( ¯Mxi,xix¯), i= 1, . . . , m.
Furthermore, the following equality constraints are also implied. ¯ c(xi, xixjx) =¯ γi(1−γj)¯c(xi, xix) +¯ γiγjπ( ¯¯ Mxi,xixjx¯) γi(1−γj) +γj(1−γi) +γiγj , i, j ∈ {1, . . . , m}, i < j (2.23) ¯ c(xj, xixjx) =¯ γj(1−γi)¯c(xj, xjx) +¯ γiγjπ( ¯¯ Mxj,xixjx¯) γi(1−γj) +γj(1−γi) +γiγj , i, j ∈ {1, . . . , m}, i < j (2.24) ¯ c0(xi, xixjx) =¯ γi0(1−γj0)¯c(xi, xix) +¯ γi0γj0π( ¯¯ Mxi,xixjx¯) γi0(1−γj0) +γj0(1−γi0) +γi0γj , i, j ∈ {1, . . . , m}, i < j (2.25) ¯ c0(xj, xixjx) =¯ γj0(1−γi0)¯c(xj, xjx) +¯ γi0γj0π( ¯¯ Mxj,xixj¯x) γi0(1−γj0) +γj0(1−γi0) +γi0γj0 , i, j ∈ {1, . . . , m}, i < j (2.26) Using the definition of inattention odds, we rewrite this equation system so that it is linear in the unknowns:
(1 +ιi+ιj)¯c(xi, xixjx) =¯ ιj¯c(xi, xix) + ¯¯ π( ¯Mxi,xixjx¯), i, j ∈ {1, . . . , m}, i < j (2.27)
(1 +ιi+ιj)¯c(xj, xixjx) =¯ ιi¯c(xj, xjx) + ¯¯ π( ¯Mxj,xixjx¯), i, j∈ {1, . . . , m}, i < j (2.28)
(1 +ι0i+ι0j)¯c0(xi, xixjx) =¯ ι0j¯c(xi, xix) + ¯¯ π( ¯Mxi,xixjx¯), i, j ∈ {1, . . . , m}, i < j (2.29)
(1 +ι0i+ι0j)¯c0(xj, xixjx) =¯ ι0i¯c(xj, xjx) + ¯¯ π( ¯Mxj,xixjx¯), i, j∈ {1, . . . , m}, i < j (2.30)
The next step is to eliminate the maximum probability terms. From (2.27) and (2.29), we can elimi- nate ¯π( ¯Mxi,xixjx¯), to obtain (2.31) below; and from (2.28) and (2.30), we can eliminate ¯π( ¯Mxj,xixjx¯),
to obtain (2.32). The result is the following system of 2 m2
equations, fori, j= 1, . . . , msuch that i < j.
[¯c(xi, xixjx)¯ −c¯0(xi, xixjx)]ι¯ j+ [¯c(xi, xix)¯ −¯c0(xi, xixjx)]∆¯ j+
[¯c(xi, xixjx)¯ −c¯0(xi, xixjx)]ι¯ i+ [−c¯0(xi, xixjx)]∆¯ i = ¯c0(xi, xixjx)¯ −c(x¯ i, xixjx)¯ (2.31)
[¯c(xj, xixjx)¯ −c¯0(xj, xixjx)]ι¯ j+ [−¯c0(xj, xixjx)]∆¯ j+ [¯c(xj, xixjx)¯ −c¯0(xj, xixjx)]ι¯ i
+[¯c(xj, xjx)¯ −¯c0(xj, xixjx)]∆¯ i = ¯c0(xj, xixjx)¯ −¯c(xj, xixjx)¯ (2.32)
These equations contain 2munknowns, so, in order to ensure that the number of equations is not less than the number of unknowns, we need to assume that the number of non-default alternativesm is
such that 2 m2
≥2m. Thus we will assume that there are at least three non-default alternatives. In contrast, for the NDO model, we had to assume that there are at least five alternatives. Nevertheless, the solution strategy is parallel to that in the NDO model. The solution is presented fully in the proof of Theorem 8, but it can be outlined as follows. First, using this linear system of equations, we solve for the unknowns as affine functions of ∆1. In particular, we find thatιi =−21−¯si∆1, i= 1, . . . , m
and ∆i = −¯ti∆1, i = 2, . . . , m, where ¯si, i = 1, . . . , m and ¯ti, i= 2, . . . , m are defined in the proof
of Theorem 8 in Appendix 1. It remains then to solve for ∆1. The proof of Theorem 8 shows that
we can do so by recourse to further rationalisation conditions, which yield the following quadratic equation, analogous to to equation (2.21) in Section 2.4:
¯
f∆21+ ¯g∆1+ ¯h= 0 (2.33)
where ¯f ,¯gand ¯hare defined in the proof of Theorem 8 in Appendix 1. This proof shows that ¯g= 0. So, from the quadratic in equation (2.33), we obtain a solution for ∆1 analogous to the solution in
equation (2.22) in Section 2.4. ¯ ∆1= −p−¯h/f¯ if ¯s1>0 p −¯h/f¯ otherwise (2.34)
As in the NDO model, our results assume that a nonzero condition obtains:
Definition 21(DO-nonzero condition)
(i) Fori, j ∈ {1, . . . , m}, i < j, ¯c0(xi, xixjx)¯ −¯c(xi, xixjx) are nonzero.¯
(ii) The following terms, which are functions of observed choice probabilities and defined in Ap- pendix 1, are nonzero: ¯f and, forj= 2, . . . , m, ¯dj1j.
The following then provides definitions, for the DO model, of the revealed inattention odds, the revealed attention probabilities and the revealed attention assignments:
Definition 22(DO revealed inattention odds, attention probabilities, attention assignments) For |X| ≥3, suppose that choice function (¯c,¯c0) satisfies the DO-nonzero condition.
(i) The DO revealed inattention odds vectors for ¯c and ¯c0 are the vectors ¯ι,¯ι0 ∈ Cm such that, for
i= 1,2, . . . , m, ¯ιi =−21 −s¯i∆¯1 and ¯ ι0i = ¯ ιi+ ¯∆1 fori= 1 ¯ιi−t¯i∆¯1 fori= 2, . . . , m
(ii) Suppose that the revealed inattention odds vectors ¯ι and ¯ι0 are real and positive. Then the
DO revealed attention probability vectors for ¯c and ¯c0 are ¯γ,¯γ0 ∈ Rm such that, for i = 1, . . . , m,
¯
γi= 1+¯1ιi and ¯γi0 = 1+¯1ι0
i.
(iii) Suppose that the revealed inattention odds vectors ¯ιand ¯ι0 are real and positive. Then theDO revealed attention assignments for ¯cand ¯c0 are the conditional attention assignments ¯α and ¯α0 that are respectively generated by ¯γ and ¯γ0.
Theorem 8 justifies the interpretation of ¯α and ¯α0 as the revealed attention assignments for the model.
Theorem 8 (Identification of attention assignments in the DO model)
For|X| ≥3, consider any frame-dependent choice function (¯c,c¯0) that satisfies the DO-nonzero con- dition, any ranking assignment ¯π on ¯X, and any pair of conditional attention assignments (α, α0). If ¯π and (α, α0) rationalise (¯c,c¯0), thenα= ¯α and α0 = ¯α0.
Appendix 5 provides a simple numerical example that illustrates how to apply the identification method presented in the proof of Theorem 8. This example also serves to verify the identification method presented in this section.
Having obtained revealed attention assignments, we adopt the strategy from the previous section to retrieve information about the ranking assignment. First, we define maximum functions and contour functions for the DO model. These can be interpreted as specifying the revealed maximum probabilities and the revealed probabilities of upper contour sets.
Definition 23(DO maximum functions and contour functions)
For|X| ≥3, suppose that a choice function (¯c,c¯0) is such that the DO-nonzero condition is satisfied and that the revealed inattention odds vectors ¯ιand ¯ι0 are real and positive.
(a) The DO maximum functions for ¯cand ¯c0 are the functions ¯µ:{(x, B) :B ∈B¯, x∈B} →Rand
¯
µ0 :{(x, B) :B ∈B¯, x∈B} →R such that: (i) ¯µ(¯x,{x¯}) = 1 and ¯µ0(¯x,{x¯}) = 1; and
(ii) for any non-emptyB ⊂X,x∈B:¯ ¯ µ(x,B) =¯ c(x,¯ ¯ B)−P A(B:x∈A¯α(A, B)¯¯ µ(x,A)¯ ¯ α(B, B) ¯ µ0(x,B) =¯ c¯ 0(x,B¯)−P A(B:x∈A¯α¯ 0(A, B)¯µ0(x,A)¯ ¯ α0(B, B)
(b) The DO contour functions for ¯c and ¯c0 are the functions ¯υ :{(x, A) :A ⊂X, x∈ X¯\A} → R
and ¯υ0:{(x, A) :A⊂X, x∈X¯\A} →R such that, for any A⊂X, x∈X¯\A:
¯ υ(x, A) = X B⊂A (−1)|A\B|µ(x,¯ X¯\B), υ¯0(x, A) = X B⊂A (−1)|A\B|µ¯0(x,X¯\B)
The following theorem says that the ¯υ and ¯υ0 can be interpreted as specifying revealed contour probabilities. If we compare this theorem to Theorem 3, it says that, in the DO model, ranking assignments are revealed to the same extent that they are revealed in the full attention version of the model. In this sense, the DO model is “fully identified”.
Theorem 9 (Identification of ranking assignments in the DO model)
For |X| ≥3, consider any choice function (¯c,¯c0) that satisfies the DO-nonzero condition, any rank- ing assignmenton ¯X , ¯π, and any pair of conditional attention assignments (α, α0). If ¯π and (α, α0) rationalise (¯c,c¯0), then for allA⊂X,x∈X¯\A, ¯π( ¯Ux,A) = ¯υ(x, A) = ¯υ0(x, A). That is, ¯π is unique up to upper contours of non-default alternatives.
Finally, we present a identification-based characterisation theorem for the DO model that is analo- gous the the characterisation result Theorem 7 in the NDO model.
Theorem 10(Characterisation in the DO model)
For |X| ≥ 3, consider any choice function (¯c,¯c0) that satisfies the DO-nonzero condition. There exists a ranking assignment on ¯X and a pair of conditional attention assignments that rationalise (¯c,¯c0) if and only if:
(i) the revealed inattention odds vectors ¯ι and ¯ι0 are real and positive; and (ii) the contour functions ¯υ and ¯υ0 are equal and non-negative.