B Proof of Theorem 1

First, let me extend the domain the utility functions to budget sets, with corresponding utility sets as value.

Definition 8. Pick anyu∈

U

andB∈P(X). Define n:=|B|and b_i, i=1, . . . ,n, such thatB={bi}^i=1,...,n. Then,u(B):={u(bi)}^i=1,...,n.

By IIA, Axiom 4 implies that for allu,u˜∈

U

, allB,B˜∈P(X), and allx∈B,y∈B,˜ u(B) =u(˜ B)˜ and u_x=u˜_y ⇔ Pr(x|u,B) =Pr(y|u,˜ B)˜ (17) Correspondingly, Axiom 5 implies that (17) holds if supu−infu=supu^′−infu^′. Proof of Point 1,⇒: By Lemma 2, Pr satisfies Axioms 1 and 2 if and only if it has a relative Luce representation. Logit satisfies Axiom 4, establishing⇒.

Proof of Point 1, ⇐: We have to show that, given Axioms 1 and 2, Axiom 4 implies logit.

Step 1 (Representation independently of x):

Pick anyu∈

U

andx,y∈X. By Axiom 4, ifu_x=u_y, then Pr(x|u,B) =Pr(y|u,B)for any B∈P(X)such thatx,y∈B, and thus

u_x=u_y ⇒ V_u(x,u_x−infu) =V_u(y,u_x−infu). (18) Thus, choice propensities in any given contextu∈U solely depend on utilities. For any u∈U, fix an inverseu⁻¹ such thatu(u⁻¹(r)) =r for all rin the image of u. Note that this inverse is not generally unique, but by the previous observation, the propensities V_u(u⁻¹(ux),u_x−infu)are independent of which inverse is chosen. Hence, we can define a function ˜V_u:R→R+ by ˜V_u(ux) =V_u(u⁻¹(ux),u_x−infu), such that

Pr(x|u,B) = V˜_u(ux)

∑_x′∈BV˜_u(ux^′) for allx∈B,(u,B)∈

D

, (19) representing propensities solely as functions of utilitiesu_x. Note that this does not rule out presentation effects; ˜Vudepends on contextu∈

U

, and the result merely states thatux

contains the information required to implicitly represent presentation effects for anyu.

Step 2 (Generalized logit representation):

Definex,y∈Xandx^′,y^′∈Xsuch that (1)u_y−u_x=r, (2)u_y′−u_x′=r, and (3)u_x′−u_x=r, for somer∈R. Hence,u^′_x=u_x′ andu^′_y=u_y′. Thus, by Axiom 4 (first equality, note that Axiom 5 actually suffices) and Axiom 2 (second equality)

Pr(x^′|u,{x^′,y^′})

Pr(y^′|u,{x^′,y^′}) = Pr(x|u^′,{x,y})

Pr(y|u^′,{x,y}) = Pr(x|u,{x,y})

Pr(y|u,{x,y}). (20)

Using the representation from Eq. (19), for allr<(supu−infu)/2 and allB∈P(X), V˜_u(ux)

∑_x′∈BV˜_u(u_x′) = V˜_u(ux+r)

∑_x′∈BV˜_u(u_x′+r) for allX ∈Band(u,B)∈

D

. (21) Hence, ˜V_u(ux+r) =V˜_u(ux)·h(r) for r ≈0 (and some function h:R→R), implying V˜_u(ux+r)/V˜_u(ux) =h(r), i.e. it is independent ofu_x and hence it is differentiable inu_x, hence log ˜Vu(ux+r)−log ˜Vu(ux) is differentiable inux, and thus ˜Vu(ux+r) and ˜Vu(ux) are differentiable inu_x. Differentiating ˜V_u(ux+r) =V˜_u(ux)·h(r)atr=0, we obtain

dV˜_u(ux)/dux=V˜_u(ux)·h^′(0) ⇒ V˜_u(ux) =exp{λ·u_x+c(x)}

as the solution of this differential equation, for some integration constant c(x). Hence, V_u(x,u_x) =exp{λ·u_x+w(x)}with w(x):=c(x) for allx∈X. As this holds separately

for allu∈

U

,V(x|u) =exp{λ_u·u_x+w_u(x)}obtains, i.e.

Pr(x|u,B) = exp{λ_u·ux+wu(x)}

∑_x′∈Bexp{λ_u·u_x′+w_u(x^′)}. (22) Finally, by narrow bracketing, this implies that we can represent Pr using λ_u=λ_u+r as well asw_u=w_u+r for allr∈

R

, as then

Pr(x|u+r,B) = exp{λ_u·(ux+r) +w_u(x)}

∑_x′∈Bexp{λ_u·(u_x′+r) +w_u(x^′)}.= exp{λ_u·u_x+w_u(x)}

∑_x′∈Bexp{λ_u·u_x′+w_u(x^′)}=Pr(x|u,B).

Step 3:

Now, pick anyu∈

U

andx,y∈X such thatux=uy. By Axiom 4, Pr(x|u,B) =Pr(y|u,B) for anyB∈P(X)such thatx,y∈B. Given that Pr satisfies Eq. (22), we thus obtain that u_x=u_y implies w_u(x) =w_u(y). Hence, it is possible to represent w_u alternatively as a function ofux, instead of x, showing that the representation Eq. (22) does not violate the result of Step 1 (that propensities may be represented solely as a function of utilities).

Step 4 (Presentation independence):

Next, take any u∈

U

, any ˜u∈

U

, and define u^′=a+b u (a,b∈R:b>0) such that infu^′≤inf ˜uand supu^′>sup ˜u; suchu^′∈

U

exists by richness (transformability). Define X^′⊆X such that for allx∈X, there is exactly onex^′∈X^′:u^′_x=u^′_x′. Define ˜X such that for eachx∈X, there is exactly one ˜x∈X˜ :u_x=u˜_x˜.

Define the function f :X^′→[infu^′,supu^′]as f(x^′) =u_x′ for allx^′∈X^′. Note that f is a bijection and thus invertible. Extend f and f⁻¹to be set functions as in Definition 8.

Pick any finite ˜B⊂X˜ and defineB^′= f⁻¹ u(˜ B)˜

. Thus, |B^′|=|B˜|and ˜u(B) =˜ f(B^′) = u^′(B^′).

For anyy∈B, if˜ x= f⁻¹ u˜_y

, then ˜u_y= f(x) =u^′_x, and by Axiom 4, Pr(y|u,˜ B) =˜ Pr(x|u^′,B^′) = exp{λ_u′·u^′_x+w_u′(x)}

∑_x′∈B^′exp{λ_u′·u^′_x′+w_u′(x^′)}.

As stated, this obtains for ally∈B˜ and all ˜B⊂X˜ (with correspondingxandB^′). Using the above result that for allx,y∈X, ˜ux=u˜yimplieswu˜(x) =wu˜(y), we thus obtain

Pr(x|u,˜ B) = exp{λ_u′·u˜_x+w_u′(f⁻¹(u˜_x))}

∑x^′∈Bexp{λ_u′·u˜_x′+w_u′(f⁻¹(u˜_x′))}

for all x∈B and all B∈P(X). Defining ˆλ=λ_u′ and ˆw:[infu^′,supu^′]→R such that

w(uˆ ^′_x) =w_u′(x)for allx∈X^′, this implies Pr(x|u,˜ B) = exp{ˆλ·u˜x+w(ˆ u˜x)}

∑_x′∈Bexp{λˆ·u˜_x′+w(ˆ u˜_x′)}. (23) Since this holds true for all ˜usuch that infu^′≤inf ˜uand supu^′≥sup ˜u, it also holds true for ˜uε=u˜+εif 0<ε≤supu^′−sup ˜u, implying

Pr(x|u˜_ε,B) = exp{ˆλ·[u˜_x+ε] +w(ˆ u˜_x+ε)}

∑x^′∈Bexp{λˆ ·[u˜_x′+ε] +w(ˆ u˜_x′+ε)} = exp{λˆ ·u˜_x+w(ˆ u˜_x+ε)}

∑x^′∈Bexp{ˆλ·u˜_x′+w(ˆ u˜_x′+ε)}. By Axiom 2, Pr(x|u,˜ B) =Pr(x|u˜_ε,B), and thus there exists a function h:R→R such that ˆw(u˜x+ε) =w(ˆ u˜x) +h(ε), i.e. εcancels out. Hence, we can represent propensities given ˜u_ε equivalently as ˆw(u˜_x+ε) =w(ˆ u˜_x)for all ε≤supu^′−sup ˜uand all x∈X. By surjectivity of ˜u (richness), it follows that ˆwis constant, which implies that w_u′ andw_u˜ are constant and cancel out. Hence, for any ˜u∈

U

, Pr(x|u,˜ B)has a logit representation withλ=λ_u˜=λ_u′.

Step 5 (Context independence):

Pick any two ˜u₁,u˜₂∈

U

, and anyu^′∈

U

such thatu^′=a+b u(a,b∈R:b>0) such that infu^′≤inf{u˜₁,u˜₂}and supu^′≤inf{u˜₁,u˜₂}. By the previous results, both Pr(x|u˜₁,B)and Pr(x|u˜1,B)have logit representations withλ_u˜1 =λ_u˜2 =λ_u′, establishing Point 1,⇐. Proof of Point 2, ⇒: By Lemma 2, Pr satisfies Axioms 1 and 3 if and only if it has a standardized Luce representation. Contextual logit satisfies Axiom 5, establishing⇒.

Proof of Point 2, ⇐: We have to show that, given Axioms 1 and 3, Axiom 5 implies contextual logit.

Steps 1–2 (Generalized contextual logit):

First, fixu∈

U

such that supu−infu=1. Hence, Pr(x|u,B) = V_u x,_supu^ux−₋^inf_inf^u_u

∑_x′∈BV_u x^′,_supu^ux′−₋^inf_infu^u = Vu(x,ux−infu)

∑_x′∈BV_u(x^′,u_x′−infu),

i.e. conditional on contextu, Pr also a relative Luce representation. Thus we may follow the arguments in the proof of Point 1 (⇐), up to Eq. (22), and obtain

Pr(x|u,B) = exp{λ_u·u_x+w_u(x)}

∑x^′∈Bexp{λ_u·u_x′+wu(x^′)} = exp _λu·ux

supu−infu+w_u(x)

∑_x′∈Bexp λu·u_x′

supu−infu+w_u(x^′) ,

withλ_u+r=λ_u andw_u+r =w_u for allr∈R. By Axiom 3, Pr(x|u,B) =Pr(x|u·r,B)for allr>0, i.e.

Pr(x|u·r,B) =Pr(x|u,B) = exp{λ_u·ux+wu(x)}

∑_x′∈Bexp{λ_u·u_x′+w_u(x^′)}= exp _λu·rux

supru−infru+wu(x)

∑_x′∈Bexp λu·ru_x′

supru−infru+wu(x^′) for allr>0,B∈P(X),x∈B; note that supru−infru=r, since supu−infu=1. Hence, usingu^′=ru,

Pr(x|u^′,B) = exp λ_u′·u^′_x

supu^′−infu^′+w_u′(x)

∑_x′∈Bexp λ_u′·u^′_x′

supu^′−infu^′+w_u′(x^′) ,

withw_u′=w_u andλ_u′ =λ_u. By above, we already knoww_r+u=w_u andλ_r+u=λ_u for allr∈R, implyingλ_u=λ_{a+b u}andw_u=w_{a+b u}for alla,b∈R:b>0 and allu∈

U

. Step 3:Next, pick anyu∈

U

and anyx,y∈Xsuch thatu_x=u_y. By Axiom 5, this implies w_u(x) =w_u(y), i.e.u_x=u_y impliesw_u(x) =w_u(y).

Step 4 (Presentation independence):

Now, pick any u^′,u˜∈

U

such that infu^′=inf ˜u=0 and supu^′ =sup ˜u=1. Note that supu^′−infu^′=sup ˜u−inf ˜u=1 initially allows me to drop the normalization by supu− infuin the choice propensities. Given this restriction of the images ofu^′and ˜u, Axiom 5 implies, simply following the proof above, up to Eq. (23),

Pr(x|u,˜ B) = exp{ˆλ·u˜_x+w(ˆ u˜_x)}

∑_x′∈Bexp{λˆ·u˜_x′+w(ˆ u˜_x′)}.

for allx∈Band allB∈P(X), with ˆλ=λ_u′=λ_u′/(supu^′−infu^′)and ˆw:[infu^′,supu^′]→ Rsuch that ˆw(u^′_x) =w_u′(x)for allx∈X^′. Again, define ˜u_ε=u˜+ε, withε>0. Noting that the image of ˜u_εis not contained in the image ofu^′, Axiom 5 applies only to optionsx:

˜

u_ε(x)≤1, but given this restriction, the arguments made in the proof of above, following Eq. (23) imply

Pr(x|u˜_ε,B) = exp{λˆ·u˜_x+w(ˆ u˜_x+ε)}

∑_x′∈Bexp{ˆλ·u˜_x′+w(ˆ u˜_x′+ε)}.

for all x∈B and all B∈P(X) such that max ˜u_ε(B)≤ 1. By Axiom 3, Pr(x|u,˜ B) = Pr(x|u˜_ε,B), which similarly to above implies ˆw(u˜_x+ε) =w(ˆ u˜_x), now only for allx∈X : u_x+ε≤1, but for allε∈(0,1), including allε≈0. Hence, ˆwis constant, implying that w_u′ andw_u˜ are constant and that givenu^′ or ˜u, Pr has a contextual logit representation withλ=λu˜=λ_u′, recalling that supu^′−infu^′=1 and sup ˜u−inf ˜u=1.

Step 5 (Weak context independence): Finally, pick any two u₁,u₂∈

U

. Define u^′ = (u1−infu₁)/(supu1−infu₁)and ˜u= (u2−infu₂)/(supu₂−infu₂). By step 2,λ_u1=λ_u′

andwu₁=w_u′as well asλ_u2=λ_u˜andwu₂=wu˜. By step 4,λ_u′=λ_u˜andw_u′=wu˜=const, and by transitivity,λ_u1 =λ_u2 and w_u1 =w_u2 =const, implying the latter cancel out and that givenu₁oru₂, Pr has a contextual logit representation with theλ_u1=λ_u2=λ. Since this obtains for allu1,u2∈

U

, Point 2,⇐is established.

Im Dokument The axiomatic foundation of logit (Seite 34-39)