Chapter II: Choosing with the Worst in Mind: A Reference-Dependent Model 35
2.3 Behavioral Foundation
2.3.1 Axioms and Representation Theorem
attraction effect. Finally, the orange shaded area is the set of all symmetrically dominated alternatives(z1,t2)that can cause a preference reversal. In other words, any alternative in the non-shaded area cannot cause a preference reversal.10
X2
Attribute 1 A
x
x∗= C(A) =C(A∪ {x})
Figure 2.8: Independence of Non-Extreme Alternatives (INEA) Axiom 7 (Transitivity) For anyx,y,z ∈ X, ifx yandy z, thenx z.11
The next two axioms, INEA and RTI, are our main axioms and weakenings of the weak axiom of revealed preference (WARP). It is well known that WARP is necessary and sufficient for choice correspondence to be consistent with utility maximization. We can phrase WARP in the following way: for any Aandx< A, if {x} ,C(A∪ {x}), thenC(A) =C(A∪ {x})\ {x}.12 INEA is a postulate that is very similar to WARP and can be stated in the following way: for any non-extremex’s, WARP is satisfied, but for extremex’s, WARP can be violated.
Axiom 8 (Independence of Non-Extreme Alternatives (INEA)) For any A∈A andx< A,
if{x} ,C(A∪ {x}) andx≥ mA, thenC(A) =C(A∪ {x})\ {x}.
INEA is illustrated in Figure 2.8. Suppose A is a triangle and x is inside the triangle. Suppose x∗ is chosen from A∪ {x}. Then INEA requires that x∗ is also chosen from A. Under, regularity and INEA, we will obtain the following representation: there existsWsuch that
11In Appendix B.4, we weaken transitivity and obtain a general representation with two different distortion functions.
12Arrow (1959) stated WARP in the following way: for anyA,B∈A withB⊂A, ifC(A)∩B,∅, thenc(A)∩B=C(B). If we state INEA in this way, then our representation theorem also works for all compact menus.
X2
X1 It
It0
t2 x2 y2
x1 y1
t1
x y
t t0
2
x0
2
y0
2
x0 y0 1
t0 1 1
x0 y0
t0
Figure 2.9: Reference Translation Invariance (RTI)
C(A) =arg max
x∈A W(x,mA).
Therefore, the last two axioms impose more structure onW.
We now define RTI, which is a modification of the standardtranslation invari- ance. The main idea of RTI is to connect two different preferencest andt0 in the following way: if we can get (x0,y0,t0) from (x,y,t) by some distance-preserving shift (to be defined later), then we have x t y iffx0 t0 y0. When we have linear utility functions, the above simply means that if(x0,y0,t0)= (x+∆,y+∆,t+∆)for some∆∈R2, thenxt yiffx0 t0 y0.
When we have nonlinear utility functions, distance-preserving shift should take account for preferences. So we formally define a notion of relative distance which takes account for nonlinear utilities.
Relative Distance: For any xi,yi,x0i, andyi0, we say arelative distance between xi
and yi is equivalent to that of x0i and yi0, denoted by [xi,yi]Di[x0i,y0i], if for any xj
andyj,x∼yif and only if(x0i,xj) ∼ (y0i,yj).
RTI is illustrated in Figure 2.9. Suppose for eachi, the relative distance between xi and yi is equivalent to that of x0i and yi0(e.g., dashed intervals) and the relative
distance between yi andti is equivalent to that of y0i andt0i (e.g., dotted intervals).
In other words, we can getx0,y0,t0fromx,y,tby a distance-preserving shift. Then RTI requires that xis indifferent withy in the presence oft (solid curve It) if and only ifx0is indifferent withy0in the presence oft0(dashed curveIt0).
Axiom 9 (Reference Translation Invariance (RTI)) For any x,y,t, x0, y0, t0 ∈ X, if for eachi ∈ {1,2},[xi,yi]Di[x0i,yi0],[yi,ti]Di[yi0,t0i], then
xtyif and only ifx0t0 t0.
RTI is much weaker than standard translation invariance. In fact, it is a weak- ening of WARP. First, note that by the definition of relative distances D1 and D2, (x1,x2)∼ (y1,y2) iff (x0
1,x2)∼ (y0
1,y2) iff (x0
1,x0
2)∼(x0
1,y0
2). Second, remember that WARP requires that the irrelevant third alternatives do not affect a comparison betweena andb. Therefore, under WARP,x∼tyif and only ifx∼yandx0∼t0y0if and only ifx0∼y0.
With the last axiom, called cancellation, in addition to transitivity, we can use existing methods to obtain additive representations. It is well known that two axioms are necessary and sufficient to have an additive representation: transitivity and cancellation (see Krantz et al. 1971, Fishburn and Rubinstein 1982, Wakker 1988, and Tversky and Kahneman 1991).13 In particular, we use cancellation for and0where0 = (0,0). Although we need to have cancellation for eacht, since the last axiomReference Translation Invariance connectst and t’ for anyt and t0, it turns out enough to have cancellation for only0.
Axiom 10 (Cancellation) For anyx,y,z ∈X,
i) if(x1,x2)∼ (y1,z2)and (y1,y2) ∼ (z1,x2), then(x1,y2) ∼ (z1,z2);
ii) if(x1,x2)∼0 (y1,z2)and(y1,y2)∼0 (z1,x2), then(x1,y2)∼0 (z1,z2).
Figure 2.10 illustrates cancellation for . Solid and dashed curves represent indifferences. Intuitively, it requires that if the relative advantage of x1 over y1 is equivalent to that of z2 over x2 (dashed intervals) (i.e., (x1,x2) ∼ (y2,z2)) and the relative advantage ofy1overz1is equivalent to that ofx2overy2(dotted intervals),
13Cancellation is sometimes called the Thomsen condition or double cancellation.
X2
X1 y2
x2 z2
x1 y1
z1
(x1,y2) y
(z1,x2)
x (y2,z2) z
Figure 2.10: Cancellation
then the relative advantage ofx1overz1is equivalent to that ofz2overy2(combined intervals).
It turns out, RTI connects t andt’ in the following way: t satisfies cancel- lation if0satisfies cancellation. Therefore, it is enough to impose cancellation on 0in cancellation (ii).
Finally, we can state our main theorem. Theorem 4 characterizes (2.2) and also provides a uniqueness result, which guarantees that utility functions have cardinal meaning.
Theorem 4 C satisfies regularity, transitivity, cancellation, INEA, and RTI if and only if there exist strictly increasing and continuous functions f,u1,u2 :R+ → R+
such that f(R+)= u1(R+) =u2(R+) =R+and for any menu A∈A, C(A) =arg max
x∈A{f u1(x1)−u1(mA
1)+ f u2(x2)−u2(mA
2)}. Moreover, for any two vectors of continuous functions (f,u1,u2) and (f0,u0
1,u0
2) such that f(1) = f0(1) and u1(1) = u0
1(1), if C = C(f,u1,u2) = C(f0,u0
1,u0
2), then (f,u1,u2) = (f0,u0
1,u0
2).
The uniqueness result can be stated in two steps. First, by Remark 1 in Section 2.2, we havex y if and only ifu1(x1)+u2(x2) ≥ u1(y1)+u2(y2). It is known
thatu1andu2are unique up to a linear transformation (e.g., see Krantz et al. 1971).
Second, for givenu1andu2, it turns out f is also unique up to a linear transformation.
In other words, after fixing f(1)andu1(1), functions f,u1, andu2are unique.