Chapter III: Theory of Decisions by Intra-Dimensional Comparisons
3.2 The Basic Model
3.2.1 Separability and Representation Theorem
In this subsection, we characterize our model. In particular, we can characterize (3.2) by an axiom calledSeparabilityin addition to standard postulates. We begin by imposing four standard properties: completeness, continuity, strong monotonicity, and richness. We concentrate all of these properties in one axiom, calledRegularity. Axiom 13 (Regularity) Letbe a binary relation on X.
1. (Completeness) For anyx,y ∈ X, eitherx yory x;
2. (Continuity) For anyx∈ X,{y∈ X|y x}and{y ∈ X|x y} ∪X0are closed sets where X0≡ {t ∈Qn
i=1[ai,bi]|∃isuch thatti= ai};
3. (Strong Monotonicity) For anyx,y,z∈ X, ifx∼yandy> z, thenx z;
4. (Richness) For any x ∈ X, i, and y−i ∈ X−i, there exists yi ∈ Xi such that x y= (yi,y−i).9
Completeness states that any two alternatives are comparable. Continuity re- quires, loosely speaking, that an upper contour set and a lower contour set of any
8Note that any IDC relation with distance-based functions f andg is consistent with the similarity relation model of Rubinstein (1988) (SRM) with the following similarity relations∼xand
∼p: x∼x yiff|f(x,y)| ≤λandp∼p qiff|g(p,q)| ≤λfor some fixedλ ∈(0,1). Here, we say that a binary relationis consistent with SRM if there are similarity relations∼xand∼p such that for anyx >yandp<q, ifp ∼p qandx x y, then(x,p) (y,q), and ifx∼x yandpp q, then (y,q)(x,p).
9In the context of choice over binary lotteries, if one wants to use the domain [0,x]×[0,1] instead of(0,x]×(0,1], then Richness is equivalent to the following simpler condition: (x,p)(y,0)and (x,p)(0,q)for any(x,p),(y,q)>>(0,0). It requires that any zero lottery is worse than non zero lotteries. For more details, see the discussion in Section 3.3.
alternative are closed. Technically, a lower contour set is closed after we take into account that each Xi = (ai,bi] is left-open. Hence the presence of X0in the axiom.
Richness states that for any vectory−i, there existsyisuch thaty= (yi,y−i)is worse than a fixed lottery x. Strong monotonicity implies the standard monotonicity ax- iom: x >yimpliesx y. Strong monotonicity also requires transitivity where one of three alternatives dominates one of the other two alternatives (y> z). We do not impose transitivity beyond this minimal case. Therefore, we allow binary relations that violate transitivity.
The key axiom in this paper is calledSeparabilityand is closely related to the IDC heuristic.10 In the IDC heuristic, the DM compares alternatives dimension- by-dimension and Separability requires that these dimension-by-dimension compar- isons are independent. For example, in the context of choice over binary lotteries, Separability requires that a comparison between prizesx andyis independent of a comparison between probabilitiesqandp. Figure 3.1 illustrates the intuition behind Separability forn= 2. In Figure 3.1, solid and dashed curves represent indifference curves. Consider two indifferent pairs of binary lotteries(x,p)and(y,q)and(x0,p) and(y0,q); i.e., (x,p) ∼ (y,q)and (x0,p) ∼ (y0,q)(two solid indifference curves).
Since these lotteries use the same probabilities, the indifferences suggest that the relative advantage ofx with respect to y is equal to that ofx0with respect to y0. If this is the case, then the same should hold even if we change the probabilities: we should have (x,p0) ∼ (y,q0) if and only if (x0,p0) ∼ (y0,q0) for any probabilities p0 and q0(two dashed indifference curves). This is the content of Separability on choice over binary lotteries. It is a simple exercise to show that any EUT preference on binary lotteries satisfies Separability.
Axiom 14 (Separability) For allx,y,x’,y’∈ X andi, if
(xi,x−i)∼ (yi,y−i), (x0i,x−i)∼ (yi0,y−i), and (xi,x’−i)∼ (yi,y’−i), then(x0i,x’−i)∼ (yi0,y’−i).
10This axiom is used in the theories of additive utility representation. For example, it is called thecorresponding tradeoff conditionin Keeney and Raiffa (1976) andtriple cancellationin Wakker (1988). When there are only two dimensions and transitivity is satisfied, Separability is implied byCancellation(sometimes it is called Double Cancellation or Thomsen condition). There are a large literature on separability (See Blackorby et al. (2008)). However, they focus on horizontal separability in line with additive utility models in which transitivity is assumed while we use vertical separability in line with IDC heuristic.
X2
X1 q0
p0 q p
x0 y0
x y
Figure 3.1: Separability whenn= 2
Separability states that for eachi, a comparison between xi and yi is indepen- dent from the other dimensions (Lemma 7). In the IDC heuristic, the DM compares alternatives dimension-by-dimension and Separability guarantees that these com- parisons can be represented by well-defined functions. Now, we are ready to state the main representation theorem.
Theorem 5 A binary relation on X satisfies Regularity and Separability if and only if it is an IDC relation with some continuous aggregatorW and continuous distance-based functions{fi}in=1such that for anyt∈ (−1,1)n,W(t)=W(t−n,0n)+tn
and fn(bn,xn)= bbnn−x−ann for any xn ∈ Xn = (an,bn]. Moreover, {fi}in=1 and W are unique.
Separability captures the idea of the IDC heuristic and guarantees dimension- specific distance-based functions f1, . . . , fnand an aggregatorW. Regularity guar- antees that these functions are well-behaved. Note that whenn =2, Theorem 5 also characterizes (3.1) sinceW(t1,t2)= t1+t2.
We can derive unique distance-based functions and an aggregator function up to a certain normalization. It is obvious that we cannot get any uniqueness without normalizing distance-based functions and an aggregator. For example, any common monotonic transformation of f andg, h(f)andh(g), satisfies (3.1). Therefore, we normalize one of the distance-based functions, e.g., fn. However, it is not necessary to specify fn on all points of Xn2. When n = 2, it is enough to normalize fn as
Xi
yi0 x0i
yi xi
Figure 3.2: Difference-i
fn(bn,xn)= bbnn−x−ann for any xn ∈ Xn. When n ≥ 3, we also need a normalization on W such that W(t) = W(t−n,0n) +tn for each t ∈ (−1,1)n. We discuss some specifications and properties of fiin the next subsection.
To gain an intuition of the proof of Theorem 5, we construct f andgfor (3.1) on choice over binary lotteries. Let X1 = (0,x] and X2 = (0,1]. For any x,y ∈ X1 with x ≥ y, set f(x,y) = 1− p whenever (x,p)∼ (y,1). By Regularity, f is a well-defined function (Lemma 6). For anyp,q ∈X2withq≥p, setg(q,p)= f(x,y) whenever(x,p)∼(y,q). Obviously, we can find two different pairs xand y and x0 andy0such that (x,p)∼(y,q)and(x0,p)∼(y0,q). Therefore, we need to show that f(x,y)= f(x0,y0). Take any two pairs xandyandx0andy0such that(x,p)∼(y,q) and (x0,p)∼(y0,q). Letr∈X2 such that (x,r)∼(y,1). By Separability, we have (x0,r) ∼ (y0,1). Then, by the construction of f, we obtain f(x,y)= f(x0,y0)=1−r which implies thatgis well-defined.