Chapter III: Theory of Decisions by Intra-Dimensional Comparisons
3.1 Introduction
Making a choice between two multidimensional alternatives is a difficult task unless one dominates the other. It is not surprising, therefore, that a decision maker adopts some procedure, or heuristic, to make this choice. One such heuristic called the Intra-Dimensional Comparison (IDC) heuristic has been documented in the experimental work of Tversky (1969). The IDC heuristic is a procedure that compares multidimensional alternatives dimension-by-dimension and makes a decision based on those comparisons. We develop an axiomatic model of the IDC heuristic and provide a general framework that is applicable to many different contexts, such as risky choice and social choice.
To illustrate the IDC heuristic, consider the following two scenarios. In the first scenario, a decision maker (henceforth, DM) has to choose a lottery from two binary lotteries. Let us denote by (x,p)the binary lottery that returns $x with probability p, and the prize $0 with probability 1−p. Suppose the DM compares binary lotteries ($11,0.43)and($10,0.45). There is no obvious answer to the question of how many dollars are equivalent to a 2% increase in the probability of winning. Instead, it is easier for the DM to compare the binary lotteries dimension-by-dimension (i.e., $11 with $10 and 43% with 45%) because the compared numbers represent the same attributes. In this scenario, the DM may prefer($11,0.43)over($10,0.45)because their winning probabilities are not very different and the former has a higher prize.
In the second scenario, a social planner (SP) needs to choose an allocation.
Suppose she wants to allocate $24 to three people and is considering two possible allocations: ($12,$4,$8)and($14,$3,$7)(person 1 gets either $12 or $14, person 2 gets either $4 or $3, and so on). Although all dimensions are expressed in dollars, it may be easier for the SP to compare the allocations dimension-by-dimension (i.e.,
$12 with $14, $4 with $3, and $8 with $7). In this scenario, the SP may choose ($12,$4,$8) over($14,$3,$7) because although person 1 is worse off, he still gets
$12, and the other two people are better off in the first allocation.
As we have seen in the above two scenarios, the IDC heuristic simplifies and
guides choice in individual decision making as well as in social choice. The IDC heuristic was documented in the experimental work of Tversky (1969). Tversky run the following experiment in the context of the first scenario. Subjects were asked to choose one lottery from each of all possible pairs froma=(5, 7
24),b=(4.5, 9
24), and c=(4, 11
24). Tversky (1969) obtained a systematic violation of transitivity: almost half of the subjects preferred the lottery with the higher payoff among adjacent pairs (aandborbandc), while on the extreme pair (aandc), they preferred the lottery with the higher winning probability.1
In the words of one of Tversky’s subjects in a post-experimental interview,
“There is a small difference between lotteries a andbor band c, so I would pick the one with higher payoff. However, there is a big difference between lotteries a andc, so I would pick the one with higher probability.” In other words, the subject compared prizes and probabilities separately and made a decision based on those comparisons; i.e., used the IDC heuristic.2
The goal of this paper is to develop an axiomatic model of the IDC heuristic and discuss its implications. The main representation theorem of the paper shows that under standard axioms, in addition to an axiom called Separability, there are dimension-specific functions that represent dimension-by-dimension comparisons and a function that aggregates those comparisons in order to make a decision.
Moreover, these functions are unique up to a certain normalization. Separability guarantees that comparisons within one dimension are independent from compar- isons within other dimensions.
Now we introduce our model of the IDC heuristic in the aforementioned two scenarios and discuss some implications of them. First, let us describe our model in the context of the first scenario: choice over binary lotteries. Consider a DM who has to choose between lotteries (x,p) and (y,q) where x>y and q>p. Our discussion above suggests that she compares xwith yandqwithpseparately. She then makes a decision based on numbers f(x,y)andg(q,p), where f(x,y)measures the advantage of x over yandg(q,p) measures the advantage ofqoverp. We can
1Therefore, we obtain a violation of transitivitya=(5, 7
24)b=(4.5, 9
24)c=(4,11
24)awhich is called theSimilarity Cycle. More generally, the similarity cycle is a triple(x,p),(y,q),(z,r) of binary lotteries such that x>y>zand(x,p)(y,q)(z,r)(x,p). While Tversky’s original experiment involved only a small number of subjects, this result was replicated by Lindman and Lyons (1978) and Budescu and Weiss (1987). Moreover, Day and Loomes (2010) produced a similar result by using different lotteries with real incentives.
2By using an eye-tracking experiment in the same setting with Tversky (1969), Arieli et al. (2011) found that when decision-making is difficult, subjects compare prize and probability separately.
imagine that if f(x,y)>g(q,p), then the prize-dimension becomes more salient, and she prefers (x,p) over(y,q)since x > y. If by contrast, f(x,y)<g(q,p), then the probability-dimension becomes more salient, and she prefers (y,q) over (x,p) sinceq > p. Finally, if f(x,y)=g(q,p), then she is indifferent between(x,p)and (y,q). More formally, we say that a binary relation on binary lotteries is anIDC relation if there are functions f andg such that for any binary lotteries (x,p) and (y,q),
(x,p) (y,q) if and only if f(x,y) ≥ g(q,p). (3.1) For example, when f(x,y) = 1− u(y)
u(x) andg(q,p) = 1− pq, (3.1) gives us Expected Utility Theory (EUT) preferences on binary lotteries. By allowing a more general form forg, the model can accommodate well-known deviations of EUT: theCommon Ratio Effect(a version of the Allais Paradoxes) and theSimilarity CycleandRegret Cycle(both violations of transitivity that have opposite directions). We are not aware of any other axiomatic model that can accommodate all deviations mentioned.3
Next, we define our general model in the context of the second scenario, social choice. A social planner (SP) for the society N = {1,2, . . . ,n} compares two allocationsx=(x1, . . . ,xn)andy=(y1, . . . ,yn), where xi andyiare the amounts of money that personireceives from the allocations. If the SP uses the IDC heuristic, then she asks each i to say how much he prefers xi over yi; in other words, she obtains a real number fi(xi,yi)that measures the advantage ofxioveryi for person i, and aggregates these real numbers. More formally, we say that a binary relation onn-vectors is anIDC relationif there exist functions{fi}ni=1andW such that for anyxandy,
x yif and only ifW f1(x1,y1), . . . , fn(xn,yn) ≥0. (3.2) Note that (3.2) generalizes (3.1). In contrast to the Bergson-Samuelson social welfare criterion4 in which the SP aggregates preferences before comparing alter- natives, in our model the SP aggregates preferences after she compares alternatives dimension-by-dimension. The IDC relations are (possibly) intransitive. For exam-
3The common ratio effect is a violation of the Independence Axiom such that(x,p)≺(y,q)and (x, αp)(y, αq)withx>yand 1> α >0. See Allais (1953) and Kahneman and Tversky (1979).
The Regret Cycle is a violation of transitivity such that(x,p)≺(y,q)≺(z,r)≺(x,p)withx>y>z, which is observed in the experiments of Loomes et al. (1991) and Day and Loomes (2010). We can generate the Similarity Cycle, the Regret Cycle, and the Common Ratio Effect with distance-based functionsg(q,p)=(1−p
q)[1−βp(q1−δ−p−µ)] and f(x,y)=1−u(y)
u(x) whereβ, µ, δ∈[0,1).
4 In the Bergson-Samuelson social welfare criterion, the SP comparesW u1(x1), . . . ,un(xn) andW u1(y1), . . . ,un(yn)
for given functionsu1, . . . ,un, andW (See Mas-Colell et al. (1995)).
ple, the model allows Condorcet Cycles in which the SP compares three allocations x,y, andz, andyis preferred tox,ztoy, butxtoz.5
The remainder of the paper is organized as follows. First, we discuss related literature in Section 3.1.1. In Section 3.2, we introduce the basic model and a behavioral foundation for (3.2). We also discuss some specifications and properties of fiin Section 3.2.2 and the proof of the main theorem is in Appendix C.
3.1.1 Related Literature
Representations similar to (3.2) are characterized in Bouyssou and Pirlot (2002).
They focus on binary aggregators, so they cannot have a uniqueness result. In other words, their representations have no cardinal meaning. Moreover, the implications of their representations for economic contexts and specifications are not worked out.
Now we discuss two axiomatic works on intransitive preferences related to our work. The first work is theSimilarity Relation Model(SRM) of Rubinstein (1988), a model of choice under risk that used an idea of the IDC heuristic.
SRM consists of two similarity relations on prizes and probabilities, denoted by
∼x and∼p respectively. Suppose a DM compares binary lotteries (x,p) and (y,q) where x > y and p< q. In SRM, the DM uses the following procedure: if she considers p and q to be similar (p ∼p q) and x and y not to be similar (x x y), then she prefers (x,p) over (y,q); if instead she considers x and y to be similar (x ∼x y) andpandqnot similar (pp q), then she prefers(y,q)over(x,p); finally, if she considers both x and y and p and q to be similar (x ∼x y and p ∼p q) or not similar (x x y and p p q), then the SRM procedure is not specified. As in the IDC heuristic, SRM compares prizes and probabilities separately and makes a decision based on these comparisons if possible. The main difference between our model and SRM is that the similarity relations∼x and∼p are exogenously given in SRM and are not unique, i.e., there could be multiple similarity relations that are compatible with one binary relation. Moreover, the procedure that is generated by the similarity relations is not complete.6 On the other hand, we fully characterize the set of IDC relations (which are complete), f and g are endogenously derived from each IDC relation, and they are unique.
5For example, suppose the SP wants to allocate $60 to three people and considers three possible allocations: x =($24,$20,$16), y =($16,$24,$20), and z =($20,$16,$24). Let fi(xi,yi)=
−fi(yi,xi)=xix−yiifor anyi andxi,yi withxi≥yi. SupposeW(f1,f2,f3)=f1+ f2+ f3. Then we obtain a Condorcet Cycle because2424−20+2020−16−24−16
24 =301 >0.
6There are cases in which the SRM procedure is almost never specified.
The second work is the Relative Discounting Model of Ok and Masatlioglu (2007) in intertemporal choice. They model a DM who compares intertemporal prospects(x,t)where(x,t)denotes an intertemporal prospect that gives $xat time t. Although their model does not build on the IDC heuristic, when there are only two dimensions, our model coincides with the Relative Discounting Model. More precisely, they have a representation similar to (3.1) in which the time-dimension corresponds to the probability-dimension in (3.1).
3.2 The Basic Model