Chapter V: The Order-Dependent Luce Model
5.5 Behavioral Phenomena
5.5.1 Consistency with Violations of IIA–the Similarity Effect and Compro- mise Effects
The similarity and compromise effects are well-known deviations from Luce’s model (see Rieskamp et al. (2006) for a survey). In this section, we demonstrate how the ODLM can capture each of these phenomena.
2In Appendix E.5, we discuss how to completeR0.
The similarity and compromise effects are defined in the same kind of exper- imental setup. An agent makes choices from the menus {x,y} and {x,y,z}. The
“effects” relate to the consequences of adding the alternativez. The Similarity Effect and Debreu’s Red Bus/Blue Bus Example
Suppose that our three alternatives are such thatxandzare somehow very similar to each other, and clearly distinct from y. This setup is discussed by Tversky (1972a), building on a well-known example of Debreu (1960a) in which the agent makes a transportation choice and x and z are a red bus and a blue bus while y is a train.
Sincez is more similar to x, adding zhurts xmore than y. This effect is called the similarity effect, and can be formalized as follows:
p(x,{x,y,z})
p(y,{x,y,z}) < p(x,{x,y}) p(y,{x,y}).
In Debreu’s example, the agent is assumed to like bus and train equally; that is, p(x,{x,y}) = p(y,{x,y}) = 12, but when there are two buses the probability of choosing blue bus halves; that is,p(x,{x,y,z}) = 14 andp(y,{x,y,z}) = 12.
Observation 6: WhenyR z R x, the ODLM is consistent with the similarity effect.
In particular, whenu(x,2) = 2,u(x,3) = 1, u(y,1) = 2, then we obtain Derbeu’s example:
p(x,{x,y,z})
p(y,{x,y,z}) = u(x,3) u(y,1) = 1
2
< p(x,{x,y})
p(y,{x,y}) = u(x,2) u(y,1) =1. The Compromise Effect
Consider again three alternatives x, y, and z. Suppose that x and z are “extreme”
alternatives, while y represents a moderate middle ground, a compromise. In the experiment studied by Simonson and Tversky (1992), x is X-370, a very basic model of Minolta camera; y is MAXXUM 3000i, a more advanced model of the same brand; andzis MAXXUM 7000i, the top of the line offered by Minolta in this class of cameras.
In Experiment 1, the menu is{x,y}and x is chosen at least as frequently asy. However, in Experiment 2, the menu is{x,y,z}andyis chosen more often thanx.3
3Essentially, it is a stochastic version of preference reversal.
Model Price ($) Choices Exp. 1 Choices Exp. 2
x (X-370) 169.99 50% 22%
y (MAXXUM 3000i) 239.99 50 % 57%
z(MAXXUM 7000i) 469.99 N/A 21%
Figure 5.1: The Compromise effect in Simonson and Tversky (1992)
Simonson and Tversky (1992) call this phenomenon thecompromise effect. As in Rieskamp et al. (2006), the compromise effect can be written as follows:
p(x,{x,y,z})
p(y,{x,y,z}) <1 ≤ p(x,{x,y}) p(y,{x,y}).
Simonson and Tversky (1992)’s explanation for the compromise effect is that subjects are averse to extremes, which helps the “compromise” optionywhen facing the problem{x,y,z}.
Observation 7: ODLM can capture the compromise effect whenyRz Rx. Moreover, we can replicate Figure 5.1 with the following numbers: u(x,2) = u(y,1) = 1, u(x,3)=2257, andu(z,2)=2157.4
5.5.2 Consistency with Violation of Regularity–Attraction Effect
The ODLM can also accommodate violations of regularity, another property that Luce’s Model satisfy. In fact, regularity is the property that all Random Utility Models satisfy which requires that adding alternative to a menu weakly decreases the probability of choosing alternatives of the original menu. Formally,
Regularity: p(a,A) ≥ p(a,A∪ {b})for any A∈A, anda,b∈ X.
We focus on the well-known attraction effect (documented by Simonson and Tversky 1992) using the following experiment. Consider our three alternatives again,x,y, andz. Suppose now that yandzare different variants of the same good:
yis a Cross pen (meaning a higher quality pen), while zis a pen of regular quality:
yclearly dominates z. We give the alternativex a monetary value ($6). Then Simonson and Tversky (1992) (p.287) asked subjects to choose betweenx and y in Experiment 1 and to choose among x,y, and z in Experiment 2. They found
4Thenp(x,xy)=u(x,2)+u(y,1)u(x,2) =1+11 =0.5,p(x,xyz)=u(x,3)+u(y,1)+u(z,2)u(x,3) =22/57+1+21/5722/57 =0.22, andp(y,xyz)=u(x,3)+u(y,1)+u(z,2)u(y,1) =22/57+1+21/571 =0.57.
Option Choices Exp. 1 Choices Exp. 2
x ($6) 64 % 52 %
y(Cross pen) 36 % 46 %
z(Other pen) N/A 2 %
Figure 5.2: Attraction effect in Simonson and Tversky (1992)
that in Experiment 2, the share of subjects who chose y becomes higher than that in Experiment 1. This effect is called the attraction effect. As in Rieskamp et al.
(2006), the effect can be described as follows:
p(y,xyz) > p(y,xy).
Observation 8: The ODLM can capture the attraction effect whenyRz Rx. More- over, we can replicate Figure 5.2 with numbersu(x,2)= 64,u(y,1)=36,u(x,3)= 401623, andu(z,2)=3623.5
5.5.3 Violation of Stochastic Transitivity
The ODLM also allows for violations of weak stochastic transitivity. Formally, weak stochastic transitivity is defined as follows.
Weak Stochastic Transitivity: For any x,y,z ∈ X, ifp(x,xy) ≥ 1
2 andp(y,yz) ≥
1
2, thenp(x,x z) ≥ 1
2.
Violations of transitivity are consistently observed in lab experiments. For ex- ample, Figure 5.3 shows observed choice probabilities in the experiment of Tversky (1969). In the experiment, subjects were asked to choose between binary lotteries x = ($5, 7
24),y= ($4.5, 9
24), andz = ($4,11
24). Here($5, 7
24)denotes a binary lottery that gives $5 with probability 247 and gives nothing with probability 1724 and so on.6
Figure 5.3: Choice Probabilities in Tversky (1969) Gambles x andy yandz x andz
x = ($5,7/24) 67.5% 36%
y= ($4.5,9/24) 32.5% 65%
z= ($4,11/24) 35% 64%
5Thenp(y,xy)=u(x,2)+u(y,1)u(y,1) =0.36 andp(y,xyz)=u(x,3)+u(y,1)+u(z,2)u(y,1) = 36
401623+36+3623 =0.46.
6Figure 5.3 is directly calculated from Tversky (1969)’s results. Tversky’s result was replicated by Lindman and Lyons (1978), Budescu and Weiss (1987), and Day and Loomes (2010).
Observation 9: ODLM allows for violations for weak stochastic transitivity when yRz Rx. Moreover, we can replicate Figure 5.3 with numbersu(x,2)=1,u(y,1)=1327, u(z,1)=169.
5.5.4 Choice Overload
Thechoice overloadis a scenario documented in both lab and field experiments, where an increase in the number of alternatives in menu leads to adverse conse- quences such as a decrease in the motivation to choose or the satisfaction with the finally chosen option (e.g., Chernev 2003 and Iyengar and Lepper 2000). One of usual explanations for the choice overload is that having too many alternatives makes it hard to choose (or find) the good alternative. Here we demonstrate that adding a new alternative into a menu may lead to a decrease in the agent’s satisfaction with the his chosen option even if added alternative does not decrease the average level of utility of the menu.
We can convey the main intuition by just considering three alternatives: x,y, and z. Take an ODLM (u,R) such thatu(a,i) = w(i)·u(a) wherew(i)is strictly decreasing in i. The following observation shows that adding alternative z into menu the{x,y}decreases the expected utility of menu even if the utility ofzis high enough.
Observation 10: Suppose x Rz Ry,u(y) > u(x), and the utility of zis equal to the expected utility of the menu{x,y}; that is,u(z) = p(x,{x,y})·u(x)+p(y,{x,y})· u(y). Ifw(3)is small enough, then
p(x,{x,y})·u(x)+p(y,{x,y})·u(y) >
p(x,{x,y,z})·u(x)+ p(y,{x,y,z})·u(y)+p(z,{x,y,z})·u(z).
Intuitively, addingzmakes it harder to choose (or find) the best alternativeybecause y is the last alternative under the ordering R. If Ris related to the search process that agents use to make consumption choices, then the intuition of Observation 10 is consistent with the usual explantation for the choice overload.