Chapter IV: The Perception-Adjusted Luce Model
4.5 Compromise and Attraction Effects
Forced Choice
Another advantage of PALM’s treatment of the outside option is that it allows us to understand forced choice. In particular, the presence of the outside option allows us to compare environments in which agents are forced to make a choice with environments in which they are not forced to make a choice. Our model is consistent with the experimental results of Dhar and Simonson (2003) on the effects of forced choice on choice (Section 4.5.3). Moreover, we show that if an agent chooses not to make choice with high probability, then utility and perception are positively correlated (Section 4.6).
same brand; andzis MAXXUM 7000i, the top of the line offered by Minolta in this class of cameras.
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 4.1: Compromise effect in Simonson and Tversky (1992)
The agent’s choice set is {x,y} in Experiment 1 and {x,y,z} in Experiment 2. The experimental data show that the probability of choosing y increases when moving from Experiment 1 to 2 (see Figure 4.1). Simonson and Tversky (1992) call this phenomenon the compromise effect. As in Rieskamp et al. (2006), the compromise effect can be written as follows:
ρ(x,{x,y,z})
ρ(y,{x,y,z}) <1 ≤ ρ(x,{x,y})
ρ(y,{x,y}). (4.7)
Proposition 8 When x y % z, ρ(u,%) exhibits thecompromise effect(i.e., (4.7)) if and only ifu(y) >u(x)and
u(z)+u(x0) > u2(x)−u2(y)+u(x)u(y)
u(y)−u(x) ≥ u(x0). (4.8)
Proposition 8 results from a straightforward calculation so the proof is omitted.
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}. PALM can capture the compromise effect when we assume thaty is “in between”xand zwith respect to priority. One rationale forx y% z is familiarity. The basic camera model may be more familiar, while the top of the line is the least familiar.
4.5.2 Attraction Effect– A Violation of Regularity
PALM can accommodate violations of regularity. We focus on the attraction effect, a well-known violation of regularity. A famous example of the attraction effect is documented by Simonson and Tversky (1992) using the following experiment.
Consider our three alternatives again, x, y and z. Suppose now that y and z are
different variants of the same good: y is a Panasonic microwave oven (meaning a higher quality and expensive good6), while z is a more expensive version of y: z is dominated by y. The alternative x is an Emerson microwave oven (meaning a lower quality and cheap good). A more recent example, which we discussed in the introduction, is due to Doyle et al. (1999). As we mentioned in the introduction, the findings in Doyle et al.’s experiments fit the story in PALM particularly well.
Option Choices Exp. 1 Choices Exp. 2
x(Emerson) 57 % 27 %
y(Panasonic I) 43 % 60 %
z(Panasonic II) N/A 13 %
Figure 4.2: Attraction effect in Simonson and Tversky (1992)
Simonson and Tversky (1992) (p. 287) asked subjects to choose betweenxand y in Experiment 1, and to choose among x,y, and z in Experiment 2 (see Figure 4.2). They found that the share of subjects who chosey in Experiment 2 is higher than in Experiment 1. This finding is called theattraction effect. As in Rieskamp et al. (2006), the effect can be described as follows:
ρ(y,{x,y,z}) > ρ(y,{x,y}). (4.9)
Proposition 9 If x y % z and u(x) is large enough, then ρ(u,%) exhibits the attraction effect(i.e., (4.9)).
Proof of Proposition9: We have
ρ(y,{x,y,z}) > ρ(y,{x,y}) ⇔
⇔ q(y,{x,y,z})(1−q(x,{x,y,z})) > q(y,{x,y})(1−q(x,{x,y}))
⇔u(x) > p
(u(y)+u(z)+u(x0))(u(y)+u(x0))
The assumptionx y % zmeans that the Emerson microwavexis more salient than the Panasonic microwaves, perhaps because of its price. The first Panasonic
6Microwave ovens were at one point expensive; see “Money for nothing” by Dire Straits.
microwave y is at least as salient as z since there are the same brands. It is also possible to tell a story of familiarity for the microwaves experiment. The Emerson microwave xis likely to be the most familiar alternative since it is the cheapest and simplest model. In Doyle et al.’s experiments (as discussed in the introduction), perception is related to the familiarity of the brand of beans.
A different, symmetric, experiment would be to add an alternativet to enhance the choice of x. Sotcould be a more expensive version of x. Heath and Chatterjee (1995) found that one isless likely to observe the attraction effect when the third alternative is dominated by the low-quality alternative (x), compared to the high- quality alternative (y). More precisely, one is more likely to have ρ(y,{x,y,z}) >
ρ(y,{x,y})compared to ρ(x,{x,y,t}) > ρ(x,{x,y}). PALM is consistent with this finding: we cannot have ρ(x,{x,y,t}) > ρ(x,{x,y})whenx y.7
4.5.3 The Effects of Forced Choice
Dhar and Simonson (2003) run choice experiments in which agents may not have to make a choice. In their design, “no-choice” and “forced choice” are two experimental treatments. Under the no-choice option, subjects can opt not to make a choice. Under the forced-choice treatment, subjects must make a choice, as in the experiments described in the two previous sections. Dhar and Simonson show that the introduction of the no choice option weakens the compromise effect and decrease the relative share of an option that is “average” on all dimensions. In our model, not making a choice corresponds to choosing the outside option x0. We proceed to illustrate how PALM can capture the evidence presented by Dhar and Simonson.
Consider two PALM models, ρ = ρ(u,%) and ρf = ρ(uf,%f). Suppose that these two models only differ inu(x0). Thus,u(x) =uf(x)for anyx ∈ X and%=%f. We assume that ρ(x0,A) > ρf(x0, A)for all A∈ A. Roughly speaking, in the PALM
ρf, a decision maker chooses the outside option less often.
In PALM, choosing the outside option more frequently is tied to a larger utility of the outside option. In particular:
Condition♠: ρ(x0,{x,y}) > ρf(x0,{x,y}) iff ρ(x0,{x,y,z}) > ρf(x0,{x,y,z})iff u(x0) >uf(x0).
7However, it is still possible that the relative probability of choosing x increases; that is,
ρ(x,{x,y,t})
ρ(y,{x,y,t}) > ρ(x,ρ(x,{x,y}){x,y}), whentis added wherex%ty.
Proposition 10 Any PALM model satisfies Condition♠.
Recall our discussion of the outside option in PALM. There are two sources behind the choice of the outside option. One is the sequential nature of choice, and the other is the utility of the outside option. Proposition 10 says that the utility of the outside option is the source behind “global” increases in the probability of choosing the outside option.
In light of Proposition 10, we can trace the probability of opting out of an experiment, and not making a choice, to the incentives provided for participation in the experiment. In particular, consider the findings of Dhar and Simonson. Fix three alternativesx,y,z ∈ X. Suppose thatx y z. Soycan be interpreted as an
“average” option. Given our assumption on ρand ρf, and under Condition ♠, we assume thatu(x0) > uf(x0).
In first place, PALM can capture Dhar and Simonson’s finding that the no-choice option decreases the relative share of an average alternative (recall thatρrepresents the case where subjects exercise the outside “no-choice” option more):
Proposition 11 Ifu(x0) >uf(x0), then ρ(y,{x,y})
ρ(x,{x,y}) > ρf(y,{x,y})
ρf(x,{x,y}) and ρ(y,{x,y,z})
ρ(x,{x,y,z}) > ρf(y,{x,y,z}) ρf(x,{x,y,z}). Proof of Proposition 11: By a direct calculation, ρ(x,{x,y})ρ(y,{x,y}) = u(x)u(y) 1+ u(y)+u(x)u(x
0)
and ρ
f(x,{x,y})
ρf(y,{x,y}) = u(x)u(y) 1+ u(y)+u(x)uf(x0)
. Since f(t) = u(x)u(y) 1+ u(y)u(x)+t
is decreasing in t, we obtain u(x0) > uf(x0) if and only if ρ(x,{x,y})ρ(y,{x,y}) < ρρff((y,{x,{x,y})x,y}). Similarly,
ρ(x,{x,y,z})
ρ(y,{x,y,z}) = u(x)u(y) 1+ u(y)+u(z)+u(x)u(x
0)
and ρ
f(x,{x,y,z})
ρf(y,{x,y,z}) = u(x)u(y) 1 + u(y)+u(z)+u(x)uf(x0)
. Since g(t) = u(x)u(y) 1+ u(y)u(x)+u(z)+t
is decreasing in t, we obtainu(x0) > uf(x0) if and only if ρ(x,{x,y,z})
ρ(y,{x,y,z}) < ρρff(x,{(y,{x,y,z})x,y,z}).
In second place, PALM can capture Dhar and Simonson’s finding that the no- choice option weakens the compromise effect as follows:
Proposition 12 Ifu(x0) >uf(x0), then ρf(x,{x,y})
ρf(y,{x,y})
.ρf(x,{x,y,z})
ρf(y,{x,y,z}) > ρ(x,{x,y}) ρ(y,{x,y})
.ρ(x,{x,y,z}) ρ(y,{x,y,z}).