Chapter V: The Order-Dependent Luce Model
5.6 Related Literature
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.
There is a non-axiomatic literature that proposes several models which can explain the similarity, compromise, and attraction effects. Rieskamp et al. (2006) is an excellent survey. Examples are Tversky (1972a), Roe et al. (2001) and Usher and McClelland (2004). The latter two papers proposedecision field theory, which allows for violations of Luce’s regularity axiom. The recent work by Natenzon (2010) presents a learning model, in which an agent learns about the utility of the different alternatives randomly and makes a choice with imperfect knowledge of these utilities. Natenzon’s model can explain all three effects. We shall not discuss these papers here, and focus instead on the more narrowly related axiomatic literature in economics. We separate literature in three categories.
1. Random Utility Models: The benchmark economic model of rational behav- ior for stochastic choice is the random utility model. Since Luce’s model is a special case of both the ODLM and random utility, the ODLM and random utility intersect.
However, the ODLM allows for the attraction effect while random utility models always satisfy regularity, so the ODLM is not a special case of random utility.
The recent paper by Gul et al. (2010) presents a random utility model in which object attributes play a key role as in Tversky (1972a). Their model has Luce’s form, but it applies sequentially, and in terms of its empirical motivation, it seeks to address the similarity effect.
2. Models with Bounded Rational Agents: A closely related paper is Echenique et al. (2013). In this paper, an order on alternatives also matters for random choice, and the model can explain the attraction and compromise effects, as well as violations of stochastic transitivity. In their paper, the source of violations of IIA is limited perception while the utilities are menu-independent.
Manzini and Mariotti (2014) study a stochastic choice model where attention is the source of randomness in choice while preferences are deterministic. Their model can explain the similarity and compromise effects as well as violations of stochastic transitivity.
The paper by Fudenberg et al. (2015) considers a decision maker who chooses a probability distribution over alternatives so as to maximize expected utility, with a cost function that ensures that probabilities are non-degenerate. One version of their model can accommodate the attraction effect, and one can accommodate the compromise effect.
3. Non-Stochastic Choices: Our model is more closely related to two lines of
research on choice theory. Before we discuss them, note that we can easily obtain a choice theoretic version of the ODLM in the following way: for any A∈A,
ais chosen from menu Aiffu a,R(a, A) >u b,R(b,A)
for eachb∈ A\ {a}. Similar to Section 5.3.1, we can also define choice-theoretic versions of increasing ODLMs and decreasing ODLMs.
The first line of research is on limited attention and consideration set. Masatli- oglu et al. (2012) attribute violations of WARP (the counterpart of IIA in determin- istic choice models) to the role of attention. They elicit revealed preference in the following way: when the choice from {x,y,z} is x and from {x,z} is z, then they conclude that x is revealed preferred to z. In contrast, we conclude that x has a higher ranking thanzin decreasing ODLMs (opposite of Observation 5). In fact, a choice-theoretic version of decreasing ODLMs is a special case of Masatlioglu et al.
(2012). However, choice-theoretic versions of increasing ODLMs are not special cases of Masatlioglu et al. (2012).
The second line of research is on framing effects. In particular, Rubinstein et al. (2006) and Salant and Rubinstein (2008) discuss the effect of different frames (e.g., different rankings over alternatives) while in our paper the ordering is fixed.
A more closely related paper is Yildiz (2012) which also discusses fixed ordering on alternatives. Since Yildiz focuses on choices in which the choice procedure is also influenced by the ordering on alternatives and an agent engage in some kind of sequential search, he obtains a very different model from ours. In particular, a random choice version of his model cannot have Luce’s Model as a special case because of the sequentiality.
Besides these two lines of research, there are several recent papers on choice theory that explain behavioral phenomena in Section 5.5. For example, Kamenica (2008) discusses model of context-dependent preferences and explains the attraction and compromise effects as well as the choice overload; Ok et al. (2014) discusses model of (endogenous) reference-dependent preferences and explains the attraction effect; De Clippel and Eliaz (2012)’s model produces the compromise and attraction effects as solutions of some bargaining problems.
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