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Discrete Choice and Rational Inattention: a General Equivalence Result
In many situations where agents must make decisions under uncertainty, information acquisition is costly (involving pecuniary, time, or psychic costs); therefore, agents may rationally choose to remain imperfectly informed about the available options. This idea underlies the theory of rational inattention, which has become an important paradigm for modeling boundedly rational behavior in many areas of economics (Sims (2003, 2010)). In this paper, our main contribution is to establish a general equivalence between additive random utility discrete choice and rational inattention models. Matejka and McKay (2015) showed that when information costs are modelled using the Shannon entropy function, the resulting choice probabilities in the rational inattention model take the multinomial logit (MNL) form. In order for the rational inattention model to generate non-MNL choice probabilities, we need to generalize the information cost function beyond the Shannon entropy function assumed in much of the existing literature. We do this by exploiting convex-analytic properties of the additive random utility model to demonstrate a duality between discrete choice and rational inattention models.\footnote{Throughout this paper, we will use the terms “additive random utility model” and “discrete choice model” interchangeably.}
Specifically, we introduce a class of Generalized Entropy Rational Inattention (GERI) models.\footnote{ This complements work by HebertWoodford16, who also consider generalizations of the information cost function.} In GERI models, the information cost functions are constructed from a class of “generalized entropy” functions; these generalized entropy functions are, essentially, “dual” to the class of random utility discrete choice models; precisely, the generalized entropy functions are the convex conjugate functions to the surplus functions in any arbitrary general additive random utility model. Hence, GERI models naturally yield choice probabilities that can equivalently be generated from general additive random utility models; the resulting choice probabilities can take forms far beyond the multinomial logit, including specifications such as nested logit, multinomial probit, and so on, which are often employed in empirical work.
Importantly, these generalized entropy functions allow for random utility models in which the random shocks are dependent across options; this corresponds to information cost functions that exhibit information spillovers across options with shared features, which may be reasonable in many decision environments. In contrast, the multinomial logit model assumes independent shocks; correspondingly, the Shannon entropy function precludes information spillovers.
The paper is organized as follows. Section 2 presents insights into the fundamental convex-analytic structure of the additive random utility discrete choice model. Using this structure, we formulate a class of generalized entropy functions and present key results about them. Section 3 introduces the rational inattention model. We show how the generalized entropy functions can be used to define the information cost function in the rational inattention model. Then we present the key result from this paper, which establishes the equivalence between choice probabilities emerging from the discrete choice model, and those emerging from the rational inattention model based on the generalized entropy functions. Section 4 discusses an example while Section 5 concludes.
Notation: Throughout this paper, for vectors $\mathbf{a}$ and $ \mathbf{b}$, we use the notation $\mathbf{a}\cdot \mathbf{b}$ to denote the vector scalar product $\sum_i a_ib_i$. $\Delta$ denotes the unit simplex in $ \mathbb{R}^{N}$.
Consider a decision-maker (DM) making discrete choices among a set of $ i=1,\ldots ,N$ options, where, for each option $i$, the utility is given by
where $\mathbf{\tilde{v}}=(\tilde{v}_{1},\ldots ,\tilde{v}_{N})$ is deterministic and $\boldsymbol{\epsilon }=(\epsilon _{1},\ldots ,\epsilon _{N})$ is a vector of random utility shocks. This is the classic additive random utility framework pioneered by McFadden78.
An important concept in this paper is the surplus function of the discrete choice model McFadden81, defined as
Under Assumption (ref), $W(\mathbf{\tilde{v}})$ is convex and differentiable\footnote{ The convexity of $W(\cdot )$ follows from the convexity of the max function. Differentiability follows from the absolute continuity of $\boldsymbol{ \epsilon }$.} and the choice probabilities coincide with the derivatives of $ W(\mathbf{\tilde{v}})$:
or, using vector notation, $\mathbf{q}(\mathbf{\tilde{v}})=\nabla W(\mathbf{ \tilde{v}})$. This is the Williams-Daly-Zachary theorem in the discrete choice literature McFadden78, McFadden81.
As a running example, we consider the familiar logit model. When the $ \epsilon _{i}$'s are distributed i.i.d. across options $i$ according to the type 1 extreme value distribution, then the resulting choice probabilities take the familiar multinomial logit form: $q_{i}(\mathbf{\tilde{v}})=e^{ \tilde{v}_{i}}/\sum_{j}e^{\tilde{v}_{j}}$. Assumption 1 above leaves the distribution of the $\epsilon $'s unspecified, thus allowing for choice probabilities beyond the multinomial logit case. Importantly, it accommodates arbitrary correlation in the $\epsilon_{i}$'s across choices, which is reasonable and realistic in applications.
We define a vector-valued function $\mathbf{H}(\cdot)=(H_1(\cdot),...,H_N( \cdot)):\mathbb{R}_+^N \mapsto \mathbb{R}_+^N$ as the gradient of the exponentiated surplus, i.e.
From the differentiability of $W$ and the Williams-Daly-Zachary theorem it follows that the choice probabilities emerging from any random utility discrete choice model can be expressed in closed-form in terms of the $ \mathbf{H}$ function as:\footnote{ By direct differentiation of ((ref)), and applying the Williams-Daly-Zachary theorem, we have $q_i(\tilde{\mathbf{v}})=H_i(e^{W( \tilde{\mathbf{v}})})/e^{W(\tilde{\mathbf{v}})}$ for all $i$. Imposing the summability restriction $\sum_i q_i(\tilde{\mathbf{v}})=1$ we have $\sum_i H_i(e^{W(\tilde{\mathbf{v}})})=e^{W(\tilde{\mathbf{v}})}$ leading to Eq. ( (ref)).}
For the multinomial logit case, the surplus function is $W(\mathbf{\tilde{v}} )=\log \left( \sum_{i=1}^N e^{\tilde{v}_{i}}\right) $, implying that $ H_{i}(e^{\tilde{\mathbf{v}}})=e^{\tilde{v}_{i}}$. Thus, for this case Eq. ( (ref)) becomes the multinomial logit choice formula.
The function $\mathbf{H}$ is globally invertible (see Lemma (ref) in the Appendix), and we introduce a function $\mathbf{S}$ defined as the inverse of $\mathbf{H}$,
For reasons that will be apparent below, we refer to $\mathbf{S}$ as a generator function. There is a close relationship between the function $ \mathbf{S}(\cdot )$ and the surplus function $W(\mathbf{\tilde{v}})$ of the corresponding discrete choice model: as the next proposition establishes, the surplus function $W(\cdot)$ and the generator function $\mathbf{S}(\cdot) $ are related in terms of convex conjugate duality Rockafellar1970.\footnote{ For a convex function $g(\mathbf{x})$, its convex conjugate function is defined as $g^*(\mathbf{y})=\max_{\mathbf{x}} \left\{\mathbf{x}\cdot \mathbf{ y} - g(\mathbf{x})\right\}$, which is also convex. Roughly speaking, the gradients (or sub-gradients, in case of non-differentiability) of $g(\mathbf{ x})$ and $g^*(\mathbf{y})$ are inverse mappings to each other.}
Parts (i) and (ii) establish a specific structure of the surplus function $W$ and its convex conjugate $W^*$; this is new in the literature on random utility models, and may be of independent interest. We use this structure to define the class of generalized entropy functions. To see how this works, consider again the multinomial logit model, for which $\mathbf{H}$ is the identity, implying that the corresponding generator function $\mathbf{S}( \mathbf{q})=\mathbf{q}$ is also just the identity. Then by Proposition (ref)(ii), the negative convex conjugate function is $-W^{\ast }( \mathbf{q})=-\mathbf{q}\cdot \log \mathbf{q}=-\sum_i q_i \log q_i$, which is just the Shannon1948 entropy function.
Generalizing from this, Proposition (ref)(ii) shows how the conjugate function for any discrete choice model can be generated by the function $\mathbf{S} $. Therefore we refer to the negative conjugate function $-W^{\ast }(\mathbf{q})=-\mathbf{q}\cdot \log \mathbf{S}(\mathbf{q} )=-\sum_i q_i \log S_i(\mathbf{q})$ for any general discrete choice model as a generalized entropy function. Comparing the generalized and Shannon entropies, the former allows for cross-effects, in the sense that the choice probability for option $j$, $q_j$, enters the entropic term for option $i$, $ S_i(\mathbf{q})$. As we will see below, these cross-effects allow for information spillovers when we use these generalized entropy functions to construct rational inattention models.
Proposition (ref)(iii) provides an alternative representation of the surplus function from a random utility model, in addition to Eq. ((ref)). It illustrates a close connection between $-W^*( \mathbf{q})$ and the joint distribution of $\boldsymbol{\mathbf{\epsilon }}$ , the random utility shocks, which aids interpretation of the generalized entropy function. Specifically, Eq. ((ref)) implies that the surplus function can be written as
Combining this with ((ref)), we obtain an alternative expression for the generalized entropy function, as a choice probability-weighted sum of expectations of the utility shocks $\boldsymbol{ \mathbf{\epsilon}}$:\footnote{ See ChiongEtAl2016. Additionally, we conjecture that $\log S_{i}( \mathbf{q})=-\mathbb{E}[\epsilon _{i}|u_{i}\geq u_{j},j\neq i]\quad \mbox{for $i=1,\ldots,N$}$. For the multinomial logit case, corresponding to $\mathbf{S}(\mathbf{q})=\mathbf{q}$, McFadden78 showed that $\gamma -\log q_{i}=\mathbb{E}[\epsilon _{i}|u_{i}\geq u_{j},j\neq i]$, for $\gamma $ being Euler's constant.}
In this way, different distributions for the utility shocks $\boldsymbol{ \mathbf{\epsilon}}$ in the random utility model will imply different generalized entropy functions, and vice versa.
We conclude this section enumerating some properties of the generator functions $\mathbf{S}(\cdot)$, which will be important in what follows.
The possibility of zero choice probabilities will play a role in what follows. We impose an additional regularity assumption on the generator functions $\mathbf{S}$.
This assumption is satisfied for the generator functions $\mathbf{S}$ corresponding to many familiar additive random utility models, including the multinomial logit and the nested logit models.\footnote{ In fact, the necessity part of Assumption (ref) arises immediately from the results in this section. As $\tilde{v}_{i}\rightarrow -\infty $, $q_{i}(\tilde{\mathbf{v}})\rightarrow 0$, which by ((ref)) implies that $H_{i}\left( e^{\tilde{\mathbf{v}}}\right) \rightarrow 0$. Then, since $\log \mathbf{S}(\mathbf{q}(\tilde{\mathbf{v}}))=\tilde{\mathbf{v }}-\log \sum_{j}H_{j}(e^{\tilde{\mathbf{v}}})$, we have $\log S_{1}(q)\rightarrow -\infty $ (by homogeneity of $\mathbf{H}$, we may suppose that $\log \sum_{j}H_{j}(e^{\tilde{\mathbf{v}}})$ is a constant).}
We now introduce the rational inattention model. The decision maker is again presented with a group of $N$ options, from which he must choose one. Each option has an associated payoff $\mathbf{v}=(v_{1},...,v_{N})$, but in contrast to the additive random utility model, the vector of payoffs is unobserved by the DM. Instead, the DM considers the payoff vector $\mathbf{V} $ to be random, taking values in a set $\mathcal{V}\subset \mathbb{R}^{N}$; for simplicity, we take $\mathcal{V}$ to be finite. The DM possesses some prior knowledge about the available options, given by a probability measure $ \mu (\mathbf{v})=\mathbb{P}(\mathbf{V}=\mathbf{v})$.
The DM's choice is represented as a random action $\mathbf{A}$ that is a canonical unit vector in $\mathbb{R}^{N}$. The payoff resulting from the action is $\mathbf{V}\cdot \mathbf{A}$, namely that value of the entry in $ \mathbf{V}$ indicated by the action $\mathbf{A}$. The problem of the rationally inattentive DM is to choose the conditional distribution $\mathbb{ P} (\mathbf{A}|\mathbf{V})$, balancing the expected payoff against the cost of information.
Denote an action by $i$ and write $p_{i}(\mathbf{v})$ as shorthand for $ \mathbb{P} \left( \mathbf{A}=i|\mathbf{V}=\mathbf{v}\right) $. Denote also the vector of choice probabilities conditional on $\mathbf{V}=\mathbf{v}$ by $\mathbf{p}(\mathbf{v})=(p_{1}(\mathbf{v}),\dots ,p_{N}(\mathbf{v}))$, and let $\mathbf{p}(\cdot )=\{\mathbf{p}(\mathbf{v})\}_{\mathbf{v}\in\mathcal{V} } $ denote the collection of conditional probabilities. The DM's strategy is a solution to the following variational problem:
The previous literature has used the Shannon entropy to specify the information cost, which connects the rational inattention model to the multinomial logit model. We review these results in the next Section (ref). Then in Section (ref) we introduce generalized entropy to the problem. This connects the rational inattention model to general additive random utility models.
The key element in the program above is the cost of information. Much of the previous literature has utilized the mutual (Shannon) information between payoffs $\mathbf{V}$ and the actions $\mathbf{A}$ to measure the information costs. Denote the Shannon entropy by $\Omega (\mathbf{q})=-\mathbf{q}\cdot \log \mathbf{q}$. Denote also the unconditional choice probabilities by $ p_{i}^{0}=\mathbb{E}p_{i}(\mathbf{V})$ and $\mathbf{p}^{0}=(p_{1}^{0},\dots ,p_{N}^{0})$. Then the mutual (Shannon) information between $\mathbf{V}$ and $\mathbf{A}$ may be written as
Accordingly, we can specify the information cost as $\lambda \kappa (\mathbf{ p},\mu )$ where $\lambda >0$ is the unit cost of information. As the distribution of payoffs is unspecified, we may take $\lambda =1$ at no loss of generality. The choice strategy of the rationally inattentive DM is the distribution of the action $\mathbf{A}$ conditional on the payoff $\mathbf{V} $\ that maximizes the expected payoff less the cost of information, which is the solution to the optimization problem
subject to
Solving this, the DM finds conditional choice probabilities
that satisfy $p_{i}^{0}=\mathbb{E}p_{i}(\mathbf{V})$. It is an important feature of the rational inattention model that some $p_{i}^{0}$ may be zero, in which case the corresponding $p_{i}\left( \mathbf{v}\right) $ are also zero. Then the rational inattention model implies the formation of a consideration set, comprising those options that have strictly positive probability of being chosen (cf. CaplinDeanLeahy16).
Under the convention that $\log 0=-\infty $ and $\exp \left( -\infty \right) =0$, we may rewrite ((ref)) as
where $\tilde{v}_{i}=v_{i}+\log p_{i}^{0}$. This may be recognized as a multinomial logit model in which the payoff vector $\mathbf{\tilde{v}}$ is $ \mathbf{v}$ shifted by $\log \mathbf{p}^{0}$. For options that are not in the consideration set, the shifted payoff is $\tilde{v}_{i}=-\infty $. From the perspective of the multinomial logit model these options have zero probability of maximizing the random utility ((ref)) and they have effectively been eliminated from the model.
In this paper we generalize the preceding equivalence result between rational inattention and multinomial logit. To achieve that, we replace the Shannon entropy by the generalized entropy introduced in Section (ref) above. Since each generalized entropy implies a corresponding discrete choice model (Proposition 2), it turns out that each RI model with an information cost derived from a generalized entropy will generate choice probabilities consistent with a corresponding discrete choice model (Proposition 4 below); this implies that any additive random utility discrete choice model can be microfounded by a corresponding rational inattention model, thus generalizing substantially the results in the previous section.
We begin by generalizing the rational inattention framework described above, using generalized entropy in place of the Shannon entropy. Specifically, we let $\mathbf{S}$ be the entropy generator corresponding to some additive random utility model satisfying Assumptions (ref) and (ref), and define $\Omega _{\mathbf{S}}\left( \mathbf{p}\right) =- \mathbf{p}\cdot \log \mathbf{S}\left( \mathbf{p}\right) $ as the corresponding generalized entropy. We define accordingly a general information cost by
A Generalized Entropy Rational Inattention{\ (GERI)} model describes a DM who chooses the collection of conditional probabilities $\mathbf{p} \left( \cdot \right) =\{\mathbf{p}(\mathbf{v})\}_{\mathbf{v}\in \mathcal{V}}$ to maximize his expected payoff less the general information cost
The following proposition characterizes the optimal solution to the GERI model.
Part (i) of the proposition shows that the solution of the GERI model involves a fixed point problem; in what follows, we assume that a solution exists. Part (iii) illustrates the close connection between convex analysis and the GERI problem. To see this, note that the GERI information cost function may be written as
Hence, given $\mathbf{p}^{0}$, the conditional choice probabilities $\mathbf{ p}(\mathbf{v})$ can be generated, for each $\mathbf{v}\in \mathcal{V}$, by the problem
the optimized value of which, by Proposition (ref)(iii), is
corresponding to Proposition (ref)(iii).
It is worth remarking that some of the optimal unconditional choice probabilities may be zero. For these options, the corresponding conditional choice probabilities will also be zero for all $\mathbf{v}$.\footnote{ To see this, consider the solution to the GERI problem given in Eq. ((ref)) and define $\mathbf{\tilde{v}}=\mathbf{v}+\log\mathbf{S}( \mathbf{p}^0)$. Let $p_i^0=0$. Then by assumption (ref) it follows that $\log S_i(\mathbf{p}^0)=-\infty$, or equivalently, $\tilde{v} _i\longrightarrow -\infty$ and hence $p_i(\mathbf{v})=0$ for all $\mathbf{v} \in \mathcal{V}$.} The rational inattention model then also describes the formation of consideration sets, i.e. the set of options that are chosen with positive probability.\footnote{ Because of the possibility of zero choice probabilities for some options, GERI models can also generate failures of the “regularity” property (adding an option to a choice set cannot increase the choice probability for any of the original choices). See section (ref) in the Appendix for an example.}
While Proposition (ref) does not characterize explicitly the optimal consideration set emerging from a GERI\ model, the following corollary describes one important feature that it has, namely that it excludes options that offer the lowest utility in all states of the world.
For the special case of Shannon entropy (when $\mathbf{S}$ is the identity function), the result can be strengthened even further. Corollary (ref) in the Appendix shows that in that case, an option that is dominated by another option in all states of the world will not be in the optimal consideration set.
We now establish the central result of this paper, namely the equivalence between additive random utility discrete choice models and rational inattention models. In particular, we show that the choice probabilities generated by a GERI model lead to the same choice probabilities as a corresponding additive random utility model and vice versa.
Combining the choice probabilities $p_{i}(\mathbf{v})$ in ((ref) ) from the GERI\ model and the choice probabilities $q_{i}(\tilde{\mathbf{v}} )$ in ((ref)) from the additive random utility model, we find that if payoffs are related by
then the two models yield the same choice probabilities
Given a GERI\ model with payoffs $\mathbf{v}\in \mathcal{V}$ and unconditional choice probabilities $\mathbf{p}^{0}$, we may then use ((ref)) to construct deterministic utility components $\tilde{ \mathbf{v}}$ for the additive random utility model. If the GERI model has some zero unconditional choice probabilities $p_{i}^{0}$, then Assumption (ref) ensures that $p_{i}(\mathbf{v})=0$ if and only if $ q_{i}(\tilde{\mathbf{v}})=0$. The additive random utility model that corresponds to the GERI\ model is then an extended additive random utility model in which some deterministic utility components are minus infinity.
Conversely, given an additive random utility model with flexible generator $ \mathbf{S}$ and a prior distribution $\tilde{\mu}$ of the deterministic utility components $\tilde{\mathbf{v}}\in \mathcal{\tilde{V}}$, define $ \mathbf{p}^{0}=\mathbb{E}\mathbf{q}(\tilde{\mathbf{v}})$ and note that all $ p_{i}^{0}>0$. Then define $\mathbf{v}$ using ((ref)) and define similarly $\mu $ and $\mathcal{V\ }$using the same location shift $ \log \mathbf{S}(\mathbf{p}^{0})$. By the same argument as before, the GERI\ model with payoffs $\mathbf{v}\in \mathcal{V}$, prior $\mu $ and flexible generator $\mathbf{S}$ for the generalized entropy has the same choice probabilities as the additive random utility model.
Hence, we have shown the following proposition.
In Section (ref), we will apply this proposition to study a GERI model in which the choice probabilities are equivalent to those from a nested logit discrete choice model.
We have shown that the generalized rational inattention model is always equivalent to an additive random utility model and conversely that the generalized rational inattention model may provide a boundedly rational foundation for any additive random utility model. The key to this result is the generalization of the information cost function $\kappa _{\mathbf{S}}( \mathbf{p}\left( \cdot \right) ,\mu )$ using generalized entropy as defined in Eq. ((ref)). It is then natural to ask whether $\kappa _{\mathbf{S}}(\mathbf{p}\left( \cdot \right) ,\mu )$ has the properties that one would desire for an information cost. In this section we show that $ \kappa _{\mathbf{S}}(\mathbf{p}\left( \cdot \right) ,\mu )$ does indeed possess two reasonable and desirable properties of cost functions that have been discussed in the existing literature (cf. deOliveira2015, HebertWoodford16), thus providing normative support for the GERI framework.
First, when $\mathbf{A}$ and $\mathbf{V}$ are independent, then the action $ \mathbf{A}$ carries no information about the payoff $\mathbf{V}$. In that case the information cost should be zero, i.e.
Second, the mutual Shannon information $\kappa (\mathbf{p}\left( \cdot \right) ,\mu )$ is a convex function of $\mathbf{p}$. This is useful as it ensures a unique solution to the problem of the rationally inattentive DM. We show that the information cost $\kappa _{\mathbf{S}}(\mathbf{p}\left( \cdot \right) ,\mu )$ has a slightly weaker property, namely that it is convex on sets where $\mathbb{E}\mathbf{p}(\mathbf{V})$ is constant.
The mutual Shannon information $\kappa (\mathbf{p}\left( \cdot \right) ,\mu ) $ satisfies these two properties. The next proposition establishes that the information cost defined in ((ref)) using the generalized entropy functions also satisfies these properties.
From an applied point of view, an important implication of Proposition (ref) is that it allows us to formulate rational inattention models that have complex substitution patterns, beyond the multinomial logit case. In this example, we consider a GERI model with an information cost function derived from a nested logit discrete choice model. The nested logit choice probabilities are consistent with a discrete choice model in which the utility shocks $\mathbf{\epsilon }$ are jointly distributed in the class of generalized extreme value distributions. Among applied researchers, the nested logit model is often preferred over the multinomial logit model because it allows some products to be closer substitutes than others, thus avoiding the \textquotedblleft red bus/blue bus\textquotedblright\ criticism. \footnote{ See, for instance, Maddalabook, and AndersondePalmaThisse1992.}
We partition the set of options $i\in \left\{ 1,\ldots ,N\right\} $ into mutually exclusive nests, and let $g_{i}$ denote the nest containing option $ i$. Let $\zeta _{g_{i}}\in (0,1]$ be nest-specific parameters. For a valuation vector $\mathbf{\tilde{v}}$, the nested logit choice probabilities are given by:
The $\mathbf{S}$ function corresponding to a nested logit model is
Using this, and applying Proposition (ref), the nested logit choice probabilities ((ref)) are also equivalent to those from a GERI model with valuations
The $\mathbf{S}$ function for the nested logit model in Eq. ((ref)) has several interesting features, relative to the Shannon entropy. First, Eq. ((ref)) allows us to write the generalized entropy $\Omega _{\mathbf{S}}(\mathbf{p})$ as
The first term in Eq ((ref)) captures the Shannon entropy within nests, whereas the second term captures the information between nests. According to this, we may interpret Eq. ((ref)) as an \textquotedblleft augmented\textquotedblright\ version of Shannon entropy.
Second, when the nesting parameter $\zeta _{g_{j}}=1$, then $\mathbf{S}$ is the identity function ($S_{j}(\mathbf{p})=p_{j}$ for all $j$), corresponding to the Shannon entropy. When $\zeta _{g_{j}}<1$, then $S_{j}(\mathbf{p})\geq p_{j}$; here, $\mathbf{S}(\mathbf{p})$ behaves as a probability weighting function which tends to overweight options $j$ belonging to larger nests. At the extreme $\zeta _{g_{j}}\rightarrow 0$, all options within the same nest effectively collapse into one aggregate option and become perfect substitutes.
From the discrete choice perspective, nested logit choice probabilities allow for correlation in the utility shocks ($\epsilon $'s) corresponding to the different choice options. Analogously, in an information cost function constructed from the nested logit $\mathbf{S}$ function in Eq. ((ref)), there will be a common cost component across all options belonging to the same nest, corresponding to the term $(\sum_{j\in g_{j}}p_{j})^{1-\zeta _{g_{j}}}$ which is common to all $S_{j}(\mathbf{p})$ for $j\in g_{j}$. From an information processing perspective, this suggests that there are spillovers in gathering information for options in the same nest. Information spillovers across choices arise in many decision environments. For example, a supermarket shopper gains information about common features of the vegetables, such as the average freshness and quality, while looking at any of them. In animal foraging, animals who have information about presence of predators in one grazing site also use that information to update about predator presence at other nearby sites.
For the Shannon entropy, in contrast, these common terms do not exist, so that there are no spillovers across options in information processing. From a behavioral point of view, then, more correlated utility shocks makes each option's signal harder to distinguish -- there is more redundant information -- implying that multinomial logit choice probabilities, which would ignore this correlation, manifest a type of correlation neglect.
To illustrate this point, we compute a GERI model using the nested-logit cost function. (This requires solving the fixed point equation ((ref)).) In this example, there are five options, in which the valuations $\mathbf{v}=(v_{1},v_{2},\ldots ,v_{5})^{\prime }$ are drawn i.i.d. uniformly from the unit interval. We assume that options (1,2,3) are in one nest, and options (4,5) are in a second nest. With this specification, all five options are a priori identical, and have equal probability of being the option with the highest valuation. Hence, we might expect that any non-uniform choice probabilities should reflect underlying asymmetries in the information cost function.
In Table (ref), we report the average choice probability for each option according for several specifications of the nested logit cost function. In the top panel, we set $\zeta _{1}=\zeta _{2}=1$, corresponding to the multinomial logit model. In the bottom panel, we set $ \zeta _{1}=\zeta _{2}=0.5$.
As we expect, we see that the average choice probabilities are identically equal to 0.2 across all five options in the multinomial logit case. As we remarked before, this reflects the feature of the Shannon-based information cost function ($S_{i}(\mathbf{p})=p_{i}$) in which information costs are separable across all five choices.\footnote{ In the nested logit case, we obtained the unconditional distribution by iterating over the fixed point relation $\mathbf{p}^{0}=\mathbb{E}\mathbf{p}( \mathbf{V})$, starting from the multinomial logit distribution.} Unlike the multinomial logit case, we see that choice probabilities are higher for the options 1,2 and 3, which constitute the larger nest, and smaller for options 4,5 which constitute the smaller nest. (However, within nest, the choice probabilities are identical.) The non-uniform choice probabilities for the nested logit model reflect the cost spillovers across options in the structure of the nested logit information cost function.
Moreover, the performance of the two models is surprisingly different. Under the multinomial logit specification, the overall efficiency -- defined as the average probability of choosing the option with the highest valuation -- is 28%. The overall efficiency for the nested logit, however, is higher, being over 35%.
This simple example demonstrates the substantive importance of allowing for information cost functions beyond the Shannon entropy, which leads to multinomial logit choice probabilities. Obviously, it makes a difference for a DM to be processing information using the nested logit cost function, as compared to the Shannon cost function, as the highest valuation option is chosen with higher probability on average using the nested logit cost function.
The central result in this paper is the observational equivalence between a random utility discrete choice model and a corresponding Generalized Entropy Rational Inattention (GERI) model. Thus the choice probabilities of any additive random utility discrete choice model can be viewed as emerging from rationally inattentive behavior, and vice-versa; we can go back and forth between the two paradigms.\footnote{ In a similar vein, Webb16 demonstrates an equivalence between random utility models and bounded-accumulation or drift-diffusion models of choice and reaction times used in the neuroeconomics and psychology literature.} Then, in order to apply an additive random utility discrete choice model, it is no longer necessary to assume that decision makers are completely aware of the valuations of all the available options. This is important, as it is clearly unrealistic to expect that decision makers to be aware of all options in a large set of options.
The underlying idea is that, by exploiting convex analytic properties of the discrete choice model, we show a \textquotedblleft duality\textquotedblright\ between the discrete choice and GERI models in the sense of convex conjugacy. Precisely, the surplus function of a discrete choice model has a convex conjugate that is a generalized entropy. Succinctly, then, GERI models are rational inattention problems in which the information cost functions are built from the convex conjugate functions of some additive random utility discrete choice model.
A few remarks are in order. First, the equivalence result in this paper is at the individual level, hence it also holds for additive random utility models with random parameters, including the mixed logit or random coefficient logit models which have been popular in applied work.\footnote{ See, for instance, BLP1995, McFaddenTrain00, FoxEtAl12.} Any mixed discrete choice model such as these is observationally equivalent to a mixed GERI model.
In addition, there is also a connection between the results here and papers in the decision theory literature. The GERI optimization problem ((ref)) bears resemblance to the variational preferences that MaccheroniEtal06 propose to represent ambiguity averse preferences, as well as to the revealed perturbed utility paradigm proposed by FudenbergEtAl16 to model stochastic choice behavior. GulNatenzonPesendorfer14 shows an equivalence between random utility and an \textquotedblleft attribute rule\textquotedblright\ model of stochastic choice. The main point in this paper is to establish a duality between rational inattention models and random utility discrete choice models, which results in observational equivalence of their choice probabilities. A similar duality might arise between random utility discrete choice models and these other models from decision theory.
Finally, there are rational inattention models outside the GERI framework; that is, rational inattention models with information cost functions outside the class of generalized entropy functions introduced in this paper.\footnote{ As an example, the function $g(\mathbf{p})=-\sum_{i=1}^N \log(p_i)$ is not a generalized entropy function; thus a rational inattention model using this as an information cost function would lie outside the GERI framework.} Obviously, choice probabilities from these non-GERI models would not be equivalent to those which can be generated from random utility discrete-choice models; it will be interesting to examine the empirical distinctions that non-GERI choice probabilities would have.
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