\section{Introduction}
An agent's behavior depends on her beliefs about which statistical relationships reflect causality and which reflect correlation.
For instance, she may observe that duration of hospitalization is positively correlated with chance of death. This relationship may be causal if time in the hospital increases one's chance of catching an unrelated infection, or spurious if both are caused by the severity of illness. She would be more  reluctant to seek treatment if she believed the former than the latter.
In this paper, we develop a theory in which an analyst can use the agent's behavior, in the form of a random choice rule, to identify her subjective causal model and test whether misperceived causality explains her choices.

We motivate our results with a pair of examples. Consider a doctor (the agent) prescribing medical treatments for Alzheimer's  disease. These treatment can affect  the patients' levels of two correlates,  amyloid plaques and neurofibrillary tangles, in addition to Alzheimer's. These levels are known only  after the patient undergoes a treatment. The doctor infers the effect of each treatment on Alzheimer's  using data from the outcomes of her patients, and performs the one that she thinks will lead to the lowest chance of disease. A drug company (the analyst) may want to know whether a misspecified causal model can explain the doctor's reluctance to prescribe its treatment, and if so, what that model is.
For instance, demonstrating that the drug decreases the chance of plaque buildup would only convince the doctor to prescribe it if she thinks  that high levels of plaque \emph{cause} Alzheimer's.
It would be ineffective if the doctor believes that the correlation is spurious, as would be the case when neurofibrillary tangles cause both  plaque buildup and Alzheimer's.
Our results show how the analyst can answer such questions based on the frequency with which each treatment is performed in different circumstances (behavior).



For a second example, consider a firm (the agent) that hires workers from distinct pools that differ in their education, ability, and productivity. The specific attributes of a given worker are revealed to the firm  only  after it hires him or her. The firm uses a causal model to predict the expected productivity of workers from a given pool, and hires from each pool in proportion to predicted productivity. Since the return to education for workers depends on the firm's behavior, the Department of Education (the analyst) may want to know the firm's model. For instance, if it believes that education is purely human capital  formation, i.e. that education is the only direct \emph{cause} of productivity,  then pushing for reforms that relax admission standards would not lower the return to education. If instead the firm believes that education is purely a signal of ability, then relaxed standards would lower its return. The analyst cannot directly observe the firm's causal model, and so must infer it from the frequency with which the firm hires from each pool. Our result allows the analyst to test whether the firm's hiring decisions (behavior) are consistent with a belief that education causes productivity. 



 This paper provides a theoretical methodology for identifying the  subjective causal model of a decision maker (DM) from her behavior. 
Following \cite{Pearl1995Causal} and \cite{Spiegler2016}, we model her perception of causality using a directed acyclic graph (DAG). 
Each node in the graph is a variable, such as the agent's action or outcome, and each edge represents a belief that one of the variables directly causes another. DAGs allow for a flexible and non-parametric representation of causal relationships. For example, they provide a language to distinguish between a doctor who believes that plaque causes Alzheimer's, represented by an edge between the two, and one who believes that both are caused by tangles, captured by an edge from tangles to both plaque and Alzheimer's.




We consider  a DM who chooses from a set of actions, each determining a probability distribution over a vector of variables. The random choice rule representing the DM's behavior has  a \emph{subjective causality representation} if she acts as if she uses a fixed DAG to predict the consequences of her actions from data generated by her past choices, and then chooses each action with a frequency proportional to its expected utility. The choices form  a personal equilibrium: how frequently she chooses each action affects her dataset, which affects her inferences about the strength of the causal relationships in her DAG, which  affects her likelihood choosing each action.%
\footnote{
While such a feedback effect occurs in many studies of agents with misspecified models, such as \cite{EspondaPouzo2016Berk}, it has typically been absent in decision theoretic work on misspecification.}
We show how to identify the DM's DAG and preferences from her behavior. Then, we turn to the question of how to test whether a random choice rule has a subjective causality representation and provide  necessary and sufficient conditions for one to exist.

The DM may mistake  correlation for causation when her causal model is misspecified.  
In particular, she neglects the effect of her choices on the dataset from which she learns about her actions. Her choices may create a correlation between two variables that she misinterprets as a causal effect, the magnitude of which  affects how likely she is to choose each action. 
 As an example, consider a firm  that thinks the pools of workers from which it hires are selected based only on education, and that education alone causes productivity, but in reality, productivity is caused only by ability. If high-ability workers in a given pool are more likely to have high education than are low-ability ones, then the perceived return to education increases with the fraction of workers hired from that pool. This incentivizes the firm to increase the frequency with which it selects that pool, reinforcing the effect. 
The firm's neglect of how its behavior affects the data on which it performs inferences poses technical challenges. For instance, its behavior may violate regularity, a necessary condition for a random utility model (RUM). However,  we show in Section \ref{sec: challenge}  that it allows the model to accommodate a number of documented  cognitive biases, including selection neglect, illusion of control, status-quo, and congruence biases. %

Our first main result identifies the causal model that explains the DM's behavior. 
Within the broad class of DAGs considered, we show that the DM's perceived causal chains identify all the  relevant variables and the causal relationships between them. We then reveal these chains from her behavior.  Our method relies on the observation that when \emph{every}  chain passes through a set of variables,  independence between those variables and the others implies indifference between all actions.
For example, the doctor may believe that treatments affect the chance of plaque buildup,
and that plaque buildup causes Alzheimer's. 
She is equally likely to choose every treatment when plaque buildup is independent of treatment, presence of tangles, and Alzheimer's.
After revealing these sets, we show they can be ordered to determine the perceived causal chains.

In contrast to the large literature that  empirically determines causality (e.g.  \cite{Card1999}), the result identifies an agent's \emph{perception} of  causal relationships. These perceptions may affect  the agent's reaction to a  policy intervention, and  thereby that policy's effectiveness, even if they are not empirically valid.
For instance, a firm that appears to offer minority workers a lower wage may do so because it  dislikes employing minorities even though they are equally or more productive  (taste-based discrimination). Alternatively, it may offer minorities a lower wage because being a minority is correlated with another attribute, such as education, that the firm thinks affects productivity (statistical discrimination, perhaps based on a wrong model and resulting in incorrect beliefs). Policies that attempt to remedy the former, such as affirmative action or awarding scholarships to minority students, may do nothing for the latter.%
\footnote{See \cite{Lang2020}  for an overview of different types of race discrimination.}

Similarly, agents' models of the macroeconomy determine how a policy change affects their expectations about macroeconomic variables.  For instance, \cite{andre2021}  show that laypeople  tend to think an increase in the federal funds rate primarily affects producer costs, leading to an increase in expected inflation. However, they also show that experts tend to  think that such an increase primarily affects demand, leading to a decrease in expected inflation. As inflation expectations have first-order importance in macroeconomics, which of the two causal models prevails determines the effect of a change in the federal funds rate.

Our second main result establishes how to test whether a misspecified causal model can explain the DM's behavior. To do so, we provide necessary and sufficient conditions for a random choice rule to have a subjective causality representation.
The axioms link the DM's perception of alternatives, as inferred from our first result, to her behavior.
Holding  her perception constant, her behavior conforms closely to Logit with an expected utility Luce index (henceforth, Logit-EU). Put another way, her choices from a pair of menus are inconsistent with Logit-EU only when the inferences she draws differ across the menus.
For example, the axioms require that the DM chooses two actions with the same relative frequency  whenever she makes the same prediction about their distribution of outcomes.
However, her evaluation of alternatives varies across menus as how she perceives them changes. As a consequence, she may violate a number of standard axioms, including regularity.

An agent with a subjective causality representation perceives her options differently from the analyst. The result places testable restrictions  on her behavior in spite of the information gap. 
Thus, it establishes that misspecified causality provides enough discipline on how her beliefs are distorted to be testable; without any restrictions on belief distortion, testing would be impossible.
More broadly, this paper adapts decision theoretic methodology to identify and test an agent's  subjective model of the world, as opposed to the usual exercise of identifying and testing her preferences with a correct, or at least an agreed upon, model of the world. We see this as a step toward providing testable implications for the growing literature studying agents with misspecified models, especially \cite{Spiegler2016},  \cite{Eliaz2018model},
\cite{Eliaz2019}, \cite{Eliaz2020}, \cite{Spiegler2020a},
and \cite{Schumacher}, which all use versions of the subjective causality representation.%
\footnote{\label{fn:misspecifed}Other models where misspecification  leads to distorted beliefs include \cite{EspondaPouzo2016Berk}, \cite{BohrenHauser2018}, \cite{Fricketal2019}, \cite{He2018}, \cite{HeidhuesKoszegiStrack2018},  \cite{SammuelsonMailath2019}, \cite{olea2021competing}, and \cite{Levyetal2021}.}

We conclude by exploring how our analysis would change with different assumptions. 
In our analysis so far, the DM makes inferences based on her own past behavior, an important feature of the recent literature on misspecified models. However, this interpretation relies on observing the long-run steady-state distribution of choice. In an experimental setting, it may be more convenient to instead provide subjects with an exogenously given dataset. We augment the random choice rule with such a dataset and show that our identification results naturally extend to such a setting. While random choice is widely used in experimental and empirical settings, much of the literature that adopts the DAG approach to causality focuses on deterministic choices. We adapt our identification results to this setting as well.

\subsection*{Related literature}
\cite{Spiegler2016} introduced the subjective causal representation, albeit without stochasticity and without axiomatic foundations.
He shows that this can capture a number of errors in reasoning, including reversed causality and omitted variables.
Taken together, our results allow us to test the underlying assumptions of existing work on the effects of causal misperception.
This growing literature has been applied to monetary policy \citep{Spiegler2020a},
 political competition \citep{Eliaz2018model},
communication
\citep{Eliaz2019}, inference \citep{Eliaz2020}, 
and contracting \citep{Schumacher}.
 The majority of these papers take the agents' DAGs as given, whereas our goal is to identify the DAG from behavior and test whether subjective causality explains choice. Consequently,  our results increase the applicability of these papers.



\cite{Pearl1995Causal} argued for using and analyzing DAGs to understand causality. A large literature \citep[e.g.,][]{cowell1999,Koller2009,Pearl2009} develops and applies this approach for probabilistic and causal inference. The typical exercise uses a DAG either to estimate the causal effect of a particular intervention  or to infer which DAG, if any, is consistent with a given dataset.%
\footnote{Recently, \cite{Imbens2020} contrasts this with the potential outcomes approach and discusses why these methods have attracted more attention outside of economics than within it.}
\cite{Schenone2020causality} introduces the DAG approach to causality into a decision theory framework. In his model, an agent expresses preferences over act-causal-intervention pairs. For instance, the DM decides which of two workers to hire; both are identical except one of them has been forced to obtain exactly 11 years of education. He provides necessary and sufficient conditions for the agent's beliefs to result from applying the ``do-operator'' to intervened variables for a fixed DAG and prior. The DAG is identified from the behavior.  His approach is complementary to the one taken by this paper. It is mainly concerned with a normative definition of causality as a manifestation of rationality. By contrast, this paper uses DAGs to capture flaws in the reasoning of a boundedly-rational DM.

More generally, our paper  is related to the  decision theory literature studying DMs who misperceive the world.
\cite{Lipman1999} studies a DM who may not understand all the logical implications of information provided to her.
\cite{EllisPiccione2017} develop a model where agents misperceive the correlation between actions.
\cite{Kochov2015} models an agent who does not accurately foresee the future consequences of her actions.
\cite{Keetal2020} study DMs who perceive lotteries through a neural network.
\cite{EllisMasatlioglu2021} consider an agent who categorizes alternatives  based on the context, and the category to which it belongs affects her evaluation (or perception) of it.
In all, the DM's perception of an alternative is unaffected by her behavior.

Finally, our paper also falls into the theoretical literature studying random choice. We fall between two strands. The first seeks to use choice data to identify features of otherwise rational behavior, such as \cite{Gul2006Random} identifying the distribution of utility indexes, \cite{Lu2016} identifying an agent's private information, and \cite{apesteguia2018monotone} studying comparative risk and time preferences.
The second interprets randomness as a result of boundedly rational behavior in abstract environments, such as the \cite{manzini2014stochastic}, \cite{brady2016menu}, and \cite{cattaneo2020randomattention} models of limited attention.
This paper uses random choice to identify features of explicitly boundedly-rational behavior.
The only other paper of which we are aware that uses stochastic choice to capture equilibrium behavior is \cite{chambersetal2021influence}.


\section{Model}
\label{sec:Model}
\subsection{Setting}
Each action $a$ determines a distribution over a payoff-relevant consequence and $n$ covariates.
The $i$th covariate takes a value in $\mathcal{X}_i$ and the consequence belongs to the set $\mathcal{X}_{n+1}$.
For a non-empty-set $S$, let $\Delta(S)$ be the set of finite support probability distributions over $S$.
Each action is a member of the set $\mathcal{X}_0= \Delta(\prod_{i=1}^{n+1} \mathcal{X}_{i})$, and it is convenient to denote $\mathcal{X}_{-0}=\prod_{i=1}^{n+1} \mathcal{X}_{i}$ and $\mathcal{X}=\mathcal{X}_0\times \mathcal{X}_{-0}$.
We require that $\mathcal{X}_{n+1}$ is a compact subset of $\mathbb{R}$ with  $|\mathcal{X}_{n+1}| \geq 2$, and take $\mathcal{X}_i = \mathbb{R}$ for simplicity.%
\footnote{We can let each $\mathcal{X}_i$, $i \leq n$, be any set with $|\mathcal{X}_i| \geq \min \{ |\mathcal{X}_{n+1}|,|\mathbb{N}|\}$ and $\mathcal{X}_{n+1}$  be a compact subset of a topological space.
This would increase the notational complexity but not substantively change the arguments or results.}

The DM's (stochastic) choice of action determines the distribution of a random vector $X=\left(X_0,X_1,\dots,X_{n+1}\right)$. 
If the DM chooses $a\in \mathcal{X}_0$, then $a(x_1,\dots,x_{n+1})$ is the probability that $X_i=x_i$ for every $i \in \{1,\dots,n+1\} \equiv N$.
We identify the distribution over actions with the $0$th random variable.
The last index $n+1$ denotes the consequence. 
The set $\{1,\dots,n\}$ indexes the  covariates.
By convention, capital letters refer to variables and lowercase letters to realizations.
We denote by $\marg_J p$ the marginal distribution of $p$ on the variables indexed by $J$.
With slight abuse of notation, we sometimes identify the action $a \in \mathcal{X}_0$ with the element of $\Delta \mathcal{X}$ that has a marginal on $\mathcal{X}_{-0}$ equal to $a$ and attaches  probability $1$ to $X_0=a$.

The DM decides between options in $S$, a finite subset of $\mathcal{X}_0$ where the support of the joint distribution of covariates is the product of their marginal supports for any available action.
Every choice problem belongs to the set \[
\mathcal{S}= \left\{ S \subset \mathcal{X}_0: \prod_{j=1}^n\text{supp}(\marg_j a) = \text{supp}(\marg_{\{1,\cdots, n\}} b) \text{ for all }a,b \in S \text{ and }|S|\in (1,\infty)\right\}.
\]
This ensures that Bayes' rule is well-defined and can be relaxed in specific examples.

A random choice rule $\rho: \mathcal{X}_0 \times \mathcal{S} \rightarrow [0,1]$ where $\sum_{a \in S}\rho(a,S)=1$ and $\rho(a,S)=0$ for every $a \notin S$ describes the DM's choices. 
The probability she chooses $a$ from $S$ is $\rho(a,S)$.
Identify $\rho^S$ with the probability distribution over $\mathcal{X}$ induced by the DM's choice probabilities, that is 
\[
\rho^S \in \Delta \mathcal{X} \text{ where } \rho^S(a,y)= \rho(a,S) a(y)\text{ for all }a\in \mathcal{X}_0\text{ and } y\in \mathcal{X}_{-0}.\]
Note $\rho(\cdot,S)$ is a distribution over actions whereas $\rho^S$ is a distribution over  $\mathcal{X}$.

For   $p \in \Delta(B)$, the qualifier ``for $p$-a.e. $z\in B$'' means ``for almost every $z\in B$ according to $p$,'' or equivalently   ``for every $z$ in the support of $p$'' since $p$ has finite support.
For a set $J \subset N$ and $x \in \mathcal{X}_{-0}$, $x_J$ denotes the event that $X_j=x_j$ for all $j\in J$.
When it will not cause confusion, we sometimes write $x_j$ instead of $x_{\{j\}}$ and $x_\emptyset$ for an arbitrary constant.
For $k\in \mathbb{R}$, $k_j$ denotes the event that $X_j=k$. 
We define the mixture between lotteries $a$ and $b$, $\alpha a +(1-\alpha) b$, in the usual way.


\subsection{Subjective Causality Representation}
A directed acyclic graph (DAG) over  a set $M$ is an  acyclic binary relation $R \subset M \times M$, where $i R j$ denotes $(i,j) \in R$. 
A DAG $R$ over $\{0,1,\dots,n+1\}$ describes the DM's perception of causality.
Here, $i R j$ indicates that the DM thinks that $X_i$ directly causes $X_j$ and corresponds to a directed edge in a graph with nodes $\{0, \dots,n+1\}$.
We often write $R(i)$ for the indexes of the variables that cause $X_i$ according to $R$, termed the \emph{parents} of $i$.
For a DAG $R$ and $p \in \Delta \mathcal{X}$, $p_R$ is the probability distribution so that $$p_R(x)=\prod_{j =0}^{n+1} p \left( x_j|x_{R(j)} \right)$$ for every $x\in \mathcal{X}$.
See \cite{Spiegler2016} for further interpretation.

A DAG $R$ is \emph{uninformed} if there is no $i \in N$ with $i R 0$.
That is, none of the other variables cause the DM to choose her action.
A DAG $R$ is \emph{nontrivial} if  $0 R i_1 R i_2 R \dots Ri_m$ for some $i_1,i_2,\dots i_m \in N$ with $i_m=n+1$.
That is, there is a channel through which the choice of action can influence the distribution over consequences.
Say that $(i,j,k)$ is an $R$-v-collider if $i Rk$, $j R k$, $j \!\not\!\! R i$, and $i \!\not\!\! R j$.
The DAG $R$ is \emph{perfect} if there are no $R$-v-colliders.
We focus on perfect DAGs because otherwise the perceived marginal distribution of some variable may diverge from its true distribution \citep{Spiegler2017}. 

\begin{defn}
The random choice rule $\rho$ has a \emph{subjective causality representation (SCR)} if there exists an uninformed, nontrivial DAG $R$ and a continuous, strictly increasing $u$ so that
\[
\rho(a,S)=\frac{\exp \left( \int_{\mathcal{X}_{n+1}} u(c)d\rho^S_{R}(c_{n+1}|a) \right) }{\sum_{a' \in S} \exp \left( \int_{\mathcal{X}_{n+1}} u(c)d\rho^S_{R}(c_{n+1}|a') \right)}
\]
for every $a\in S$ and $S \in \mathcal{S}$;
then, we say $\rho$ has an SCR $(R,u)$  and that $R$ represents $\rho$.
An SCR is perfect if its DAG is perfect.
\end{defn}


The representation corresponds to the following ``as if'' procedure.
The DM maximizes expected utility but with a potentially incorrect prediction of the consequence distribution resulting from her actions. 
She believes that taking an action only directly affects the variables caused by it, which  in turn affects the variables caused by them, and so on and so forth. 
Each of these causal effects are calculated from the ``dataset'' $\rho^S$ that contains every realization of the random vector $X$ with a frequency determined by her choices. 
She begins by  predicting the distribution of a variable caused only by  her action.
Then, she forms an overall prediction by recursively extending her (interim) predicted distribution to subsequent variables using her estimates of the causal effects.
Ultimately, she expects to receive consequence $c$ with probability $\rho^S_R(c_{n+1}|a)$ if she takes action $a$.

For comparison, a $\rho$ has a \emph{Logit-EU} representation if there is a continuous, increasing $u$ so that 
for every $a\in S$ and $S \in \mathcal{S}$, \[
\rho(a,S)=\frac{\exp \left( \int_{\mathcal{X}_{n+1}} u(c)da(c_{n+1}) \right) }{\sum_{a' \in S} \exp \left( \int_{\mathcal{X}_{n+1}} u(c)da'(c_{n+1}) \right)}
=\frac{\exp \left( \int_{\mathcal{X}_{n+1}} u(c)d\rho^S(c_{n+1}|a) \right) }{\sum_{a' \in S} \exp \left( \int_{\mathcal{X}_{n+1}} u(c)d\rho^S(c_{n+1}|a') \right)}.
\]
The SCR replaces the Bayesian update $\rho^S(\cdot|a)$ with the one generated by her causal model $\rho^S_R(\cdot|a)$.

An SCR is a personal equilibrium \citep{KR06}: the DM maximizes expected utility given her beliefs that depend on her choices.
It is easy to show that an equilibrium exists for any $S \in \mathcal{S}$ using Brouwer's fixed point theorem.
For menus with more than one, we place no restrictions on which is selected.

\begin{rem}
We adopt the exponential function for concreteness and applicability. Our results adapt to any other strictly increasing and positive function.
See Footnote \ref{fn:IA explain}.
\end{rem}

\subsection{Running Example}
\label{sec:running example}

Throughout, we illustrate our framework with the following running example. 
The random choice rule represents frequencies with which a doctor performs  medical treatments for Alzheimer's disease on observationally identical patients.%
\footnote{While we focus on the individual interpretation of random choice for expositional purposes, our results apply equally well to a group interpretation provided that the group has a sufficiently similar causal model. Here, the DM is a ``representative'' patient choosing their own treatment on the basis of their inferences from other patients' results. This would apply to vaccination, for instance.}
 An analyst, e.g. a drug company, observes the doctor's choices, and wants to know whether the decisions can be explained by an SCR, and if so, what the doctor's causal model is.
Each treatment leads to a realization of three variables: two correlates,    amyloid plaque   buildup (``plaque'' for short, indexed by $P=1$) and the presence of neurofibrillary tangles (``tangles'' for short, indexed by $T=2$), and long-term health status, i.e., whether the patient gets Alzheimer's (indexed by $H=3$).
Each of the variables is binary and takes values in $\{0,1\}$.
 The vector $(0,1,1)$ represents a patient who does not have plaque buildup, has tangles, and is in good health (i.e., does not have Alzheimer's).  The probability that a patient undergoing treatment $a$ has those characteristics is $a(0,1,1)$.
The doctor only cares about the long-term health of her patients. 
When choosing the treatment, the outcome of the correlates as well as health is unknown. However, she observes the dataset of joint realizations of treatments, correlates, and health.

Figure \ref{fig:example DAGs} gives some possible DAGs.
Each DAG represents a different theory of causation.%
\footnote{DAGs are common tools in applied health research; see \cite{Tennat2020DAGHealth} for a survey.}
A doctor represented by $R_P$ believes that the treatment directly influences the plaque buildup, and that plaque, and  plaque alone, causes Alzheimer's. 
By contrast, one represented by $R_T$ believes that the treatment only directly influences tangles, and that tangles cause both plaque and Alzheimer's. 
A doctor represented by $R_{Rat}$ always correctly predicts the outcome distribution of each action because it  is a complete graph and therefore does not embed any conditional independence assumptions.
However, a DM represented by any of the other DAGs in Figure \ref{fig:example DAGs} potentially makes incorrect predictions. For instance, if she is represented by $R_P$, then she must believe that Alzheimer's is unaffected by the treatment conditional on level of plaque, regardless of whether this holds empirically.

\begin{figure}
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{slides/DAG_example_noa}

	\subcaption[first caption.]{$R_{P}$}
	\end{minipage}
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{slides/DAG_example_nop}

	\subcaption[first caption.]{$R_{Both}$}
	\end{minipage}
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{slides/DAG_example_line2}

	\subcaption[first caption.]{$R_{PT}$}
	\centering
	\end{minipage}
	
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{slides/DAG_example_noe}

	\subcaption[first caption.]{$R_{T}$}
	\end{minipage}
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{slides/DAG_example_base}

	\subcaption[first caption.]{$R_{Rat}$}
	\end{minipage}
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{slides/DAG_example_line}

	\subcaption[first caption.]{$R_{TP}$}
	\end{minipage}
	
\caption{Possible DAGs in the Running Example}
\label{fig:example DAGs}
\end{figure}

\subsection{Interpretations of the model}
Our main interpretation of an SCR is that it describes a DM who predicts the outcome of her action using a causal model.
It may also describe a DM with limited data access \citep{Spiegler2017}.
In this interpretation, she only considers or observes the distributions of several overlapping subsets of variables.
She then extrapolates to form a distribution over all variables using the principle of insufficient reason.
Formally, she uses the distribution that maximizes entropy subject to matching the marginal distributions over the subsets she considers.
Identifying her DAG corresponds to identifying the considered subsets.
In the running example, a doctor represented by $R_P$ only observes, or only has access to, two datasets:
one that keeps track of the efficacy of the treatments on plaque and another one that tracks the impact of plaque on Alzheimer's.
  Recently (and controversially), the FDA approved the Alzheimer's medication aducanumab on the basis of its reduction in plaque buildup despite limited evidence of its effects on the disease itself.%
\footnote{ See \href{http://www.theatlantic.com/health/archive/2021/07/americas-drug-approval-system-unsustainable/619422/}{www.theatlantic.com/health/archive/2021/07/americas-drug-approval-system-unsustainable/619422/} for a description of the controversy and 
\href{http://fda.report/media/143503/PCNS-20201106-CombinedFDABiogenBackgrounder0.pdf}{fda.report/media/143503/PCNS-20201106-CombinedFDABiogenBackgrounder\_%
0.pdf} for the evidence submitted and the FDA's evaluation thereof.
}


Alternatively, it may describe a DM   learning the distribution of the random vector $X$ from the dataset $\rho^S$ with the aid of a DAG $R$.
 The model factorizes the distribution $\rho^S$  into a perceived distribution $\rho^S_R$, and because  $R$ embeds conditional independence assumptions, it may reduce the number of moments needed to reconstruct the distribution.
In the running example, the DM can store all the relevant information according to $R_P$ using only $6$ parameters; for $S=\{a,b\}$, the numbers $p(a)$, $p(1_P\mid a)$, $p(1_T)$, $p(1_H\mid 1_P)$ and $p(1_H\mid 0_P)$ fully determine $p_R$.
In contrast, it would require $2^4-1=15$ parameters to record the probability of each possible realization without this assumption.
Such an interpretation is agnostic as to why the DM infers the overall distribution rather than just the relationship between action and consequence.
She may (incorrectly) anticipate the arrival of additional information or the possibility of taking other actions.
She may think it is easier or quicker to learn the stronger correlations between the covariates that make up a causal chain than to learn the weaker correlation between her action and the outcome.



For a final interpretation, we note that when $\rho$ has an SCR, $\rho^S_R$ minimizes Kullback--Liebler divergence from $\rho^S$ among all the probability distributions on $\mathcal{X}$ that are consistent with $R$.
Then, $\rho$ represents a single agent Berk--Nash equilibrium \citep{EspondaPouzo2016Berk} with extreme-value errors.
As in that model, we can interpret the behavior as the steady state of a learning process with a set of parameters (probability distributions) that do not include the  ``true'' one.


\section{Behavioral Implications of Misspecification}
\label{sec: challenge}
The DM's behavior may endogenously create correlations that she misinterprets as causation.
Fundamentally,  the DM neglects the effect of her choices on her data.%
\footnote{\cite{esponda2018endogenous} experimentally document selection neglect, and   \cite{denrell2020sampling} provides a recent survey of evidence for it in managers.}
This leads to two key technical challenges.
First, an SCR random choice rule may violate \emph{regularity}, a property necessary for representation by a random utility model (RUM). 
Second, the agent's behavior may be \emph{self-confirming}:
because she chooses $a$ frequently, she thinks it is better than $b$, but  would reverse her thinking if she chose $b$ more frequently.
However, these features allow the model to accommodate a number of biases documented in the psychology literature, including violations of independence of irrelevant alternatives,   illusion of control, status-quo bias, and congruence bias.
In this section, we illustrate these challenges and relate them to the behavior permitted.


\subsection{Regularity violation}
\label{sec: reg violation}
A choice rule with an SCR may violate regularity, the requirement that  $\rho(a,S) \geq \rho(a,S')$ whenever  $a \in S \subseteq S'$. 
Consequently, the class of choice rules with an SCR and those with a RUM do not coincide.
This poses a challenge for identifying the model as usual techniques do not directly apply.
By contrast, many behaviors interpreted as irrational can be represented by a RUM, such as the model of limited attention due to \cite{manzini2014stochastic}. 

To illustrate why regularity may be violated, consider a doctor in the running example whose behavior has a SCR $(R_{P},u)$. 
There are three treatment options, $\iota,\pi,\nu$, that are equally likely to lead to good  health. Plaque and health are independent after undergoing treatment $\iota$, positively correlated under treatment $\pi$, and negatively correlated under treatment  $\nu$.
When the doctor decides between only $\iota$ and $\pi$,  plaque buildup is necessarily positively correlated with good health.
As she mistakes the correlation for causation, this  makes treatments that are more likely to lead to plaque buildup more attractive. 
However, when she chooses between all three treatments, the patients to whom she prescribes $\nu$ may cancel out or even reverse the perceived positive effect of plaque on health.
When this effect is strong enough, it can lead to an increase in the probability of choosing the treatment with a lower probability of plaque buildup.

Formally, suppose that $\iota (1_P,1_H)=\iota (1_P,0_H)=\frac12$, $\pi (0_P,0_H)=\pi (1_P,1_H)=\frac12$, $\nu (1_P,0_H)=\nu (0_P,1_H)=\frac12$, $u(L)=0$, and $u(H)=6$.%
\footnote{The distribution of tangles does not affect behavior, so we leave it unspecified.}
Consider menus $S=\{\iota,\pi\}$ and $S'=\{\iota,\nu,\pi\}$.%
\footnote{We note that $S,S' \notin \mathcal{S}$, so this example, and that in the next subsection, are technically outside our domain. At the cost of complicating the algebra and obscuring the logic, they can be made consistent with our assumptions by replacing each action $c$ with $c'=(1-\epsilon)c+\epsilon d$ where $\epsilon>0$ is small enough and $d(y)=\frac18$ for each $y\in \{0,1\}^3$.} If $\rho(\iota,S)=z$, then 
\[ 
\rho^S(1_H|0_P)=0<\rho^S(1_H|1_P)=\frac{z \frac12 +(1-z) \frac12}{z(1) +(1-z)\frac12}=\frac{1}{1+z},
\]
and since $\iota(1_P)>\pi(1_P)>0$, we have $1>z>\frac12$. 
Then, $\frac12<\rho^S(1_H|1_P)<\frac23$, so $\rho^S_R(1_H|\pi)<\frac13$, while $\rho^S_R(1_H|\iota)>\frac12$. 
Hence 
\[
\frac{\rho(\pi,S)}{\rho(\iota,S)}<\frac{\exp[\frac13 6+\frac23 0]}{\exp[\frac12 6+\frac12 0]}= \exp [-1]<\frac12
\]
and $\rho(\pi,S)<\frac13$.
Because health is independent of the treatment conditional on plaque according to $R_P$, the doctor is indifferent between two treatments with the same probability of plaque buildup.
Therefore, $\rho(\nu,S')=\rho(\pi,S')=\gamma$. Then,
\[ 
\rho^{S'}(1_H|1_P)=\frac{(1-2\gamma) \frac12+ \gamma \frac12 + \gamma(0)}{(1-2\gamma)+ 2 \gamma \frac12}=\frac12=
\frac{(1-2\gamma) (0) +\gamma(0) + \gamma\frac12}{(1-2\gamma)(0)+2\gamma\frac12 }=\rho^{S'}(1_H|0_P),
\]
so $\rho(\iota,S')=\rho(\nu,S')=\rho(\pi,S')=\frac13>\rho(\pi,S)$, violating regularity.

While the violations of regularity allow the model to accommodate phenomena like the decoy effect, the failure stems from faulty reasoning.
The above doctor overestimates her ability to control events, or exhibits illusion of control \citep{Langer1975illusion}.
Her choices do not affect long-term health, yet she would be willing to pay a premium to choose one treatment over another. 
Moreover, she also exhibits ``patternicity'' \citep{Shermer1998} in that she perceives a pattern, namely that using $\iota$ leads to better long term-health outcomes, where none exist.

\subsection{Self-confirming choices}
As illustrated above, the outcome that  the DM expects to get from an action may depend on how frequently she chooses it. She may perceive one action to be better than another only if she chooses it sufficiently frequently; this can lead to multiple personal equilibria.
For instance, consider a doctor who thinks that plaque alone causes health.
Action $b$ represents prescribing a drug that prevents the disease but often leads to plaque buildup.
Action $a$ represents not intervening, and such patients often get the disease but rarely  have plaque buildup and then only when the patient gets the disease.
When the doctor usually does not prescribe the drug, low plaque buildup is mistakenly believed to increase the chance of good health. Consequently, prescribing the drug, which raises their plaque, seems like a bad idea. Symmetrically, when she usually prescribes the drug, not intervening to decrease plaque buildup seems counterproductive.

Formally, the DM has an SCR $(R_{P},u)$ and chooses between two treatments, $S=\{a,b\}$, such that $b(1,1,1)=1$, $a(0,0,1)=\frac12 q$, $a(0,0,0)=\frac12 (1-q)$, and $a(1,1,0)=\frac12$ where $q \in (0,1)$.
One can compute that
\[
\rho^S \left(1_H|1_P \right)=\frac{2-2\rho(a,S)}{2-\rho(a,S)}\text{ and }\rho^S \left( 1_H|0_P \right)= q.
\] 
That is, whether the doctor thinks that plaque has a positive or negative effect on Alzheimer's depends on the fraction  who take action $a$.
Setting $u(1)=\lambda$ and $u(0)=0$ for $\lambda>0$, we must have
$$
\lambda \frac12 \left( q- \frac{2-2\rho(a,S) }{2-\rho(a,S)}  \right) = \ln \rho(a,S) - \ln (1-\rho(a,S)).$$
There are multiple solutions for large enough $\lambda$.
For instance when $\lambda = 30$ and $q=\frac34$, the solutions are $\rho(a,S) \approx .34$, $\rho(a,S) \approx 0.02$, and $\rho(a,S) \approx 0.99$.


With data that contains a small enough fraction of patients who received the drug, the doctor often does not intervene because her misspecified model concludes that plaque buildups lead to good health. 
While this is true in that dataset,    taking the drug is unambiguously superior, as she would realize  if a larger fraction took it. 
Interpreting the choices as the steady state of a learning process, such a DM exhibits status quo bias \citep{SamuelsonZeckhauser88}, a tendency toward ``maintaining one's current or previous decision,''  and congruence bias \citep{Wason1960Congruence} by failing to test the alternative hypothesis that the drug is better than not intervening.


\section{Revealing the Subjective Causal Model}
\label{sec:ID}
In this section, we reveal the DM's subjective causal model from her choice behavior.
In a DAG $R$, information flows along the chains of causal relations from actions to consequences.
We show that we can reveal these causal chains from the DM's behavior, and that they pin down the subjective causal model.

Formally, a causal chain is an \emph{active path  from $0$ to $n+1$ in $R$} (an \emph{$R$-AP} for short): a finite sequence of variables $(i_0,i_1,i_2,\dots, i_m)$ with $i_0=0$, $i_m=n+1$, and $i_j R i_{j+1}$ for every $j<m$.
This represents a chain of causal reasoning: according to $R$, variable $0$ causes $i_1$, which in turn causes $i_2$, and so on, ending with a cause of the outcome variable.
A \emph{minimal $R$-AP, or $R$-MAP,} is an $R$-AP that cannot be made shorter.
That is,  $(i_0,\dots, i_m)$ is an $R$-MAP if it is a $R$-AP and $i_j \!\not\!\! R i_{j'}$ whenever $j' \neq j+1$.
The main result of this section shows that  the  $R$-MAPs suffice to identify the DAG.
\begin{thm}
\label{thm: RMAP characterize DAG}
Let $\rho$ have a perfect  SCR $(R,u)$ and $R'$ be a perfect, uninformed, nontrivial DAG.
Then, $\rho$ has an SCR $(R',u')$ if and only if the set of  $R'$-MAPs coincides with the set of $R$-MAPs
and there exists $\beta$ so that $u(c)=u'(c)+\beta$ for every $c\in \mathcal{X}_{n+1}$.
\end{thm}
The result shows that two DMs  behave identically if and only if they agree on the channels through which their actions affect their payoffs and they have the same tastes.
In particular, the $R$-MAPs capture the parts of the causal model relevant for predicting the expected consequence of each action.
This has two immediate corollaries.
First, only relationships between variables that appear in at least one $R$-MAP affect the DM's behavior.
 In particular, any variables caused by the consequence are inconsequential for the her behavior and can therefore be ignored.
Second, the chains of causality involving such variables determine all other causal relationships.
While there may be causal links not in an $R$-MAP, their existence and direction can be determined from the causal chains or are immaterial to the DM's choices.

To see the consequences of the first implication, note that the presence of tangles does not belong to any $R_P$-MAPs.
A doctor represented by $R_P$ can also be represented by a DAG $R'$ that  contains the links in $R_P$ and adds links to and from tangles  that do not create a cycle, an $R'$-MAP, or a v-collider.
For the second, a DM represented by $R_{both}$ can also be represented by a DAG that reverses the link between plaque and tangles, but no other DAG. Any representation must have the same $R_{both}$-MAPs ($(0,P,H)$ and $(0,T,H)$) and must have some link between plaque and tangles to rule out a v-collider. 

The next two subsections illustrate how the analyst can reveal the $R$-MAPs from choice behavior, establishing the necessity of the condition in Theorem \ref{thm: RMAP characterize DAG}.
We begin by identifying the sets of variables through which every causal chain must pass.
Then, we provide an ordering on these sets that reveals the $R$-MAPs.

\subsection{Revealing relevant variables}
This subsection identifies the sets of covariates that intersect every minimal causal chain from the DM's choices.
These sets reveal the relevant variables for her causal model, in that any relationship involving one of the others can be dropped from her model without affecting her behavior.

Given disjoint $I,J \subseteq N$, we say that \emph{$X_I$ is independent of $X_J$ within  $S \in \mathcal{S}$}, written  $X_{I} \perp_S X_{J}$, if $\marg_I a=\marg_I b$ for any $a,b\in S$ and $a(x_I,x_{J})=a(x_I)a(x_{J})$ for disjoint every $x\in \mathcal{X}_{-0}$ and every $a \in S$.
If $X_I$ is independent of $X_J$ within  $S$, then regardless of how the DM chooses from $S$, the random vector $X_I$ is independent of $X_J$ in the resulting joint distribution $\rho^S$.

\begin{defn}
\label{def: separates}
The set \emph{$I \subseteq N$ separates}
if $\rho(a,\{a,b\})=\frac12$ whenever $X_{I} \perp_{\{a,b\}} X_{N\setminus I}$. 
\end{defn}

A set $I$ of covariates  separates if the DM is indifferent between her available actions whenever $X_I$ is independent of all other variables within $\{a,b\}$. 
That is, she believes that there is no relationship between the outcome and  action whenever there is no relationship between the variables indexed by $I$ and the others.
An experimenter can create this independence by intervening to set their values without changing the others in a randomized controlled trial.
Then, $I$ separates if the subject does not think she can affect the outcome  after the experimenter intervenes to set the value of $X_I$. 
Alternatively, we explore allowing for an exogenous dataset in Section \ref{sec: discuss}  where independence can be induced using unavailable actions.
 
\begin{lemma}
\label{lem: I contains AP iff block}
If $\rho$ has an SCR $(R,u)$, then $I \subseteq N$ separates if and only if every $R$-MAP intersects $I$.
\end{lemma}

To illustrate, consider a doctor who is equally likely to prescribe every treatment whenever plaque buildup is independent of the other variables, i.e. $\{P\}$ separates.
In particular, she is indifferent between such treatments even when one provides a lower chance of tangles and better health.
Then, she must believe that the correlation between treatments, tangles, and health is spurious. Hence, her choices reveal that she believes that every causal chain includes plaque.

In general, the set $I$ separates when every causal chain in the DM's model passes through $I$.
Intuitively, making a variable independent shuts down any causal chain containing it.
If $I$ intersects every chain, then the independence of $X_I$ within $S$ shuts down all the channels through which the DM thinks her choice can affect the outcome, causing her to be indifferent.
Observe that $\{n+1\}$ separates, as does any set containing $n+1$. 
In any menu where the consequence is independent of all other variables, all causal channels to it are shut down, so the DM is equally likely to choose both actions. 
Whenever $I$ does not intersect some causal chain, one can construct a menu within which  $X_I$ is independent and yet the DM is not indifferent between all available options.
The construction of the menu is independent of $R$ and $u$, so given that $\rho$ has an SCR, it suffices to observe choice from a single menu to assess whether $I$ separates.  


The set of all \emph{minimal separators},
$$\mathcal{A}= \{ I \subset N: I\text{ is a minimal set that separates}\},$$
has a tight connection to the DM's subjective causal model. 
Consider an $R$-MAP $(i_0,...,i_m)$.
For any $A,B\in \mathcal{A}$, $(i_0,...,i_m)$ must intersect both $A$ and $B$.
If $A\cap B =\emptyset$, it contains exactly one member of each of $A$ and $B$.
If $A\cap B \neq \emptyset$, then either it contains $k \in A \cap B$ and no other member of $A\cup B$, or exactly one member from each of $A \setminus B$ and $B\setminus A$.
Moreover, the contrapositive of Lemma \ref{lem: I contains AP iff block} says that if $N \setminus I$ does not separate, then some $R$-MAP involves only variables in $I$.
That is, if $I$ intersects every $A \in \mathcal{A}$, then $I$ contains every covariate in some $R$-MAP. 

Determining $\mathcal{A}$ immediately rules out many DAGs.
In the running example, the minimal separators suffice to distinguish between populations with any of the DAGs in Figure \ref{fig:example DAGs} except $R_{PT}$ and $R_{TP}$.
If $\rho$ has an SCR $(R,u)$, then $\mathcal{A}$ equals $\{\{P\},\{H\}\}$ when $R=R_{P}$, $\{\{T\},\{H\}\}$ when $R=R_{T}$,  $\{\{P,T\},\{H\}\}$ when $R=R_{Both}$, and $\{\{H\}\}$ when $R=R_{Rat}$.
By contrast, $\mathcal{A}=\{\{P\},\{T\},\{H\}\}$ reveals that $R$ has exactly one $R$-MAP that contains both plaque and tangles.
That is, the analyst can infer that either $R=R_{PT}$ or $R=R_{TP}$ but not which.

\subsection{Revealing causal chains}
To distinguish between causal chains containing more than one covariate, we define an ordering on $\mathcal{A}$ from $\rho$ that reveals the perceived direction of causality

\begin{lemma}
\label{lem: rmap astar}
If $\rho$ has a perfect SCR $(R,u)$, then $\mathcal{A}$ can be ordered $\mathcal{A}=\{A^*_1,\dots,A^*_{|\mathcal{A}|}\}$ so that $(i_0,\dots,i_m)$ is a $R$-MAP if and only if for every $j \in \{0,\dots,m-1\}$, there exists $k$ so that $i_j \in A^*_{k} \setminus A^*_{k+1}$ and $i_{j+1} \in A^*_{k+1} \setminus A^*_k$ when $A^*_0=\{0\}$.
\end{lemma}
If all perceived causes of the variables indexed by $B \in \mathcal{A}$ are contained in $A$, then the DM believes any relationship between her action and the consequence is spurious whenever the variables indexed by $A$ are independent of those indexed by $B$. Intuitively, independence shuts down all the causal chains through which she thinks her action can affect the consequence. 
We order $\mathcal{A}$ using this observation and that the consequence must come last.

To illustrate, consider a doctor represented by $R_{PT}$.
On the one hand, if tangles are independent of health, then since she believes that tangles are the only cause of Alzheimer's, the doctor thinks that each treatment is equally effective.
She is equally likely to prescribe either, regardless of the true relationship between her action and health. 
On the other hand, the doctor may think one drug is superior  when plaque is  independent of Alzheimer's.
For instance, when tangles occur if and only if health is bad and there is no plaque buildup, then tangles are negatively correlated with both the other variables.
Then, the doctor thinks the drug that leads to a lower chance of plaque buildup is best, even if plaque is independent of  health. 
Hence, her choices reveal that the presence of tangles causes Alzheimer's and that plaque causes tangles, so $A^*_3=\{H\}$, $A^*_2=\{T\}$, and $A^*_1=\{P\}$.

We construct the general ordering recursively, starting with the last.
\begin{defn}\label{def: reveal}
	 Let  $A^*_{|\mathcal{A}|}=\{n+1\}$. Recursively define $A^*_{i} \in  \mathcal{A} \setminus \left\{A^*_{i+1},\dots,A^*_{|\mathcal{A}|} \right\}  $  so that 
	 $ A^*_{i} \cap A^*_{i+1}  \supseteq A \cap A^*_{i+1}$ for all $A \in  \mathcal{A} \setminus \left\{A^*_{i+1},\dots,A^*_{|\mathcal{A}|} \right\}  $
	  and $\rho(a,\{a,b\})=\frac12$ for any $\{a,b\} \in \mathcal{S}$ so that $$X_{A^*_{i} \cap A^*_{i+1}} \perp_{\{a,b\}} X_{N \setminus [A^*_{i} \cap A^*_{i+1}]}\ \&\  X_{A^*_{i+1} \setminus A^*_{i}} \perp_{\{a,b\}} X_{ A^*_{i} \setminus A^*_{i+1} }.$$
\end{defn}


Consider first a  DM whose subjective causal model consists of a single minimal causal chain, $(j_0=0,j_1,\dots,j_m=n+1)$. Then, $\mathcal{A}=\left\{ \{j_1\},\{j_2\},\dots,\{j_{m}\} \right\}$, so   $A \cap B = \emptyset$ and $A \setminus B=A$ for any distinct $A,B \in \mathcal{A}$. The independence condition suffices to determine the ordering. 
Since $j_i$ is the only cause of $j_{i+1}$, independence of $X_{j_i}$ from $X_{j_{i+1}}$ within $\{a,b\}$ shuts down the causal chain. This leads the DM to dismiss any actual relationship between her action and the consequence as spurious.
However, the DM may perceive a correlation between $X_{j_k}$ and $X_{j_{i+1}}$ for any $k<i$ even when they are actually independent, as illustrated above.
Consequently, independence between $X_{j_k}$ and $X_{j_{i+1}}$ suffices for indifference if and only if $k=i$.
Therefore,  we must have $A^*_i=\{j_{i}\}$ when $A^*_{k}=\{j_{k}\}$ for every $k\geq i+1$.

Consider now a DM whose model consists of more than one chain. Then, for $A,B \in \mathcal{A}$, $A $ may not be a singleton and $A \cap B$ may not be empty.
The analyst can find $A^*_i$ by applying the same logic as that in the single-chain case with three modifications. 
First, $A^*_{i}$ must have the largest intersection with $A^*_{i+1}$. If $A \in \mathcal{A}\setminus \left\{A^*_{i+1},\dots,A^*_{m} \right\} $ has $A \cap A^*_{i+1} \not \subseteq A^*_{i} \cap A^*_{i+1}$, then  there is a cycle: for any $j \in \left( A^*_{i+1} \cap A \right) \setminus A^*_{i}$  and $k \in A^*_{i} \setminus A^*_{i+1}$, there must be an $R$-MAP $(\dots,j,\dots,k,j\dots)$. 
Therefore, any $A,B \in \mathcal{A}\setminus \left\{A^*_{i+1},\dots,A^*_{m} \right\}$ that have the largest intersection with $A^*_{i+1}$ must satisfy $ A^*_{i+1} \cap A = A^*_{i+1} \cap B = A^{**}$.  
Second, the actions available must shut down the  chains that intersect $A^{**}$.
To shut them down, we consider menus in which the variables indexed by $A^{**}$ are independent of all others. 
Third, if $A$ precedes $A^*_{i+1}$ then every $j \in A \setminus A^{**}$ must come before every $k \in A^*_{i+1} \setminus A^{**}$ in some $R$-MAP since each $A \in \mathcal{A}$ contains one variable from each causal chain.
Therefore, the DM must be indifferent between $a$ and $b$ only if $X_{A^{**}} \perp_{\{a,b\}} X_{N \setminus A^{**}}$ and \emph{every} variable indexed by $A_i^* \setminus A^{**}$ is independent of \emph{every} variable indexed by  $ A^*_{i+1} \setminus A^{**}$ within $\{a,b\}$.



\begin{proposition}\label{prop: Finite choices}
If $\rho$ has a perfect SCR, then $\mathcal{A}$ and its ordering as per Definition \ref{def: reveal} can be determined by observing $\rho$ in a finite number of binary menus.
\end{proposition}

For $I\subset N$, we can construct a binary menu $\{a,b\}$ so that the DM is indifferent between $a$ and $b$ if and only if $I$ separates.
Then, $X_I \perp_{\{a,b\}} X_{N \setminus I}$ and  $a$ ($b$) almost always leads to high (low) realizations, so all the variables indexed by $N\setminus I$  are almost perfectly correlated.  The DM realizes the relationship between her action and outcome if and only if $I$ does not separate.
The ordering of the minimal separators is revealed recursively as above. 
To reveal whether $A\in\mathcal{A}\setminus \{A^*_{i+1},\cdots, A^*_{|\mathcal{A}|}\}$ equals $A^*_i$, we construct a menu $\{a,b\}$ so that the DM  strictly prefers $a$ to $b$ if and only if $A\neq A^*_i$. 
Combining the two observations establishes the result.

\subsection{Revealed causes}
A perfect SCR may be represented by more than one DAG, provided that both have the same minimal causal chains (Theorem \ref{thm: RMAP characterize DAG}).
We turn to the question of which links must belong to any representation of the DM's subjective causal model.

\begin{defn}\label{def: revealed cause}
For $j,k \in \{0,1,\dots,n+1\}$, \emph{$j$ is revealed to cause $k$ for $\rho$}, written $j\hat{R}^\rho k$, if $j R k$ for every SCR $(R,u)$ that represents $\rho$.
\end{defn}

That is, $j$ is revealed to cause $k$ if $j$ causes $k$ in every representation of $\rho$. This definition is analogous to how \cite{MNO12} define revealed preference with limited attention.  When $i$ directly precedes $j$ in an $R$-MAP, then $i \hat{R}^\rho j$.
However, there may be other revealed causes.
Proposition \ref{prop: fundamental links} characterizes all of them.

\begin{proposition}
\label{prop: fundamental links}
If $\rho$ has a perfect SCR, then $j \hat{R}^\rho k$ if and only if there exists $i$ so that $j \in A^*_{i} $ and $k \in A^*_{i+1} \setminus A^*_i$ when $A^*_0=\{0\}$.
\end{proposition}

The result extends Lemma \ref{lem: rmap astar}.
The ordering of separators reveals not only the $R$-MAPs but also every causal relationship that belongs to every representation of $\rho$.
Moreover, $\hat{R}^\rho$ includes but is not limited to the links in $R$-MAPs.
The causal links revealed by the ordering are the only ones whose direction is uniquely determined by choice behavior.
Other links, such as that between $P$ and $T$ when $\rho$ is represented by $R_{Both}$, need to be included in the DAG, but their direction is not pinned down.
Determining the direction of causality between such variables requires additional data, such as choices following interventions as studied by \cite{Schenone2020causality}. 

\begin{rem}\label{rem: revealed DAG}
The DAG $\hat{R}^\rho$ may not be perfect. One can construct a perfect \emph{revealed DAG} $R^{\rho}$ by setting
$j R^{\rho} k$ if and only if  either $j \hat{R}^\rho k$ or there exists $i$ so that $j,k \in A^*_{i+1} \setminus A^*_{i}$ and $j<k$ with $A^*_0=\{0\}$. When $\rho$ has a perfect SCR,  $R^\rho$ represents $\rho$. Moreover,\[
 \rho^S_R \left(x_{\cup_{i = 1}^{|\mathcal{A}|} A^*_i } |a\right) =a \left(x_{A^*_1} \right)\prod_{i = 1}^{|\mathcal{A}|-1} \rho^S \left(x_{A^*_{i+1} \setminus A^*_i}|x_{A^*_{i}} \right)
\]
for any perfect DAG $R$ that represents $\rho$, any $a\in S$, and $a$-a.e. $x \in \mathcal{X}_{-0}$.
\end{rem}


\section{Behavioral Foundations for Subjective Causality}
\label{sec:Foundation}
This section characterizes the random choice rules that have a subjective causality representation. 
Throughout, the properties are illustrated in the context of the running example.
For these purposes, we consider a doctor for which $\mathcal{A}=\{\{P\},\{H\}\}$, and so, using results in Section \ref{sec:ID}, can be represented by $R_{P}$.
That is, the DM believes that the treatments can influence plaque build-up, and  plaque is the sole determinant of  health. 

The first axiom is standard, requiring simply that the DM chooses every available action with positive probability.

\begin{ax}[Full-support]
\label{ax: positivity}
\label{ax: first SCR}
\label{ax: nontrivial}
For any $S \in \mathcal{S}$ and $a\in S$, $\rho(a,S)>0$.
\end{ax}

The next axiom limits the perceived difference between any two options. 

\begin{ax}[Bounded Misperception]
\label{ax: bounded}
The Luce ratio is bounded: $\sup_{S, a,b\in S} \frac{\rho(a,S)}{\rho(b,S)}<\infty$.
\end{ax}

The relative frequency with which the DM takes two actions, their \emph{Luce ratio}, indicates the strength of her preference.
Since the set of consequences is compact, there is a best and worst outcome. 
These provide a natural limit to how much she prefers one action to another, which bounds the Luce ratio. 
The axiom thus  bounds the size of the mistakes that the DM can make.
In the running example, the doctor's Luce ratio for any two treatments is bounded above by that of the one she believes will always lead to a good health outcome and that of the one she believes will always leads to a bad health. 

The third axiom ensures that the minimal separators  can be ordered as in Section \ref{sec:ID}.

\begin{ax}[Consistent Revealed Causes]
\label{ax: Consistent Revealed Causes}
The set $\{n+1\} \in \mathcal{A}$, and for $i=1,\dots,|\mathcal{A}|$, $A^*_i$ exists.
\end{ax}

This axiom implies that the DM does not think that the realization of a variable can simultaneously influences and be influenced by the realization of another variable.
The DM may still believe that the current inflation level influence the current unemployment level, and that both influence future inflation levels.
However, current and future inflation levels are regarded as two distinct variables.
Thus, it is possible to give a consistent account of which variables influence each other variable.
As per Proposition \ref{prop: Finite choices}, it can be falsified by observing the DM's choices from a finite number of menus.

The fourth axiom requires that if the DM predicts two actions lead to the same outcome distribution, then she chooses each with the same probability.

\begin{ax}[Indifferent If Identical Immediate Implications, I5]
\label{ax: same pr if first clique}For $a,b \in S \in \mathcal{S}$, if $\marg_{A^*_1} a= \marg_{A^*_1} b$, then $\rho(a,S)=\rho(b,S)$.
\end{ax}

The covariates directly caused by the DM's action are a sufficient statistic for her prediction of the outcome distribution.
That is, if two actions have the same distribution over these covariates, then she believes that both lead to identical probabilities of every consequence.
She is therefore indifferent between any two actions with identical immediate implications according to her subjective causal model.
In the running example, it implies that the doctor is equally likely to choose any treatments with the same probability of plaque buildup, regardless of any other differences between them.
Note that this axiom only considers the marginal distributions on $X_{A^*_1}$ of $a$ and $b$, remaining agnostic about their distribution on other variables and the available actions.

The following axiom relates similarities in the DM's inferences to her choices.

\begin{ax}[Luce's Choice Axiom Given Inferences, LCI]
\label{ax: luce axiom inference}\ \\
	For    any $S,S_1,S_2, \dots \in \mathcal{S}$ with $a,b \in S_m \cap S$ for each $m$, 	if $$\rho^{S_m} \left( y_{A^*_{i+1} \setminus A^*_i}|y_{A^*_{i} } \right) \rightarrow \rho^{S} \left(y_{A^*_{i+1} \setminus A^*_i}|y_{A^*_{i} } \right)$$    for $\rho^S$-a.e. $y \in \mathcal{X}_{-0}$ and every $i =1,\dots,|\mathcal{A}|$, 
	then $\frac{\rho(a, S_m)}{\rho(b,S_m)}\rightarrow \frac{\rho(a,S)}{\rho(b, S)}$.
\end{ax}

The Logit model is characterized by Luce's Choice Axiom \citep{luce1959individual}, which requires that $\frac{\rho(a, S')}{\rho(b,S')}=\frac{\rho(a,S)}{\rho(b, S)}$ whenever $a,b \in S \cap S'$.
LCI requires that the choice axiom holds when the DM's inferences about variables conditional on their revealed causes are close.
That is, it implies that if $S,S'\in \mathcal{S}$, $a,b\in S\cap S'$, and $\rho^{S}(y_{A^*_{i+1} \setminus A^*_i}|y_{A^*_{i} })=\rho^{S'}(y_{A^*_{i+1} \setminus A^*_i}|y_{A^*_{i} })$ for each $i$, then $\frac{\rho(a, S')}{\rho(b,S')}=\frac{\rho(a,S)}{\rho(b, S)}$.
Moreover, the choice axiom is ``close'' to holding whenever these conditional probabilities are ``close.''
In the running example, suppose that given the doctor's choices when facing $\{a,b\}$ and $\{a,b,c\}$, the statistical relationship between plaque and health is (almost) the same.
Then, the predicted long-term health prospect from treatment $a$ or $b$ is (almost) the same across menus.
Thus, the relative frequency with which she chooses each of these treatments should be (almost) the same.

The next definition identifies a set of menus for which the DM's predictions about the consequence of each action is correct.

\begin{defn}\label{def: correct perception}
A menu $S \in \mathcal{S}$ is \emph{(revealed to be) correctly perceived} if
\[
b \left( y_{\bigcup_{i = 1}^{|\mathcal{A}|} A^*_i} \right)=b \left(y_{A^*_1} \right)\prod_{i = 1}^{|\mathcal{A}|-1} a \left(y_{A^*_{i+1} \setminus A^*_i}|y_{A^*_{i}} \right) ,
\]
for every $a,b \in S$ and $a$-a.e. $y\in \mathcal{X}_{-0}$.
\end{defn}

Recall from Remark \ref{rem: revealed DAG}  that only the set of variables indexed by $\bigcup_{i = 1}^{|\mathcal{A}|+1} A^*_i$ is relevant for the DM's choices and that she thinks that the variables indexed by $A^*_i$ cause those indexed by $A^*_{i+1} \setminus A^*_i$.
In particular, she thinks that any variable in $A^*_{i+1} \setminus A^*_i$ is independent of any in $\bigcup_{k = 1}^{i-1} A^*_k$ conditional on $A^*_i$.
A correctly perceived menu satisfies this conditional independence, and her prediction of the distribution of all relevant variables, including the consequence, is correct.

The remaining axioms ensure that $\rho$ has a Logit-EU representation for menus where the DM's predictions are correct.

\begin{ax}[Correctly Perceived Independence]
\label{ax: IA R markov}
For any $\alpha \in (0,1)$, if $\{\alpha p +(1-\alpha)r,r\},\{\beta p +(1-\beta)r,r\} \in \mathcal{S}$ are  correctly perceived, then \[
\beta \ln \frac{\rho(\alpha p +(1-\alpha)r),\{\alpha p +(1-\alpha)r,r\})}{\rho(r,\{\alpha p +(1-\alpha)r,r\})}= \alpha \ln \frac{\rho(\beta p +(1-\beta)r,\{\beta p +(1-\beta)r,r\})}{\rho(r,\{\beta p +(1-\beta)r,r\})}.
\]
\end{ax}

\begin{ax}[Correctly Perceived Dominance]
\label{ax: markov domininace}
If $\{p,q\} \in \mathcal{S}$ is  correctly perceived and  $\marg_{n+1} p$ strictly first-order stochastically dominates $ \marg_{n+1}q$, then $\rho(p,\{p,q\})>\frac12$.
\end{ax}

\begin{ax}[Correctly Perceived   Continuity]
\label{ax: n+1 c}
\label{ax: last SCR}
If $S,\{p_1,q_1\},\{p_2,q_2\},\dots \in \mathcal{S}$ are  correctly perceived, $p,q \in S$,  $\marg_{n+1} p_m  \rightarrow \marg_{n+1}p$, and $\marg_{n+1} q_m  \rightarrow \marg_{n+1}q$, then \[
\frac{\rho(q_m,\{p_m,q_m\})}{\rho(p_m,\{p_m,q_m\})}\rightarrow \frac{\rho(q,S)}{\rho(p,S)}.\]
\end{ax}

Combined, the axioms imply that the DM's behavior is suitably well-behaved  when she correctly predicts the outcomes of each action.  
Axiom \ref{ax: IA R markov}  guarantees that the independence axiom holds, and that the relative probability of choosing $\alpha p +(1-\alpha)r$ to $r$  is log linear in $\alpha$ because of the Logit functional form.%
\footnote{This is the only axiom that needs be adapted to replace $\exp$ with a different strictly increasing, positive function.
For instance, dropping ``$\ln$'' from Axiom \ref{ax: IA R markov} replaces it with the identity function. \label{fn:IA explain}}
Axioms \ref{ax: markov domininace} and \ref{ax: n+1 c} guarantee monotonicity and continuity over the distribution of consequences.

The main result of this section characterizes the rules with a perfect SCR.

\begin{thm}
\label{thm: proper SCR}
A random choice rule $\rho$ has a  perfect subjective causality representation if and only if  $\rho$ satisfies Full-support,  Bounded Misperception, I5, Consistent Revealed Causes, LCI, 
and Correctly Perceived Independence, Dominance, and Continuity.
\end{thm}

The result highlights the connection between SCR and the Logit-EU model.
Notice that if Axioms \ref{ax: nontrivial} and \ref{ax: luce axiom inference}-\ref{ax: n+1 c} hold when the part of their hypotheses involving the minimal separators are dropped, the choice rule has a Logit-EU representation.
Bounded Misperception gives a maximum deviation in the relative choice frequencies.
The axioms thus indicate the circumstances under which the choice rule does not diverge from Logit. 
I5 says that two alternatives are chosen with same probability whenever they coincide on the distribution of variables that the action is revealed to cause, whereas Logit-EU requires coincidence on the consequence distribution.
LCI restricts violations of Luce's Choice Axiom to when inferences about causal effects change.  
Correctly Perceived Independence, Dominance, and Continuity show that choice is well-behaved whenever the DM's predictions about actions match reality. 

We outline the proof for sufficiency here, and defer a formal proof to the appendix.
We first show that the choice rule has a Logit-EU representation when restricted to correctly perceived menus.
By Axiom \ref{ax: Consistent Revealed Causes},  the DAG $R^\rho$ from Remark \ref{rem: revealed DAG} is well-defined and  a  natural candidate to represent $\rho$. 
For every menu $S$, we construct a correctly perceived version of it, $S'_1$.
That is, for every $a \in S$, there is an $a' \in S'_1$ so that $a'(\cdot)=\rho^S_{R^\rho}(\cdot |a)$ and $S'_1$ is correctly perceived.
Our goal is to show that for any $a,b \in S$, $a$ and $b$ are chosen with the same relative frequency in $S$ as $a'$ and $b'$ are in $S'_1$.
To do so, we add distinct alternatives to $S'_1$ to form a nested sequence of menus $(S'_m)_{m=1}^\infty$ while maintaining that each $S'_m$ is correctly perceived.
Bounded Misperception implies that the probability of choosing anything in $S$ from $S'_m \cup S$ goes to zero as the number of alternatives in $S'_m$ goes to infinity. 
In particular, the inferences that the DM makes from $S'_m \cup S$ approach those she makes from $S'_1$, which are in turn equal to those she makes from $S$.
LCI implies that the relative frequency with which $a'$ and $b'$ are chosen from $S'_m \cup S$ converges to that for $a'$ and $b'$ in $S'_1$.
Moreover, $a$ and $a'$ (as well as $b$ and $b'$) are chosen from $ S'_m \cup S$ with the same probability by I5.
Applying LCI another time, we see that $a$ and $b$ are chosen with the same relative frequency in $S$ as $a'$ and $b'$ are in $S'_1$, completing the proof. 

We conclude by further clarifying the relationship between SCR and  Logit-EU.

\begin{cor}
A random choice rule $\rho$ has a  Logit-EU representation if  and only if
$\mathcal{A}=\{\{n+1\}\}$ and $\rho$ satisfies Full-support 
and Correctly Perceived Independence, Dominance, and Continuity.
\end{cor}

If $\rho$ has an SCR $(R,u)$ and  $\mathcal{A}=\{\{n+1\}\}$, then $0 R (n+1)$.
Moreover, every menu is correctly perceived. Consequently, the DM acts as if Logit-EU on all menus.


\section{Discussion and Extensions}
\label{sec: discuss}
This section concludes the paper by looking at some implications of the model and considering how our modeling decisions affect our results.
We compare the behavior of two DMs with nested causal models
Then, we examine how to extend our analysis to eliminate stochasticity and the endogeneity of the dataset.


\subsection{Comparative Coarseness}
A coarser causal model leaves out some variables or relationships relative to another.
Authors often explain ``irrational'' behavior in situations with adverse selection via coarseness.
For instance, \cite{EysterRabin2005}, \cite{Jehiel2008}, and \cite{Esponda2008} argue that the winner's curse reflects bidders who do not fully take into account the relationship between others' actions and signals.%
\footnote{Section 5 of \cite{Spiegler2016} discusses how and to what extent these models fit into the DAG framework.}
In this subsection, we compare DMs in terms of the coarseness of their model.
In particular, how can an analyst separate two DMs who differ in that one's model contains more variables than the other's?

\begin{defn}
Say that $\rho_2$ \emph{has a coarser model than} $\rho_1$ if  $\rho_1(\cdot,S)=\rho_2(\cdot,S)$ whenever $X_i \perp_S X_{N \setminus \{i\}}$ for all $i \in N$ that are not in any minimal   $\rho_2$-separator.
\end{defn}

Consider DM1 represented by $\rho_1$ and DM2 represented by $\rho_2$.
As revealed by Theorem \ref{thm: RMAP characterize DAG}, DM2 considers the variables that do not belong to a $\rho_2$-MAP irrelevant for determining the consequence of their action.
The condition says that whenever those variables are actually irrelevant when choosing from $S$, i.e., they are independent of the other variables, then the two DMs behave in the same way.
This ensures that if DM2 thinks a variable is relevant, so does DM1.


\begin{proposition}
\label{prop:coarseness}
Let $\rho_i$  have a perfect SCR $(R_i,u_i)$ for $i=1,2$.
If  $\rho_2$ has a coarser model than $\rho_1$, then  $\rho_2$ has a perfect SCR $\left ( R_1 \cap N' \times N',u_2 \right)$ for some $N' \subset \{0,1,\dots,n+1\}$ and $u_2=u_1+\beta$.
The converse holds up to the selection of a personal equilibrium.
\end{proposition}

The result shows that the comparison reveals when the models of two DMs are nested.
Specifically, they agree on the causal relationship between any two variables that both consider relevant and on the desirability of outcomes.
However, they may disagree on which variables are relevant, with DM1 considering more of them relevant than DM2. 


\subsection{Welfare}
There is no general relationship between the DM's DAG and an analyst's evaluation of her choices.
For instance, Section 4.3 of \cite{Spiegler2016} shows that a DM may achieve higher objective payoff with a coarser model.
Similar examples apply to this setting as well.
With a perfect DAG, the DM and the analyst agree on the overall distribution of consequences.
This implies that a refinement such as preferred personal equilibrium can be defined from either the agent's or analyst's perspective without changing conclusions.

\subsection{Exogenous dataset}
We can extend our results to a setting where the dataset used by the DM is exogenously given and does not depend on her behavior.
This setting provides rich variation in the DM's inferences.
It is particularly applicable to an experimental implementation of our result.
Most of the insights from our analysis with an endogenous dataset are readily applicable.
Indeed, it guarantees uniqueness of the personal equilibrium and ensures that the DM conforms to Logit holding the dataset fixed.

Formally, we consider behavior in an environment $(S,q)$ where the DM's choice set $S \in \mathcal{S}$ and the DM's dataset $q \in \Delta \mathcal{X}$ has $q(a)>0$ for each $a \in S$ and 
\[\prod_{j=0}^n\text{supp}(\marg_j q) = \text{supp}(\marg_{\{0,\cdots,n\}} q).\]
Let $\mathcal{E}$ be the set of such pairs.
The DM's behavior is given by the augmented random choice rule $\rho^*:\mathcal{X}_0 \times \mathcal{E} \rightarrow [0,1]$ with $\sum_{a\in S} \rho^*(a;S,q)=1$ and $\rho^*(a;S,q)>0$ only if $a \in S$.
The frequency that they choose $a$ in the environment $(S,q)$ is  $\rho^*(a;S,q)$.

\begin{defn}
The augmented random choice rule $\rho^*$ has an \emph{Exogenous SCR (ESCR)} if there exists an uninformed, nontrivial DAG $R$ and a continuous, nonconstant $u$ so that
\[
\rho^*(a;S,q)=\frac{\exp \left( \int_{\mathcal{X}_{n+1}} u(c)dq_{R}(c_{n+1}|a) \right) }{\sum_{a' \in S} \exp \left( \int_{\mathcal{X}_{n+1}} u(c)dq_{R}(c_{n+1}|a') \right)}
\]
for every $a\in S$ and $S \in \mathcal{S}$.
\end{defn}

It is easy to adapt our identification results to this setting.
Theorem \ref{thm: RMAP characterize DAG} holds as stated.
To establish that the behavior on $R$-MAPs is uniquely pinned down, we apply the conditions to the dataset directly rather than to the options in the menu.
For instance, Lemma \ref{lem: I contains AP iff block} says that a subset of covariates contains a $R$-AP if and only if its complement separates.
The result continues to hold after we modify Definition \ref{def: separates} to say that $K \subset N$ separates if $\rho^*(a;\{a,b\},q)=\frac12$ whenever $X_K$ is independent of the other variables according to $q$: $q(x)=q(x_K)q(x_{N \setminus K})$ for $q$-a.e. $x \in \mathcal{X}$.
That is, independence is required for the dataset, not the menu.
If the dataset is easily manipulable, as in an experiment, then the condition may be substantially easier to test. 


\subsection{Inference from multiple choice sets}
We have implicitly assumed that the modeler sees the steady-state distribution of choices from a fixed menu (or alternatively an exogenous dataset).
Real-world agents face different choice sets on different occasions and learn from their decisions in all of them.
Our framework generalizes to accommodate this, provided that the distribution of choice sets is fixed across time.
Consider a distribution $\mu \in \Delta \mathcal{S}$ with $\mu(S)$ indicating the frequency with which the agent faces menu $S$.
Then, we take as a primitive $\hat{\rho}:\mathcal{X}_0 \times \Delta(\mathcal{S}) \times \mathcal{S} \rightarrow [0,1]$ so that $\rho(\cdot; \mu, \cdot)$ is a random choice rule for fixed $\mu$.
The DM chooses $a$ with probability $\rho(a,\mu,S)$ when they face menu $S$.
The above analysis is the special case $\rho(\cdot,S)=\hat{\rho}(\cdot; (1, S), S)$.
To take into account how learning spills over across menus, we replace the dataset $\rho^S$ with $\hat{\rho}^\mu$ where 
$$\hat{\rho}^\mu(a,y)=\sum_{\mu-a.e.\ S} \mu(S) \hat{\rho}(a;\mu,S)a(y)$$
for each $(a,y) \in \mathcal{X}$.
With such a substitution, our results generalize naturally.
As an additional feature, the analyst can reveal the agent's causal model from its behavior when facing a single, suitably chosen distribution over menus.


\subsection{Deterministic choice}
The SCR is derived from \cite{Spiegler2016}, where choice is deterministic.
We have adopted a stochastic choice framework throughout the paper. 
The stochastic setting is closer to that typically used in empirical and experimental work.
It also deals with some technical issues.
For instance, it pins down beliefs about the consequence distribution of every alternative.
Moreover, it applies when only one of potentially many personal equilibria is observed.
Our insights apply to a deterministic choice model, once suitably adapted.
We discuss how to apply them in this subsection.

Formally, we suppose that the DM's behavior is described by a choice correspondence $c: \mathcal{S} \rightrightarrows \Delta(\mathcal{X}_0)$ where $p(S)=1$ for all $p \in c(S)$ and $c(S)\neq \emptyset$ for each $S \in \mathcal{S}$.%
\footnote{
As shown in \cite{Spiegler2016}, there may not exist a personal equilibrium that does not mix.}
For any $p \in \Delta(\mathcal{X}_0)$, write $p^X \in \Delta(\mathcal{X})$ for the resulting dataset, i.e. $p^X$ is the lottery so that $p^X(a,y)=p(a)a(y)$ for every $(a,y)\in \mathcal{X}$.

\begin{defn}[\cite{Spiegler2016}]
For $\epsilon>0$, the lottery $p \in \Delta(B)$ is a \emph{$(R,u,\epsilon)$-personal equilibrium} for $B\in \mathcal{S}$ if $p(a)>0$ for all $a \in B$ and
\[ p(a) \geq \epsilon \implies a \in \arg\max_{a' \in B} \int_{\mathcal{X}_{n+1}} u(c)d p^X_R(c_{n+1}|a').
\]
The lottery $p\in \Delta(B)$ is a \emph{$(R,u)$-personal equilibrium} for $B\in \mathcal{S}$ if there exists a sequence $(p_t)_{t=1}^\infty$ so that $p_t$ is a $(R,u,1/t)$-personal equilibrium for $B$ and $p_t \rightarrow p$.
\end{defn}

The choice correspondence $c$ has a \emph{Deterministic SCR (DSCR)}  if there exists a uninformed, nontrivial DAG $R$ and a nonconstant, continuous $u: \mathcal{X}_{n+1} \rightarrow \mathbb{R}$ so that for every $B\in \mathcal{S}$, $ p \in c(B)$ if and only if $p$ is a $(R,u)$-personal equilibrium for $B$.
A DSCR $(R,u)$ is perfect if $R$ is perfect.
Observe that limiting cases of SCR are personal equilibria. Formally, let $\rho^\lambda$ be a random choice rule having a perfect SCR $(R,\lambda u)$ for $\lambda>0$.
If $\rho^{\lambda_n}(a,S)\rightarrow p(a)$ for every $a \in S$ and $\lambda_n \rightarrow \infty$, then $p$ is an $(R,u)$-personal equilibrium for $S$.

A version of Theorem \ref{thm: RMAP characterize DAG} holds in this setting.
For instance, Lemmas \ref{lem: I contains AP iff block} and \ref{lem: rmap astar} require only minor alterations. Specifically, one must replace ``$\rho(\{a,b\})(a)=\frac12$'' with ``$c(\{a,b\})=\Delta\{a,b\}$'' in Definitions \ref{def: separates} and \ref{def: reveal}. 


\subsection{Directions for future research}
Our results apply only to perfect subjective DAGs and settings where the agent has no access to information.
Relaxing those two features is a natural direction for future research.
An imperfect DAG allows for v-colliders, which provide a natural way to model an agent who theorizes the exogeneity of a variable.
One direction of Theorem \ref{thm: RMAP characterize DAG} fails for imperfect DAGs.
Although any other DAG representing $\rho$ must agree on the minimal active paths from the action to the consequence, this is not sufficient for it  to represent $\rho$ when $\rho$ has an imperfect SCR.
They must also agree, for instance, on the v-colliders with a path to the consequence. 

Informed agents provide additional challenges, not only about how a primitive should be defined, but also how to change the model itself.
In the setting we studied in this paper, our uninformed, nontrivial assumption implies that $p_R(\cdot|a)$ equals the update after applying the ``do-operator'' of \cite{Pearl1995Causal}.
If the DM thinks signals cause actions but that she can also change her action, this no longer holds.
Since the ``do-operator'' severs links pointing to the intervened variable, it can effectively create v-colliders.
This suggests spillovers between the study of informed DMs and those with an imperfect causal model.

We also restrict attention to situations where the DM takes a single action and cares about the realization of a single variable.
While without loss in a rational model, modeling multiple outcomes as a many individual variables or as a single vector affects behavior in this model.
For instance, when each outcome is modeled as a separate variable, we can consider a DM who thinks the outcomes are independent given the action.
By contrast, if both outcomes are modeled a single, vector-valued variable the DM considers them independent conditional on a common parent only if they actually are.

DAGs provide a tractable, non-parametric, and flexible approach to modeling misspecified causality.
Many other forms of misspecification have interesting economic applications (see Footnote \ref{fn:misspecifed}).
While unrestricted misspecified beliefs are untestable, our results show that exploiting the regularities implied by a particular model of misspecification may render it testable.
The techniques and setup introduced herein  may prove useful for identifying and testing some of these models.

































\section{Introduction}

\begin{frame}<3>[label=motivation]
\frametitle{Motivation}
\begin{itemize}[<1->]
\item<1-> Correlation does not imply causation
\item Correlation only requires data,  causation  also needs model
\begin{itemize}
\item Say's law? Phillips curve? Signaling vs. Human Capital?
\end{itemize}
\bigskip

\item \alert{Misspecified} model distorts agent's beliefs about her actions

\bigskip

\item<3-> Can analyst \alert{reveal} model generating agent's distorted beliefs? 
\smallskip

even invalid causal models affect reactions to policy changes

\end{itemize}
\end{frame}

\frame{
\frametitle{Running Example}
\begin{itemize}
\item Patients (collectively, \emph{DM}) decide between treatments
\begin{itemize}
\item Leads to data on health outcomes  and several correlates (e.g. antibodies, Covid, bloodclots, microchips, magnetism)

\item Sufficiently homogeneous
\end{itemize}
\bigskip
\item \emph{Analyst}, e.g. public health authority, sees treatment choices
\begin{itemize}
\item Wants to know why (did not) chose treatment
\item Can misspecified causal model explain decisions?
\end{itemize}

\end{itemize}

}

\frame{
\frametitle{Other examples}

\begin{itemize}
\item Do firms believe more education causes higher productivity or is just a signal of ability?
\item Does central bank governor believe Phillips curve is causal? 
 If so,  does inflation cause employment or unemployment cause deflation? 
\item Is employer taste-based or statistically discriminating? Does it believe trait causes productivity or signals education?
\item What does monopolist think is its demand curve? I.e., what is the causal effect of increasing its price on quantity demanded? 
\end{itemize}
}

\frame{
\frametitle{Objectives}
Following Pearl (1995) and Spiegler (2016),  model perception of causality with a \alert{directed acyclic graph (DAG)} 

  {
	\centering
	\include{DAG_example_noa_word_2}
}
\vspace{-1em}

\begin{itemize}
\item Arrows indicate  \emph{causes}, otherwise non-parametric
\begin{itemize}

\item Variable independent of ancestors given its direct causes

\item Health  independent of Treatment  \& Antibodies given Covid 
\end{itemize}
\end{itemize}
}
\frame{
\begin{center}

\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_noa_word_2}
	\end{minipage}
	\qquad\qquad\qquad 
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_noe_word_2}
	\end{minipage}
	
	
	\begin{minipage}{0.2\textwidth} 
	\centering
	\include{DAG_example_line_word_2} 
	\end{minipage}
	\qquad\qquad\qquad 
	\begin{minipage}{0.2\textwidth} 
	\centering
	\include{DAG_example_both_word_2} 
	\end{minipage}

  \end{center}
  }


\frame{
\begin{center}
\begin{tikzpicture}[scale=1.6]
\node (v0) at (-2.06,-1.72) {1};
\node (v1) at (0.776,0.806) {2};
\node (v2) at (1.07,1.28) {3};
\node (v3) at (1.30,0.423) {4};
\node (v4) at (-2.33,2.08) {5};
\node (v5) at (-2.07,1.10) {6};
\node (v6) at (0.492,-0.301) {7};
\node (v7) at (-1.57,0.703) {8};
\node (v8) at (-1.19,0.506) {9};
\node (v9) at (0.337,0.478) {10};
\node (v10) at (-1.08,-0.583) {11};
\node (v11) at (-1.15,1.19) {12};
\node (v12) at (-0.520,-1.16) {13};
\node (v13) at (0,-0.817) {14};
\node (v14) at (0.195,1.68) {15};
\node (v15) at (-1.74,1.60) {16};
\node (v16) at (-2.00,-0.126) {17};
\node (v17) at (-1.56,-1.00) {18};
\node (v18) at (-2.34,-2.16) {\color{red}{0}};
\node (v19) at (-2.38,-1.17) {19};
\node (v20) at (1.62,-2.17) {\color{red}{21}};
\node (v21) at (1.58,2.04) {20};
\draw [->] (v0) edge (v3);
\draw [->] (v0) edge (v4);
\draw [->] (v0) edge (v5);
\draw [->] (v0) edge (v6);
\draw [->] (v0) edge (v7);
\draw [->] (v0) edge (v8);
\draw [->] (v0) edge (v9);
\draw [->] (v0) edge (v11);
\draw [->] (v0) edge (v13);
\draw [->] (v0) edge (v15);
\draw [->] (v0) edge (v20);
\draw [->] (v1) edge (v3);
\draw [->] (v1) edge (v20);
\draw [->] (v2) edge (v20);
\draw [->] (v3) edge (v20);
\draw [->] (v4) edge (v1);
\draw [->] (v4) edge (v5);
\draw [->] (v4) edge (v7);
\draw [->] (v4) edge (v14);
\draw [->] (v4) edge (v15);
\draw [->] (v4) edge (v21);
\draw [->] (v5) edge (v1);
\draw [->] (v5) edge (v7);
\draw [->] (v5) edge (v16);
\draw [->] (v6) edge (v20);
\draw [->] (v7) edge (v16);
\draw [->] (v7) edge (v17);
\draw [->] (v7) edge (v18);
\draw [->] (v8) edge (v1);
\draw [->] (v8) edge (v6);
\draw [->] (v8) edge (v7);
\draw [->] (v8) edge (v20);
\draw [->] (v9) edge (v20);
\draw [->] (v10) edge (v12);
\draw [->] (v10) edge (v20);
\draw [->] (v11) edge (v2);
\draw [->] (v11) edge (v3);
\draw [->] (v11) edge (v20);
\draw [->] (v12) edge (v20);
\draw [->] (v13) edge (v12);
\draw [->] (v13) edge (v20);
\draw [->] (v14) edge (v1);
\draw [->] (v14) edge (v7);
\draw [->] (v14) edge (v11);
\draw [->] (v14) edge (v15);
\draw [->] (v14) edge (v21);
\draw [->] (v15) edge (v3);
\draw [->] (v15) edge (v20);
\draw [->] (v16) edge (v10);
\draw [->] (v16) edge (v17);
\draw [->] (v16) edge (v18);
\draw [->] (v17) edge (v12);
\draw [->] (v17) edge (v20);
\draw [->] (v18) edge (v12);
\draw [->] (v18) edge (v20);
\draw [->] (v19) edge (v3);
\draw [->] (v19) edge (v4);
\draw [->] (v19) edge (v5);
\draw [->] (v19) edge (v9);
\draw [->] (v19) edge (v14);
\draw [->] (v19) edge (v15);
\draw [->] (v19) edge (v20);
\draw [->] (v19) edge (v21);
\draw [->] (v21) edge (v3);
\draw [->] (v21) edge (v20);
\end{tikzpicture}
\end{center}

DAG used by Evandt et al. (2017, \textit{Environmental Health})
}
\frame{
\frametitle{Objectives}
A DM has a \emph{subjective causality representation (SCR)} if she 
\begin{itemize}
\item forms beliefs about her actions using an endogenous dataset and a subjective causal model described by a \alert{DAG} 

\item uses those beliefs to evaluate lotteries
\item maximizes perceived expected utility (plus Logit shock)
\end{itemize}

\uncover<2>{
Her choices form a \emph{personal equilibrium}:

  \qquad choices affect data affect inferences affect choices affect ...
  
  \qquad when model \alert{misspecified}}
}

\frame{
\frametitle{Objectives}


\begin{itemize}

\item Analyst observes only the DM's behavior, a \emph{random choice rule $\rho$} over lotteries, not her model
\bigskip

\item \alert{Result 1}: Identifies subjective causal model from $\rho$
\bigskip

\item \alert{Result 2}: Provides test of model in terms of $\rho$ (axioms)
\end{itemize}
}

\frame{
\frametitle{Implications}
\begin{itemize}
\item DM's behavior endogenously creates correlations that she misinterprets as causation

\item Leads to two technical challenges in analyzing model:
\begin{itemize}

\item violation of regularity (stochastic WARP)
\item self-confirming behavior (multiple equilibria)
\end{itemize}

\item But allows model to accommodate:
\begin{itemize}

\item selection neglect
\item status quo bias
\item congruence bias
\item illusion of control
\item patternicity
\end{itemize}
\end{itemize}
}




\frame{
\frametitle{Big Picture}



\begin{itemize}

\item Growing literature studying \emph{misspecified} models:


\begin{itemize}
\item \alert{Spiegler (2016)}, Eliaz \& Spiegler (2018),
Eliaz et al. (2019), Schumacher \& Thysen (2020), Spiegler (2020),  etc.

\item Esponda \& Pouza (2016), He (2018), Bohren and Hauser (2018), Heidhues et al. (2018),   Samuelson \& Mailath (2019), Frick et al. (2020), Fudenberg et al. (2021), Levy et al. (2021), Montiel Olea et al. (2021), etc.
\end{itemize}


\item Given that the model is \emph{misspecified}, how can we identify what the agent believes it is?

\begin{itemize}
\item  Lipman (1999), Ahn \& Ergin (2010), Ellis \& Piccione (2017), Kochov (2018), etc.
\item \alert{this paper}
\end{itemize}

\end{itemize}

}

\frame{
\frametitle{Outline}
\begin{enumerate}

\item Model
\item Identification 
\item Foundations
\end{enumerate}
}

\section{Model}

\frame{
\frametitle{Setup}
\begin{itemize}
\item DM  takes actions  -- full support \emph{lotteries}   over $\{Y,N\}^{n+1}$

\begin{itemize}
\item e.g. ``given type of patient undergoes treatment'' 
\medskip 

lottery captures uncertain effects of treatment
\end{itemize}
\bigskip

\item \alert{Single, payoff-relevant} consequence, indexed $n+1$, 

and $n$ \alert{covariates}, indexed $1$, $2$, ... , $n$
\begin{itemize}
\item e.g.  two correlates $C_1,C_2$ (Covid, Antibodies) and \textbf{H}ealth
\medskip

DM only cares about health ($H=3$)
\end{itemize}
\bigskip

\item Analyst observes \textbf{random choice rule}  indicating
how often  DM chooses each action $a$ from every menu $S$: \emph{$\rho(a,S)$}  

\begin{itemize}
\item e.g. how often patients undergo treatment $a$

\end{itemize}
\end{itemize}
}


\frame{
\frametitle{Model}
\begin{itemize}
\item DM wants to learn effects of treatments
\bigskip

\item Choices $\implies$ distribution \emph{$\rho^S$} over  actions, $C_1,C_2$,  and Health
\medskip

(endogenous) dataset generated from $\rho(\cdot,S)$

$$\rho^S(a,x_1,y_2,z_H)=\rho(a,S)a(x_1,y_2,z_H)$$


\end{itemize}

}

\frame{
\frametitle{Model}

  
\vspace{-1em}
  {
	\centering
	\include{DAG_example_noe}
}
\vspace{-2em}
\begin{itemize}
\item  when deciding which treatment, uses both $\rho^S$ and \alert{DAG $R$} 

(action labeled $0$)

\item each variable \emph{conditioned only on its causes},\\ 
i.e. conditions Health on only $C_2$ and leaves out others
\bigskip
\pause

\begin{itemize}
\item \textbf{Pros:} less storage, quicker to learn, adjusts for endogeneity
\item \textbf{Cons:} may be \textbf{misspecified}, so might either

 mistake spurious \alert{correlation for causality }
 
 or \alert{omit} important explanatory variables
\end{itemize}
\end{itemize}
}

\begin{frame}[label=factorization]
\frametitle{Model}
\vspace{-1em}
  {
	\centering
	\include{DAG_example_noe}
}
\hypertarget{specialDAGformula}{``as if''}  uses dataset \emph{$\rho^S_R$} instead of $\rho^S$ where
\begin{align*}
\rho^S_R(Y_3|a)=&\sum_{x=Y,N} \rho^S(x_2|a)\Bigg[\sum_{z=Y,N}\rho^S(z_1|x_2\hbox{\color{red}\sout{$,a$}}) \rho^S(Y_3|x_2\hbox{\color{red}\sout{$,z_1,a$}})\Bigg]\\
=&\sum_{x=Y,N} \rho^S(x_2|a) \Bigg[\qquad\qquad\qquad\qquad \rho^S(Y_3|x_2 \qquad) \Bigg]
\end{align*}
 
\hyperlink{generalDAGformula}{\beamergotobutton{General formula}}
\end{frame}



\frame{
\begin{center}
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_noa}
	\end{minipage}
	\qquad
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_nop}
	\end{minipage}
		\qquad
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_line2}
	\centering
	\end{minipage}
	
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_noe}
	\end{minipage}	
	\qquad
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_base}
	\end{minipage}
	\qquad
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_line}
	\end{minipage}

\end{center}	
\frametitle{Running Example: DAGs}
}



\frame{

\begin{defn}
Random choice rule $\rho$ has  \emph{subjective causality representation (SCR)}  if there exists ``well-behaved'' DAG $R$ and continuous, increasing  $u$ so that for every menu $S $ and action $a\in S$:
\[
\rho(a,S)=\frac{\exp \left( \sum_{x=Y,N} u(x)\rho^S_{\alert{R}}(x_{n+1}|a) \right) }{\sum_{a' \in S} \exp \left( \sum_{x=Y,N} u(x)\rho^S_{\alert{R}}(x_{n+1}|a') \right)}
\]

\end{defn}
\medskip


Equivalent to maximizing $$\sum_{x=H,L} u(x)\rho^S_{\alert{R}}(x_{n+1}|a) +\epsilon_a$$ when $\epsilon_a$ $\sim$ extreme value
}
\frame{
DAG is well-behaved if
\begin{enumerate}
\item Nothing causes action (uninformed)

\item Path from action to outcome (nontrivial)


\item Common causes are linked (perfect)

gets marginal distribution of each variable correct

\end{enumerate}

}



\subsection{Identification}

\frame{
\emph{Identify DAG} by revealing its 
 minimal paths 
 
 each path represents a chain of causal relationships
\bigskip

	{\centering
	\begin{minipage}{0.3\textwidth}
	\centering
	\include{DAG_example_noa}
	
	Paths: 	\alert{$(0,1,3)$}
	\end{minipage}
	\qquad
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_nop}
	
	\alert{$(0,1,3),(0,2,3)$}
	\end{minipage}
	\qquad
	\begin{minipage}{0.3\textwidth}
	\centering
	\include{DAG_example_line}
	
	\alert{$(0,2,1,3)$}
	\end{minipage}
	
	\begin{minipage}{0.3\textwidth}
	\centering
	\include{DAG_example_noe}
	
	Paths: \alert{$(0,2,3)$}
	\end{minipage}
	\qquad
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_base}
	
	\alert{$(0,3)$}
	\end{minipage}
	\qquad
	\begin{minipage}{0.3\textwidth}
	\centering
	\include{DAG_example_line2}
	
	\alert{$(0,1,2,3)$}
	\end{minipage}
}

}

\frame{
\begin{thm}
Let $\rho$ have a SCR $({\color{red}R},u)$ and \emph{$R'$} be a well-behaved DAG
\medskip

Then, $\rho$ also has a SCR $({\color{blue}R'},u')$ \textbf{if and only if}

\emph{every $R'$ minimal path} is a \alert{$R$ minimal  path} \& vice versa

and there is $\beta$ so that $u(x)=u'(x)+\beta$ for all  $x\in \{Y,N\}$
\end{thm}
\bigskip
two DMs behave same iff  same causal chains
\bigskip
\uncover<2>{ }
\begin{enumerate}[<2->]
\item Any DAGs that represent $\rho$ agree on minimal paths
\item Only the variables in some minimal path matter  
\begin{itemize}
\item 
The rest can be dropped without loss
\end{itemize}
\item Links in minimal paths capture most causal relationships 

\begin{itemize}
\item 
Only exceptions are those between common parents

\end{itemize}
\end{enumerate}
}

\frame{
\frametitle{Illustration}
\begin{center}
\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_noe}
	
	$R$
	\end{minipage}
	\qquad
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_noe2}
	
	$\hat{R}^\rho$
	\end{minipage}
	\qquad
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_noe3}
	
	$R'$
	\end{minipage}
\end{center}

Given $\rho$ with  SCR $(R,u)$, Theorem tells us:
\begin{enumerate}
\item  $\hat{R}^\rho \subset R$

\item $\rho$ can also be represented by any other well-behaved DAG containing $\hat{R}^\rho$ as long as it has no new minimal paths 
\begin{itemize}
\item e.g. $\hat{R}^\rho$ or $R'$
\end{itemize}
\end{enumerate}
}



\frame{
\frametitle{Proof: Separates}

\begin{defn}
A subset of covariates $I$ \emph{separates} if $\rho(a,\{a,b\})=\frac12$, i.e. DM is indifferent, whenever the variables indexed by $I$ are  \textbf{independent} of all others, regardless of how DM chooses from $\{a,b\}$
\end{defn}
\bigskip\pause

\begin{lemma}
\label{prop: I contains AP iff block}
A subset of covariates \alert{$I $  separates} \textbf{if and only if}

\emph{every path} from $0$ to $n+1$ in the DM's DAG \emph{intersects $I$}
\end{lemma}

Notation: $\mathcal{A}= \{ I \subset N: I\text{ is a minimal set that separates}\} $
}

\frame{
\frametitle{Illustration: Separates}

\begin{itemize}
\item Suppose that  $C_1$ is \textbf{independent} of all other variables regardless of how DM chooses from $\{a,b\}$
\item If DM \alert{doesn't believe} either action  or $C_2$ causes Health, then  \alert{infers} that all treatments lead to same expected health
\item Hence indifferent: $\rho(a, \{a,b\})=\rho(b, \{a,b\})$  \medskip
\pause

\item $\{1\}$ \emph{separates} if indifferent for all such $\{a,b\}$;\\
\alert{reveals all} paths go through $1$
\end{itemize}
{\centering
\include{DAG_example_noa}
}
}

\frame{
\frametitle{Illustration: Separates}
	\centering
	\begin{minipage}{0.25\textwidth}
	\centering
	\include{DAG_example_noa}
	
	$\mathcal{A}$: 	\alert{$\{\{1\}\}$}
	\end{minipage}
	\qquad
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_nop}
	
	\alert{$\{\{1,2\}\}$}
	\end{minipage}
	\qquad
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_line}
	
	\alert{$\{\{1\},\{2\}\}$}
	\end{minipage}
	
	\begin{minipage}{0.25\textwidth}
	\centering
	\include{DAG_example_noe}
	
	$\mathcal{A}$: \alert{$\{\{2\}\}$}
	\end{minipage}
	\qquad
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_base}
	
	\alert{$\emptyset$}
	\end{minipage}
	\qquad
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_line2}
	
	\alert{$\{\{1\},\{2\}\}$}
	\end{minipage}
}




\frame{
\frametitle{Proof: Ordering}
\begin{itemize}
\item in illustration, if $\mathcal{A} \neq \{\{1\},\{2\}\}$, done; 


otherwise, need to know whether DM believes treatment \textbf{directly} causes $C_1$ or $C_2$
\item If we  know direction of causality, done; \\
otherwise, we need to reveal it



\end{itemize}
\bigskip

\bigskip
\begin{center}
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_line2}
	\end{minipage}
	\qquad
	\qquad
	\begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_line}
	\end{minipage}

\end{center}
}

\frame{\frametitle{Proof: Ordering}
\begin{lemma}
\label{lem: rmap astar}
 $\mathcal{A}$ can be (uniquely) ordered $\mathcal{A}=\{A^*_1,\dots,A^*_{|\mathcal{A}|}\}$ so that 
 
 \emph{$(i_0,\dots,i_m)$ is a minimal path} from $0$ to $n+1$ \textbf{if and only if} 
 
 for every $j $ there exists $k$ so that \alert{$$i_j \in A^*_{k} \setminus A^*_{k+1}\ \&\ i_{j+1} \in A^*_{k+1} \setminus A^*_k$$} where $A^*_0=\{0\}$ and $A^*_{|\mathcal{A}|+1}=\{n+1\}$.
\end{lemma}

 \textbf{Intuition: }
if a variable is \emph{independent} of the next variable in the chain, then the causal chain is \alert{``broken''} so DM is \emph{indifferent}
}
\frame{
\frametitle{Proof: Ordering}
\begin{defn}\label{def: reveal1}
	 When $\mathcal{A} \neq \emptyset$, $A \in \mathcal{A}$  equals $A^*_1$
	   if $\rho(a,\{a,b\})=\frac12$ whenever $\marg_{A} a = \marg_{A} b$.
\end{defn}

when $\marg_{I} a = \marg_{I} b$,  the variables indexed by $I$ are independent of action regardless of how DM chooses
\bigskip

\begin{itemize}
\item 
if \emph{equally likely} to choose either when  both treatments lead to  \emph{same distribution of $C_1$}, then $0 R 1$
\item 
if \alert{equally likely} to choose either when  both treatments lead to  \alert{same distribution of $C_2$}, then $0 R 2$
\end{itemize}
}

\frame{
\frametitle{Proof: Ordering}
When $|\mathcal{A}|>2$, we also need:
\begin{defn}\label{def: reveal2}
The set $A^*_{i+1} \in \mathcal{A}\setminus \{A^*_1,\dots,A^*_{i}\}$ follows $A^*_i$ if  
\begin{itemize}
\item 
$A^*_{i+1}$ has $\subseteq$-largest intersection with $A^*_i$ and
\item $\rho(a,\{a,b\})=\frac12$  whenever
the variables indexed by $  A^*_{i+1}$ are independent of  those indexed by $A^*_i$ regardless of how DM chooses from $\{a,b\}$
\end{itemize}
\end{defn}
\bigskip
\begin{itemize}
\item 
if first condition failed, then we would have a cycle
\item 
second condition generalizes single chain logic

\begin{itemize}
\item note this requires that the variables indexed by $ A^*_i \cap A^*_{i+1} $ are independent of all others
\end{itemize}
\end{itemize}


}

\frame{
\frametitle{Implications}
\begin{defn}\label{def: revealed cause}
For $j,k \in \{0,1,\dots,n+1\}$, \emph{$j$ is revealed to cause $k$ for $\rho$}, written \emph{$j\hat{R}^\rho k$}, if $j R k$ \textbf{whenever} $\rho$ has SCR $(R,u)$
\end{defn} 
\begin{itemize}
\item If more than one DAG represents $\rho$, $\hat{R}^\rho$ common to all
\item 
Similar to Masatlioglu et al (2012)
\end{itemize}
\pause
\begin{proposition}
\label{prop: fundamental links}
If $\rho$ has a SCR, then $j \hat{R}^\rho k$ if and only if there exists $i$ so that $j \in A^*_{i} $ and $k \in A^*_{i+1} \setminus A^*_i$ when $A^*_0=\{0\}$ and $A^*_{|\mathcal{A}|+1}=\{n+1\}$.
\end{proposition}
} 

\frame{
\frametitle{Implications}
\begin{proposition}
If $\rho$ has a SCR $(R,u)$, then \[
 \rho^S_R \left(x_{\cup_{i = 1}^{|\mathcal{A}|+1} A^*_i } |a\right) =a \left(x_{A^*_1} \right)\prod_{i = 1}^{|\mathcal{A}|} \rho^S \left(x_{A^*_{i+1} \setminus A^*_i}|x_{A^*_{i}} \right)
\]
for ever $x\in \{Y,N\}^{n+1}$ where $A^*_{|\mathcal{A}|+1}=\{n+1\}$
\end{proposition}

($x_E$ for $E \subset \{1,...,n+1\}$ is projection of $x$ onto $E$)
} 

\frame{
\frametitle{Lessons so far}

When $\rho$ has a SCR $(R,u)$, then the DM's choices will reveal the minimal causal chains in her model
\bigskip

This can be done by ordering the set of minimal separators
\bigskip

Reveals every causal relationships that must be in model
\bigskip

Allows analyst to recover DAG(s) representing $\rho$
\bigskip

Which testable predictions can be used to refute model?
}

 \section{Axiomatization}
 \frame{
 \frametitle{Behavioral Characterization}
 \begin{itemize}

 \item 
\hypertarget{axioms}{Provide axioms equivalent to $\rho$ having SCR}
 
 \item Illustrations: running example with $\mathcal{A}=\{\{2\}\}$
 \end{itemize}
 
 \begin{center}
 \begin{minipage}{0.2\textwidth}
	\centering
	\include{DAG_example_noe2}
	\end{minipage}
 
 \end{center}
	}
 
 \frame{
 \begin{ax}[Full-Support]
\label{ax: positivity}
\label{ax: first SCR}
\label{ax: nontrivial}
For any menu $S $ and action $a\in S$, $\rho(a,S)>0$. 
\end{ax}

\begin{itemize}
\item every treatment taken with positive probability


\end{itemize}
 }
 \frame{

\begin{ax}[Bounded Misperception]
\label{ax: bounded}
The Luce ratio is bounded: $$\sup_{S; a,b\in S} \frac{\rho(a,S)}{\rho(b,S)}<\infty$$
\end{ax}

\begin{itemize}

\item relative frequency of taking either of a pair of treatments

\medskip
is \alert{bounded above} by that of 

\medskip
treatment \textbf{known} to lead to good health and one \textbf{known} to lead to bad health
\bigskip

\item limits how much DM misperceives distribution

\medskip
similar to Weak Monotonicity in Ellis and Piccione (2017)
\end{itemize}}

\frame{
\begin{ax}[Consistent Revealed Causes]
\label{ax: Consistent Revealed Causes}
For $i=1,\dots,|\mathcal{A}|$, $A^*_i$ exists.
\end{ax}
\begin{itemize}
\item Possible to order $\mathcal{A}$
\end{itemize}

}
 \frame{
 \frametitle{Indifferent If Identical Immediate Implications}
 \begin{ax}[I5]
\label{ax: same pr if first clique}For $A^*_{|\mathcal{A}|+1}=\{n+1\}$ and $a,b \in S $, 

if $\marg_{A^*_1} a= \marg_{A^*_1} b$, then $\rho(a,S)=\rho(b,S)$.
\end{ax}
\bigskip
\begin{itemize}
\item 
equally likely to undergo any  pair of treatments  leading to  same distribution of $C_2$
\end{itemize}
}



\frame{
\frametitle{Luce's Choice Axiom Given Inferences}
\begin{itemize}

\item Recall: variables in $A^*_i$ revealed to cause those in $A^*_{i+1}\setminus A^*_i $
\item Reveals menus for which DM makes same or similar inferences
\pause

\item if for all $i$ and $x$ $$\rho^{S'}(x_{A^*_{i+1} \setminus A^*_i}|x_{A^*_{i} }) {\color{blue}{=}} \rho^{S}(x_{A^*_{i+1} \setminus A^*_i}|x_{A^*_{i} }),$$ then DM makes the \emph{same inferences} about health outcome of each treatment when facing $S$ as when facing $S'$
\pause

\item and if for all $i$ and $x$ $$\rho^{S'}(x_{A^*_{i+1} \setminus A^*_i}|x_{A^*_{i} }) \alert{\approx } \rho^{S}(x_{A^*_{i+1} \setminus A^*_i}|x_{A^*_{i} }),$$  then DM makes \alert{almost the same inferences} about health outcome of each treatment  when facing $S$ as when facing $S'$
\end{itemize}
}


\frame{
\frametitle{Luce's Choice Axiom Given Inferences}
\begin{ax}[LCI]
\label{ax: luce axiom inference}
For    any $S,S_1,S_2, \dots \in \mathcal{S}$ with $a,b \in S_m \cap S$ for each $m$: if $$\rho^{S_m}(x_{A^*_{i+1} \setminus A^*_i}|x_{A^*_{i} }) \rightarrow \rho^{S}(x_{A^*_{i+1} \setminus A^*_i}|x_{A^*_{i} })$$    for every $i =1,\dots,|\mathcal{A}|$ and   $x \in \{Y,N\}^{n+1}$  and where $A^*_{|\mathcal{A}|+1}=\{n+1\}$,
	then $\frac{\rho(a, S_m)}{\rho(b,S_m)}\rightarrow \frac{\rho(a,S)}{\rho(b, S)}$
\end{ax}


if relationship between $C_2$ and Health is same, satisfies LCA: 
\[
\rho^{S'}(x_{H}|z_{2 }) = \rho^{S}(x_{H}|z_{2 })\forall x,z 
\implies \frac{\rho(a, S')}{\rho(b,S')}=\frac{\rho(a,S)}{\rho(b, S)}\]

\pause
if relationship between $C_2$ and Health is \textbf{close enough}, \textbf{almost} satisfies LCA
}



\frame{\begin{defn}\label{def: correct perception}
A menu $S \in \mathcal{S}$ is \emph{(revealed to be) correctly perceived} if
\[ b(x_{\cup_{i = 1}^{|\mathcal{A}|+1} A^*_i })=b(x_{A^*_1})\prod_{i = 1}^{|\mathcal{A}|} a(x_{A^*_{i+1} \setminus A^*_i}|x_{A^*_{i}})
\]
for any $a,b \in S$ and  $x\in \{Y,N\}^{n+1}$, where $A^*_{|\mathcal{A}|+1}=\{n+1\}$
\end{defn}
In illustration, $S$ correctly perceived if 
$a(x_{H}|z_{2 }) =b(x_{H}|z_{2 })$ for all $x,z\in \{Y,N\}$ and $a,b \in S$
}

\frame{
\begin{ax}[Correctly Perceived Logit] 
There exists a increasing $u:\{Y,N\}\rightarrow \mathbb{R}$  so that 
$$\rho(a,S)= \frac{\exp \left(  \sum_{c=Y,N} u(c)a(c_{n+1}) \right)}
{ \sum_{b\in S} \exp \left(  \sum_{c=Y,N} u(c)b(c_{n+1})\right)}$$
for every correctly perceived menu $S$
\end{ax}

\begin{itemize}
\item 
When DM perceives  menus correctly,  acts ``rationally''
\bigskip

\item 
Three axioms: ``independence,'' ``monotonicity,'' and ``continuity''
\end{itemize}
}
\frame{
\begin{thm}
A random choice rule $\rho$ has a SCR \textbf{if and only if} 

$\rho$ satisfies Full-Support,  Bounded Misperception, Consistent Revealed Causes, I5, LCI, and Correctly Perceived Logit
\end{thm}
\bigskip

Axioms  make clear when \& why $\rho$ differs from Logit-EU

\begin{itemize}
\item BM -- maximum deviation from Luce rule
\item  LCI -- only violates Luce rule when inferences change
\item I5 -- same (incorrect) perception implies same frequency
\item Correctly Perceived Logit
\end{itemize}
}

\frame{
\begin{cor}
A random choice rule $\rho$ is  Logit-EU \textbf{if and only if} 

 $\rho$ has a SCR and $\mathcal{A}=\emptyset$\textbf{ if and only if}
 
 
 $\mathcal{A}=\emptyset$ and $\rho$ satisfies Basic, I5, Bounded Misperception,  LCI, and Correctly Perceived Logit

\end{cor}
When $0\  R\   n+1$:
\begin{itemize}
\item $A^*_1=\{n+1\}$ so coincidence of $a$ and $b$'s consequence  distribution implies equal probability of choosing either
\item hypothesis of LCI holds vacuously,  so choice axiom holds
\end{itemize}


}

\frame{
\frametitle{Proof Outline}
\begin{enumerate}
\item For any $S$, let $S'_1$ be a correctly perceived copy of $S$:
 
for each $a \in S$ there is $a' \in S'_1$ so that $a'(x)=a(x_{A^*_1}) \prod_i \rho^S(x_{A^*_{i+1} \setminus A^*_i}|x_{A^*_i})$

\item Form a nested sequence of menus $(S'_m)_m$ so that each $S'_m$ is correctly perceived and $|S'_m|\rightarrow\infty$; 

$\rho^{S'_m}(y_{A^*_{i+1} \setminus A^*_i}|y_{A^*_{i} }) = \rho^{S}(y_{A^*_{i+1} \setminus A^*_i}|y_{A^*_{i} })$ for all $i,y$


\medskip

Pick any $a,b \in S$
 \medskip
 
\item BM implies $\rho(S,S \cup S'_m) \rightarrow 0$ and so 

$\rho^{S'_m \alert{ \cup S}}(y_{A^*_{i+1} \setminus A^*_i}|y_{A^*_{i} }) \rightarrow \rho^{S}(y_{A^*_{i+1} \setminus A^*_i}|y_{A^*_{i} })$ for all $i,y$


\item I5 implies $\frac{\rho(a',S'_m \cup S)}{\rho(b',S'_m \cup S)}=\frac{\rho(a,S'_m \cup S)}{\rho(b,S'_m \cup S)} $ for each $m$

\item LCI implies that  $\frac{\rho(a',S'_m \cup S)}{\rho(b',S'_m \cup S) }=\frac{\rho(a,S'_m \cup S)}{\rho(b,S'_m \cup S)}\rightarrow \frac{\rho(a',S'_1)}{\rho(b', S'_1)}$ 


\item LCI also implies $\frac{\rho(a,S'_m \cup S)}{\rho(b,S'_m \cup S)}=\frac{\rho(a',S'_m \cup S)}{\rho(b',S'_m \cup S) }\rightarrow\frac{ \rho(a, S)}{\rho(b,S)} $ 
\bigskip

 Applying Correctly Perceived Logit  completes proof
\end{enumerate}

}


\frame{
\frametitle{Other Literature}
\begin{itemize}
\item \textbf{DAG as causal model:} Pearl (2009)

\item \textbf{Axioms for normative model of causality:} Schenone (2020)


\item \textbf{Boundedly Rational Stochastic Choice: }Manzini \& Mariotti (2014), Brady \& Rehbeck (2016), Cattaneo et al. (2020), etc.

\item \textbf{Stochastic Choice with rich alternatives:} Gul \& Pesendorfer (2006), Lu (2016), etc.
\end{itemize}

}


\frame{
\frametitle{Wrap up}
\begin{itemize}
\item Provided identification and axiomatization result for model of subjective causality

\item Extensions:

\begin{itemize}
\item Relationships to cognitive biases

\item Comparative statics on dropping nodes/links in DAG

\item Deterministic vs. random choice

\item Exogenous datasets

\end{itemize}

\item Directions for future research:

\begin{itemize}
\item Information

\item Imperfect DAGs

\item Empirical/experimental implementation


\end{itemize}


\end{itemize}

}
\frame{

{\huge Thank you}}

\frame{
\frametitle{General formula}
\hypertarget{generalDAGformula}{}
Let $R(i) = \{j: j R i\}$, called parents of $i$
\bigskip

Given $p \in \Delta \mathcal{X}$, \alert{$p_R\in \Delta \mathcal{X}$} is such that 
$$p_{\color{red}R}(x)=p(x_0)\prod_{j =1}^{n+1} p(x_j|x_{\alert{ R(j)} }) $$
when nothing causes $0$
\medskip

For comparison, $$p(x)=p(x_0)\prod_{j =1}^{n+1} p(x_j|x_{{\color{blue}\{0,\dots,j-1\}}})$$

\hyperlink{specialDAGformula}{\beamerreturnbutton{Back}}
}

\end{document}