Who Is Liable If Your Child Is Injured at Daycare?
These are scary injuries that can have life-long impact. When a child is injured at daycare, here is what to expect and how you as a parent can keep your family protected.
Any licensed daycare center you take your child to in the Valley has a legal obligation to keep your child safe from undue harm. Parents may sign a liability waiver that lifts the liability off a daycare center when a child experiences an accident not directly caused by the daycare provider's negligence, such as if a child is injured during a field trip due to circumstances outside the daycare provider's control. However, a daycare center may still be held liable for an injury in which their negligence was the contributing factor.
They don't remove hazards that pose a threat to a child.
The provider neglects to safely monitor children.
They place a child in harm's way.
The provider engages in careless or reckless actions that contributed to the child's injury.
The provider abuses a child.
Proving this negligence is essential to secure compensation after your child is personally injured at a daycare center. In some cases, a daycare center may be properly following all legal requirements to protect children they're caring for, and even so, another child may cause serious harm to yours. In a case like this, the parents of that child may be deemed the negligent party.
To pursue any case against a child care provider who may be negligent, you will need to have proof that the daycare provider had accepted responsibility for watching your child. Some families take the risk of having an informal babysitter look after their child at a home. If this is the type of daycare you use for your child, make sure to have an agreement in writing about the services they provide.
You may also want to work with a Phoenix personal injury lawyer to draft up a formal agreement or look over any one you are considering signing before handing your child over for care. In any case where your child experiences a serious injury at daycare, you should take the following steps.
To properly assess the situation and the severity of the injuries, it's vital that your child get medical attention. The daycare worker may be the one to call 911 (and hopefully you right after). If you find out your child has been injured before they've received medical attention, take them to a hospital or medical care provider.
Toddlers and young children usually don't have the ability to clearly state the condition of any injuries they're experiencing. Only a licensed medical professional can determine the extent of injuries, which may not be immediately apparent.
Retain all medical records. They're going to be necessary if you decide to pursue legal action against the provider or another child's family.
Just like you'd gather witness statements at the scene of a car accident, it's vital to document details about the state of the daycare center when the personal injury accident occurred. Were there hazards that posed a threat to the child? Did the worker have too many children to watch after, which prevented her or him from properly monitoring the environment and preventing accidents? Ask if the daycare facility has CCTV camera footage you can access.
Take photos or videos if they would be helpful. At the very least, write down and date an immediate account of the accident, and ask for a report from the daycare center. Note anything that other children might say about the accident, too.
When your child sustains serious personal injuries while in the care of someone else, you should contact a Phoenix personal injury attorney. An attorney can help you gather evidence and build a case against a daycare provider if they were the negligent party, or help you navigate how to deal with another family whose child caused serious personal injuries to yours.
If your child has been seriously injured at daycare, contact the Millea Law Firm at (480) 462-5540 for a free consultation.


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Note: The Mobility Bit (the T-shank adapter that connects the Papa Bit to your jigsaw) is not included and can be purchased through our site separately. If you already have our Mobility Bit, then you're good to go!
The papa bit was great addition to my MobilityBit set. I have the intro set which I love, but would sometimes go over a bone and that would hurt a bit. The papa bit provides the need pressure and also is forgiving if going over an area that has bone. I have been using the Papa bit every day after my workouts for recovery.
I had trouble with one bit and a new one was delivered cross country overnight!


Some recent news stories have featured falling prices for real estate in some of Canada's larger markets. Vancouver, in particular, has been singled out. However, lower prices are nothing new for the Sunshine Coast. While larger markets have seemed immune to downward pressure on prices, our market has been adjusting for some time now. We think much of this adjustment on the Coast is already behind us.
And remember, our business is built on great service and happy clients relationships – we're always on the look-out for your referrals to friends and family.
For real estate on the Sunshine Coast, 2012 looked a lot like 2011 - good stretches of activity and sales, followed by slower periods, followed again by more good activity. All in all, we're happy with the past year. We did see continued downward pressure on prices, mainly because sales still lagged below traditional levels. However, one thing has remained constant: for the 17th year in a row, RE/MAX Oceanview Realty was the top selling office on the Sunshine Coast. We know of no other real estate office anywhere that has experienced this level of success. Our goal is to do it again in 2013!


Embracing Creativity or running a creative thinking session is not unlike running a theatrical performance. It needs both stage management and direction to manage creativity.
Such activities can of course vary widely in scale. You may be working on your own or be chatting informally with one or two others. It is possible that you are running a classic small-group brainstorming session. In all cases you must pay attention to the following components.
People – the 'cast' of the drama. If you have the luxury of choosing people to work with then this may present a dilemma. On the one hand, you want as much diversity as possible. This is where new insights and connections will come from.
On the other hand, they must work well together. Creative work requires a high level of personal openness and risk taking, and a lot of interaction.


text:
Illawong is a suburb in southern Sydney, in the state of New South Wales, Australia. Illawong is located 27 kilometres south of the Sydney Central Business District, in the local government area of the Sutherland Shire in the area commonly called Menai. The post code is 2234, which is also assigned to the contiguous suburbs of Menai, Bangor Barden Ridge and Alfords Point.

Illawong sits between the southern shore of the Georges River and the northern shore of the Woronora River. Illawong consists mainly of residential homes, a nursing home, a primary school and a small shopping centre.

History
 is an Aboriginal word meaning between two waters, referring to the Georges and Woronora Rivers. Illawong was originally inhabited by the Tharawal and/or Eora tribes who left remnants of their lives in many middens, rock carvings and cave paintings. The Illawong Nature Reserve also lies "between two waters". In this case, the reserve lies between a system of 2 wetlands, which in combination with the heavily vegetated ridgeland, provide habitat which is particularly suited to the long necked turtle.

Captain John Hunter (1731–1821) was the first European to explore the area. In 1789, Hunter sailed through the Georges and Woronora Rivers. In 1795–6, with Matthew Flinders and George Bass, Hunter explored further up the Georges River and declared the area of Bankstown.

Construction of Old Illawarra Road took place between 1843 and 1845 and was inspected by Major Thomas Mitchell, it being one of the last public works completed by convicts. The construction allowed the new route to be about 32 kilometres shorter towards the South Coast. At the southern end of the district Major Mitchell took the road across the ford at the head of the Woronora River, naming the ford "Pass of Sabugal", possibly due to the resemblance in scenery of the north-eastern frontier of Portugal.

In the 1890s Menai was a farming area and Illawong, which was then known as East Menai, was an area where wealthy people of Sydney built holiday homes and weekenders.

During the 1970s the Menai region was suburbanised, and Illawong was chosen to have historic nautical street names. (Bangor, New South Wales having Aboriginal names, Alfords Point, New South Wales having botanic names and Barden Ridge, New South Wales having names to do with Australian pioneers.

This area was ravaged in the 1994 Eastern seaboard fires, which also caused destruction in the suburbs of Alfords Point, Bangor, Menai and Como. Illawong is considered to be a bushfire prone area.

Demographics
At the 2016 census, there were 7,271 residents in Illawong. The most common ancestries in Illawong were English 22.9%, Australian 22.0%, Irish 7.6%, Scottish 5.9% and Lebanese 5.5%. 75.7% of people were born in Australia. The next most common countries of birth were England 2.7%, Lebanon 2.1% and China 1.9%. 74.4% of people spoke only English at home. Other languages spoken at home included Arabic 5.3%, Greek 4.2%, Cantonese 1.8%, Macedonian 1.6% and Mandarin 1.6%. The most common responses for religion were Catholic 32.3%, Anglican 18.2%, No Religion 17.3% and Eastern Orthodox 9.6%. The median weekly household income was $2,466 and this was considerably higher than the national figure of $1,438. Most dwellings were separate houses and these tended to be large, with 72.4% of dwellings having 4 or more bedrooms.

Commercial area
Illawong Village shopping centre sits between Fowler Road and Hobart Place. Originally a small collection of shops, the shopping centre was rebuilt in 2007. Illawong Village features a Coles supermarket, Liquorland and many specialty shops including butcher, chemist, gift shops and several cafes and restaurants.

Transport
Major roads include Fowler Road and Moreton Road. Transdev NSW has regular bus services which links Illawong to surrounding centres such as Menai, Sutherland and Bankstown.

Schools
 Illawong Primary School
 Tharawal Primary School
 Menai High School is the major public secondary co-educational high school servicing the Menai Area.

Churches
The church within Illawong is the Menai Illawong Uniting Church, near the Illawong shopping centre.

Sport and recreation
The Scout group in Illawong is 1st Wearne Bay Sea Scouts. They have all five youth sections (Joeys, Cubs, Scouts, Venturers and Rovers) represented. The group meets in the scout hall at Wearne Bay, on the Georges River. This hall was previously the house of the former ferry master, Mr Maurie Chegwidden. The Sea Scout group was started in 1966 by Mr Ron Robertson,(first Scoutmaster), Mr William Arthur (Skip) Ross and his wife Mrs Yvonne Marlene Ross (aka Akela) Cubmaster.
In the early days of the group invaluable assistance was provided by Mr Steve Hobkirk in a group leadership role.
The original hall was located in a boatshed on the waterfront nearby to the present day scout hall.

Emergency Services
The Menai/Illawong Rural Fire Service (RFS) is located on the Old Illawarra Road roundabout intersection with Fowler Road.

References

Suburbs of Sydney
Sutherland Shire

redpajama_set_name: RedPajamaWikipedia


How do I make my honey runny again?
What does 'free from Antibiotics & Pesticides' mean?
Is your honey processed or Pasteurised?
Why is Raw Honey not suitable for children under 1 year of age?
Does your Raw Honey contain Sugar?
Is Raw Honey the same as Manuka Honey?
How should I store my Honey?
Overview: Celebrated all over Europe and renowned for its unique complex after taste, our Raw Chestnut honey is one of the most robust honey in our selection. For centuries Chestnut Honey has been reputed to aid digestion and boost the immune system and has been used in many traditional remedies. It is not an overly sweet honey but yet it's subtle herbal aroma and slightly bitter after taste makes it a fantastic addition to cheeses and salad dressings. Slightly varying taste, colour and texture from season to season, it is raw, natural and unpasteurized as nature intended.
Aroma: Herbal, Floral Aroma, with a clear component of dry wood.
Taste: A sweet caramel, tempered by clear salty notes, and often, particularly if Heather is present, slightly bitter notes; while if bramble is present, slight sharp notes and fruity aromas.
Floral Source: The bees have foraged on the blossoms of the Sweet Spanish Chestnut Tree (Castanea Sative).
Geography: Harvested from the deep forests of North of the Peninsula and the diverse terrains of Andalusia.
Chestnut Honey Uses: A teaspoon in the morning for that added boost. Excellent addition to cheeses, salad dressings and fruits. Also great with a luke warm drink. A great way to compliment your healthy diet.
Certified by the Animal Health Service, Chamber of Agriculture and the National Government of Spain.
Please note: If the honey sets (crystallizes) please don't panic as it has not gone off. It is a natural characteristic of Raw Honey. Simply place your jar of raw honey in a glass bowl, quarter fill a pan with boiling water then place the bowl into the pan. The steam should heat the bowl, which heats the jar with out over heating it. Keep the water boiling by turning up the heat on the hob. Your honey should be runny again in 10 minutes or so (depending on size).


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text:
\section{Introduction} \label{introduction}

A \emph{counterfactual} states how the world should be modified so that a given outcome occurs. For instance, the statement \emph{had you been a woman, you would have gotten half your salary} is a counterfactual relating the \emph{intervention} \say{had you been a woman} to the \emph{outcome} \say{you would have gotten half your salary}. Counterfactuals have been used to define causation \citep{lewis1973} and hence have attracted attention in the fields of explainability and robustness in machine learning, as such statements are tailored to explain black-box decision rules. Applications abound, including algorithmic recourse \citep{joshi2019realistic,poyiadzi2020face,karimi2020algorithmic}, defense against adversarial attacks \citep{ribeiro2016should,moosavi2016deepfool} and fairness \citep{kusner2017counterfactual, asher2020adequate, black2020fliptest, plecko2020fair}.

State-of-the-art models for computing counterfactuals have mostly focused on the \emph{nearest counterfactual instances} principle \citep{wachter2017counterfactual}, according to which one finds minimal translations, minimal changes in the features of an instance that lead to a desired outcome. However, as noted by \cite{black2020fliptest} and \cite{poyiadzi2020face}, this simple distance approach generally fails to describe realistic alternative worlds, as it implicitly assumes the features to be independent. Changing just the sex of a person in such a translation might convert from a typical male into an untypical female rendering out-of-distribution counterfactuals like the following: {\em if I were a woman I would be 190cm tall and weigh 85 kg}. According to intuition, such counterfactuals are false and rightly so because they are not representative of the underlying statistical distributions. As a practical consequence, such counterfactuals typically hide biases in machine learning decision rules \citep{lipston2018does,besse2020survey}.

The link between counterfactual modality and causality motivated the use of Pearl's causal modeling \citep{pearl2009causality} to address the aforementioned shortcoming \citep{kusner2017counterfactual,joshi2019realistic,karimi2020algorithmic,mahajan2020preserving}. Pearl's do-calculus, by enforcing a change in a set of variables while keeping the rest of the causal mechanism untouched, provides a rigorous basis for generating intuitively true counterfactuals. The cost of this approach is fully specifying the causal model, namely specifying not only the Bayesian network (or graph) capturing the causal links between variables, but also the structural equations relating them, and the law of the latent, exogenous variables. The reliance on such a strong prior makes the causal approach appealing in theory, but limited for systematic implementation.

To sum-up, research has mostly focused on two divergent frameworks to compute counterfactuals: one that proposes an easy-to-implement model that leads, however, to intuitively untrue counterfactuals; another rigorously takes into account the dependencies between variables to produce realistic counterfactuals, but at the cost of feasibility. Our contribution addresses a third way. Extending the work of \cite{black2020fliptest}, who first suggested substituting causality-based counterfactual reasoning with optimal transport, we define \emph{transport-based counterfactual models}. Such models, by characterizing a counterfactual operation as a coupling, a mass transportation plan between two observable distributions, ensures that the generated counterfactuals are in-distribution, hence realistic. In addition, they remedy to the impracticability issues of causal modeling as they can be computed through any mass transportation techniques, for instance optimal transport. As a consequence, this renders many counterfactual frameworks for fairness feasible.

\subsection{Outline of contributions}

We propose a mass transportation framework for counterfactual reasoning, and point out its similarities to the causal approach. Additionally, we use these framework to derive new transport-based counterfactual fairness criteria.

\begin{enumerate}
    \item In Section \ref{surrogate}, we introduce a general causality-free framework for the computation of counterfactuals through mass transportation techniques (not only optimal transport), leading to the definition of \emph{transport-based} counterfactual models.
    \item In Section \ref{structural}, we present the causal approach of counterfactual inference from a mass transportation perspective. The interest of this viewpoint is two-fold. First, it offers an elegant representation of counterfactual operations and common causal assumptions. In particular, we study the implications of two general causal assumptions in this formalism. Under the \emph{single-world} assumptions, we show that the pairing between factual and counterfactual instances is given by a closed-form deterministic map; when the intervened feature is \emph{relative exogenous}, we show that the distribution of counterfactual instances coincide with an observational distribution. Second, it provides a common formalism with transport-based counterfactual models.
    \item On the basis of the formalism and assumptions presented in Section \ref{structural}, we demonstrate in Section \ref{coincide} that optimal transport maps for the quadratic cost generates the same counterfactual instances as a large class of causal models, including linear additive models. We argue that this makes transport-based counterfactual models relevant surrogate models in the absence of a known causal model.
    \item In Section \ref{applications}, we apply the mass transportation viewpoint of structural counterfactuals by recasting the \emph{counterfactual fairness} criterion into a transport-like one. Then, we propose new causality-free criteria by substituting the causal model by transport-based models in the original criterion.
\end{enumerate}

\subsection{Problem setup} \label{question}

Let $(\Omega, \mathcal{A}, \P)$ be a probability space, and set $d \geq 1$. Define the random vector $V := (X,S) \in \R^{d+1}$, where the variables $X : \Omega \to \mathcal{X} \subset \R^d$ represent some observed features, while the variable $S: \Omega \to \mathcal{S} \subset \R$ can be subjected to interventions. For simplicity, we assume that $\S$ is finite such that for every $s \in \S$, $\P(S=s)>0$. For every $s \in \S$, set $\mu_s := \mathcal{L}(X \mid S=s)$ the \emph{factual} or \emph{observational} probability distribution of $s$-instances, and denote by $\X_s$ its support. We consider the problem of computing the potential outcomes of $X$ when intervening on $S$. Typically, $S$ represents the treatment variable in treatment modeling, or the sensitive, protected attribute in fairness settings. Suppose for instance that the event $\{X=x,S=s\}$ is observed, and set $s' \neq s$. We aim at answering the counterfactual question: \emph{had $S$ been equal to $s'$ instead of $s$, what would have been the value of $X$?} As aforementioned, because of correlations between the variables, computing the alternative state does not amount to change the value of $S$ while keeping the features $X$ equal.

\section{Transport-based counterfactuals}\label{surrogate}

We firstly introduce the necessary background on mass (or measure) transportation. Then, we present the notion of transport-based counterfactual model. Finally, we discuss the specific case of optimal transport.

\subsection{Background in mass transportation}

In probability theory, the problem of mass transportation amounts to constructing a joint distribution namely a \emph{coupling}, between two marginal probability measures. Suppose that each marginal distribution is a sand pile in the ambient space. A coupling is a \emph{mass transportation plan} transforming one pile into the other, by specifying how to move each elementary sand mass from the first distribution so as to recover the second distribution. Alternatively, we can see a coupling as a random matching which pairs start points to end points between the respective supports with a certain weight. Formally, let $P, Q$ be both probabilities on $\R^d$, whose respective supports are denoted by $\text{supp}(P)$ and $\text{supp}(Q)$. A coupling between $P$ and $Q$ is a probability $\pi$ on $\R^d \times \R^d$ admitting $P$ as first marginal and $Q$ as second marginal, precisely  $\pi(A\times\R^d)=P(A)$ and  $\pi(\R^d\times B)=Q(B)$ for all measurable sets $A,B\subseteq \R^d$. Throughout the paper, we denote by $\Pi(P,Q)\subset \mathcal{P}(\R^d\times \R^d)$ the set of joint distributions whose marginals coincide with $P$ and $Q$ respectively.

A coupling $\pi \in \Pi(P,Q)$ is said to be \emph{deterministic} if each instance from the first marginal is paired with probability one to an instance of the second marginal. Such a coupling can be identified with a measurable map $T : \R^d \to \R^d$ that \emph{pushes forward} $P$ to $Q$, that is $Q(B) := P (T^{-1}(B))$ for any measurable set $B\subset \R^d$. This property, denoted by $T_\sharp P = Q$, means that if the law of a random variable $Z$ is $P$, then the law of $T(Z)$ is $Q$. To make the relation with random couplings, we also introduce the action of couples of functions on probability measures. For any pairs of functions $T_1, T_2 : \R^d \to \R^d$, we define $(T_1 \times T_2) : \R^d \to \R^d \times \R^d, x \mapsto (T_1(x),T_2(x))$. As such, $(T_1 \times T_2)_\sharp P$ denotes the law of $\big(T_1(Z),T_2(Z)\big)$ where $Z \sim P$. Thus, a push-forward operator $T$ satisfying $T_\sharp P = Q$ characterizes the coupling $\pi = (I \times T)_\sharp P$ where $I$ is the identity function on $\R^d$. This coupling matches every instance $x \in \text{supp}(P)$ to $T(x) \in \text{supp}(Q)$ with probability 1.

\subsection{Definition}

In \citep{black2020fliptest}, the authors mimicked the structural account of counterfactuals by computing alternative instances using a deterministic optimal transport map. Extending their idea, we propose a more general framework where the counterfactual operation $s \mapsto s'$ can be see as a coupling, a mass transportation plan between the two observable distributions $\mu_s$ and $\mu_{s'}$. We define \emph{counterfactual models} as models that assign a probability to all the cross-world statements on $X$ w.r.t. interventions on $S$

\begin{definition}\label{TCFM}

\begin{enumerate}
    \item A \emph{transport-based counterfactual model} is a collection $\Pi := \{\pi_{ \langle s'|s \rangle}\}_{s,s' \in \S}$ of couplings belonging to $\Pi(\mu_s,\mu_{s'})$, such that $\pi_{ \langle s|s \rangle} = (I \times I)_\sharp \mu_s$. An element of $\Pi$ is called a \emph{counterfactual coupling}. We say that $\Pi$ is a \emph{random counterfactual model} if at least one coupling for $s \neq s'$ is not deterministic.
    \item A \emph{deterministic transport-based counterfactual model} is a collection $\T := \{T_{\mathsmaller{ \langle s'|s \rangle}}\}_{s,s' \in \S}$ of mappings from $\X$ to $\X$ satisfying for any $s,s' \in \S$, ${T_{\mathsmaller{ \langle s'|s \rangle}}}_\sharp \mu_s = \mu_{s'}$ and $T_{\mathsmaller{ \langle s|s \rangle}} = \text{I}$. An element of $\T$ is called a \emph{counterfactual operator}.
\end{enumerate}

\end{definition}

In this formalism, the quantity $d \pi_{\langle s' | s\rangle}(x,x')$ is the elementary probability of the counterfactual statement \emph{had $S$ been equal to $s'$ instead of $s$ then $X$ would have been equal to $x'$ instead of $x$}. The conditions in Definition \ref{TCFM} translate the intuition that a realistic counterfactual operation on $S$ should morph the non-intervened variables $X$ so that their values fit the targeted distribution. The adjective \emph{deterministic} refers to the fact that the model assigns to each factual instance a unique counterfactual counterpart. Formally, the counterfactual counterpart of some observation $x \in \text{supp}(\mu_s)$ for a change $s \mapsto s'$ is given by $x' = T_{\mathsmaller{\langle s' | s \rangle}}(x) \in \text{supp}(\mu_{s'})$. In contrast, a \emph{random} model allows possibly several counterparts with probability weights.

One challenge for the transport-based approach is to choose the model appropriately in order to define a relevant notion of counterpart. There possibly exists an infinite number of admissible counterfactual models in the sense of Definition \ref{TCFM}, many of them being inappropriate. As an illustration, consider the family of trivial couplings, namely $\{\mu_s \otimes \mu_{s'}\}_{s,s'\in \S}$ where $\otimes$ denotes the factorization of measures. Though it is a well-defined transport-based counterfactual model, it is not intuitively justifiable as it completely decorrelates factual and counterfactual instances. A transport-based counterfactual model must be both \emph{intuitively justifiable} and \emph{computationally feasible}. We justify next that \emph{optimal-transport-based} counterfactual models fit these conditions.

\subsection{The case of optimal transport}\label{OT}

We recall here some basic knowledge on optimal transport theory, and refer to \citep{villani2003topics,villani2008optimal} for further details. Optimal transport restricts the set of feasible couplings between two marginals by isolating ones that are optimal in some sense.

The \emph{Monge formulation} of the optimal transport problem with general cost $c : \R^d \times \R^d \to \R$ is the optimization problem
\begin{equation}\label{monge}
    \min_{T :\ T_\sharp P = Q} \int_{\R^d} c(x,T(x)) dP(x).
\end{equation}
We refer to solutions to \eqref{monge} as \emph{optimal transport maps} between $P$ and $Q$ with respect to $c$. One difficulty is that the push-forward constraint renders the problem unfeasible in many general settings, in particular when $P$ and $Q$ are not absolutely continuous w.r.t. the Lebesgue measure or have unbalanced numbers of atoms.

This issue motivated the following \emph{Kantorovich relaxation} of the optimal transport problem with cost $c$,
\begin{equation}\label{kanto}
    \min_{\pi\in \Pi(P,Q)} \int_{\R^d\times \R^d} c(x,x') d\pi(x,x').
\end{equation}
Solutions to \eqref{kanto} are \emph{optimal transport plans} between $P$ and $Q$ with respect to $c$. In contrasts to optimal transport maps, they exist under very mild assumptions, like the non-negativeness of the cost.

As argued by \cite{black2020fliptest}, generating a counterfactual operation by solving \eqref{monge} leads to intuitively relevant counterfactuals, as they are obtained by minimizing a metric between paired instances while preserving the probability distributions. The framework we propose generalizes the one of \cite{black2020fliptest}, restricted to optimal transport maps, by enabling random couplings. As mentioned before, using a random coupling rather than a deterministic map has philosophical implications since it renders non necessarily unique the counterfactual counterparts of a single instance. More practically, this relaxation is also crucial for dealing with non-continuous variables and for efficient implementations. In concrete settings, as practitioners do not completely know the \emph{true} measures $\mu_s$ and $\mu_{s'}$ but have access to empirical samples, they must estimate the model from data. The computational complexity of solving \eqref{kanto} between an $n$-sample and an $m$-sample is in $\mathcal{O}\big((n+m)nm \log(n+m)\big)$, but one can substantially improve on this to reach $\mathcal{O}\big(nm\big)$ with entropy-regularized versions \citep{cuturi2013sinkhorn,peyre2019computational}. However, such regularization necessarily entails non-deterministic couplings, making difficult the implementation of deterministic counterfactual models.

Interestingly, \cite{black2020fliptest} observed that optimal transport maps generated nearly identical counterfactuals to the ones based on causal models. In the following, we show that the transport-based approach can be seen as an approximation of Pearl's computation of counterfactuals. Specifically, we reframe the generation of structural counterfactuals as a problem of mass transportation in Section \ref{structural}, and prove that it identifies (possibly deterministic) transport-based counterfactual models in several settings. In Section \ref{coincide}, we demonstrate that a relevant choice of the transportation cost $c$ makes optimal-transport counterfactuals coincide with a large class of causal counterfactuals.

\section{Structural counterfactuals, revisited}\label{structural}

Pearl's causal reasoning provides a natural framework to address the counterfactual problem introduced in Section \ref{question}. We consider a causal framework where $V = (X,S) \in \R^{d+1}$ is the unique solution to an acyclic \emph{structural causal model} (SCM). More precisely, each \emph{endogenous} variable $V_i$ is defined (up to sets of probability zero) by the structural equation
\begin{equation}\label{SEM}
    V_i \stackrel{\P-a.s.}{=} G_i(V_{\text{Endo}(i)},U_{\text{Exo}(i)}),
\end{equation}
where $G_i$ is a real-valued measurable function, $U$ is a set of \emph{exogenous} variables, while $V_{\text{Endo}(i)}$ and $U_{\text{Exo}(i)}$ denote respectively the endogenous and exogenous parents of $V_i$. Throughout the paper, we denote by $U_X$ and $U_S$ the exogenous parents of respectively $X$ and $S$. The word \say{parent} refers to the fact that causal dependencies are generally represented by a \emph{directed acyclic graph} (DAG).

We write $X_{S=s}$ the random vector defined as the solution to \eqref{SEM} under the do-intervention $do(S=s)$, that is after replacing the structural equation on $S$ by $S=s$ while keeping the rest of the causal mechanism equal.  This framework enables to define counterfactual statements as \emph{had $S$ been equal to $s$, then $X$ would have been equal to $X_{S=s}$}. Following \cite{karimi2020algorithmic}, we will refer to these counterfactuals as \emph{structural} counterfactuals throughout the paper. In this section, we underline the mass transportation formalism of this computation of counterfactuals.

\subsection{Definition}

We introduce the following  notations to formalize the contrast between interventional, counterfactual and factual outcomes. For $s,s' \in \S$ we define three probability distributions. Firstly, we recall that $\mu_s := \mathcal{L}(X \mid S=s)$ is the distribution of the \emph{factual} $s$-instances. This observable measure describes the possible values of $X$ such that $S=s$. Secondly, we denote by $\mu_{S=s} := \mathcal{L}(X_{S=s})$ the distribution of the \emph{interventional} $s$-instances. It describes the alternative values of $X$ in a world where $S$ is forced to take the value $s$. On the contrary to the factual distribution, the interventional distribution is in general not observational, in the sense that we cannot draw empirical observations from it. Finally, we define by $\mu_{\langle s'|s \rangle} := \mathcal{L}(X_{S=s'} \mid S=s)$ the distribution of the \emph{counterfactual} $s'$-instances given $s$. It describes what would have been the factual instances of $\mu_s$ \emph{had $S$ been equal to $s'$ instead of $s$}. According to the \emph{consistency rule} \citep{pearl2016causal}, the factual and counterfactual distributions coincide when $s=s'$, that is $\mu_s = \mu_{\langle s|s \rangle}$. However, when $s \neq s'$, the counterfactual distribution $\mu_{\langle s'|s \rangle}$ is generally not observable. This is a key difference compared to the transport-based approach which relates observable distributions.

As for the transport-based approach, we need a counterfactual model on $X$ w.r.t. interventions on $S$ to characterize the distribution of all the cross-world statements. We refer to the model derived from Pearl's causal reasoning as the \emph{structural counterfactual model}.

\begin{definition}\label{ctp} For every $s,s' \in \S$, the \emph{structural counterfactual coupling} between $\mu_s$ and $\mu_{\langle s'|s \rangle}$ is given by

$$
\pi^*_{\langle s'|s \rangle} := \mathcal{L}\big((X, X_{S=s'}) \mid S=s\big).
$$
\noindent
We call the collection of couplings $\Pi^* := \{\pi^*_{\langle s'|s \rangle}\}_{s,s'\in \S}$ the \emph{structural counterfactual model}.

\end{definition}

This definition shares similarities with Definition \ref{TCFM} by characterizing a counterfactual model as a collection of joint probability distributions. However, a structural couterfactual model is \emph{a priori} not a transport-based counterfactual model since a coupling $\pi^*_{\langle s'|s \rangle}$ belongs to $\Pi(\mu_s,\mu_{\langle s'|s \rangle})$ and not $\Pi(\mu_s,\mu_{s'})$. By convention, we use the superscript $^*$ to denote the \emph{structural} counterfactual models defined a causal model, and no superscript for the \emph{transport-based} counterfactual models. Moreover, the structural counterfactuals couplings are not necessarily deterministic. This is due to the fact that, in general, there are several values of $X_{S=s}$ consistent with an evidence $\{X=x,S=s\}$.

In this section, we introduced Pearl's counterfactuals from a joint probability distribution perspective, which concurs with the work of \cite{bongers2020foundations}. However, some readers may be more familiar with the definition based on the so-called \emph{three-step procedure} \citep{pearl2016causal}. This is for instance how \cite{kusner2017counterfactual} and \cite{barocas-hardt-narayanan} presented them. Crucially, these two viewpoints characterize the exact same counterfactual statements. We provide further insight on this topic in Appendix \ref{link3steps}.

In what follows, we study two specific scenarios of this counterfactual framework: first, when the counterfactuals are deterministic--- then the computation can be written as an explicit push-forward operation; second, when $S$ can be considered exogenous---then the counterfactual distribution is observable.

\subsection{The deterministic case}\label{SW}

We show that when the SCM deterministically implies the counterfactual values of $X$, then the counterfactual coupling is deterministic, and can be identified with a push-forward operator. To reformulate structural counterfactuals in deterministic transport terms, we first highlight the functional relation between an individual and its intervened counterparts.

From the acyclicity of the causal model, we can recursively substitute for the $X_i$ their functional form to obtain a measurable function $\F$ such that $\P$-almost surely $X = \F(S,U_X)$ and $X_{S=s} = \F(s,U_X)$ for any $s \in \S$. Now, let us define for every $s \in \S$ the function $f_s : u \mapsto \F(s,u)$. Note that the $s'$-counterfactual counterparts of an $s$-instance $x$ belong to the set $f_{s'} \circ f^{-1}_s(\{x\})$. This means that the counterfactual quantities are uniquely determined when the following assumption holds.

\begin{description}
    \item[Assumption (SW)\namedlabel{Invertibility}{\textbf{(SW)}}] \textit{The functions $\{f_s\}_{s \in \S}$ are injective.}
\end{description}

While the unique solvability of acyclic models ensures that $(X,S)$ is deterministically determined by $U$, \ref{Invertibility} states that, conversely, $U_X$ is deterministically determined by $\{X=x,S=s\}$. This \emph{single-world} assumption implies that all the couplings between the factual and counterfactual distributions are deterministic, as written in the next proposition.

\begin{proposition}\label{oto} Let \ref{Invertibility} hold, and define for any $s,s' \in \S$, $T^*_{\mathsmaller{ \langle s'|s \rangle}} := f_{s'} \circ f_s^{-1} \restr{\X_s}$.\footnote{$f_s^{-1} \restr{\X_s}$ denotes the restriction of $f_s^{-1}$ to $\X_s$.} The following properties hold:

\begin{enumerate}
    \item $\mu_{\langle s'|s \rangle} = {T^*_{\mathsmaller{ \langle s'|s \rangle}}}_\sharp \mu_s$;
    \item $\pi^*_{\langle s'|s \rangle} = (I \times T^*_{\mathsmaller{ \langle s'|s \rangle}})_\sharp \mu_s$.
\end{enumerate}

We say that $T^*_{\mathsmaller{ \langle s'|s \rangle}}$ is a \emph{structural counterfactual operator}, and identify $\T^* := \{T^*_{\langle s'|s \rangle}\}_{s,s'\in \S}$ to the deterministic structural counterfactual model $\Pi^* = \{(I \times T^*_{\langle s'|s \rangle})_\sharp \mu_s\}_{s,s'\in \S}$.
\end{proposition}

Similarly to the structural counterfactual couplings, the operators in $\T^*$ describe the effect of causal interventions on factual distributions. We highlight that in our setting, they are well-defined without any knowledge on $\mathcal{L}(U)$.

Assumption \ref{Invertibility} is frequent in practice. It holds in particular for every \emph{additive noise models}, models where the exogenous variables are additive terms of the structural equations, which accounts for most of the state-of-the-art SCMs. In \citep{karimi2020algorithmic} for example, the authors considered such a setting for the purpose of computing algorithmic recourse. Note also that assumption \ref{Invertibility} imposes constraints on the variables and their laws to enable a deterministic correspondence between $X$ and $U_X$. First, it entails that $X$ and $U_X$ have the same dimension, meaning that one exogenous variable $U_i$ corresponds to exactly one endogenous variable $X_i$. Second, it requires for every pair $(X_i,U_i)$ that both variables are either continuous or discrete with same-size domains. Remark that we framed \ref{Invertibility} so that it implies that all the counterfactuals instances for \emph{any} changes on $S$ are deterministic, leading to a fully deterministic counterfactual model. However, it suffices that one $f_s$ be injective for some $s \in \S$ to render all the counterfactual couplings $\{\pi^*_{\langle s'|s \rangle}\}_{s' \in \S}$ deterministic. Therefore, when \ref{Invertibility} does not hold, the structural counterfactual model possibly contains both random and deterministic couplings.

\subsection{The exogenous case}\label{RE}

We discuss the counterfactual implications of the position of $S$ in the causal graph. More specifically, we focus on the case where $S$ can be considered as a root node. This entails that the structural counterfactual model is a transport-based counterfactual model.

Let $\independent$ denote the independence between random variables. The variable $S$ is said to be \emph{exogenous relative to} $X$ \citep{galles1998axiomatic} if the following holds:

\begin{description}
    \item[Assumption (RE)\namedlabel{Exogeneity}{\textbf{(RE)}}] \textit{$U_S \independent U_X$ and $X_{\text{Endo}(S)} = \emptyset$.}
\end{description}

\begin{figure}[h!]
    \centering
    \begin{tikzpicture}[-latex ,auto ,node distance =2 cm ,on grid ,
    semithick ,
    state/.style ={circle ,top color =white,
    draw, minimum width =1 cm}]
    \node[state] (S) {$S$};
    \node[state] (X)[right= of S] {$X$};
    \node[state] (Ux)[above=of X]{$U_X$};
    \node[state] (Us)[above=of S]{$U_S$};
    \path (S) edge (X);
    \path (Ux) edge (X);
    \path (Us) edge (S);
    \end{tikzpicture}
    \caption{DAG satisfying \ref{Exogeneity}}
    \label{causalmodel}
\end{figure}

This represents a scenario where: (1) there is no hidden confounder between $X$ and $S$, (2) no variable in $X$ is a direct cause of $S$. Note that \ref{Exogeneity} entails that $S \independent U_X$. Then, it is easy to see that at the distributional level, intervening on $S$ amounts to conditioning $X$ by a value of $S$.

\begin{proposition}\label{conditioning}
If \ref{Exogeneity} holds, then for every $s,s' \in \S$ we have $\mu_{S=s'} = \mu_{s'} = \mu_{\langle s'|s \rangle}$.
\end{proposition}

Relative exogeneity is a critical assumption. Recall that the structural counterfactual coupling $\pi^*_{ \langle s'|s \rangle}$ represents an intervention transforming an observable distribution $\mu_s$ into an \emph{a priori} non-observable counterfactual distribution $\mu_{\langle s'|s \rangle}$. According to Proposition \ref{conditioning}, \ref{Exogeneity} renders the causal model otiose for the purpose of generating the counterfactual distribution, as the latter coincides with the observable factual distribution $\mu_{s'}$. However, the coupling is {\em still required} to determine how each instance is matched at the individual level. We note that \ref{Exogeneity} provides transitivity properties to our counterfactual operators.
\begin{proposition}\label{cff} Suppose that \ref{Exogeneity} and \ref{Invertibility} hold. Then, for any $s,s',s'' \in \S$:

\begin{enumerate}
    \item The operator $T^*_{\mathsmaller{ \langle s'|s \rangle}}$ is invertible, such that $\mu_{s'}$-almost everywhere ${T^*}^{-1}_{\mathsmaller{ \langle s'|s \rangle}} = T^*_{\mathsmaller{ \langle s|s' \rangle}}$;
    \item $\mu_s$-almost everywhere, $T^*_{\mathsmaller{ \langle s''|s' \rangle}} \circ T^*_{\mathsmaller{ \langle s'|s \rangle}} = T^*_{\mathsmaller{ \langle s''|s \rangle}}$.
\end{enumerate}

\end{proposition}

In terms of real-world modeling, \ref{Exogeneity} is intuitively satisfied in many scenarios. Let $X$ represent the socio-economics features of individuals. Figure out a situtation where $\S = \{0,1\}$ such that $S=0$ stands for \textit{female} while $S=1$ stands for \textit{male}, and assume for simplicity that \ref{Invertibility} holds. In this presumably exogenous model, any factual woman described by $x$ is the counterfactual counterpart of her counterfactual male counterpart described by $T^*_{\mathsmaller{ \langle 1|0 \rangle}}(x)$, and changing all the factual women into their counterfactual male counterparts recovers the factual male population. As noted by \cite{plecko2020fair}, a practical consequence of \ref{Exogeneity} is that it enables to link observational and causal notions of fairness, as  addressed in Section \ref{applications}. We remark that when both \ref{Invertibility} and \ref{Exogeneity} hold, then $\Pi^*$ is a deterministic transport-based counterfactual model.

\subsection{The example of linear additive SCMs}

We illustrate how our notation and assumptions apply to the case of \emph{linear additive} structural models, which account for most of the state-of-the-art models.

\begin{example}\label{ex} Under \ref{Exogeneity}, a linear additive SCM is characterized by the structural equations

\begin{align*}
    X &= MX + wS + b + U_X,\\
    S &= U_S,
\end{align*}

\noindent where $w,b \in \R^d$ and $M \in \R^{d \times d}$ are deterministic parameters. Acyclicity implies that $I-M$ is invertible, so that $X = (I-M)^{-1}(wS+b+U_X) =: \F(S,U_X)$. Note that \ref{Invertibility} holds such that for any $s \in S$, $f^{-1}_s(x) = (I-M)x-ws-b$. Then, for any $s,s' \in \S$, $T^*_{\mathsmaller{ \langle s'|s \rangle}}(x) = x + (I-M)^{-1}w(s'-s)$.

\end{example}

Remarkably, in the specific case of linear additive SCMs fitting \ref{Exogeneity}, computing counterfactual quantities amounts to applying translations between factual distributions. Therefore, should an oracle reveal that the SCM belongs to this class without providing the structural equations, it would suffice to compute the mean translation between sampled points from $\mu_s$ and $\mu_{s'}$ to obtain an estimator of the counterfactual operator $T^*_{\mathsmaller{ \langle s'|s \rangle}}$. For more complex SCMs satisfying both \ref{Invertibility} and \ref{Exogeneity}, it is presumably difficult to infer the counterfactual model from data. We address this issue in Section \ref{brenier}.

\subsection{Shortcomings}

We conclude Section \ref{structural} by discussing important drawbacks of the causal approach to counterfactual reasoning.

The main limitation, as for any causal-based framework, is its feasibility. Assuming a known causal model, in particular a fully-specified causal model, is a too strong assumption in practice. It requires experts to reach a consensus on the causal graph, the structural equations, the distribution of the input exogenous variables, and to test the validity of their model on available data. This is not a realistic scenario, especially when dealing with a high-number of features and possibly complex structural relations. Besides, this is not practical as a causal model must be designed and tested for each possible dataset. A more straighforward approach is to directly infer the causal graph from observational data. While there exist sound techniques to do so, they suffer from being NP-hard, with an exponential worst-case complexity with respect to the number of nodes \citep{cooper1990computational,chickering2004large,scutari2019learning}. In addition, this is not enough to compute counterfactual quantities, as the structural equations would still be lacking.

A related issue is causal uncertainty. There exist several causal models corresponding to a same data distribution, leading to possibly different counterfactual models (see Example 4.2 in \citep{bongers2020foundations}). It cannot be tested whether the adjusted model is the \emph{true} one, making the modeling inherently uncertain. Moreover, for non-deterministic structural counterfactual models, the computation of counterfactual quantities requires to know the law of the exogenous variables, which is not observable. While it is common to assume a prior distribution on $U$, this also adds uncertainty in the causal modeling, hence on the induced counterfactuals.

Perhaps more surprisingly, counterfactual quantities are sometimes nonexistent in Pearl's causal framework. The causal modeling we introduced is very general: we do not assume the exogenous variables to be mutually independent, and only suppose that the equations are acyclic. The DAG assumption is very common for both practicality and interpretability reasons. In general, however, observational data can be generated though an acyclical mechanism. Critically, (solvable) acyclic models do not always admit solutions under do-interventions, implying that $X_{S=s}$ may not be defined. We refer to Example 2.17 in \citep{bongers2020foundations} for an illustration. As a consequence, counterfactual quantities are ill-defined in such settings.

Note that the mass transportation viewpoint of structural counterfactuals we introduced in this section does not resolve these impracticability issues. This viewpoint is a formulation, not a trick. Specifically, we still need a fully-specified causal model to determine $\{\pi^*_{\langle s'|s \rangle}\}_{s,s' \in \S}$. Nevertheless, this perspective suggests that transport-based methods can be natural substitutes for structural counterfactual reasoning, as they remedy to the aforementioned issues. Moreover, basic optimal-transport counterfactual models can sometimes coincide with structural counterfactuals as explained in the following section.

\section{When optimal transport meets causality}\label{coincide}

Firstly, we introduce the key properties of optimal transport for the squared euclidean cost. Then, we show that it generates the same counterfactuals as a large class of causal models.

\subsection{Squared euclidean optimal transport}\label{brenier}

Suppose that the ground cost $c$ is the squared euclidean distance $c(x,x'):=\norm{x-x'}^2$ on $\R^d \times \R^d$, that $P$ is absolutely continuous with respect to the Lebesgue measure in $\R^d$, and that both $P$ and $Q$  have finite second order moments. Theorem~2.12 in \cite{villani2003topics}, Brenier's theorem, states that there exists a unique solution to Kantorovich's formulation of optimal transport \eqref{kanto}, whose form is $(I \times T )_\sharp P$ where $T : \R^d \to \R^d$ solves the corresponding squared Monge problem,

\begin{equation}\label{square_monge}
    \min_{T:\ T_\sharp P = Q} \int_{\R^d} \norm{x-T(x)}^2 dP(x).
\end{equation}

Although it may not be unique, this optimal transport map $T$ is uniquely determined $P$-almost everywhere, and we will abusively refer to it as \emph{the} solution to \eqref{square_monge}. Crucially, this map coincides $P$-almost everywhere with the gradient of a convex function. Moreover, according to \cite{mccann1995}, under the sole assumption that $P$ is absolutely continuous with respect to the Lebesgue measure, there exists only one (up to $P$-negligible sets) gradient of a convex function $\nabla \phi$ satisfying the push-forward condition $\nabla \phi_\sharp P = Q$. We combine Brenier's and McCann's theorems into the following lemma, which simplifies the search for the solutions to \eqref{square_monge}.

\begin{lemma}\label{connect}
Assume that $P$ is absolutely continuous w.r.t. the Lebesgue measure, and that both $P$ and $Q$ have finite second order moments. If $T : \text{supp}(P) \to \text{supp}(Q)$ is a measurable map satisfying the two following conditions,
\begin{enumerate}
    \item $T_\sharp P = Q$,
    \item there exists a convex function $\phi : \R^d \to \R$ such that $T = \nabla \phi$ $P$-almost everywhere,
\end{enumerate}
then $T$ is the solution to \eqref{square_monge}.
\end{lemma}

This result will play a key role in the next subsection.

\subsection{Main theorem}

We focus on the deterministic transport-based counterfactual model defined by the collection of solutions to \eqref{square_monge} between the pairs of measures $\{(\mu_s,\mu_{s'})\}_{s,s' \in \S}$. This model gives an elegant interpretation to the obtained counterfactual statements, as they are defined by minimizing the squared euclidean distance between paired instances, and can be seen as \emph{non-decreasing} models. Note that being the gradient of a convex function generalizes the notion of non-decreasing function to several dimensions. Moreover, it recovers structural counterfactuals in specific cases.

\begin{theorem}\label{monotone}Consider an SCM satisfying \ref{Invertibility} and \ref{Exogeneity}. Suppose that the factual distributions are absolutely continuous w.r.t. Lebesgue measure and have finite second order moments. If for $s,s' \in \S$, the structural counterfactual operator $T^*_{\mathsmaller{ \langle s'|s \rangle}}$ is the gradient of some convex function, then it is the solution to \eqref{square_monge} between $\mu_s$ and $\mu_{s'}$.
\end{theorem}
The mass transportation formalism of Pearl's counterfactual reasoning introduced in Section \ref{structural} renders the proof of this theorem straightforward. We underline that it does not require any prior knowledge on optimal transport theory except what we summarized in Lemma \ref{connect}. Thus, for the sake of illustration and clarity, we reproduce the demonstration directly below.

\begin{proof}
According to \ref{Invertibility} and Proposition \ref{oto}, the SCM defines a structural counterfactual operator $T^*_{\mathsmaller{ \langle s'|s \rangle}}$ between $\mu_s$ and $\mu_{\langle s'|s \rangle}$. Additionally, \ref{Exogeneity} implies through Proposition \ref{conditioning} that $\mu_{\langle s'|s \rangle} = \mu_{s'}$. Therefore, ${T^*_{\mathsmaller{ \langle s'|s \rangle}}}_\sharp \mu_s = \mu_{s'}$. Assume now that $\mu_s$ is absolutely continuous w.r.t. the Lebesgue measure, and that both $\mu_s$ and $\mu_{s'}$ have finite second order moments. If $T^*_{\mathsmaller{ \langle s'|s \rangle}}$ is the gradient of some convex function, then according to Lemma \ref{connect} it is the solution to \eqref{square_monge} between $\mu_s$ and $\mu_{s'}$.
\end{proof}

Understanding the strength and limits of Theorem \ref{monotone} requires understanding how rich is the class of SCMs fitting its assumptions. The larger the class, the more likely optimal transport maps for the squared eculidean cost will provide (nearly) identical counterfactuals to causality. Finding explicit conditions on $f_s$ and $f_{s'}$ so that $f_{s'} \circ f_s^{-1}$ is the gradient of a convex potential requires tedious computations as soon as $d>1$, which renders the identification of the relevant SCMs difficult. Nevertheless, we can find specific sub-classes of causal models fitting Theorem \ref{monotone}. For instance, as the structural counterfactual operator from Example \ref{ex} is the gradient of a convex function, we obtain the following corollary.

\begin{corollary}\label{linear}

Consider a linear additive SCM satisfying \ref{Exogeneity} (see Example \ref{ex}). If the factual distributions are absolutely continuous w.r.t. Lebesgue measure and have finite second order moments, then for any $s,s' \in \S$, the structural counterfactual operator $T^*_{\mathsmaller{ \langle s'|s \rangle}}$ is the solution to \eqref{square_monge} between $\mu_s$ and $\mu_{s'}$.

\end{corollary}

The scope of Theorem \ref{monotone} goes beyond linear additive SCMs, as shown in the following non-linear non-additive example.

\begin{example}
Consider the following SCM,
$$
\begin{cases}
X_1 = \alpha(S) U_1 + \beta_1(S),\\
X_2 = - \alpha(S) \ln^2\big(\frac{X_1-\beta_1(S)}{\alpha(S)}\big) U_2 + \beta_2(S),\\
S = U_S,
\end{cases}
$$
where $\alpha, \beta_1,\beta_2$ are $\R$-valued functions such that $\alpha > 0$, and $U_1 > 0$. It satisfies \ref{Invertibility} and \ref{Exogeneity}, such that for any $s,s' \in \S$, the associated structural counterfactual operator is given by,
$$
T^*_{\mathsmaller{ \langle s'|s \rangle}}(x) = \frac{\alpha(s')}{\alpha(s)} x + \big[\beta(s')-\beta(s)\big],
$$
where $\beta = (\beta_1,\beta_2)$ is $\R^2$-valued. This is the gradient of the convex function $x \mapsto \frac{\alpha(s')}{2 \alpha(s)} \norm{x}^2 + \big[\beta(s')-\beta(s)\big]^T x$. Then, if the factual distributions are absolutely continuous w.r.t. the Lebesgue measure, $T^*_{\mathsmaller{ \langle s'|s \rangle}}$ is the solution to \eqref{square_monge} between $\mu_s$ and $\mu_{s'}$.
\end{example}

On the contrary, one can effortlessly find counterexamples to Theorem \ref{monotone}. This comes from the fact that many functions (even continuous ones) cannot be written as gradients when $d>1$.

\begin{example}
Consider the following SCM,
$$
\begin{cases}
X_1 = U_1,\\
X_2 = S X^2_1 + U_2,\\
S = U_S.
\end{cases}
$$
It satisfies \ref{Invertibility} and \ref{Exogeneity}, such that for any $s,s' \in \S$, the associated structural counterfactual operator is given by,
$$
T^*_{\mathsmaller{ \langle s'|s \rangle}}(x_1,x_2) = (
x_1,\ x_2 + (s'-s) x^2_1).
$$
Note that this cannot be written as the gradient of a function. Consequently, it is not a solution to \eqref{square_monge}.
\end{example}

Theorem \ref{monotone} and Corollary \ref{linear} support the intuition that substituting $\Pi^*$ with a surrogate $\Pi$ from optimal transport provides a decent approximation of causal changes. Using a model close to $\Pi^*$ would be ideal for interpretability reasons, but an expert can always propose and defend a different notion of similarity $\Pi$ (built with optimal transport or not). The main interest lies in the feasibility of transport-based solutions, as they do not require any causal assumptions.

\section{Application to fairness}\label{applications}

We now turn to the application of the mass transportation viewpoint of counterfactual reasoning to fairness in machine learning. Suppose that the random variable $S$ encodes a so-called \emph{sensitive} or \emph{protected attribute} (for example race or gender) which divides the population into different classes in a machine learning prediction task. We denote by $h : \R^d \times \S \mapsto \R$ an arbitrary predictor defining the random variable of predicted output $\hat{Y} := h(X,S)$. Fairness addresses the question of the dependence of $\hat{Y}$ on the protected attribute $S$. The most classical fairness criterion is the so-called \emph{demographic} or \emph{statistical parity}, which is achieved when
$\hat{Y} \independent S$.

However, this criterion is notoriously limited, as it only gives a notion of \emph{group fairness}, and does not control discrimination at a subgroup or an individual level: a conflict illustrated by \cite{dwork2012fairness}. The counterfactual framework, by capturing the structural or statistical links between the features and the protected attribute, allows for sharper notions of fairness. We first use the mass transportation formalism introduced in Section \ref{structural} to reformulate the accepted \textit{counterfactual fairness} condition \citep{kusner2017counterfactual}. On the basis of the reformulation, we then propose new fairness criteria derived from transport-based counterfactual models.

\subsection{Generalizing counterfactual fairness}

Counterfactual fairness is achieved when individuals and their structural counterfactual counterparts are treated equally.

\begin{definition}\label{cf} A predictor $\hat{Y}=h(X,S)$ is \emph{counterfactually fair} if for every $s,s' \in \S$ and $\mu_s$-almost every $x$,

$$
   \mathcal{L}(\hat{Y}_{S=s} \mid X=x,S=s) = \mathcal{L}(\hat{Y}_{S=s'} \mid X=x,S=s),
$$
where $\hat{Y}_{S=s} := h(X_{S=s},s)$.

\end{definition}

The above definition does not emphasize the pairing between factual and counterfactual values. Interestingly, the structural counterfactual transport plans allow for pair-wise characterizations of counterfactual fairness.

\begin{proposition}\label{rcf}

\begin{enumerate}
    \item A predictor $h(X,S)$ is counterfactually fair if and only if for every $s,s' \in \S$ and $\pi^*_{ \langle s'|s \rangle}$-almost every $(x,x')$,
    $$
    h(x,s) = h(x',s').
    $$
    \item If \ref{Invertibility} holds, then a predictor $h(X,S)$ is counterfactually fair if and only if for every $s,s' \in \S$ and $\mu_s$-almost every $x$,
    $$
    h(x,s) = h(T^*_{\mathsmaller{ \langle s'|s \rangle}}(x),s').
    $$
    \item If \ref{Invertibility} and \ref{Exogeneity} hold, then a predictor $h(X,S)$ is counterfactually fair if and only if for every $s,s' \in \S$ such that $s<s'$ and $\mu_s$-almost every $x$,
    $$
    h(x,s) = h(T^*_{\mathsmaller{ \langle s'|s \rangle}}(x),s').
    $$
\end{enumerate}

\end{proposition}

Assumption \ref{Invertibility} highlights the deterministic relationship between factual and counterfactual quantities. In addition, it makes unnecessary the knowledge of $\mathcal{L}(U)$ to test counterfactual fairness. Note that, if \ref{Exogeneity} holds, then counterfactual fairness is a stronger criterion than the statistical parity across groups.

\begin{proposition}\label{stronger} Suppose that \ref{Exogeneity} holds. If the predictor $h(X,S)$ satisfies \textit{counterfactual fairness}, then it satisfies \textit{statistical parity}. The converse does not hold in general.
\end{proposition}

One can think of being counterfactually fair as being invariant by counterfactual operations w.r.t. the protected attribute. In order to define SCM-free criteria, we generalize this idea to the models introduced in Section \ref{surrogate}.

\begin{definition}\label{tcounter}

\begin{enumerate}
    \item Let $\Pi = \{\pi_{\langle s'|s \rangle}\}_{s,s' \in \S}$ be a (random) transport-based counterfactual model. A predictor $h(X,S)$ is \emph{$\Pi$-counterfactually fair} if for every $s,s' \in \S$ and $\pi_{\langle s'|s \rangle}$-almost every $(x,x')$,
    $$
    h(x,s) = h(x',s').
    $$
    \item Let $\T = \{T_{\langle s'|s \rangle}\}_{s,s' \in \S}$ be a deterministic transport-based counterfactual model. A predictor $h(X,S)$ is \emph{$\T$-counterfactually fair} if for every $s,s' \in \S$ and $\mu_s$-almost every $x$,
    $$
    h(x,s) = h(T_{\mathsmaller{ \langle s'|s \rangle}}(x),s').
    $$
\end{enumerate}

\end{definition}

Because the proof of Proposition \ref{stronger} only relies on the assumption that the couplings have factual distributions for marginals, the following proposition holds.

\begin{proposition}\label{Tcf} Let $\Pi$ be a transport-based counterfactual model (deterministic or not). If a predictor $h(X,S)$ satisfies $\Pi$-counterfactual fairness, then it satisfies statistical parity. The converse does not hold in general.
\end{proposition}

This proposition has interesting consequences. Consider that, for the purpose of computing counterfactual quantities, some practitioners designed a candidate SCM fitting the data and satisfying \ref{Exogeneity}. Even if the candidate is misspecified, it would still characterize a transport-based counterfactual model controlling statistical parity. The fair data processing transformation proposed by \cite{plecko2020fair} is an illustrative example.

More generally, the conceptual interest of transport-based fairness criteria is the same as the original counterfactual fairness criterion: they offer notions of individual fairness while still controlling for discrimination against protected groups. The added value is their feasibility. In contrast to Definition \ref{cf} and Proposition \ref{rcf}, Definition \ref{tcounter} relies on computationally feasible counterfactual models that obviate any assumptions about the data-generation process. In addition, as Definition \ref{cf} amounts to $\Pi^*$-counterfactual fairness (when \ref{Exogeneity} holds), one can as well think of Definition \ref{tcounter} as an approximation of counterfactual fairness.

Crucially, these new criteria can naturally be applied in classical explainability and fairness machine learning frameworks based on counterfactual reasoning. We note that the explanability framework of \cite{black2020fliptest} can be seen as controlling for a transport-based notion of counterfactual fairness.

\subsection{Ethical risk}

We conclude this section by discussing a potentially negative impact of our work. As aforementioned, the transport-based approach allows for many counterfactual models, but they do not all define legitimate notions of counterparts. Consequently, transport-based notions of counterfactual fairness could be used for unethical fair-washing. The next proposition formalizes this risk.

\begin{proposition}\label{attack}
If $h(X,S)$ is a classifier satisfying statistical parity, then there exists a transport-based counterfactual model $\Pi$ such that $h(X,S)$ satisfies $\Pi$-counterfactual fairness.
\end{proposition}

Practitioners could take advantage of the weak notion of statistical parity to construct counterfactual models such that their trained classifiers are counterfactually fair, while still discriminating at the subgroup or individual level. This is why we argue that practitioners must always be able to justify the counterfactual models when not imposed by legal experts of the prediction task.

\section{Conclusion}

In this paper, we focused on the challenge of designing sound and feasible counterfactuals. Our work showed that the causal account for counterfactual modeling can be written in a mass transportation formalism, where implying either deterministic or random counterfactuals has a direct formulation in terms of the deterministic or random nature of the coupling. This perspective enabled us to generalize sharp but unfeasible causal criteria of fairness by transport-based ones. In doing so, we hope to have shed a new light on counterfactual reasoning, and fairness techniques.

redpajama_set_name: RedPajamaArXiv


Background Since their introduction in 2007, electronic cigarette ('e-cigarette') awareness and use has grown rapidly. Little is known about variation in e-cigarette availability across areas with different levels of tobacco taxes and smoke-free air policies. This paper looks at US retail availability of e-cigarettes and factors at the store, neighbourhood and policy levels associated with it.
Methods In-person store audit data collected in 2012 came from two national samples of tobacco retailers in the contiguous US. Study 1 collected data from a nationally representative sample of tobacco retailers (n=2165). Study 2 collected data from tobacco retailers located in school enrolment zones for nationally representative samples of 8th, 10th and 12th grade public school students (n=2526).
Results In 2012, e-cigarette retail availability was 34% in study 1 and 31% in study 2. Tobacco, pharmacy and gas/convenience stores were more likely to sell e-cigarettes than beer/wine/liquor stores. Retail availability of e-cigarettes was more likely in neighbourhoods with higher median household income (study 1), and lower percent of African–American (studies 1 and 2) and Hispanic residents (study 2). Price of traditional cigarettes was inversely related to e-cigarette availability. Stores in states with an American Lung Association Smoke-Free Air grade of F (study 1) or D (study 2) compared with A had increased likelihood of having e-cigarettes.
Conclusions Currently, e-cigarette availability appears more likely in areas with weak tax and smoke-free air policies. Given the substantial availability of e-cigarettes at tobacco retailers nationwide, states and localities should monitor the sales and marketing of e-cigarettes at point of sale (POS).
At the neighbourhood level, sales and marketing of traditional tobacco products show disparities in low-income and minority neighbourhoods. There is increased tobacco retailer density in minority and low-income communities, increased advertising and marketing in such communities, and lower cigarette prices in minority communities.8 However, marketing of e-cigarettes may follow an opposite pattern. African-American smokers are less likely to have heard about,3 ,9 or used,9 e-cigarettes than white smokers. Use of e-cigarettes also increases with education level.9 Tobacco retailer density and advertising near schools is associated with higher smoking rates at those schools.8 Use of e-cigarettes among middle and high school students doubled between 2011 and 2012,2 warranting examination of the availability of e-cigarettes near schools.
We conducted this analysis to identify retail availability of e-cigarettes and to determine factors at the store, neighbourhood and policy levels associated with availability across the USA. We hypothesise retail availability of electronic cigarettes to at least double from 2011 estimates to 14%.5 We expect more e-cigarette availability in store types that rely on tobacco product revenue and less availability in low-income and minority communities. We expect more availability in areas with stronger smoke-free air policies and higher tobacco taxes/cigarette prices. As a non-directional hypothesis, we expect a significant difference in odds of e-cigarette availability at stores near schools compared with stores further away from schools but cannot project whether e-cigarette availability will be higher or lower near schools.
Study 1 collected data from a nationally representative sample of retailers in the 48 contiguous US. We employed a two-stage probability proportionate to population size (PPS) sampling design. First, we randomly selected 100 counties with replacement, resulting in 97 unique counties.
Second, we randomly selected tobacco retailers within each county. A PPS sample is an equal probability sampling method in which the first stage of sampling selects counties with PPS. The second stage of sampling selects stores with probability inversely proportionate to size by selecting a fixed number of establishments out of a variable number. This yields probabilities of selecting establishments in initially selected urban counties substantially lower than in rural counties. The probabilities offset yielding equal probability of selection at the store level, thus resulting in a nationally representative sample of retailers. To obtain comparable lists across counties, we purchased lists from two commercial sources, ReferenceUSA and Dun & Bradstreet (D&B). We selected the 10 establishment types representing 98% of tobacco sales in the USA.7 After cleaning and de-duplication, we phone-verified up to 55 randomly sampled stores in each county, or all stores in small counties. Research assistants called stores to ask whether cigarettes were sold. We selected the first 24 eligible stores, or all eligible stores if fewer, for in-person audits. Our final sample had 2346 stores.
The sampling frame for study 2 derived from retail food stores in the Bridging the Gap Community Obesity Measures Project (BTG-COMP). School enrolment zones surrounding nationally representative samples of 8th, 10th and 12th grade public high schools in the contiguous US participating in the Monitoring the Future study14 provided a national sample of communities. We identified tobacco retailers in these communities by purchasing InfoUSA and D&B lists associated with SIC codes for food, liquor/wine/beer and tobacco-only retail stores, and stores with 'dollar' or '99¢' in their name. We supplemented these lists with retail stores 'discovered' in the field, based on the expected number of stores for each of our sampling strata.15 Research assistants called each food store from the list and confirmed eligibility. Beer/wine/liquor stores also were called to verify cigarette sales, and as with tobacco-only stores, to confirm location and hours. From this phone-verified list, we used a half-interval sampling approach16 to calculate an expected sampling frame, splitting the sample by store type into a business list group and a field discovery group. We randomly selected the appropriate number of stores from the business list. For field discovery, while travelling around the community, field staff audited a specific number of stores of each type not on the list but that met eligibility criteria. For this analysis, food stores selling tobacco products were identified and classified as tobacco retail stores. Field staff attempted to observe 2492 food stores and 196 tobacco stores, for a total of 2688 tobacco retailers.
In both studies, we trained data collectors using didactic and field experience in conducting the assessment. Data collectors conducted all audits in person in retail stores. For study 1, data collection occurred from June to October, and April to July 2012 for study 2. We completed interior and exterior audits for 2165 retailers out of 2346 sampled stores in study 1. Reasons for the 181 incomplete audits included 109 ineligible stores (outside of study county, not found or not at address listed, did not sell tobacco, out of business, duplicate of another store on the list or misclassified as a store) and 72 that were not started (refused, temporarily closed or under construction).
In study 2, data collectors completed 2603 observations at 2688 retailers known to sell tobacco products. For the 85 incomplete audits, 63 were ineligible (permanently closed, did not exist, outside the community boundaries, misclassified as an eligible store type), and 22 were not started (temporary inaccessible/outside business hours; not safe; asked to leave).
For study 2, store auditors introduced themselves at their discretion, often depending on the size, and thus, their visibility, within the store. If e-cigarettes or Marlboro Red prices were not clearly visible in the store, auditors were instructed to ask a store employee. Auditors used a paper and pencil instrument, and took an average of 10 min to complete the tobacco observations. Both measures had been previously tested in a reliability study in November 2009 in Chicago. e-cigarette availability had a κ score of 1.0, and the ICC for the advertised price of Marlboro Red cigarettes was .97, indicating near perfect agreement.
Both studies collected information on e-cigarette availability, cigarette price and store location. We linked store location with neighbourhood or community demographics and tobacco control policy variables. We used comparable measures across the two studies except as noted.
Store auditors determined whether or not e-cigarettes were available in the store (y/n).
Both studies collected information on store types, grouped in six or seven categories: (1) groceries/supermarkets; (2) gas, gas convenience and other gas stations; (3) pharmacy and drug stores; (4) tobacco stores; (5) beer, wine and liquor stores; (6) warehouse clubs and supercenters (study 1 only); and (7) other store types (including discount department stores, newsstands and general merchandise stores).
For study 1, we created a binary measure to code retailers as more than 1000 feet from or within 1000 feet of a K-12 public school based on a straight-line distance from the school centroid position from the National Center for Education Statistics 2009–2010. In study 2, by design, all retailers were within the school enrolment zone of the selected high school. In study 2, data were unavailable for store distance from all K-12 public schools, therefore distance from school is only included in multivariate analysis for study 1.
We examined retailer neighbourhood or community characteristics of per cent of non-Hispanic Black residents, per cent of Hispanic residents and median household income. In study 1, we used census tract as the retailer neighbourhood while study 2 mapped all the census block groups in the school enrolment zone to determine retailer neighbourhood. Both studies linked race/ethnicity variables and median household income from the American Community Survey 5-year estimates, 2007–2011. Given county sampling, study 1 found minimal clustering of stores by census tract; over 80% of census tracts had only one store. Study 2 aggregated block group data to determine neighbourhood demographics.
Both studies examined variation in smoke-free air policy by applying the 2012 American Lung Association (ALA) state smoke-free air grade19 to the county level (study 1) or store level (study 2). ALA assigns annual grades of A to F to states based on the overall level of clean air restrictions in government workplaces, private workplaces, schools, child care facilities, restaurants, retail stores and recreational/cultural facilities, taking into account level of penalties and enforcement.
For study 1, to account for clustering of stores within counties, we used weighted multilevel logistic regression models with full maximum likelihood estimation and adaptive quadrature (HLM 7) to estimate e-cigarette availability. In the null model, we found that 6% of the variance in the underlying availability of e-cigarettes at the store level was accounted for by county-level variance (intraclass correlation=.06). We entered store-level variables and then added county-level variables in successive models.
For study 2, because of the complex sampling design and the need to account for the probability of selection of the school enrolment zone and the store, we ran logistic regression models in STATA V.12.1, applying appropriate cluster, stratum and weight variables to estimate maximum likelihood.
Study 1 included 2165 stores with interior audit data from 97 counties in 40 states. Study 2 included 2526 stores in 160 communities in 38 states. Table 1 shows the sample characteristics for stores in studies 1 and 2. Store-type composition was generally similar across the two study samples. In both studies, convenience stores with and without gas stations comprised over 50% of the sample. Beer, wine and liquor stores (the reference category in multivariate analyses) comprised 10% of the sample for both studies. Average advertised price for Marlboro Red cigarettes was comparable ($6.25 in study 1 vs $6.15 in study 2). In study 1, prices ranged from $3.39 to $15.00 and in study 2 from $3.29 to $13.00. In both studies, we found the highest average priced cigarettes in New York State. In study 1, almost 13% of stores were within 1000 feet of a public K-12 school; in study 2, all stores were within the sampled school's enrolment zone.
In study 1, we found retail e-cigarette availability of 34.1% (95% CI 32.0 to 36.3). Study 2 identified e-cigarettes in 31% of retail stores (95% CI 26.7 to 35.1). In study 1, e-cigarettes were significantly more available in tobacco, warehouse and supercenters, pharmacy/drug, and gas/convenience than in beer, wine and liquor stores (table 2, models 1 and 2). Similarly, in study 2, we found higher odds of availability of e-cigarettes in tobacco, pharmacy/drug, and convenience compared with beer, wine and liquor stores (table 3, models 1 and 2). In both studies, odds of availability of e-cigarette sales in tobacco stores was substantially higher than in other store types; in study 1, 79% of tobacco stores sold e-cigarettes and 73% sold in study 2. Only 10% (study 1) or 12% (study 2) of beer, wine or liquor stores sold e-cigarettes. In study 1, 31% of stores near schools had e-cigarettes, but availability did not significantly differ compared with stores further away from schools (35%) (table 2, model 2) (adjusted OR (AOR) 0.94, 95% CI 0.69 to 1.28).
In study 1, the odds of a store having e-cigarettes available was 1.5 times greater for every one unit increase in log median household income (95% CI 1.15 to 2.04) (table 2, model 2). In relation to the raw scale, for example, this corresponds to 34% greater odds of e-cigarette availability (OR=1.34) for stores in neighbourhoods with a median household income of $20 000 compared with $10 000. Whereas at the higher end of income, this same OR (1.34) is found for stores in neighbourhoods of median household income of $100 000 compared with $50 000. Study 2 did not find this association. In final models, both studies found that odds of e-cigarette availability were 1% less, for every percentage point increase in the proportion of African–American residents in store neighbourhoods (AOR 0.99, 95% CI 0.98 to 1.00) (tables 2 and 3, model 2). Study 2, but not study 1, found a similar inverse relationship for Hispanic residents (AOR 0.98, 95% CI 0.97 to 0.99) (table 3, model 2).
Both studies found significant results regarding the relationship of e-cigarette availability and tobacco policy variables. In studies 1 and 2, in models that did not account for smoke-free air policy (tables 2 and 3, model 1), we found a significant inverse relationship between advertised cigarette price as a proxy for tax and e-cigarette availability. For every one unit increase in the log price of Marlboro Red cigarettes, we found almost a 50% decrease in the odds of e-cigarette availability (tables 2 and 3, model 1). For comparison in dollars, in study 2, a store with a $4 Marlboro Red pack compared with a store with a $3 pack has an 18% decrease in the odds of e-cigarette availability (OR=0.82). At a higher pack price, this same OR (0.82) corresponds to a store with a $12 pack compared with a store with a $9 pack. Once smoke-free air policy was included in the model, this effect diminished to non-significance in study 1 (table 2, model 2), but remained significant in study 2 (table 3, model 2).
We also found in study 1 that stores in states with a smoke-free air grade of F were 1.6 times (95% CI 1.18 to 2.27) more likely to have e-cigarettes available than those in states with an A grade (41% compared with 33%) (table 2, model 2). We also note a rough gradient in odds of availability across smoke-free air grades in study 1. This effect of higher availability with weaker regulations was replicated in study 2, which found stores with D grades were 3.7 times (95% CI 2.41 to 5.60) more likely to have e-cigarettes than stores in A grade states (table 3, model 2).
These two studies represent the first published US national data on the availability of e-cigarettes in retail stores. We only found one other published study of e-cigarette availability; a study of London retailers in 2013 found 57% availability.21 In our paper, both studies found e-cigarettes available at approximately one-third of US tobacco retailers in 2012 (31% in Study 2 and 34% in Study 1). This finding may simply reflect differences in sampling frame between the two studies.
A second possibility is that the slightly higher estimate in Study 1 (34%) compared with Study 2 (31%) may reflect increased national availability from the later time of data collection in Study 1 (June–October) versus Study 2 (April–July). Prior years of the BTG Study (Study 2) found that e-cigarette availability increased from 3% in 2010 to 7% in 2011 to 31% in 2012.5Additionally, there is evidence to suggest rapid growth of the number of outlets selling e-cigarettes during the time period covered by the two studies. For example, Lorillard purchased blu e-cigarettes in April 2012 at the beginning of Study 2 data collection. Then, news reports indicated the number of outlets carrying blu was 13 000.22 At the end of the third quarter 2013, Lorillard reported the number of outlets was 127 000.23 Additional years of data collection will be needed to assess time trends.
Consistent with expectations, e-cigarettes are generally more available in stores that derive a higher proportion of their revenue from tobacco product sales: tobacco stores, warehouse/supercenters, and convenience/gas stations.7 One exception to this pattern is the high level of availability in pharmacy and drug stores. Pharmacy/drug stores derive less than 2% of sales from tobacco products.7 However, nicotine replacement products are highly available in pharmacies,24 perhaps sparking availability of e-cigarettes as cessation devices or as products for more health-conscious smokers.
Neighbourhood demographic effects were consistent with our hypotheses based on current use patterns among consumers. e-cigarettes were slightly less available in stores in African–American (studies 1 and 2) and Hispanic neighbourhoods (study 2). We also found greater availability with higher median household income (study 1).
States with stronger tax and smoke-free air policies are associated with greater declines in smoking rates than states with weaker policies.27 Tobacco retailers are likely selling e-cigarettes where there is a demand for them (ie, where there are more smokers, in part, due to weaker tobacco control regulations). Our findings suggest that stronger tobacco control policies are not the predominant or most direct factors driving retail availability of e-cigarettes.
We also identify a number of limitations. First, study 1 included only traditional tobacco product retailers; the sampling frame of study 2 was built upon a frame for retail food outlets, supplemented by tobacco-only stores. e-cigarettes may be sold in other store types not captured in these frames. In study 2, we added stores in the field to improve our sampling frame; however, non-random 'discovery' stores may have differed from the business list sample. In both studies, we measured only e-cigarettes availability; future waves of study 1will collect data on e-cigarette promotions, placement and price. Finally, other factors influencing e-cigarette availability, such as proprietary contracts between stores and e-cigarette or tobacco distributors, were not measured. These industry practices may be the most proximal influence on patterns of e-cigarette distribution seen in this study.
As a final limitation, both studies are cross-sectional, corresponding to rapid increase in the use of e-cigarettes in the USA. Given both potential increases in marketing of e-cigarettes with tobacco industry promotion and for regulation of these products by the FDA,6 we expect fluctuation in trends in retail e-cigarette availability. In the future, we can provide longitudinal data about retail availability and how correlates of availability change over time.
We found sizable retail availability of e-cigarettes only 6 years after their introduction to the US market. e-cigarette availability is generally higher in store types with a higher market share of tobacco products. Encouragingly, unlike traditional cigarettes, we did not find retail e-cigarettes sales targeted to minority racial/ethnic communities, low-income neighbourhoods or near schools. We found more e-cigarette availability in areas with weak tax and smoke-free air policies, perhaps suggesting supply to smokers. We anticipate the impact of tobacco control policies, neighbourhood characteristics and store factors on availability will change over time given the rapidly shifting marketing and regulatory landscape for these products. Given these changes, states and localities should monitor the sales and marketing of e-cigarettes at POS.
The current study, with findings from two national studies of tobacco retailers, demonstrates substantial retail availability of electronic cigarettes in the USA.
Retail availability of e-cigarettes, unlike cigarettes, does not show targeting to minority and low-income communities, or near schools.
E-cigarette availability correlates with weaker tobacco control tax and smoke-free air policies, suggesting supply to smokers.
These findings provide future directions for states and localities to monitor e-cigarettes at point of sale.
Christopher Quinn, M.S., was instrumental in developing the analytic data set for study 2. J. Michael Bowling, PhD, created the sampling approach for study 1.
Centers for Disease Control and Prevention. Notes from the field: electronic cigarette use among middle and high school students—United States, 2011–2012. MMWR Morb Mortal Wkly Rep 2013;62:729–30.
. Awareness and ever-use of electronic cigarettes among U.S. Adults, 2010–2011. Nicotine Tob Res 2013;15:1623–7.
. Changes in E-Cig Availability Over Time in the United States, 2010–2012—A BTG Research Brief. Chicago, IL: Bridging the Gap Program, Health Policy Center, Institute for Health Research and Policy, University of Illinois at Chicago, 2014.
. E-cigarette sales surpass $1 billion as Big Tobacco moves in. Forbes. 2013.
U.S. Census Bureau; Economic Census; generated by H. D'Angelo. American Factfinder, 2010. http://factfinder2.census.gov.
US Department of Health and Human Services. Preventing tobacco use among youth and young adults: a report of the Surgeon General. Atlanta, GA: US Department of Health and Human Services, Centers for Disease Control and Prevention, Office on Smoking and Health, 2012.
. Options for state and local governments to regulate non-cigarette tobacco products. Ann Health Law 2012;21:407–45.
. Monitoring the future occasional paper 76: the monitoring the future project after thirty-seven years: design and procedures. Ann Arbor, MI: University of Michigan, Institute for Social Research, 2011.
. Bridging the gap community obesity project: methodology, bridging the gap research paper series, paper 1. Chicago, IL: Bridging the Gap, 2014.
. Comparing the coverage of a household sampling frame based on mailing addresses to a frame based on field enumeration. Proceedings of the American Statistical Association, Section on Survey Research Methods, 2007.
. Answers to 20 questions about interrater reliability and interrater agreement. Organ Res Methods 2008;11:815–52.
Orzechowski W, Walker R. Tax burden on tobacco, Historic compilation, Vol. 46. Arlington, VA: Orzechowski and Walker, 2011.
American Lung Association. American Lung Association State of Tobacco Control 2012. Washington, DC: ALA, 2012.
U.S. Census Bureau. State and County QuickFacts. Washington, DC: Population Estimates, American Community Survey, Census of Population and Housing, State and County Housing Unit Estimates, County Business Patterns, Nonemployer Statistics, Economic Census, Survey of Business Owners, Building Permits, 2012. http://quickfacts.census.gov/qfd/states/00000.html (accessed 29 Oct 2013).
. An observational study of retail availability and in-store marketing of e-cigarettes in London: potential to undermine recent tobacco control gains? BMJ Open 2013;3:e004085.
. Got a light – er charger? Big Tobacco's latest buzz. Wall Street J 201225 April 2012.
. Lorillard, Inc. reports third quarter results. Greensboro, NC: Lorillard, 2013.
. Availability and range of tobacco products for sale in Massachusetts pharmacies. Tob Control 2013; 22:372–5.
. Tracking the rise in popularity of electronic nicotine delivery systems (electronic cigarettes) using search query surveillance. Am J Prev Med 2011;40:448–53.
. The Impact of Prices and Tobacco Control Policies on the Demand for Electronic Nicotine Delivery Systems. Tob Control doi:10.1136/tobaccocontrol-2013-051515. In press.
. Changes in smoking prevalence among U.S. adults by state and region: estimates from the Tobacco Use Supplement to the Current Population Survey, 1992–2007. BMC Public Health 2011;11:512.
. Retailer adherence to family smoking prevention and Tobacco Control Act, North Carolina, 2011. Prev Chronic Dis 2013;10:E47.
Contributors SWR wrote the manuscript. SWR and HD'A conducted statistical analyses for study 1. DCB drafted methods related to study 2. DCB and TK conducted statistical analyses for study 2. DCB, JH, FJC and KMR contributed significantly to the conception and design of the study. All authors provided review and substantive revisions for important intellectual content and provided final approval of the manuscript.
Funding This work was supported by the National Cancer Institute, grant numbers U01 CA154281, U01 CA154248 and R25 CA57726.
Competing interests SWR, HD'A, DCB, TK, JH and FJC declare no competing interests. KMR is a special government employee for the FDA Center for Tobacco Products and a member of the Tobacco Products Scientific Advisory Committee; the views expressed in this article are his and not those of the FDA.
Data sharing statement Data from study 1 are not yet publically available, but will be available in the future. The data from study 2 are not publicly available—the sample of communities is based on schools participating in the Monitoring the Future surveys and the schools/respondents' confidentiality is protected as part of their agreement to participate.


Tom Kostopoulos spent the first five seasons of his professional hockey player pulling on a Penguins sweater. When he left the organization following the 2003-04 campaign, Kostopoulos had appeared in a team-record 318 games as a member of the Wilkes-Barre/Scranton Penguins.
Rob Scuderi got close to the mark in 2005-06, when he appeared in his 305th game with Wilkes-Barre/Scranton. Tim Wallace crept up on the record during his five-year stay with the Pens (2006-11), suiting up for 304 regular season games with the club.
When Kostopoulos returned to Wilkes-Barre/Scranton during the 2012-2013 season and embarked on his second stint with the AHL Penguins, his games played record remained intact.
But, for a minute, let's imagine Kostopoulos never played for the Penguins during the team's early years. Assume that his first game with the club came on January 25, 2013, against the Rochester Americans. Any guesses as to where he would rank on the team's all-time games played list?
When he suited up against the Lehigh Valley Phantoms on Friday night, Kostopoulos appeared in the 306th regular season game of his second stint with the team. If you were to look at his two stays with the Pens as separate careers, he would hold the top two spots on the club's games register.
As a matter of fact, splitting his Penguins stays in two gives Kostopoulos the top two spots in assists (149/119) and points (246/203) as well. His goals totals would put him in first (97) and third (84) places, with Tomas Surovy holding down the two spot with 89 tallies.
But let's get back to the real numbers now. At 38-years old, Kostopoulos holds every significant offensive record in Penguins history, with 625 games played, 181 goals, 268 assists and 449 points. He also paces the Pens with 69 power play goals and 22 game-winners, while ranking tied for second with six career short-handed scores.
Wilkes-Barre/Scranton's next home game isn't until Wednesday, Apr. 12, and its opponent will once again be the Lehigh Valley Phantoms.
Individual game tickets for Apr. 12's game and all Penguins home games are available at the Mohegan Sun Arena at Casey Plaza box office, online at TicketMaster.com, or by calling 800-745-3000.