# Quantity Limits on Addictive Goods

Eric Gao Department of Economics, Stanford University<EMAIL_ADDRESS>
###### Abstract

Addiction is a major societal issue leading to billions in healthcare losses
per year. Policy makers often introduce ad hoc quantity limits—limits on the
consumption or possession of a substance—something which current economic
models of addiction have failed to address. This paper enriches Bernheim and
Rangel (2004)’s model of addiction driven by cue-triggered decisions by
incorporating endogenous choice of how much of the addictive good to consume,
instead of just whether or not consumption happens. Stricter quality limits
improve welfare as long as they do not preclude the myopically optimal level
of consumption.

JEL Codes: D01, D11, I18.

##  1 Introduction and Motivation

Misuse of addictive substances costs the United States billions in healthcare
spending (Peterson et al. (2021)) and over a hundred thousand deaths every
year (National Institute on Drug Abuse (2023)). While there has been much work
in economics dedicated to analyzing this issue and evaluating policy
proposals, such inquires have focused on full criminalization versus
legalization and the impact of taxes or subsidies. In the real world, however,
policies often feature limits on the quantity an individual can possess:
California allows possession of up to 28.5 grams of recreational cannabis,
while most states where cannabis is legal have possession limits of around 1-3
ounces.111See Pacula et al. (2021) for a more detailed breakdown of policies
by area. Similarly, California recently passed laws prohibiting bartenders
from serving alcohol “to anyone who is obviously intoxicated” (California
Department of Alcoholic Beverage Control (2024)). Not only that, but the
California law also prohibits alcohol from being served to anyone “who has
lost control over his or her drinking”.222This can be thought of as a “higher-
order” or “proactive” quantity limit, but falls slightly beyond the scope of
this paper.

What are the behavioral implications of such limits, and how does that spill
over to impact public welfare? While limits on the quantity of addictive goods
someone can possess restricts how much they consume at once, it can also
create a perception that addiction “isn’t too bad”, since the worst that can
happen is consumption at the limit. As such, quantity limits may may counter-
productively increase overall usage by increasing individuals’ rationally
optimal quantity consumed (while decreasing compulsive consumption). For
instance, consider a recovering alcoholic who doesn’t go to bars due to fear
of losing control, drinking too much, and ending up in the hospital. If a law
were passed preventing bartenders from serving drunk customers, they may now
venture into bars, thinking that even if they lose control, the bartender will
exercise restraint on their behalf.

With this in mind, suppliers of addictive goods may find it beneficial to
impose their own limits on how much a customer can purchase at once to
increase the number of customers (at the cost of each individual customer
consuming less if the limit is binding). One example of this could be table
limits in casinos restricting how much an individual can bet at a time (but
the common explanation of why table limit exists is to protect the casino
itself from risk). If quantity limits both increase welfare and are profitable
for suppliers (perhaps for goods where continuous low usage has negligible
harms, but overuse leads to enormous costs), supply-side restrictions at the
store-by-store level can be a useful policy tool. The remainder of this paper
formalizes these ideas and proposes an experimental evaluation of quantity
limits.

###  1.1 Related Literature

This work is most closely related to Bernheim and Rangel (2004). Their seminal
paper is the first to incorporate a psychological foundation of rational
consumption of addictive goods. Addictive substances are addictive because
they directly trigger a response in the brain’s mesolimbic dopamine system
which regulates feelings of happiness. When certain environmental cues are
presented in conjunction with stimulation of the mesolimbic dopamine system,
laboratory experiments have shown that the brain learns to associate dopamine
with the cues themselves opposed to eventual rewards. As such, Bernheim and
Rangel (2004) model the process of compulsive consumption through individuals
entering a “hot” state after being triggered by some environmental.

In addition to introducing the novel model, the authors derive comparative
statics on how lifestyle choices that bring different levels of utility, but
also different levels of exposures to cues, change as model primitives change.
Instead of having consumption of the addictive good being a binary choice, the
individual’s action space is enriched to being a continuous interval. Being
less interested in the dynamics associated with lifestyle choice, my model
only has two possible lifestyle activities opposed to three. Unfortunately,
there is not much work utilizing models with cue-triggered decision-making
beyond Bernheim and Rangel (2004).

Alternative models of rational addiction exist. Gul and Pesendorfer (2007)
considers a similar model where decision-makers sometimes make compulsive
choices. However, “compulsive consumption” in their model comes from goods
having some temptation utility opposed to triggered decisions due to
environmental cues. The main drawback of such a model is that it becomes
difficult to conduct positive welfare analysis: when preferences are not
consistent, should temptation utility be considered part of the decision-
maker’s well-being?

Another class of rational addiction models stems from Becker and Murphy
(1988); recently, Allcott et al. (2022) studies overuse of social media as
addiction to digital goods while Hussam et al. (2022) investigates habit
formation behind regular hand-washing using a similar model. Addiction in
these models comes from two sources: temptation utility as discussed
previously, and higher marginal utility from consuming the good as prior
consumption increases. However, someone with a higher history of past
consumption might not necessarily enjoy consumption more today (except through
being more tempted). Furthermore, this paper does not rule out interdependence
between current utility from consumption and past consumption; higher
sensitivity to quantity consumed as prior consumption increases can be viewed
as a special case of my model. While Allcott et al. (2022) considers the
impact of a quantity limit experimentally, the limit is self-prescribed
(participants could choose how much screen time to give themselves). This
paper focuses on the impact of externally-imposed limits.

Other papers such as Irvine and Nguyen (2011) study the welfare implications
of smoking and bans on smoking. Their theoretical model of how smoking
influences smoker utility can help inform the shape of utility functions in my
model while their empirical estimations can help inform population parameters
that can come into play when looking at aggregate welfare or total profits of
firms selling addictive goods. DeCicca et al. (2022) provides an overview of
different policies that seek to regulate tobacco, while Levy et al. (2018)
analyzes how regulatory agencies have analyzed consumer surplus under
different policies. Odermatt and Stutzer (2015) empirically analyzes the
impact of taxes on smoking, finding that higher prices reduce life
satisfaction of habitual smokers but increases the well-being of smokers who
are trying to quit.

Many reviews of the different behavioral components that influence gambling
decisions exist, such as Gainsbury et al. (2018). Similarly, Stetzka and
Winter (2021) evaluates how rational gambling behavior is, evaluating multiple
different potential explanations of why individuals choose to gamble. Cameron
et al. (2022) develops a two-period model of gambling behavior, where gambling
brings positive utility in the first period but cravings for gambling take
over and gambling becomes detrimental in the second period. In terms of
policies aimed at mitigating the harms of compulsive gambling, Broda et al.
(2008) analyzes the impact of self-imposed betting limits on an online
gambling site, finding that (1) most users do not violate their self-imposed
limits, and (2) those that do go above self-imposed limits do not suffer from
poor outcomes and bet rationally. Similarly, Badji et al. (2023) investigates
the welfare effects of increased gambling availability and find that financial
and mental well-being decrease as proximity to gambling venues increase.

##  2 Model

The model largely follows Bernheim and Rangel (2004). There is a decision-
maker (DM) that operates in an infinite-horizon discrete time setting who
discounts the future at rate $\delta$. At every period the DM observes their
addictive state $s\in\mathbf{R}$ and choose a lifestyle activity from
$A=\\{E,R\\}$ where $E$ denotes exposure to the addictive good and $R$ denotes
rehabilitation. If $R$ is chosen, the DM needs to pay a monetary cost of
$r(s)$ for some continuous $r$, no amount of the addictive good is consumed,
and the period ends. If $E$ is chosen, the DM can then choose to purchase some
quantity $q\in[0,Q]$ of the addictive good at a price $p$ per unit, spending a
total of $pq$.

With some probability depending on the history of past consumption, the DM may
enter a triggered state if $E$ is chosen and compulsively consumes $q=Q$ of
the addictive good. To model this, the DM’s past consumption determines their
addictive state $s$. At state $s$, consumption of $q$ leads to state $\lambda
s+q$ in the next period, for some $\lambda\in(0,1)$. There are two
consequences of this setup:

  1. (1)

Addictive state is upper bound by $\overline{S}=Q/(1-\lambda)$: If $s\leq
Q/(1-\lambda)$, then consumption of $Q$ leads to an addictive state of

$\lambda
s+Q\leq\lambda\frac{Q}{1-\lambda}+Q=\frac{(\lambda+(1-\lambda))Q}{1-\lambda}=\frac{Q}{1-\lambda}.$

  2. (2)

If the DM ever leaves the state $s=0$, they can never return: At any $s>0$, we
have that $\lambda s+q\geq\lambda s>0$.

With (1) in mind, given some arbitrary upper bound on upper bounds
$\overline{Q}$, let the set of possible state-quantity limit pairs be

$\mathcal{D}=\\{(s,Q):0\leq Q\leq\overline{Q},0\leq s\leq Q/(1-\lambda)\\}.$

For finite $\overline{Q}$, we have that $\mathcal{D}$ is a compact subset of
$\mathbb{R}^{2}$. For all intents and purposes, $\overline{Q}$ can be taken to
be large enough to encompass the relevant region of quantity limits.
Incorporating the quantity limit into the state space and restricting
attention to a a compact subset will be useful for applying dynamic
programming results when deriving comparative statics.

The DM’s perceived attractiveness of consuming the addictive good additionally
depends on some cue (such as advertising) $\omega\in\Omega$ drawn by nature
according to distribution $\mu$. Then, $M:A\times S\times\Omega\to\mathbb{R}$
measures the perceived attractiveness of consuming the addictive good. We
assume that:

  1. (1)

No attractiveness in rehabilitation: $M(R,s,\omega)=0$ for all $s,\omega$.

  2. (2)

No attractiveness if no history: $M(a,0,\omega)=0$ for all $a,\omega$.

  3. (3)

Higher prior use leads to additional perceived attractiveness: $M(E,s,\omega)$
is increasing in $s$ for $s\in[0,\overline{S}]$.

The DM enters the triggered state if $M(a,s,\omega)$ is larger than some
threshold $M^{T}$. Let $\mathcal{C}(a,s)=\\{\omega\in\Omega:M(a,s,\omega)\geq
M^{T}\\}$ be the set of cues that leads to the triggered state and
$\mathcal{P}(a,s)=\mu(\mathcal{C}(a,s))$ be the induced probability of
entering into the triggered state. A direct consequence of our three
assumptions on $M$ is the following:

######  Lemma 1 (Properties of $\mathcal{P}$).

The probability of entering the triggered state satisfies:

  1. (1)

No trigger in rehabilitation: $\mathcal{P}(R,s)=0$ for all $s$.

  2. (2)

No trigger if no history: $\mathcal{P}(a,0)=0$ for all $a$.

  3. (3)

Higher prior use leads to higher chance of trigger: $\mathcal{P}(E,s)$ is
increasing in $s$ for $s\in[0,\overline{S}]$.

In each period, the DM receives income $y(s)$ that depends continuously on
their addictive state. Their payoff in each state is determined by their
addictive state, income, quantity of addictive good purchased, and activity
captured in the function $u:S\times\mathbb{R}\times[0,Q]\times
A\to\mathbb{R}$. In a given period, if the DM chooses rehabilitation, their
payoff is

$u(s,y(s)-r(s),0,R).$

If the DM chooses exposure and consumption of $q$ units of the addictive good,
with probability $1-\mathcal{P}(E,s)$ they do not enter the triggered state
and collect a payoff of

$u(s,y(s)-pq,q,E)$

and with probability $\mathcal{P}(E,s)$ they do enter the triggered state and
collect a payoff of

$u(s,y(s)-pQ,Q,E).$

Let $V:\mathcal{D}\to\mathbb{R}$ be the DM’s value function of their
stochastic programming problem given their current addictive state is $s$ and
the quantity limit $Q$ (recall that $\mathcal{D}$ is the relevant domain of
possible additive state-quantity limit pairs). With this, we can define

$\begin{split}q^{*}(s,Q)=\operatorname*{\arg\\!\max}_{q\in[0,Q]}&\Big{\\{}(1-\mathcal{P}(E,s))[u(s,y(s)-pq,q,E)+\delta
V(\lambda s+q,Q)]\\\ &+\mathcal{P}(E,s)[u(s,y(s)-pQ,Q,E)+\delta V(\lambda
s+Q,Q)]\Big{\\}}.\end{split}$

to be the DM’s optimal level of consumption of the addictive good conditional
on choosing exposure and at addictive state $s$. As

$1-\mathcal{P}(E,s)\text{ and }\mathcal{P}(E,s)[u(s,y(s)-pQ,Q,E)+\delta
V(\lambda s+Q,Q)]$

do not depend on $q$, we can equivalently define

$q^{*}(s,Q)=\operatorname*{\arg\\!\max}_{q\in[0,Q]}u(s,y(s)-pq,q,E)+\delta
V(\lambda s+q,Q).$

If the DM decides to choose an activity of $E$, they will always choose
$q^{*}(s)$ as their level of consumption of the addictive good. Thus, the
value function solves

$\begin{split}V(s,Q)=\max\Bigg{\\{}&(1-\mathcal{P}(E,s))[u(s,y(s)-pq^{*}(s),q^{*}(s),E)+\delta
V(\lambda s+q^{*}(s),Q)]\\\ &+\mathcal{P}(E,s)[u(s,y(s)-pQ,Q,E)+\delta
V(\lambda s+Q,Q)],\\\ &u(s,y(s),0,R)+\delta V(\lambda
s,Q)\Bigg{\\}}.\end{split}$

A crucial note about the model is that addictive dynamics only necessarily
depend on the probability the DM compulsively consumes. In particular, this
means that we do not require DMs who have a higher period of consumption to be
more sensitive to the quantity of the addictive good consumed. There are many
cases of individuals who want to stop using an addictive substance and
recognize that usage is harmful, yet are unable to stop. However, our model
can incorporate this case: If more addicted DM’s are more sensitive to the
quantity of the good consumed, that can be modeled by assuming that

$\left|\frac{\partial}{\partial q}u(s,y(s)-pq,q,E)\right|$

is increasing in $s$.

Going forward, let

$\begin{split}\mathcal{P}(s)&=\mathcal{P}(E,s);\\\
u(s,q)&=u(s,y(s)-pq,q,E);\\\ u(s,Q)&=u(s,y(s)-pQ,Q,E);\\\
u(s,R)&=u(s,y(s)-r(s),0,R)\end{split}$

to simplify notation whenever there is no ambiguity.

##  3 Results

The main result is that if the DM dislikes being at a higher addictive state,
then tightening the quantity limit increases their utility, as long as it
never infringes on myopically optimal consumption. This extends the standard
behavioral intuition that helping the DM commit to not taking actions they do
not want to take may make them better off into this setting. We will work
towards the following formal result:

######  Proposition 1 (Properties of the Value Function).

The value function satisfies the following:

  1. (1)

If $u$ is continuous in $s$ and $q$, then $V$ is continuous in $s$ and $Q$;

  2. (2)

If $u$ is decreasing in $s$, then $V$ is decreasing in $s$ for all $Q$;

  3. (3)

Suppose that in addition to (2), there exists $\overline{q}$ such that
$u(s,y(s)-pq,q,E)$ is decreasing in $q$ for all $q\geq\overline{q}$. Then, $V$
is increasing in $Q$ for all $s$ and $Q\geq\overline{q}$;

  4. (4)

If $u$ is (strictly) concave in $s$, then $V$ is (strictly) concave in $s$.

All three results follow from standard dynamic programming techniques. Given
any function $W:\mathcal{D}\to\mathbf{R}$, define the “value iteration”
functional as follows:

$\begin{split}(TW)(s,Q)=\max\Bigg{\\{}&\max_{q\in[0,Q]}\Big{\\{}(1-\mathcal{P}(s))[u(s,q)+\delta
W(\lambda s+q,Q)]\\\ &\hskip 30.00005pt+\mathcal{P}(s)[u(s,Q)+\delta W(\lambda
s+Q,Q)]\Big{\\}},\\\ &u(s,R)+\delta W(\lambda s,Q)\Bigg{\\}}.\end{split}$

Then, the value function $V$ is a fixed point of $T$, so $V=TV$. Value
iteration preserves each of the three properties, so any fixed point must also
have the three properties.

From a policy perspective, this result establishes that if society is
confident that no rational individual would optimally consume above some
quantity of good, then a quantity limit at that level can only increase
welfare. A shortcoming of the current analysis is that we have only considered
a single decision-maker: What if there were many individuals with
heterogeneous preferences over the addictive good? If there were a
(potentially non-binding, for some individuals who do not like the addictive
good very much) uniform bound on the rationally optimal quantity consumed over
all individuals, such a quantity limit would still be welfare-improving.

###  3.1 Quantity Consumed

When the DM decides how much of the addictive substance to consume, they trade
off between three competing forces: Less income today due to more spending on
the addictive substance, increased experiential utility from consumption
itself, and lowered utility tomorrow from being at a higher addictive state.
Changing the consumption limit leaves the first two effects unchanged, but can
shift continuation payoffs. While it is intuitively tempting to say that a
looser quantity limit magnifies the harms of addiction, this holds regardless
of the addictive state. Instead, the DM cares about how changes in the
consumption limit changes the marginal harms from a higher addictive state.

######  Proposition 2 (Quantity Consumed as Limit Changes).

If $Q$ is greater than the myopic optimal level of consumption at state $s$,
then $q^{*}(s,Q)$ is (weakly) increasing as $Q$ decreases if $V$ has strictly
decreasing differences in $s,Q$. In particular, a sufficient condition is for

$\frac{\partial^{2}}{\partial q\partial Q}V(\lambda s+q,Q)<0.$

This corresponds to the intuition that external commitment from the quantity
limit is a substitute for internal commitment from choosing a lower level of
rational consumption. When the quantity limit is sufficiently extreme, we are
able to offer a sharper characterization. In this case, the restrictiveness of
external commitment completely replaces the need for any internal commitment.

######  Proposition 3 (Consistent Consumption).

For every model specification, there exists $\underline{Q}>0$ such that the DM
chooses exposure at every addictive state.

Unfortunately, comparative statics as to how quantity consumed changes as
addictive state changes are harder to pin down. In particular, the curvature
of the DM’s value function with respect to addictive state influences their
choice of how much to consume. As such, it is difficult to pin down how
consumption changes as addictive state changes purely in terms of model
primitives.

###  3.2 Numerical Solution

Consider the following parameterization of the model. Take $\lambda=0.8$. Let
the DM’s within-period utility from choosing exposure and consuming $q$ be

$u(s,q)=\sqrt{100-s}-q+q(10-q).$

This could potentially represent a DM with utility that is quasilinar in
income and addictive state, enjoys experiential utility of $\sqrt{100-s}$ from
being at a given addictive state, pays a price of one for the addictive good,
and consumption utility of $q(10-q).$ If continuation payoffs are
inconsequential, then the in-period optimum is to consume $q^{*}=4.5$
regardless of the addictive state. The probability the DM enters the hot state
is $\mathcal{P}(s)=\sqrt{s}/15$.333A range of denominators were tried. If the
probability is too low, the DM always consumes. If the probability is too
high, the DM always chooses rehabilitation. In general, these functional forms
were chosen to provide a clean visualization of general results.

The simplified model is solved using value iteration in Matlab. Value
functions are as follows, and confirms the result that tightening the quantity
limit improves the DM’s utility, as long as it does not restrict the
myopically optimal level of consumption:

Figure 1: Value Function at Different Quantity Limits

Next, we plot the DM’s policy function, where a quantity choice of $-1$
corresponds to rehabilitation. Trembles in the DM’s consumption choice are
most likely due to the solution being numeric, as opposed to being a
consequence of the model.

Figure 2: Policy at Different Quantity Limits

When the quantity limit decreases from $10$ to $9$ to $8$, quantity consumed
at every state increases. However, that does not hold at other quantity
levels: When the quantity limit decreases in the range of $15$ to $11$, the
set of states in which the DM chooses to consume increases, yet the quantity
consumed decreases.

In this case, there is a “last hurrah” effect: Valenzky and Jacobsen (2023)
describes how often times, drinkers think that it is fine to let loose one
last time before entering rehabilitation. If the problem is going to be solved
tomorrow anyways, then why not drink again today? In this particular model
specification, this effect dominates the disutility from a higher addictive
state, leading to consumption close to the myopically optimal level right
before rehabilitation.

##  4 Conclusion

Consumption of addictive goods is a pressing concern that the economics
literature does not have a good answer for. In particular, the impact on
limits of how much of an addictive substance one can consume at a time is
severely understudied, yet policy regulations in the real world often takes
this form. This paper takes a first step at analyzing such quantity limits.

Our main result shows that if a quantity limit does not infringe on anyone’s
immediate optimal level of consumption, it can only increase welfare. For
instance, if the quantity limit is set at the quantity where individuals start
overdosing, society can be quite confident that no one’s optimal level of
consumption is above that.

Further research can extend results in several directions. First is a sharper
characterization of what the optimal quantity limit is. In general, different
quantity limits can have differing effects on people in different addictive
states: Looser limits might be more beneficial if there is a large mass of
people at low addictive states, while sharper limits might be more beneficial
if there is a large mass of people at high addictive states. Finding ways to
aggregate welfare over a while population would produce interesting (and more
applicable) results. Bernheim and Rangel (2004) pilot this analysis,
considering a large economy of many short-lived DM’s that transition through
different addictive states. Second, if it is indeed the case that sharper
quantity limits lead to increased rational consumption, quantity limits may
also help sellers of such goods. Characterizing when a sharper quantity limit
both increases welfare and producer surplus would be a useful policy tool. In
these cases, forcing sellers to commit to only supplying a certain amount at
most would benefit both sides of the market, providing another implementation
mechanism for quantity limits. Finally, compulsive consumption may not respect
the quantity limit (for instance, people still get arrested for public
intoxication or possessing more than the legal amount of marijuana). To
account for this, instead of consuming $\overline{Q}$ in the triggered state,
DMs may instead consume some $\hat{q}(s)$, which could potentially be above
$\overline{Q}$. For instance, someone who has a long history of consumption
may build up a tolerance for the addictive good, leading to larger and larger
amounts needed to achieve the same previous levels of happiness. Enriching the
model in these dimensions would help garner a better understanding of how
addiction works, and in turn, spur more effective policy responses.

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## Appendix: Omitted Proofs

### PROOF OF PROPOSITION 1

###### Proof.

We will prove the proposition through a series of Lemmas. Our first result
establishes that value iteration maps continuous functions to continuous
functions:

######  Lemma 2.

If $W$ is continuous in $s$ and $Q$, then so is $TW$.

###### Proof.

The maximum of continuous functions is continuous and

$u(s,R)+\delta W(\lambda s,Q)$

is clearly continuous by preceding assumptions, so it suffices to show that

$\begin{split}&\max_{q\in[0,Q]}\Big{\\{}(1-\mathcal{P}(s))[u(s,q)+\delta
W(\lambda s+q,Q)]+\mathcal{P}(s)[u(s,Q)+\delta W(\lambda
s+Q,Q)]\Big{\\}}\end{split}$

is continuous. Let this function be $h(s,Q)$.

Let $X=\mathcal{D}$ and $Y=\mathbb{R}$. Let $\Gamma:X\to Y$ be the
correspondence defined by $\Gamma(s,Q)=[0,Q]$. Then, $\Gamma$ is compact-
valued and continuous. Let

$\begin{split}f(s,Q,q)=&(1-\mathcal{P}(s))[u(s,q)+\delta W(\lambda
s+q,Q)]+\mathcal{P}(s)[u(s,Q)+\delta W(\lambda s+Q,Q)].\end{split}$

By Berge’s Theorem of the Maximum, we have that

$h(s,Q)=\max_{q\in[0,Q]}f(s,Q,q)=\max_{q\in\Gamma(s,Q)}f(s,Q,q)$

is continuous as desired. ∎

Our next result establishes that if utility is decreasing in addictive state
at a proposed value function, then utility is still decreasing in state after
applying value iteration to it:

######  Lemma 3.

If $W(s,Q)$ is non-increasing in $s$ for any $Q$, then $TW$ is non-increasing
in $s$ for any $Q$ as well.

###### Proof.

Suppose $W$ is non-increasing in $s$. Then,

$u(s,R)+\delta W(\lambda s,Q)$

is non-increasing in $s$ so it suffices to show that

$\begin{split}&\max_{q\in[0,Q]}\Big{\\{}(1-\mathcal{P}(s))[u(s,q)+\delta
W(\lambda s+q,Q)]+\mathcal{P}(s)[u(s,Q)+\delta W(\lambda
s+Q,Q)]\Big{\\}}\end{split}$

is non-increasing in $s$. The above is equivalent to

$\begin{split}&(1-\mathcal{P}(s))\max_{q\in[0,Q]}\Big{\\{}(u(s,q)+\delta
W(\lambda s+q,Q)\Big{\\}}+\mathcal{P}(s)[u(s,Q)+\delta W(\lambda
s+Q,Q)].\end{split}$

As $q=Q$ is in the choice set,

$\max_{q\in[0,Q]}\Big{\\{}(u(s,q)+\delta W(\lambda s+q,Q)\Big{\\}}\geq
u(s,Q)+\delta W(\lambda s+Q,Q).$

Thus, if $s^{\prime}>s$, we have that
$\mathcal{P}(E,s^{\prime})\geq\mathcal{P}(E,s)$ so

$\begin{split}&(1-\mathcal{P}(s))\max_{q\in[0,Q]}\Big{\\{}(u(s,q)+\delta
W(\lambda s+q,Q)\Big{\\}}+\mathcal{P}(s)[u(s,Q)+\delta W(\lambda s+Q,Q,Q)]\\\
\geq&(1-\mathcal{P}(s^{\prime}))\max_{q\in[0,Q]}\Big{\\{}(u(s,q)+\delta
W(\lambda s+q,Q)\Big{\\}}+\mathcal{P}(s^{\prime})[u(s,Q)+\delta W(\lambda
s+Q,Q)].\\\ \end{split}$

As $u$ and $W$ are non-increasing in $s$, we also have that

$\begin{split}&(1-\mathcal{P}(s^{\prime}))\max_{q\in[0,Q]}\Big{\\{}(u(s,q)+\delta
W(\lambda s+q,Q)\Big{\\}}+\mathcal{P}(s^{\prime})[u(s,Q)+\delta W(\lambda
s+Q,Q)]\\\
\geq&(1-\mathcal{P}(E,s^{\prime}))\max_{q\in[0,Q]}\Big{\\{}(u(s^{\prime},q)+\delta
W(\lambda
s^{\prime}+q,Q)\Big{\\}}+\mathcal{P}(s^{\prime})[u(s^{\prime},Q)+\delta
W(\lambda s^{\prime}+Q,Q)].\\\ \end{split}$

∎

A similar result holds to show (3): If a proposed value function is increasing
in $Q$ for $Q\geq\overline{q}$, then performing value iteration preserves that
property:

######  Lemma 4.

Suppose there exists $\overline{q}$ such that $u(s,y(s)-pq,q,E)$ is decreasing
in $q$ for all $q\geq\overline{q}$ and $s$. If $W(s,Q)$ is non-increasing in
$s$ for all $Q$ and non-increasing in $q$ for all $q\geq\overline{q}$ and $s$,
then $(TW)(s,Q)$ is non-increasing in $s$ for all $Q$ and non-increasing in
$Q$ for all $Q\geq\overline{q}$ and $s$.

###### Proof.

Lemma 3 gives that $(TW)(s,Q)$ is non-increasing in $s$ for all $Q$.

By assumption of $W$ being non-increasing in $Q$ for $Q\geq\underline{Q}$ and
$u(s,y(s),0,R)$ being independent of $Q$, we have that

$u(s,R)+\delta W(\lambda s,Q)$

is non-increasing in $Q$ for $Q\geq\underline{Q}$.

Similarly, by assumption of $W$ being non-increasing in $Q$ for
$Q\geq\underline{Q}$, $W$ being non-increasing in $s$, and
$Q\geq\underline{Q}$ combined with $u(s,q)$ being non-increasing in $q$ for
$q\geq\underline{Q}$, we have that

$u(s,Q)+\delta W(\lambda s+Q,Q)$

is non-increasing in $Q$ as well.

As such, it suffices to show that

$\max_{q\in[0,Q]}\Big{\\{}(u(s,q)+\delta W(\lambda s+q,Q)\Big{\\}}$

is non-increasing in $Q$. Fix $s$ and $\underline{Q}\leq Q<Q^{\prime}$ with
$(s,Q),(s,Q^{\prime})\in\mathcal{D}$ noting that if no such $Q,Q^{\prime}$
exist, then the lemma is vacuously true. Recall that

$q^{*}(s,Q)=\operatorname*{\arg\\!\max}_{q\in[0,Q]}\Big{\\{}(u(s,q)+\delta
W(\lambda s+q,Q)\Big{\\}}.$

For any $Q\geq\underline{Q}$, we have that $q^{*}(s,Q)\leq\underline{Q}$ since
both $u$ and $W$ are decreasing past $\underline{Q}$. Then,

$\begin{array}[]{>{\displaystyle}r>{\displaystyle}l>{\displaystyle}l}\hfil&\max_{q\in[0,Q]}\Big{\\{}(u(s,q)+\delta
W(\lambda s+q,Q)\Big{\\}}&\\\ =&u(s,q^{*}(s,Q)+\delta W(\lambda
s+q^{*}(s,Q),Q)&\text{by definition of }q^{*}\\\
\geq&u(s,q^{*}(s,Q^{\prime})+\delta W(\lambda
s+q^{*}(s,Q^{\prime}),Q)&q^{*}(s,Q^{\prime})\in[0,\underline{Q}]\subseteq[0,Q]\\\
\geq&u(s,q^{*}(s,Q^{\prime})+\delta W(\lambda
s+q^{*}(s,Q^{\prime}),Q^{\prime})&W\text{ non-increasing in }Q\\\
=&\max_{q\in[0,Q^{\prime}]}\Big{\\{}(u(s,q)+\delta W(\lambda
s+q,Q^{\prime})\Big{\\}}&\end{array}$

as desired. ∎

######  Lemma 5.

Suppose $u$ is (strictly) concave in $s,q$ and $W$ is concave. Then, $TW$ is
(strictly) concave in $s$.

###### Proof.

We have that for any $Q$,

$\begin{split}&TW(\gamma s+(1-\gamma)s^{\prime},Q)\\\ =&(1-\mathcal{P}(\gamma
s+(1-\gamma)s^{\prime}))\Big{[}u(\gamma s+(1-\gamma)s^{\prime},q^{*}(\gamma
s+(1-\gamma)s^{\prime}),Q))\\\ &+\delta W(\lambda(\gamma
s+(1-\gamma)s^{\prime})+q^{*}(\gamma s+(1-\gamma)s^{\prime}),Q)\Big{]}\\\
&+\mathcal{P}(\gamma s+(1-\gamma)s^{\prime})\left[u(\gamma
s+(1-\gamma)s^{\prime},Q)+\delta W(\lambda(\gamma
s+(1-\gamma)s^{\prime})+Q,Q)\right]\\\ \geq&(1-\mathcal{P}(\gamma
s+(1-\gamma)s^{\prime}))\Big{[}u(\gamma s+(1-\gamma)s^{\prime},\gamma
q^{*}(s,Q)+(1-\gamma)q^{*}(s^{\prime},Q))\\\ &+\delta W(\lambda(\gamma
s+(1-\gamma)s^{\prime})+\gamma
q^{*}(s,Q)+(1-\gamma)q^{*}(s^{\prime},Q),Q)\Big{]}\\\ &+\mathcal{P}(\gamma
s+(1-\gamma)s^{\prime})\left[u(\gamma s+(1-\gamma)s^{\prime},Q)+\delta
W(\lambda(\gamma s+(1-\gamma)s^{\prime})+\gamma Q+(1-\gamma)Q,Q)\right]\\\
\geq&(1-\mathcal{P}(\gamma s+(1-\gamma)s^{\prime}))\Big{[}\gamma
u(s,q^{*}(s,Q))+(1-\gamma)u(s^{\prime},q^{*}(s^{\prime},Q))\\\ &+\delta(\gamma
W(\lambda s+q^{*}(s,Q),Q)+(1-\gamma)W(\lambda
s^{\prime}+q^{*}(s^{\prime},Q),Q))\Big{]}\\\ &+\mathcal{P}(\gamma
s+(1-\gamma)s^{\prime})\left[\gamma
u(s,Q)+(1-\gamma)u(s^{\prime},Q)+\delta(\gamma W(\lambda
s+Q,Q)+(1-\gamma)W(\lambda s^{\prime}+Q,Q))\right]\\\
=&\gamma\Big{\\{}(1-\mathcal{P}(\gamma
s+(1-\gamma)s^{\prime}))[u(s,q^{*}(s,Q))+\delta W(\lambda s+q^{*}(s,Q),Q)]\\\
&+\mathcal{P}(\gamma s+(1-\gamma)s^{\prime})[u(s,Q)+\delta W(\lambda
s+Q,Q)]\Big{\\}}\\\ &+(1-\gamma)\Big{\\{}(1-\mathcal{P}(\gamma
s+(1-\gamma)s^{\prime}))[u(s^{\prime},q^{*}(s^{\prime},Q))+\delta W(\lambda
s^{\prime}+q^{*}(s^{\prime},Q),Q)]\\\ &+\mathcal{P}(\gamma
s^{\prime}+(1-\gamma)s^{\prime})[u(s^{\prime},Q)+\delta W(\lambda
s^{\prime}+Q,Q)]\Big{\\}}\end{split}$

∎

Finally, we show that underlying space of possible value functions is complete
and value iteration is a contraction mapping to guarantee that value iteration
converges to a unique fixed point:

######  Lemma 6.

The space of $L_{\infty}$-uniformly bounded functions
$W:\mathcal{D}\to\mathbb{R}$ that have any subset of the following properties:

  1. (1)

non-increasing in $s$;

  2. (2)

non-increasing in $Q$ for all $Q\geq\overline{q}$ and all $s$;

  3. (3)

concave;

is complete.

###### Proof.

The space of functions $W:\mathcal{D}\to\mathbb{R}$ that are bounded is
complete in the sup norm. Then, the set of functions that satisfy condition
(1), (2), or (3) is closed. Intersections of closed sets are closed, which
implies that the set of functions that satisfy any subset of these conditions
is closed. Finally, closed subspaces of complete spaces are complete. ∎

######  Lemma 7.

Value iteration is a contraction mapping.

###### Proof.

By Blackwell’s Sufficient Conditions for a functional to be a contraction
mapping, it suffice to show that

  1. (1)

$W\leq U$ implies $TW(s,Q)\leq TU(s,Q)$;

  2. (2)

There exists $\beta\in(0,1)$ such that $(T[W+a])(s,Q)\leq TW(s,Q)+\beta a$ for
$a\geq 0$.

The first point is immediate. Then,

$\begin{split}(T[W+a])(s,Q)=\max\Bigg{\\{}&\max_{q\in[0,Q]}\Big{\\{}(1-\mathcal{P}(s))[u(s,q)+\delta(W(\lambda
s+q,Q)+a)]\\\ &\hskip 30.00005pt+\mathcal{P}(s)[u(s,Q)+\delta W(\lambda
s+Q,Q)]\Big{\\}},\\\ &u(s,R)+\delta(W(\lambda s,Q)+a)\Bigg{\\}}\\\
=\max\Bigg{\\{}&\max_{q\in[0,Q]}\Big{\\{}(1-\mathcal{P}(s))[u(s,q)+\delta
W(\lambda s+q,Q)]\\\ &\hskip 30.00005pt+\mathcal{P}(s)[u(s,Q)+\delta W(\lambda
s+Q,Q)]\Big{\\}},\\\ &u(s,R)+\delta W(\lambda s,Q)\Bigg{\\}}+\delta a\\\
&=TW(s,Q)+\delta a\end{split}$

where $\delta a$ is constant and hence affects neither maximization problem.
Thus, taking $\beta=\delta$ suffices. ∎

To prove the Theorem, we can start by taking $W(s,Q)=0$ for all $s,Q$. This is
continuous in $s,Q$, non-increasing in $s$ for any $Q$, and non-increasing in
$q$ for any $s$. Then, $TW$ satisfies all these properties as well. Finally,
the sequence $\\{T^{k}W\\}_{k=1}^{\infty}$ converges as the space of all
functions is complete and value iteration is a contraction mapping.
Furthermore, it converges to a unique value function $V$ that inherits all
desired properties.

∎

### PROOF OF PROPOSITION 2

###### Proof.

Recall that

$q^{*}(s,Q)=\operatorname*{\arg\\!\max}_{q\in[0,Q]}u(s,q)+\delta V(\lambda
s+q,Q).$ (1)

We will show that strictly decreasing differences of $V$ in $s,Q$ is a
sufficient condition for the overall maximand to have strictly decreasing
differences in $q,Q$. Then, by the Monotone Selection Theorem of Milgrom and
Shannon (1994), the DM’s optimal selection of $q$ is decreasing as $Q$
increases.

By strict decreasing differences in $V$, for any
$\lambda,s,q,q^{\prime},Q,Q^{\prime}$ with
$(s,Q),(s^{\prime},Q^{\prime}),(\lambda s+q,Q),(\lambda
s+q^{\prime},Q^{\prime})\in\mathcal{D}$, we have that

$V(\lambda s+q^{\prime},Q^{\prime})-V(\lambda s+q,Q^{\prime})>V(\lambda
s+q^{\prime},Q)-V(\lambda s+q,Q).$

Adding

$u(s,q^{\prime})-u(s,q)$

to both sides gives that

$\begin{split}&\big{(}V(\lambda
s+q^{\prime},Q^{\prime})+u(s,q^{\prime})\big{)}-\big{(}V(\lambda
s+q^{\prime},Q^{\prime})+u(s,q)\big{)}\\\ >&\big{(}V(\lambda
s+q^{\prime},Q)+u(s,q)\big{)}-\big{(}V(\lambda
s+q^{\prime},Q)+u(s,q^{\prime})\big{)}\end{split}$

Thus, the maximand in Equation 1 has decreasing differences. ∎

### PROOF OF PROPOSITION 3

###### Proof.

For exposure to be chosen at all addictive states, it must be that

$\begin{split}&\max_{q\in[0,Q]}\Big{\\{}(1-\mathcal{P}(s))[u(s,q)+\delta
V(\lambda s+q,Q)]\\\ &\hskip 30.00005pt+\mathcal{P}(s)[u(s,Q)+\delta V(\lambda
s+Q,Q)]\Big{\\}}\\\ >&u(s,y(s)-r(s),0,R)+\delta V(\lambda s,0)\end{split}$

at all $s$ for some $Q$. Then, for any $Q$

$\begin{split}&\max_{q\in[0,Q]}\Big{\\{}(1-\mathcal{P}(s))[u(s,q)+\delta
V(\lambda s+q,Q)]\\\ &\hskip 30.00005pt+\mathcal{P}(s)[u(s,Q)+\delta V(\lambda
s+Q,Q)]\Big{\\}}\\\ >&u(s,Q)+\delta V(\lambda s+Q,Q)\end{split}$

so it suffice to show that there exists $Q$ such that

$u(s,Q)+\delta V(\lambda s+Q,Q)>u(s,y(s)-r(s),0,R)+\delta V(\lambda s,0)$

for all $s$.

Now, suppose that the quantity limit is zero. Then,

$u(s,y(s),0,E)+\delta V(\lambda s,0)>u(s,y(s)-r(s),0,R)+\delta V(\lambda s,0)$

for all $s$. By continuity of $u$ in $s$ and the fact that the set of all
possible addictive states $[0,\overline{Q}/(1-\lambda]$ is closed, we have
that

$\min_{s}\left\\{u(s,y(s),0,E)-u(s,y(s)-r(s),0,R)\right\\}=\varepsilon>0.$

As such, it suffice to find $Q$ such that

$u(s,Q)+\delta V(\lambda s+Q,Q)>u(s,y(s),0,E)+\delta V(\lambda
s,0)-\varepsilon.$ (2)

Let

$h(s,Q)=u(s,y(s),0,E)-u(s,Q)+\delta(V(\lambda S,0)-V(\lambda s+Q,Q)).$

Then, Equation 2 is equivalent to

$\varepsilon>h(Q,s).$

By continuity of $u$ and $V$, we have that $h$ is continuous in $s,Q$. As
such, $h^{-1}([0,\varepsilon))$ is an open set (where $[0,\varepsilon)$ is the
half-open interval from $0$ to $\varepsilon$) in
$[0,\overline{Q}/(1-\lambda]\times[0,\overline{Q}]$. As such, the complement
of $h^{-1}((0,\varepsilon))$ is closed.

Towards a contradiction, suppose there does not exist $Q>0$ such that
$(s,Q)\in h^{-1}([0,\varepsilon))$ for all $s$. Then, there must exist a
sequence $\\{s_{k},Q_{k}\\}_{k=1}^{\infty}$ such that
$(s_{k},Q_{k})\in(h^{-1}([0,\varepsilon)))^{C}$ for all $k$ and
$\lim_{k\to\infty}Q_{k}=0$. Then, letting $\hat{s}=\lim_{k\to\infty}s_{k}$, it
must be that $(\hat{s},0)\in(h^{-1}([0,\varepsilon)))^{C}$. However,
$h(\hat{s},0)=0$, a contradiction. ∎