# Public policy for management of forest pests within an ownership mosaic

Andrew R. Tilman  Robert G. Haight  USDA Forest Service, Northern Research
Station, St. Paul, MN, USA

(December 8, 2023)

###### Abstract

The benefits of investments in sustainability are often public goods that
accrue at broad scales and to many people. Urban forests exemplify this; trees
supply ecosystem service benefits from local (in the case of shade) to global
(in the case of carbon sequestration) scales. The complex mosaic of public and
private ownership that typically defines an urban forest makes the public
goods problem of investing in forest sustainability especially acute. This
results in incentives for private tree owners to invest in tree care that
typically fall short of those of a public forest manager aiming for the social
optimum. The management of a forest pest, such as emerald ash borer, provides
a salient focus area because pests threaten the provision of public goods from
urban forests and pest management generates feedback that alters pest spread
and shapes future risks. We study how managers can design policies to address
forest pest outbreaks and achieve uniform management across a mosaic of
ownership types. We develop a game theoretic model to derive optimal subsidies
for the treatment of forests pests and evaluate the efficacy of these policies
in mitigating the spread of forest pests with a dynamic epidemiological model.
Our results suggest that a combination of optimal treatment subsidies for
privately owned trees and targeted treatment of public trees can be far more
effective at reducing pest-induced tree mortality than either approach in
isolation. While we focus on the management of urban forests, designing
programs that align private and public incentives for investment in public
goods could advance sustainability in a wide range of systems.

_Keywords_ Public goods $\cdot$ Forest pests $\cdot$ Subsidy policy $\cdot$
Urban forests $\cdot$ Private landowners $\cdot$ Emerald ash borer

The UN’s 2030 Agenda for Sustainable Development defines a set of
interconnected global goals which includes transitioning toward renewable
energy and creating resilient and green cities (United Nations,, 2015).
Realizing these objectives will require major investments by governments,
companies, and individuals (Sachs et al.,, 2019). A challenge at the heart of
spurring sustainability transitions, especially in urban areas, is that the
benefits of individual investments often accrue at broad scales and thus to
many people. In other words, there are public goods problems embedded within
many potential sustainability solutions and private incentives for action will
often fall short (Olson Jr,, 1965; Bergstrom et al.,, 1986). Therefore, a
general challenge for policy makers aiming for urban sustainability is
crafting cost effective policies that support the private provision of public
goods (Wichman,, 2016).

This challenge arises in many contexts. For example, increasing the
sustainability of transportation could come from public investment in mass-
transit systems or from individuals’ adoption of electric vehicles (Sultana et
al.,, 2021). In the case of electric vehicle adoption, the climate benefit of
reduced greenhouse gas emissions is a global public good that individuals are
not compensated for. Renewable electricity program adoption follows a similar
pattern where households pay a premium for electricity with a reduced carbon
footprint, making these investments a public good (Kotchen and Moore,, 2007).
Urban forests, comprised of the public and private trees within urban areas,
contain 5.5 billion trees in the US and contribute to sustainability goals by
providing benefits to the over 80% of Americans who live in urban areas (Dwyer
et al.,, 1992; Nowak and Greenfield,, 2018; Pataki et al.,, 2021). However, in
the US only about 20 percent of urban trees are publicly owned, such as in
parks or on rights-of-way where planting strips are commonly used for street
trees (Arbor Day Foundation,, 2018). Since most of the urban forest grows on
private land, including both residential and industrial properties, the
investments needed to expand and steward urban forests are impure public
goods. This is a defining feature of the complex mosaic of public and private
land ownership in cities. Thus, moving toward cities that are sustainable,
resilient, and green may benefit from carefully designed public policies that
support individuals in taking actions that collectively contribute to
sustainability.

In the United States, the Inflation Reduction Act, signed into law in 2022, is
making historic sustainability and climate mitigation investments (Bistline et
al.,, 2023). This is accomplished through a portfolio of approaches including
expanding tax credits for electric vehicle purchases and residential solar
installations, as well as through grant programs, such as the $1 billion
awarded in September 2023 by the USDA Forest Service for urban and community
forestry 111USDA Press Release: USDA invests $1 billion for nearly 400
projects to expand access to trees and green spaces in communities and
neighborhoods nationwide. These funds support tree planting and tree care in
cities, across public and private land. For example, the Pittsburgh Canopy
Alliance received $8 million to plant and maintain trees along streets, in
parks and public greenspaces, and residential and institutional property in
neighborhoods where investments in greening will contribute to social justice
222Urban and Community Forestry Grants - 2023 Grant Awards . The city of
Minneapolis received an $8 million grant for treatments to protect ash trees
(Fraxinus spp.) from emerald ash borer (EAB) (Agrilus planipennis) as well as
removal of infested ash trees and diverse species replanting 333Press Release:
Minneapolis Equitable Tree Canopy Management Program. Both approaches could
contribute to more resilient future forests. We focus on the case of forest
pest management in urban areas, as it bears many of the hallmarks of the
broader challenge of designing policies to promote private investment in
public goods to aid in transitions toward sustainable and resilient futures.
At the same time, given the historic levels of funding for urban forestry,
made possible by the Inflation Reduction Act, studying polices that could cost
effectively sustain urban forests in the face of threats from pests is
especially salient.

Urban forests provide myriad ecosystem services, including carbon
sequestration, erosion mitigation, and urban cooling (Roy et al.,, 2012;
Livesley et al.,, 2016). At the same time, urban forests grow in complex
mosaics of public and private land comprised of many individually managed
properties with a variety of uses (Dwyer et al.,, 2003; Epanchin-Niell et
al.,, 2010). In these management mosaics a majority of trees are privately
owned (Clark et al.,, 1997) and many of the benefits provided by these trees
are public goods that accrue from local to global scales. However, in the case
of privately owned trees, the costs of sustaining these public benefits fall
on individual landowners, making investment in management an impure public
good. Therefore, under-investment in management actions that sustain ecosystem
services on private land is expected (Bergstrom et al.,, 1986), but public
subsidy and cost-share programs hold promise for closing this gap (Andreoni
and Bergstrom,, 1996; Wichman,, 2016; Jacobsen et al.,, 2017; Chan and
Dinelli,, 2020; Chizmar et al.,, 2021).

An objective for public urban forest managers is to design and implement
policies that align public and private management actions to achieve socially-
optimal ecosystem service provision across this mosaic landscape. The
management of forest pests is an exemplar of this problem because each
manager’s control decisions affect the local impact and spread of the pest or
disease and these effects alter the provision of ecosystem services across the
landscape. Oak tree mortality caused by the non-native forest pathogen,
Phytophthora ramorum, on a single property reduces the values of other
properties in the vicinity (Kovacs et al.,, 2011). This persistent property
value decline for homes located near an infested property reflects people’s
willingness to pay for tree cover and associated ecosystem services across the
landscape (e.g., beauty, wildlife habitat, shade and comfort). A landowner’s
control decision can also affect the risk of pest spread and damage to
neighboring landowners. Insecticide treatments, such at trunk injections with
emamectin benzoate, provide effective EAB control to treated trees (McCullough
et al.,, 2011). In most jurisdictions these treatments must be administered by
licensed tree care professionals and therefore policies to support treatment
must take into account the role of tree care firms as intermediaries between
municipalities and tree owners. This mirrors the structure of residential
solar adoption where homeowners hire firms to install solar arrays on their
property. Insecticide treatments protect treated ash trees while also
mitigating the risk of infestation of neighboring ash trees (McCullough and
Mercader,, 2012). This spillover effect results in a spatial-dynamic
externality (Wilen,, 2007) that has been studied in the context of pest spread
within management mosaics (Sims et al.,, 2010; Kovacs et al.,, 2014).

Subsidies and transfer payments are proposed as mechanisms to align public and
private incentives for pest management in urban forests. Chen et al., (2022)
use a principal-agent modeling framework to design programs in which a local
government reimburses landowners for pest management costs based on numbers of
infested or treated trees on the property. Cobourn et al., (2019) show that
transfer payments between municipalities can induce levels of pest control
that make each city better off and approach a central planner’s goal of
maximizing net social benefit.

Building on this work, we construct a game theoretic model to study optimal
subsidy policies for treatments to address forest pests and apply the results
of the game theoretic model to a epidemiological model of pest infestation
dynamics to explore the implications of treatment subsidies on the dynamics of
pest spread. We construct our game theoretic model based on a focal privately-
owned tree. First, a municipality offers tree-health-dependent subsidies to
tree care firms for treatment of the tree to protect it from pest damage.
Next, the tree care firms offer bids for tree protection to the tree owner.
Finally, the tree owner will accept or reject the bids in order to maximize
their net benefits. We derive the subgame-perfect Nash equilibrium behavior of
tree owners and firms in order to reveal the subsidy policy that results in
socially-optimal management of private trees. We determine the conditions
under which subsidies are warranted from the perspective of a municipality
aiming to maximize the social benefits of trees. Whereas the game theoretic
model results apply to a single tree, we apply these results at the community
scale with a dynamic epidemiological model of forest pests that tracks pest
spread and tree mortality through time under a range of subsidy and treatment
scenarios constructed from the game theoretic model.

We apply our model to the management of urban ash trees that are threatened by
emerald ash borer, a non-native forest insect that attacks and kills North
American ash (Herms and McCullough,, 2014). Ash species are an important part
of urban forests throughout the United States, having been planted for decades
because of their tolerance to drought and soil compaction (MacFarlane and
Meyer,, 2005). Ash trees provide ecological benefits such as wildlife habitat
and stormwater absorption, social benefits such as pleasing environments for
people, and human health benefits such as reduced stress and cooling (Dwyer et
al.,, 1992; Pataki et al.,, 2021). One highly vulnerable ash species, Black
ash (Fraxinus nigra), is a cultural keystone species for many Native Peoples
in North America with spiritual, cultural, and utilitarian significance
(Costanza et al.,, 2017; Siegert et al.,, 2023). Since EAB was discovered near
Detroit, Michigan, and Windsor, Ontario, in 2002, the insect has spread to 36
states in the U.S. and five Canadian provinces (Michigan State University and
USDA Forest Service,, 2022), killed millions of ash trees (Herms and
McCullough,, 2014), and cost homeowners and cities hundreds of millions of
dollars in losses of property value and damage mitigation (Kovacs et al.,,
2010; Aukema et al.,, 2011). In response, cities have developed EAB management
plans for public trees, including surveillance of tree health, application of
systemic insecticides for tree protection, and preemptive removal of infested
or unhealthy trees (Herms and McCullough,, 2014). Simulation and optimization
studies of urban ash populations show that emphasizing insecticide treatment
rather than tree removal can result in greater net social benefits (McCullough
and Mercader,, 2012; Kovacs et al.,, 2014; Kıbış et al.,, 2021; Sadof et al.,,
2023). The model that we study is consistent with EAB management plans in
cities that strategically treat their publicly owned trees and negotiate with
tree care firms to offer discounted prices for insecticide treatment to
private tree owners as an incentive to protect their ash trees and sustain the
ecosystem service benefits they provide. The EAB Management Plan of the city
of Burnsville, Minnesota exemplifies this approach. The City contracts with a
tree care firm to treat public trees and part of this contract stipulates that
private ash tree owners receive discounted pricing from the firm for
insecticide treatment.

We use our model to explore how subsidy levels impact treatment choices and
tree survival of privately owned trees. By deriving optimal subsidy levels,
our model could help refine public programs that support pest management on
private land and increase their cost effectiveness. We apply the model to a
case study using parameter values for the effectiveness of insecticide
treatment, costs of treatment and tree removal, and values of healthy trees
obtained from the literature on EAB management (Herms and McCullough,, 2014).
The model suggests that subsidies should target the treatment of healthy and
infested trees early in a community’s EAB outbreak. Late in an outbreak when
many ash trees have died and relatively few infested trees remain, the model
suggests that treatment of healthy but not infested trees should be
subsidized. While our game theoretic model defines optimal subsidies across a
range of infestation states, it does not model the spread of EAB though time.
We explore the impact of these subsidy levels on the spread of EAB with a
compartmental model of infestation dynamics. This dynamical model allows us to
compare long-term ash survival under various scenarios, including the case
where there is no treatment to slow the spread of EAB on public or private
land, the case where there is no public action to treat EAB, and the case
where optimal subsidies are employed for private trees and optimal treatment
decisions are made for public trees. These simulations illustrate that a
combined approach where subsides are used as a tool to align pest management
across public and private land can be far more effective at reducing pest-
induced tree mortality and sustaining the ecosystem service benefits of urban
tree cover than approaches targeted solely at either public or private land.

## Game theoretic model

We construct our model around the treatment of a single ash tree growing on
private land. The model includes strategic interactions between three
stakeholders: a municipal forester with an objective of maximizing forest
ecosystem service benefits, two tree care firms with the objective of
maximizing their profits, and an ash tree owner whose objective is to maximize
the benefits they receive from their tree, taking into account management
costs. Henceforth, we call these three stakeholders the municipality, the
firms, and the tree owner. The structure of the game is shown in Figure 1. The
model begins with a focal ash tree that falls into one of three health
(infestation) states: healthy, infested, or dying / dead. We assume that
community-level prevalence of EAB is known by the stakeholders and used to the
infer the likelihood of a tree being in each state. Next an arborist conducts
a tree health assessment by observing the tree’s crown condition. The
relationship between an ash tree’s crown condition and its EAB infestation
state has been quantified by Flower et al., (2013) and Knight et al., (2014)
and used in another model of ash tree management by Chen et al., (2022).
Depending on the level of crown dieback, the arborist classifies the tree as
healthy (no visible crown dieback), infested (visible crown dieback), or dead
/dying (insufficient live foliage for recovery) and shares this information
with the municipality and tree owner. Canopy assessment is an imperfect
predictor of infestation status. For example, a tree with no visible crown
dieback may have either no infestation or a low level of infestation that has
yet to cause visible damage. Conversely, crown dieback observed in a tree
could be due to a different cause. Based on the relationship between crown
condition and infestation state identified by Flower et al., (2013) and Knight
et al., (2014), we can estimate the conditional probabilities of observing
each crown condition given some true underlying infestation state of a tree.
These probabilities allow stakeholders to estimate the likelihood of each
underlying tree health state, given the assessment of crown dieback, using
Bayes’ theorem.

We consider the municipally subsidized treatment of a privately owned ash tree
by a tree care firm as a dynamic game of incomplete information in three
stages. We analyze the game that follows from each possible assessed health
status of the tree. Figure 1 highlights the structure of the game for a tree
that is assessed as infested. In each case, the municipality selects non-
negative subsidy levels that are available to the tree care firms, with the
potential for these subsidy levels to vary between the firms. The firms then
simultaneously submit bids to the tree owner defining the price they would
charge for treating the tree. Finally, the tree owner decides whether to
accept the bid of either firm, or opt not to treat their tree. The tree owner
has a unique ‘type’ which we define as the value that they assign to saving
their tree. While the tree owner knows this value, the municipality and the
firms only know the distribution from which this value is drawn. This
introduces incomplete information for the firms and the municipality.
Therefore, we will assume that the municipality and the firms aim to maximize
their expected utility given their beliefs about the distribution of tree
owner values. These assumptions allow us to solve for the Perfect Bayesian
Equilibrium (a refinement of the Nash equilibirum) of the game, as illustrated
in Figure 1.

Figure 1: The structure of the game from assessment of a tree to treatment. At
the beginning, a focal tree can fall into one of three states with likelihoods
given by community level EAB prevalence. Next, the health status of the tree
is assessed by an arborist, with assessment errors possible. This splits the
game into three branches, one for each assessed tree state. Each branch can be
analyzed independently. We highlight the case of a tree that has been assessed
as being in an infested state. First, the municipality sets subsidy levels
that each firm will receive if they treat the tree with this assessed state.
The arc in the figure at this stage represents the continuous range of subsidy
pairs that the municipality can set for the firms. Next, knowing the assigned
subsidy levels, the firms simultaneously submit bids for treatment of a focal
tree, à la Bertrand competition. Finally, the tree owner will decided whether
to treat their tree and which firm’s bid to select. Here we show a case where
firm 1 offers a lower bid, and the tree owner decides to have their tree
treated by firm 1.

We proceed to introduce key aspects of tree health, assessment, and treatment
that underlie the (expected) payoffs of the tree owner and the municipality.
Then we will derive the benefits of treatment from the perspectives of the
tree owner and municipality. Using these benefits, we construct the utility
functions of each player in the game, and describe how this game can be
analyzed. Finally, we will define the equilibrium behavior of each player.

### Tree health assessment

We assume an arborist will assess the health status of a tree in one of three
states,

$\hat{\phi}\in\\{\hat{h},\hat{i},\hat{d}\\},$ (1)

with the three possible assessed states being: healthy, $\hat{h}$, infested,
$\hat{i}$, or dead / dying, $\hat{d}$. A tree’s health status with respect to
EAB is only partially observable, with the true underlying state

$\phi\in\\{h,i,d\\}$ (2)

of the tree possibly diverging from the assessed state. We assume that
surveillance data are available at the municipality scale, with publicly known
frequencies of each underlying tree state given by $P(h)$, $P(i)$, and $P(d)$.

Since we assume that assessment techniques are imperfect predictors of the
true underlying health status of a tree, we define the accuracy of assessment
with probabilities. For example, $P(\hat{h}\mid i)$ is the probability that a
tree is assessed as healthy given that it actually is infested. There are nine
of these probabilities, which can, in general, be written as

$P(\hat{\phi}\mid\phi)$ (3)

for some assessed state, $\hat{\phi}$ and some true underlying state $\phi$.

When a tree owner is deciding whether to treat their tree and when the
municipality is setting subsidy policies for trees, they use the information
in the assessment of the tree’s health to update their beliefs about the
likelihood of the tree having each of the three possible underlying states.
Their prior belief about the state of the tree is determined by the community
level surveillance probabilities, and this is integrated with the information
in the assessment via Bayes rule. This results in updated beliefs about the
likelihoods of the each underlying state. For example, given a tree is
assessed as being healthy, the probability that it is actually infested is
given by

$P(i\mid\hat{h})=\frac{P(\hat{h}\mid i)P(i)}{P(\hat{h}\mid
h)P(h)+P(\hat{h}\mid i)P(i)+P(\hat{h}\mid d)P(d)}.$ (4)

The likelihoods of all underlying health states of a tree given its assessed
health can be similarly computed (there are nine of these probabilities in
total) and will be pivotal in defining the utility functions of the
municipality and tree owner.

### Impact of treatment

The effectiveness of treatment with insecticides on maintaining or recovering
a tree’s health depends on the underlying state of the tree. For a tree with a
healthy or infested state, the probabilities of the tree staying or becoming
healthy depends on the treatment choice. We have the probabilities

$\displaystyle P(h\mid t\cap\phi)$ $\displaystyle=h_{t\phi}$ $\displaystyle
P(h\mid u\cap\phi)$ $\displaystyle=h_{u\phi}$

which define the likelihood of a tree remaining (becoming) healthy, $h$, given
the treatment decision ($t:$ treated, $u:$ untreated) and the underlying
infestation state of the tree, $\phi\in\\{h,i,d\\}$. If a tree is already
dying or dead, we assume it cannot be saved, and therefore $P(h\mid u\cap
d)=P(h\mid t\cap d)=0$. We assume that following the decision to treat or not,
a tree will either remain (become) healthy or die. In other words, we ignore
the outcome of a tree staying in an infested state, as this is thought to be a
transient state that either results in tree death or recovery. In our
analyses, we find that the benefit of treatment to a tree owner and the
municipality is driven by the increase in the likelihood of a tree staying
(becoming) healthy in response to treatment, which we define with the
quantities

$\displaystyle t_{h}$ $\displaystyle=h_{th}-h_{uh}$ $\displaystyle t_{i}$
$\displaystyle=h_{ti}-h_{ui}$ $\displaystyle t_{d}$ $\displaystyle=0.$

### Value of treatment

The tree owner can break their payoff down into six cases: treatment or not,
given an assessment that a tree is healthy, infested or dying. We assume that
the value of a healthy tree to its owner is $V_{o}$ and that the cost
associated with a dead / dying tree is $W_{o}$ for the owner. Together with
the likelihoods of becoming (staying) healthy given a treatment decision and
the state of the tree, the value of a treated or untreated tree to its owner
is

$\displaystyle\pi_{t\hat{\phi}}$
$\displaystyle=P(h\mid\hat{\phi})\left(h_{th}V_{o}-\left(1-h_{th}\right)W_{o}\right)+P(i\mid\hat{\phi})\left(h_{ti}V_{o}-\left(1-h_{ti}\right)W_{o}\right)-P(d\mid\hat{\phi})W_{o}$
$\displaystyle\pi_{u\hat{\phi}}$
$\displaystyle=P(h\mid\hat{\phi})\left(h_{uh}V_{o}-\left(1-h_{uh}\right)W_{o}\right)+P(i\mid\hat{\phi})\left(h_{ui}V_{o}-\left(1-h_{ui}\right)W_{o}\right)-P(d\mid\hat{\phi})W_{o}$

for some assessed state $\hat{\phi}$. We find that the private net benefit of
saving a single tree that would otherwise die,

$\Delta_{o}=V_{o}+W_{o},$

shapes the benefit of treatment from the perspective of a tree owner. We
define the benefit of treatment from the owners perspective as the difference
in payoff between treatment and non-treatment for a tree with a particular
assessed state. For an arbitrary assessed state, the benefit of treatment is

$\displaystyle b_{t\hat{\phi}}$
$\displaystyle=\pi_{t\hat{\phi}}-\pi_{u\hat{\phi}}$
$\displaystyle=(V_{o}+W_{o})\left(P(i\mid\hat{\phi})(P(h\mid t\cap i)-P(h\mid
u\cap i))+P(h\mid\hat{\phi})(P(h\mid t\cap h)-P(h\mid u\cap h))\right)$
$\displaystyle=\Delta_{o}\left(P(i\mid\hat{\phi})(h_{ti}-h_{ui})+P(h\mid\hat{\phi})(h_{th}-h_{uh})\right)$
$\displaystyle=\Delta_{o}\left(P(i\mid\hat{\phi})t_{i}+P(h\mid\hat{\phi})t_{h}\right)$
$\displaystyle=\Delta_{o}k_{\hat{\phi}}$

where

$k_{\hat{\phi}}=P(i\mid\hat{\phi})t_{i}+P(h\mid\hat{\phi})t_{h}$ (5)

is the probability that treating a tree assessed in a particular state will
save it and $\Delta_{o}$ is the value to the tree owner of saving the tree.
This can be applied to the three assessed states, resulting in benefits of
treatment of

$\displaystyle b_{t\hat{h}}$ $\displaystyle=\Delta_{o}k_{\hat{h}},$ (6)
$\displaystyle b_{t\hat{i}}$ $\displaystyle=\Delta_{o}k_{\hat{i}},$ (7)
$\displaystyle b_{t\hat{d}}$ $\displaystyle=\Delta_{o}k_{\hat{d}},$ (8)

which shows that while the value of saving a tree remains the same across the
three cases, the likelihood that treatment will save a tree assessed in each
state can vary. As a result, the benefit of treatment depends of the assessed
state of the tree.

Since a tree provides benefits to communities and society as a whole, whereas
the costs of care are primarily shouldered by the tree owner, it is likely
that a tree owner’s valuation will not align with the value of the tree in the
eyes of a socially-minded municipality. Consequently, we define the social
value of a healthy tree as $V_{m}$ and the social cost of a dying tree as
$W_{m}$. From the perspective of the municipality, the net benefit of saving a
single tree is thus

$\Delta_{m}=V_{m}+W_{m}.$

These values and costs, together with the treatment effectiveness, assessment
accuracy, and community level infestation rates can be used to construct the
social payoffs of treatment or non-treatment of a tree, with

$\displaystyle\Pi_{t\hat{\phi}}$
$\displaystyle=P(h\mid\hat{\phi})\left(h_{th}V_{m}-\left(1-h_{th}\right)W_{m}\right)+P(i\mid\hat{\phi})\left(h_{ti}V_{m}-\left(1-h_{ti}\right)W_{m}\right)-P(d\mid\hat{\phi})W_{m}$
$\displaystyle\Pi_{u\hat{\phi}}$
$\displaystyle=P(h\mid\hat{\phi})\left(h_{uh}V_{m}-\left(1-h_{uh}\right)W_{m}\right)+P(i\mid\hat{\phi})\left(h_{ui}V_{m}-\left(1-h_{ui}\right)W_{m}\right)-P(d\mid\hat{\phi})W_{m}$

defining the social value of a treated and untreated tree that is assessed to
be in some state $\hat{\phi}$. The benefit of treatment from the socially
oriented perspective of the municipality, given the assessment of the tree’s
health state can be written as

$\displaystyle B_{t\hat{h}}$
$\displaystyle=\Pi_{t\hat{h}}-\Pi_{u\hat{h}}=\Delta_{m}k_{\hat{h}}$ (9)
$\displaystyle B_{t\hat{i}}$
$\displaystyle=\Pi_{t\hat{i}}-\Pi_{u\hat{i}}=\Delta_{m}k_{\hat{i}}$ (10)
$\displaystyle B_{t\hat{d}}$
$\displaystyle=\Pi_{t\hat{d}}-\Pi_{u\hat{d}}=\Delta_{m}k_{\hat{d}}$ (11)

for the three possible health assessment outcomes. These equations show that
private and municipal values of treatment across the scenarios share
$k_{\hat{\phi}}$ which describes the effectiveness of treatment given the
information that is available. However, $B_{t\hat{\phi}}$ and
$b_{t\hat{\phi}}$ will not be equal to each other when
$\Delta_{m}\neq\Delta_{o}$.

### Utility functions

Given the value of treated and untreated trees, we construct the utility
function for the municipality. We assume that subsidies, $s_{1\hat{\phi}}\geq
0$ and $s_{2\hat{\phi}}\geq 0$ are paid to firms for treatment at a cost to
the municipality. This leads to a utility function for the municipality given
by

$U_{m\hat{\phi}}=\begin{cases}\Pi_{t\hat{\phi}}-s_{1\hat{\phi}},&\text{if tree
assessed as $\hat{\phi}$ is treated by firm 1}\\\
\Pi_{t\hat{\phi}}-s_{2\hat{\phi}},&\text{if tree assessed as $\hat{\phi}$ is
treated by firm 2}\\\ \Pi_{u\hat{\phi}}&\text{if tree assessed as $\hat{\phi}$
is untreated}\end{cases}$ (12)

The utility function for firm 1 depends on the the cost of administering the
treatment, $c$, the price they bid, $p_{1\hat{\phi}}$ and the level of the
subsidy that they receive, $s_{1\hat{\phi}}$ if their bid is selected by the
tree owner. This leads to a utility function given by

$U_{f_{1}\hat{\phi}}=\begin{cases}p_{1\hat{\phi}}+s_{1\hat{\phi}}-c,&\text{if
firm 1 treats tree}\\\ 0,&\text{if firm 1 does not}\end{cases}$ (13)

for firm 1. A similar function holds for firm 2, since we assume that the
costs for each firm are the same. This leads to a utility function of

$U_{f_{2}\hat{\phi}}=\begin{cases}p_{2\hat{\phi}}+s_{2\hat{\phi}}-c,&\text{if
firm 2 treats tree}\\\ 0,&\text{if firm 2 does not.}\end{cases}$ (14)

Finally, we consider the utility of a private tree owner, which will depend on
their valuation of the tree. These valuations can be used to construct the
utility function of the tree owner as

$U_{o\hat{\phi}}=\begin{cases}\pi_{t\hat{\phi}}-p_{1\hat{\phi}},&\text{if firm
1 treats the tree}\\\ \pi_{t\hat{\phi}}-p_{2\hat{\phi}},&\text{if firm 2
treats the tree}\\\ \pi_{u\hat{\phi}},&\text{if the tree is
untreated}\end{cases}$ (15)

where the owners benefit from their private valuation of a treated or
untreated tree, and if the tree is treated, they bear the cost of treatment.
We use these utility functions to derive the optimal course of action for each
player assuming that they are aiming to maximize their (expected) utility.

### Game structure

Once a tree has had its health status assessed, we subsequently consider a
game in three stages. First, the municipality chooses subsidy levels,
$s_{1\hat{\phi}}$ and $s_{2\hat{\phi}}$ that are available to firm 1,
($f_{1}$) and firm 2, ($f_{2}$) for the treatment of a tree assessed in state
$\hat{\phi}$. Next, the tree firms simultaneously submit bids to the tree
owner for the price they will charge for treating the tree, with the bids of
firm 1 and 2 being $p_{1\hat{\phi}}$ and $p_{2\hat{\phi}}$, respectively.
Finally, the tree owner decides whether to accept the bid of either firm, or
opt not to treat their tree. While the tree owner knows that their value of
treatment given the assessed status of the tree is
$b_{t\hat{\phi}}=\Delta_{o}k_{\hat{\phi}}$, the firms and municipality only
know that $\Delta_{o}$ is drawn from a uniform distribution between $a$ and
$b$, denoted

$\Delta_{o}\sim U[a,b].$ (16)

The strategy space of the municipality is defined as the set of subsidy level
pairs that they offer, given that subsidy levels are are restricted to be non-
negative. This can be written as

$\sigma_{m\hat{\phi}}=\\{(s_{1\hat{\phi}},s_{2\hat{\phi}})\mid
s_{1\hat{\phi}}\geq 0,s_{2\hat{\phi}}\geq 0\\}.$ (17)

The strategy space of firm 1 is the price that they offer for treatment of the
tree,

$\sigma_{f_{1}\hat{\phi}}=\\{p_{1\hat{\phi}}\mid p_{1\hat{\phi}}\geq 0\\},$
(18)

with a similar strategy space for firm 2, given by

$\sigma_{f_{2}\hat{\phi}}=\\{p_{2\hat{\phi}}\mid p_{2\hat{\phi}}\geq 0\\}.$
(19)

Finally, the strategy space of the tree owner is given by a mapping between
their type, $\Delta_{o}$, which is drawn from a uniform distribution and the
probability that they choose any of their actions

$A_{o\hat{\phi}}=\\{\text{treat w/}f_{1},\text{treat
w/}f_{2},\text{untreated}\\}.$ (20)

We denote a mapping that forms their strategy as

$\sigma_{o\hat{\phi}}:[a,b]\to P(A_{o\hat{\phi}})$ (21)

so that a strategy $\sigma_{o\hat{\phi}}$ will assign a probability
distribution over the set $A_{o\hat{\phi}}$ for every possible value of
$\Delta_{o}\in[a,b]$.

## Analysis of game theoretic model

The analysis of the game will proceed via backward induction, starting with
the tree owner’s optimal strategy for responding to an arbitrary pair of bids.
Next, we consider the simultaneous bids that firms will submit to
independently maximize their expected utility, given the subsidy levels that
they would receive if they treat the tree and their beliefs about what the
tree owner will do in response to their bids. Finally, we consider the optimal
subsidy policy of the municipality. We assume that the municipality maximizes
their expected utility, given the bidding behavior they expect from the firms
and their beliefs about how the tree owner will respond to the bids. In total,
this will yield strategies for all the players that satisfy the conditions of
a Perfect Bayesian Equilibrium, which implies that no player will have an
incentive to unilaterally deviate from their strategy.

We begin from the perspective of a tree owner who knows the assessed state of
their tree is $\hat{\phi}\in\\{\hat{h},\hat{i},\hat{d}\\}$. They
simultaneously receive a pair of bids from the tree care firms for treatment
of the tree, $(p_{1\hat{\phi}},p_{2\hat{\phi}})$. When the price of the lowest
bid is less than the benefit of treatment, then the tree owner will maximize
their utility by having that bidder treat their tree. Any type of tree owner
with $\Delta_{o}\in[a,b]$ will maximize their utility by following the actions
defined by the optimal strategy mapping

$\sigma_{o\hat{\phi}}^{*}\left[\Delta_{o}\right]=\begin{cases}\text{treat
w/}f_{1}&\text{if }p_{1\hat{\phi}}<p_{2\hat{\phi}}\text{ and
}p_{1\hat{\phi}}\leq\Delta_{o}k_{\hat{\phi}}\\\ \text{treat w/}f_{2}&\text{if
}p_{2\hat{\phi}}<p_{1\hat{\phi}}\text{ and
}p_{2\hat{\phi}}\leq\Delta_{o}k_{\hat{\phi}}\\\ \text{flip coin}&\text{if
}p_{1\hat{\phi}}=p_{2\hat{\phi}}\text{ and
}p_{2\hat{\phi}}\leq\Delta_{o}k_{\hat{\phi}}\\\ \text{don't treat}&\text{if
}\Delta_{o}k_{\hat{\phi}}<\text{min}\\{p_{1\hat{\phi}},p_{2\hat{\phi}}\\}\end{cases}$
(22)

where the ‘flip coin’ action corresponds to the case where the tree owner
treats their tree but randomly selects one of the firms. This strategy assumes
that the tree owner knows their own valuation of saving a tree, $\Delta_{o}$,
and also knows the impact that treatment will have, $k_{\hat{\phi}}$, given
the assessment of the tree’s health. Together, these define the benefit of
treatment of a tree, $b_{t\hat{\phi}}=\Delta_{o}k_{\hat{\phi}}$, and only when
this benefit is at least as great as the cost of the lowest bid will the owner
choose to have their tree treated.

Given this strategy is followed by the tree owner, we can now consider the
bidding strategies that the firms will follow if they both independently
maximize their expected utility. This resembles Bertrand competition because
the firms simultaneously offer bids to the tree owner (Bertrand,, 1883). We
assume that the firms consider their beliefs about the value of treatment to
the tree owner that result from the assessed state of the tree and the level
of subsidy offered to their firm for treatment. Given this information, we
derive the best response of each firm to an arbitrary bid by the opposing
firm. Pairs of bids that are mutual best responses will be Bayesian Nash
equilibria for this stage of the game.

The utility functions of the firms have the same structure, so we focus on the
expected utility function of firm 1. Expected utility depends on how
profitable treatment is if the firm wins the bid and how likely it is that
they win the bid. This leads to

$E[U_{f_{1}}]=\left(p_{1\hat{\phi}}+s_{1\hat{\phi}}-c\right)P(f_{1}\text{ wins
bid})$ (23)

for the expected utility of firm 1 which can be further broken down by
defining the probability function of firm 1 winning the bid. The probability
of winning the bid depends on the bid of the other firm and the beliefs of the
firm about the benefit of treatment to the tree owner.

There are several cases for this probability function. First, if
$p_{1\hat{\phi}}>p_{2\hat{\phi}}$ then firm 1 will never win the bid. If
$p_{1\hat{\phi}}<p_{2\hat{\phi}}$ then firm 1 will win the bid if
$p_{1\hat{\phi}}<b_{t\hat{\phi}}$. If the two bids are equal, we assume for
simplicity that the odds that firm 1 wins the bid are half of what they would
be if they had the lowest bid. Recalling that it is common knowledge that the
value of saving a tree to its owner, $\Delta_{o}$, is drawn from a uniform
distribution, then the firm’s expectations about the distribution of the
benefit of treatment to the tree owner, given the assessed health of tree is

$b_{t\hat{\phi}}=\Delta_{o}k_{\hat{\phi}}\sim
U[ak_{\hat{\phi}},bk_{\hat{\phi}}]$ (24)

which can be used to calculate the probability that the firm’s bid would be
low enough to have a chance of being accepted. This probability function is

$P(p_{1\hat{\phi}}\leq b_{t\hat{\phi}})=\begin{cases}1&p_{1\hat{\phi}}\leq
ak_{\hat{\phi}}\\\
\frac{bk_{\hat{\phi}}-p_{1\hat{\phi}}}{k_{\hat{\phi}}(b-a)}&ak_{\hat{\phi}}<p_{1\hat{\phi}}<bk_{\hat{\phi}}\\\
0&bk_{\hat{\phi}}\leq p_{1\hat{\phi}}.\end{cases}$ (25)

This can be used to construct the expected utility function of firm 1, which
is

$E[U_{f_{1}}]=\begin{cases}p_{1\hat{\phi}}+s_{1\hat{\phi}}-c&p_{1\hat{\phi}}\leq
ak_{\hat{\phi}}\text{ and }p_{1\hat{\phi}}<p_{2\hat{\phi}}\\\
\left(p_{1\hat{\phi}}+s_{1\hat{\phi}}-c\right)\frac{bk_{\hat{\phi}}-p_{1\hat{\phi}}}{k_{\hat{\phi}}(b-a)}&ak_{\hat{\phi}}<p_{1\hat{\phi}}<bk_{\hat{\phi}}\text{
and }p_{1\hat{\phi}}<p_{2\hat{\phi}}\\\
1/2\left(p_{1\hat{\phi}}+s_{1\hat{\phi}}-c\right)&p_{1\hat{\phi}}\leq
ak_{\hat{\phi}}\text{ and }p_{1\hat{\phi}}=p_{2\hat{\phi}}\\\
1/2\left(p_{1\hat{\phi}}+s_{1\hat{\phi}}-c\right)\frac{bk_{\hat{\phi}}-p_{1\hat{\phi}}}{k_{\hat{\phi}}(b-a)}&ak_{\hat{\phi}}<p_{1\hat{\phi}}<bk_{\hat{\phi}}\text{
and }p_{1\hat{\phi}}=p_{2\hat{\phi}}\\\ 0&bk_{\hat{\phi}}\leq
p_{1\hat{\phi}}\text{ or }p_{1\hat{\phi}}>p_{2\hat{\phi}}\end{cases}$ (26)

If firm 1 believes that they have a chance to profitably treat the tree and
are uncontested by firm 2, then they follow the bidding strategy given by

$p_{1\hat{\phi}}^{\prime}=\max\left\\{\frac{1}{2}\left(bk_{\hat{\phi}}+c-s_{1\hat{\phi}}\right),ak_{\hat{\phi}}\right\\}.$
(27)

We use this to write the best response as to any bid by firm 2 as

$\hat{p}_{1\hat{\phi}}\left(p_{2\hat{\phi}}\right)=\begin{cases}p_{1\hat{\phi}}\geq
c-s_{1\hat{\phi}}&s_{1\hat{\phi}}<c-bk_{\hat{\phi}}\text{ or
}s_{1\hat{\phi}}<c-p_{2\hat{\phi}}\\\
p_{2\hat{\phi}}-\epsilon&s_{1\hat{\phi}}\geq c-bk_{\hat{\phi}}\text{ and
}c-s_{1\hat{\phi}}\leq p_{2\hat{\phi}}\leq p_{1\hat{\phi}}^{\prime}\\\
p_{1\hat{\phi}}^{\prime}&s_{1\hat{\phi}}\geq c-bk_{\hat{\phi}}\text{ and
}p_{1\hat{\phi}}^{\prime}<p_{2\hat{\phi}}\end{cases}$ (28)

which shows that if the subsidy is so low that either firm 1 cannot profitably
win the bid or profitably undercut firm 2, then any large enough bid is a best
response. On the other hand, if the bid of firm 2 is not competitive, then
firm 1 will act like a monopolist and select their bid,
$p_{1\hat{\phi}}^{\prime}$ which aligns with the bidding policy we calculated
as $p_{1\hat{\phi}}^{*}$. For the intermediate cases, firm 1 will undercut the
bid of firm 2 to guarantee that they win the bid because winning the bid will
lead to positive profit, whereas losing the bid leads to nothing and bidding
equal to the other firm leads to half as much profit. In these cases, expected
profit decreases as their bid price decreases because the loss of profit of
treating a tree at a lower price is not offset by the increased likelihood
that the tree owner will accept the bid. Therefore the firm will seek to
minimally undercut its competitor. (See S.I. 2.2 for a detailed derivation).

We focused on the best response of firm 1 to an arbitrary bid of firm 2.
However, since the firm’s have the symmetric utility functions and therefore
the best response of firm 2 will exactly mirror that of firm 1 with subscripts
switched, given by

$\hat{p}_{2\hat{\phi}}\left(p_{1\hat{\phi}}\right)=\begin{cases}p_{2\hat{\phi}}\geq
c-s_{2\hat{\phi}}&s_{2\hat{\phi}}<c-bk_{\hat{\phi}}\text{ or
}s_{2\hat{\phi}}<c-p_{1\hat{\phi}}\\\
p_{1\hat{\phi}}-\epsilon&s_{2\hat{\phi}}\geq c-bk_{\hat{\phi}}\text{ and
}c-s_{2\hat{\phi}}\leq p_{1\hat{\phi}}\leq p_{2\hat{\phi}}^{\prime}\\\
p_{2\hat{\phi}}^{\prime}&s_{2\hat{\phi}}\geq c-bk_{\hat{\phi}}\text{ and
}p_{2\hat{\phi}}^{\prime}<p_{1\hat{\phi}}\\\ \end{cases}$ (29)

where we define

$p_{2\hat{\phi}}^{\prime}=\max\left\\{\frac{1}{2}\left(bk_{\hat{\phi}}+c-s_{2\hat{\phi}}\right),ak_{\hat{\phi}}\right\\}$
(30)

as the price the firm 2 would bid if they have a chance to profitably treat
the tree and expect the other firm not to submit a competitive bid.
Intersections of these best responses in the
$(p_{1\hat{\phi}},p_{2\hat{\phi}})$ plane correspond to a Bayesian Nash
equilibrium of the this stage of the game. Figure 2 illustrates this for a
case with equal subsidies available to each firm,
$s_{\hat{\phi}}=s_{1\hat{\phi}}=s_{2\hat{\phi}}$. In such cases, the
equilibrium bids follow a simple pattern and are

$\left(p_{1\hat{\phi}}^{*},p_{2\hat{\phi}}^{*}\right)=\begin{cases}(c-s_{\hat{\phi}},c-s_{\hat{\phi}})&s_{\hat{\phi}}\geq
c-bk_{\hat{\phi}}\\\ (p_{1\hat{\phi}}\geq c-s_{\hat{\phi}},p_{2\hat{\phi}}\geq
c-s_{\hat{\phi}})&s_{\hat{\phi}}<c-bk_{\hat{\phi}}\end{cases}$ (31)

with the details of the second case being irrelevant to outcomes because these
bids will never lead a tree owner to opt for treatment. When subsidies for the
two firms are different, then a broad range of possible types of equilibria
are possible. Some of these possibilities are illustrated in figure 3.

Figure 2: The best response of firm 1, to an arbitrary price set by firm 2,
$\hat{p}_{1}(p_{2})$, is plotted along with the best response of firm 2 to an
arbitrary price set by firm 1, $\hat{p}_{2}(p_{1})$. Nash equilibria occur at
intersections of the best response relations. The willingness to pay for
treatment of a tree by its owners is uniformly distributed between
$ak_{\hat{\phi}}$ and $bk_{\hat{\phi}}$ and if the low bid by one of the firms
is less than the realized willingness to pay then the tree owner will accept
that bid. Each firm receives a subsidy, $s_{\hat{\phi}}$ when they treat a
tree assessed in health state $\hat{\phi}$. In the absence of competition,
each firm would act as a monopolist and charge a price $p^{\prime}$ for
treatment. Also, neither firm will bid less than $c-s_{\hat{\phi}}$ because
$c$ is the marginal cost of treatment for the firm. If they bid less than
their subsidized marginal cost they would be worse off treating the tree. The
shaded regions are those regions where a firm is indifferent among all bids in
the region. In this case the shaded regions do not intersect and therefore
bids in these regions will not be part of a Nash equilibrium. For intermediate
bids of their competitor, a firm has an incentive to undercut the other bid.
Putting this together allows one to construct the best response relations for
each firm and show that, in this case, there is a unique Nash equilibrium:
these best response relations only intersect at only one point, marked on this
plot. ($ak_{\hat{\phi}}=1.5$, $bk_{\hat{\phi}}=7$ $c=10$,
$s_{1\hat{\phi}}=s_{2\hat{\phi}}=s_{\hat{\phi}}=7$)

(a) Firm 2’s bid undercuts firm 1 ($ak_{\hat{\phi}}=3$, $bk_{\hat{\phi}}=7$
$c=10$, $s_{1\hat{\phi}}=6$, $s_{2\hat{\phi}}=8$)

(b) Firm 1 acts as a monopolist ($ak_{\hat{\phi}}=3$, $bk_{\hat{\phi}}=7$
$c=10$, $s_{1\hat{\phi}}=5.5$, $s_{2\hat{\phi}}=1.5$)

(c) A range of equilibrium bids ($ak_{\hat{\phi}}=3$, $bk_{\hat{\phi}}=6$
$c=10$, $s_{1\hat{\phi}}=3$, $s_{2\hat{\phi}}=1.5$)

Figure 3: When the municipality sets unequal subsidy levels for the two firms,
a much broader range of equilibrium bid pairs and outcomes can arise. Three
such cases are illustrated here. First, we show a case where firm 2 receives a
higher subsidy than firm 1 and bids a price that marginally undercuts the
lowest bid that firm 1 can make without losing money on treating a tree. The
second case corresponds to an outcome where the subsidy level of firm 2 is
significantly lower than firm 1. In this case, firm 1 can bid as if they are a
monopolist. Finally, we illustrate a case where neither firm receives a
subsidy high enough to win a bid and make a profit. In this case the firms are
largely indifferent about their bidding, and the tree is never treated. While
the outcomes under unequal subsidies are more varied than those that arise
under equal subsidy levels, we show that it is never in a municipality’s
interest to choose unequal subsidy levels for the two firms.

When the municipality sets subsidy levels for the two firms, we assume that
they do so to maximize the expected value of their utility function under the
assumption that subsequent players will follow their optimal strategies
outlined above. In supplementary information section S.I. 2.3 we show that the
municipality never finds it advantageous to set different subsidies for each
firm. Therefore we will focus on the cases with equal subsidy levels for the
firms. This results in the following expected utility function for the
municipality:

$E[U_{m}]=\Pi_{u\hat{\phi}}+\left(B_{t\hat{\phi}}-s_{\hat{\phi}}\right)P(\text{treated})$
(32)

where the final term is the probability that the tree will be treated, given
the subsidies the municipality offers. This probability of treatment depends
on the valuations of trees by their owners and the bids of the firms in
response to the subsidy level. This probability can be broken down as

$P(\text{treated})=P\left(c-s_{\hat{\phi}}\leq
b_{t\hat{\phi}}\right)=\begin{cases}1&c-ak_{\hat{\phi}}\leq s_{\hat{\phi}}\\\
\frac{bk_{\hat{\phi}}-c+s_{\hat{\phi}}}{k_{\hat{\phi}}(b-a)}&c-bk_{\hat{\phi}}<s_{\hat{\phi}}<c-ak_{\hat{\phi}}\\\
0&s_{\hat{\phi}}\leq c-bk_{\hat{\phi}}\end{cases}$ (33)

With this, we can rewrite the municipality’s expected utility function as

$E[U_{m}]=\begin{cases}\Pi_{u\hat{\phi}}+B_{t\hat{\phi}}-s_{\hat{\phi}}&c-ak_{\hat{\phi}}\leq
s_{\hat{\phi}}\\\
\Pi_{u\hat{\phi}}+\left(B_{t\hat{\phi}}-s_{\hat{\phi}}\right)\frac{bk_{\hat{\phi}}-c+s_{\hat{\phi}}}{k_{\hat{\phi}}(b-a)}&c-bk_{\hat{\phi}}<s_{\hat{\phi}}<c-ak_{\hat{\phi}}\\\
\Pi_{u\hat{\phi}}&s_{\hat{\phi}}\leq c-bk_{\hat{\phi}}\end{cases}$ (34)

where the first case corresponds to a subsidy level which is high enough to
induce all tree owners to treat their tree. The second corresponds to the case
where there is a positive probability that the tree will be treated and the
final case is when subsidies are set to a level which is too low to induce
treatment by any tree owners. Figure 4 shows how expected utility depends of
the subsidy choice of the municipality across three representative cases.
Expected utility is maximized at peaks of these curves. In general, utility
maximizing behavior by the municipality leads to the following optimal subsidy
policy

$s^{*}_{\hat{\phi}}=\begin{cases}0&c\leq ak_{\hat{\phi}}\\\
c-ak_{\hat{\phi}}&c>ak_{\hat{\phi}}\;\text{ and
}\;c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\leq B_{t\hat{\phi}}\\\
1/2\left(B_{t\hat{\phi}}+c-bk_{\hat{\phi}}\right)&c>ak_{\hat{\phi}}\;\text{
and }\;\mid
c-bk_{\hat{\phi}}\mid<B_{t\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\\\
0&c>ak_{\hat{\phi}}\;\text{ and }\;B_{t\hat{\phi}}\leq\mid
c-bk_{\hat{\phi}}\mid\end{cases}$ (35)

which shows that positive subsidies are only used when they will increase
treatment of trees, and only when the resulting treatments increase the
welfare of society (for a complete analysis, see S.I. 2.3). This will lead to
different subsidy policies for each assessed tree state. The exact policies
for each tree state will depend on the prevalence of the disease, the accuracy
of assessment and the effectiveness of treatment at saving trees that
otherwise would die. We will explore several case studies in a later section.

Figure 4: The expected utility of the municipality is shown as a function of
the subsidy level, $s_{\hat{\phi}}$, that they set for a tree assessed in
health state $\hat{\phi}$ for both firms. When the subsidy level chosen by the
municipality $s_{\hat{\phi}}<c-bk_{\hat{\phi}}$, then there is no chance that
a tree in assessed in state $\hat{\phi}$ will be treated. When
$s_{\hat{\phi}}\geq c-ak_{\hat{\phi}}$ then a focal tree will be treated with
probability 1. Here utility curves are shown for three different social
benefit values, $B_{t\hat{\phi}}$, of treating a tree assessed in state
$\hat{\phi}$. For low levels of social benefit from treating a tree, the
municipality has its highest expected utility when subsidies do not lead to
treatment of a tree. For intermediate social value of treatment, expected
municipal utility is maximized at an intermediate subsidy level that leads to
a positive probability of treatment. For high social values of treatment,
expected utility is maximized when the subsidy is set just high enough to
guarantee that a tree assessed in state $\hat{\phi}$ will be treated by its
owner. Subsidy levels in excess of $c-ak_{\hat{\phi}}$ lead to declining
expected utility in all cases. ($c=10$, $ak_{\hat{\phi}}=7$,
$bk_{\hat{\phi}}=9$, $\Pi_{u\hat{\phi}}=4$)

## Equilibrium game outcomes

Here we explore the outcomes that results from owner, firms, and
municipalities adhering to the optimal behaviors identified in preceding
sections. Given the subsidy policy in equation 35, the firms’ equilibrium bids
will be

$p_{1\hat{\phi}}^{*}=p_{2\hat{\phi}}^{*}=\begin{cases}c&c\leq
ak_{\hat{\phi}}\\\ ak_{\hat{\phi}}&c>ak_{\hat{\phi}}\;\text{ and
}\;c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\leq B_{t\hat{\phi}}\\\
1/2\left(c+bk_{\hat{\phi}}-B_{t\hat{\phi}}\right)&c>ak_{\hat{\phi}}\;\text{
and }\;\lvert
c-bk_{\hat{\phi}}\rvert<B_{t\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\\\
c&bk_{\hat{\phi}}>c>ak_{\hat{\phi}}\;\text{ and }\;B_{t\hat{\phi}}\leq
bk_{\hat{\phi}}-c\\\ p\geq c&c>bk_{\hat{\phi}}\;\text{ and
}\;B_{t\hat{\phi}}\leq c-bk_{\hat{\phi}}\end{cases}$ (36)

These bids will result in a likelihood of treatment of a focal tree being

$P^{*}(\text{treated})=\begin{cases}1&c\leq ak_{\hat{\phi}}\;\text{ or
}\;c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\leq B_{t\hat{\phi}}\\\
\frac{B_{t\hat{\phi}}-c+bk_{\hat{\phi}}}{2k_{\hat{\phi}}(b-a)}&c>ak_{\hat{\phi}}\;\text{
and }\;\mid
c-bk_{\hat{\phi}}\mid<B_{t\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\\\
\frac{bk_{\hat{\phi}}-c}{k_{\hat{\phi}}(b-a)}&bk_{\hat{\phi}}>c>ak_{\hat{\phi}}\;\text{
and }\;B_{t\hat{\phi}}\leq bk_{\hat{\phi}}-c\\\ 0&c\geq
bk_{\hat{\phi}}\;\text{ and }\;B_{t\hat{\phi}}\leq
c-bk_{\hat{\phi}}\end{cases}$ (37)

which leads to expected utility for the municipality of

$E[U_{m}^{*}]=\begin{cases}\Pi_{u\hat{\phi}}+B_{t\hat{\phi}}&c\leq
ak_{\hat{\phi}}\\\
\Pi_{u\hat{\phi}}+B_{t\hat{\phi}}-c+ak_{\hat{\phi}}&c>ak_{\hat{\phi}}\;\text{
and }\;c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\leq B_{t\hat{\phi}}\\\
\Pi_{u\hat{\phi}}+\frac{\left(B_{t\hat{\phi}}-c+bk_{\hat{\phi}}\right)^{2}}{4k_{\hat{\phi}}(b-a)}&c>ak_{\hat{\phi}}\;\text{
and }\;\mid
c-bk_{\hat{\phi}}\mid<B_{t\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\\\
\Pi_{u\hat{\phi}}+B_{t\hat{\phi}}\frac{bk_{\hat{\phi}}-c}{k_{\hat{\phi}}(b-a)}&bk_{\hat{\phi}}>c>ak_{\hat{\phi}}\;\text{
and }\;B_{t\hat{\phi}}\leq bk_{\hat{\phi}}-c\\\ \Pi_{u\hat{\phi}}&c\geq
bk_{\hat{\phi}}\;\text{ and }\;B_{t\hat{\phi}}\leq
c-bk_{\hat{\phi}}\end{cases}$ (38)

and an expected utility for the firms of

$E\left[U_{f_{1}}^{*}\right]=E\left[U_{f_{2}}^{*}\right]=0$ (39)

because under all cases where treatment can happen, the firms bid
competitively and receive 0 profit.

A representative tree owner knows their own valuation of their tree.
Therefore, rather than consider their expected utility, we can instead
calculate their utility as a function of their true underlying valuation of
the tree, $\Delta_{o}$ that other players in the game know is drawn from a
uniform distribution. Recalling that
$b_{t\hat{\phi}}=\Delta_{o}k_{\hat{\phi}}\in[ak_{\hat{\phi}},bk_{\hat{\phi}}]$
and $B_{t\hat{\phi}}=\Delta_{m}k_{\hat{\phi}}$, this results in

$U_{o}^{*}[\Delta_{o}]=\begin{cases}\pi_{u\hat{\phi}}+b_{t\hat{\phi}}-c&c\leq
ak_{\hat{\phi}}\\\
\pi_{u\hat{\phi}}+b_{t\hat{\phi}}-ak_{\hat{\phi}}&c>ak_{\hat{\phi}}\;\text{
and }\;c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\leq B_{t\hat{\phi}}\\\
\pi_{u\hat{\phi}}+b_{t\hat{\phi}}-\frac{1}{2}\left(c+bk_{\hat{\phi}}-B_{t\hat{\phi}}\right)&c>ak_{\hat{\phi}}\;\text{
and }\;\mid
c-bk_{\hat{\phi}}\mid<B_{t\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\\\
&\hskip 40.0pt\text{ and
}\;\Delta_{o}\geq\frac{1}{2}\left[c/k_{\hat{\phi}}+b-\Delta_{m}\right]\\\
\pi_{u\hat{\phi}}&c>ak_{\hat{\phi}}\;\text{ and }\;\mid
c-bk_{\hat{\phi}}\mid<B_{t\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\\\
&\hskip 40.0pt\text{ and
}\;\Delta_{o}<\frac{1}{2}\left[c/k_{\hat{\phi}}+b-\Delta_{m}\right]\\\
\pi_{u\hat{\phi}}+b_{t\hat{\phi}}-c&bk_{\hat{\phi}}>c>ak_{\hat{\phi}}\;\text{
and }\;B_{t\hat{\phi}}\leq bk_{\hat{\phi}}-c\;\text{ and }\;\Delta_{o}\geq
c/k_{\hat{\phi}}\\\
\pi_{u\hat{\phi}}&bk_{\hat{\phi}}>c>ak_{\hat{\phi}}\;\text{ and
}\;B_{t\hat{\phi}}\leq bk_{\hat{\phi}}-c\;\text{ and
}\;\Delta_{o}<c/k_{\hat{\phi}}\\\ \pi_{u\hat{\phi}}&c\geq
bk_{\hat{\phi}}\;\text{ and }\;B_{t\hat{\phi}}\leq
c-bk_{\hat{\phi}}\end{cases}$ (40)

where the top case assumes the cost of treatment is low enough that all types
of tree owners will treat their trees in the absence of subsidies, the second
case arises when the social benefit of treatment is great enough that the
municipality chooses a subsidy level to assure that all types of tree owners
will treat their trees. The next two cases occur when the municipality’s best
option is an intermediate subsidy that leads tree owners with high valuations
of their tree to treat and low valuations not to treat. This cutoff is
calculated in terms of the tree owner’s true valuation of saving one tree,
$\Delta_{o}$. The next two cases arise when the municipality chooses a subsidy
of zero, but there are still some types of tree owners that will treat their
tree. The final case occurs when the social benefit of treatment does not
justify a positive subsidy level, and the private benefit is small enough that
no types of tree owners opt for treatment.

We have shown that optimal subsidies are always equal for both firms. Given
that the two firms are competing on price (Bertrand competition) even a
duopoly leads to a competitive market among the firms where the entire subsidy
is passed on to a tree owner in the form of lower prices. Given this, the
result that equal subsidies for all firms is most cost effective should
generalize to situations where more tree care firms are present. While we find
that subsidy levels should be equal for both firms, the optimal subsidy does
depend on the observed health status of the tree and the prevalence of EAB in
the community. This is because the likelihood of saving a tree with treatment
depends on its health status. Interestingly, the optimal subsidy policy
described by equation 35 shows that subsidies should not be applied in cases
where the benefit of treatment is very large (all types of private owners
would chose treatment anyway) or very small (treatment not cost effective for
society). It is in the intermediate cases that subsidy policies can be used to
boost treatment uptake and improve ecosystem service provision and benefit
society.

Variable | Definition | Description
---|---|---
$\phi\in\\{h,i,d\\}$ |  | Underlying tree state: healthy, infested, dead/dying
$\hat{\phi}\in\\{\hat{h},\hat{i},\hat{d}\\}$ |  | Assessed tree state: healthy, infested, dead/dying
$P(\phi)$ |  | Probability of a tree being in state $\phi$
$P(\hat{\phi}\mid\phi)$ |  | Probability of a tree in some state $\phi$ being assessed as $\hat{\phi}$
$P(\phi\mid\hat{\phi})$ | via Bayes Theorem | Probability of a tree being in state $\phi$ given that it was assessed as $\hat{\phi}$
$P(h\mid t\cap h)$ | $h_{th}$ | Probability that a healthy treated tree will remain healthy
$P(h\mid u\cap h)$ | $h_{uh}$ | Probability that a healthy untreated tree will remain healthy
$P(h\mid t\cap i)$ | $h_{ti}$ | Probability that an infested treated tree will become healthy
$P(h\mid u\cap i)$ | $h_{ui}$ | Probability that an infested untreated tree will become healthy
$P(h\mid t\cap d)$ | 0 | Probability that an dead/dying treated tree will become healthy
$P(h\mid u\cap d)$ | 0 | Probability that an dead/dying untreated tree will become healthy
$t_{h}$ | $h_{th}-h_{uh}$ | Marginal impact of treatment on health of a healthy tree
$t_{i}$ | $h_{ti}-h_{ui}$ | Marginal impact of treatment on health of an infested tree
$t_{d}$ | $0$ | Marginal impact of treatment on health of a dead/dying tree
$V_{o}$ |  | Value of a healthy tree to its owner
$V_{m}$ |  | Value of a healthy tree to society
$W_{o}$ |  | Cost of a dead/dying tree to its owner
$W_{m}$ |  | Cost of a dead/dying tree to society
$\Delta_{o}$ | $V_{o}+W_{o}\sim U[a,b]$ | Net private value of saving a tree is drawn from a uniform distribution
$\Delta_{m}$ | $V_{m}+W_{m}$ | Net social value of saving a tree
$\pi_{t\hat{\phi}}$ |  | Payoff to a tree owner of treating a tree assessed as $\hat{\phi}$
$\pi_{u\hat{\phi}}$ |  | Payoff to a tree owner of not treating a tree assessed as $\hat{\phi}$
$\Pi_{t\hat{\phi}}$ |  | Payoff to society of treating a tree assessed as $\hat{\phi}$
$\Pi_{u\hat{\phi}}$ |  | Payoff to society of not treating a tree assessed as $\hat{\phi}$
$k_{\hat{\phi}}$ | $P(i\mid\hat{\phi})t_{i}+P(h\mid\hat{\phi})t_{h}$ | Probability that treating a tree assessed in state $\hat{\phi}$ will save it
$b_{t\hat{\phi}}$ | $\pi_{t\hat{\phi}}-\pi_{u\hat{\phi}}=\Delta_{o}k_{\hat{\phi}}$ | Marginal benefit of treating a tree assessed in state $\hat{\phi}$ to its owner
$B_{t\hat{\phi}}$ | $\Pi_{t\hat{\phi}}-\Pi_{u\hat{\phi}}=\Delta_{m}k_{\hat{\phi}}$ | Marginal social benefit of treating a tree assessed in state $\hat{\phi}$
$c$ |  | Cost to a firm of treating a tree
$p_{j}$ |  | Bid price to treat a tree by firm $j$
$s_{j}$ |  | Treatment subsidy offered to firm $j$ (may depend on assessed tree state)
$U_{m}$ |  | Utility of municipality
$U_{f_{j}}$ |  | Utility of firm $j$
$U_{o}$ |  | Utility of tree owner
Table 1: Variables and parameters in the model, their definitions or values,
and meanings.

## Epidemiological model

Using the optimal subsidy policy for a representative privately owned tree
derived above, we will now build a dynamic epidemiological model to test the
impact of treatment policies on the dynamics of EAB within a community. With
this model we will integrate municipally owned trees as well as privately
owned trees. We assume that municipally and privately owned trees experience
distinct management practices, but the spread of EAB occurs across this mosaic
of public and private land. We will use the model to illustrate the
performance of the optimal subsidy and treatment policies for public and
private trees.

We project the dynamics of an ash population using a compartmental model of
disease dynamics (Kermack and McKendrick,, 1927). In these models, populations
are categorized into classes with distinct properties that influence the
spread of a pathogen. We study the dynamics of EAB spread through a population
of ash trees by subdividing the population into healthy, infested, and dying
classes for municipally owned and privately owned trees. Rates of transfer
between compartments describe EAB spread, and tree death or recovery. We model
a population of municipal ash trees divided into three classes described by
the fraction of healthy trees, $H_{m}$, infested trees, $I_{m}$, and dead or
dying trees, $D_{m}$ (we use capital letters to distinguish the population of
trees from individual tree states considered earlier). The municipally owned
trees interact with three classes of privately owned trees, $H_{o}$, $I_{o}$,
and $D_{o}$ ($o$ subscripts denote that the ownership of these classes of
trees is private as before). Since we measure the fraction of trees in each
category, we have $H_{m}+I_{m}+D_{m}+H_{o}+I_{o}+D_{o}=1$.

We denote the rate of change of the fraction of trees as $\dot{H}_{m}$ for
healthy municipally owned trees and follow similar notation for the rates of
change of other classes of trees. We describe the dynamics in EAB infestation
with the following system of equations

$\displaystyle\dot{H}_{m}=-\beta\left(1-h_{th}P_{thm}\right)H_{m}(I_{m}+I_{o})+\alpha\left(h_{ui}+t_{i}P_{tim}\right)I_{m}$
(41)
$\displaystyle\dot{H}_{o}=-\beta\left(1-h_{th}P_{tho}\right)H_{o}(I_{m}+I_{o})+\alpha\left(h_{ui}+t_{i}P_{tio}\right)I_{o}$
(42)
$\displaystyle\dot{I}_{m}=\beta\left(1-h_{th}P_{thm}\right)H_{m}(I_{m}+I_{o})-\gamma\left(1-h_{ui}-t_{i}P_{tim}\right)I_{m}-\alpha\left(h_{ui}+t_{i}P_{tim}\right)I_{m}$
(43)
$\displaystyle\dot{I}_{o}=\beta\left(1-h_{th}P_{tho}\right)H_{o}(I_{m}+I_{o})-\gamma\left(1-h_{ui}-t_{i}P_{tio}\right)I_{o}-\alpha\left(h_{ui}+t_{i}P_{tio}\right)I_{o}$
(44) $\displaystyle\dot{D}_{m}=\gamma\left(1-h_{ui}-t_{i}P_{tim}\right)I_{m}$
(45) $\displaystyle\dot{D}_{o}=\gamma\left(1-h_{ui}-t_{i}P_{tio}\right)I_{o}$
(46)

where $P_{tio}$, $P_{tho}$, $P_{tim}$, and $P_{thm}$ correspond to treatment
probabilities of trees that are infested or healthy and privately or
municipally owned. A complete derivation of the model structure can be found
in Section S.I. 5. In Figure 7, we show the simulation results of this model
under six different treatment scenarios that build on the results of the game
theoretic model developed earlier. As before, the $h_{th}$ term denotes the
odds that a healthy treated tree will remain healthy and $h_{ui}$ ($h_{ti}$)
describes the odds that an untreated (treated) infested tree recovers from
infestation. The marginal impact of treatment on the odds of an infested tree
becoming healthy remains $t_{i}=h_{ti}-h{ui}$, which measures the impact of
treatment on the likelihood of a tree staying ($t_{h}$) or becoming ($t_{i}$)
healthy. The propensity for EAB to spread in the absence of treatment is given
by $\beta$, which integrates the spread rate of EAB to new hosts and the
propensity for infestations to become established. The recovery rate for ash
trees is given by $\alpha$ and this baseline rate is boosted by treatment. The
baseline death rate of infested trees is $\gamma$. With this model
formulation, the fraction of municipally and privately owned trees will remain
constant throughout a simulation, as desired. For privately owned trees, we
consider the cases where no treatment occurs, where no public subsidies are
provided for treatment of privately owned trees and where the optimal subsidy
policy is in place for privately owned trees. For municipally owned trees, we
consider two cases: non-treatment and optimal treatment rates which are
derived in Supplementary Information Section S.I. 4. This creates six possible
treatment scenarios that illustrate the impact of treatment and subsidy
policies on the spread of EAB within a management mosaic of interacting public
and private ownerships.

## Case study

In this section, we explore a case study that integrates subsidy policies and
treatment scenarios with the dynamic model of EAB outbreak to assess the
consequences of subsidy policies, public land treatment decisions, and
intervention timing on ash mortality. We examine the relative outcomes of a
focus on public land, private land or both in designing treatment programs and
find that implementing policies early in an EAB outbreak is critical for
maintaining ash populations.

We draw on the literature for estimates of the parameter values (summarized in
Table S.I. 1) that shape our analyses. In our model, a tree owner’s value of
saving their tree is the sum of the present value of a healthy tree and the
cost of removing a dying tree, $\Delta_{o}=V_{o}+W_{o}$. We assume that
$\Delta_{o}$ is a random variable uniformly distributed between $600 and $900.
This range of values reflects annual tree values estimated from hedonic
property value studies of $25 to $100 (Sander et al.,, 2010; Kovacs et al.,,
2022) and costs of tree removal of $575 to $800. In our analysis below, we
assume the municipality’s value of saving a tree, $\Delta_{m}=V_{m}+W_{m}$,
which is the sum of the social value of a healthy tree and the social cost of
a dying tree, can range up to $1000. This reflects the finding that the
municipal value of a percentage change in tree cover in a neighborhood can
exceed the tree owner’s value by a factor of five or more (Kovacs et al.,,
2022). We assume that the municipality does not consider private tree owners
tree removal costs in their assessment of the benefit of saving a tree, but
this could further enhance the value of saving a privately owned tree from the
perspective of the municipality.

We rely on studies of ash health assessment under EAB risk to come to
reasonable values for assessment accuracy probabilities (Flower et al.,, 2013;
Knight et al.,, 2014). For ash trees with no EAB present, we assume there is a
very small chance that the tree will be classified as dying (1%) with a higher
chance of being misclassified as infested (10%) and most EAB-free trees
accurately assessed as healthy (89%). For trees with a treatable infestation
of EAB, the visual characteristics of early stages crown dieback can be seen
but only once a sufficiently large population size of EAB larva are present.
Therefore there is a chance that such a tree will be categorized as healthy
(49%) or infested (50%) with only a small possibility of a treatably infested
tree being assessed as dying (1%). For trees with late-stage untreatable EAB
infestations, advanced crown dieback can signal that it may be too late for
treatments to be effective. We assume that in these cases there is only a 1%
chance that the tree will be assessed as healthy, a 19% chance it will be
assessed as infested, and a 80% chance it will accurately be asses as dying.

Consistent with studies on the effectiveness of insecticide treatments for
different health classes of ash trees (McCullough et al.,, 2011), we assume
there is a 100% chance that an EAB-free treated tree will remain EAB-free and
that the likelihood of recovery of a treatably infested ash tree is $90\%$. We
assume that a dying / dead ash tree will not recover in response to treatment.
Similarly, if an untreated tree is infested or dying, we assume there is no
chance that it will recover. If an EAB-free tree is untreated, we assume there
is a 70% chance it will remain EAB-free.

Using these parameter values, we compute the optimal subsidy level, treatment
probabilities, and survival probability for each assessed tree state as a
function of the social value of saving the tree, $\Delta_{m}$. These levels
are shown in Figure 5, for the case when the background rates of infestation
and tree mortality are $(P(h),P(i),P(d))=(0.7,0.25,0.05)$. In this case, trees
assessed as infested or dying never receive subsidies. Trees assessed as
healthy receive subsidies when the value of saving a tree is greater than
about $200. An infested tree never receives a subsidy because the tree owner’s
value of treatment is high enough that they will treat the tree with certainty
even in the absence of a subsidy. A dying tree also does not receive a
subsidy. However, in this case the absence of a subsidy is because the social
value of treating a dying tree is not high enough to justify a subsidy. The
lower right-hand plot in Figure 5 shows the survival probability of trees in
each assessed state. This illustrates that increasing social values of
treating trees leads to policies that increase tree survival.

We also compute the optimal subsidy policies and treatment probabilities for
each assessed tree state for all possible background infestation states, where
$P(h)+P(i)+P(d)=1$, and with a fixed value of saving a tree, $\Delta_{m}=550$
(Figure 6). Each triangular plot is a two-dimensional simplex with color
coding according to the outcomes across all possible background infestation
states. Each row corresponds to a different assessed tree health state. The
first column shows how optimal subsidy levels depend on the state of the pest
outbreak. The second column illustrates the probability that a tree will be
treated given the outbreak state and optimal subsidies. The third column
illustrates the likelihood of treatment in the absence of subsidies. Thus,
differences between the second and third columns of plots show the impact that
optimal subsidies have on treatment likelihoods. The plots in these columns
show that trees assessed as being healthy will be subsidized when infested
trees are rare. These subsidies increase treatment probabilities where private
tree owners wouldn’t otherwise treat healthy trees. Trees assessed as healthy
are treated across most of the state space even in the absence of subsidies.
However, when infestation rates are low and few trees are dying, subsides
should be used to boost the likelihood of treatment. On the other hand, if
most trees are dying and few are infested, subsidies for infested trees are
not warranted. The intuition for this result is that in this part of the state
space, it may be too late to save many of the trees that are assessed as
infested. The far right-hand column shows the optimal treatment decision for
publicly owned trees (see Supplementary information Section S.I. 4 for a
derivation). For these parameters, optimal subsidy polices nearly align
treatment of public and private trees. This need not be the case, however.
There exist cases where privately owned trees are more likely to be treated
because the municipality can leverage the willingness to pay of private tree
owners and get these trees treated at a lower cost than public trees due to
cost sharing between the parties.

We have considered optimal policies for treatment of privately owned trees and
seen that these policies can help to align the management of public and
private trees. Next, we will use the epidemiological model introduced above,
to project the dynamics of EAB spread within a community with a population of
ash trees that are 60% privately owned and 40% public trees. We start the
simulations when nearly all trees are EAB-free but a few have treatable EAB
infestations. In Figure 7, we show six scenarios of public and private
treatment decisions that were constructed based on the analysis of the game
theoretic model. If treatments are not administered to public or private
trees, the model predicts that over 80% of trees will die within 15 years
(Figure 7, top left panel), a finding that is consistent with EAB literature
(McCullough and Mercader,, 2012; Knight et al.,, 2013; Burr and McCullough,,
2014; Sadof et al.,, 2017; Kappler et al.,, 2019). When subsidies are not
implemented and public trees are untreated, ash mortality is still greater
than 80%; however, reaching this level mortality takes almost 50 years. The
optimal subsidy policy for private trees combined with optimal treatment
choices for public trees prevents the establishment and spread of EAB within
the community and trees remain healthy throughout the projection. The
population dynamics under these scenarios are represented in a simplex plot in
the first panel of Figure 8. The dynamics in the ‘no treatment’ case are shown
with arrows and by a black arc that wends toward the state where nearly all of
the ash trees die. For the scenario where there is ‘no public action’ (public
trees untreated, private trees unsubsidized), there is significant treatment
undertaken by tree owners and this results in significantly lower peak
infestation levels, illustrated by the dark grey arc. In this plot, we also
show the dynamics that would result after the introduction of ‘optimal
policies’ that guide treatment of public trees and subsidies for privately
owned trees. Several cases are shown with different intervention timings. Once
the municipality implements an optimal subsidy policy and treatment plan, the
dynamics are altered and the spread of EAB is halted (as shown by the light
grey lines heading toward the bottom of the simplex). The final two panels in
Figure 8 show that earlier implementation of the optimal policy results in
greatly increased long-run healthy ash populations.

In sum, this case study shows how subsidy polices can be designed to enhance
the stewardship of privately owned ash trees, support the provision of
ecosystem services in communities, and limit the spread of EAB. However, in
our case study, achieving these outcomes required coordinated action on both
public and private land. While the implementation of treatment and subsidy
programs is costly, our approach assures that subsidies and treatment are only
used when the benefits (via increased ecosystem services) exceed the costs of
the programs. Through a synthesis of the game theoretic and epidemiological
models we show that early implementation of treatment on public land and
support programs from private tree owners increases ash survival over a 50
year time horizon.

Figure 5: These plots illustrate optimal subsidy levels and several relevant
outcomes across a range of social values for saving a tree, $\Delta_{m}$, and
for fixed background rates of infestation and tree mortality,
$(P(h),P(i),P(d))=(0.7,0.25,0.05)$. The first plot shows the optimal subsidy
levels for each assessed health class of trees. The second plot shows
treatment probabilities for each class of tree and the final plot shows tree
survival probabilities for each category of tree as a function of the social
value of saving a tree. In this case, trees assessed as infested or dying
never receive subsidies. Trees assessed as healthy receive subsidies when the
value of saving a tree is high enough. Infested trees never receive a subsidy
because the value of treating these trees is so high that they will be treated
with certainty even in the absence of a subsidy. Dying trees also do not
receive subsidies. However, in this case the absence of subsidy is because the
social value of treating these trees is never high enough to justify a
subsidy. The final plot shows the survival probability of trees in each
assessed state showing that increasing social values of treating trees leads
to policies which increase tree survival. Figure 6: In these plots we fix the
social value of saving a tree, with $\Delta_{m}=550$. We allow the background
state of the outbreak to vary across all possible states where
$P(h)+P(i)+P(d)=1$, illustrated by a two-dimensional simplex. Each row of
plots corresponds to a different assessed tree health state. The first column
shows how optimal subsidy levels depend on the state of the pest outbreak. The
second column illustrates the probability that a tree will be treated given
the outbreak state and optimal subsidies. The third column illustrates the
likelihood of treatment in the absence of subsidies and thus differences
between the second and third columns of plots show the impact that optimal
subsidies have on treatment likelihoods. Finally, the last column shows the
likelihood of treatment that would occur if the tree were publicly owned. For
these parameter values, trees assessed as dying will never be subsidized and
never be treated. Trees assessed as being healthy will be subsidized when
infested trees are rare and this increases treatment probabilities for those
cases where private tree owners wouldn’t otherwise treat healthy trees. Trees
assessed as healthy are treated across most of the state space even in the
absence of subsidies; however, when infestation rates are low and not too many
trees are dying, subsides should be used to boost the likelihood of treatment.
On the other hand, if most trees are dying and few are infested, subsidies for
infested trees are not warranted under our model. The intuition for this
result is that in this part of the state space, it may be too late to save
many of the trees that are assessed as infested. Figure 7: Here we show the
dynamics of EAB spread within a community using a compartmental model using
parameter values for spread rates drawn from the literature. In each plot the
horizontal axis corresponds to time in years and the vertical axis represents
the fraction of trees in a given health and ownership class. We illustrate the
expected dynamics of EAB in a mixed landscape of 40% public and 60% private
ash tree ownership over time across six treatment scenarios. The first row of
plots shows cases where publicly owned trees are not treated and the second
row shows cases where public trees are treated optimally. The first column of
plots shows simulations where privately owned trees are untreated, the second
column shows the dynamics of EAB spread when there are no subsidies for the
treatment of privately owned trees, and the final column shows cases where
optimal subsidy levels are set for the treatment of private trees. If no
treatments are administered, this model predicts that over 80% of trees will
be dying or dead within 15 years, a finding that is consistent with EAB
literature. When subsidies are not implemented and public trees are untreated,
ash mortality is still greater than 80%, however, reaching this level
mortality takes over 30 years. For the parameter values we consider, the
optimal subsidy policy combined with optimal treatment of public trees
prevents the establishment and spread of EAB. The curves show the distinct EAB
spread dynamics within public and private trees. Supplementary Materials
Figure S.I. 2 shows the joint dynamics averaged across public and private
trees. Figure 8: Using the same parameter values, we illustrate the spread of
EAB as a function of intervention timing. Since the sum of the fractions of
healthy, infested and dying/dead trees always equals one, we can plot the
dynamics of trees on a two-dimensional simplex where each corner corresponds
to all trees being in the labeled health state. On the simplex we show the
expected dynamics of EAB under a case where no trees are treated with black
arrows that show the direction of dynamics. The thick black curve shows a
representative infestation trajectory starting from a mostly healthy ash
population. Rather than using the no treatment case as a baseline, consider a
‘no public action’ baseline scenario where there are no subsidies for the
treatment of private trees and no publicly owned trees are treated (shown in
dark grey). Given the parameter values used in this simulation, private tree
owners choose to treat often enough to significantly lower the peak
infestation rate even in the absence of public actions to address EAB. From
this no public action scenario, we simulate the introduction of ‘optimal
policies’ for EAB management. Under our ‘optimal policy’ scenario, we assume
optimal treatment of public trees (described in Supplementary Information
Section S.I. 4) and optimal subsidy policies for privately owned trees. We
consider the impact of intervention timing by simulating the introduction of
optimal policies at 3.5 year intervals (shown in light grey). To illustrate
the impact of intervention timing on 50-year ash tree survival rates, in the
second and third panels we show outcomes under cases where public policies are
enacted 3.5 years after EAB arrival and 10.5 years after arrival. In these two
panels, dashed lines correspond to the ‘no public action’ case while solid
lines correspond to the ‘optimal policy’ scenario. The increased fraction of
remaining healthy trees after 50 years shows a significant benefit from early
intervention.

## Discussion

Urban forests and trees provide a wide range of ecosystem services with
benefits that accrue from local to global scales. This can be seen in the
relationship between tree cover and home values. Trees on a property increase
a home’s value, but so does having higher tree cover within the neighborhood
(Kovacs et al.,, 2022). This illustrates that some ecosystem service benefits
of trees are local public goods. Carbon sequestration by trees provides an
even starker example of this phenomenon because the benefit of reduced
atmospheric carbon dioxide is a global public good (Buchholz and Sandler,,
2021). Therefore, sustaining the ecosystem service benefits of privately owned
urban trees resembles a public goods problem. This has been central to the
challenge of achieving sustainability because public goods result in a
conflict between the collective good and our individual self-interest (Levin,,
2010, 2014). In the case of urban forest management, this framing implies that
private tree owners may under-invest in management interventions that sustain
the ecosystem service benefits of their trees. At the same time, urban forests
are embedded in complex ownership mosaics of public and private land with many
uses and distinct management aims. This creates a unique challenge for urban
landscape-scale management of forest pests.

The management of EAB provides a salient example of this phenomenon. EAB has
spread widely across the United States and Canada since its discovery in the
region in 2002 (Michigan State University and USDA Forest Service,, 2022) and
has caused hundreds of millions of dollars in losses (Kovacs et al.,, 2010;
Aukema et al.,, 2011). However, numerous studies have found that insecticide
treatments are effective (McCullough et al.,, 2011) and can provide net social
benefits by protecting the flow of ecosystem services from ash trees
(McCullough and Mercader,, 2012; Kovacs et al.,, 2014; Kıbış et al.,, 2021).
However, given that investing in treatment for ash trees is a public good, we
expect private tree owners to under-invest in treatment in the absence of
public policies that support treatment. Therefore, achieving effective forest
pest management across public and private land may require programs that
support private tree owners.

We address this problem with a game theoretic model of the interactions
between a municipality, tree care firms and a tree owner. Our model considers
the health status of the tree as well as the imperfect assessment of tree
health based on visible characteristics. We incorporate uncertainty of a tree
owner’s valuation of their tree into our analyses. We use this game theoretic
model to explore optimal subsidy policies from the perspective of an urban
forest manager who aims to sustain forest ecosystem service benefits through
their actions and programs.

We derive optimal subsidy levels as a function of the health status of the
tree and the community prevalence of EAB infestation and tree mortality. These
results could help establish or hone public programs that invest in pest
management on private land. The optimal subsidy policies are nuanced in that
they take into consideration an individual’s expected willingness to pay for
treatment. Optimal subsidies are often zero when treatment is the most
beneficial because in these cases a tree owner is expected to treat their tree
regardless. Subsidies are targeted at increasing treatment in cases where the
social value of treatment is high but the private value to a tree owner does
not always justify treatment. The result is a subsidy policy that cost-
effectively increases treatment rates. Our model shows that subsidies may
require regular revision because we find that the optimal subsidy policy
depends on the state of EAB infestation in a community. For example, our case
study suggests that early in an EAB outbreak when infested trees are rare it
makes sense to subsidize treatment of healthy and infested trees. On the other
hand, when the fraction of infested trees is high subsidies are not warranted
for any trees. This context dependence of the optimal subsidy policy is
central to its efficiency.

While our game theoretic model defines optimal subsidies across a range of
infestation states, it does not model the spread of EAB though time. We
develop a compartmental model of EAB spread that integrates the impacts of
treatment. This dynamic model allows us to explore how subsidy policies and
treatment decisions alter the trajectory of EAB within a community. Even if
the spread of EAB cannot be stopped, slowing its spread within a community can
still be valuable. If the ash trees in a community are decimated over a 5-15
year period, this can significantly disrupt the supply of forest ecosystem
services. On the other hand, if these same losses are extended across a
50-year period, then the rate of tree mortality would be closer to the
baseline rate within urban areas and the impact of EAB on the provision of
ecosystem services would be far less pronounced. We use the dynamic model to
compare long-term ash survival under a range of assumptions about treatment.
We consider the cases where public trees are either untreated or optimally
treated and the treatment of private trees is either optimally subsidized,
unsubsidized, or the trees remain untreated. This results in six scenarios,
including the case where there is no treatment, the case where there is no
public action to treat EAB, and the case where optimal policies are employed
for both public and private trees. The model shows that while a focus on only
public or private land can slow the spread of EAB and increase 50-year ash
survival, early implementation of a unified approach that addresses pest
management on public and private land is most effective at reducing pest-
induced mortality and sustaining the benefits of urban tree cover.

The game theoretic and dynamic models that we study are simple enough to make
their analysis tractable yet detailed enough to reflect critical features of
EAB management including health assessment of trees, competition among tree
care firms, and uncertainty about the values of tree owners. The dynamic model
is not spatially explicit and assumes that all the trees within the model
interact. This assumption would be unrealistic if the model were applied at
large spatial scales, but may be reasonable at the scale of a community or
municipality. A benefit of this simplicity is that it yields intuition into
what determines optimal subsidy policies and testing their impact with the
dynamical model is not overly computationally intensive. As such, these models
could form the basis of a tool for considering policies for EAB treatment in
urban areas.

Our analysis was aimed at deriving subsidy policies to optimize the provision
of ecosystem services. The distributional effect of these policies was not
considered by our analysis. However, recent investments in urban forestry have
emphasized environmental justice 444USDA Press Release: USDA invests $1
billion for nearly 400 projects to expand access to trees and green spaces in
communities and neighborhoods nationwide and it is therefore important to
consider how the management approach studied in this paper might hinder (or
advance) these objectives. The optimal subsidy levels derived in our game
theoretic analysis fall into three categories, two of which should not impact
inequality. When no subsidy is optimal, then there will be no impact on
inequality because no one receives a subsidy. Similarly, when the optimal
subsidy is high enough to ensure that all individuals treat their trees we
also need not worry about the impact on inequality of the policy because it
impacts all ash tree owners the same way. In this case, assuring that the
program is being administered equitably could advance environmental justice.

However, there are cases that could generate feedback effects between
inequality and the environment (Hamann et al.,, 2018) that impede
environmental justice. When an intermediate subsidy level is optimal,
individuals with high willingness to pay for treatment opt to treat and those
with low willingness to pay do not. Research has found that the relationship
between socioeconomic factors and urban forests is complex, context dependent,
and shaped by legacy effects (Roman et al.,, 2018; Grove et al.,, 2020; Koo et
al.,, 2023). However, positive relationships between income and demand for
urban forests have been documented (Zhu and Zhang,, 2008; Le Tran et al.,,
2017). If we assume a positive correlation between socioeconomic status and
willingness to pay for treatment of ash trees, intermediate subsidy levels
could exacerbate inequality because the subsidy will flow to those with higher
socioeconomic status. In addition to the direct effect of the subsidy on
inequality, there is also the potential for diverging treatment decisions to
impact EAB spread and ash decline. For example, in communities with high
levels of residential segregation (Massey and Denton,, 1988; Logan and
Parman,, 2017), this could manifest as spatial EAB hotspots that
disproportionately impact historically marginalized communities and reinforce
inequitable patterns of urban tree canopy cover (Schwarz et al.,, 2015; Locke
et al.,, 2021). In these cases, it may be worth considering alternative
objective functions for the municipality that explicitly weigh distributional
effects and equity so that harmful positive feedback effects between
inequality and the environment can be avoided.

The management of forests pests in an urban mosaic of public and private land
poses unique public goods problems that can be addressed with carefully
designed management interventions. We find that subsidy policies for treatment
of private trees can align treatment actions by private tree owners with those
that a municipality may choose for public trees. Such alignment helps slow the
spread of forest pests and sustain the public benefits that urban trees
provide in the form of ecosystem services. This general approach of designing
programs to align private and public incentives for investment in public goods
could be employed across a range of management challenges that arise in
ownership mosaics and could contribute to urban sustainability.

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## Acknowledgements

The findings and conclusions in this publication are those of the authors and
should not be construed to represent any official USDA or U.S. Government
determination or policy. Code that generates figures is available at $\href
https://github.com/atilman/EAB-treatment_code$

Supplementary Information for:

## Public policy for management of forest pests within an ownership
mosaic

Emerald ash borer (EAB) is an invasive insect that has had devastating impacts
on ash trees in North American forests. In years since the arrival of EAB,
treatments have been developed which dramatically reduce the risk of
infestation and death of treated ash trees. These treatments may also slow the
spread of EAB at landscape scales if deployed strategically. However,
achieving coordinated deployment of these treatments is challenging due to the
mosaic of forest ownership types in the region, the costs of treatment, and
the public nature of many of the ecosystem services that mature ash trees
provide.

Due to these factors, in the absence of active intervention on the part of
municipalities or land management agencies, private market forces are likely
to lead to sub-optimal outcomes from the perspective of society. We develop
and analyze a model to understand how municipal subsidy programs for ash
treatment can be optimized to achieve stewardship of ash populations at least
cost.

##  Appendix S.I. 1 Game theoretic model

We consider the municipally subsidized treatment of a privately owned ash tree
by a tree care firm as a dynamic game of incomplete information in three
stages. While we focus on a single tree, our analysis can be apply to a
population of trees as well. The health status of the tree is independently
assessed and this status is know by the municipality, the tree care firms and
the owner of the tree in question. Next, the municipality chooses subsidies
that are available to the tree care firms, with the potential for these
subsidy levels to vary across firms and tree health status. The firms then
simultaneously submit bids to the tree owner defining the price they will
charge for treating the tree. Finally, the tree owner decides whether to
accept the bid of either firm, or opt not to treat their tree. We assume that
the value of the tree to its owner is drawn from a distribution and that the
while the tree owner knows their personal valuation (referred to as their
‘type’), the municipality and tree care firms only know the distribution from
which this valuation is drawn. This introduces incomplete information.
However, since the tree owner is the last player to take an action in the
game, we do not need to update other players beliefs about the type of the
tree owner. Instead, we can simply analyze this dynamic game via backward
induction with players at each stage choosing a strategy to maximize their
expected utility given the distribution of types of tree owners. First, we
introduce key aspects of tree health, assessment, and treatment that underlie
the payoffs of the tree owner and the municipality. Then we will derive the
benefits of treatment from the tree owner and municipal perspective.

###  S.I. 1.1 Tree health assessment

We assume that that arborists affiliated with or independent from the tree
care firms or the municipality assess the health of a tree and share this
information with the tree care firms, tree owner, and municipality. In
practice, arborists are able to assess infestation with reasonable accuracy by
visually observing crown density. If tree assessment is undertaken by the tree
care firms, we assume that all firms use the same assessment methodology and
thus all firms that assess a tree’s health, will reach the same conclusion
about a tree’s health state state. While errors in assessment can occur, we
will assume that these errors result from deficiencies of the assessment
methods rather than of the assessor. This implies that repeatedly assessing
the health of a tree will yield no additional information about the true
underlying health state of the tree.

We assume an arborist will assess the health status of a tree in one of three
states,

$\hat{\phi}\in\\{\hat{h},\hat{i},\hat{d}\\},$ (47)

with the three possible assessed states being: healthy, $\hat{h}$, infested,
$\hat{i}$, or dead / dying, $\hat{d}$. We denote a tree’s true underlying
state

$\phi\in\\{h,i,d\\}$ (48)

where the true state of the tree can ether be healthy, $h$, infested, $i$, or
dead / dying, $d$.

We assume that surveillance data are available at the municipality scale, with
publicly known frequencies of each underlying tree state given by $P(h)$,
$P(i)$, and $P(d)$.

We allow for assessment techniques to be imperfect predictors of the true
underlying health status of a tree. We define the accuracy of assessment with
probabilities. For example,

$P(\hat{h}\mid d)$ (49)

is the probability that an dying tree is determined to be healthy by a tree
care specialist ($\hat{h}$ corresponds to the assessed state, whereas $d$
without a hat corresponds to the true underlying state of the tree). There are
several of these probabilities, which can be written as

$P(\hat{\phi}\mid\phi)$ (50)

for some assessed state, $\hat{\phi}$ and some true underlying state $\phi$.
These accuracy probabilities have been estimated by researchers and are known
to be quite high. A tree’s underlying infestation state can correlates closely
with the extent and severity of crown die back.

###  S.I. 1.2 Inferring underlying tree health

When a tree owner is deciding whether to treat their tree or when a
municipality is setting subsidy policies for trees assessed in state
$\hat{i}$, for example, they need to know how likely it is that the true
health of the tree matches or diverges from this assessment. These
probabilities can be computed with Bayes Theorem. For example,

$P(h\mid\hat{h})=\frac{P(\hat{h}\mid h)P(h)}{P(\hat{h}\mid
h)P(h)+P(\hat{h}\mid i)P(i)+P(\hat{h}\mid d)P(d)}$ (51)

is the probability that a tree is healthy given that it was assessed as such
by an arborist. There is also a chance that a tree given a healthy assessment
is actually infested, with this probability given by

$P(i\mid\hat{h})=\frac{P(\hat{h}\mid i)P(i)}{P(\hat{h}\mid
i)P(i)+P(\hat{h}\mid h)P(h)+P(\hat{h}\mid d)P(d)}.$ (52)

The likelihoods of all true underlying health states of a tree given its
assessed health can be similarly computed and will be pivotal in defining the
expected utility functions of the municipality, firms and tree owner.

###  S.I. 1.3 Impact of treatment

We now break down the impact of treatment or non-treatment on tree health into
cases where the tree has a healthy, infested, or dying underlying state. We
assume that a dying tree cannot be saved. For a tree with a healthy or
infested state, the probabilities of the tree staying (or becoming healthy)
depends on the treatment choice. We have the probabilities

$\displaystyle P(h\mid t\cap h)$ $\displaystyle=h_{th}$ (53) $\displaystyle
P(h\mid u\cap h)$ $\displaystyle=h_{uh}$ (54) $\displaystyle P(h\mid t\cap i)$
$\displaystyle=h_{ti}$ (55) $\displaystyle P(h\mid u\cap i)$
$\displaystyle=h_{ui}$ (56) $\displaystyle P(h\mid t\cap d)$ $\displaystyle=0$
(57) $\displaystyle P(h\mid t\cap d)$ $\displaystyle=0$ (58)

which define the likelihood of a tree remaining (becoming) healthy, $h$, given
the treatment decision ($t:$ treated, $u:$ untreated) and the underlying
infestation state of the tree, $\phi\in\\{h,i,d\\}$. We assume that following
the decision to treat or not, a tree will either remain (become) healthy or
die. In other words, we ignore the outcome of a tree staying in an infested
state, as this is thought to be a transient state that either results in tree
death or possibly recovery. In our analyses, we find that the benefit of
treatment to a tree owner and the municipality is driven by the increase in
the likelihood of a tree staying (becoming) healthy in response to treatment,
which we define with the quantities

$\displaystyle t_{h}$ $\displaystyle=h_{th}-h_{uh}$ (59) $\displaystyle t_{i}$
$\displaystyle=h_{ti}-h_{ui}$ (60) $\displaystyle t_{d}$ $\displaystyle=0.$
(61)

###  S.I. 1.4 Private value of treatment

Given the assessed health of the tree by an arborist, the tree owner can now
break their payoff down into six cases: treatment or not, given an assessment
that a tree is healthy, infested or dying. The value of the tree to its owner
under these four cases is

$\displaystyle\pi_{t\hat{h}}$
$\displaystyle=P(h\mid\hat{h})\left(h_{th}V_{o}-\left(1-h_{th}\right)W_{o}\right)+P(i\mid\hat{h})\left(h_{ti}V_{o}-\left(1-h_{ti}\right)W_{o}\right)-P(d\mid\hat{h})W_{o}$
(62) $\displaystyle\pi_{t\hat{i}}$
$\displaystyle=P(h\mid\hat{i})\left(h_{th}V_{o}-\left(1-h_{th}\right)W_{o}\right)+P(i\mid\hat{i})\left(h_{ti}V_{o}-\left(1-h_{ti}\right)W_{o}\right)-P(d\mid\hat{i})W_{o}$
(63) $\displaystyle\pi_{t\hat{d}}$
$\displaystyle=P(h\mid\hat{d})\left(h_{th}V_{o}-\left(1-h_{th}\right)W_{o}\right)+P(i\mid\hat{d})\left(h_{ti}V_{o}-\left(1-h_{ti}\right)W_{o}\right)-P(d\mid\hat{d})W_{o}$
(64) $\displaystyle\pi_{u\hat{h}}$
$\displaystyle=P(h\mid\hat{h})\left(h_{uh}V_{o}-\left(1-h_{uh}\right)W_{o}\right)+P(i\mid\hat{h})\left(h_{ui}V_{o}-\left(1-h_{ui}\right)W_{o}\right)-P(d\mid\hat{h})W_{o}$
(65) $\displaystyle\pi_{u\hat{i}}$
$\displaystyle=P(h\mid\hat{i})\left(h_{uh}V_{o}-\left(1-h_{uh}\right)W_{o}\right)+P(i\mid\hat{i})\left(h_{ui}V_{o}-\left(1-h_{ui}\right)W_{o}\right)-P(d\mid\hat{i})W_{o}$
(66) $\displaystyle\pi_{u\hat{d}}$
$\displaystyle=P(h\mid\hat{d})\left(h_{uh}V_{o}-\left(1-h_{uh}\right)W_{o}\right)+P(i\mid\hat{d})\left(h_{ui}V_{o}-\left(1-h_{ui}\right)W_{o}\right)-P(d\mid\hat{d})W_{o}$
(67)

where $V_{o}$ is the value of a healthy tree and $W_{o}$ is the cost of a dead
or dying tree to its owner. Since a tree provide benefits to communities and
society as a whole, whereas the costs of care are primarily shouldered by the
tree owner, it is likely that a tree owners valuation will not align with the
value of the tree in the eyes of a socially-minded municipality. We find that
the private net benefit of saving a single tree that would otherwise die,

$\Delta_{o}=V_{o}+W_{o},$

shapes the benefit of treatment from the perspective of a tree owner. We
define the value of treatment from the owners perspective as the difference in
payoff between treatment and non-treatment for a tree with a particular health
assessment. If the tree’s assessment indicates infestation, then the benefit
of treatment is

$\displaystyle b_{t\hat{i}}$ $\displaystyle=\pi_{t\hat{i}}-\pi_{u\hat{i}}$
$\displaystyle=(V_{o}+W_{o})\left(P(i\mid\hat{i})(P(h\mid t\cap i)-P(h\mid
u\cap i))+P(h\mid\hat{i})(P(h\mid t\cap h)-P(h\mid u\cap h))\right)$
$\displaystyle=\Delta_{o}\left(P(i\mid\hat{i})(h_{ti}-h_{ui})+P(h\mid\hat{i})(h_{th}-h_{uh})\right)$
$\displaystyle=\Delta_{o}\left(P(i\mid\hat{i})t_{i}+P(h\mid\hat{i})t_{h}\right)$
$\displaystyle=\Delta_{o}k_{\hat{i}}$

where $k_{\hat{i}}=P(i\mid\hat{i})t_{i}+P(h\mid\hat{i})t_{h}$ is the
probability that treating a tree assessed in an infested state will save it
and $\Delta_{o}$ is the value to the tree owner of saving the tree. Similarly,
if the tree owner is told their tree is healthy, this benefit function is

$\displaystyle
b_{t\hat{h}}=\pi_{t\hat{h}}-\pi_{u\hat{h}}=\Delta_{o}\left(P(i\mid\hat{h})t_{i}+P(h\mid\hat{h})t_{h}\right)=\Delta_{o}k_{\hat{h}},$
(68)

where $k_{\hat{h}}=P(i\mid\hat{h})t_{i}+P(h\mid\hat{h})t_{h}$ is the
probability that treating a tree assessed as healthy will save it. Finally, we
can write benefit of treating a tree reported to be dead or dying as

$\displaystyle
b_{t\hat{d}}=\pi_{t\hat{d}}-\pi_{u\hat{d}}=\Delta_{o}\left(P(i\mid\hat{d})t_{i}+P(h\mid\hat{d})t_{h}\right)=\Delta_{o}k_{\hat{d}},$
(69)

where $k_{\hat{d}}=P(i\mid\hat{d})t_{i}+P(h\mid\hat{d})t_{h}$ which shows that
the only benefit from treating a tree assessed as dying is if this assessment
is incorrect.

###  S.I. 1.5 Social value of treatment

We define the social value of a healthy tree as $V_{m}$ and the social cost of
a dying tree as $W_{m}$. From the perspective of the municipality, the net
benefit of saving a single tree is thus

$\Delta_{m}=V_{m}+W_{m}.$

These values and costs, together with the treatment effectiveness, assessment
accuracy, and community level infestation rates can be used to construct the
social payoffs of treatment or non-treatment for each assessed tree state,
with

$\Pi_{u\hat{h}}=P(h\mid\hat{h})\left(h_{uh}V_{m}-\left(1-h_{uh}\right)W_{m}\right)+P(i\mid\hat{h})\left(h_{ui}V_{m}-\left(1-h_{ui}\right)W_{m}\right)-P(d\mid\hat{h})W_{m}\\\
$ (70)

defining the social value of an untreated tree that is assessed as healthy,
for example. The benefit of treatment from the socially oriented perspective
of the municipality, given the assessment of the tree’s health state can be
written as

$\displaystyle B_{t\hat{h}}$
$\displaystyle=\Pi_{t\hat{h}}-\Pi_{u\hat{h}}=\Delta_{m}\left(P(i\mid\hat{h})t_{i}+P(h\mid\hat{h})t_{h}\right)=\Delta_{m}k_{\hat{h}}$
(71) $\displaystyle B_{t\hat{i}}$
$\displaystyle=\Pi_{t\hat{i}}-\Pi_{u\hat{i}}=\Delta_{m}\left(P(i\mid\hat{i})t_{i}+P(h\mid\hat{i})t_{h}\right)=\Delta_{m}k_{\hat{i}}$
(72) $\displaystyle B_{t\hat{d}}$
$\displaystyle=\Pi_{t\hat{d}}-\Pi_{u\hat{d}}=\Delta_{m}\left(P(i\mid\hat{d})t_{i}+P(h\mid\hat{d})t_{h}\right)=\Delta_{m}k_{\hat{d}}$
(73)

for the three possible health assessment outcomes.

###  S.I. 1.6 Game structure

Once as tree has had its health status assessed, we subsequently consider a
game in three stages. First, the municipality chooses subsidy levels,
$s_{1\hat{\phi}}$ and $s_{2\hat{\phi}}$ that are available to firm 1,
($f_{1}$) and firm 2, ($f_{2}$) for the treatment of a tree assessed in as
being in state $\hat{\phi}$. Next, the tree firms simultaneously submit bids
to the tree owner for the price they will charge for treating the tree, with
the bids of firm 1 and 2 being $p_{1\hat{\phi}}$ and $p_{2\hat{\phi}}$,
respectively. Finally, the tree owner decides whether to accept the bid of
either firm, or opt not to treat their tree. While the tree owner knows that
their value of treatment given the assessed status of the tree is
$b_{t\hat{\phi}}=\Delta_{o}k_{\hat{\phi}}$, the firms and municipality only
know the distribution from which $\Delta_{o}$ is drawn, with

$\Delta_{o}\sim U[a,b].$ (74)

###  S.I. 1.7 Strategy spaces

The strategy space of the municipality is defined by the non-negative subsidy
levels that they offer,

$\sigma_{m\hat{\phi}}=\\{(s_{1\hat{\phi}},s_{2\hat{\phi}})\mid
s_{1\hat{\phi}}\geq 0,s_{2\hat{\phi}}\geq 0\\}.$ (75)

The strategy space of firm 1 is the price that they offer for treatment of the
tree,

$\sigma_{f_{1}\hat{\phi}}=\\{p_{1\hat{\phi}}\mid p_{1\hat{\phi}}\geq 0\\},$
(76)

with a similar strategy space for firm 2, given by

$\sigma_{f_{2}\hat{\phi}}=\\{p_{2\hat{\phi}}\mid p_{2\hat{\phi}}\geq 0\\}.$
(77)

Finally, the strategy space of the tree owner is given by a mapping between
their type, $\Delta_{o}$, which is drawn from a uniform distribution and the
probability that they choose any of their actions

$A_{o\hat{\phi}}=\\{\text{treat w/}f_{1},\text{treat
w/}f_{2},\text{untreated}\\}.$ (78)

We denote a mapping that forms their strategy as

$\sigma_{o\hat{\phi}}:[a,b]\to P(A_{o\hat{\phi}})$ (79)

so that a strategy $\sigma_{o\hat{\phi}}$ will assign a probability
distribution over the set $A_{o\hat{\phi}}$ for every possible value of
$\Delta_{o}\in[a,b]$.

###  S.I. 1.8 Utility functions

Given the value treated and untreated trees, we construct the utility
functions for the municipality. We assume that subsidies are paid at a cost to
the municipality. This leads to a utility function for the municipality given
by

$U_{m\hat{\phi}}=\begin{cases}\Pi_{t\hat{\phi}}-s_{1\hat{\phi}},&\text{if tree
assessed as $\hat{\phi}$ is treated by firm 1}\\\
\Pi_{t\hat{\phi}}-s_{2\hat{\phi}},&\text{if tree assessed as $\hat{\phi}$ is
treated by firm 2}\\\ \Pi_{u\hat{\phi}}&\text{if tree assessed as $\hat{\phi}$
is untreated}\end{cases}$ (80)

The utility function for firm 1 depends on the the cost of administering the
treatment, $c$, the price they bid, $p_{1\hat{\phi}}$ and the level of the
subsidy that they receive, $s_{1\hat{\phi}}$. This leads to a utility function
given by

$U_{f_{1}\hat{\phi}}=\begin{cases}p_{1\hat{\phi}}+s_{1\hat{\phi}}-c,&\text{if
firm 1 treats tree}\\\ 0,&\text{if firm 1 does not}\end{cases}$ (81)

for firm 1. A similar function holds for firm 2, since we assume that the
costs for each firm are the same. This leads to a utility function of

$U_{f_{2}\hat{\phi}}=\begin{cases}p_{2\hat{\phi}}+s_{2\hat{\phi}}-c,&\text{if
firm 2 treats tree}\\\ 0,&\text{if firm 2 does not.}\end{cases}$ (82)

Finally, we consider the utility of a private tree owner, which will depend on
their valuation of the tree. These valuations can be used to construct the
utility function of the tree owner as

$U_{o\hat{\phi}}=\begin{cases}\pi_{t\hat{\phi}}-p_{1\hat{\phi}},&\text{if firm
1 treats the tree}\\\ \pi_{t\hat{\phi}}-p_{2\hat{\phi}},&\text{if firm 2
treats the tree}\\\ \pi_{u\hat{\phi}},&\text{if the tree is
untreated}.\end{cases}$ (83)

##  Appendix S.I. 2 Analysis

The analysis of the game in this case will proceed via backward induction,
starting from the tree owners optimal strategy for responding to an arbitrary
pair of bids and working back to the subsidy policy which maximizes the
expected utility of the municipality.

###  S.I. 2.1 Tree owner behavior

We assume that the tree owner knows the assessed state of their tree is some
$\hat{\phi}\in\\{\hat{h},\hat{i},\hat{d}\\}$. They simultaneously receive a
pair of bids from the tree care firms for treatment of the tree,
$(p_{1\hat{\phi}},p_{2\hat{\phi}})$. Any type of tree owner with type
$\Delta_{o}\in[a,b]$ will maximize their utility by following the actions
defined by the strategy mapping

$\sigma_{o\hat{\phi}}^{*}\left[\Delta_{o}\right]=\begin{cases}\text{treat
w/}f_{1}&\text{if }p_{1\hat{\phi}}<p_{2\hat{\phi}}\text{ and
}p_{1\hat{\phi}}\leq\Delta_{o}k_{\hat{\phi}}\\\ \text{treat w/}f_{2}&\text{if
}p_{2\hat{\phi}}<p_{1\hat{\phi}}\text{ and
}p_{2\hat{\phi}}\leq\Delta_{o}k_{\hat{\phi}}\\\ \text{flip coin}&\text{if
}p_{1\hat{\phi}}=p_{2\hat{\phi}}\text{ and
}p_{2\hat{\phi}}\leq\Delta_{o}k_{\hat{\phi}}\\\ \text{don't treat}&\text{if
}\Delta_{o}k_{\hat{\phi}}<\text{min}\\{p_{1\hat{\phi}},p_{2\hat{\phi}}\\}\end{cases}$
(84)

where the ‘flip coin’ action corresponds to the case where the tree owner
treats their tree but randomly selects one of the firms. As desired, this
strategy mapping assumes that the tree owner knows their own valuation of
saving a tree, $\Delta_{o}$, and also knows the impact that treatment will
have, given the assessment of the tree’s health that they receive,
$k_{\hat{\phi}}$. Together, these define the benefit of treatment of a tree,
$b_{t\hat{\phi}}=\Delta_{o}k_{\hat{\phi}}$, and only when this benefit is at
least as great as the cost of the lowest bid will the owner choose to have
their tree treated.

###  S.I. 2.2 Firm behavior

We assume that the firms have the comparable utility functions and therefore
we can compute the best response for firm 1 and then apply this to both firms.
We assume that each firm seeks to maximize their expected utility given the
bid of the opposing firm, their beliefs about the value of treatment to the
tree owner, based on the assessed state of the tree, and the level of subsidy
offered to their firm for treatment.

First, we will construct the expected utility function of firm 1. The firms
utility depends on how profitable treatment is if they win the bid and how
likely it is that they win the bid. This leads to

$E[U_{f_{1}}]=\left(p_{1\hat{\phi}}+s_{1\hat{\phi}}-c\right)P(f_{1}\text{ wins
bid})$ (85)

for the expected utility of firm 1 which can be further broken down by
defining the probability function of firm 1 winning the bid. The probability
of winning the bid depends on the bid of the other firm and the beliefs of the
firm about the benefit of treatment to the tree owner.

There are several cases for this probability function. First, if
$p_{1\hat{\phi}}>p_{2\hat{\phi}}$ then firm 1 will never win the bid. If
$p_{1\hat{\phi}}<p_{2\hat{\phi}}$ then firm 1 will win the bid if
$p_{1\hat{\phi}}<b_{t\hat{\phi}}$. If the two bids are equal, we assume for
simplicity that the odds that firm 1 wins the bid are half of what they would
be if they had the lowest bid. However, the results of the analysis still hold
for any case where the tree owner has a non-zero probability of selecting each
of the equal bids. Putting this together, we have

$P(f_{1}\text{ wins bid})=\begin{cases}P(p_{1\hat{\phi}}\leq
b_{t\hat{\phi}})&\text{if }p_{1\hat{\phi}}<p_{2\hat{\phi}}\\\
1/2\;P(p_{1\hat{\phi}}\leq b_{t\hat{\phi}})&\text{if
}p_{1\hat{\phi}}=p_{2\hat{\phi}}\\\ 0&\text{if
}p_{1\hat{\phi}}>p_{2\hat{\phi}}\end{cases}$ (86)

which can be further broken down by calculating the likelihood that a bid
would be acceptable to the tree owner. First, recalling that it is common
knowledge that the net value of saving a tree to its owner, $\Delta_{o}\sim
U[a,b]$, follows a uniform distribution, then the firm’s expectations about
the distribution of the benefit of treatment to the tree owner, given the
assessed health of tree is

$b_{t\hat{\phi}}=\Delta_{o}k_{\hat{\phi}}\sim
U[ak_{\hat{\phi}},bk_{\hat{\phi}}]$ (87)

which can be used to calculate the probability that the firms bid would be low
enough to have a chance of being accepted. This probability function is

$P(p_{1\hat{\phi}}\leq b_{t\hat{\phi}})=\begin{cases}1&p_{1\hat{\phi}}\leq
ak_{\hat{\phi}}\\\
\frac{bk_{\hat{\phi}}-p_{1\hat{\phi}}}{k_{\hat{\phi}}(b-a)}&ak_{\hat{\phi}}<p_{1\hat{\phi}}<bk_{\hat{\phi}}\\\
0&bk_{\hat{\phi}}\leq p_{1\hat{\phi}}.\end{cases}$ (88)

This can be used to construct the expected utility function of firm 1, which
is

$E[U_{f_{1}}]=\begin{cases}p_{1\hat{\phi}}+s_{1\hat{\phi}}-c&p_{1\hat{\phi}}\leq
ak_{\hat{\phi}}\text{ and }p_{1\hat{\phi}}<p_{2\hat{\phi}}\\\