\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}$ (89)

We will first restrict our consideration to those cases where the second
firm’s bid is higher, and see what firm 1’s optimal behavior is in these
cases. The optimal bid can be computed by taking the partial derivative of the
expected utility with respect to $p_{1\hat{\phi}}$, which yields

$\frac{\partial E[U_{f_{1}}]}{\partial
p_{1\hat{\phi}}}=\begin{cases}1&p_{1\hat{\phi}}\leq ak_{\hat{\phi}}\\\
\frac{bk_{\hat{\phi}}-s_{1\hat{\phi}}+c-2p_{1\hat{\phi}}}{k_{\hat{\phi}}(b-a)}&ak_{\hat{\phi}}<p_{1\hat{\phi}}<bk_{\hat{\phi}}\\\
0&bk_{\hat{\phi}}\leq p_{1\hat{\phi}}\end{cases}$ (90)

and shows that when firm 1 knows they have the lowest bid they will never set
$p_{1\hat{\phi}}<ak_{\hat{\phi}}$ since this only reduces their profitability.
Also, if there is to be an optimal bid that falls in the middle case above, it
would be

$p_{1\hat{\phi}}=1/2\left(bk_{\hat{\phi}}+c-s_{1\hat{\phi}}\right)$ (91)

assuming that firm 1 knows they are the lowest bidder, because this is the bid
where $\frac{\partial E[U_{f_{1}}]}{\partial p_{1\hat{\phi}}}=0$ given that
$ak_{\hat{\phi}}<p_{1\hat{\phi}}<bk_{\hat{\phi}}$. However, care must be taken
to assure that this bid does indeed fall within the relevant range of the
state space.

In sum, an uncontested firm 1, which doesn’t have to worry about competing
with firm 2 has an optimal bid policy of

$p_{1\hat{\phi}}^{*}=\begin{cases}ak_{\hat{\phi}}&c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}<s_{1\hat{\phi}}\\\
1/2\left(bk_{\hat{\phi}}+c-s_{1\hat{\phi}}\right)&c-bk_{\hat{\phi}}\leq
s_{1\hat{\phi}}\leq c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\\\ p_{1\hat{\phi}}\geq
c-s_{1\hat{\phi}}&s_{1\hat{\phi}}<c-bk_{\hat{\phi}}\\\ \end{cases}$ (92)

This optimal bid policy will be useful later when considering the case where
only one firm receives a subsidy. It also will contribute to the analysis of
how firm 1 should respond to an arbitrary bid placed by firm 2.

In order to construct the best response relation for firm 1, first consider
the case where the subsidy to firm 1 is so low that they can not profitably
win the bid, regardless of what the other firm does. This is the case where
$s_{1\hat{\phi}}<c-bk_{\hat{\phi}}$ and in this case we will assume that the
firm chooses some bid $p_{1\hat{\phi}}\geq c-s_{1\hat{\phi}}$. While any bid
greater than that of the second firm will lead to the same payoff of zero, we
assume that the first firm will limit their bids to those that would not lead
to negative utility should they win the bid.

Now we will turn our attention to the cases where subsidies, costs of
treatment and the benefits of treatment to tree owners are such that it is
feasible for a firm to win a profitable bid. These are the cases where
$s_{1\hat{\phi}}\geq c-bk_{\hat{\phi}}$. If firm 2’s bid is high, then firm 1
won’t need to competitively bid. Instead, firm 1 will follow their uncontested
optimal bidding policy defined as $p_{1\hat{\phi}}^{*}$. Because we are in the
case where subsidies are sufficient to induce potentially winning bids, we
know that one of the first two conditions on $p_{1\hat{\phi}}^{*}$ will
result. Therefore, for $s_{1\hat{\phi}}\geq c-bk_{\hat{\phi}}$ and
$p_{1\hat{\phi}}^{*}<p_{2\hat{\phi}}$, the best response of firm 1 is
$p_{1\hat{\phi}}^{*}$.

If the bid of firm 2 is so low that firm 1 can not bid less than firm 2 and
have a chance of a positive payoff, then any bid greater than that of firm 2
will lead to the same outcome of 0 payoff, whereas a bid lower than that of
firm 2 will lead to a negative expected payoff. We further restrict the bids
of firm 1 in these cases to bids that would not lead to negative payoffs
should they for some reason end up winning the bid. Therefore, if
$p_{2\hat{\phi}}<c-s_{1\hat{\phi}}$, the best response of firm 1 will be any
bid $p_{1\hat{\phi}}\geq c-s_{1\hat{\phi}}$.

For intermediate bids of firm 2, that fall between $c-s_{1\hat{\phi}}$ and
$p_{1\hat{\phi}}^{*}$, there is the potential for competition among the firms.
If firm 1 bids higher than firm 2 in this case, they will receive a payoff of
zero. On the other hand, any bid greater than $c-s_{1\hat{\phi}}$ but strictly
less than $p_{2\hat{\phi}}$ will lead to a positive expected payoff. These are
bids that are less than that which firm 1 would bid were there no competition
from firm 2, and therefore firm 1 will want to choose the highest price is
that falls within this range. However, if they match the bid of firm 2, they
will win the bid with half the probability that they would if they just
marginally decreased their bid. Therefore, in this intermediate case, firm 1’s
best response is to slightly undercut the bid of firm 2 by setting their price
at $p_{2\hat{\phi}}-\epsilon$. In this setting there always exists some very
small $\epsilon>0$ that leads to higher expected utility than $\epsilon=0$.

These three cases can be aggregated and used to construct the best response of
firm 1 to an arbitrary bid by firm 2. We will define

$p_{1\hat{\phi}}^{\prime}=\max\left\\{\frac{1}{2}\left(bk_{\hat{\phi}}+c-s_{1\hat{\phi}}\right),ak_{\hat{\phi}}\right\\}$
(93)

as the uncontested bid of firm 1, that they would choose if they were able to
act as a monopolist. We use this to write the best response as

$\hat{p}_{1\hat{\phi}}\left(p_{2\hat{\phi}}\right)=\begin{cases}p_{1\hat{\phi}}\geq
c-s_{1\hat{\phi}}&s_{1\hat{\phi}}<c-bk_{\hat{\phi}}\text{ or
}s_{1\hat{\phi}}<c-p_{2\hat{\phi}}\\\
p_{2\hat{\phi}}-\epsilon&s_{1\hat{\phi}}\geq c-bk_{\hat{\phi}}\text{ and
}c-s_{1\hat{\phi}}\leq p_{2\hat{\phi}}\leq p_{1\hat{\phi}}^{\prime}\\\
p_{1\hat{\phi}}^{\prime}&s_{1\hat{\phi}}\geq c-bk_{\hat{\phi}}\text{ and
}p_{1\hat{\phi}}^{\prime}<p_{2\hat{\phi}}\end{cases}$ (94)

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.

We focused on the best response of firm 1 to an arbitrary bid of firm 2.
However, 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

$\hat{p}_{2\hat{\phi}}\left(p_{1\hat{\phi}}\right)=\begin{cases}p_{2\hat{\phi}}\geq
c-s_{2\hat{\phi}}&s_{2\hat{\phi}}<c-bk_{\hat{\phi}}\text{ or
}s_{2\hat{\phi}}<c-p_{1\hat{\phi}}\\\
p_{1\hat{\phi}}-\epsilon&s_{2\hat{\phi}}\geq c-bk_{\hat{\phi}}\text{ and
}c-s_{2\hat{\phi}}\leq p_{1\hat{\phi}}\leq p_{2\hat{\phi}}^{\prime}\\\
p_{2\hat{\phi}}^{\prime}&s_{2\hat{\phi}}\geq c-bk_{\hat{\phi}}\text{ and
}p_{2\hat{\phi}}^{\prime}<p_{1\hat{\phi}}\\\ \end{cases}$ (95)

where we define

$p_{2\hat{\phi}}^{\prime}=\max\left\\{\frac{1}{2}\left(bk_{\hat{\phi}}+c-s_{2\hat{\phi}}\right),ak_{\hat{\phi}}\right\\}$
(96)

as the price the firm 2 would bid if they have a chance to treat the tree and
expect the other firm not to submit a competitive bid.

Nash equilibrium bids occur when pairs of strategies are mutual best
responses. In other words, at intersections of the two firms best response
correspondences. There are many possible Nash equilibria in cases where the
subsidy levels provided to the two firms by the municipality differ. Some of
these are illustrated in figure 3, where sometimes the Nash equilibrium is
unique and other times there is a continuum of Nash equilibria. However, the
Nash equilibrium bids that the firms will select when subsidies are equal,
$s_{\hat{\phi}}=s_{1\hat{\phi}}=s_{2\hat{\phi}}$, follow a simple pattern and
are

$\left(p_{1\hat{\phi}}^{*},p_{2\hat{\phi}}^{*}\right)=\begin{cases}(c-s_{\hat{\phi}},c-s_{\hat{\phi}})&s_{\hat{\phi}}\geq
c-bk_{\hat{\phi}}\\\ (p_{1\hat{\phi}}\geq c-s_{\hat{\phi}},p_{2\hat{\phi}}\geq
c-s_{\hat{\phi}})&s_{\hat{\phi}}<c-bk_{\hat{\phi}}\end{cases}$ (97)

with the details of the second case being irrelevant to outcomes because these
bids will never lead to treatment of the tree555All that work for such a
simple answer???. We decline to compute the conditions that for all possible
Nash equilibria under unequal subsidies, however, we will show that these
cases are of little importance as it is never in a municipality’s best
interest to select unequal subsidies for the firms.

###  S.I. 2.3 Municipality behavior

When the municipality sets subsidy levels for the two firms, we assume that it
does so to maximize the expected value of its utility function. First we will
show that the municipality will never find it advantageous to set different
subsidies for each firm. After we show this we will present the analysis of
optimal subsidy choice assuming ex-ante that equal subsidies will be chosen.

#### Unequal subsidies

When subsidies are unequal, we cannot rely on the equilibrium bids derived in
the previous section, because these bids corresponded to the case of equal
subsidies. First, we consider the case where
$s_{1\hat{\phi}}<c-bk_{\hat{\phi}}$ and $s_{2\hat{\phi}}<c-bk_{\hat{\phi}}$.
From the best responses of the firms we know that subsidy levels this low will
lead to bids of $p_{1\hat{\phi}}\geq c-s_{\hat{\phi}}$ and ,
$p_{2\hat{\phi}}\geq c-s_{\hat{\phi}}$, which will not lead to treatment of
the tree. Therefore, from the perspective of the municipality, in this case
their expected utility is $E[U_{m}]=\Pi_{u\hat{\phi}}$, which does not depend
on the subsidy levels. Therefore for low subsidy levels that do not lead to
treatment, there is no incentive to set subsidy levels that are unequal. The
municipality might as well choose $s_{1\hat{\phi}}=s_{2\hat{\phi}}$.

Therefore, if there is a situation where unequal subsidies are preferable, it
will have to occur when either $s_{1\hat{\phi}}\geq c-bk_{\hat{\phi}}$ or
$s_{2\hat{\phi}}\geq c-bk_{\hat{\phi}}$. We will proceed by way of
contradiction and assume without loss of generality that there exists a
setting in which the municipality finds it optimal to choose
$s_{1\hat{\phi}}\geq s_{2\hat{\phi}}$.

Given that $s_{1\hat{\phi}}\geq s_{2\hat{\phi}}$ and $s_{1\hat{\phi}}\geq
c-bk_{\hat{\phi}}$, we can write the best responses of the firms as

$\hat{p}_{1\hat{\phi}}\left(p_{2\hat{\phi}}\right)=\begin{cases}c-s_{1\hat{\phi}}&p_{2\hat{\phi}}<c-s_{1\hat{\phi}}\\\
p_{2\hat{\phi}}-\epsilon&c-s_{1\hat{\phi}}<p_{2\hat{\phi}}<p_{1\hat{\phi}}^{\prime}\\\
p_{1\hat{\phi}}^{\prime}&p_{1\hat{\phi}}^{\prime}<p_{2\hat{\phi}}\end{cases}$
(98)

with

$p_{1\hat{\phi}}^{\prime}=\max\left\\{\frac{1}{2}\left(bk_{\hat{\phi}}+c-s_{1\hat{\phi}}\right),ak_{\hat{\phi}}\right\\}$
(99)

and

$\hat{p}_{2\hat{\phi}}\left(p_{1\hat{\phi}}\right)=\begin{cases}c-s_{2\hat{\phi}}&s_{2\hat{\phi}}<c-bk_{\hat{\phi}}\text{
or }p_{1\hat{\phi}}<c-s_{2\hat{\phi}}\\\
p_{1\hat{\phi}}-\epsilon&s_{2\hat{\phi}}\geq c-bk_{\hat{\phi}}\text{ and
}c-s_{2\hat{\phi}}\leq p_{1\hat{\phi}}\leq p_{2\hat{\phi}}^{\prime}\\\
p_{2\hat{\phi}}^{\prime}&s_{2\hat{\phi}}\geq c-bk_{\hat{\phi}}\text{ and
}p_{2\hat{\phi}}^{\prime}<p_{1\hat{\phi}}\\\ \end{cases}$ (100)

where we have

$p_{2\hat{\phi}}^{\prime}=\max\left\\{\frac{1}{2}\left(bk_{\hat{\phi}}+c-s_{2\hat{\phi}}\right),ak_{\hat{\phi}}\right\\}$
(101)

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. This can
lead to two types of equilibrium bid pairs of,

$\left(p_{1\hat{\phi}}^{*},p_{2\hat{\phi}}^{*}\right)=\begin{cases}(p_{1\hat{\phi}}^{\prime},c-s_{2\hat{\phi}})&c-s_{2\hat{\phi}}>p_{1\hat{\phi}}^{\prime}\\\
(c-s_{2\hat{\phi}}-\epsilon,c-s_{2\hat{\phi}})&c-s_{2\hat{\phi}}\leq
p_{1\hat{\phi}}^{\prime}\end{cases}$ (102)

given our assumptions. This makes clear that
$p_{1\hat{\phi}}^{*}<p_{2\hat{\phi}}^{*}$ so if the tree is treated it will be
by firm 1. Therefore we can write the expected utility of the municipality as

$E[U_{m}]=\Pi_{u\hat{\phi}}+\left(B_{t\hat{\phi}}-s_{1\hat{\phi}}\right)P(p_{1\hat{\phi}}^{*}\leq
b_{t\hat{\phi}})$ (103)

which breaks into four cases based on the two types of equilibrium bid pairs
and the two different values of $(p_{1\hat{\phi}}^{\prime}$ that can arise.
Writing expected utility in terms of these four cases yields

$E[U_{m}]=\begin{cases}\Pi_{u\hat{\phi}}+\frac{\left(B_{t\hat{\phi}}-s_{1\hat{\phi}}\right)\left(bk_{\hat{\phi}}-c+s_{1\hat{\phi}}\right)}{2k_{\hat{\phi}}(b-a)}\par&c-p_{1\hat{\phi}}^{\prime}>s_{2\hat{\phi}}\text{
and }c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\geq s_{1\hat{\phi}}\\\
\Pi_{u\hat{\phi}}+\frac{\left(B_{t\hat{\phi}}-s_{1\hat{\phi}}\right)\left(bk_{\hat{\phi}}-c+\epsilon+s_{2\hat{\phi}}\right)}{k_{\hat{\phi}}(b-a)}\par&c-p_{1\hat{\phi}}^{\prime}\leq
s_{2\hat{\phi}}\text{ and }c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\geq
s_{1\hat{\phi}}\\\
\Pi_{u\hat{\phi}}+B_{t\hat{\phi}}-s_{1\hat{\phi}}\par&c-p_{1\hat{\phi}}^{\prime}>s_{2\hat{\phi}}\text{
and }c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}<s_{1\hat{\phi}}\\\
\Pi_{u\hat{\phi}}+B_{t\hat{\phi}}-s_{1\hat{\phi}}\par&c-p_{1\hat{\phi}}^{\prime}\leq
s_{2\hat{\phi}}\text{ and
}c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}<s_{1\hat{\phi}}\end{cases}$ (104)

where the first case corresponds to equilibrium where
$p_{1\hat{\phi}}^{*}=\frac{1}{2}\left(bk_{\hat{\phi}}+c-s_{1\hat{\phi}}\right)$,
the second case corresponds to the case where the subsidy level for firm 2 is
high enough to induce some competition and thus
$p_{1\hat{\phi}}^{*}=c-s_{2\hat{\phi}}-\epsilon$. The final two case are where
the subsidy level of firm 1 is high enough that the tree will be treated with
certainty. In case three, $p_{1\hat{\phi}}^{*}=ak_{\hat{\phi}}$ and in the
final case the equilibrium bid of firm 1 is again,
$p_{1\hat{\phi}}^{*}=c-s_{2\hat{\phi}}-\epsilon$. But for these final two
cases, this difference in price does not impact the municipality. Both cases
lead to equivalent expected utility.

The municipality chooses subsidy levels that maximize expected utility.
Subsidy pairs that lead to either of the final two cases will never be
optimal, because in both these cases, expected utility is decreasing in
$s_{1\hat{\phi}}$. Lower values of $s_{1\hat{\phi}}$ will be chosen and this
will move the bid pair out of the parameter space of these cases as they are
defined by having large values of $s_{1\hat{\phi}}$. Therefore, we know that
an optimum with $s_{1\hat{\phi}}>s_{2\hat{\phi}}$ must fall within the first
two cases of equation 104.

If we look at the first case in equation 104, the expected utility of the
municipality does not depend on the subsidy level that firm 2 would receive if
they treat the tree. Furthermore, on the boundary between case 1 and case 2,
when $s_{2\hat{\phi}}=c-p_{1\hat{\phi}}^{\prime}$, the utility in both of the
first two cases is equivalent. Therefore any level of utility that can be
attained within case 1 can also arise in case 2. Therefore and optimum with
$s_{1\hat{\phi}}>s_{2\hat{\phi}}$ must arise in the region of subsidy space
where $c-p_{1\hat{\phi}}^{\prime}\leq s_{2\hat{\phi}}\text{ and
}c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\geq s_{1\hat{\phi}}$. Across this region,
the expected utility of the municipality is decreasing in $s_{1\hat{\phi}}$
and increasing in $s_{2\hat{\phi}}$. Decreasing the value of $s_{1\hat{\phi}}$
and increasing the value of $s_{2\hat{\phi}}$ will always stay within the
parameter region that defines case 2. However, this implies that there cannot
be an optimum where $s_{1\hat{\phi}}>s_{2\hat{\phi}}$ because we showed that
such an optimum would have to occur within case 2 of the expected utility
function and that within this case the municipality will always want to
decrease $s_{1\hat{\phi}}$ and $s_{2\hat{\phi}}$ given
$s_{1\hat{\phi}}>s_{2\hat{\phi}}$. We conclude that a municipality will never
find it optimal to set unequal subsidies for the two firms.

(a) The best response of firm 1 for $s_{1\hat{\phi}}=7.5$ plotted against four
best responses of firm 2 with $s_{2\hat{\phi}}\in\\{0.5,3,5.75,7.5\\}$.
Progressively darker curves correspond to higher subsidy levels for firm 2.
Equilibrium bid pairs are marked with circles. Notice that for
$s_{2\hat{\phi}}<s_{1\hat{\phi}}$ the circles are in the region of the plane
where $p_{1}<p_{2}$ and therefore firm 2 will never win the bid. This means
that whenever the bid is awarded the municipality pays firm 1
$s_{1\hat{\phi}}=7.5$. Increasing $s_{2\hat{\phi}}$ toward $s_{1\hat{\phi}}$
leads to lower equilibrium prices, increased probability of treatment and no
change to the per-tree subsidy paid. Therefore municipalities have an
incentive to select equal subsidies for the two firms.

(b) The expected utility of the municipality is shown as a function of the
subsidy level of firm 2, $s_{2\hat{\phi}}$, given three different values of
$B_{t\hat{\phi}}$. Each of these curves illustrate that for a fixed value of
$s_{1\hat{\phi}}$, expected utility peaks at
$s_{2\hat{\phi}}=s_{1\hat{\phi}}$. In this case, for
$s_{2\hat{\phi}}<c-p^{\prime}_{1\hat{\phi}}$ firm 1 can behave like a
monopolist and set their bid price at $p^{\prime}_{1\hat{\phi}}$. For values
of $s_{2\hat{\phi}}>s_{1\hat{\phi}}$ the municipality pays increasing
subsidies to firm 2 but this doesn’t result in lower prices. This reduces the
expected utility of the municipality. For intermediate subsidy levels for firm
2, increasing $s_{2\hat{\phi}}$, increases expected utility of the
municipality because it leads to increased treatment likelihoods without
impacting marginal costs to the municipality.

Figure S.I. 1: ($s_{1\hat{\phi}}=7.5$, $ak_{\hat{\phi}}=1.5$,
$bk_{\hat{\phi}}=9.5$, $c=10$, $\Pi_{u\phi}=1$)

Intuition for the result can be understood by looking at the best response
relations for gradually increasing values of $s_{2\hat{\phi}}$. At first,
increasing $s_{2\hat{\phi}}$ has no effect, but eventually, this increase
leads firm 1 to have to lower their price. This increases the likelihood that
the tree will be treated but has no impact on the subsidy that the
municipality will have to pay out. Therefore, the municipality will want to
keep increasing $s_{2\hat{\phi}}$ up to the point that the two subsidy levels
are equal, because when they are both equal, the likelihood of treatment can
be maximized at the least cost due to competition among the firms. We now turn
to the analysis of the optimal equal subsidy level.

#### Analysis given equal subsidies

We will begin by constructing the expected utility function of the
municipality when $s_{1\hat{\phi}}=s_{2\hat{\phi}}=s_{\hat{\phi}}$, which is

$E[U_{m}]=\Pi_{u\hat{\phi}}+\left(B_{t\hat{\phi}}-s_{\hat{\phi}}\right)P(\text{treated})$
(105)

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 tree’s by their owners, the bids of the firms, as well
and how a firm’s bidding is impacted by the subsidy level that the
municipality chooses. This probability can be broken down as

$P(\text{treated})=P\left(\min\left\\{p_{1\hat{\phi}}^{*},p_{2\hat{\phi}}^{*}\right\\}\leq
b_{t\hat{\phi}}\right)$ (106)

which, given that $s_{\hat{\phi}}=s_{1\hat{\phi}}=s_{2\hat{\phi}}\geq 0$, is
equivalent to

$P(\text{treated})=P\left(c-s_{\hat{\phi}}\leq
b_{t\hat{\phi}}\right)=\begin{cases}1&c-ak_{\hat{\phi}}\leq s_{\hat{\phi}}\\\
\frac{bk_{\hat{\phi}}-c+s_{\hat{\phi}}}{k_{\hat{\phi}}(b-a)}&c-bk_{\hat{\phi}}<s_{\hat{\phi}}<c-ak_{\hat{\phi}}\\\
0&s_{\hat{\phi}}\leq c-bk_{\hat{\phi}}\end{cases}$ (107)

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

$E[U_{m}]=\begin{cases}\Pi_{u\hat{\phi}}+B_{t\hat{\phi}}-s_{\hat{\phi}}&c-ak_{\hat{\phi}}\leq
s_{\hat{\phi}}\\\
\Pi_{u\hat{\phi}}+\left(B_{t\hat{\phi}}-s_{\hat{\phi}}\right)\frac{bk_{\hat{\phi}}-c+s_{\hat{\phi}}}{k_{\hat{\phi}}(b-a)}&c-bk_{\hat{\phi}}<s_{\hat{\phi}}<c-ak_{\hat{\phi}}\\\
\Pi_{u\hat{\phi}}&s_{\hat{\phi}}\leq c-bk_{\hat{\phi}}\end{cases}$ (108)

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. We will analyze the optimal behavior of the
municipality across three cases and subsequently synthesize the results.

#### Case 1: $c<ak_{\hat{\phi}}$

In this case, the cost of treatment is low enough that firms could profitably
treat trees will all values even in the absence of subsidies. The competition
among firms will lead them to bid in a way that will lead to the tree being
treated with certainty for any subsidy level. This is the first case from the
expected utility function of a municipality. In this case, utility declines in
the level of the subsidy, and therefore the municipality will set their
subsidy, $s^{*}_{\hat{\phi}}=0$.

#### Case 2: $c\in\left[ak_{\hat{\phi}},bk_{\hat{\phi}}\right]$

In this second case the cost of treatment is such that in the absence of any
subsidy, there will be a chance that a tree will be treated, given the optimal
bidding policies of the firms. Since we only consider non-negative subsidy
levels, only the first two cases of the expected utility function can occur.
The optimal subsidy can be determined from the partial derivative of the
utility function with respect to the subsidy level, which is

$\frac{\partial E[U_{m}]}{\partial
s_{\hat{\phi}}}=\begin{cases}-1&c-ak_{\hat{\phi}}\leq s_{\hat{\phi}}\\\
\frac{B_{t\hat{\phi}}-bk_{\hat{\phi}}+c-2s_{\hat{\phi}}}{k_{\hat{\phi}}(b-a)}&0\leq
s_{\hat{\phi}}<c-ak_{\hat{\phi}}.\end{cases}$ (109)

This sets an upper bound on the optimal subsidy level of $c-ak_{\hat{\phi}}$.
The lower case has an optimum at
$s_{\hat{\phi}}=1/2\left(B_{t\hat{\phi}}-bk_{\hat{\phi}}+c\right)$ and this
falls within the range of the lower case for intermediate values of
$B_{t\hat{\phi}}$. In sum, the optimal subsidy policy for the municipality is

$s^{*}_{\hat{\phi}}=\begin{cases}c-ak_{\hat{\phi}}&c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\leq
B_{t\hat{\phi}}\\\
1/2\left(B_{t\hat{\phi}}-bk_{\hat{\phi}}+c\right)&bk_{\hat{\phi}}-c<B_{t\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\\\
0&B_{t\hat{\phi}}\leq bk_{\hat{\phi}}-c\end{cases}$ (110)

which shows how the optimal municipal subsidy depends on the relationship
between social and private benefits of treatment and the cost of treatment.

#### Case 3: $c>bk_{\hat{\phi}}$

This corresponds to a situation where firms cannot profitably treat a private
tree in the absence of subsidies. Now, all three cases from the expected
utility function can result. The relevant partial derivative is

$\frac{\partial E[U_{m}]}{\partial
s_{\hat{\phi}}}=\begin{cases}-1&c-ak_{\hat{\phi}}\leq s_{\hat{\phi}}\\\
\frac{B_{t\hat{\phi}}+c-bk_{\hat{\phi}}-2s_{\hat{\phi}}}{k_{\hat{\phi}}(b-a)}&c-bk_{\hat{\phi}}<s_{\hat{\phi}}<c-ak_{\hat{\phi}}\\\
0&0\leq s_{\hat{\phi}}\leq c-bk_{\hat{\phi}}\end{cases}$ (111)

The optimum at the middle case is once again
$s_{\hat{\phi}}=1/2\left(B_{t\hat{\phi}}+c-bk_{\hat{\phi}}\right)$, and this
will fall within the middle case for intermediate values of $B_{t\hat{\phi}}$,
following

$s^{*}_{\hat{\phi}}=\begin{cases}c-ak_{\hat{\phi}}&c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\leq
B_{t\hat{\phi}}\\\
1/2\left(B_{t\hat{\phi}}+c-bk_{\hat{\phi}}\right)&c-bk_{\hat{\phi}}<B_{t\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\\\
0&B_{t\hat{\phi}}\leq c-bk_{\hat{\phi}}\end{cases}$ (112)

where the final case corresponds to the case where a subsidy program could
lead treatment of trees would not be in collective best interest of society.
In this bottom case any subsidy
$s_{\hat{\phi}}\in\left[0,c-bk_{\hat{\phi}}\right]$ will lead to the same
result. However, for simplicity we assume a $0$ subsidy will result since the
municipality is deliberately choosing a subsidy that won’t lead to treatment,
so a positive subsidy in this case wouldn’t be particularly wise. The middle
case leads to a chance of treatment and the top case is when the social
benefits are high enough that a municipality should choose a subsidy that
leads everyone to treat their trees.

#### Synthesis

Combining the results of the three possible cases leads to the following
optimal subsidy policy across all conditions. The municipality should choose

$s^{*}_{\hat{\phi}}=\begin{cases}0&c\leq ak_{\hat{\phi}}\\\
c-ak_{\hat{\phi}}&c>ak_{\hat{\phi}}\;\text{ and
}\;c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\leq B_{t\hat{\phi}}\\\
1/2\left(B_{t\hat{\phi}}+c-bk_{\hat{\phi}}\right)&c>ak_{\hat{\phi}}\;\text{
and }\;\mid
c-bk_{\hat{\phi}}\mid<B_{t\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\\\
0&c>ak_{\hat{\phi}}\;\text{ and }\;B_{t\hat{\phi}}\leq\mid
c-bk_{\hat{\phi}}\mid\end{cases}$ (113)

so that positive subsidies are only used when they will increase treatment of
trees, and only when the resulting treatments increase the welfare of society.
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.

###  S.I. 2.4 Outcomes under optimal behavior

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 above, the firms Nash equilibrium bids will
be

$p_{1\hat{\phi}}^{*}=p_{2\hat{\phi}}^{*}=\begin{cases}c&c\leq
ak_{\hat{\phi}}\\\ ak_{\hat{\phi}}&c>ak_{\hat{\phi}}\;\text{ and
}\;c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\leq B_{t\hat{\phi}}\\\
1/2\left(c+bk_{\hat{\phi}}-B_{t\hat{\phi}}\right)&c>ak_{\hat{\phi}}\;\text{
and }\;\lvert
c-bk_{\hat{\phi}}\rvert<B_{t\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\\\
c&bk_{\hat{\phi}}>c>ak_{\hat{\phi}}\;\text{ and }\;B_{t\hat{\phi}}\leq
bk_{\hat{\phi}}-c\\\ p\geq c&c>bk_{\hat{\phi}}\;\text{ and
}\;B_{t\hat{\phi}}\leq c-bk_{\hat{\phi}}\end{cases}$ (114)

The likelihood of treatment of a focal tree under the optimal behaviors
defined above is

$P^{*}(\text{treated})=\begin{cases}1&c\leq ak_{\hat{\phi}}\\\
1&c>ak_{\hat{\phi}}\;\text{ and }\;c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\leq
B_{t\hat{\phi}}\\\
\frac{B_{t\hat{\phi}}-c+bk_{\hat{\phi}}}{2k_{\hat{\phi}}(b-a)}&c>ak_{\hat{\phi}}\;\text{
and }\;\mid
c-bk_{\hat{\phi}}\mid<B_{t\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\\\
\frac{bk_{\hat{\phi}}-c}{k_{\hat{\phi}}(b-a)}&bk_{\hat{\phi}}>c>ak_{\hat{\phi}}\;\text{
and }\;B_{t\hat{\phi}}\leq bk_{\hat{\phi}}-c\\\ 0&c\geq
bk_{\hat{\phi}}\;\text{ and }\;B_{t\hat{\phi}}\leq
c-bk_{\hat{\phi}}\end{cases}$ (115)

and this leads to expected utility for the municipality of

$E[U_{m}^{*}]=\begin{cases}\Pi_{u\hat{\phi}}+B_{t\hat{\phi}}&c\leq
ak_{\hat{\phi}}\\\
\Pi_{u\hat{\phi}}+B_{t\hat{\phi}}-c+ak_{\hat{\phi}}&c>ak_{\hat{\phi}}\;\text{
and }\;c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\leq B_{t\hat{\phi}}\\\
\Pi_{u\hat{\phi}}+\frac{\left(B_{t\hat{\phi}}-c+bk_{\hat{\phi}}\right)^{2}}{4k_{\hat{\phi}}(b-a)}&c>ak_{\hat{\phi}}\;\text{
and }\;\mid
c-bk_{\hat{\phi}}\mid<B_{t\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\\\
\Pi_{u\hat{\phi}}+B_{t\hat{\phi}}\frac{bk_{\hat{\phi}}-c}{k_{\hat{\phi}}(b-a)}&bk_{\hat{\phi}}>c>ak_{\hat{\phi}}\;\text{
and }\;B_{t\hat{\phi}}\leq bk_{\hat{\phi}}-c\\\ \Pi_{u\hat{\phi}}&c\geq
bk_{\hat{\phi}}\;\text{ and }\;B_{t\hat{\phi}}\leq
c-bk_{\hat{\phi}}\end{cases}$ (116)

and an expected utility for the firms of

$E\left[U_{f_{1}}^{*}\right]=E\left[U_{f_{2}}^{*}\right]=0$ (117)

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

$U_{o}^{*}[\Delta_{o}]=\begin{cases}\pi_{u\hat{\phi}}+b_{t\hat{\phi}}-c&c\leq
ak_{\hat{\phi}}\\\
\pi_{u\hat{\phi}}+b_{t\hat{\phi}}-ak_{\hat{\phi}}&c>ak_{\hat{\phi}}\;\text{
and }\;c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\leq B_{t\hat{\phi}}\\\
\pi_{u\hat{\phi}}+b_{t\hat{\phi}}-\frac{1}{2}\left(c+bk_{\hat{\phi}}-B_{t\hat{\phi}}\right)&c>ak_{\hat{\phi}}\;\text{
and }\;\mid
c-bk_{\hat{\phi}}\mid<B_{t\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\;\text{
and }\;\Delta_{o}\geq\frac{1}{2}\left[c/k_{\hat{\phi}}+b-\Delta_{m}\right]\\\
\pi_{u\hat{\phi}}&c>ak_{\hat{\phi}}\;\text{ and }\;\mid
c-bk_{\hat{\phi}}\mid<B_{t\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\;\text{
and }\;\Delta_{o}<\frac{1}{2}\left[c/k_{\hat{\phi}}+b-\Delta_{m}\right]\\\
\pi_{u\hat{\phi}}+b_{t\hat{\phi}}-c&bk_{\hat{\phi}}>c>ak_{\hat{\phi}}\;\text{
and }\;B_{t\hat{\phi}}\leq bk_{\hat{\phi}}-c\;\text{ and }\;\Delta_{o}\geq
c/k_{\hat{\phi}}\\\
\pi_{u\hat{\phi}}&bk_{\hat{\phi}}>c>ak_{\hat{\phi}}\;\text{ and
}\;B_{t\hat{\phi}}\leq bk_{\hat{\phi}}-c\;\text{ and
}\;\Delta_{o}<c/k_{\hat{\phi}}\\\ \pi_{u\hat{\phi}}&c\geq
bk_{\hat{\phi}}\;\text{ and }\;B_{t\hat{\phi}}\leq
c-bk_{\hat{\phi}}\end{cases}$ (118)

where the top case corresponds to the setting where 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
municipalities best option is an intermediate subsidy which 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 owners 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 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.

##  Appendix S.I. 3 Special case where only firm 1 is subsidized

Here we assume that $s_{2\hat{\phi}}=0$, and $bk_{\hat{\phi}}<c$. These
restrictions imply that firm 2 does not receive a subsidy, that no private
owners value treatment enough to offset its cost in the absence of a subsidy.
We use this setting as a simplified setting within which to compare outcomes
under the optimal policy of equal subsidies with outcomes that arise when only
one firm receives a subsidy.

###  S.I. 3.1 Tree owner best response

As before, given a pair of bids, $(p_{1\hat{\phi}},p_{2\hat{\phi}})$, the tree
owner maximizes their utility by following the strategy defined by

$\sigma_{o\hat{\phi}}^{*}\left[\Delta_{o}\right]=\begin{cases}\text{treat
w/}f_{1}&\text{if }p_{1\hat{\phi}}<p_{2\hat{\phi}}\text{ and
}p_{1\hat{\phi}}\leq\Delta_{o}k_{\hat{\phi}}\\\ \text{treat w/}f_{2}&\text{if
}p_{2\hat{\phi}}<p_{1\hat{\phi}}\text{ and
}p_{2\hat{\phi}}\leq\Delta_{o}k_{\hat{\phi}}\\\ \text{flip coin}&\text{if
}p_{1\hat{\phi}}=p_{2\hat{\phi}}\text{ and
}p_{2\hat{\phi}}\leq\Delta_{o}k_{\hat{\phi}}\\\ \text{don't treat}&\text{if
}\Delta_{o}k_{\hat{\phi}}<\text{min}\\{p_{1\hat{\phi}},p_{2\hat{\phi}}\\}\end{cases}$
(119)

Which has not changed from earlier cases because subsidies have no direct
impact on the tree owner.

###  S.I. 3.2 Behavior of the firms

In this case, firms will maximize their expected utility, given the bid of the
opposing firm and their beliefs about the value of treatment to the tree
owner.

Previously, we derived best response correspondences for unequal subsidy
levels. We will apply these to the special case where $s_{2\hat{\phi}}=0$, and
$bk_{\hat{\phi}}<c$.

Firm 1 has the same best response as in the general case, which given

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

is

$\hat{p}_{1\hat{\phi}}\left(p_{2\hat{\phi}}\right)=\begin{cases}p_{1\hat{\phi}}\geq
c-s_{1\hat{\phi}}&s_{1\hat{\phi}}<c-bk_{\hat{\phi}}\text{ or
}s_{1\hat{\phi}}<c-p_{2\hat{\phi}}\\\
p_{2\hat{\phi}}-\epsilon&s_{1\hat{\phi}}\geq c-bk_{\hat{\phi}}\text{ and
}c-s_{1\hat{\phi}}\leq p_{2\hat{\phi}}\leq p_{1\hat{\phi}}^{\prime}\\\
p_{1\hat{\phi}}^{\prime}&s_{1\hat{\phi}}\geq c-bk_{\hat{\phi}}\text{ and
}p_{1\hat{\phi}}^{\prime}<p_{2\hat{\phi}}.\end{cases}$ (120)

However, due to the cost of treatment and the lack of subsidy, the best
response of firm 2 becomes

$\hat{p}_{2\hat{\phi}}\left(p_{1\hat{\phi}}\right)=\begin{cases}p_{2\hat{\phi}}\geq
c\end{cases}$ (121)

because firm 2 is unsubsidized and the cost of treatment is high enough that
no owners will accept the minimum profitable bid that they could submit. To
further simplify our analysis, we assume that

$\hat{p}_{2\hat{\phi}}\left(p_{1\hat{\phi}}\right)=c.$

These best responses can lead to three types of equilibrium bid pairs, given
our assumptions. We can aggregate the Nash equilibria across the three sub-
cases and use this to determine the best subsidy level from the perspective of
the municipality. We can define the cases in terms of the subsidy level set by
the municipality, $s_{1\hat{\phi}}$ as

$(p_{1\hat{\phi}}^{*},p_{2\hat{\phi}}^{*})=\begin{cases}(c-s_{1\hat{\phi}},c)&s_{1\hat{\phi}}<c-bk_{\hat{\phi}}\\\
\left(\frac{1}{2}\left(bk_{\hat{\phi}}+c-s_{1\hat{\phi}}\right),c\right)&c-bk_{\hat{\phi}}<s_{1\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\\\
(ak_{\hat{\phi}},c)&c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\leq
s_{1\hat{\phi}}\end{cases}$ (122)

where the first case implies that the subsidy for firm 1 is not high enough to
lead to a positive probability of treatment, so the details here are not
important. The second case arises when the subsidy level is high enough to
allow for a chance of firm 1 profitably treating the tree. The final case
arises when the subsidy level is high enough for firm 1 to bid a price that
will lead to treatment with certainty. Across all cases, firm 2 never has a
chance at winning the bid.

###  S.I. 3.3 Optimal municipal subsidy

Taking into account the above responses to their subsidy levels, the
municipality will seek to choose a subsidy level, $s_{1\hat{\phi}}\geq 0$ that
maximizes their expected utility. The expected utility function of the
municipality in this case is

$E\left[U_{m}(s_{1\hat{\phi}})\right]=\begin{cases}\Pi_{u\hat{\phi}},&s_{1\hat{\phi}}<c-bk_{\hat{\phi}}\\\
\Pi_{u\hat{\phi}}+\left(B_{t\hat{\phi}}-s_{1\hat{\phi}}\right)\frac{bk_{\hat{\phi}}-c+s_{1\hat{\phi}}}{2k_{\hat{\phi}}(b-a)},&c-bk_{\hat{\phi}}<s_{1\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\\\
\Pi_{u\hat{\phi}}+B_{t\hat{\phi}}-s_{1\hat{\phi}},&c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}<s_{1\hat{\phi}}\end{cases}$
(123)

where the first case implies that subsidies are too low for trees to get
treated, the middle case corresponds to subsidy levels which lead a positive
probability of the tree getting treated and the final case corresponds to the
expected utility that the municipality receives when subsidies are set high
enough to assure that the tree will be treated.

We can find the optimal subsidy by setting the partial derivative

$\frac{\partial}{\partial
s_{1\hat{\phi}}}E\left[U_{m}(s_{1\hat{\phi}})\right]=\begin{cases}0,&s_{1\hat{\phi}}<c-bk_{\hat{\phi}}\\\
\frac{B_{t\hat{\phi}}-bk_{\hat{\phi}}+c-2s_{1\hat{\phi}}}{2k_{\hat{\phi}}(b-a)},&c-bk_{\hat{\phi}}<s_{1\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\\\
-1,&c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}<s_{1\hat{\phi}}\end{cases}$ (124)

equal to zero. For the middle case this implies an optimal subsidy level of

$s^{\prime}_{1\hat{\phi}}=\frac{B_{t\hat{\phi}}+c-bk_{\hat{\phi}}}{2}$ (125)

which will lead to a positive probability of a tree being treated.

Recalling that we assumed at the outset of this section that
$bk_{\hat{\phi}}<c$, we can show that an optimal subsidy that leads to
treatment, $s^{\prime}_{1\hat{\phi}}>c-bk_{\hat{\phi}}$ will arise when
$B_{t\hat{\phi}}>c-bk_{\hat{\phi}}>0$. Thus the optimal subsidy policy for the
municipality is

$s^{*}_{1\hat{\phi}}(B_{t\hat{\phi}})=\begin{cases}0,&B_{t\hat{\phi}}<c-bk_{\hat{\phi}}\\\
\frac{1}{2}(B_{t\hat{\phi}}+c-bk_{\hat{\phi}}),&c-bk_{\hat{\phi}}\leq
B_{t\hat{\phi}}<c+3bk_{\hat{\phi}}-4ak_{\hat{\phi}}\\\
c+bk_{\hat{\phi}}-2ak_{\hat{\phi}},&c+3bk_{\hat{\phi}}-4ak_{\hat{\phi}}\leq
B_{t\hat{\phi}}\end{cases}$ (126)

with the first implying that a subsidy is not in the municipal best interest,
the second case leading to a positive probability that the tree will be
treated and the third case corresponding to the minimum subsidy that assures
that the tree will be treated.

This subsidy policy leads to equilibrium firm bids of

$(p_{1\hat{\phi}}^{*},p_{2\hat{\phi}}^{*})=\begin{cases}(c,c),&B_{t\hat{\phi}}<c-bk_{\hat{\phi}}\\\
\left(\frac{c+3bk_{\hat{\phi}}-B_{t\hat{\phi}}}{4},c\right),&c-bk_{\hat{\phi}}\leq
B_{t\hat{\phi}}<c+3bk_{\hat{\phi}}-4ak_{\hat{\phi}}\\\
\left(ak_{\hat{\phi}},c\right),&c+3bk_{\hat{\phi}}-4ak_{\hat{\phi}}\leq
B_{t\hat{\phi}}\end{cases}$ (127)

Which implies that the tree owner will never contract with firm 2 and firm 1
will effectively have a local monopoly. These bids will lead to a probability
function that the tree owner will treat their tree with firm 1 given by

$P^{*}(\text{treated})=\begin{cases}0,&B_{t\hat{\phi}}<c-bk_{\hat{\phi}}\\\
\frac{B_{t\hat{\phi}}-c+bk_{\hat{\phi}}}{4k_{\hat{\phi}}(b-a)},&c-bk_{\hat{\phi}}\leq
B_{t\hat{\phi}}<c+3bk_{\hat{\phi}}-4ak_{\hat{\phi}}\\\
1,&c+3bk_{\hat{\phi}}-4ak_{\hat{\phi}}\leq B_{t\hat{\phi}}\end{cases}$ (128)

With the strategy choices in hand, we can proceed to consider a welfare
analysis.

###  S.I. 3.4 Welfare analysis

Now we will examine the utility of each player that results when the all
players adhere to the Nash equilibrium strategies determined above. For firm 1
their Nash equilibrium expected utility is

$E\left[U_{f_{1}}^{*}\right]=\begin{cases}0,&B_{t\hat{\phi}}<c-bk_{\hat{\phi}}\\\
\frac{(B_{t\hat{\phi}}-c+bk_{\hat{\phi}})^{2}}{16k_{\hat{\phi}}(b-a)},&c-bk_{\hat{\phi}}\leq
B_{t\hat{\phi}}<c+3bk_{\hat{\phi}}-4ak_{\hat{\phi}}\\\
k_{\hat{\phi}}(b-a),&c+3bk_{\hat{\phi}}-4ak_{\hat{\phi}}\leq
B_{t\hat{\phi}}\end{cases}$ (129)

which is clearly at least as great as that which occurs under equal subsidies
because there is a chance for positive utility for the firm. Firm 2 still has
an expected utility of zero in all cases. The municipality has an expected
utility of

$E\left[U_{m}^{*}\right]=\begin{cases}\Pi_{u\hat{\phi}},&B_{t\hat{\phi}}<c-bk_{\hat{\phi}}\\\
\Pi_{u\hat{\phi}}+\frac{(B_{t\hat{\phi}}-c+bk_{\hat{\phi}})^{2}}{8k_{\hat{\phi}}(b-a)},&c-bk_{\hat{\phi}}\leq
B_{t\hat{\phi}}<c+3bk_{\hat{\phi}}-4ak_{\hat{\phi}}\\\
\Pi_{u\hat{\phi}}+B_{t\hat{\phi}}-c-bk_{\hat{\phi}}+2ak_{\hat{\phi}},&c+3bk_{\hat{\phi}}-4ak_{\hat{\phi}}\leq
B_{t\hat{\phi}}\end{cases}$ (130)

which is never greater than the utility that results from an optimal equal
subsidy policy. When treatment occurs with certainty, the expected utility of
the municipality is equivalent to the equal subsidy case only when $a=b$. In
other words, when there is no uncertainty about the value of saving a tree to
its owner. As $a$ decreases and/or $b$ increases, which leads to uncertainty
about the valuation that people have for their trees, then the utility
declines for the municipality below that which would be achieved under equal
subsidies. The intermediate case con only arise when there is some uncertainty
about the value tree owners place on their trees ($b>a$) and when this
intermediate case occurs, it leads to lower expected utility than that which
would result under equal subsidies. Further, the range of values of
$B_{t\hat{\phi}}$ for which this case arises is broader in this case where
$s_{2\hat{\phi}}=0$ than it is the the case where
$s_{1\hat{\phi}}=s_{2\hat{\phi}}=s_{\hat{\phi}}$.

A tree owner knows their own type, $\Delta_{o}$ and therefore they can
calculate they utility rather than expected utility, which is

$U_{o}^{*}[\Delta_{o}]=\begin{cases}\pi_{u\hat{\phi}},&B_{t\hat{\phi}}<c-bk_{\hat{\phi}}\\\
\pi_{u\hat{\phi}}+b_{t\hat{\phi}}-1/4(c+3bk_{\hat{\phi}}-B_{t\hat{\phi}}),&c-bk_{\hat{\phi}}\leq
B_{t\hat{\phi}}<c+3bk_{\hat{\phi}}-4ak_{\hat{\phi}}\text{ and }\Delta_{o}\geq
1/4(c/k_{\hat{\phi}}+3b-\Delta_{m})\\\
\pi_{u\hat{\phi}},&c-bk_{\hat{\phi}}\leq
B_{t\hat{\phi}}<c+3bk_{\hat{\phi}}-4ak_{\hat{\phi}}\text{ and
}\Delta_{o}<1/4(c/k_{\hat{\phi}}+3b-\Delta_{m})\\\
\pi_{u\hat{\phi}}+b_{t\hat{\phi}}-ak_{\hat{\phi}},&c+3bk_{\hat{\phi}}-4ak_{\hat{\phi}}\leq
B_{t\hat{\phi}}\end{cases}$ (131)

and which is never greater than the utility that results when the municipality
chooses equal optimal subsidies.

In sum, the analysis of the case where $s_{2\hat{\phi}}=0$ and
$c>bk_{\hat{\phi}}$ showed two main results. First, when there is no
uncertainty about the value of saving a tree to its owner, $\Delta_{o}$, then
the outcomes for this case exactly correspond to those that would occur for
equal optimal subsidy levels for both firms. However, when the municipality
and the firms are uncertain about the value of saving a tree, and
$\Delta_{o}\sim U[a,b]$ for $b>a$, then the outcomes that arise when
$s_{2\hat{\phi}}=0$ diverge from those that occur under equal subsidies. Firm
1 is better off, because the fact that the other firm is unsubsidized means
that they can behave like a monopolist and achieve positive profits. This
comes at a cost to the expected utility of the municipality and the tree
owner. Firm 1 does not pass all of the subsidy on to the tree owner and
therefore for any particular subsidy level, the likelihood of treatment will
typically be lower (or equivalent in some cases) to that which would result
under equal subsidies. Therefore it is more costly for the municipality to
treat the same number of trees. For tree owners, for any given subsidy level,
there is a decreased chance that they will find a suitable bid from firm 1 (or
firm 2) if they do choose to treat their tree they will tend to have a lower
utility that they would have under an equal subsidy policy.

This shows that firms might find it in their interest to advocate for
exclusive subsidies for the treatment of private trees. Should this arise,
then it would be in the interest of tree owners and the municipality to reduce
uncertainty around the value of treatment, because less uncertainty leads to
higher (expected) utility for the municipality and the tree owner.

##  Appendix S.I. 4 Treatment of municipally owned trees

This paper focuses on the management of forest pests within a mosaic of
private and public ownership of trees. This pattern of ownership is
characteristic of urban forests, where parcels of private and public land
intermingle to create a complex management setting where uniform management
across these ownership types can be difficult to achieve. In the main text, we
present an analysis of public subsidy programs that can result in socially
optimal treatment actions taken by private tree owners. In this section, we
consider a simpler problem: the optimal treatment choices for publicly owned
trees. Much of the setup and notation remains the same from previous sections.
Here we assume that public trees can be treated at a cost, $c$, by the
municipality. The problem for a forest manager is to identify the subset of
trees for which the social benefit of treatment is greater than the public
cost. Previously we defined the social value of a healthy tree as $V_{m}$ and
this value remains unchanged. However, the social cost of a dying municipally
owned tree will be higher than the social cost of a privately owned tree
because in the case of a dying municipally owned tree, the social cost also
includes the removal cost of the tree. Thus we define $W^{\prime}_{m}$ as the
social cost of a dying tree, including removal costs. Therefore the net social
benefit of saving a public tree is $\Delta^{\prime}_{m}=V_{m}+W^{\prime}_{m}$.
As before, denote the probability that treating a tree assessed in state
$\hat{\phi}$ will save it as $k_{\hat{\phi}}$ and with this we can define the
social benefit of treating a municipally owned tree as

$B^{\prime}_{\hat{\phi}}=\Delta^{\prime}_{m}k_{\hat{\phi}}$ (132)

which can be used to define the cases where a municipal tree should be treated
with the optimal probability of treating a public tree given by

$P^{*}_{tm}(\hat{\phi})=\begin{cases}1&c\leq B^{\prime}_{t\hat{\phi}}\\\
0&c\geq B^{\prime}_{t\hat{\phi}}\end{cases}$ (133)

this equation provides a strikingly simple proscription for treatment of
public trees. However, $B^{\prime}_{t\hat{\phi}}$ depends on $k_{\hat{\phi}}$,
which in turn depends on treatment effectiveness, assessment accuracy, and the
state of EAB infestation within the community.

##  Appendix S.I. 5 Compartmental model

Here, we derive a dynamic model of an urban ash tree population that is
subject to an EAB infestation and management. We assume that municipally and
privately owned trees are separate classes of trees with distinct management
practices, yet the spread of EAB does not distinguish these differences. We
will use the model to illustrate the performance of the optimal subsidy and
treatment policies for public (municipally owned) 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 across a population
of municipal and private 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 disease 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, $D_{m}$. The municipally owned trees
interact with three classes of trees with private owners, $H_{o}$, $I_{o}$,
and $D_{o}$. Since we measure the fraction of trees in each category, we have
$H_{m}+I_{m}+D_{m}+H_{o}+I_{o}+D_{o}=1$.

### Infestation rate

We assume that a infested tree emits EAB at a rate of $\lambda$ regardless of
whether or not it has been treated and regardless of whether it is publicly or
privately owned (however, treatment can result in the tree becoming healthy
which would stop the spread of EAB). The probability that a healthy private
tree (for example) is exposed to EAB will therefore be $\lambda
H_{o}(I_{m}+I_{o})$ following the mass action kinetics assumptions of simple
compartmental models. However, not all exposures to EAB result in infestation.
We assume the probability of the establishment of an infestation given
exposure is $\delta$ for untreated trees and $\delta_{t}=\delta(1-h_{th})$ for
treated trees, so that treatment lessens the likelihood of the establishment
of an infestation. This results in an infestation rate for healthy treated
municipal trees of $P_{thm}\delta_{t}\gamma H_{m}(I_{m}+I_{o})$, where
$P_{thm}$ is the probability of treatment for healthy municipally owned trees.
Following similar approaches for other infestation terms we can write the
infestation rate for municipal trees as

$\left(P_{thm}\delta_{t}+(1-P_{thm})\delta\right)\gamma H_{m}(I_{m}+I_{o})$

and the infestation rate of private trees as

$\left(P_{tho}\delta_{t}+(1-P_{tho})\delta\right)\gamma H_{o}(I_{m}+I_{o})$

which shows that the rate at which public and private trees become infested
varies mainly through differences in treatment probabilities, but these
differences can lead to downstream changes in the prevalence of healthy trees,
which also shapes the rate of new infestations. We can simplify our expression
for the infestation rate as

$\left(1-h_{th}P_{thm}\right)\beta H_{m}(I_{m}+I_{o})$ (134)

for municipally owned trees where $\beta=\delta\gamma$ is the net infestation
rate parameter that combines spread and establishment under baseline
conditions. This rate of spread is modified by the impact and likelihood of
treatment taking into account the differential infestation states that could
arise among public and private trees. The expression for private trees can be
similarly simplified.

### Recovery Rate

In our model, we allow for recovery of infested trees and let the likelihood
of recovery depend on treatment and we assume that the effectiveness of
treatment is equal across municipal and private trees. The recovery rate for
treated trees is $\alpha_{t}$ and the recovery rate for untreated trees is
$\alpha_{u}$ resulting is a total recovery rate of

$\left(\alpha_{t}P_{tio}+\alpha_{u}(1-P_{tio})\right)I_{o}$

for private trees, and

$\left(\alpha_{t}P_{tim}+\alpha_{u}(1-P_{tim})\right)I_{m}$

for municipally owned trees, where $P_{tio}$ is the probability of treatment
for infested trees with private owners and $P_{tim}$ is the probability of
treatment for infested municipally owned trees. Further, we assume that
$\alpha_{t}=\alpha h_{ti}$ where $h_{ti}$ is the probability that a treated
infested tree will recover and $\alpha$ is the rate at which recovery occurs.
Similarly, we let $\alpha_{u}=\alpha h_{ui}$ where $h_{ui}$ is the probability
that an untreated tree recovers from infestation. In the case of EAB, these
odds are very low. We define $t_{i}=h_{ti}-h_{ui}$ as the marginal impact of
treatment on the likelihood of tree recovery given an active infestation.
Using this, we can simplify our expression for the recovery rate as

$\left(h_{ui}+t_{i}P_{tim}\right)\alpha I_{m}$ (135)

for municipally owned trees.

### Death rate

While some infested trees can recover, others will eventually die. The death
rate for a treated tree is $\gamma_{t}$ and for an untreated tree is
$\gamma_{u}$. Therefore the rate at which infested municipal trees die is

$\left(\gamma_{u}(1-P_{tim})+\gamma_{t}P_{tim}\right)I_{m}$

and the rate at which infested private trees die is

$\left(\gamma_{u}(1-P_{tio})+\gamma_{t}P_{tio}\right)I_{o}$

where we further assume that the death rate parameters can be expressed in
terms of the probability that treatment (or non-treatment) fails to cause a
tree to recover and the baseline death rate for infested trees, with
$\gamma_{t}=\gamma(1-h_{ti})$ and $\gamma_{u}=\gamma(1-h_{ui})$. We can again
use $t_{i}=h_{ti}-h_{ui}$ to describe the marginal impact of treatment on the
likelihood of tree recovery given an active infestation. Using this, we can
simplify our expression for the death rate as

$\left(1-h_{ui}-t_{i}P_{tim}\right)\gamma I_{m}$ (136)

for municipally owned trees.

### Dynamical system

Given these descriptions of the infestation, recovery and death rates for
municipal and private trees, we can define a system of equations that
describes the dynamics of each health class of tree given its ownership
status. We denote the rate of change of the fraction of healthy municipal
trees, for example, as

$\frac{dH_{m}}{dt}=\dot{H}_{m}$

for simplicity of notation. Using this notation can write a complete
description of the dynamical system as

$\displaystyle\dot{H}_{m}=-\beta\left(1-h_{th}P_{thm}\right)H_{m}(I_{m}+I_{o})+\alpha\left(h_{ui}+t_{i}P_{tim}\right)I_{m}$
(137)
$\displaystyle\dot{H}_{o}=-\beta\left(1-h_{th}P_{tho}\right)H_{o}(I_{m}+I_{o})+\alpha\left(h_{ui}+t_{i}P_{tio}\right)I_{o}$
(138)
$\displaystyle\dot{I}_{m}=\beta\left(1-h_{th}P_{thm}\right)H_{m}(I_{m}+I_{o})-\gamma\left(1-h_{ui}-t_{i}P_{tim}\right)I_{m}-\alpha\left(h_{ui}+t_{i}P_{tim}\right)I_{m}$
(139)
$\displaystyle\dot{I}_{o}=\beta\left(1-h_{th}P_{tho}\right)H_{o}(I_{m}+I_{o})-\gamma\left(1-h_{ui}-t_{i}P_{tio}\right)I_{o}-\alpha\left(h_{ui}+t_{i}P_{tio}\right)I_{o}$
(140) $\displaystyle\dot{D}_{m}=\gamma\left(1-h_{ui}-t_{i}P_{tim}\right)I_{m}$
(141) $\displaystyle\dot{D}_{o}=\gamma\left(1-h_{ui}-t_{i}P_{tio}\right)I_{o}$
(142)

### Municipal tree treatment scenarios

We have described the dynamics of EAB infestation on a mosaic of interacting
public and private land given the probabilities of treatment across these
categories of trees. Here we construct treatment scenarios based on the game
analysis described earlier in this manuscript. For municipally owned trees, we
consider two scenarios. First, we explore the case where no trees are treated.
Next, we consider the case where trees are optimally treated, given by

$P^{*}_{tm}(\hat{\phi})=\begin{cases}1&c\leq B^{\prime}_{t\hat{\phi}}\\\
0&c\geq B^{\prime}_{t\hat{\phi}}\end{cases}$ (143)

which was derived in Supplementary Information Section S.I. 4. The assessment
of trees is imperfect, so there is not perfect alignment between assessment
and the actual state of a trees health. We can write the probability that a
healthy public tree is treated as

$P^{*}_{thm}=P(\hat{h}\mid h)P^{*}_{tm}(\hat{h})+P(\hat{i}\mid
h)P^{*}_{tm}(\hat{i})+P(\hat{d}\mid h)P^{*}_{tm}(\hat{d})$ (144)

and an infested public tree’s treatment probability is

$P^{*}_{tim}=P(\hat{h}\mid i)P^{*}_{tm}(\hat{h})+P(\hat{i}\mid
i)P^{*}_{tm}(\hat{i})+P(\hat{d}\mid i)P^{*}_{tm}(\hat{d}).$ (145)

The treatment rates for dead trees have no impact of the dynamics of
infestation in our model.

### Private tree treatment scenarios

For privately owned trees, we consider three scenarios. First, we examine the
case where there is no treatment of any type of privately owned trees and
$P_{tho}=P_{tio}=P_{tdo}=0$. The second case is where treatment is subsidized,
and thus the treatment subsidy received by each firm are
$s_{\hat{h}}=s_{\hat{i}}=s_{\hat{d}}=0$. Finally, we consider the case of
optimal subsidies for treatment of privately owned trees. Optimal subsidies
result in treatment probabilities of trees assessed in state
$\hat{\phi}\in\\{\hat{h},\hat{i},\hat{d}\\}$ of

$P^{*}_{to}(\hat{\phi})=\begin{cases}1&c\leq ak_{\hat{\phi}}\;\text{ or
}\;c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\leq B_{t\hat{\phi}}\\\
\frac{B_{t\hat{\phi}}-c+bk_{\hat{\phi}}}{2k_{\hat{\phi}}(b-a)}&c>ak_{\hat{\phi}}\;\text{
and }\;\mid
c-bk_{\hat{\phi}}\mid<B_{t\hat{\phi}}<c+bk_{\hat{\phi}}-2ak_{\hat{\phi}}\\\
\frac{bk_{\hat{\phi}}-c}{k_{\hat{\phi}}(b-a)}&bk_{\hat{\phi}}>c>ak_{\hat{\phi}}\;\text{
and }\;B_{t\hat{\phi}}\leq bk_{\hat{\phi}}-c\\\ 0&c\geq
bk_{\hat{\phi}}\;\text{ and }\;B_{t\hat{\phi}}\leq
c-bk_{\hat{\phi}}\end{cases}$ (146)

and recalling that assessments are imperfect we can use this expression to
write the probability of a tree that is healthy being treated as

$P^{*}_{tho}=P(\hat{h}\mid h)P^{*}_{to}(\hat{h})+P(\hat{i}\mid
h)P^{*}_{to}(\hat{i})+P(\hat{d}\mid h)P^{*}_{to}(\hat{d})$ (147)

and an infested tree’s treatment probability is

$P^{*}_{tio}=P(\hat{h}\mid i)P^{*}_{to}(\hat{h})+P(\hat{i}\mid
i)P^{*}_{to}(\hat{i})+P(\hat{d}\mid i)P^{*}_{to}(\hat{d}).$ (148)

Combining the municipal and private treatment scenarios yields six distinct
cases for consideration. In our case study, we project the ash population
under a baseline scenario where there is no treatment of public or private
trees or other actions to slow the spread of EAB. Then, we project the ash
population under a more realistic baseline where there is no municipal subsidy
for treatment of private trees but some tree owners may nonetheless treat
their trees. Finally, we compare outcomes of these baseline cases with the
trajectory of the ash population under the optimal treatment policies for
municipal and private trees.

Parameter | Figure 5 Values | Figure 6 Values | Figure 7-8,S.I. 2 Values
---|---|---|---
$a$ | 600 | 600 | 600
$b$ | 900 | 900 | 900
$c$ | 300 | 300 | 300
$P(h)$ | 0.7 | 0-1 | 0-1
$P(i)$ | 0.25 | 0-1 | 0-1
$P(d)$ | 0.05 | 0-1 | 0-1
$P(\hat{h}\mid h)$ | 0.89 | 0.89 | 0.89
$P(\hat{i}\mid h)$ | 0.1 | 0.1 | 0.1
$P(\hat{d}\mid h)$ | 0.01 | 0.01 | 0.01
$P(\hat{h}\mid i)$ | 0.49 | 0.49 | 0.49
$P(\hat{i}\mid i)$ | 0.5 | 0.5 | 0.5
$P(\hat{d}\mid i)$ | 0.01 | 0.01 | 0.01
$P(\hat{h}\mid d)$ | 0.01 | 0.01 | 0.01
$P(\hat{i}\mid d)$ | 0.19 | 0.19 | 0.19
$P(\hat{d}\mid d)$ | 0.8 | 0.8 | 0.8
$h_{th}$ | 1 | 1 | 1
$h_{uh}$ | 0.7 | 0.7 | 0.7
$h_{ti}$ | 0.9 | 0.9 | 0.9
$h_{ui}$ | 0 | 0 | 0
$h_{td}$ | 0 | 0 | 0
$h_{ud}$ | 0 | 0 | 0
$\Delta_{m}$ | 0-1000 | 550 | 550
$\Delta^{\prime}_{m}$ | N/A | N/A | 1000
$\Delta_{o}$ | 600-900 | 600-900 | 600-900
$\beta$ | N/A | 1 | 1
$\alpha$ | N/A | 1 | 1
$\gamma$ | N/A | 0.3 | 0.3
Table S.I. 1: Case study parameter values, which apply to Figures 5-S.I. 2
Figure S.I. 2: Using parameter values for spread rates drawn from the
literature we illustrate the expected dynamics of EAB in a mixed landscape of
public and private tree ownership over time across six treatment scenarios.
These plots show aggregate dynamics of EAB spread averaged across public and
private ownership. The first row shows cases where publicly owned trees are
not treated and the second row shows cases where public trees are treated
optimally. The first column 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 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 almost 50 years. For the parameter values we consider, optimal
subsidy policy combined with optimal treatment of public trees is required to
prevent the establishment and spread of EAB within the community.


# IMPROVED SPEECH PRE-TRAINING WITH Supervision-ENHANCED ACOUSTIC UNIT

###### Abstract

Speech pre-training has shown great success in learning useful and general
latent representations from large-scale unlabeled data. Based on a well-
designed self-supervised learning pattern, pre-trained models can be used to
serve lots of downstream speech tasks such as automatic speech recognition. In
order to take full advantage of the labed data in low resource task, we
present an improved pre-training method by introducing a supervision-enhanced
acoustic unit (SEAU) pattern to intensify the expression of comtext
information and ruduce the training cost. Encoder representations extracted
from the SEAU pattern are used to generate more representative target units
for HuBERT pre-training process. The proposed method, named SeHuBERT, achieves
a relative word error rate reductions of 10.5% and 4.9% comared with the
standard HuBERT on Turkmen speech recognition task with 500 hours and 100
hours fine-tuning data respectively. Extended to more languages and more data,
SeHuBERT can aslo achieve a relative word error rate reductions of
approximately 10% at half of the training cost compared with HuBERT.

Index Terms—  self-supervised learning, supervision-enhanced acoustic unit,
low resource, speech recognition

##  1 Introduction

The research on automatic speech recognition (ASR) system has seen great
achievement in the recent decades [1, 2, 3, 4, 5]. However, ASR has suffered
from the reliance on human annotations, which is too costly or impractical for
the popularization and application of most languages around the world.
Therefore, there has been a lot of interest in how to better use unlabeled
speech data [6, 7, 8],especially in low resource languages.

The most widely used method in this study can be classified into two
categories: classical self-training and self-supervised learning
(SSL).Classical self-training method achieves releative considerable
improvement [2, 9, 10, 11, 12] by pseudo-labeling unannotated audio data and
then retraining the final system with the additional labeled data. Self-
supervised learning typically utilizes speech pre-training to learn better
representations. Based on a pre-trained model, the recognition system only
needs to fine-tune on a small amount of labeled data [13] and can achieve a
competitive performance compared with the supervised models trained with
quantities of labeled data. Moreover, speech pre-training is easier to operate
compared with the complex cycle training process and careful selection
strategy of pseudo labels in classical self-training method. Therefore, speech
pre-training has received a lot of attention and achieved great success [13,
14, 15, 16, 17, 18, 19, 20].

As one of the effective pre-training methods in recent studies, Hidden-Unit
BERT (HuBERT) [18] benefits from an offline clustering step to provide target
labels for a BERT-like prediction loss [21]. However, like most other pre-
training methods, HuBERT targets at learning a model that is able to produce
the universal speech representations, which can be used for multiple
downstream tasks. Instead, the labeled data used in the downstream tasks may
not provide more help to the pre-training process. In terms of ASR task, We
argue that the performance could be further improved if the network focus on
learning content information with some guidance information such as labeling
information. Meanwhile, as the standard configuration option, multiple
iteration pattern of the pre-training model may increase the difficulty of
promotion. Obviously, better initial target labels from supervised model can
reduce the training cost and improve the specific representation while most
methods do not fully exploit these advantages in low resource application
tasks.

To this end, we use a supervised model trained with the labeled data which can
be used in the downstream task, to guide the generation of target labels for
HuBERT pre-training. We call the newly designed paradigm supervision-enhanced
hidden unit BERT (SeHuBERT) which is better suited to the low resource
application task. The experiments are mainly conducted on a Turkmen speech
recognition task. Compared with the standard HuBERT, the results show that the
proposed SeHuBERT method achieves a significant improvement for different
scales of data in just one iteration. Extended to more languages and more
data, SeHuBERT also has stable performance improvement at half of the training
cost.

The rest of this paper is organized as follows. Section 2 gives a brief
description of the related work. The paradigm of the proposed SeHuBERT pre-
training method is introduced in Section 3. Section 4 mainly shows our
experimental setup and other details, including the experiment results and
ablations. Finally, the discussion and conclusion is presented in Section 5.

Fig. 1: Illustration of SeHuBERT.

##  2 Related Works

HuBERT [18] is a typical approach for self-supervised speech representation
learning, which utilizes an offline clustering step to generate aligned target
labels for a BERT-like pre-training. Researchers argue that wav2vec 2.0 only
explores quantizing the waveform encoder output, which may not be the best
feature for quantization due to the limited capacity of the convolutional
feature extractor. Different from wav2vec 2.0, HuBERT benefits from the
offline clustering step due to the consistency of the targets, not just their
correctness. The generated target labels are also called acoustic units or
target codes. What is more, the prediction loss is only applied over the
masked regions, forcing the model to learn a combined acoustic and language
model over the continuous inputs.

HuBERT pre-training adopts an iterative re-clustering and re-training process:
For the first iteration, the discrete targets are obtained by clustering on
the MFCC features extracted from the raw audio data. For the second iteration,
the training targets are updated by clustering the hidden representations of
the HuBERT model after the first iteration. According to the authors’ studies,
acoustic units generated from the middle to top layers have much higher mutual
information score than other layers, indicating that they are better at
phonetic information learning.

##  3 Proposed Method

The standard HuBERT paradigm provides more universal speech representations
suited for different speech tasks. In specific low-resource recognition tasks,
a small amount of labeled data is still readily available. Utilizing the prior
information from a few labeled data in the pre-training stage is of great
interest and worth exploring. Combining some guidance information from
downstream labeled data and pre-training strategy, we could take full
advantage of self-training advantages to improve the performance of
representation at a lower training cost.

Starting with the enhancement of these guidance information, we propose an
improved speech pre-training paradigm named supervise-enhanced acoustic unit
HuBERT (SeHuBERT) for these low resource application task.The architecture of
SeHuBERT is illustrated as Figure 1. We firstly train a supervised model with
a certain amount of labeled speech data from the downstream tasks. Then the
unsupervised speech-only data are fed to the supervised model encoder to
generate the continuous deep representations. After that, simple discrete
latent variable models such as k-means are used to generate discrete target
units with the deep representations for HuBERT pre-training. Both supervised
and discretized models together constitute the SeHuBERT paradigm.

###  3.1 Supervision-enhanced acoustic unit pattern (SEAU)

In standard HuBERT pre-training process, the discrete targets are assigned by
clustering on the MFCC features for the first iteration. However, MFCC
features are artificial designed, which may have certain limitations in
modeling acoustic contextual information. Furthermore, the model quality of
the first iteration directly influences the effect of the second iteration.
Therefore, instead of generating acoustic units from MFCC features, we use the
deep representations extracted from a supervised model to strengthen the
content presentation and reduce the training cost.

We use the attention-based encoder-decoder (ED) model as the baseline
architecture in this work. For SEAU pattern, firstly, a speech recognition
model is trained with the labeled data in downstream task to minimize the
cross entropy loss as follows:

$L(\theta;\boldsymbol{X},\boldsymbol{Y})=-\sum_{i=1}^{N}logp(\boldsymbol{Y}|\boldsymbol{X},\theta)$
(1)

Where $\boldsymbol{X}$ is the sequence of input speech frames,
$\boldsymbol{Y}$ is the corresponding sequence of labels, which contains N
tokens. $\theta$ are the model parameters to be optimized.

Once speech recognition model has converged, it can be used to extract the
deep context representations. To get the frame-level aligned target units, we
choose the output of deep encoder layer from the supervised model as
clustering features. Then, for the speech-only pre-training data, the deep
representations can be obtained from the forward inference through the chosen
encoder layers. Finally, the continuous deep representations are discretized
by a clustering model to obtain the pre-training targets. The SEAU pattern is
denoted with
$\boldsymbol{Z}=h(\boldsymbol{Encoder}(\boldsymbol{X}))=[\boldsymbol{z}_{1},\boldsymbol{z}_{2},\dots,\boldsymbol{z}_{T}]$,
where $\boldsymbol{z}_{t}\in[\boldsymbol{C}]$ is a class categorical variable,
and $h$ is a clustering model such as k-means.

The deep features of speech recognition network contain higher-dimensional
acoustic information, and thus are more robust to speaker variations and noise
interferences compared to the MFCC features. More importantly, since the model
is trained on labeled data, the deep features contain rich accurate contextual
information. Therefore, the final clustering acoustic units force the pre-
training model concentrate on learning content information, which is helpful
to the ASR task.

Obviously, we can also use the forced alignment from a hybrid model [22] or a
connectionist temporal classification (CTC) [23] model as the target units.
Therefore, the proposed SEAU is generic for different acoustic models.
Moreover, the process of clustering is similar to the decision tree clustering
and state binding in hybrid models, which can merge similar tokens together
and get more robust target units.

###  3.2 HuBERT Pre-training

After the supervision-enhanced acoustic units are obtained in an offline
clustering step, we use them to pre-train the HuBERT model. The backbone of
HuBERT in our work includes a feature extractor and a context representation
encoder. During pre-training process, the feature extractor module takes the
frame-level speech as the input to generate a higher dimensional features.
Then some frames are randomly selected and furtuer masked. The corrupted
features $\boldsymbol{\tilde{X}}$ are then fed into the encoder and
transformed into the hidden states $\boldsymbol{H}$ at the top layer. The
network is optimized to predict the discrete target units $\boldsymbol{Z}$.
The distribution over target units is parameterized as:

$p(c|\boldsymbol{\tilde{X}},t)=\frac{exp(sim(\boldsymbol{W}h_{t}),e_{c}/\tau)}{\sum_{c^{{}^{\prime}}=1}^{C}exp(sim(\boldsymbol{W}h_{t}),e_{c}^{{}^{\prime}}/\tau)}$
(2)

Where $\boldsymbol{W}$ is a projection matrix, $h_{t}$ is the hidden state at
frame $t$, $e_{c}$ is the embedding for unit c, $sim(a,b)$ computes the cosine
similarity between two vectors and $\tau=0.1$ scales the logit. A key
ingredient of HuBERT is the prediction loss is only applied over the masked
regions, forcing the model to learn high-level representations of unmasked
inputs to infer the targets of masked ones correctly.

In this work, we replace transformer block [22] in standard HuBERT with
conformer [4] beacuse conformer has presented higher effectiveness in ASR
task. The major component of Conformer is a stack of conformer blocks, each of
which is a series of multi-headed self-attention, depth-wise convolution and
feed-forward layers. On the other hand, to improve computational efficiency,
we use filter-bank features as input of network instead of raw waveforms. The
feature extractor that acts as a convolutional sub-sampling block is replaced
with two 2D-convolution layers, both with strides $(2,2)$, resulting in a 4x
reduction in the acoustic sequence length.

###  3.3 Fine-tuning

For simplicity, we keep the ASR network structure the same as supervised model
in section 3.1 during fine-tuning stage. The difference is that the encoder is
initialized with the pre-trained model. Simple yet effective, transformer
decoder is kept. We insert a linear layer between the pre-trained HuBERT model
and the decoder layer as the projection block. The whole ASR network is
optimized to minimize the cross entropy loss with labeled data.

##  4 Experiments

###  4.1 Datasets

We mainly evaluate the proposed methods on a Turkmen speech recognition task
with low resource. The entire dataset has only 500 hours of training data. We
randomly select 100 hours as a low-resource subset. In addition, we use
another dataset as out-domain data to evaluate the ability of SEAU to learn
cross-lingual representations. The out-domain data contains 476 hours of
Hungarian, a language quite distinct from Turkmen. We also randomly choose 100
hours as a subset. The test set has 1000 utterances of Turkmen, total in about
3 hours. All the speech files are sampled at 16K Hz with 16 bits.Furthermore,
a Portuguese recognition task with 1600 hours training data and a French
recognition task with 4300 hours training data are introduced to prove the
practicability and generalization of SeHuBERT.

###  4.2 Model Setups

The main framework of the supervised model in both SEAU and fine-tuning stage
are kept consistent for simplicity. The whole ASR encoder consists of a
feature extractor and a representation encoder. Feature extractor has 2
convolutional layers with ?lter size (3, 3) and strides (2, 2). Representation
encoder consists of 16 Conformer blocks, each with encoder embedding 512,
hidden dimension 2048, 8 attention heads and convolution kernel size 15. The
final projection layer dimension is 768. In total, the whole encoder including
feature extractor has about 100 million trainable parameters. The decoder
consists of 6 transformer decoder layers, each with decoder embedding 512,
hidden dimension 2048, 8 attention heads. We take specaugment [28] with mask
parameter (F=15, T=40) as a basic configuration. The inputs of the model are
the 40-dimensional log Mel-scale filter-bank features. Modeling units are 15k
BPE sub-words [24] trained from the labeled data.

###  4.3 Training Details

We train a standard HuBERT model with our modified architecture as the pre-
training baseline. We follow the configuration of the acoustic unit pattern
and pre-training configures described in [18]. All subsequent experiments were
performed by the fairseq toolkit [25] and run on a server equipped with 4 v100
GPUs with adam optimizer [26].

SEAU pattern. For in-domain evaluation, we train the supervised model on 100
hours and 500 hours of Turkmen data respectively. While for out-domain
evaluation, we train the supervised model on 100-hours subset of Hungarian
data. For all the supervised models, the entire 500 hours of Turkmen speech
are forwarded through the model encoder and produce deep representations. We
run k-means clustering with 1000 clusters on the supervised deep
representations for the HuBERT training. As a whole, We train the supervised
model for 15 epochs which is a small price compared with the pre-training and
fine-tuning process.

Pre-training. We train the SeHuBERT model for one iteration with 400k steps.
Dropout 0.1 was used in the conformer and at the output of the feature
extractor. Layers were dropped at a rate of 0.2. We optimize with Adam,
warming up the learning rate for the first 8000 updates to a peak of 0.0005,
and then linearly decay it. We also regularize the model by applying a L2
penalty to the activations of the final layer of the feature extractor and
scale down the gradients for the extractor by a factor of 10.

Fine-tuning. The fine-tuning step is similar to the supervised training except
that the model encoder is initialized from the pre-trained SeHuBERT model. The
learning rate warms up linearly from 0 to the peak learning rate 0.0007 for
the first 8000 training steps, decays to the 15-th epoch and then is halved
for each epoch from the 16-th epoch. During fine-tuning, the feature extractor
parameters were fixed.

###  4.4 Results and Discussion

Table 1 shows the word error rate (WER) of the no pre-training, HuBERT and
SeHuBERT model on Turkmen test set with various amount of fine-tuning data
while the training data used in SEAU and HuBERT are all fixed 500 hours.
Iter-1 and iter-2 stands for the first and second iteration of HuBERT
respectively. It can be seen that HuBERT significantly outperforms the model
without pre-training, achieving a relative 20.3% and 61.9% WER reduction
respectively, which shows the importance of pre-training to the performance of
low-resource speech recognition task. Furthermore, the proposed SeHuBERT
method achieves a relative 4.9% WER reduction compared with the final result
of standard HuBERT on a 500 hours data set and a relative 10.5% WER reduction
on a 100 hours data set. Along with the significant recognition performance
improvement, the new paradigm cut their training cost in half.

Table 1: WER (%) results of the proposed method on Turkmen  Method |  fine-tuning data
---|---
100h |  500h
No pre-training |  53.63 |  16.20
HuBERT-iter1 |  35.86 |  16.29
HuBERT-iter2 |  20.41 |  12.91
SeHuBERT |  18.26 |  12.28

We also presented the results on Portuguese and French task. The results shown
in Table 5 are consistent with the results on Turkmen task. The proposed
SeHuBERT method achieves a relative 8.7% WER on Portuguese task and a relative
10.5% WER reduction over the standard HuBERT model. In terms of both
effectiveness and efficiency, the proposed SeHuBERT may be more advisable.

Table 2: WER (%) results of SeHuBERT on different languages  Method |  Portuguese |  French
---|---|---
HuBERT-iter2 |  7.14 |  6.29
SeHuBERT |  6.52 |  5.63

###  4.5 Ablation Studies

Amount of supervised data in SEAU Pattern: Intuitively, the quality of the
supervised model in SEAU affects the effectiveness of the target units used
for pre-training. Therefore, we first explore the impact of the amount of
supervised data in SEAU, which is closely related to the performance of the
supervised model. We use the 100h subset of Turkmen to train the supervised
model in SEAU and compare it with the entire 500h set. The results are shown
in Table 3. Overall, the performance of SEAU with 500h data is better than
that of 100h data. But the gap narrows as the model is fine-tuned with more
labeled data. The quality of the targets generated from the SEAU pattern with
100h labeled data are worse than that with 500h labeled data. Therefore, with
a well-trained supervised model for SEAU, HUBERT can achieve good performance
with only one iteration.

Table 3: WER (%) results of different amount of supervised data used in SEAU pattern fine-tuning data based on SeHuBERT |  data used in SEAU
---|---
100h |  500h
100h |  20.45 |  18.26
500h |  13.02 |  12.28

Source of supervised data in SEAU Pattern:We then investigate the effect of
the data from different source for the supervised model in SEAU pattern. We
use a 100h subset of Hungarian to train the supervised model to generate the
target units. It is expected that the SEAU can learn cross-lingual speech
information. The Hungarian subset is denoted as out-domain, while the
corresponding 100h Turkmen subset is denoted as in-domain. As shown in Table
4, out-domain case achieves a competitive performance compared with the in-
domain source on 100h fine-tuning data. Though on 500h fine-tuned data, out-
domain performs worse than in-domain, it is comparable to the standard HuBERT
model shown in table Table 1.

Table 4: WER (%) results of different source of supervised data used in SEAU pattern fine-tuning data based on SeHuBERT | domains used in SEAU/100h
---|---
In-domain | Out-domain
100h | 20.45 | 20.21
500h | 13.02 | 13.65

Clustering Number in SEAU Pattern: The clustering number is another factor
that influence the pre-training performance. We compare the impact of
different clustering number in the case of different amount of supervised data
used in SEAU pattern. The results are summarized in Table 5, $C$ refers to the
clustering number. We find that a larger number of clusters achieves better
performance when using 100h supervised data in SEAU pattern. However, for 500h
in SEAU, the conclusion is the opposite. It is speculated that the result is
in a normal fluctuation range.

Table 5: WER (%) results of different cluster numbers used in SEAU pattern Clustering number of SEAU | data used in SEAU
---|---
100h | 500h
C500 | 20.62 | 17.94
C1000 | 20.45 | 18.26

##  5 Conclusion

In this work, we propose a novel speech pre-training method named SeHuBERT, in
which a supervised model is utilized to guide the generation of the acoustic
units for HuBERT pre-training. The idea of the proposed method is to force the
model concentrate on learning content information during pre-training.
Experimental results show that based on a supervised model without detailed
design, the proposed SeHuBERT achieves relative word error rate reductions of
10.5% and 4.9% comared with the standard HuBERT on 500h and 100h Turkmen fine-
tuning data. Extended to more languages and more data, SeHuBERT also has about
relative 10% performance improvement at half of the training cost.

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