Concealment and verification over environmental regulations: a game-theoretic analysis

Abstract

We consider a strategic situation in which a firm may conceal the illegal activity of violating environmental regulations and a regulator seeks to verify the illegality to punish the firm. We study two main factors, fines and social monitoring, that influence the firm’s decision in that situation. First, we find all the possible equilibria of our model and examine conditions of those two factors that lead to each equilibrium. Using the equilibrium conditions, we then study the optimal enforcement policies that induce the most socially desirable equilibrium and improve social welfare within each equilibrium. Our main findings are as follows. First, the two factors have a complementary relationship in getting the most desirable equilibrium: Certain high levels of fines and social monitoring are both needed. Second, if making the social monitoring above the certain critical level is impossible, setting the level of the fines as high as possible may be the optimal enforcement policy. Finally, if setting the fines above the certain critical level is not available, either, setting the level of the fines as low as possible might be optimal, and the higher level of the social monitoring does not necessarily bring higher social welfare.

Appendices

Appendix 1: Notation used in the paper

	Definition
B	Activity of the firm’s violating the regulations
G	Activity of the firm’s complying with the regulations
b	Message generated when the firm has chosen B and not concealed it
g	Message generated when the firm has chosen G or has chosen B and concealed it
$e_{F}$	Firm’s concealment effort (level)
$e_{R}$	Regulator’s verification effort (level)
Q	Level of social monitoring
k	Level of the fines
M	Private gains the firm obtains when it violates the regulations
m	Common gains both the firm and the citizens obtain when the firm complies with the regulations
D	Social damages imposed on citizens when the firm violates the regulations
$\theta$	Regulator’s relative effectiveness in the verification process
p	Probability that the firm’s activity is verified
$\mu (B|g)(\equiv \beta )$	Regulator’s belief (subjective probability) that, given message g, the firm has chosen B
$\alpha$	Probability that the firm chooses B and conceals it

Appendix 2: Lemma 4 in Equilibrium $Bg\oplus Gg$ and the other Equilibria of the game

$\square$ Equilibrium $Bg \oplus Gg$

Lemma 4

• (a) When activity B is chosen

  • If $0<Q\le \frac{\theta \alpha kD}{(1+\theta \alpha )^2}$, then $e_F^{Bg\oplus Gg}=\frac{\theta \alpha kD}{(1+\theta \alpha )^2}$, $e_R^{Bg\oplus Gg}=\frac{\theta \alpha ^2 kD}{(1+\theta \alpha )^2}$, and $\Pi _F^{Bg\oplus Gg}=M-kD+\frac{kD}{(1+\theta \alpha )^2}$.

  • If $\frac{\theta \alpha kD}{(1+\theta \alpha )^2}<Q\le \theta \alpha kD$, then $e_F^{Bg\oplus Gg}=Q$, $e_R^{Bg\oplus Gg}=\frac{1}{\theta } (\sqrt{\theta \alpha kDQ}-Q)$, and $\Pi _F^{Bg\oplus Gg}=M-kD+\sqrt{Q}\left( \sqrt{\frac{kD}{\theta \alpha }}-\sqrt{Q}\right)$.

  • If $\theta \alpha kD <Q$, then $e_F^{Bg\oplus Gg}=Q$, $e_R^{Bg\oplus Gg}=0$, and $\Pi _F^{Bg\oplus Gg}=M-Q$.

• (b) When activity G is chosen

  • If $0<Q\le \frac{\theta \alpha kD}{(1+\theta \alpha )^2}$, then $e_F^{Bg\oplus Gg}=0$, $e_R^{Bg\oplus Gg}=\frac{\theta \alpha ^2 kD}{(1+\theta \alpha )^2}$, and $\Pi _F^{Bg\oplus Gg}=m$.

  • If $\frac{\theta \alpha kD}{(1+\theta \alpha )^2}<Q\le \theta \alpha kD$, then $e_F^{Bg\oplus Gg}=0$, $e_R^{Bg\oplus Gg}=\frac{1}{\theta } (\sqrt{\theta \alpha kDQ}-Q)$, and $\Pi _F^{Bg\oplus Gg}=m$.

  • If $\theta \alpha kD <Q$, then $e_F^{Bg\oplus Gg}=0$, $e_R^{Bg\oplus Gg}=0$, and $\Pi _F^{Bg\oplus Gg}=m$.

$\square$ Equilibrium $Bb\oplus Gg$

In this equilibrium, the firm chooses Bb and Gg with positive probabilities. Let us denote the probability the firm assigns to Bb by $\alpha \in (0,1)$ and the probability the firm assigns to Gg by $1-\alpha$. Note that each different message for each activity chosen is generated in this equilibrium. Because the regulator’s belief should be consistent with the firm’s strategy in the equilibrium, the regulator forms his belief

$$\begin{aligned} \mu (B|s=g)=0 \end{aligned}$$

when he receives message g. Therefore, after receiving message g, the regulator doesn’t exert any verification effort. Similarly, in the equilibrium, the firm chooses its concealment effort level. That is, if Bb is chosen, the firm doesn’t expend any effort and then gets its payoff $M-kD$. If Gg is chosen, the firm doesn’t make any effort, either, and obtains its payoff m. Lemma 5 summarizes these outcomes in the equilibrium.

Lemma 5

• (a) When activity B is chosen

  • $e_F^{Bb\oplus Gg}=0$, $e_R^{Bb\oplus Gg}=0$, and $\Pi _F^{Bb\oplus Gg}=M-kD$.

• (b) When activity G is chosen

  • $e_F^{Bb\oplus Gg}=0$, $e_R^{Bb\oplus Gg}=0$, and $\Pi _F^{Bb\oplus Gg}=m$.

Using Lemma 5, we look for the equilibrium value of $\alpha$ and the conditions that guarantee the existence of the equilibrium. In equilibrium $Bb \oplus Gg$, the firm chooses Bb and Gg with positive probability, which means that the firm should be indifferent between choosing Bb and choosing Gg. Otherwise, it is not optimal for the firm to choose both Bb and Gg randomly. Therefore, for this equilibrium to exist, the payoff the firm obtains when activity B is chosen in Lemma 5 should be equal to the one when activity G is chosen. In addition to this condition, there shouldn’t be incentive for the firm to deviate from this equilibrium and choose Bg. The firm gets its payoff $M-Q$ when deviating from the equilibrium to Bg, because it is optimal for the firm to choose its minimum concealment effort level Q which generates message g, given the regulator’s belief, $\mu (B|s=g)=0$. So, the payoff $M-Q$ should be less than or at most equal to the payoff obtained in the equilibrium. Proposition 6 describes these conditions for the existence of equilibrium $Bb \oplus Gg$ and the equilibrium value of $\alpha$.

Proposition 8

Equilibrium $Bb \oplus Gg$ exists if $Q \ge kD$ and $k=\frac{M-m}{D}$, and the equilibrium value of $\alpha$, $\alpha ^*$, is any value that belongs to (0, 1).

In this equilibrium, the firm never chooses Bg since the level of social monitoring is so high that the concealment cost exceeds the amount of the fines (the cost of the violation without concealment) and the cleaning cost (the cost of the compliance). The equilibrium conditions are presented in Fig. 2b. The set of (Q, k) satisfying the equilibrium conditions in the proposition corresponds to the set of all points on the vertical dotted lines.

$\square$ Equilibrium $Bb \oplus Bg$

In this equilibrium, the firm mixes over Bb and Bg. Let us denote the probability the firm assigns to Bb by $\alpha \in (0,1)$ and the probability the firm assigns to Bg by $1-\alpha$. Since the regulator’s belief should be consistent with the firm’s strategy in the equilibrium, the regulator has his belief

$$\begin{aligned} \mu (B|s=g)=1 \end{aligned}$$

when message g is received. Hence, after receiving message g, the regulator chooses his verification effort level

$$\begin{aligned} e_R (s=g)=\text {arg} \max _{e_R \ge 0} -D + \frac{\theta e_R}{\theta e_R + e_F}(kD) -e_R. \end{aligned}$$

(18)

In the equilibrium the firm chooses its concealment effort level $e_F$, which is contingent on the strategy chosen. If Bb is chosen, the firm doesn’t make any concealment effort and has its payoff $M-kD$. On the other hand, if Bg is chosen, the firm chooses its concealment effort level

$$\begin{aligned} e_F (B, s=g) = \text {arg}\max _{e_F \ge Q} M-\frac{\theta e_R}{\theta e_R + e_F} (kD)-e_F . \end{aligned}$$

(19)

By solving the players’ incentive-compatibility conditions (18) and (19) together with respect to $e_F$ and $e_R$, we can find the equilibrium effort levels of the firm and the regulator in case where Bg is chosen. The equilibrium outcomes are summarized in the following lemma.

Lemma 6

• (a) When Bb is chosen

  • $e_F^{Bb\oplus Bg}=0$, $e_R^{Bb\oplus Bg}=0$, and $\Pi _F^{Bb\oplus Bg}=M-kD$.

• (b) When Bg is chosen

  • If $0<Q\le \frac{\theta kD}{(1+\theta )^2}$, then $e_F^{Bb\oplus Bg}=e_R^{Bb\oplus Bg}=\frac{\theta kD}{(1+\theta )^2}$ and $\Pi _F^{Bb\oplus Bg}=M-kD+\frac{kD}{(1+\theta )^2}$.

  • If $\frac{\theta kD}{(1+\theta )^2}<Q\le \theta kD$, then $e_F^{Bb\oplus Bg}=Q$, $e_R^{Bb\oplus Bg}=\frac{1}{\theta } (\sqrt{\theta kDQ}-Q)$, and $\Pi _F^{Bb\oplus Bg}=M-kD+\sqrt{Q}\left( \sqrt{\frac{kD}{\theta }}-\sqrt{Q}\right)$.

  • If $\theta kD <Q$, then $e_F^{Bb\oplus Bg}=Q$, $e_R^{Bb\oplus Bg}=0$, and $\Pi _F^{Bb\oplus Bg}=M-Q$.

We now figure out the equilibrium value of $\alpha$ and the conditions that guarantee the existence of the equilibrium. In equilibrium $Bb \oplus Bg$, the firm chooses Bb and Bg with positive probability, which means that the firm should be indifferent between choosing Bb and choosing Bg. Therefore, for this equilibrium to exist, the payoff the firm obtains when Bb is chosen in Lemma 6 should be equal to the one when Bg is chosen. In addition to this condition, there shouldn’t be incentive for the firm to deviate from this equilibrium and choose Gg. The firm gets its payoff m when deviating from the equilibrium to Gg. So, the payoff m should be less than or at most equal to the payoff obtained in the equilibrium. Proposition 9 describes these conditions for the existence of equilibrium $Bb \oplus Bg$ and the equilibrium value of $\alpha$.

Proposition 9

Equilibrium $Bb \oplus Bg$ exists if $Q = \frac{kD}{\theta }$ and $k \le \frac{M-m}{D}$, and the equilibrium value of $\alpha$, $\alpha ^*$, is any value that belongs to (0, 1).

Gg is not chosen by the firm in this equilibrium. The equilibrium conditions are shown in Fig. 2b. The set of (Q, k) satisfying the equilibrium conditions in the proposition corresponds to the set of all points on the dotted line with a positive slope.

$\square$ Equilibrium $Bb \oplus Bg \oplus Gg$

Finally, let us consider equilibrium $Bb \oplus Bg \oplus Gg$ in which the firm chooses Bb, Bg, and Gg with respectively positive probabilities. Denote the probability the firm assigns to Bb by $\alpha _1 (>0)$, the probability the firm assigns to Bg by $\alpha _2 (>0)$, and the probability the firm assigns to Gg by $\alpha _3 (>0)$, where $\alpha _1 + \alpha _2 + \alpha _3 =1$. Since the belief of the regulator should be consistent with the firm’s strategy in the equilibrium, the regulator forms his belief

$$\begin{aligned} \mu (B|s=g)=\frac{\alpha _2}{\alpha _2 +\alpha _3} \end{aligned}$$

when he receives message g. So, when the regulator receives message g, he chooses his verification effort level

$$\begin{aligned} e_R (s=g)=\text {arg} \max _{e_R \ge 0} \gamma \big (-D + \frac{\theta e_R}{\theta e_R + e_F}(kD) \big ) + (1-\gamma ) (m) -e_R , \end{aligned}$$

(20)

where $\gamma \equiv \frac{\alpha _2}{\alpha _2 +\alpha _3}$.

In the equilibrium, the firm chooses its concealment effort level $e_F$, which is contingent on the strategy chosen. If Bb is chosen in the equilibrium, the firm makes no effort and consequently obtains its payoff $M-kD$. If Bg is chosen, the firm chooses its concealment effort level

$$\begin{aligned} e_F (B, s=g) = \text {arg}\max _{e_F \ge Q} M-\frac{\theta e_R}{\theta e_R + e_F} (kD)-e_F . \end{aligned}$$

(21)

Solving the incentive-compatibility conditions (20) and (21) simultaneously with respect to $e_F$ and $e_R$, we can find how much effort the firm and the regulator expend when Bg is chosen in the equilibrium. Lastly, if Gg is chosen, the firm gets its payoff m by exerting zero effort. Lemma 7 summarizes these.

Lemma 7

• (a) When Bb is chosen

  • $e_F^{Bb\oplus Bg\oplus Gg}=0$, $e_R^{Bb\oplus Bg\oplus Gg}=0$, and $\Pi _F^{Bb\oplus Bg\oplus Gg}=M-kD$.

• (b) When Bg is chosen

  • If $0<Q\le \frac{\theta \gamma kD}{(1+\theta \gamma )^2}$, then $e_F^{Bb\oplus Bg\oplus Gg}=\frac{\theta \gamma kD}{(1+\theta \gamma )^2}$, $e_R^{Bb\oplus Bg\oplus Gg}=\frac{\theta \gamma ^2 kD}{(1+\theta \gamma )^2}$, and $\Pi _F^{Bb\oplus Bg\oplus Gg}=M-kD+\frac{kD}{(1+\theta \gamma )^2}$.

  • If $\frac{\theta \gamma kD}{(1+\theta \gamma )^2}<Q\le \theta \gamma kD$, then $e_F^{Bb\oplus Bg\oplus Gg}=Q$, $e_R^{Bb\oplus Bg\oplus Gg}=\frac{1}{\theta } (\sqrt{\theta \gamma kDQ}-Q)$, and $\Pi _F^{Bb\oplus Bg\oplus Gg}=M-kD+\sqrt{Q}\left( \sqrt{\frac{kD}{\theta \gamma }}-\sqrt{Q}\right)$.

  • If $\theta \gamma kD <Q$, then $e_F^{Bb\oplus Bg\oplus Gg}=Q$, $e_R^{Bb\oplus Bg\oplus Gg}=0$, and $\Pi _F^{Bb\oplus Bg\oplus Gg}=M-Q$.

• (c) When Gg is chosen

  • $e_F^{Bb\oplus Bg\oplus Gg}=0$, $e_R^{Bb\oplus Bg\oplus Gg}$ is the same as in (b), and $\Pi _F^{Bb\oplus Bg\oplus Gg}=m$.

Let us now find out the equilibrium values of $\alpha _1$, $\alpha _2$, and $\alpha _3$, and the conditions that make equilibrium $Bb \oplus Bg \oplus Gg$ exist. In the equilibrium, the firm mixes over Bb, Bg, and Gg with positive probabilities, which means that the firm should be indifferent among choosing Bb, choosing Bg, and choosing Gg. Therefore, for this equilibrium to exist, the payoffs the firm gets when each strategy is chosen should be equivalent. This requirement and the equilibrium values of $\alpha _1$, $\alpha _2$, and $\alpha _3$ are presented in the following proposition, and the equilibrium conditions are seen in Fig. 2b. (Q, k) satisfying the conditions for this equilibrium corresponds to a point.

Proposition 10

Equilibrium $Bb \oplus Bg \oplus Gg$ exists if $Q =kD \cdot \text {min} \left\{ \frac{1}{\theta \gamma ^*}, 1 \right\}$ and $k = \frac{M-m}{D}$, and the equilibrium values of $\alpha _1$, $\alpha _2$, and $\alpha _3$ are any values that satisfy $\alpha _1^*$, $\alpha _2^*$, $\alpha _3^* >0$ and $\alpha _1^* +\alpha _2^* +\alpha _3^* =1$, where $\gamma ^* = \frac{\alpha _2^*}{\alpha _2^* +\alpha _3^*}$.

Appendix 3: Proofs of Propositions

Proof of Proposition 1

In equilibrium Bb the regulator’s belief is not restricted by the consistency condition. At first, suppose that the regulator has his belief $\beta =0$ off the equilibrium path. Then, when receiving message g, the regulator chooses $e_R (g)=0$ with the belief. Having perfect foresight about this, the firm chooses $e_F (B, s=g)=Q$ if it deviates from equilibrium Bb and chooses Bg, and gets its payoff $M-Q$. So, the firm will not deviate in this way if $Q \ge kD$. If the firm deviates from equilibrium Bb and chooses Gg, it gets its payoff m by expending no concealment effort. Hence, the firm will not deviate to Gg if $M-kD \ge m$ or $k \le \frac{M-m}{D}$. In summary, the non-deviation condition for the firm, given the regulator’s belief $\beta =0$, is

$$\begin{aligned} Q\ge kD\quad \text {and}\quad ~k\le \frac{M-m}{D}. \end{aligned}$$

(*)

Next, suppose that the regulator has his belief $\beta \in (0,1]$. Then, when receiving message g, the regulator chooses his verification effort level

$$\begin{aligned} e_R (s=g)=\text {arg} \max _{e_R \ge 0}~ \beta \big (-D + \frac{\theta e_R}{\theta e_R + e_F}(kD) \big ) + (1-\beta ) \big (m \big ) -e_R. \end{aligned}$$

(A1)

Similarly to the above, for equilibrium Bb to exist, the firm shouldn’t have any incentive to deviate to Bg or Gg, given the regulator’s belief off the equilibrium path. If the firm deviates from the equilibrium and chooses Bg, it chooses its concealment effort level

$$\begin{aligned} e_F (B, s=g)=\text {arg} \max _{e_F \ge Q} M+\frac{\theta e_R}{\theta e_R +e_F} (-kD) -e_F , \end{aligned}$$

(A2)

while considering the regulator’s response in (A1). By solving the incentive-compatibility constraints (A1) and (A2) simultaneously, we have the following outcomes when the firm deviates from equilibrium Bb and chooses Bg:

• if $0<Q\le \frac{\theta \beta kD}{(1+\theta \beta )^2}$, $e_F^{Bb^{'}}=\frac{\theta \beta kD}{(1+\theta \beta )^2}$, $e_R^{Bb^{'}}=\frac{\theta \beta ^2 kD}{(1+\theta \beta )^2}$, $\Pi _F^{Bb^{'}}=M-kD+\frac{kD}{(1+\theta \beta )^2}$

• if $\frac{\theta \beta kD}{(1+\theta \beta )^2}<Q\le \theta \beta kD$, $e_F^{Bb^{'}}=Q$, $e_R^{Bb^{'}}=\frac{1}{\theta } (\sqrt{\theta \beta kDQ}-Q)$, $\Pi _F^{Bb^{'}}=M-kD+\sqrt{Q}\left( \sqrt{\frac{kD}{\theta \beta }}-\sqrt{Q}\right)$

• if $\theta \beta kD <Q$, $e_F^{Bb^{'}}=Q$, $e_R^{Bb^{'}}=0$, $\Pi _F^{Bb^{'}}=M-Q$.

Comparing $\Pi _F^{Bb}$ in Lemma 1 with $\Pi _F^{Bb^{'}}$ above, we see that the firm will not deviate to Bg if $Q \ge kD \cdot \text {min} \left\{ \frac{1}{\theta \beta }, 1 \right\}$. On the other hand, given the belief, if the firm deviates from equilibrium Bb and chooses Gg, then it is the best interest for the firm to exert zero effort, and consequently gets its payoff m. Hence, the firm will not deviate in this way if $k\le \frac{M-m}{D}$. In summary, the non-deviation condition for the firm, given the regulator’s belief $\beta \in (0,1]$, is

$$\begin{aligned} Q \ge kD \cdot \text {min} \left\{ \frac{1}{\theta \beta }, 1 \right\} ~{and}~k\le \frac{M-m}{D}. \end{aligned}$$

(**)

Finally, from $(*)$ and $(**)$, we have the conditions that result in the emergence of equilibrium Bb:

$$\begin{aligned} Q \ge \frac{kD}{\theta }~\text {and}~k \le \frac{M-m}{D}. \end{aligned}$$

$\square$

Proof of Proposition 2

In equilibrium Gg, the regulator has his belief $\mu (B|s=g)=0$. Given this belief, it is optimal for the firm to choose its concealment effort level Q, which is the minimum effort level which generates message g, if it deviates from the equilibrium to Bg. By deviating in this way, the firm gets its payoff $M-Q$. Comparing this and $\Pi _F^{Gg}$ in Lemma 2, we have the first non-deviation condition: $m \ge M-Q.$

If the firm deviates from the equilibrium to Bb, then it is optimal for the firm to choose zero effort level, and hence obtains its payoff $M-kD$. Comparing this and $\Pi _F^{Gg}$ in Lemma 2, we have the second non-deviation condition: $m \ge M-kD$. $\square$

Proof of Proposition 3

Given the regulator’s belief, $\mu (B|s=g)=1$, the firm gets its payoff $M-kD$ by deviating from equilibrium Bg to Bb. So, the firm doesn’t have incentive to deviate in this way if $\Pi _F^{Bg} \ge M-kD$. From Lemma 3, we find the following condition under which $\Pi _F^{Bg} \ge M-kD$ holds: $Q \le \frac{kD}{\theta }.$

If the firm deviates from the equilibrium to Gg, then it obtains its payoff m. So, $\Pi _F^{Bg} \ge m$ should be held for the firm not to deviate. Considering both the condition obtained above and the results in Lemma 3, we can find the following conditions that make $\Pi _F^{Bg} \ge \max \left\{ M-kD , m \right\}$ hold:

$$\begin{aligned} Q \le \frac{kD}{\theta }~\text {and}~k \le \frac{M-m}{D} \end{aligned}$$

or

$$\begin{aligned}&Q \le \frac{kD}{\theta }, k>\frac{M-m}{D}, M-kD+\frac{kD}{(1+\theta )^2} \ge m~\text {for}~0<Q \le \frac{\theta kD}{(1+\theta )^2},\\&\qquad \text {and}~ M-kD+\sqrt{Q} \left( \sqrt{\frac{kD}{\theta }}-\sqrt{Q} \right) \ge m~\text {for}~\frac{\theta kD}{(1+\theta )^2} < Q \le \frac{kD}{\theta }. \end{aligned}$$

$\square$

Proof of Proposition 4

From Lemma 4 we know that the firm’s expected payoff when activity B is chosen in the equilibrium, $\Pi _F^{Bg\oplus Gg}$, has different values according to the value of Q. So, we compute the value of $\alpha$ which equates the firm’s payoff when activity B is chosen with m for each different values of Q. For example, if $0<Q \le \frac{\theta \alpha k D}{(1+\theta \alpha )^2}$, then $\Pi _F^{Bb\oplus Gg} = M-kD+\frac{kD}{(1+\theta \alpha )^2}$. Equating this with m and solving it in terms of $\alpha$, we have

$$\begin{aligned} \alpha ^* = \frac{1}{\theta } \left( \sqrt{\frac{kD}{kD-M+m}}-1 \right) . \end{aligned}$$

Substituting $\alpha ^*$ into $0<Q \le \frac{\theta \alpha k D}{(1+\theta \alpha )^2}$, we have the equilibrium range of Q under which the firm chooses activity B with probability $\alpha ^*$ computed above:

$$\begin{aligned} 0< Q \le \sqrt{kD(kD-M+m)}-(kD-M+m). \end{aligned}$$

With the similar way, we find out each equilibrium value of $\alpha$ for $\frac{\theta \alpha k D}{(1+\theta \alpha )^2}<Q < \theta \alpha kD$ and for $\theta \alpha kD \le Q$, respectively, and specify the equilibrium range of Q for each equilibrium value of $\alpha$. The range of Q here is also the condition which is required for the existence of the equilibrium.

In addition to the equilibrium range of Q, for the equilibrium to exist, there shouldn’t be any incentive for the firm to deviate from the equilibrium. That is, if the firm deviates from the equilibrium and chooses Bb, its payoff $M-kD$ is obtained. So, we have the following non-deviation condition: $m \ge M-kD$. $\square$