An Evolutionary Approach to Pollution Control in Competitive Markets

Abstract

We consider a large population of firms in a market environment. The firms are divided into a finite set of types, with each type being characterized by a distinct private cost function. Moreover, the firms generate an external cost like pollution in the production process. As a result, the Nash equilibrium outcome is not socially optimal. We propose an evolutionary implementation mechanism to achieve the socially optimal outcome. In contrast to the classical VCG mechanism, evolutionary implementation does not require the planner to know or elicit any private information from firms. By imposing a tax equal to the current external damage being imposed by a firm, the planner can guide the evolution of the society toward the social optimum. The imposition of the tax generates a potential game whose potential function is the social welfare function of the model. Evolutionary dynamics converge to the maximizer of this function thereby evolutionarily implementing the social welfare maximizer.

A Appendix

Proof of Proposition 3.3

See Proposition 5.3 in Lahkar [14] for the proof that F defined by (3) is a potential game with potential function 4. For the proof of concavity of the potential function, let $\mu ,\nu \in \Delta$, $\mu \ne \nu$, be two states. Fix $\lambda =(0,1)$. Clearly,

$$\begin{aligned}&\sum _{p\in \mathcal {P}}\int _\mathcal {S}c_p(x)(\lambda \mu _p+(1-\lambda )\nu _p)(dx)\nonumber \\&\quad =\lambda \sum _{p\in \mathcal {P}}\int _\mathcal {S}c_p(x)\mu _p(dx)+(1-\lambda )\sum _{p\in \mathcal {P}}\int _\mathcal {S}c_p(x)\nu _p(dx). \end{aligned}$$

(28)

Recall that $\beta$ is strictly decreasing on $[\underline{x},\bar{x}]$. Hence, $\int _{\underline{x}}^\alpha \beta (z)dz$ is strictly concave on $[\underline{x},\bar{x}]$. Further, $A(\cdot )$ is linear and since $\mu ,\nu \in \Delta$, $A(\lambda \mu +(1-\lambda )\nu )\in \Delta$. Combining these facts, we obtain

$$\begin{aligned} \int _{\underline{x}}^{A(\lambda \mu +(1-\lambda )\nu )}\beta (z)dz&=\int _{\underline{x}}^{\lambda A(\mu )+(1-\lambda )A(\nu )}\beta (z)dz\nonumber \\&\ge \lambda \int _{\underline{x}}^{A(\mu )}\beta (z)dz+(1-\lambda )\int _{\underline{x}}^{A(\nu )}\beta (z)dz, \end{aligned}$$

(29)

with $\ge$ holding with equality only if $\mu ,\nu$ such that $A(\mu )=A(\nu )$. Combining (28) and (29), we obtain

$$\begin{aligned} f(\lambda \mu +(1-\lambda )\nu )\ge \lambda f(\mu )+(1-\lambda )f(\nu ). \end{aligned}$$

Thus, the potential function f defined by (4) is concave but not strictly concave. Strict concavity fails because even if $\mu \ne \nu$, it is possible that $A(\mu )=A(\nu )$. $\square$

Proof of Lemma 3.4

Notice that the aggregate strategy for all firms is $A(\mu )=\sum _pa(\mu _p)=\sum _pm_p\alpha _p$. Therefore, any difference between f defined by (4) and g defined by (7) can only arise due to the difference between $\sum _{p\in \mathcal {P}}\int _{\mathcal {S}}c_p(x)\mu _p(dx)$ and $\sum _pm_p\alpha _p$.

• 1. If $\mu _p$ is monomorphic for all p, then $\alpha _p=\frac{a(\mu _p)}{m_p}$ implies all firms of type p are playing $\alpha _p$. Therefore, $\sum _{p\in \mathcal {P}}\int _{\mathcal {S}}c_p(x)\mu _p(dx)=\sum _pm_p\alpha _p$. Hence, $f(\mu )=g(\alpha _1,\alpha _2,\cdots ,\alpha _n)$.

• 2. Suppose $\mu _q$ is polymorphic. By definition, $\alpha _p=\frac{a(\mu _p)}{m_p}$ implies $\int _\mathcal {S}x\frac{\mu _q}{m_q}(dx)=\alpha _q$. Recall $c_q$ is strictly convex. Hence,

  $$\begin{aligned}&\int _\mathcal {S}c_q(x)\frac{\mu _q}{m_q}(dx)> c_q\left( \int _\mathcal {S}x\frac{\mu _q}{m_q}(dx)\right) =c_q(\alpha _q)\\ \Rightarrow&\int _\mathcal {S}c_q(x)\mu _q(dx)>m_qc_q(\alpha _q). \end{aligned}$$

  Thus, if $\mu _q$ is polymorphic, $\int _\mathcal {S}c_q(x)\mu _q(dx)>m_qc_q(\alpha _q)$ while part 1 implies that if $\mu _p$ is monomorphic, then $\int _\mathcal {S}c_p(x)\mu _p(dx)>m_pc_p(\alpha _p)$. Thus, if the state of at least one type is polymorphic, then $\sum _{p\in \mathcal {P}}\int _{\mathcal {S}}c_p(x)\mu _p(dx)>\sum _pm_p\alpha _p$. Hence, $g(\alpha _1,\alpha _2,\cdots ,\alpha _n)>f(\mu )$. $\square$

Proof of Lemma 3.5

Consider the quasi-potential function (7). Take two points $(\hat{\alpha }_1,\cdots ,\hat{\alpha }_n) \ne (\tilde{\alpha }_1,\cdots ,\tilde{\alpha }_n)\in [\underline{x},\bar{x}]$. We need to show that

$$\begin{aligned}&\int _{\underline{x}}^{\sum _qm_q(\lambda \hat{\alpha }_q+(1-\lambda )\tilde{\alpha }_q)}\beta (z)dz-\sum _{q\in \mathcal {P}}m_qc_q(\lambda \hat{\alpha }_q+(1-\lambda )\tilde{\alpha }_q)\nonumber \\&\quad >\lambda \left( \int _{\underline{x}}^{\sum _qm_q\hat{\alpha }_q}\beta (z)dz-\sum _{q\in \mathcal {P}}m_qc_q(\hat{\alpha }_q)\right) \nonumber \\&\qquad +(1-\lambda )\left( \int _{\underline{x}}^{\sum _qm_q\tilde{\alpha }_q}\beta (z)dz-\sum _{q\in \mathcal {P}}m_qc_q(\tilde{\alpha }_q)\right) . \end{aligned}$$

(30)

Since $\beta$ is a strictly decreasing function, $\int _{\underline{x}}^y\beta (z)dz$ is a strictly concave function, for $y\in \mathbf {R}$. Hence,

$$\begin{aligned} \int _{\underline{x}}^{\sum _qm_q(\lambda \hat{\alpha }_q+(1-\lambda )\tilde{\alpha }_q)}\beta (z)dz\ge \lambda \int _{\underline{x}}^{\sum _qm_q\hat{\alpha }_q}\beta (z)dz+(1-\lambda )\int _{\underline{x}}^{\sum _qm_q\tilde{\alpha }_q}\beta (z)dz, \end{aligned}$$

(31)

with equality holding only if $\sum _qm_q\hat{\alpha }_q=\sum _qm_q\tilde{\alpha }_q$. Further, the strict convexity of $c_q$ implies

$$\begin{aligned} c_q(\lambda \hat{\alpha }_q+(1-\lambda )\tilde{\alpha }_q)<\lambda c_q(\hat{\alpha }_q)+(1-\lambda )c_q(\tilde{\alpha }_q). \end{aligned}$$

(32)

Since (32) holds for all $q\in \mathcal {P}$, it together with (31) implies (30). $\square$

Proof of Theorem 3.6

Consider the potential function f defined by (4) and the quasi-potential function g defined by (7). Recall the notation from Lemma 3.4 that $a(\mu _p)=\int _\mathcal {S}x\mu _p(dx)$. Note that if $\mu ^{*}=\left( m_1\delta _{\alpha _1^{*}},m_2\delta _{\alpha _2^{*}},\cdots ,m_n\delta _{\alpha _n^{*}}\right)$ is as described in the theorem, then $\frac{a(\mu ^{*}_p)}{m_p}=\alpha _p^{*}$ for all $p\in \mathcal {P}$. Therefore, by part 1 of Lemma 3.4, $f(\mu ^{*})=g(\alpha _1^{*},\cdots ,\alpha _n^{*})$.

Now consider $\mu \ne \mu ^{*}$ such that $\mu _p$ is monomorphic for every p. Hence, $\mu =(m_1\delta _{\alpha _1},\cdots ,m_n\delta _{\alpha _n})$ for some $(\alpha _1,\cdots ,\alpha _n)\in \prod _{p=1}^n[\underline{x},\bar{x}]$, with $\alpha _p\ne \alpha _p^{*}$ for at least one $p\in \mathcal {P}$. Note that in this case, $\frac{a(\mu _p)}{m_p}=\alpha _p$, $\alpha _p\in [\underline{x},\bar{x}]$. Therefore, by Lemma 3.4(1), $f(\mu )=g(\alpha _1,\cdots ,\alpha _n)$. But then, by applying Lemma 3.5, we obtain

$$\begin{aligned} f(\mu ^{*})=g(\alpha _1^{*},\cdots ,\alpha _n^{*})>g(\alpha _1,\cdots ,\alpha _n)=f(\mu ). \end{aligned}$$

(33)

Next, consider $\mu$ such that $\mu _p$ is polymorphic for at least one $p\in \mathcal {P}$. For any $p\in \mathcal {P}$, define $\alpha _p\in [\underline{x},\bar{x}]$ such that $\frac{a(\mu _p)}{m_p}=\alpha _p$. Then,

$$\begin{aligned} f(\mu ^{*})=g(\alpha _1^{*},\cdots ,\alpha _n^{*})\ge g(\alpha _1,\cdots ,\alpha _n)>f(\mu ), \end{aligned}$$

(34)

where the weak inequality holds if $(\alpha _1^{*},\cdots ,\alpha _n^{*})=(\alpha _1,\cdots ,\alpha _n)$ and the strict inequality follows from Lemma 3.4(2).

Combining (33) and (34), we conclude that $\mu ^{*}$ is the unique global maximizer of the potential function f. Since f is concave (by Proposition 3.3), this implies $\mu ^{*}$ is the unique Nash equilibrium of F. $\square$

Proof of Proposition 4.1

Recall that $c_p(x)$ is strictly convex for all p while s(x) is convex. Therefore, $\left( c_p(\alpha _p)+s(\alpha _p)\right)$ is also strictly convex. Hence, the concavity of $\hat{f}$ follows from similar argument as in Proposition 3.3 once we replace $c_p(x)$ in (4) with $\left( c_p(\alpha _p)+s(\alpha _p)\right)$. To establish the existence of a unique maximizer $\mu ^{**}$, we construct (12) and note that it is analogous to the quasi-potential function (7) except for the presence of the $\sum _p m_ps(\alpha _p)$ term. But since $\left( c_p(\alpha _p)+s(\alpha _p)\right)$, $\hat{g}$ is strictly convex by the same argument as in Lemma 3.5. Hence, $\hat{g}$ has a unique maximizer $(\alpha _1^{**},\alpha _2^{**},\cdots ,\alpha _n^{**})$.

The strict convexity of $\left( c_p(\alpha _p)+s(\alpha _p)\right)$ also implies that $\hat{f}$ and $\hat{g}$ share the same relationship as f and g as characterized in Lemma 3.4. The identification of the social welfare maximizer $\mu ^{**}$ such that each type state $\mu _p^{**}=m_p\delta _{\alpha _p^{**}}$ then follows from Theorem 3.6. The conclusion that $\alpha _p^{**}\le \alpha _p^{*}$ follows from the maximization of (7) and (12) and the fact that for every $x\in \mathcal {S}$, $c_p^{\prime }(x)+s^{\prime }(x)>c_p^{\prime }(x)$. $\square$

1.1 A.1 Standard Evolutionary Dynamics

To briefly describe the standard evolutionary dynamics, we consider a population game F with strategy set $\mathcal {S}$ and let the payoff of a population p agent playing strategy x at social state $\mu$ be $F_{x,p}(\mu )$. Let $\bar{F}_p(\mu )=\frac{1}{m_p}\int _\mathcal {S}F_{x,p}(\mu )\mu _p(dx)$ be the average payoff in population p at $\mu$. The excess payoff of a strategy x in population p at $\mu$ is then $F_{x,p}(\mu )-\bar{F}_p(\mu )$. We require this notion of the excess payoff to define the replicator dynamic and the BNN dynamic.

For the logit dynamic, we need to define the probability measure $L_{\eta ,p}(\mu )$ on $\mathcal {S}$, known as the logit choice measure, as $L_{\eta ,p}(\mu )(B)=\int _B\frac{\exp (\eta ^{-1}F_{x,p}(\mu ))}{\int _\mathcal {S}\exp (\eta ^{-1}F_{y,p}(\mu ))dy}dx$, $B\subseteq \mathcal {S}$, $\eta >0$. The parameter $\eta$ is a measure of perturbation. The logit choice is generated when agents best respond to a perturbed version of payoffs, where the perturbation depends upon $\eta$ (Lahkar and Riedel [12]). For $\eta$ small, it puts most of the probability mass on the set of best responses to $\mu$. Hence, intuitively, the logit choice measure is an approximation of the best response.

The replicator dynamic, the BNN dynamic, the pairwise comparison dynamic and the logit dynamic, respectively, in F are now defined as follows.

$$\begin{aligned} \dot{\mu }_p(B)&=\int _B\left( F_{x,p}(\mu )-\bar{F}_p(\mu )\right) \mu _p(dx), \end{aligned}$$

(35)

$$\begin{aligned} \dot{\mu }_p(B)&=m_p\int _B\left[ F_{x,p}(\mu )-\bar{F}_p(\mu )\right] _+dx-\mu _p(B)\int _\mathcal {S}\left[ F_{y,p}(\mu )-\bar{F}_p(\mu )\right] _+dy, \end{aligned}$$

(36)

$$\begin{aligned} \dot{\mu }_p(B)&=\int _{\mathcal {S}}\int _B\left[ F_{x,p}(\mu )-F_{y,p}(\mu )\right] _+dx\mu _p(dy)\nonumber \\&\quad -\int _{\mathcal {S}}\int _B\left[ F_{y,p}(\mu )-F_{x,p}(\mu )\right] _+\mu _p(dx)dy, \end{aligned}$$

(37)

$$\begin{aligned} \dot{\mu }_p(B)&=m_pL_{\eta ,p}(\mu )(B)-\mu _p(B),\text { where }\eta >0. \end{aligned}$$

(38)

In each of these dynamics, $\dot{\mu }_p(B)$ is the direction and magnitude of change in the mass of agents in population p who are playing strategies in $B\subseteq \mathcal {S}$. The replicator dynamic (35) and the BNN dynamic (36) depends upon the excess payoff. If the aggregate excess payoff of strategies in B is positive, then the replicator dynamic increases the mass of agents playing strategies in B. Under the BNN dynamic, agents adopt strategy x with probability proportional to the positive part of the excess payoff $F_{x,p}(\mu )-\bar{F}_p(\mu )$ of that strategy (note that $[a-b]_+=\max (a-b,0)$). The pairwise comparison dynamic involves agents abandoning strategy y and adopting strategy x with probability proportional to $\left[ F_{x,p}(\mu )-F_{y,p}(\mu )\right] _+$. The logit dynamic moves the social state $\mu$ toward the logit choice measure $L_{\eta ,p}(\mu )$.

While the four dynamics (35)–(38) are well defined for all population games, the best response dynamic for aggregative games, as the name suggests, is valid only in such aggregative games where every social state $\mu$ generates a unique best response. Denote the aggregate strategy level $A(\mu )$ at $\mu$ in such a game as $\alpha$ and let $b_p(\alpha )$ be the unique best response to $\mu$. Under the best response dynamic, the change in the population state $\mu _p$ is given by

$$\begin{aligned} \dot{\mu }_p=m_p\delta _{b_p(\alpha )}-\mu _p. \end{aligned}$$

(39)

The reason why it may be difficult to define this dynamic in more general games is that the best response may not be uniquely defined or, due to the continuous structure of the strategy set, may not even exist.

In Sect. 4.2, we analyzed the best response dynamic for the externality adjusted game $\hat{F}$ defined by (18)–(19) generated by Example 2.1. We can also derive the other dynamics described in (35)–(38) for $\hat{F}$. For example, to write down the replicator dynamic for type 1 at a state $\mu =(\mu _1,\mu _2)$, we need to derive the average payoff $\bar{\hat{F}}_1(\mu )$ for population 1. Using (18) and recalling that $m_1=0.4$ in Example 2.1, we can calculate that average payoff as

$$\begin{aligned} \bar{\hat{F}}_1(\mu )&=\frac{1}{0.4}\int _{\mathcal {S}}\hat{F}_{x,1}(\mu )\mu _1(dx)\nonumber \\&=2.5\left[ (97-\alpha )\int _{\mathcal {S}}x\mu _1(dx)-\int _{\mathcal {S}}x^2\mu _1(dx),\right] \end{aligned}$$

(40)

where $\alpha =A(\mu )$ is the aggregate output at state $\mu$. Using (18), (35) and (40), we can then write the replicator dynamic for type 1 as

$$\begin{aligned} \dot{\mu }_1(B)=\int _{B}\left( x(97-\alpha )-x^2-2.5\left[ (97-\alpha )\int _{\mathcal {S}}x\mu _1(dx)-\int _{\mathcal {S}}x^2\mu _1(dx)\right] \right) \mu _1(dx)\nonumber \\ \end{aligned}$$

(41)

for $B\subseteq \mathcal {S}$. As can be seen, this dynamic cannot be reduced entirely to the aggregate output $\alpha$ and, hence, cannot be analyzed using the elementary techniques used for the best response dynamic in Sect. 4.2, hence the need for the potential game method in Proposition 4.3 on evolutionary implementation. The other dynamics for $\hat{F}$ can also be derived using their definitions in (35)–(38) and the payoffs (18)–(19). But doing so in not helpful for our analysis. Hence, we refrain from presenting them here.