Two amendments of the benchmark model are sketched here. The first one introduces the incentive to free-ride within the environmental pressure group. The second one deals with the situation where both lobbies know the damage at the time of the contest game.

We depart from the benchmark model by assuming that the environmental lobby brings together \(n>1\) identical individual members, whereas the industrial lobby still represents a single innovative firm. Each member \(i\) of lobby \(E\) exerts a nonnegative effort \(y_{i}\). Given the aggregate efforts of lobbies \(I\) and \(E\), \(x\) and \(y=\sum_{i=1}^{n}y_{i}\) respectively, the regulator approves the new product with the probability \(\pi\left(x,y\right)\). If the product is marketed, each member \(i\) of lobby \(E\) bears a damage \(\left(1/n\right)\delta\) and receives in court a compensation \(\mbox{$\left(1/n\right)$min}\left(\delta,k\right)\) from lobby \(I\).

A Nash equilibrium of the contest game, where lobby \(I\) with type \(\delta\) plays \(x^{n}\left(\delta\right)\) and each member \(i\) of lobby \(E\) plays \(y_{i}^{n}\), can easily be adapted from Section 3, yielding¹¹There is an infinity of Nash equilibria, each allocating the same aggregate effort within lobby \(E\).

From this, we can calculate the ex ante expected social surplus²²The calculus is not different from the one used in Section 5. See the Appendix. Clearly, we verify that \(s^{*}\left(b,k\right)=s^{1}\left(b,k\right)\).

Pursuing the last section, we wish to know whether the incentive to free-ride should prompt a broader use of the contest as a social decision scheme. Comparing (LABEL:eq:s*) and (\ref{eq:sn}), we see that the free-riding behaviors do not harm the efficiency of the contest (i.e., \(s^{*}\left(b,k\right)\leq s^{n}\left(b,k\right)\) for all \(n\)) if and only if \(1/2-2b+k+b^{2}f\left(k/b\right)/3\leq 0\). However, this also implies that \(s^{*}\left(b,k\right)\leq s^{n}\left(b,k\right)\leq b-1/2\leq\max\left(0,b-1/2\right)\), meaning that the cost-benefit analysis is a better decision scheme in both settings. These findings imply Proposition 2 below.

Proposition 2. Compared with the benchmark model, the incentive to free-ride within the environmental pressure group should prompt to restrict further the use of the contest as a tool for social decision.

We can explain it as follows, using (\ref{eq:xn}) and (\ref{eq:yn}). The free-riding behaviors reduce the socially wasteful efforts of lobbying. The members of lobby \(E\) reduce their efforts, in the prospect of benefiting from the other’s contributions, and lobby \(I\) replies by limiting his own effort in turn. However, the free-riding behaviors increase the probability of approving the new product, no matter the level of harm. This is because lobby \(I\) proportionally reduces his bid less than lobby \(E\) does. In general, any of the two effects can dominate the other. Proposition 2 holds true because the former dominates the latter when the cost-benefit analysis is a better instrument.

Up to here, we have assumed that only the industrial lobby observes the damage ex ante. This is appealing if the latter, having registered a patent, is in a privileged position to conduct the laboratory experiments necessary to assess the detrimental effects of his product. However, the alternative assumption, where both lobbies are informed, is also relevant and should be considered.

Given that both players observe \(\delta\) before playing the contest, a Nash equilibrium of the contest game, where lobby \(I\) plays \(X^{*}\left(\delta\right)\) and lobby \(E\) plays \(Y^{*}\left(\delta\right)\), for all \(\delta\), is standard to derive (Nti, 1999), giving

From this, we can calculate the ex ante expected social surplus³³The calculus can be found in the Appendix.

Again, we wish to know whether the public information about the damage should encourage a larger use of the contest as a social decision scheme. Proposition 3 below states that this is indeed the case. The proof in the Appendix shows that the condition such that the contest is socially better is satisfied on a larger set of the parameters here than in the benchmark model.

Proposition 3. Compared with the benchmark model, the setting with public information should prompt to expand further the use of the contest as a tool for social decision.

We can understand this using the equilibrium outcome (4) and (5). In contrast with the benchmark model, we see here that the pressure groups exert zero effort when the damage is smaller than the level of assets of the industrial lobby (i.e., \(\delta\leq k\)). Therefore, not only less resources are wasted in lobbying activities, but also the new product is approved with probability one (which is the socially optimal outcome since \(k<b\)). Both reasons make the contest game more efficient compared with the benchmark model with private information. When \(\delta>k\), the equilibrium outcome remains inefficient here.

The main message that emerges from all settings is that the outcome of the contest is socially preferable than that of the cost-benefit analysis, provided that the industrial lobby risks in court a sufficient fraction of his benefit. Moreover, in the limit case where \(k\rightarrow b\), our results even imply that the contest always implements the first-best allocation at no cost (i.e., \(x+y\rightarrow 0\), for all \(\delta\), and \(\pi\left(x,y\right)\rightarrow 1\) if \(\delta<b\) and \(\pi\left(x,y\right)\rightarrow 0\) if \(\delta\geq b\)).

This suggests to implement legal rules ensuring that the maximum amount of damages that lobby \(I\) can pay in court never be less than the benefit earned from marketing the new product. In our setting, this can be done as follows. If \(k\) represents a statutory limit on damages, the latter should simply be set equal to \(b\). If \(k\) represents lobby \(I\)’s level of assets, lobby \(I\) should be prohibited to distribute to the shareholders more dividends than \(k\) before the judgment. Clearly, both policies would secure a payment capacity of lobby \(I\) in court at least equal to \(b\) and would thus implement the first-best optimal allocation.⁴⁴The payment capacity of lobby \(I\) at the time of the judgment is equal to his initial assets, plus his benefit, less the dividends distributed to his shareholders.