Suppose there is a set of agents causing loss $l_i$ to themselves and aggregate loss $L$ to the society, both of which can be minimised by their investments $c_i$.

The social planner who is concerned with minimising $L(c_i)$ wants to incentivise players to invest more. Players have utility $u_i=-f(l_i)-c_i$ and they decide on whether to invest more based on utilities.

What are some mechanisms which can incentivise agents in this framework? Can VCG-type mechanisms be adapted to this situation?

(PS: Sorry if the question is vague; please feel free to add assumptions)

  • $\begingroup$ is L just the $\sum l_i$? If not shouldn't it appear somewhere in the utility function? $\endgroup$
    – CarrKnight
    Commented Nov 27, 2014 at 11:19
  • $\begingroup$ @CarrKnight: Well, $L$ could be some function $g(l_i,i=1,2,\ldots)$, but Ubiquitous has assumed it is the sum. It does not appear in the utility function because every individual cares only about his own loss $l_i$. $\endgroup$
    – Bravo
    Commented Nov 27, 2014 at 13:34

1 Answer 1


You say that losses are reduced by investment, so we can write the (private) loss as a function of own investment and others' investment: $l^i(c^i,\mathbf{c}^{-i})$ with $l^i_1(c_i,,\mathbf{c}^{-i})<0$.

Player $i$'s utility is $U^i(c^i,\mathbf{c}^{-i})=-f(l^i(c^i,\mathbf{c}^{-i}))-c^i$ and he invests until the first-order condition is satisfied: $$-f'[l^i(c^i,\mathbf{c}^{-i})]l^i_1(c^i,\mathbf{c}^{-i})-1=0.$$ The social loss is $$\sum_i\left[-f_i(l^i(c^i,\mathbf{c}^{-i}))-c_i\right]$$ so that the first-order condition for the optimal $c^i$ is $$-f'[l^i(c^i,\mathbf{c}^{-i})]l^i_1(c^i,\mathbf{c}^{-i})-\left[\sum_{j\neq i}f'[l^j(c^j,\mathbf{c}^{-j})]\frac{\partial}{\partial c^i}l^j(c^j,\mathbf{c}^{-j})\right]-1=0.$$

This now looks like a standard externality model, where the external benefit of $i$'s investment is $$b^i(\mathbf{c}^{-i})=-\sum_{j\neq i}f'[l^j(c^j,\mathbf{c}^{-j})]\frac{\partial}{\partial c^i}l^j(c^j,\mathbf{c}^{-j})>0.$$

If $l^i(\cdot)$ are publicly known then the problem can easily be solved as by imposing upon $i$ a standard Pigouvian subsidy of $b^i(\mathbf{c}^{-i})$ per unit of investment. This means that $i$'s payoff (including the subsidy) when $j\neq i$ choose $\mathbf{c}^{-i}$ is $$\underbrace{-f(l^i(c^i,\mathbf{c}^{-i}))-c^i}_{U^i(c^i,\mathbf{c}^{-i})}+b^i(\mathbf{c}^{-i})c^i.$$ A rational $i$ will maximise this subsidy-augmented payoff; his first order condition is $$-f'[l^i(c^i,\mathbf{c}^{-i})]l^i_1(c^i,\mathbf{c}^{-i})\underbrace{-\left[\sum_{j\neq i}f'[l^j(c^j,\mathbf{c}^{-j})]\frac{\partial}{\partial c^i}l^j(c^j,c^{-j})\right]}_{b^i(\mathbf{c}^{-i})}-1=0,$$ which is exactly the same as the first-order condition used to maximise social welfare. It is therefore a Nash equilibrium for each $i$ to choose the socially efficient level of investment given that everyone else does the same.

If the $l$s are heterogeneous and private information then things get more complicated because we have to get people to reveal their private information. Assume there is a benevolent central planner with the power to monitor agents' investments. One way to do this is then, indeed, to use a VCG mechanism. It would work like this: first, the central planner asks each agent to report their $l^i(\cdot)$. Write $\widetilde{l}^i$ for the reports. He then solves the first-order condition for the maximum of social welfare given the (possibly dishonest) reports. Thus, he maximises $$\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})=\sum_i\left[-f_i(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c_i\right].$$ Next the planner instructs all players to invest up to the optimal level. In return, he promises to pay to each $i$ a subsidy equal to $$S^i=\max_{\mathbf{c}^{-i}}\left[\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})-(-f(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c^i)|c^i\right]-\max_{\mathbf{c^{-i}}}\left[\widetilde{W}(\widetilde{\mathbf{l}},c^1,\ldots,c^{i-1},0,c^{i+1},\ldots,c^n)-(-f(\widetilde{l}^i(0,\mathbf{c}^{-i})))\right].$$ Woah, what is that ugly thing!? The first set of square brackets is the welfare everyone other than $i$ gets. The second is the welfare everyone other than $i$ would get if $i$ were to drop out of the market all together. Thus, the difference between the two is $i$'s positive externality on everyone else.

Now let's look at $i$'s payoff:


When evaluated at the equilibrium $\mathbf{c}^{-i}$ some terms cancel and we have


The second square bracket here does not depend on $c^i$ (remember, it is the surplus if $i$ were to drop out of the market) so $i$'s problem collapses to choosing $c^i$ to maximise $\max_{\mathbf{c}^{-i}}\left[\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})|c^i\right]$. Thus, $i$ has the right incentive to choose $c^i$ to maximise social welfare with respect to the reported $l$s.

What about the incentive to report $l$ honestly? The social planner is going to chose the optimal $c$s to maximise $$\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})=\left[\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})-(-f(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c^i)\right]+(-f(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c^i)$$ (this is just $\widetilde{W}$ with $i$'s utility added on and then subtracted again). $i$ would like to maximise $$U^i=S^i+(-f(l^i(c^i,\mathbf{c}^{-i}))-c^i)$$ which is $$U^i=\underbrace{\left[\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})-(-f(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c^i)\right]+z}_{S^i}+(-f(l^i(c^i,\mathbf{c}^{-i}))-c^i)$$ where $z$ is the second term in $S^i$ (does not depend on $\widetilde{l}^i$). Thus we see that the social planner's choice will coincide with $i$'s optimum exactly when $l^i=\widetilde{l}^i$ so that $i$ has an incentive to report $l^i$ truthfully.

  • $\begingroup$ I wrote this a little hastily, so there may be an error. But the main pieces of a solution should be there. $\endgroup$
    – Ubiquitous
    Commented Nov 27, 2014 at 11:37
  • $\begingroup$ I have a question on the first part of the solution. There are essentially two different optimal cost vectors, say $\mathbf{c_1}$ and $\mathbf{c_2}$ for individual and social optimum. In your last claim, you say that adding a Pigouvian subsidy to $\mathbf c_1$ will help you attain the optimum of $\mathbf c_2$. Could you elaborate how? $\endgroup$
    – Bravo
    Commented Nov 27, 2014 at 14:05
  • 1
    $\begingroup$ @Bravo I edited the answer to give a bit more detail on how the Pigouvian subsidy would align the private and social incentives. $\endgroup$
    – Ubiquitous
    Commented Nov 27, 2014 at 15:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.