# Mechanisms to incentivise social welfare

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)

• is L just the $\sum l_i$? If not shouldn't it appear somewhere in the utility function? – CarrKnight Nov 27 '14 at 11:19
• @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 $l_i$. – Bravo Nov 27 '14 at 13:34

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:

$$U^i=S^i-f(\widetilde{l}^i(c^i,\mathbf{c}^{-i}))-c^i$$

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

$$U^i=\max_{\mathbf{c}^{-i}}\left[\widetilde{W}(\widetilde{\mathbf{l}},\mathbf{c})|c^i\right]-\max_{\mathbf{c}^{-i}}\left[\widetilde{W}(\widetilde{\mathbf{l}},c^1,\ldots,c^{i-1},0,c^{i+1},\ldots,c^n)-(-f(\widetilde{l}^i(0,\mathbf{c}^{-i})))\right]$$

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.

• I wrote this a little hastily, so there may be an error. But the main pieces of a solution should be there. – Ubiquitous Nov 27 '14 at 11:37
• 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? – Bravo Nov 27 '14 at 14:05
• @Bravo I edited the answer to give a bit more detail on how the Pigouvian subsidy would align the private and social incentives. – Ubiquitous Nov 27 '14 at 15:05