<p>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 $l^i_1(c_i,,\mathbf{c}^{-i})&lt;0$.</p>

<p>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
$$-f'[l^i(c^i,\mathbf{c}^{-i})]l^i_1(c^i,\mathbf{c}^{-i})-\left[\sum_{j\neq i}f'[l^j(c^j,\mathbf{c}^{-j})]\frac{\partial}{\partial c^i}l^j(c^j,\mathbf{c}^{-j})\right]-1=0.$$</p>

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

<p>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.</p>

<hr>

<p>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.</p>

<p>Now let's look at $i$'s payoff:</p>

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

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

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

<p>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.</p>

<p>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.</p>

CC BY-SA 3.0
2014-11-27T11:37:56.137Z
I wrote this a little hastily, so there may be an error. But the main pieces of a solution should be there.
Ubiquitous
CC BY-SA 3.0
2014-11-27T14:05:57.767Z
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
CC BY-SA 3.0
2014-11-27T15:05:07.653Z
@Bravo I edited the answer to give a bit more detail on how the Pigouvian subsidy would align the private and social incentives.
Ubiquitous
CC BY-SA 3.0
2014-11-27T11:37:25.427Z
2014-11-27T15:05:13.077Z
2014-11-27T15:05:13.077Z
Ubiquitous
<p>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 $c_i$.</p>

<p>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.</p>

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

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

CC BY-SA 3.0
2014-11-27T11:19:29.993Z
is L just the $\sum l_i$? If not shouldn't it appear somewhere in the utility function?
CarrKnight
CC BY-SA 3.0
2014-11-27T13:34:48.537Z
@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
<game-theory><social-welfare>
Mechanisms to incentivise social welfare
Bravo
# Mechanisms to incentivise social welfare

(Asked by: username_0 on 2014-11-27)

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)

**Tags:** game-theory, social-welfare

> **Comments:**
> **username_1**: is L just the $\sum l_i$? If not shouldn't it appear somewhere in the utility function?  
> **username_0**: @username_1: Well, $L$ could be some function $g(l_i,i=1,2,\ldots)$, but username_2 has assumed it is the sum. It does not appear in the utility function because every individual cares only about his own loss $l_i$.  

---

## 1 answers

### Answer by username_2 on 2014-11-27. Score: 5 | Accepted answer

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.

> **Comments:**
> **username_2**: I wrote this a little hastily, so there may be an error. But the main pieces of a solution should be there.  
> **username_0**: 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?  
> **username_2**: @username_0 I edited the answer to give a bit more detail on how the Pigouvian subsidy would align the private and social incentives.  

---

