<p>In your example, the playstation is a public good to be used by both roommates. Note that  <span class="math-container">$\theta_1,\theta_2 &lt; \theta_3$</span>, but <span class="math-container">$\theta_1+\theta_2&gt;\theta_3$</span>, that is, one roommate alone would not buy it, but together it is efficient to buy.
The idea of your VCG mechanism is that people only pay when they are &quot;pivotal,&quot; that is, when they have some impact on the collective decision and their VCG payment reflects this impact.</p>
<p>You have <span class="math-container">$v_i(\theta_i,k) = k \theta_i$</span> for <span class="math-container">$i\in\{1,2\}$</span> and <span class="math-container">$v_3(\theta_3,k) = (1-k) \theta_3$</span>.
Now, you can now calculate the VCG transfers <span class="math-container">$t_i (\theta_i,\theta_{-i}) = \sum_{j\neq i} v_j(\theta,k^*(\theta) ) - \sum_{j\neq i} v_j(\theta_{-i},k^*(\theta_{-i}) )$</span>.
Hence, <span class="math-container">$t_1 =t_2 = (300+0) - (0+400) = -100$</span>, and <span class="math-container">$t_3 = (300+300)-(300+300)=0$</span>. So the mechanism collects in total 200 from agents 1 and 2, while the cost to agent 3 is 400. This is a budget-balance problem.</p>

CC BY-SA 4.0
2021-08-04T09:31:10.120Z
2021-08-04T09:31:10.120Z
Bayesian
<p>I have a general question on how we state that in VCG (which doesn't balance the budget) we can't distribute the money as it will destroy incentives. I don't understand how for ex. this would affect my payment if I knew that at the end of the finished project I would possibly get money back. Why would that harm my incentives?</p>
<p>Also, I have the answer to the following problem but I'm missing the logic.
Two roommates would like to purchase PlayStation 4 with individual evaluations
<span class="math-container">$\theta_1 = \theta_2 = 300$</span>. Cost (which is the evaluation of the seller) of it is <span class="math-container">$\theta_3 = 400$</span>. Compute the VCG payment associated to the efficient allocation.</p>
<p>so for <span class="math-container">$-t_i=\sum v_j(k(\hat{\theta_j}),\hat{\theta_j})-\sum v_j(k_{-i}(\hat{\theta_{-i}}),\hat{\theta_j})$</span> where first sum is with <span class="math-container">$i$</span> and the second without <span class="math-container">$i$</span></p>
<p>For the first case we would have <span class="math-container">$t_1=t_2=300-400=100$</span></p>
<p>For the second case, we would have <span class="math-container">$t_3=600-600=0$</span> but why so?</p>

<microeconomics><mechanism-design><information>
VCG mechanism and budget balance
Maybeline Lee
# VCG mechanism and budget balance

(Asked by: username_0 on 2021-08-02)

I have a general question on how we state that in VCG (which doesn't balance the budget) we can't distribute the money as it will destroy incentives. I don't understand how for ex. this would affect my payment if I knew that at the end of the finished project I would possibly get money back. Why would that harm my incentives?

Also, I have the answer to the following problem but I'm missing the logic.
Two roommates would like to purchase PlayStation 4 with individual evaluations
$\theta\_1 = \theta\_2 = 300$. Cost (which is the evaluation of the seller) of it is $\theta\_3 = 400$. Compute the VCG payment associated to the efficient allocation.

so for $-t\_i=\sum v\_j(k(\hat{\theta\_j}),\hat{\theta\_j})-\sum v\_j(k\_{-i}(\hat{\theta\_{-i}}),\hat{\theta\_j})$ where first sum is with $i$ and the second without $i$

For the first case we would have $t\_1=t\_2=300-400=100$

For the second case, we would have $t\_3=600-600=0$ but why so?

**Tags:** microeconomics, mechanism-design, information

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## 1 answers

### Answer by username_1 on 2021-08-04. Score: 4

In your example, the playstation is a public good to be used by both roommates. Note that $\theta\_1,\theta\_2 < \theta\_3$, but $\theta\_1+\theta\_2>\theta\_3$, that is, one roommate alone would not buy it, but together it is efficient to buy.
The idea of your VCG mechanism is that people only pay when they are "pivotal," that is, when they have some impact on the collective decision and their VCG payment reflects this impact.

You have $v\_i(\theta\_i,k) = k \theta\_i$ for $i\in\{1,2\}$ and $v\_3(\theta\_3,k) = (1-k) \theta\_3$.
Now, you can now calculate the VCG transfers $t\_i (\theta\_i,\theta\_{-i}) = \sum\_{j\neq i} v\_j(\theta,k^\*(\theta) ) - \sum\_{j\neq i} v\_j(\theta\_{-i},k^\*(\theta\_{-i}) )$.
Hence, $t\_1 =t\_2 = (300+0) - (0+400) = -100$, and $t\_3 = (300+300)-(300+300)=0$. So the mechanism collects in total 200 from agents 1 and 2, while the cost to agent 3 is 400. This is a budget-balance problem.


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