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# Econ经济作业代写Economics代考|The Clarke-Groves mechanism

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## Public-goods problems and functions|Econ经济作业代写Economics代考

the Clarke-Groves tax and it is used to induce player $i$ to reveal his true willingness to pay, $m_{i}=t_{i}$.

DEFINITION XVIII.8 (PUBLIC-GOOD PROBLEM). The social choice problem $(N, T, \tau, Z, u)$ is called a public-good problem where

• $N$ is the set of consumers of the public good,
• $T$ is defined by $T_{i}=\mathbb{R}$ for all $i \in N$,
• $\tau$ is a probability distribution on $T$,
• $Z={0,1} \times \mathbb{R}^{n}$ is the outcome set, and
• $u=\left(u_{i}\right){i \in N}$ is the tuple of utility functions $u{i}: Z \times T_{i} \rightarrow \mathbb{R}, i \in N$, given by $\left(b, z_{1}, \ldots, z_{n}\right) \mapsto \begin{cases}t_{i}-\frac{C}{n}-z_{i}, & b=1 \ -z_{i}, & b=0\end{cases}$

The public good should be provided if and only if the aggregate willingness to pay is not smaller than the cost of providing the good. Thus, efficiency has nothing to do with who contributes how much to the cost $C$.
DEFINITION XVIII.9. Let $(N, T, \tau, Z, u)$ be a public-good problem.
$$f: T \rightarrow Z$$
is called a public-good (social choice) function if
$$\sum_{i \in N} t_{i} \geq C \Leftrightarrow b=1$$
holds.

## The definition of the Clarke-Groves mechanism|ECON经济作业代写ECONOMICS代考

Definition XVIII.10 (CLARKE-GROVES MECHANISM). Let $T$ and $Z$ be defined as above. The direct mechanism $(T, \zeta)$ with tuple of message sets $M:=T$ is called the Clarke-Groves mechanism if $\zeta: M \rightarrow Z$ is defined by
$$\zeta(m)=\left(b(m), \zeta_{1}(m), \ldots, \zeta_{n}(m)\right)$$
with
$$b(m)= \begin{cases}1, & \sum_{i \in N} m_{i} \geq C \ 0, & \sum_{i \in N} m_{i}<C\end{cases}$$
and, for every player $i \in N$,
$$\zeta_{i}(m)= \begin{cases}\frac{n-1}{n} C-\sum_{j \in N \backslash{i}} m_{j}, & \sum_{j \in N} m_{j} \geq C \text { and } \sum_{j \in N \backslash{i}} m_{j}<\frac{n-1}{n} C \ \sum_{j \in N \backslash{i}} m_{j}-\frac{n-1}{n} C, & \sum_{j \in N} m_{j}<C \text { and } \sum_{j \in N \backslash{i}} m_{j} \geq \frac{n-1}{n} C \ 0, & \text { otherwise }\end{cases}$$
Note that the additional payment as defined by $\zeta_{i}$ is always non-negative. If the other players prefer not to have the public good (and pay for it), the damage they suffer is $\frac{n-1}{n} C-\sum_{j \in N \backslash{i}} m_{j}$, their share of the cost burden minus their (announced) willingness to pay.

## Lemma and proof|Econ经济作业代写Economics代考

Let us begin with the first case and consider player 1. The truthful announcement of his type leads to the provision of the public good and $z_{1}=0$. His utility is
$$u_{1}(1,0, \ldots)=t_{1}-\frac{C}{n}$$
Player 1 has no incentive to overstate his willingness to pay (that will not change anything). Indeed, a utility change occurs only if player 1 understates his willingness to pay so that the public good is not provided. Then, player 1 has to pay $\sum_{j \in N \backslash{1}} m_{j}-\frac{n-1}{n} C$ (third case). Player 1 is harmed by this understatement:
\begin{aligned} u_{1}(1,0, \ldots) &=t_{1}-\frac{C}{n} \ & \geq\left(C-\sum_{j \in N \backslash{1}} m_{j}\right)-\frac{C}{n}(\text { case } 1) \ &=-\left(\sum_{j \in N \backslash{1}} m_{j}-\frac{n-1}{n} C\right) \ &=u_{1}\left(0, \sum_{j \in N \backslash{1}} m_{j}-\frac{n-1}{n} C, \ldots\right) \end{aligned}

## Discussion|ECON经济作业代写ECONOMICS代考

The reader will note a similarity to the second-price auction. The announced willingness to pay determines

• who obtains the object in the second-price auction and
• whether the public good will be provided according to the ClarkeGroves mechanism
but has no effect
• on how much the successful bidder pays for the object (second-price auction) and
• on the size of the externality payment (Clarke-Groves mechanism).

## PUBLIC-GOODS PROBLEMS AND FUNCTIONS|ECON经济作业代写ECONOMICS代考

• ñ是公共物品的消费者集合，
• 吨定义为吨一世=R对所有人一世∈ñ,
• τ是一个概率分布吨,
• 和=0,1×Rn是结果集，并且
• $u=\left(u_{i}\right) {i \in N}一世s吨H和吨你p一世和○F你吨一世一世一世吨和F你nC吨一世○nsu {i}: Z \times T_{i} \rightarrow \mathbb{R}, i \in N,G一世v和nb和\left(b, z_{1}, \ldots, z_{n}\right) \mapsto{吨一世−Cn−和一世,b=1 −和一世,b=0$

F:吨→和

∑一世∈ñ吨一世≥C⇔b=1

## THE DEFINITION OF THE CLARKE-GROVES MECHANISM|ECON经济作业代写ECONOMICS代考

G(米)=(b(米),G1(米),…,Gn(米))

b(米)={1,∑一世∈ñ米一世≥C 0,∑一世∈ñ米一世<C

G一世(米)={n−1nC−∑j∈ñ∖一世米j,∑j∈ñ米j≥C 和 ∑j∈ñ∖一世米j<n−1nC ∑j∈ñ∖一世米j−n−1nC,∑j∈ñ米j<C 和 ∑j∈ñ∖一世米j≥n−1nC 0, 否则

## DISCUSSION|ECON经济作业代写ECONOMICS代考

• 谁在第二价拍卖中获得了标的物，以及
• 公共物品是否会按照 ClarkeGroves 机制提供
但没有效果
• 中标人为该物品支付的金额（次价拍卖）和
• 关于外部性支付的规模（克拉克-格罗夫斯机制）。

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