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(LR is) as flexible and responsive as the market, but avoids free-rider problems.

• Flexible and responsive: any one can propose a new public good project, and the project can get enough fundings even when only a small community funds it.
• Subsidies create incentives for citizens to fund projects.

## Background

(Free-rider problem is) due to the expense or inefficiency involved in excluding individuals from access.

### Cons of Existing Solutions

• 1p1v: oppress minorities.
• Capitalism: inefficiently exclude potential users.
• Charitable organization: difficult to closely align reliably with the common good.
• QV: it doesn’t solve the problem of flexibility, a.k.a., it requires a curated projects list.

## Model

### Assumptions

• We can verifiably distinguish among and identify these citizens.
• Any citizen may at any time propose a new public good.
• Our interest here is in maximization of dollar-equivalent value rather than achieving an equitable distribution of value.
• Utility function V is concave (国内的叫法一般是反的，即我们平常说的凸函数), smooth, increasing.
• The deficit is not bounded, a.k.a, the funding solution can collect unbounded taxes.

### Optimization Target Function

Given:

• Let $V_i^p(F^p)$ be the currency-equivalent utility function. It is the benefit that the citizen $i$ can get when the project $p$ is funded with $F^p$ units of currency.
• The $i \times p$ contributions matrix $\left\{c_i^p\right\}_i^p$, where $c_i^p$ is the contribution citizen $i$ makes to project $p$.

We need to find a funding distribution solution $\left\{F^p\right\}^p$ which is a p-dimention vector. $F^p$ is the funds allocated to project $p$. The solution maximize:

$\sum _ { p } \left(V _ { i } ^ { p } \left( F ^ { p } \right) - c _ { i } ^ { p }\right) - t _ { i }$


where

$\sum _ { i } t _ { i } = \sum _ { p } \left( F ^ { p } - \sum _ { i } c _ { i } ^ { p } \right)$


Since V is concave, smooth and increasing, it is easy to find the maximum using the first order derivative, which gives

$V ^ { p ^ { \prime } } = 1$


### Capitalism

$F ^ { p } = \sum _ { i } c _ { i } ^ { p }$


Result

$V ^ { p ^ { \prime } } = N$


### 1p1v

$N \cdot \operatorname{Median}_{i} V_{i} ^ { p ^ { \prime } } \left( F ^ { P } \right) = 1$


The optimal solution requires mean, where median is absolutely different with mean.

### LR

$F ^ { p } = \left( \sum _ { i } \sqrt { c _ { i } ^ { p } } \right) ^ { 2 }$


(We) assume that citizens ignore their impact on the budget and costs imposed by it.

After incorporating the deficit:

$V ^ { p ^ { \prime } } \approx 1 + \Lambda$


It is assumed that $\Lambda$ is on the order of $1/N$.

## Extensions

### Budgeted matching funds

CLR: linear combine LR and Capitalism until the deficit is under the budget.

$F ^ { p } = \alpha \left( \sum _ { i } \sqrt { c _ { i } ^ { p } } \right) ^ { 2 } + ( 1 - \alpha ) \sum _ { i } c _ { i } ^ { p }$


LR allows every project gets the optimal funding by incentivize citizens via the deficit. CLR is more practically, because in real world, there is always a budget.

### Collusion

Coercion Resistance: A voter cannot prove to anyone else who they voted for (or even, ideally, whether or not they voted) even if they wanted to.

### Negative contributions

More broadly, negative contributions may be a quite powerful way to deter collusive schemes as they offer a way for any citizen to be a “vigilante enforcer” against fraud and abuse.

On the other-side, it also can be used to attack and threaten other communities.