(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.