paper source

(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.