# Quadratic Funding has a 'marketing problem'
*Or why we need impact evaluation to make QF cool again*

## Part I: Core Mechanism Design

### 1. Quadratic Funding Framework

**Economic Formula:**
$$
M = \left(\sqrt{\sum_{i=1}^n c_i}\right)^2 - \sum_{i=1}^n c_i
$$
where:
- $M$ is matching amount
- $c_i$ is contribution from participant $i$
- $n$ is number of participants

**Plain English:**
The matching formula works like this:
1. Take everyone's contributions
2. Find the square root of each contribution
3. Add all these square roots together
4. Square that sum
5. Subtract the original contributions

This creates a multiplier effect that favors many small donations over a few large ones.

### 2. Value and Contribution Theory

#### 2.1 Value Components
**Economic Formula:**
$$
v_{perceived} = v_{true} + m(e)
$$
where:
- $v_{true}$ is actual impact/utility
- $m(e)$ is marketing effect from effort $e$

**Plain English:**
What people think a project is worth ($v_{perceived}$) is made up of two parts:
1. The actual value it creates ($v_{true}$)
2. How well it's marketed ($m(e)$)
For example, a project might create $100 worth of value but with $50 worth of marketing effect, making people think it's worth $150.

#### 2.2 Contribution Function
**Economic Formula:**
$$
c_i = f(v_{perceived}) = f(v_{true} + m(e))
$$

**Plain English:**
People decide how much to contribute based on what they think the project is worth, not its true value. If marketing makes something look twice as valuable, people might contribute twice as much.

## Part II: The Marketing Problem

### 3. Marketing Distortion Analysis

#### 3.1 Quadratic Amplification of Marketing
**Economic Formula:**
$$
M = \left(\sqrt{\sum_{i=1}^n f(v_{true} + m(e))}\right)^2 - \sum_{i=1}^n f(v_{true} + m(e))
$$

**Plain English:**
The quadratic funding formula accidentally makes the marketing problem worse. When marketing increases perceived value:
1. It increases initial contributions
2. These increased contributions get amplified by the matching formula
3. This creates a double-boost for good marketing

#### 3.2 Welfare Analysis

**Optimal Social Welfare Formula:**
$$
W_{optimal} = \sum_{j=1}^m \left(v_{true,j} \cdot F_j\right)
$$
Subject to budget constraint:
$$\sum_{j=1}^m F_j = B$$

**Plain English:**
In a perfect world, projects would receive funding proportional to their true value. The total welfare would be the sum of each project's true value multiplied by its funding.

**Actual Welfare Formula:**
$$
W_{actual} = \sum_{j=1}^m \left(v_{true,j} \cdot F_j(v_{perceived,j})\right)
$$
where:
$$F_j(v_{perceived,j}) = \sum_{i=1}^n c_{i,j} + \left(\sqrt{\sum_{i=1}^n c_{i,j}}\right)^2 - \sum_{i=1}^n c_{i,j}$$
$$c_{i,j} = f(v_{true,j} + m_j(e_j))$$

**Plain English:**
In reality, funding is based on perceived value, which includes marketing effects. This means even low-value projects can get lots of funding if they market well.

#### 3.3 Marketing Effect Model
**Economic Formula:**
$$
m_j(e_j) = k \cdot \log(1 + e_j)
$$
where:
- $k$ is marketing multiplier
- $e_j$ is marketing effort/spend

**Plain English:**
Marketing effectiveness follows a logarithmic curve:
- Initial marketing efforts have big effects
- Additional marketing has diminishing returns
- The multiplier $k$ determines how powerful marketing is in your community

#### 3.4 Value Loss Calculation

**Deadweight Loss Formula:**
$$
DWL = W_{optimal} - W_{actual}
$$

**Misallocation Ratio:**
$$
\eta = \frac{W_{actual}}{W_{optimal}} = \frac{\sum_{j=1}^m \left(v_{true,j} \cdot F_j(v_{true,j} + k\log(1 + e_j^*))\right)}{\sum_{j=1}^m \left(v_{true,j} \cdot F_j^{optimal}\right)}
$$

**Plain English:**
We can measure how much value is lost by comparing:
1. The value we'd get if funding matched true impact
2. The value we actually get with marketing distortions
The difference is pure waste - money that could have created more value if allocated better.

## Part III: Marketing Problem Example

### 4. Two-Project Case Study
**Economic Formula:**
$$\eta_{two} = \frac{v_{true,1}F_1(v_{true,1} + k\log(1 + e_1)) + v_{true,2}F_2(v_{true,2} + k\log(1 + e_2))}{v_{true,1}F_1^{optimal} + v_{true,2}F_2^{optimal}}$$

**Example Parameters:**
- High-impact project: $v_{true,1} = 100, e_1 = 1$
- Low-impact project: $v_{true,2} = 20, e_2 = 10$
- Marketing multiplier: $k = 30$
Results in $\eta \approx 0.6$

**Plain English:**
In a real-world test case:
1. A high-impact project with low marketing gets underfunded
2. A low-impact project with high marketing gets overfunded
3. About 40% of potential value is lost

This matches what we see in actual quadratic funding rounds.

### 5. Practical Implications

The analysis reveals four key mechanism design failures:

1. **Incentive Compatibility Violation**
- Economic: Projects optimize $e_j$ instead of $v_{true}$
- Plain English: Projects focus on marketing instead of impact

2. **Pareto Inefficiency**
- Economic: Resources spent on $m(e)$ could have increased $v_{true}$
- Plain English: Marketing budgets could have been spent on actual impact

3. **Strategic Manipulation**
- Economic: $\exists s_i \in S_i$ such that $u_i(s_i) > u_i(θ_i)$
- Plain English: Being strategic (marketing) beats being honest

4. **Information Asymmetry**
- Economic: Participants cannot distinguish $v_{true}$ from $m(e)$
- Plain English: It's hard to tell real impact from good marketing

## Part IV: Solutions

### 6. Single-Dimension Solutions

#### 6.1 Impact Verification
**Economic Formula:**
$$
M_{impact} = \left(\sqrt{\sum_{i=1}^n (c_i \cdot I(x))}\right)^2 - \sum_{i=1}^n c_i
$$
where $I(x)$ is impact verification score

**Plain English:**
Multiply each contribution by an impact score before calculating the match. This means marketing alone isn't enough - you need to prove real impact.

#### 6.2 Time-Delayed Release
**Economic Formula:**
$$
M(t) = \left(\sqrt{\sum_{i=1}^n c_i}\right)^2 \cdot \frac{I(t)}{I(0)}
$$

**Plain English:**
Release funding over time based on measured impact:
- Initial funding based on promises
- Continued funding based on delivered results

This creates accountability.

#### 6.3 Expertise Weighting
**Economic Formula:**
$$
M_{expert} = \left(\sqrt{\sum_{i=1}^n (w_i \cdot c_i)}\right)^2 - \sum_{i=1}^n c_i
$$

**Plain English:**
Give more weight to contributions from experts who:
- Have domain knowledge
- Have good track records
- Can evaluate true impact

### 7. Multi-Dimensional Solutions

#### 7.1 Multi-Round Learning
**Economic Formula:**
$$
w_n = g(|v_{predicted_{n-1}} - v_{true_{n-1}}|)
$$

**Plain English:**
Update how much we trust each contributor based on how well they predicted impact in previous rounds:
- Good predictors get more weight
- Poor predictors get less weight

This creates a "wisdom of the crowds" effect.

#### 7.2 Hybrid Oracle System
**Economic Formula:**
$$
M_{hybrid} = \alpha\left(\sqrt{\sum_{i=1}^n c_i}\right)^2 + (1-\alpha)O(x)
$$

**Plain English:**
Combine two funding sources:
1. Community quadratic funding (weighted $\alpha$)
2. Expert oracle decisions (weighted $1-\alpha$)

This provides checks and balances.

#### 7.3 Comprehensive Solution
**Economic Formula:**
$$
M_{complete} = \left(\sqrt{\sum_{i=1}^n (w_i \cdot c_i \cdot I(x))}\right)^2 \cdot R(t)
$$

**Plain English:**
The complete solution combines:
- Expert weights ($w_i$)
- Impact verification ($I(x)$)
- Time-based release ($R(t)$)

This addresses multiple failure modes simultaneously.

## Conclusion

Quadratic Funding's core innovation - making the number of contributors matter more than the amount - is powerful. However, its current implementation creates incentives for marketing over impact. By measuring impact, we can help align incentives with true value creation while maintaining the democratic spirit of the original mechanism.

## Appendix: The Marketing Arms Race: A Prisoner's Dilemma Analysis

### 3.1 Game Theoretic Model

Quadratic Funding creates an unintended prisoner's dilemma in marketing decisions. Projects must choose between allocating resources to impact creation or marketing efforts, leading to a collectively suboptimal equilibrium.

#### 3.1.1 Basic Model
Consider two projects competing for a matching pool $M$. Each project $i$ has budget $B_i$ and must decide on marketing expenditure $e_i$. The payoff function for project $i$ is:

$$
\pi_i(e_i, e_j) = \frac{(v_{true,i} + k\log(1 + e_i))^2}{\sum_j(v_{true,j} + k\log(1 + e_j))^2}M - e_i
$$

where:
- $v_{true,i}$ is project i's true impact
- $k$ is the marketing effectiveness parameter
- $M$ is the matching pool size

#### 3.1.2 Strategic Choices
Each project can choose between:
1. Low Marketing (L): $e_i = 0.05B_i$
2. High Marketing (H): $e_i = 0.30B_i$

The resulting payoff matrix (normalized to 100 units):

$$
\begin{matrix}
& \text{L} & \text{H} \\
\text{L} & (400,400) & (200,500) \\
\text{H} & (500,200) & (300,300)
\end{matrix}
$$

### 3.2 Equilibrium Analysis

#### 3.2.1 Nash Equilibrium
The game has a unique Nash equilibrium where both projects choose High Marketing (H,H), demonstrable through best response analysis:

$$
\text{BR}_i(L) = H \text{ since } 500 > 400 \\
\text{BR}_i(H) = H \text{ since } 300 > 200
$$

#### 3.2.2 Social Optimum
The socially optimal outcome is (L,L), which maximizes total value creation:

$$
\text{Social Welfare}(L,L) = 800 > \text{Social Welfare}(H,H) = 600
$$

This divergence between individual rationality and collective optimality defines the prisoner's dilemma.