# King Protocol

## Token Dynamics Governance Research

**Update 1: Partner Allocation Mechanism**
**January 30, 2025**

### Overview

This note examines the mechanism used for allocating token rewards to partner protocols.

**Highlights:**
* Initial research finds three potential issues in which the proposed token distribution mechanism can be gamed.
* We offer some modifications to the mechanism that would address the concerns.
* Overall, there doesn't seem to be much risk of direct governance capture through gaming the mechanism. This is mostly due to the fact that the partner protocol token allocations for any epoch are relatively small.
* There is more risk of second order social effects: partner protocols who are unpleasantly surprised by certain outcomes. This could lead to social unrest: wasted time in discussion forums, or protocols ragequitting based on unexpected outcome.

### Background

One way to prevent whales from overtaking governance in a token supply is to create a sub-linear redistribution. In other words: take the initial distribution, apply a concave power function based on $x^p$ for $0 < p < 1$, and re-weight *pro rata* to obtain a new distribution. Many popular allocation mechanisms, such as quadratic funding in GitCoin, follow this approach.

King Protocol is currently considering such a mechanism. The initial mechanism we considered has the following form.
0. A power $p$ is chosen, with $p$ between 0 and 1.
1. There are $N$ partner protocols: Protocol 1, Protocol 2, ..., Protocol N.
2. Protocol $k$ receives a grade $g_k$, based on available data. Grades are ultimately translated to numbers, e.g. $A = 1.0, B = 0.8, etc.$
3. Protocol $k$ receives a *value* $v_k$, which is translated to a proportion between 0 and 1.
4. Protocol $k$ is given the normalized sum of values raised to the $p$. This can be written as:
$$weightToProtocol_{k} = \displaystyle\frac{gradeWeight_{k} * value_{k}^{power}}{Sum_{all} \left( gradeWeight * value^{power} \right)}$$
or
$$w_k = \displaystyle\frac{g_k \cdot v_k^p}{\displaystyle\sum_{j=1}^N g_j \cdot v_j^p}.$$

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### Concern 1: Concave Power Functions Can Be Too Effective at Redistributing Weight

**Scenario 1**:
* We use the proposed mechanism with power $p = 0.5$ (square root)
* There are five partner protocols, each with **A** grades
* One protocol has 80% of value, the other four each have 5%

In this scenario, the final redistribution has the top 80% provider receiving 50% of the allocation. This "one-vs-all" scenario leads to greater loss as more protocols join.


As the power $p$ gets closer to 1 (e.g. $p = 2/3$ or $p = 3/4$), this effect is mitigated -- but it is impossible to completely prevent, since it is just an extension of the mechanism's intended effect.

**Discussion of Risk:** Governance capture in this scenario would be unlikely; a "coup" would require all of the smaller protocols to effectively cooperate. A more likely concern is that the dominant protocol becomes uneasy once its loss-of-allocation is quantified, straining social relations.

**Possible Mitigation:** Using the original $p=2/3$ curve is less harsh in terms of dominant redistribution


**Key Question:** How much should the dominant player in the industry receive, relevant to the others?

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### Concern 2: Low-Performance Protocols Get Exaggerated Return on Investment

Let's assume for a moment that partner protocols can control the value $v_k$ being assigned. A reasonable "bang-for-your-buck" metric would be Return on Value: the ratio of final assigned weight to value given:
$$Return_k = \displaystyle\frac{weightToProtocol_k}{value_k}.$$

For partner protocols that produce extremely small value, their **Return** increases without bound.

**Scenario 2:** We modify Scenario 1 slightly so that a small protocol generates 0.01% of the value. In this case, it can receive 1% of the allocations -- a 100X return.

**Discussion of Risk:** Just like Concern 1, there is little risk of governance capture. The low-value protocol is still receiving a small allocation.

The greater issue is that this return may drive noncooperative strategies. such as "quiet quitting" on the King Protocol. A Protocol which starts to die can receive KING, which it rush-dumps on the market.

**Potential Mitigation:** Decide on a maximum return multiplier $c$ that should be allowed. Instead of basing on the mechanism on $$f(x) = x^p,$$ use $$f(x) = \min(cx, x^p),$$ whic has a small linear part at the beginning. This is sufficient to guarantee that no protocols receives more than $c$ times its value.



If this route is chosen, the values of $c$ and $p$ should be set together to achieve the desired effect. (Choosing $p$ and $c$ determines where the cutoff begins.)

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### Concern 3: Concave Power Functions are Vulnerable to Collusion

Again we assume that protocols have some ability to control their value through their own decisions.

There are many scenarios where two or more protocols would be incentivized to redistribute their value allocations. One is to get over a temporary governance threshold.

The table below illustrates a potential collusion scenario:


Without collusion, Protocol 1 and Protocol 5 combined contribute 0.801 of value and receive a combined 0.684 of the redistributed value.

With collusion, Protocol 1 and Protocol 5 combine to contribute 0.801 of the value and receive a combined 0.706621 of the redistributed value.

**Discussion of Risk:** Relatively Low. This is an extreme edge case that would likely require either a.) extensive cooperation between protocols, which creates risk, or b.) sufficient incentive to cross a threshold.