Curve Pledge Benefit
Abstract
Modifying the current rewards calculation equation by substituting a nroot curved relationship between pledge and pledge benefit rewards for the current linear relationship will better achieve the original design goal of incentivizing pledge to help prevent Sybil attacks. This also reduces the unfortunate side effect in the current equation that over rewards private pools which provide no additional security benefit.
Motivation: why is this CIP necessary?
There are two main reasons for changing the current linear a0 pledge benefit factor in the rewards equation.

Pools pledging less than 1 million ADA see very little reward benefit. This is not a strong incentive for pool operators as at current prices that is approximately $150,000 USD.

Private pools get massive reward benefit without providing any additional protection against Sybil attacks. Why should a private pool make 29% more rewards than a pool with 5m ADA pledge while doing the same work?
Specification
This is a modification of the maxPool function defined in section 11.8 Rewards Distribution Calculation of “A Formal Specification of the Cardano Ledger”.
maxPool = (R / (1 + a0)) * (o + (s * a0 * ((o  (s * ((z0  o) / z0))) / z0)))
where: R = ((reserve * rho) + fees) * (1  tau) o = min(pool_stake / total_stake, z0) = z0 for fully saturated pool s = pledge / total_stake z0 = 1 / k and the following are current protocol parameters: k = 150 rho = 0.0022 a0 = 0.3 tau = .05
The idea is to replace s in the above equation with an nroot curve expression of pledge rather than the linear pledge value.
We use an expression called crossover to represent the point where the curve crosses the line and the benefit in the new and original equations is identical. Because the a0 pledge benefit is spread over the pledge range from 0 to saturation there is a dependence on k and total_stake. Since k and total_stake will likely change over time it is best to express crossover in terms of k and total_stake as follows:
crossover = total_stake / (k * crossover_factor)
where crossover_factor is any real number greater than or equal to 1. So crossover_factor is essentially a divisor of the pool saturation amount. For example, setting crossover_factor to 20 with k = 150 and total_stake = 31 billion gives a crossover of approximately 10.3 million.
Also, we can parameterize the nroot curve exponent. This gives us:
s = pow(pledge, (1 / curve_root)) * pow(crossover, ((curve_root  1) / curve_root)) / total_stake
The curve_root could be set to any integer greater than 0 and when set to 1 produces the current rewards equation. The curve_root is n in nroot. For example, 1 = linear, 2 = square root, 3 = cube root, 4 = fourth root, etc.
By making this modification to the rewards equation we introduce two new protocol parameters, crossover_factor and curve_root, that need to be set thoughtfully.
Test Cases
See rewards.php for some simple PHP code that allows you to try different values for crossover_factor and curve_root and compare the resulting rewards to the current equation. For usage, run "php f rewards.php help".
An interesting set of parameters as an example is:
curve_root = 3 crossover_factor = 8
Running "php f rewards.php 3 8" produces:
Assumptions Reserve: 14b Total stake: 31.7b Tx fees: 0 Fully Saturated Pool Rewards available in epoch: 29.3m Pool saturation: 211.3m
Curve root: 3 Crossover factor: 8 Crossover: 26.4m
Pledge Rewards Benefit Alt Rwd Alt Bnft 0k 150051 0% 150051 0% 10k 150053 0% 150458 0.27% 50k 150062 0.01% 150747 0.46% 100k 150073 0.01% 150928 0.58% 200k 150094 0.03% 151156 0.74% 500k 150158 0.07% 151551 1% 1m 150264 0.14% 151941 1.26% 2m 150477 0.28% 152432 1.59% 5m 151116 0.71% 153282 2.15% 10m 152181 1.42% 154122 2.71% 20m 154311 2.84% 155180 3.42% 50m 160702 7.1% 157012 4.64% 100m 171352 14.2% 158821 5.84% 211.3m 195067 30% 161305 7.5%
As you can see this gives meaningful pledge benefit rewards to pools pledging less than 1m ADA.
Rationale: how does this CIP achieve its goals?
Using the nroot curve pledge benefit shows a much more reasonable distribution of pledge related rewards which will encourage meaningful pledges from more pool operators thus making the network more secure against Sybil attacks. It also provides higher rewards for higher pledge without disproportionately rewarding a very few private pool operators who provide no additional security value to the network. This modification maintains the general principles of the current rewards equation and does not introduce any hard limits. It improves the incentives that were originally designed to make them more meaningful for the majority of pool operators.
Backward Compatibility
This proposal is backwards compatible with the current reward function by setting the curve_root parameter to 1.
Path to Active
Acceptance Criteria
 The new equation is implemented in the ledger and enacted through a hardfork.
Implementation Plan

Agreement by the Ledger team as defined in CIP0084 under Expectations for ledger CIPs including "expert opinion" on changes to rewards & incentives.

Author has offered to produce an implementation of this change as a pull request if shown where the current maxPool reward equation is implemented in the code.
Copyright
2020 Shawn McMurdo. This CIP is licensed under Apache2.0.