Additive Increase Multiplicative Decrease (AIMD) EIP-1559

Overview

Definitions:

• Target Block Gas: This is the target block gas consumption.
• Max Block Gas: This is the maximum block gas consumption, fetched from the x/consensus module.

This plugin implements the AIMD (Additive Increase Multiplicative Decrease) EIP-1559 dynamic fee pricing as described in this AIMD EIP-1559 research publication.

The AIMD EIP-1559 dynamic fee pricing is a slight modification to Ethereum's EIP-1559 dynamic fee pricing. Specifically it introduces the notion of an adaptive learning rate which scales the base gas price more aggressively when the network is congested and less aggressively when the network is not congested. This is primarily done to address the often cited criticism of EIP-1559 that it's base fee often lags behind the current demand for block space. The learning rate on Ethereum is effectively hard-coded to be 12.5%, which means that between any two blocks the base fee can maximally increase by 12.5% or decrease by 12.5%. Additionally, AIMD EIP-1559 differs from Ethereum's EIP-1559 by considering a configured time window (number of blocks) to consider when calculating and comparing target block gas and current block gas.

Parameters

Ethereum EIP-1559

Base EIP-1559 currently utilizes the following parameters to compute the base fee:

• PreviousBaseGasPrice: This is the base gas price from the previous block. This must be a value that is greater than 0.
• TargetBlockSize: This is the target block size in bytes. This must be a value that is greater than 0.
• PreviousBlockSize: This is the block size from the previous block.

The calculation for the updated base fee for the next block is as follows:

currentBaseGasPrice := previousBaseGasPrice * (1 + 0.125 * (currentBlockSize - targetBlockSize) / targetBlockSize)

AIMD EIP-1559

AIMD EIP-1559 introduces a few new parameters to the EIP-1559 dynamic fee pricing:

• Alpha: This is the amount we additively increase the learning rate when the target gas is less than the current gas i.e. the block was more full than the target gas. This must be a value that is greater than 0.0.
• Beta: This is the amount we multiplicatively decrease the learning rate when the target gas is greater than the current gas i.e. the block was less full than the target gas. This must be a value that is greater than 0.0.
• Window: This is the number of blocks we look back to compute the current gas. This must be a value that is greater than 0. Instead of only utilizing the previous block's gas, we now consider the gas of the previous Window blocks.
• Gamma: This determines whether you are additively increase or multiplicatively decreasing the learning rate based on the target and current block gas. This must be a value that is between [0, 1]. For example, if Gamma = 0.25, then we multiplicatively decrease the learning rate if the average ratio of current block gas to max block gas over some window of blocks is within (0.25, 0.75) and additively increase it if outside that range.
• MaxLearningRate: This is the maximum learning rate that can be applied to the base fee. This must be a value that is between [0, 1].
• MinLearningRate: This is the minimum learning rate that can be applied to the base fee. This must be a value that is between [0, 1].

The calculation for the updated base fee for the next block is as follows:

// sumBlockGasInWindow returns the sum of the block gas in the window.
blockConsumption := sumBlockGasInWindow(window) / (window * maxBlockGas)

if blockConsumption <= gamma || blockConsumption >= 1 - gamma {
    // MAX_LEARNING_RATE is a constant that is configured by the chain developer
    newLearningRate := min(MaxLearningRate, alpha + currentLearningRate)
} else {
    // MIN_LEARNING_RATE is a constant that is configured by the chain developer
    newLearningRate := max(MinLearningRate, beta * currentLearningRate)
}

newBaseGasPrice := currentBaseGasPrice * (1 + newLearningRate * (currentBlockGas - targetBlockGas) / targetBlockGas)

The expected behavior is the following: when the current block gas is close to the targetBlockGas (in other words, when blockConsumption is in the gamma range), then the base gas price is close to the right value, so the algorithm reduces the learning rate to reduce the size of the oscillations. By contrast, if the current block gas is too small or too high (blockConsumption is out of gamma range), then the base fee is apparently far away from its equilibrium value, and the algorithm increases the learning rate.

Examples

Assume the following parameters:

• TargetBlockGas = 50
• MaxBlockGas = 100
• Window = 1
• Alpha = 0.025
• Beta = 0.95
• Gamma = 0.25
• MAX_LEARNING_RATE = 1.0
• MIN_LEARNING_RATE = 0.0125
• Current Learning Rate = 0.125
• Previous Base Fee = 10.0

Block is Completely Empty

In this example, we expect the learning rate to additively increase and the base fee to decrease.

blockConsumption := sumBlockGasInWindow(1) / (1 * 100) == 0
newLearningRate := min(1.0, 0.025 + 0.125) == 0.15
newBaseGasPrice := 10 * (1 + 0.15 * (0 - 50) / 50) == 8.5

As we can see, the base fee decreased by 1.5 and the learning rate increases.

Block is Completely Full

In this example, we expect the learning rate to additively increase and the base fee to increase.

blockConsumption := sumBlockGasInWindow(1) / (1 * 100) == 1
newLearningRate := min(1.0, 0.025 + 0.125) == 0.15
newBaseGasPrice := 10 * (1 + 0.15 * ((100 - 50) / 50)) == 11.5

As we can see, the base fee increased by 1.5 and the learning rate increases.

Block is at Target Gas

In this example, we expect the learning rate to multiplicatively decrease and the base fee to remain the same.

blockConsumption := sumBlockGasInWindow(1) / (1 * 100) == 0.5
newLearningRate := max(0.0125, 0.95 * 0.125) == 0.11875
newBaseGasPrice := 10 * (1 + 0.11875 * (50 - 50) / 50) == 10

As we can see, the base fee remained the same and the learning rate decreased.

Default EIP-1559 With AIMD EIP-1559

It is entirely possible to implement the default EIP-1559 dynamic fee pricing with the AIMD EIP-1559 dynamic fee pricing. This can be done by setting the following parameters:

• Alpha = 0.0
• Beta = 1.0
• Gamma = 1.0
• MAX_LEARNING_RATE = 0.125
• MIN_LEARNING_RATE = 0.125
• Window = 1