SharesAccounting.md

Shares Accounting

This document outlines the changes to the staker and operator Shares accounting resulting from the Slashing Upgrade. There are several introduced variables such as the deposit scaling factor ($k_n$), max magnitude ($m_n$), and beacon chain slashing factor ($l_n$). How these interact with the operator and staker events like deposits, slashing, withdrawals will all be described below.

Prior Reading

• ELIP-002: Slashing via Unique Stake and Operator Sets

Pre-Slashing Upgrade

We'll look at the "shares" model as historically defined prior to the Slashing upgrade. Pre-slashing, stakers could receive shares for deposited assets, delegate those shares to operators, and withdraw those shares from the protocol. We can write this a bit more formally:

Staker Level

$s_n$ - The amount of shares in the storage of the StrategyManager/EigenPodManager at time n.

Operator Level

$op_n$ - The operator shares in the storage of the DelegationManager at time n which can also be rewritten as
$op_n = \sum_{i=1}^{k} s_{n,i}$ where the operator has $k$ number of stakers delegated to them.

Staker Deposits

Upon each staker deposit of amount $d_n$ at time $n$, the staker's shares and delegated operator's shares are updated as follows:

$$ s_{n+1} = s_{n} + d_{n} $$

$$ op_{n+1} = op_{n} + d_{n} $$

Staker Withdrawals

Similarly for staker withdrawals, given an amount $w_n$ to withdraw at time $n$, the staker and operator's shares are decremented at the point of the withdrawal being queued:

$$ s_{n+1} = s_{n} - w_{n} $$

$$ op_{n+1} = op_{n} - w_{n} $$

Later after the withdrawal delay has passed, the staker can complete their withdrawal to withdraw the full amount $w_n$ of shares.

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Slashing Upgrade Changes

The remaining portions of this document will assume understanding of Allocations/Deallocations, Max Magnitudes, and Operator Sets as described in ELIP-002.

Terminology

The word "shares" in EigenLayer has historically referred to the amount of shares a staker receives upon depositing assets through the StrategyManager or EigenPodManager. Outside of some conversion ratios in the StrategyManager to account for rebasing tokens, shares roughly correspond 1:1 with deposit amounts (i.e. 1e18 shares in the beaconChainETHStrategy corresponds to 1 ETH of assets). When delegating to an operator or queueing a withdrawal, the DelegationManager reads deposit shares from the StrategyManager or EigenPodManager to determine how many shares to delegate (or undelegate).

With the slashing release, there is a need to differentiate "classes" of shares.

Deposit shares:

Formerly known as "shares," these are the same shares used before the slashing release. They continue to be managed by the StrategyManager and EigenPodManager, and roughly correspond 1:1 with deposited assets.

Withdrawable shares:

When an operator is slashed, the slash is applied to their stakers asynchronously (otherwise, slashing would require iterating over each of an operator's stakers; this is prohibitively expensive).

The DelegationManager must find a common representation for the deposit shares of many stakers, each of which may have experienced different amounts of slashing depending on which operator they are delegated to, and when they delegated. This common representation is achieved in part through a value called the depositScalingFactor: a per-staker, per-strategy value that scales a staker's deposit shares as they deposit assets over time.

When a staker does just about anything (changing their delegated operator, queueing/completing a withdrawal, depositing new assets), the DelegationManager converts their deposit shares to withdrawable shares by applying the staker's depositScalingFactor and the current slashing factor (a per-strategy scalar primarily derived from the amount of slashing an operator has received in the AllocationManager).

These withdrawable shares are used to determine how many of a staker's deposit shares are actually able to be withdrawn from the protocol, as well as how many shares can be delegated to an operator. An individual staker's withdrawable shares are not reflected anywhere in storage; they are calculated on-demand.

Operator shares:

Operator shares are derivative of withdrawable shares. When a staker delegates to an operator, they are delegating their withdrawable shares. Thus, an operator's operator shares represent the sum of all of their stakers' withdrawable shares. Note that when a staker first delegates to an operator, this is a special case where deposit shares == withdrawable shares. If the staker deposits additional assets later, this case will not hold if slashing was experienced in the interim.

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Each of these definitions can also be applied to the pre-slashing share model, but with the caveat that for all stakers, withdrawable shares equal deposit shares. After the slashing upgrade this is not necessarily the case - a staker may not be able to withdraw the amount they deposited if their operator got slashed.

Now let's look at these updated definitions in detail and how the accounting math works with deposits, withdrawals, and slashing.

Stored Variables

Note that these variables are all defined within the context of a single Strategy. Also note that the concept of "1" used within these equations is represented in the code by the constant 1 WAD, or 1e18.

Staker Level

$s_n$ - The amount of deposit shares in the storage of the StrategyManager/EigenPodManager at time $n$. In storage: StrategyManager.stakerDepositShares and EigenPodManager.podOwnerDepositShares

$k_n$ - The staker's “deposit scaling factor” at time $n$. This is initialized to 1. In storage: DelegationManager.depositScalingFactor

$l_n$ - The staker's "beacon chain slashing factor" at time $n$. This is initialized to 1. For any equations concerning non-native ETH strategies, this can be assumed to be 1. In storage: EigenPodManager.beaconChainSlashingFactor

Operator Level

$m_n$ - The operator magnitude at time n. This is initialized to 1.

$op_n$ - The operator shares in the storage of the DelegationManager at time n. In storage: DelegationManager.operatorShares

Conceptual Variables

$a_n = s_n k_n l_n m_n$ - The withdrawable shares that the staker owns at time $n$. Read from view function DelegationManager.getWithdrawableShares

Note that $op_n = \sum_{i=1}^{k} a_{n,i}$.

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Deposits

For an amount of newly deposited shares $d_n$,

Staker Level

Conceptually, the staker's deposit shares and withdrawable shares both increase by the deposited amount $d_n$. Let's work out how this math impacts the deposit scaling factor $k_n$.

$$ a_{n+1} = a_n + d_n $$

$$ s_{n+1} = s_n +d_n $$

$$ l_{n+1} = l_n $$

$$ m_{n+1} = m_n $$

Expanding the $a_{n+1}$ calculation

$$ s_{n+1} k_{n+1} l_{n+1} m_{n+1} = s_n k_n l_n m_n + d_n $$

Simplifying yields:

$$ k_{n+1} = \frac{s_n k_n l_n m_n + d_n}{s_{n+1} l_{n+1} m_{n+1}}=\frac{s_n k_n l_n m_n + d_n}{(s_n+d_n)l_nm_n} $$

Updating the slashing factor is implemented in SlashingLib.update.

Operator Level

For the operator (if the staker is delegated), the delegated operator shares should increase by the exact amount the staker just deposited. Therefore $op_n$ is updated as follows:

$$ op_{n+1} = op_n+d_n $$

See implementation in:

• StrategyManager.depositIntoStrategy
• EigenPodManager.recordBeaconChainETHBalanceUpdate

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Slashing

Given a proportion to slash $p_n = \frac {m_{n+1}}{m_n}$ ,

Operator Level

From a conceptual level, operator shares should be decreased by the proportion according to the following:

$$ op_{n+1} = op_n p_n $$

$$ => op_{n+1} = op_n \frac {m_{n+1}} {m_n} $$

Calculating the amount of $sharesToDecrement$:

$$ sharesToDecrement = op_n - op_{n+1} $$

$$ = op_n - op_n \frac {m_{n+1}} {m_n} $$

This calculation is performed in SlashingLib.calcSlashedAmount.

Staker Level

From the conceptual level, a staker's withdrawable shares should also be proportionally slashed so the following must be true:

$$ a_{n+1} = a_n p_n $$

We don't want to update storage at the staker level during slashing as this would be computationally too expensive given an operator has a 1-many relationship with its delegated stakers. Therefore we want to prove $a_{n+1} = a_n p_n$ since withdrawable shares are slashed by $p_n$.

Given the following:

$l_{n+1} = l_n$
$k_{n+1} = k_n$
$s_{n+1} = s_n$

Expanding the $a_{n+1}$ equation:

$$ a_{n+1} = s_{n+1} k_{n+1} l_{n+1} m_{n+1} $$

$$ => s_{n} k_{n} l_{n} m_{n+1} $$

We know that $p_n = \frac {m_{n+1}}{m_n}$ => $m_{n+1} = m_n p_n$

$$ => s_n k_n l_n m_n p_n $$

$$ => a_n p_n $$

This means that a staker's withdrawable shares are immediately affected upon their operator's maxMagnitude being decreased via slashing.

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Queue Withdrawal

Withdrawals are queued by inputting a depositShares amount $x_n <= s_n$. The actual withdrawable amount $w_n$ corresponding to $x_n$ is given by the following:

$$ w_n = x_n k_n l_n m_n $$

This conceptually makes sense as the amount being withdrawn $w_n$ is some amount <= $a_n$ which is the total withdrawable shares amount for the staker.

Operator Level

When a staker queues a withdrawal, their operator's shares are reduced accordingly:

$$ op_{n+1} = op_n - w_n $$

Staker Level

$$ a_{n+1} = a_n - w_n $$

$$ s_{n+1} = s_n - x_n $$

This means that when queuing a withdrawal, the staker inputs a depositShares amount $x_n$. The DelegationManager calls the the EigenPodManager/StrategyManager to decrement their depositShares by this amount. Additionally, the depositShares are converted to a withdrawable amount $w_n$, which are decremented from the operator's shares.

We want to show that the total withdrawable shares for the staker are decreased accordingly such that $a_{n+1} = a_n - w_n$.

Given the following:

$l_{n+1} = l_n$
$k_{n+1} = k_n$
$s_{n+1} = s_n$

Expanding the $a_{n+1}$ equation:

$$ a_{n+1} = s_{n+1} k_{n+1} l_{n+1} m_{n+1} $$

$$ => (s_{n} - x_n) k_{n+1} l_{n+1} m_{n+1} $$

$$ = (s_{n} - x_n) k_n l_n m_n $$

$$ = s_n k_n l_n m_n - x_n k_n l_n m_n $$

$$ = a_n - w_n $$

Note that when a withdrawal is queued, a Withdrawal struct is created with scaled shares defined as $q_t = x_t k_t$ where $t$ is the time of the queuing. The reason we define and store scaled shares like this will be clearer in Complete Withdrawal below.

See implementation in:

• DelegationManager.queueWithdrawals
• SlashingLib.scaleForQueueWithdrawal

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Complete Withdrawal

Now the staker completes a withdrawal $(q_t, t)$ which was queued at time $t$.

Operator Level

If the staker completes the withdrawal as tokens, any operator shares remain unchanged. The original operator's shares were decremented when the withdrawal was queued, and a new operator does not receive shares if the staker is withdrawing assets ("as tokens").

However, if the staker completes the withdrawal as shares, the shares are added to the staker's current operator according to the formulae in Deposits.

Staker Level

Recall from Queue Withdrawal that, when a withdrawal is queued, the Withdrawal struct stores scaled shares, defined as $q_t = x_t k_t$ where $x_t$ is the deposit share amount requested for withdrawal and $t$ is the time of the queuing.

And, given the formula for calculating withdrawable shares, the withdrawable shares given to the staker are $w_t$:

$$ w_t = q_t m_t l_t = x_t k_t l_t m_t $$

However, the staker's shares in their withdrawal may have been slashed while the withdrawal was in the queue. Their operator may have been slashed by an AVS, or, if the strategy is the beaconChainETHStrategy, the staker's validators may have been slashed/penalized.

The amount of shares they actually receive is proportionally the following:

$$ \frac{m_{t+delay} l_{now} }{m_t l_t} $$

So the actual amount of shares withdrawn on completion is calculated to be:

$$ sharesWithdrawn = w_t (\frac{m_{t+delay} l_{now}}{m_t l_t} ) $$

$$ = x_t k_t l_t m_t (\frac{m_{t+delay} l_{now}}{m_t l_t} ) $$

$$ = x_t k_t m_{t+delay} l_{now} $$

Now we know that $q_t = x_t k_t$ so we can substitute this value in here.

$$ = q_t m_{t+delay} l_{now} $$

From the above equations the known values we have during the time of queue withdrawal is $x_t k_t$ and we only know $m_{t+delay} l_{now}$ when the queued withdrawal is completable. This is why we store scaled shares as $q_t = x_t k_t$. The other term ($m_{t+delay} l_{now}$) is read during the completing transaction of the withdrawal.

Note: Reading $m_{t+delay}$ is performed by a historical Snapshot lookup of the max magnitude in the AllocationManager while $l_{now}$, the current beacon chain slashing factor, is done through the EigenPodManager. Recall that if the strategy in question is not the beaconChainETHStrategy, $l_{now}$ will default to "1".

The definition of scaled shares is used solely for handling withdrawals and accounting for slashing that may have occurred (both on EigenLayer and on the beacon chain) during the queue period.

See implementation in:

• DelegationManager.completeQueuedWithdrawal
• SlashingLib.scaleForCompleteWithdrawal

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Handling Beacon Chain Balance Decreases in EigenPods

Beacon chain balance decreases are handled differently after the slashing upgrade with the introduction of $l_n$ the beacon chain slashing factor.

Prior to the upgrade, any decreases in an EigenPod balance for a staker as a result of completing a checkpoint immediately decrements from the staker's shares in the EigenPodManager. As an edge case, this meant that a staker's shares could go negative if, for example, they queued a withdrawal for all their shares and then completed a checkpoint on their EigenPod showing a balance decrease.

With the introduction of the beacon chain slashing factor, beacon chain balance decreases no longer result in a decrease in deposit shares. Instead, the staker's beacon chain slashing factor is decreased, allowing the system to realize that slash in any existing shares, as well as in any existing queued withdrawals. Effectively, this means that beacon chain slashing is accounted for similarly to EigenLayer-native slashing; deposit shares remain the same, while withdrawable shares are reduced:

Now let's consider how beacon chain balance decreases are handled when they represent a negative share delta for a staker's EigenPod.

Added Definitions

$welw$ is withdrawableExecutionLayerGwei. This is purely native ETH in the EigenPod, attributed via checkpoint and considered withdrawable by the pod (but without factoring in any EigenLayer-native slashing). DelegationManager.getWithdrawableShares can be called to account for both EigenLayer and beacon chain slashing.

$before\text{ }start$ is time just before a checkpoint is started

$after\text{ }complete$ is the time just after a checkpoint is completed

As a checkpoint is completed, the total assets represented by the pod's native ETH and beacon chain balances before and after are given by:

$g_n = welw_{before\text{ }start}+\sum_i validator_i.balance_{before\text{ }start}$
$h_n = welw_{after\text{ }complete}+\sum_i validator_i.balance_{after\text{ }complete}$

Staker Level

Conceptually, the above logic specifies that we decrease the staker's withdrawable shares proportionally to the balance decrease:

$$ a_{n+1} = \frac{h_n}{g_n}a_n $$

We implement this by setting

$$ l_{n+1}=\frac{h_n}{g_n}l_n $$

Given:

$m_{n+1}=m_n$ (staker beacon chain slashing does not affect its operator's magnitude) $s_{n+1} = s_n$ (no subtraction of deposit shares) $k_{n+1}=k_n$

Then, plugging into the formula for withdrawable shares:

$$ a_{n+1} = s_{n+1}k_{n+1}l_{n+1}m_{n+1} $$

$$ =s_nk_n\frac{h_n}{g_n}l_nm_n $$

$$ = \frac{h_n}{g_n}a_n $$

Operator Level

Now we want to update the operator's shares accordingly. At a conceptual level $op_{n+1}$ should be the following:

$$ op_{n+1} = op_n - a_n + a_{n+1} $$

We can simplify this further

$$ =op_{n}-s_nk_nl_nm_n + s_nk_nl_{n+1}m_n $$

$$ = op_{n}+s_nk_nm_n(l_{n+1}-l_n) $$

See implementation in:

• EigenPodManager.recordBeaconChainETHBalanceUpdate
• DelegationManager.decreaseDelegatedShares

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Implementation Details

In practice, we can’t actually have floating values so we will substitute all $k_n, l_n, m_n$ terms with $m_n$/1e18 $\frac{k_n}{1e18},\frac{l_n}{1e18} ,\frac{m_n}{1e18}$ respectively where $k_n, l_n, m_n$ are the values in storage, all initialized to 1e18. This allows us to conceptually have values in the range $[0,1]$.

We make use of OpenZeppelin's Math library and mulDiv for calculating $floor(\frac{x \cdot y}{denominator})$ with full precision. Sometimes for specific rounding edge cases, $ceiling(\frac{x \cdot y}{denominator})$ is explicitly used.

Multiplication and Division Operations

For all the equations in the above document, we substitute any product operations of $k_n, l_n, m_n$ with the mulWad pure function.

function mulWad(uint256 x, uint256 y) internal pure returns (uint256) {
    return x.mulDiv(y, WAD);
}

Conversely, for any divisions of $k_n, l_n, m_n$ we use the divWad pure function.

function divWad(uint256 x, uint256 y) internal pure returns (uint256) {
    return x.mulDiv(WAD, y);
}