Summary
The bonus parameter in Aave V3, and its generalization to multiple parameters in V4, is a key dial for adjusting the incentives for timely liquidation of unhealthy positions. This note accomplishes the following
- Introduces the mathematics describing liquidations in Aave V3 and V4
- Defines a notion of protocol welfare and the trade-off between accounting costs to the protocol and strength of liquidation incentives
- Proposes a decision rule, maximization of conditional protocol equity, for the selection of protocol welfare-optimal liquidation parameters
- Describes the simulation framework for the evaluation of counterfactual bonus parameter settings
- Applies the methodology we develop in this note, combining the above analytical insights and the simulation framework, to historical user position data and simulated price trajectory data, to help us understand the scale and qualitative nature of possible improvements to protocol welfare
Liquidations
Liquidation parameters
Aave V4 represents an upgrade over V3, with significant changes to the architecture of liquidity aggregation and markets. In Aave V3, liquidations are governed by two parameters. The threshold parameter, jointly with value of supplied tokens and the value of outstanding debt, determines the Health Factor (HF) at which a position becomes eligible for liquidation. The bonus parameter determines the discount on the collateral transferred to the liquidator. In V4, there is no longer a single bonus parameter, but rather a baseline bonus, a maximum bonus, and an HF threshold where the bonus reaches its maximum.
Our methodology applies directly to both V3 and V4, with historic V3 data and parameter mechanics used in our empirical analysis.
Mathematics of liquidations in V3 & V4
A user position can be associated with multiple collateral and debt tokens, but an atomic liquidation call always âtradesâ a particular collateral token for a particular debt token. This observation enables a straightforward characterization of the fundamental mathematics underlying liquidation accounting. In our formal notation, we will assume that every position is associated with only a pair of a reserve token and a debt token, for simplicity.
Let us introduce some symbolic quantities:
s, ratio of dollar value of the selected collateral token to the dollar value of the selected debt token
T, liquidation threshold parameter
f, relative size of the debt value redeemed by a liquidation
b, fraction of debt value transferred to the liquidator, above the value of the redeemed debt
\underline{b}, \overline{b}, the minimal and maximal possible in V4
p_i, x_i, p_j, x_j, the dollar prices and quantities of some collateral and debt tokens i, j, respectively
The familiar Health Factor quantity is Ts. In V3, f is fixed at 1/2 or 1, depending on the state of the position and its liquidation history, but we can derive the minimum relative liquidation size necessary, under certain conditions, to restore the position to an HF of 1
The changes to bonus parameters in V4 make b a function of HF. Additionally, the liquidation size is dynamically calculated and generally equal to the f_{\min} quantity we derived. Within the range where the bonus varies with the HF, we can characterize the effective relative size of the liquidation
In our V3 simulations, f follows existing protocol rules, not being subject to any decision rule, although we may sometimes treat it as a choice variable for purposes of exposition.
Protocol welfare optimization
Protocol welfare and liquidation parameters
To discuss the economics of liquidations, we must have some notion of the economic objective of the protocol. Taking the perspective of a planner, we may consider some of the following
- Future value accrual from interest payments
- Risk of bad debt accrual from unhealthy borrower positions
- User experience with the protocol and user retention
- MEV recapture (for Aave, via Smart Value Recapture, or SVR) proceeds
These considerations are not exhaustive, but we can join them to characterize the protocol objective in choosing bonus parameters (particularly, in V3) as
The expectation is taken over some stochastic price process, a complication that we will address later in this note. This characterization captures a tradeoff between maximizing revenue by keeping borrower positions active, the risk of positions accruing bad debt and the value leakage to liquidators. We will set aside the impact of risk parameter choice on user experience and concentrate on the tradeoff between bonus payouts and bad debt accrual under several decision rules for parameter determination.
In principle, the functional we apply to some measure of realized protocol welfare, does not have to be an unconditional expectation. The typical functional considered in managing risk is the conditional value at risk (CVaR), which needs to be minimized, rather than maximized, giving us an alternative protocol welfare objective, informally expressed below
CVaR minimization considers only bad debt beyond some set quantile and does not incorporate the costs incurred by the protocol to eliminate bad debt either âon average,â or in the tails of the outcome distribution. While the CVaR minimization problem can be formulated with cost constraints, however difficult these may be to align with the real operation of the protocol, it is more productive to use it as a supplemental decision tool, instead of a protocol objective. The implicit decision rule associated with CVaR is a benchmark we will use to evaluate the decision rule proposed in the next two sections.
Conditional protocol equity
To operationalize the notion of protocol welfare deriving from future income and losses, suggested above, we will first develop a notion of protocol equity. An informal definition can be given by
Assets of the protocol are both ERC-20 tokens initially held as collateral, as well as ERC-20 tokens and Aave asset tokens accrued due to liquidations. The debt is accounted for by variableDebtTokens. Note that this is an accounting formula, not a definition of protocol valuation or a protocol welfare function. However, using this formula, we can define conditional protocol equity, by restricting it to encompass only the assets and debts provided or accrued by positions eligible for liquidations.
Let use introduce a few more symbolic quantities:
C, dollar value of the collateral
D, dollar value of the debt
E, protocol equity
R, accrued repayments from liquidations
M, accrued recaptured value from SVR
\rho, liquidator revenue
r, recapture factor
k, liquidation costs given liquidity conditions and amount of collateral seized
We characterize the effects of a liquidation at some given close factor and bonus in the following table
The condition where a protocol equity delta is realized is simply
Given the above, we can see that maximization of aggregate conditional protocol equity uplift \Delta E (we will simply it equity uplift in subsequent sections) across multiple potentially profitable liquidation hinges on the trade-off between realizing the maximal possible liquidation volume and minimizing value leakage to the liquidators.
Another interesting property of conditional protocol equity uplift is that it is a linearization of bad debt elimination, if we are conditioning on positions that are not just eligible for liquidation, but already underwater. In this case, E is already negative. Minimizing bad debt then amounts to
However, over the same set of positions, maximizing aggregate conditional protocol equity uplift is equivalent to
However, maximizing aggregate conditional protocol equity uplift captures the impact of bonus parameter decisions on the balance sheet even when conditioning on positions where no position is underwater.
We propose that the decision rule for selecting welfare optimal protocol parameters should be based on maximizing aggregate conditional protocol equity uplift, with guardrails to make the rule less sensitive to outlier outcomes without adding excessive bad debt risk in the long tail. The rationale for this is that deriving optimal parameters from minimizing CVaR can overpay for elimination of tail bad debt risk on average, because the CVaR rule is insensitive to value leakage and is indiscriminate in market regimes where no bad debt is likely to be accumulated.
Protocol welfare objective
We make the assumption that the âmarketâ for liquidations is highly competitive, with any \epsilon of gross profit picked up within the block associated with the relevant price oracle update. This assumption matches the actual design of SVR and the observed liquidator behavior, since such price updates drive most liquidation activity by marking positions to market, with the right to backrun the price updates being a subject to the SVR auction. Additionally, the availability of flashloans in DeFi greatly reduces the capital costs of atomic liquidation operations, which means that the gross profit subject to SVR, which handles most liquidation transactions, is bid away to zero. This assumption of allows us to describe the fundamental protocol designerâs problem as a simple static optimization problem, for exposition, using a simple gross profitability condition.
In the simulations we run, however, the model is not static and the liquidation costs are driven by the availability of liquidity in simulated DEX markets, which is path-dependent in any given run of a simulation. The objectives below are meant only to illustrate the mathematical logic of the objective functions implicit in our description of conditional protocol equity. For more information on modeling liquidator behavior, see the appendix.
We will let g denote the joint distribution of prices.
Aave V3
From the table characterizing liquidation effects on equity, we can readily derive the V3 objective
The profitability condition is
Aave V4
Recall that at a given health factor, the close factor is a function of the minimal and maximal bonus parameters in V4. We do not consider the recapture factor here, for clarity.
The profitability condition in V4 becomes more complex
Visualizing the leakage-volume tradeoff
While we have a complete simulation of liquidation dynamics, described in the following section, we can illustrate the mathematics of equity uplift using a much more simplified simulation model, featuring similar price dynamics, but no DEX liquidity modeling. Using the Aave V3 objective and this alternative simulation model, where positions can be liquidated only at a single point in time, with all eligible positions liquidated at once, we can illustrate the dynamics of of the tradeoff between leakage and liquidated collateral volume.
In this chart, the close factor is always set to 0.5, while a fixed liquidation cost is varied. The âsawtoothâ pattern is a direct illustration of the tradeoff, where raising a bonus can only increase equity uplift because a new position becomes profitable. We can also observe that increasing the bonus is more impactful the closer the baseline is to 0. This is because, with fixed costs, the easiest positions to make profitable are the largest ones. However, in the full-scale simulation, this effect is obscured, because of the explicit modeling of DEX liquidity, costs and the concomitant path dependence.
Comparative statics of parameter choice
Simulation framework
Our full simulation model treats evolution of user positions, and therefore protocol equity, as a dynamic process governed by counterfactual price trajectories. The joint GARCH price process generating these trajectories is parametrized using regularly updated estimates, derived from historically observed prices for all tokens supported by Aave V3. The price process captures correlation in price movements, and can be further adjusted to express higher or lower volatility than what the empirical estimates suggest.
The model explicitly captures liquidation frictions, which our prior discussion abstracted as k, with an instance of a unitary synthetic AMM. The simulated AMM, representing an instance of Uniswap V3, is instantiated using liquidity estimates from regular onchain snapshots via fitting. This synthetic Uniswap-like DEX allows a faithful approximation of market dynamics, with liquidators evaluating profitability based on simulated slippage, liquidations depleting AMM pool liquidity on the debt side and the liquidity distribution âmovingâ across ticks in response to price movements.
The liquidator model solves a constrained optimization problem of selecting liquidations to undertake, given the current state of user positions. The liquidity constraints are reified by the AMM model, while, for performance reasons, the number of liquidations undertaken is directly restricted. We impose the assumption that the market for liquidations is highly competitive, with gross profits driven down to a negligible net amount after SVR auction participation and bidding, with 65% of the profits treated as accruing directly to the Aave protocol.
The dynamics of the model are straightforward. Each pair of a price trajectory and a candidate parameter (in our case, a delta to the currently set liquidation bonus) defines a simulation instance that evolves for a fixed number of iterations, interpretable as blocks, with the model executing the following steps for each iteration:
- Adjust positions in accordance with the latest predicted prices
- Scan positions for liquidation opportunities
- Assess whether a liquidation is profitable using the synthetic AMM model and Aave V3 close factor rules
- Liquidate profitable positions and continue
Robust parameter selection
The application of our proposed rule in practice requires a solution to a problem frequently encountered in our numerical experiments. Average equity uplift can stay relatively flat in many scenarios, making it sensitive to outlier trajectories and liquidation path dependencies in the AMM model. For example, we can observe this objective flatness in a parameter sweep for WETH with historical volatility conditions.
Let m be some positive number. To stabilize optimal bonus selection, we apply the following rule:
- Select the raw optimal candidate bonus
- Calculate the standard error of equity uplift pooled across all candidates and trajectories
- Construct a m\cdot\text{SE} length band of eligible candidate values that lies between the raw optimal candidate and the current bonus, inclusive of the raw optimal candidate value
- Select candidate bonus within this band that is closest to the current bonus
In short, we select a parameter value that is closest to existing settings, while producing equity uplift gains similar to what is attainable with the raw optimal parameter value, given the simulated outcomes.
The empirical results below use m=0.1 for illustrative purposes.
Distributional effects in simulations
The observed difference between different decision rules generally attenuates in the far tail of the bad debt distribution, but can be more important closer to the 99th percentile. Below are charts representing the bad debt tail distribution for the LINK sweep asset, with elevated and high volatility relative to historical estimates, respectively. Note the small scale of potential debt at elevated volatility, which makes it irrelevant for the equity uplift rules, but nevertheless causes the CVaR rule to select a large negative adjustment to the bonus. At higher volatility, robust equity uplift and CVaR rules disagree, although the distributional impact is mild.
Below, we have the breakdowns of our accounting terms, with decision spaces being altered radically by much higher volatility
We can observe the distributional attenuation across many tokens, but the relationship between volatility and attenuation is complex. Below are charts from simulations where WETH was the sweep asset at two elevated volatility levels
The decision spaces can look similar across volatility regimes, but different across tokens, and vice versa. Here, for WETH, the decision space at very high volatility is qualitatively similar to that at elevated volatility
Benchmarking decision rules
Parameter choices can move equity uplift from liquidations and tail bad debt, relative to the current setting, and the nature of these choices can be summarized as follows
| Bad debt â | Bad debt â | |
|---|---|---|
| Equity uplift â | Every choice is a Pareto improvement over currently set parameter value | Equity uplift gain compensates for accrued bad debt |
| Equity uplift â | Equity uplift loss pays for reducing bad debt | Every choice is strictly worse than currently set parameter value |
We will number these âquadrants,â counterclockwise, starting from the left upper quadrant, 1, 2, 3 and 4. A concern with the state of the protocol balance sheet, but also tail bad debt, suggests a partial order over these sets of decisions
In principle, we can apply our proposed decision rule and ignore this heterogeneity in types of choices we make. The rule will avoid picking anything in quadrant 3 by construction, and we can speak of it imposing a total order on the sets of possible decisions
The proposed rule explicitly prioritizes equity uplift, and so breaks the ambiguity over quadrants 2 and 4 in favor of 4. Note that the CVaR minimization decision rule would break the ambiguity too, but in favor of 2. However, protocol equity is an abstract construct meant to capture some drivers of long-run protocol welfare, not a direct reflection of the operational realities of Aave. The proposed decision rule only reflects the tradeoff between reducing unhealthy position and value leakage. It is insensitive to the fact that bad debt has to be offset by dollars that are controlled by the DAO directly, or by protocol safety mechanisms, such as Umbrella. In protocol balance sheet terms, the DAO cannot directly use a dollar of \Delta R to offset a dollar of accumulated bad debt. It can, however, do this with \Delta M.
The above suggests a method of benchmarking the equity uplift rule against the CVaR rule. Let B denote bad debt, which we construct empirically by summing the individual shortfalls across all sampled positions. We can define a quantity \lambda_v for some candidate parameter v
If \lambda_v > 1, we maintain a favorable exchange ratio between dollars of bad debt and dollars recaptured through SVR. This is because, in quadrant 4, \lambda_v represents dollars recaptured per dollars of bad debt accrued, while in quadrant 2, it represents dollars of bad debt avoided per recaptured dollar forgone. We can say that choices in quadrants 2 and 4 that satisfy this criterion are feasible.
Now we have a benchmark for any one parameter choice relative to another. The value \lambda_v quantifies affordability and safety of the deltas to equity and bad debt induced by v. This enables comparison between choices made under the equity uplift and CVaR rules.
This benchmark also provides a refinement to the robust aggregate conditional protocol equity uplift rule, since we can simply exclude any choices that would not be safely affordable, before applying the rule, reducing the tension between the two induced orders on sets of decisions.
Benchmarking rules in simulations
Below, we have a chart of sets of decisions (changes to the liquidation bonus parameter) available under different scenarios. Each decision is a dot or a triangle, and the coordinates are determined by the deltas, relative to the existing parameter setting, in average equity uplift and conditional mean tail bad debt (CVaR at the 99th percentile). The chart captures several characteristics for each decision using the following devices
Asset - color differentiates the three assets we observe
Volatility - hue, from dark to light, captures the increase in volatility across our scenarios
Feasibility - in quadrants 2 and 4, triangles represent infeasible choices
Optimality for some rule - a green halo represents the optimal choice under the average equity uplift rule, a blue halo represents the robust optimal choice under the average equity uplift rule and a red halo represents the optimal choice under the CVaR rule (sometimes several rules can yield the same optimal choices; this is represented as split halos)
Exclusion - all choices in quadrant 3 are represented as smaller dots
Note that we do not explicitly represent the choice of existing parameter (which sits at the center of the âcrosshairsâ for all asset/volatility combinations), to avoid crowding the chart. Note that a number of choices are clustered in the vertical line, because many choices under some volatility regimes simply never generate any bad debt.
Here, we can observe both CVaR and equity uplift rules failing to clear the feasibility benchmark in certain scenarios. The CVaR rule forgoes too much recaptured revenue to justify the reduction in tail bad debt in quadrant 2, for a low volatility LINK scenario, while the raw optimal equity uplift rule takes on too much bad debt in quadrant 4, again for the same LINK scenario. However, both rules turn out to be empirically well-aligned with each other, as evidence by multiple overlapping choices in quadrant 1. The robust rule decisions âpull backâ towards the origin, with many choices being simply to leave the liquidation bonus unchanged.
Use of decision rules and benchmarking in V4
Although the simulations we report rely on the specific liquidation parameter feature set of V3, the methodology translates directly to V4. The methodology applies to a decision space of parameter choices and over the induced distribution of simulation outcomes under each such choice. Whether the choice is a scalar, a vector, or some other object makes no conceptual difference. In V3, the relevant decision variable is a scalar adjustment to the liquidation bonus. In V4, the relevant decision variable is a parameter bundle. The same simulation engine generates outcome distributions for candidate choices; the same equity-uplift functional ranks those choices; the same CVaR-based benchmark remains available; and the same feasibility filter can be imposed before a final selection is made.
Appendix: simulating liquidator behavior
Introduction
We formulate a liquidation-execution model suitable for Monte Carlo simulation over random price trajectories when collateral must be sold into on-chain liquidity. Execution quality is endogenized by explicitly simulating AMM pool states and strategic agent interactions: liquidators unwind seized collateral through onchain swap functions, arbitrageurs restore pool prices toward the external market, and liquidity providers adjust active liquidity in response to volatility. To remain computationally tractable, we restrict liquidations to a finite set of pre-mapped DEX routes for each collateralâdebt pair and initialize pool states from time-weighted on-chain liquidity snapshots. Conditional on the realized AMM states, a liquidator solves a constrained optimization problem to choose the repayment size (and hence liquidation fraction) maximizing net surplus after slippage, swap fees, and gas costs, subject to close-factor constraints. We then impose an âefficient biddingâ assumption (via priority fees or protocol-native SVR) enabled by flashloans: competitive bidding drives the winning liquidatorâs ex-post profit to a small residual \varepsilon, with the remaining surplus paid as an effective bid, of which a fixed share (65%) accrues to the Aave protocol.
Borrower position, prices, and protocol constraints
At a given time (t), consider a single borrower position indexed by a debt asset (D) and a collateral asset (C). Let (d_t>0) denote the borrowerâs outstanding debt balance (in units of (D)) and (c_t>0) the posted collateral balance (in units of (C)). We express values in a common numeraire (N) (e.g., USD) via oracle price processes (P_t^D\in\mathbb{R}{>0}) and (P_t^C\in\mathbb{R}{>0}), where (P_t^D) has units ((N/D)) and (P_t^C) has units ((N/C)). These prices are taken as exogenous to the liquidation subproblem at time (t) (though they may evolve stochastically across the simulated trajectory).
Liquidation incentives are parameterized by a liquidation bonus (b\ge 0) and, where applicable, an explicit protocol haircut (\phi\in[0,1)) applied to seized collateral. Conditional on the oracle prices, these parameters induce an oracle-implied collateral-per-unit-debt conversion factor
\kappa_t:= (1-\phi)(1+b)\frac{P_t^D}{P_t^C},
which maps a repaid debt amount (in units of (D)) into the corresponding collateral entitlement (in units of (C)) prior to accounting for collateral exhaustion. In other words, repaying (x) units of (D) entitles the liquidator to (\kappa_t x) units of (C).
In V3, feasible liquidation size is constrained by protocol-enforced close-factor rules that depend on the borrowerâs health factor (HF_t). We model the close factor as the piecewise-constant functions
such that the liquidatorâs repayment decision (x) (in units of (D)) must satisfy
0 \le x \le \bar x_t := \mathrm{CF}(HF_t), d_t.
This constraint encodes the protocolâs partial-liquidation regime near the solvency boundary ((HF_t\in[0.95,1))) and the unconstrained regime under deeper insolvency ((HF_t<0.95)), while ruling out liquidation when the position is solvent ((HF_t\ge 1)).
In V4, however, such close factor parameterization becomes non-static and inherent to the optimization problem.
The liquidatorâs decision problem and objective function
At time (t), conditional on the realized borrower state (c_t,d_t,HF_t), oracle prices ((P_t^C,P_t^D)), and the contemporaneous AMM states that pin down route execution functions ({\Psi_{r,t}}{r\in\mathcal{R}}), the liquidator faces a one-shot constrained execution problem. A liquidation with repayment size (x) (in units of (D)) generates seized collateral (y(x)) (in units of (C)), which is immediately unwound along a chosen route (r) into the debt asset. Let (\Psi{r,t}(y)) denote the deterministic (given state at (t)) route output in units of (D) from swapping (y) units of (C), and let (G_{r,t}) denote auxiliary execution costs (base gas, etc) expressed in units of (D). The route-specific pre-bid surplus is
\Pi_{r,t}(x):=\Psi_{r,t}(y(x))-x-G_{r,t},
i.e., realized unwind proceeds net of the debt repayment and execution costs.
The liquidator selects both the liquidation route and the repayment size to maximize this surplus, subject to protocol feasibility. In particular, the repayment amount is constrained by the close-factor rule, (x\in[0,\bar x_t]) with (\bar x_t=\mathrm{CF}(HF_t),d_t), and by collateral availability through (y(x)=\min{c_t,\kappa_t x}). The resulting discreteâcontinuous optimization problem is
S_t^*:=\max_{r\in\mathcal{R},0\le x\le \bar x_t}\Big(\Psi_{r,t}(y(x)) - x - G_{r,t}\Big),\qquad (r_t^*,x_t^*)\in\arg\max_{r,x}\Pi_{r,t}(x).
We interpret the optimal repay decision as the endogenous liquidation intensity, (\alpha_t^*:=x_t^*/d_t). Participation is governed by non-negativity of surplus: if (S_t^*\le 0), the liquidator optimally abstains and we set (x_t^*=0) (equivalently (\alpha_t^*=0)).
When multiple borrower positions are simultaneously eligible for liquidation at time t, we induce a total order on the candidate set \mathcal{I}_t by ranking each position i by its optimal pre-bid surplus S_{i,t}^* (descending), yielding a well-defined priority ordering for liquidation execution within the positional system.
Efficient bidding / Clearing layer
Conditioning on the filtration generated by the simulated price path and AMM states at time t, let \tilde S_t^* be the \mathcal{F}_t-measurable raw liquidation surplus induced by the optimal route and repayment decision, net of endogenous price impact, AMM fees, and gas. We assume the liquidation marketplace is approximately efficient in the sense that the set of competing searchers is sufficiently large and flashloan-enabled that private rents are competed down to a small deterministic residual \varepsilon, with the remainder âextractedâ via an inclusion-payment equivalent (priority fee/bribe) or via protocol-native SVR, in the form of a first-price sealed bid auction. Accordingly, we model the equilibrium transfer as \tilde B_t=(\tilde S_t^*-\varepsilon)^+, implying realized private profit \Pi_t^{\mathrm{liq}}=\tilde S_t^*-\tilde B_t=\min\{\tilde S_t^*,\varepsilon\}.
SVR execution feasibility under incomplete MEV-Boost adoption
SVR liquidations are implemented as private bundles routed through proposerâbuilder infrastructure (e.g., MEV-Boost / relay-based PBS). Because this middleware is not universally used by all proposers, there is a strictly positive mass of slots in which SVR liquidation bundles are not technically executable in that block. Empirically, analyses of MEV-Boost datasets note that publicly observable MEV-Boost payment data omits âroughly 10%â of nonâMEV-Boost blocks, consistent with MEV-Boost being dominant but not universal.
We model this by introducing a Bernoulli slot-availability indicator
M_t \in \{0,1\},\qquad \mathbb{P}(M_t=1)=p_t,\qquad \mathbb{P}(M_t=0)=1-p_t,
where M_t=1 denotes that the current proposer is operating the relevant PBS/relay infrastructure required for SVR bundle inclusion, and M_t=0 denotes a âvanillaâ slot in which SVR execution is infeasible. In calibration, one may set p_t\approx 0.88\text{â}0.90 (equivalently 1-p_t\approx 0.10\text{â}0.12) to reflect the typical MEV-Boost slot share (Ethereum Research).
Operationally, this imposes a feasibility constraint on the liquidatorâs action set:
such that the executed repayment is
\hat x_t^* \;=\; M_t\,x_t^*,\qquad \hat \alpha_t^* \;=\; \frac{\hat x_t^*}{d_t}=M_t\,\alpha_t^*.
Equivalently, one can think of the liquidator solving the same (r_t^*,x_t^*) problem as before, but the liquidation only clears in-slot when M_t=1; when M_t=0, the attempt is deferred to subsequent slots, capturing the fact that the liquidation cannot occur in that given block. In accordance with the cutoff time associated with SVR, after 5 blocks the protocol âunlocksâ the instantaneous oracle price from 5 blocks prior.
Revenue allocation
With this execution constraint, SVR proceeds (the endogenous âbidâ) becomes
\hat{\tilde B}_t \;:=\; M_t\,(\tilde S_t^*-\varepsilon)^+,
and the Aave take-rate s_A=0.65 implies realized revenues
\tilde R_t^{\mathrm{Aave}} \;=\; 0.65\,M_t\,(\tilde S_t^*-\varepsilon)^+,
Taking conditional expectations (and assuming M_t is independent of the within-slot liquidation surplus given state),