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Anders Elowsson pfp
Anders Elowsson
@anderselowsson
1. Minimum viable issuance is an important principle in staking economics. This thread will further advance the concept by analyzing how issuance level affects Ethereum’s equilibrium staking conditions, guiding us to a utility-maximizing reward curve. https://notes.ethereum.org/@anderselowsson/MinimumViableIssuance
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Anders Elowsson pfp
Anders Elowsson
@anderselowsson
2. This thread is 116 tweets long and has 32 figures, so I am here linking to it as a post as well. There is a second part forthcoming and I will also provide an ethresearch post covering the topic shortly for those more comfortable with that format. https://notes.ethereum.org/@anderselowsson/HyUIqjo_6
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Anders Elowsson
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3. Before I start I would like to thank @barnabe, @fradamt, @vitalik.eth, @soispoke and @justindrake for fruitful discussions and feedback for this thread, as well as @ansgar.eth, @davidecrapis, @caspar and @julianma for fruitful discussions. I also wish to thank Flashbots for providing the data used for this analysis.
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Anders Elowsson
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4. The demand curve shifted upwards after The Merge when stakers started to receive MEV and priority fees. Reservation yields fell after Shapella due to improved liquidity, and the supply curve shifted downwards. Both changes pushed up the equilibrium quantity of stake.
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Anders Elowsson
@anderselowsson
5. Because of this, Ethereum has arguably entered a phase of overpaying for security. To what extent can we stop overpaying? Can we reduce issuance while still retaining consensus stability, proper incentives, and acceptable conditions for solo staking?
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Anders Elowsson
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6. Can we adopt a reward curve that lets the issuance yield go negative past some specific staking deposit size D, or target some specific desirable D by simply adapting the yield to enforce it? Otherwise, should a more moderate approach be adopted?
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Anders Elowsson
@anderselowsson
7. We will take a closer look at these questions and review features of staking economics that affect consensus incentives and reward variability—including how they vary across deposit size. This will guide our approach to minimum viable issuance (MVI).
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Anders Elowsson pfp
Anders Elowsson
@anderselowsson
RELATIONSHIP BETWEEN THE EQUILIBRIUM STAKING AND F 8. The base reward factor F is the big knob for adjusting the total issuance level under the current reward curve. It affects all consensus rewards and penalties and provides an issuance yield under idealized performance of y_i=cF/D^0.5. The constant c is around 2.6.
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Anders Elowsson
@anderselowsson
9. The total yield provided by the protocol to stakers implies its demand for stake and it is y=y_i+y_v, where y_v is the yield from realized extractable value (REV). Denote yearly REV as V (currently around 300k ETH). We then have y_v=V/D.
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Anders Elowsson
@anderselowsson
10. Note that y is the endogenous yield derived exclusively from staked participation in the consensus process. Yield from DeFi/restaking that is exogenous to staking is not modeled here; it can under competitive equilibrium also be derived by non-stakers. https://x.com/weboftrees/status/1710720809797407063
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Anders Elowsson
@anderselowsson
11. The colormap and y-axis both capture y, with the colormap restricted to F=0-75. Currently, F=64. At equilibrium, the demand curve will intersect the supply curve that captures how prospective ETH holders’ inclination to stake varies with yield.
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Anders Elowsson
@anderselowsson
12. The shape of the supply curve is unknown. The two examples in blue have a yield elasticity of supply of 2, with the lower also used in the last thread. Plots will be provided covering a broader range so that different assumptions can be mapped to various outcomes. https://x.com/weboftrees/status/1710706011185545671
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Anders Elowsson
@anderselowsson
13. The upper curve could for example represent the supply curve underpinning a medium-run equilibrium (within a year or two), whereas the lower curve could be the supply curve after a few years of improvements to the the staking experience and better financial integrations.
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Anders Elowsson
@anderselowsson
14. The left dashed blue line indicates the revealed preference of The Merge at D=14M ETH. By staying to the right of this line, we are operating at a deposit size that has been previously regarded as sufficiently safe by the Ethereum community. https://x.com/weboftrees/status/1710717027520831749
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Anders Elowsson
@anderselowsson
15. The right dashed line indicates 2^25 ETH (33.6M ETH), which has been used as a reference point capturing when network conditions (and economics) start to degrade. By staying to the left of the line, Ethereum is better positioned on these issues. https://notes.ethereum.org/@vbuterin/single_slot_finality
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Anders Elowsson
@anderselowsson
16. Potential equilibria under the current issuance policy (F=64) and REV are indicated by blue circles, and the equilibria if issuance is halved (F=32) are indicated by blue squares. Such a reduction brings the deposit size closer to a previously suggested desirable range.
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Anders Elowsson
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17. It is not possible to ascertain the exact effect of a reduction in F, but we can be rather certain that the yield elasticity of supply for the medium run is not 0 (a vertical supply curve). Reducing F will therefore always reduce the quantity of stake, ceteris paribus.
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Anders Elowsson
@anderselowsson
18. Note that the full reduction in yield from a change in F (white downwards arrow) will not remain at the new equilibrium, because some stakers will presumably leave (blue leftwards arrow), bringing the yield for remaining stakers back up a bit (white leftwards arrow).
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Anders Elowsson
@anderselowsson
19. For the specified supply and demand curves, the equilibrium yield initially falls from 2.89% to 1.74% when reducing F from 64 to 32 (a reduction of around 2/5), but then comes back up to 2.33% as some stakers leave. The equilibrium yield is thus reduced by around 1/5.
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Anders Elowsson
@anderselowsson
20. How will this dynamic affect the solo staker and delegating staker? The outcome over shorter time horizons will depend on variations in cost structures and frictions affecting the decision to stake or de-stake.
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Anders Elowsson
@anderselowsson
21. A solo staker who will not buy new hardware at some low yield may still stake over the lifetime of their current hardware. Delegating stakers dissatisfied with the yield may keep their savings in the LST until the next time they wish to spend their money, or leave directly.
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Anders Elowsson
@anderselowsson
22. Solo stakers’ upfront costs and illiquidity presumably give them a lower yield elasticity of supply in the short run. This is comforting, because a temporarily lower-than-equilibrium yield (if F is reduced in a hard fork) may not push them out forever. https://x.com/weboftrees/status/1710719952968106177
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Anders Elowsson
@anderselowsson
23. It seems likely that the supply curve will gradually shift downwards over time as the staking experience simplifies and DeFi integrations improve. The outlined dynamic in §18 may therefore not fully materialize, as a lowering supply curve can nullify any de-staking process.
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Anders Elowsson
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24. The equilibrium quantity of stake will however still be lower with a reduction in F than if F is kept fixed. We must evaluate each possible outcome at the medium-run equilibrium. The effect of a gradually lowering supply curve is a gradually increasing deposit size/ratio.
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Anders Elowsson
@anderselowsson
25. This figure has F on the y-axis instead of yield. You may think of it as dragging down and straightening the bent colormap in §11 such that it becomes a rectangle. The colors encode the same yield as previously (also indicated by black lines).
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Anders Elowsson
@anderselowsson
26. This viewpoint is convenient as we will now further explore the effect of a change to the issuance level. We will often provide both alternative graphs to the reader and indicate the same two supply curves as guidelines.
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Anders Elowsson
@anderselowsson
INCENTIVE STRUCTURE 27. When contemplating a change to the issuance policy, it is important to consider the effects on consensus stability, in particular how incentives may change for different consensus roles that validators will be assigned to.
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Anders Elowsson
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28. This figure shows the expected rewards stemming from issuance at various settings for the base reward factor F. The hypothetical equilibria are once again indicated. Naturally, the lower F is set, the lower the proportion of rewards that come from issuance.
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Anders Elowsson
@anderselowsson
29. Right now at the prevailing level of REV, more than 2/3 of rewards come from issuance. Since y_v changes by the reciprocal of D whereas y_i changes by the reciprocal of D^0.5 under the current reward curve, a higher proportion of rewards will stem from issuance at a higher D.
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Anders Elowsson
@anderselowsson
30. This figure instead shows the yield that comes from attester duties (y_a) in proportion to the total yield (y_a/y). Since the proposer gets 1/8 of the issued rewards, the situation under this viewpoint worsens somewhat.
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Anders Elowsson
@anderselowsson
31. If F is reduced to 32, almost half the rewards will come from the sparse chances of proposing a block. There is no well-defined proportion of y that must be provided for attester duties, but higher is generally better, and providing at least 0.5 for them seems healthy to me.
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Anders Elowsson
@anderselowsson
32. When y_i/y and y_a/y fall too low, the consensus mechanism breaks down. Consensus rewards and penalties stop providing correct incentives for stakers. Honest attestation is less compelling. The only thing that matters is to collect REV and to not get slashed.
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Anders Elowsson
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33. Ignoring attester duties comes at little to no cost as long as the inactivity leak is not triggered, and instigating reorgs will be more tempting. If REV rises relative to the proposer reward, timing games also become a little more attractive.
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Anders Elowsson
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34. Moving back to having staking yield on the y-axis gives another perspective on how various hypothetical changes to issuance policy may affect the proportion of rewards awarded for attestation duties. This time, the x-axis extends across the full circulating supply.
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Anders Elowsson
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35. In red, we contemplate the various stricter issuance policies that can be attempted, and the adverse effects they may bring before MEV burn is in place. For example, to target D=24M ETH (dashed red line) while keeping y_a/y>0.5, the yield must be around 3% (red square).
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Anders Elowsson
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36. This seems unreasonably high given that the supply curve slopes upwards. Therefore, to enforce D=24M ETH, an even lower proportion of rewards must be given for attester duties. Indeed, it may not be possible to give these duties any issuance rewards at all (i.e., 0 issuance).
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Anders Elowsson
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37. Such an outcome is manifested by an equilibrium in the white region of the figure. This could be the outcome if the supply curve over time drifts lower—which seems like a very reasonable assumption—or simply because of a not particularly unlikely rise in REV.
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Anders Elowsson
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38. The same type of problem can be encountered when adopting a reward curve that goes negative to enforce a deposit size below some specific level. The red line indicates the reward curve previously suggested by Buterin. https://notes.ethereum.org/@vbuterin/single_slot_finality
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Anders Elowsson
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39. At many reasonable equilibria with such strict reward curves, the rewards for attester duties will be very low (red circles), or non-existent. The breakdown in accordance with §32-33 is then complete.
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Anders Elowsson
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40. Note that the supply curves in the last three figures were adjusted slightly to illustrate the notion that the final fraction of the circulating supply would not be staked in the medium run until the yield becomes very high (see the post-version for details).
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Anders Elowsson
@anderselowsson
41. As previously outlined, when D rises, the white region representing y_v becomes smaller and smaller. This happens because y_v falls by the reciprocal with a rise in D. I highlight this because it is currently not very well understood.
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Anders Elowsson
@anderselowsson
42. For example, as a motivation for “EIP-7514: Add Max Epoch Churn Limit”, it is implied that y_v could be “much higher” than y_i=1.6% at 120M ETH staked, but this is unreasonable. At the prevailing level of REV, the expected y_v would be just 0.25 % at 120M ETH staked.
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Anders Elowsson
@anderselowsson
43. Some of the issues of a lower issuance here outlined can be remedied by taking out a staking fee each epoch and increasing the base reward correspondingly. To prevent consensus breakdown, the fee must be introduced already at positive yields.
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Anders Elowsson
@anderselowsson
44. However, introducing a fee challenges long-standing tenets promoted to solo stakers ("you can go offline X% of the time and still break even", etc.). Trying to push through these far-reaching changes when MEV burn eventually can make them obsolete therefore seems undesirable.
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Anders Elowsson
@anderselowsson
45. Furthermore, a staking fee will not resolve other issues of a very low issuance, such as a rise in the relative and equilibrium variability in rewards for stakers that do not pool their MEV income.
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Anders Elowsson
@anderselowsson
VARIABILITY IN REWARDS FOR SOLO STAKERS 46. The variability in rewards is higher for solo stakers than delegating stakers under the current consensus mechanism, because delegating stakers can in a frictionless manner rely on pooling of rewards from a large number of validators. This affects solo stakers negatively.
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Anders Elowsson
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47. A change in issuance policy could further widen the gap in variability, and it is therefore necessary to model that. The most prominent research on reward variability has been done by @pintail, with analysis up until and just after The Merge. https://pintail.xyz/
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48. The reader is also encouraged to study writings on this matter by @benjaminion. In fact, the whole chapter on the incentive layer is fundamental reading to any consensus layer developer. https://eth2book.info/capella/part2/incentives/rewards/#individual-validator-rewards-vary
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Anders Elowsson
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49. This thread models variability for solo validators over one year in a rather simple fashion, with the distribution in proposer and sync-committee duties assigned according to the probabilities given from the consensus spec at each modeled deposit size.
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50. The focus is on the greatest source of variability, namely that of variation in REV. To this end, block proposers are assigned REV using sampling with replacement from the roughly 2.7 million block-level sample points provided by Flashbots. https://flashbots-data.s3.us-east-2.amazonaws.com/index.html
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51. A probability density function (PDF) of the REV in Ethereum shows a positive skew, with a mode of around 0.025. The mean of around 0.12, indicated by a dashed vertical line, is higher due to the occasional blocks with very high MEV.
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52. To discern more details, the cumulative distribution function (CDF) is here plotted with a log-scaled x-axis and a logit-scaled y-axis. The median MEV is around 0.045 ETH and less than 20 % of the slots are above the mean. Note the occasional >10 ETH outcome.
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53. Blocks have been missed with a probability of around 0.96 % since The Merge, a condition that is also included in the model. Missed sync-committee assignments are not precisely modeled and instead assigned to have the same probability as missed blocks.
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Anders Elowsson
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54. It is presumably less common to fully miss the sync-committee assignment (which spans over more than a day), and much more likely to partially miss it, but modeling this exactly is beyond the scope of this thread.
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Anders Elowsson
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55. Notably, blocks are to a higher proportion missed by solo stakers than professional stakers, as shown by @toniw.eth, something that further degrades conditions for solo stakers when y_i is reduced relative to y_v. This specific feature is not included in the model. https://x.com/nero_eth/status/1742534075343061363
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Anders Elowsson
@anderselowsson
56. Finally, attesters are assumed to perform their duties correctly and are set to receive the full rewards (a slight overestimate). Attestations will produce much less variability that is out of the control of the operator, so they are less relevant to this analysis.
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Anders Elowsson
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Effect of pooling 57. The figure plots the influence of pooling on CDFs of yield at the current deposit size of D=29M ETH. Annualized validator rewards were simulated 30M times, sampling REV with replacement (s. w. r.). Pools were created from these distributions, also s. w. r. 30M times.
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Anders Elowsson
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58. As evident, variability is gradually reduced with more staked ETH (the curve becomes steeper). A solo staker running 2 or 5 validators can already reduce variance quite a bit; a small pool managing a couple of thousand ETH will still not be able to fully remove variance; etc.
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