Optimal voting for ve(3,3) holders
Voting for your own pools is generally profitable, even if the DEX UI overlooks the benefits of directing emissions in their APR calculations.
Abstract
Protocols incentivize their liquidity on Solidly forks rather than Uniswap V2 pools because the value of emissions far exceeds the cost of the bribes they spend. They also vote for their own pools for the same reason.
Despite the higher value of emissions, retail investors typically only chase bribes, which is highly inefficient and leads to a deceptive experience.
Optimisation shows that anyone LPing a greater proportion of a pool than his proportion of the overall voting power — this is the typical retail experience— should primarily vote its own pools.
Numerical example
While Dexes voters’ dashboards highlight bribe and fee rewards, voters with significant LPing positions can achieve far greater benefits by voting for their own pools.
In the example below, a typical ve(3,3) Dex suggests voting equally across all pools to earn an average of $100 in bribe and fee rewards. However, by voting for their own pools, voters can redirect emissions and gain significantly more.
Instead of the standard $100 in voting returns, voters could realise total benefits of $210 (or $320) by allocating 60% (or 100%) of their votes to their own pools, thanks to the emissions boost.
Aligning voters’ and farmer’s interest benefits not only users, but also Dexes themselves. When Dex users do not realise the benefits of holding the farming tokens, it’s no wonder they perceive Dex tokenomics as a race to the bottom .
Table 1: Consider a small holder (e.g., 1%) LPing in a pool that represents 5% of the Dex’s TVL and is expected to receive 5% of the votes (assuming bribes are equally distributed and voters are ‘naive’). The emission-to-bribe multiplier is 2.
If you hold 10% of the pool, voting 60% instead of 5% reduces your bribe returns by $10 but generates an boosts your share of emissions to $120. If you hold 20% of the pool, voting 100% instead of 5% reduces your bribe returns by $20 but boosts your share of emissions to $240.
Rationale
Although ve(3,3) protocols base their business development on the fact that their emission-to-bribe ratio is significantly greater than 1, they ignore the value of directing emissions when communicating to retail investors.
A better use of the protocol can only enhance the experience of farmer-voters. Taking the value of emissions into account would show retail users that the value from using the protocol is much greater than what Dexes usually show. It could make them stick longer to protocols.
Voting for emissions is also much less deceiptive than voting for bribes. After all:
- the early voters’ share of bribes and emissions is diluted by late voters. Early retail voters’ experience is deceptive in the current setting
- by contrast, late votes increase emissions to your pools — taking emissions into account leads to a much more positive voter experience
Simplified maths
Naive bribe-chasing optimisation
The ith pool has bribe b(i), and known external votes u(i). For a naive optimiser chasing only voting rewards b(i) will vote v(i) each eligible pools, where v(i) obeys:
which can also be written as
this shows that a naive optimiser will ensure that total vote to each pool are proportional to the square root of exiting votes u(i) times voting rewards b(i).
here, λ is a constant such that total the total voting power V is spent ( Σvi=V ), and eligible pools can be found recursively by eliminating any negatively voted pool.
Note that in equilibrium, ui+vi:=ui, and naive voter just vote all pools as a proportion of their expected returns bi. Deviating from equilibrium has no aggregate benefits, it just shifts away voting from retail to sophisticated investors.
The farmer-voter
The farmer-voter sees total returns as the combination of voting returns (the share of b(i) proportional to his vote on any given pool) plus the additional emissions directed to his share l(i)/L(i) of any pool.
The benefits of redirecting emissions to one’s own pools quickly outweigh the benefits of capturing bribes.
Qualitatively speaking:
- If voters are naive, bribe APR tends to be similar across all pools at the end of the voting period.
- The value of emissions is greater than the value of bribes — protocols would directly bribe the pool rather than bribing voters otherwise.
- If you own 5% of a pool, you’d want to increase the total votes to this pool by more than 5% relative to pools you don’t LP in. And if your voting power is no more than 5%, it means you’d need to vote 100% for your pool to achieve this result.
Quantitatively speaking:
The exact solution is
The first-order expansion (for a small fraction of the pool owned by the retail farmer) reads:
Here, E/P is another version of the multiplier, typically comprised between 2 and 4 — its says that emissions must outweigh bribes for a Dex to be sustainable.
This formula indicates that even a small fraction of pool ownership can shift the desired total votes by a few percentages! And when your voting share is small, this can mean putting your whole woting power into one single pool.
Numerical examples:
Consider an emission to bribe ratio of 2, and two pools receiving equal bribes and equal outside votes ( the rest of the voters are naive).
You as an optimal farmer voter hold 2% of votes. Without taking the value of emissions into account, you’d vote 50% for either pool.
- if you own 2% of the first pool (the same proportion as your voting power), you want to vote 75% for the first pool
- if you own 3% of the first pool, you want to vote 88%;
- at 3.5% you vote 95% for you own pool, and you’d vote 100% if you own 4% of the pool
Note that if your pool voting rewards are only half that of the other pool, you’d still massively outvote it as soon as your onwership of the pool is greater than your voting power!
Conclusion
Poor voting guidelines and decisions are detrimental to both Dexes and their users: they undermine the utility of the protocol, leading uninformed and deceipt users to hop from one dex to another.
Generally speaking, voters should largely overvote their own pools — the benefit of directing emissions to their own pools usually outweighs the rewards captured by chaing bribe rewards.
The strategic approach presented here would benefit both individual users and the overall health of the protocol.
Detailed maths
The voting rewards are r(i)=(v(i)/(u(i)+v(i)) * b(i), so the Lagrangian of the naive optimiser is
with first order condition
Note that λ has a closed form solution which leads to the optimal voting v(i) accross eligible pools where P is the total voting power
By. contrast, the farmer-voter sees total returns as the combination of voting returns (the share of b(i) proportional to his vote on any given pool) plus the additional emissions that are directed to his share l(i)/L(i) of any pool.
The retail farmer-voter’s expected total returns from voting are:
The Lagrangian reads
and the first order condition
The exact solution has the first-order expansion (for a small fraction of the pool owned by the retail farmer):
A retail user who owns a small fraction li/Li of the pool will attempt to shift the total pool votes up by an amount the same order of magnitude than his ownership li/Li in the pool.
E/P, the emissions by vote, are another version of the multiplier, and can typically be normalised to 2 to 4 here). E/P is not the inflation — inspect the overall return equation, and see that emissions returns can quickly outweigh the bribe and fee capture.
