# A Discussion On Direct Incentives

Over the last couple of days I’ve been thinking a lot about direct incentives. As far as I can tell, the idea has lost favor with a large part of the community. And I think it’s important that we determine whether or not they’re effective.

I’ve been having a discussion with @fei.saver on another thread that’s convinced me that direct incentives fail in at least one important scenario. That conversations starts with this post: Reweight Without Reweights (using cumulative deviation) - #6 by Inaniel

I’ve learned that to have any meaningful discussion about whether direct incentives are effective we need to be more precise about what we’re looking to achieve with them. So I’m making this post in the hopes that we can have a discussion about what a “good pair” of direct incentives is.

I’ve already discovered that some of the definitions that I first thought of are impossible. For example, I thought a “good pair” of direct incentives would at least (1) mint less than it burns, (2) reward buyers and punish sellers relative to an unincentivized market, regardless of the sequence of purchases and sales in the market, and (3) prevent drainage of PCV under extreme sell pressure.

It turns out this definition is too strong. @fei.saver showed me that any pair of direct incentives that satisfies (1) fails (2). Here’s why:

Suppose there’s a perfectly elastic sell pressure up to marginal price p. That is, sellers are willing to sell any FEI until the marginal price of FEI on the market is p.

In an unincentivized market, sellers will sell until the marginal price of FEI on the market is p. In this case, this marginal price of FEI is both the price to buy and the price to sell. So, after all the sell pressure has been released, buyers can buy FEI at p.

In a market incentivized by direct incentives satisfying (1), sellers will sell until the marginal selling price of FEI on the market is p. If b is the marginal burn punishment for selling FEI, this means sellers will sell until the unincentivized marginal buying price of FEI on the market is p + b. If m is the marginal mint reward for buying FEI, the only way for the marginal buying price to be better than in the unincentivized case is for b < m. But m < b by (1), so (2) cannot be satisfied.

The immediate variation that came to mind is to tweak the definition to give probabilistic guarantees. That is, instead of (2), we’d want a “good pair” to satisfy (2’): reward buyers and punish sellers relative to an unincentivized market with probability at least 1/2 + \epsilon for some \epsilon > 0 over the space of all sequences of purchases and sales in the market.

I know: this starts looking exceedingly abstract. But I think the question of whether “good pairs” that satisfy (2’) exist is important. If the answer’s yes, then it’s possible we can worry about finding “good pairs” with larger and larger \epsilon. We can try to determine how close to \epsilon = 1/2 we can get. If the answer’s no for every choice of \epsilon, then we can lay the quest for direct incentives to bed.

In any case, I know there’s still a lot of work to be done to make the ideas I’m describing precise. But I wanted to share what I’ve been thinking to see if anyone wants to join me on this exploration.

Any ideas? Better definitions? Counterarguments to the fact that the original definition is too strong?

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It sounds interesting to introduce a probabilistic guarantee. To have a precise setting to work with, I suppose we could start by specifying the probability space, i.e. describing where the randomness comes from.

My first idea is this.

In plain english, the probability is taken over the pseudorandomness of the good pair of direct incentives and all choices of purchase and sale sequences. The rest of this post just makes this idea more precise.

For sufficiently large N and K, denote by S_N^K the set of all rational numbers smaller than N with precision 1/K. Precisely,

S_N^K = \{\frac{m}{K} \in \mathbb{Q} \mid m \in \mathbb{Z}, |m| < NK\}

We can model activity on the market by an ordered list of elements in S_{N}^K. That is, we can model activity on the market by an element of (S_{N}^K)^n, for sufficiently large n.

For instance,

(-5000000, 100, -5, 1000000, 0, 0, \ldots, 0).

means, starting at peg, someone sold 5M FEI, then someone bought 100 FEI, then someone sold -5 FEI, then someone bought 1M FEI. There’s a sufficiently large n precisely because reweights are always in the finite horizon. I think it’s fair to call each of these vectors in (S_N^K)^n a path.

If we think about a constant product AMM, there exists a function p_{cp}: (S_N^K)^n \to \mathbb{R} that takes any path to the marginal price at the end of the path, assuming the market started at peg. The fact that a constant product AMM is path independent is precisely the statement that there exists a function p_{cp}^*: \mathbb{R} \to \mathbb{R} such that, for all x \in (S_N^K)^n,

p_{cp}(x) = p_{cp}(x_1, x_2, \ldots, x_n) = p_{cp}^*(\sum_{i=1}^n x_i)

Now, if we allow direct incentives to be pseudorandom with seed k \in \{0, 1\}^m, for any directly incentivized constant product AMM, there exist two keyed functions s_{di}^k:(S_N^K)^n \to \mathbb{R} and b_{di}^k:(S_N^K)^n \to \mathbb{R}. The first takes any path to the marginal selling price at the end of the path, assuming the market starts at peg. The second takes any path to the marginal buying price at the end of the path, assuming the market starts at peg.

This allows us to state more precisely what the probabilistic guarantee is. Let \Omega = (S_N^K)^n \times \{0, 1\}^m, let \mathcal{F} be the set of all subsets of \Omega, and let P be the counting function. Then (\Omega, \mathcal{F}, P) is a probability space with random variables p_{cp}, s_{di}^k and b_{di}^k. The probabilistic guarantee I have in mind is

P\left( s_{di}^k < p_{cp} - \delta_s, b_{di}^k < p_{cp} - \delta_b \right) \geq \frac{1}{2} + \epsilon

for some \epsilon > 0 and adequate choices of \delta_s, \delta_b > 0. Perhaps we can weaken this condition further to

P\left( s_{di}^k < p_{cp}, b_{di}^k < p_{cp} \right) \geq \frac{1}{2} + \epsilon

And, thinking about it a little more, it occurs to me that what we really want is a lower bound on the conditional probabilities we get when we fix x \in (S_N^K)^n. Otherwise, there are going to be paths where the condition doesn’t hold often enough. And that’s obviously undesirable. So

\forall x \in (S_N^K)^n, P\left( s_{di}^k < p_{cp}, b_{di}^k < p_{cp} \mid x \right) \geq \frac{1}{2} + \epsilon

Hmm. I have to think about this more. The model above doesn’t actually cover what we care about, because it implicitly assumes demand and supply won’t be affected by the choice of incentives.

Agree, and there is also a deeper question: How do we make FEI at least as desirable as other stablecoins?

Even if we can develop incentives that offer a probabilistic guarantee that the purchase / sale price of FEI will average to $1, why would I choose to take that risk over using another stable coin where the price is “always$1 (== backed up by collateral)?”

Moreover, how likely would a market maker be to arb a stablecoin with a probabilistic guarantee?

At a very high level, this sort of structure seems strictly worse than changing to a collatearlized stablecoin structure because a market maker would have to accrue a LOT of gas fees for his transactions to converge in expectation to a level close enough to justify transacting.

Assuming this logic is right (for ETH now), I actually don’t think a probabilistic structure is likely to work without the payouts draining the PCV. The problem is … transaction costs! Large transactions that are gas efficient won’t converge quickly to $1. Small transactions that converge quickly will cost a ton of gas. So why deal with this? This also makes usage REALLY ugly because anyone taking the coin as collateral will have to deal with the same problem. i.e., Why would I want to take FEI as collateral if I can’t sell it for$1? I’d just haircut it by the max burn

i.e., Why would I want to buy FEI to use it as collateral if it will be haircut?

Here, the problem is both gas and the fact that we want FEI to BEAT OR MATCH alternatives

The ultimate goal is that FEI will almost always be within a fraction of a percent from $1. If that’s not the case, then it won’t attract users seeking stability. Any question of probabilistic incentives and guarantees is about behavior below the peg. The problem at the moment is that there are scenarios where it’s under the peg and the price is worse for buyers than it would have been had an unincentivized AMM accommodated the very same sell pressure. If someone is trying to sell under the peg, they’re rolling dice: while the expected price of their sale with be considerably lower than the unincentivized price, it’s possible that they’ll get a price as good as the unincentivized price, and it’s possible they’ll get a price that’s much lower than the expected price of their sale. This should have the effect of strongly disincentivizing sales below the peg. Not only is it a bad deal to begin with because you’re selling something that’s worth a dollar below a dollar; you could be unlucky and get an awful deal. No whale will dump$1,000,000 in one sale anywhere below the peg if there’s a remote chance the pool will buy it from them at $0.50 when the listed price was$0.98. In the worst case, whales will dump \$1,000,000 in many small sales. The only way to know that they’ll get something close to the expected price is to do arbitrarily many arbitrarily small sales. But at that point the cost of each transaction’s gas might be higher than the potential profit from each transaction. In net, the idea is that bots and whales couldn’t dump the way they have been recently without risking massive losses.

If someone’s trying to buy under the peg, they’re also rolling the dice. But now it’s in their favor. No matter what, they’ll get a price that’s at least as good as the unincentivized price. Yet there’s a chance they’ll get a better price. So everyone that arbitrages below the peg stands a chance to make a profit when the peg is stably restored. But some will win a lottery and make an even larger profit.

I do think all of this needs to be combined with a system of systematically time locking rewards as well as a system of sound reweights so that it’s clear to buyers that they represent long term profits, rather than something they’ll be able to sell as soon as the peg is restored. And the time locks should be such that on any given day, only as many rewards are released as would not significantly affect the peg if all of it was sold at once.

An ideal system would have the following properties:

1. Only sellers that are willing to take arbitrarily large losses would sell under the peg.

2. Any arbitrageur with a big wallet would be willing to buy under the peg and stand a chance of winning at least the same as they would if they were to arbitrage for any other stable coin, but potentially more with nontrivial probability.

3. The prospect of reweights would prevent reweights from ever happening.

4. The cost to the protocol of minting rewards to stabilize the peg is, with high probability, lower than the cost of a reweight.

5. 1-4 conspire so that FEI is barely ever off peg.

Buried somewhere in my response is my answer to your concern about gas fees. In short, I think you’re right that a probabilistic system would freeze sales under the peg because the only reasonable approach for sellers would incur exorbitantly high gas fees. But it would encourage buts under the peg because the randomness can only favor them, not punish them.

None of these off-peg mechanisms should affect the viability of FEI to be used as collateral. The right set of off-peg mechanisms would be sufficiently effective at returning FEI to peg that FEI would never be off peg for too long. Any mechanism that doesn’t manage to do that should be dispensed with in the long term.

I now think the approach I described in A Discussion On Direct Incentives - #3 by Inaniel is completely misguided.

The question that started all of this was “how do we know whether direct incentives worked effectively in the few days during which they were turned on post launch?” @fei.saver made some compelling points that if you fix the economic profile of sellers and buyers, direct incentives as they work at the moment can conspire in some situations to make the price under the peg worse for buyers than it would have been without direct incentives. At this point, there’s no doubt in my mind about this. But after thinking about what the right way to define direct incentives is, I’ve realized that whether or not the price is under the peg is better or worse for buyers and sellers with than without direct incentives is at best an intermediary goal. In the middle of my shower this morning I remembered that the entire point of direct incentives is to restore the peg without resorting to reweights! What the prices are for buyers and sellers when the price is below the peg is only consequential in so far as it affects how quickly the peg is restored. So here’s yet another attempt at a definition.

Intuitively speaking, a pair of direct incentives is “good” if, whenever we fix the economic profile of sellers and buyers, the peg is restored more quickly with direct incentives than without.

Notice that under this definition, it’s not clear whether the pair of direct incentives in the current implementation is not “good.” After all, given the economic profile of sellers and buyers post launch, it’s likely that the peg would never have been restored without direct incentives. In fact, the behavior of the market after the direct incentives were deactivated is partial evidence that the peg would never have been restored without direct incentives. So the fact that it was never restored with direct incentives doesn’t mean they weren’t “good.”

It’s worth pointing out that direct incentives played at least one other positive role in those days: they discouraged sales and in turn ameliorated a predatory bank run.

To make this definition precise, we need a good model of the “profile of sellers and buyers.” I’m not sure how to do this beyond modeling simple corner cases like perfectly inelastic and perfectly elastic demand and supply. Do any of you know how to?

For the sake of clarity, suppose we are able to encapsulate the economic profile of sellers and buyers in some vector E \in \mathbb{R}^n. And suppose, for seeds s \in \{0, 1\}^k, the deterministic functions T_{AMM}:\mathbb{R}^n \to \mathbb{N} \cup \infty and the randomized function T_{DI}^s: \mathbb{R}^n \to \mathbb{N} \cup \infty give the time until the peg is restored, assuming the buyers and sellers act rationally and efficiently, under an unincentivized market and a directly incentivized market, respectively. Then a good pair would guarantee that for any E \in \mathbb{R}^n,

P_{s \leftarrow \{0, 1\}^k}[T_{DI}^s (E) \leq T_{AMM}(E)] \geq \frac{1}{2} + \epsilon

It’s likely that this formalization is completely impractical! On the one hand, it seems hard to pick the right model for E. On the other hand, T_{DI}^s and T_{AMM}^s are probably difficult to describe in any concise expression. I’m writing it here at least to express more clearly the idea that’s captured in the bold text above.

I’ll repeat another key takeaway of viewing direct incentives in this light: it’s no longer clear whether the pair of direct incentives in the current implementation is not “good”. To prove it isn’t, someone would have to produce an economic profile of sellers and buyers under which the directly incentivized market would take longer to stabilize than the unincentivized market. If you know of one, please share it!