Fair LP Allocation

[jalextowle]

Overview

The current LP scheme is safe but isn't fair. In the case where an LP joins immediately after a long is opened, the LP will receive more LP shares than an LP that joined prior to the long. When the long is closed, the new LP will benefit more from the closure than an LP that joined before the long. The exact opposite occurs with shorts. These games can lead to new LPs getting significant rewards without taking on risk and vice-versa.

This PR implements a scheme that interpolates between two schemes that are fair in special cases. Since the flat+curve model linearly matures a long or short position, a linear interpolation seems like a good initial approximation; however, more testing should be done to ensure that the scheme works in general.

Methodology

Let $c$ be the share price, $z$ be the share reserves, $l$ be the total supply of LP shares, $\Delta z$ be the amount of shares the new LP supplied to the pool, and $\Delta l$ be the amount of LP shares awarded to the LP. The amount of shares $z'$ that can be immediately withdrawn by an LP with $l'$ shares:

$$ z' = (z - \tfrac{o_l}{c}) \cdot \tfrac{l'}{l} $$

Furthermore, let $o_l$ denote the outstanding longs, $o_s$ denote the outstanding shorts, $v_l$ denote the base volume for longs, and $v_s$ denote the base volume for shorts. The two LP schemes that we interpolate between are:

$$ \Delta l = \frac{\Delta z \cdot l}{z - \tfrac{o_l}{c} + \tfrac{o_s}{c}} $$

and

$$ \Delta l = \frac{\Delta z \cdot l}{z - \tfrac{v_l}{c} + \tfrac{v_s}{c}} $$

Both of these schemes are unfair for LPs in different situations (they are flipped for longs and shorts). Having said this, there are examples of when each is fair for LPs. In particular, the first scheme is fair for LPs when a short is opened. The second scheme is fair for LPs when a long is opened. To get the best of both worlds, we linearly interpolate between the two versions.

[coveralls]

Pull Request Test Coverage Report for Build 4235537235

• 10 of 16 (62.5%) changed or added relevant lines in 2 files are covered.
• No unchanged relevant lines lost coverage.
• Overall coverage increased (+5.0%) to 57.706%

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Changes Missing Coverage	Covered Lines	Changed/Added Lines	%
contracts/libraries/HyperdriveMath.sol	0	6	0.0%

Totals
Change from base Build 4235299611:	5.0%
Covered Lines:	322
Relevant Lines:	558

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💛 - Coveralls

[jrhea]