This is an analysis of the protocol fee and the limits it creates on protocol-fee-based governance extractible value.
We start by a game-theoretical competitive analysis of network effects. Based on it, we derive an attack made by Uniswap users on Uniswap’s governance. This attack profits LPs and swapper ****as long as their utility as users of the service are more important to them than their UNI holdings.
We will aim at producing a set of smart contracts that enable these participants to coordinate to produce the attack, and thus keep the value extracted out by UNI tokenholders in check. As shown in the analysis below, the effect of such contracts is useful to them, thus it is expected they interim-rationally participate in them, notably by using the fork and funding the subsidy.
Finally we observe the consequences on welfare from a mechanism design standpoint, revealing that the attack is a myopic game whereas investors interests need to be protected within the R&D investment game.
Automated Market Makers (AMMs) are modeled as strict smart contract equivalent to Uniswap v2. The reasoning can be extended to multiple pools and to Uniswap v3 tick-per-tick.
Consider two AMMs with reserves $R_1$ and $R_2$ in two tokens $\alpha$ and $\beta.$
Using the formalism from [1], consider fees to be equal in both AMMs to $1-\gamma$.
For a given swapper, the payoff or value for a trade $\alpha \rarr \beta$ can be modeled as:
$$ v_{swapper} = v_s = \textsf{TradeValue} - \textsf{LPFees}
Presented with a choice to make a trade in any of the two AMMs or both, a swapper’s utility will be
$$ u_s(\Delta_\alpha) = \tau(\Delta_\alpha) -\textsf{PriceImpactCost}1(\Delta{\alpha,1}) - \textsf{PriceImpactCost}2(\Delta{\alpha,2}) - \textsf{TxCost}(\Delta_{\alpha,1},\Delta_{\alpha,2}) $$
with $\Delta_\alpha = \Delta_{\alpha, 1} + \Delta_{\alpha, 2}$ with each term representing the amount traded on each AMM, $\tau$ the trade value minus LP fees (which is independent of the distribution of the amount traded among AMM 1 and AMM 2).
The $TxCost$ term represents blockchain transaction costs and can be modeled as
$$ \textsf{TxCost}(\Delta_{\alpha,1},\Delta_{\alpha,2}) = \begin{cases} c & \text{if}\ \Delta_{\alpha,1} =0\ \text{or}\ \Delta_{\alpha,2} =0 \\ 2c & \text{otherwise} \end{cases} $$
with $c > 0$ the cost to write a swap transaction on-chain, considered constant for simplicity.
Using [1, (7)], we can thus model the utility as