# Analysis of dual-ve

## Overview

We introduce the innovative dynamics of the USDFI system, a pioneering model that operates akin to a financial institution, albeit in DeFi. This system allows participants to exercise control over specific parameters with the objective of minimizing their anticipated costs, akin to strategically selecting a financial product that optimizes returns while minimizing fees.

A cornerstone of the USDFI system is its commitment to issuing a low-volatility asset, a digital currency designed to maintain a stable value. This stability is crucial in fostering a reliable and robust system that can withstand the inherent volatility of the digital currency market.

We further delve into the "Principle of Weak Monotonicity," a concept that posits that over reasonable time periods, the controllable mean component in the model formulation remains positive. This principle suggests a system designed to appreciate in value over time, providing a steady return for its participants.

Our exploration introduces the concept of a "dual-ve system", a system where two distinct assets, in this case, STABLE and USDFI, each have their own price processes. This system can be likened to having two distinct financial products, each with its unique return rates and fees.

In a significant departure from existing ve models, we argue that this dual-ve system represents a non-zero-sum game. Unlike a zero-sum game, where one participant's gain equates to another's loss, a non-zero-sum game allows for mutual benefit among all participants. We posit that in the USDFI system, market participants can collaborate to create a mutually beneficial environment, thereby enhancing the system's long-term stability and usability.

The USDFI design represents a novel approach to decentralized finance, employing advanced modeling to create a stable, beneficial, and cooperative environment for its users. This work stands as a testament to the potential of DeFi to reshape our understanding of economic systems and financial interactions.

## Introduction

We introduce the dynamics of USDFI system as a controlled stochastic price process, in which players execute strategies $u_{i}$ in order to control parameters $\overline{\delta}$ to minimize their expected costs $J(\cdot)$ in the Ito process:

$dP(t) = (\delta_{1}P(t), \delta_{2}[\epsilon(t), \rho(t)]^{\intercal})dt + (\delta_{3}P(t), \delta_{4}[\epsilon(t), \rho(t)]^{\intercal})dW_t$

With costs to player i:

$J_{i}(u_{i},\overline{u}_{-i}) = \limsup\limits_{T \to \infty} \frac{1}{T} \ {\int}_{t}^{T} \alpha|P(t) - \psi(t, P_{i}(t), P_{-i}(t))|^2 + \beta u_{i}(t)^2 dt$

where $\alpha, \beta \in \mathbb{R}^{+}$ are predefined weights, $\psi$ is a convex trajectory function, $ρ$ gives the protocol rewards, and $ε$ giving protocol emissions.

## Principle of Stability

One of the central points of this system is to issue a low volatility asset relative to a widely liquid numeraire. Demonstrating that USDFI satisfies this condition is equivalent to showing that:

$\forall t \in R \frac{\partial }{\partial t}\mu_{usdfi}(t, \cdot) \leq 0 \leftrightarrow P(t) > 1$

and

$\forall t \in R \frac{\partial }{\partial t}\mu_{usdfi}(t, \cdot) \geq 0 \leftrightarrow P(t) < 1$

Where $R \subseteq \mathbb{R}^{+}$, some reasonable subset of all time.

In other words, the drift component of USDFI’s price process trends toward 1, irrespective of whether USDFI is at peg, above or below. Formal arguments for this will be presented in the paper to be released in late 2023.

## Principle of weak monotonicity

Given the dynamics of liquidity restrictions on STABLE posed previously, we proceed with the restriction that

$\forall t \in R \frac{\partial }{\partial t}\mu_{stable}(t, \cdot) \geq 0$

Or in other words, for reasonable periods of time, the controllable mean component in the model formulation of $dP(t)$ is positive. Formal arguments for this will be presented in the paper to be released in late 2023.

## Principle of rewards

We define the strategy of holding USDFI as $u_{hold}(t)$ and $u_{locked}(t)$

We establish that the return of locking USDFI in vote escrow is greater than that of holding:

$\forall t \in R \frac{\partial }{\partial t}\mu_{locked}(t, \cdot) \geq \frac{\partial }{\partial t}\mu_{held}(t, \cdot)$

This is a reasonable assumption because so long as there is a non-zero treasury, returns are positive. The treasury can never be zero since base funds are locked and minted.

Given the stability principle, it follows that $J_{1}(P(0), u_{locked}(t), u_{x}(t)) \geq J_{1}(P(0), u_{locked}(t))$ due to the reasoning above, and the fact that escrow rewards for an individual are independent of exogenous market participants.

## Optimal controls of the USDFI dual-ve system

We formalize our dual-ve system in terms of a stochastic price processes, governed by linear quadratic regulators from players 1 and 2 with strategies $u_1(t)$and $u_2(t)$, posed originally by Chitra et. al. (2022). One central difference between the USDFI and Curve Finance's type single-ve systems, is that within the zero-sum bribing game played to govern the single-ve system, and subsequently regulate the underlying price process, we have a single asset’s price process granting governance authority to users of the system:

$dP_{crv}(t) = \mu_{crv}(P(t), u_1(t), u_2(t))dt + \sigma_{crv}(P(t), u_1(t), u_2(t))dW_t$

an extended form of:

$dP(t) = (\delta_{1}P(t), \delta_{2}[\epsilon(t), \rho(t)]^{\intercal})dt + (\delta_{3}P(t), \delta_{4}[\epsilon(t), \rho(t)]^{\intercal})dW_t$

Where $P(t)$ gives the price as a function of time, $μ$ the controlled mean component of the system, $σ$ the controlled variance component, and $Wt$ a standard Weiner process.

The costs written as: $J_{1}(P, u_1, u_2)$ and $J_{2}(P, u_1, u_2)$

The central difference between the Curve type single-ve and USDFI systems are that rather than gaming the system to manipulate a fixed allocation of emissions in terms of a single price process, we extend this notion to both STABLE and USDFI which both have stochastic price processes of their own. By the stability principle outlined in the introduction, and the weak monotonicity principle outlined above, we demonstrate firstly that the game is non-zero sum. We formalize this system in terms of two processes:

$dP_{usdfi}(t) = \mu_{usdfi}(P(t), u_1(t), u_2(t))dt + \sigma_{usdfi}(P(t), u_1(t), u_2(t))dW_t$

and

$dP_{stable}(t) = \mu_{stable}(P(t), u_1(t), u_2(t))dt + \sigma_{stable}(P(t), u_1(t), u_2(t))dW_t$

with costs $J(P, u_1, u_2) = \limsup\limits_{T \to \infty} \frac{1}{T} {\int}_{t}^{T} \alpha|P(t) - \psi(t, P_{1}(t), P_{2}(t))|^2 + \beta u_{1}(t)^2 dt$

To demonstrate the non-zero-sum nature of this system, it is sufficient to show:

$\exists u_{1}, u_{2}:J_{1}(P(0), u_{1}(t), u_{2}(t)) - J_{2}(P(0), u_{1}(t), u_{2}(t)) \neq 0$

Although transaction fees create some market friction, we assume this to be trivial (a reasonable assumption given the non churning nature of the vote escrow process) such that moving between USDFI and vote escrow USDFI is frictionless (zero cost).

It follows directly from the above assumption, the formalization of vote escrow, and the principal of stability that players may arbitrarily construct strategies with costs $J(·)$ such that

$|J_{locked}(P(0), u_{locked}(t), u_{held}(t))| > |J_{held}(P(0), u_{locked}(t), u_{held}(t))|$

The extension to games with arbitrarily many players follows directly from the formalizing of escrow rewards above. While this is not a concrete proof, it is a sufficient basis by which to outline the dynamics of the system. We show above that because of the dual-ve dynamics, the near-frictionless vote escrow environment, and the comprehensive finance platform that USDFI substantiates, that we have constructed a non-zero-sum game in which market participants may cooperate to yield greater benefit to all players, subsequently enforcing the long-term usability and stability of the system.

We leave concrete proofs of the above theorems, as well as the existence of $\epsilon$-Nash Equilibria to a separate paper.

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