The Holy Grail of a stablecoin[Her18], an asset with all the benefits of decentralisation but none of the volatility, remains the most elusive single-horned creature of the cryptocurrency market. In fact, price stability is the most wanted feature of a cryptocurrency: in a recent survey[BCC19], hedging against depreciation risk (i.e., price stability) was the most important attribute and it has a much higher feature than anonymity (40% vs. 1%) or illiquidity risk; however, subjects of the survey assigned to the anonymous medium-of-payment a value on average only 1.44% higher than to the non-anonymous medium-of-payment.
In monetary economics, monetary policy rules refer to a set of rule of thumb that the central bank is committed to, so it can maintain the price stability of a currency (Taylor rule, McCallum rule, inflation targeting, fixed exchange rate targeting, nominal income targeting, etc). However, the fixed rules for the emission of most cryptocurrencies[Mou19] cannot maintain price stability: the inflexibility of their emission rules and their inelasticity of supply provoke part of the high volatility of the cryptocurrency market; their lack of good monetary rules preclude their wide used as money[Cac18] as they lack clear a clear focus on monetary equilibrium; instead, they feature technical rules for stabilising the difficulty of mining[NOH19], but not monetary rules. Stablecoins[MIOT19, BKP19, PHP19] were born to explicitly solve the volatility problem of cryptocurrencies: however, their current formulation relies on heuristics[Mak19, KKMP19, Lee14, SI19, IKMS14] without a general mathematical framework within which advantageous properties can be mathematically proven such as stability and convergence. Stablecoins lacking stability regimes and/or convergence guarantees suffer from the instabilities of unstable domains and deleveraging spirals that cause illiquidity during crises[KMM19]: these shortcomings cause price volatility, making cryptocurrencies unusable as short-term stores of value and means of payment, increasing barriers to adoption.
This paper introduces the novel combination of multiple mathematical frameworks in order to design a decentralised stablecoin by inheriting multiple useful properties of said frameworks: stability, convergence, truthfulness, faithfulness, and malicious-security.
The main and novel contributions are:
first formal treatment of decentralised stablecoin within which multiple mathematical properties can be proven: stability, convergence, truthfulness, faithfulness, and malicious-security.
dynamical models of economic systems: currency prediction with deep learning, and stabilisation and emission of stablecoins.
decomposition of Model Predictive Controllers with consensus-ADMM for their implementation in decentralised networks (i.e., blockchains).
protection against malicious adversaries in said decentralised networks.
from mechanism design, proofs to guarantee truthfulness for all the parties involved and faithfulness of the execution for the decentralised implementation.
Previous cryptocurrencies with a controlled money suppy similar to a central bank currency were centralised[DM15, She16, HLX17, WKCC18]: for first time, this paper solves the decentralisation of the monetary policy, achieving a fully decentralised cryptocurrency when combined with a public permissionless blockchain.
Most stablecoins are centralised: the few ones that are decentralised (e.g., [Mak19]), rely on heuristics without a general mathematical framework within which advantageous properties can be mathematically proven such as stability and convergence.
This section provides a brief introduction to the main technologies of the decentralised stablecoin: blockchains, model predictive control, alternating direction method of multipliers (ADMM), mechanism design, secure multi-party computation, and zero-knowledge proofs. A high-level and conceptual rendering of the interrelationship between these techniques can be found in Figure 1.
A blockchain is a distributed ledger that stores a growing list of unmodifiable records called blocks that are linked to previous blocks. Blockchains can be used to make online secure transactions, authenticated by the collaboration of the P2P nodes allowing participants to verify and audit transactions. Blockchains can be classified according to their openness. Open, permissionless networks don’t have access controls and reach decentralised consensus through costly Proof-of-Work calculations over the most recently appended data by miners. Permissioned blockchains have identity systems to limit participation and do not rely on Proofs-of-Work. Blockchain-based smart contracts are computer programs executed by the nodes and implementing self-enforced contracts. They are usually executed by all or many nodes (on-chain smart contracts), thus their code must be designed to minimise execution costs. Lately, off-chain smart contracts frameworks are being developed that allow the execution of more complex computational processes.
Model Predictive Control
Advanced method of process control including constraint satisfaction: a dynamical model of a system is used to predict the future evolution of state trajectories while bounding the input to an admissible set of values determined by a set of constraints, in order to optimise the control signal and account for possible violation of the state trajectories; at every time step, the optimal sequence over steps in determined but only the first element is implemented. Model Predictive Control is widely used in industrial settings, and its large literature contains proofs of feasibility, stability, convergence, robustness and many other useful properties that could be reused in many other settings.
In this paper, multiple dynamic systems for Model Predictive Control will be introduced: 4.1 Economic Model for an Algorithmic Stablecoin and 4.2 Economic Model for a Collaterised Stablecoin; 4.3 Economic Model for a Central-Banked Currency; 5 Decentralised Prediction of Currency Prices through Deep Learning; 6 Decentralised Stabilisation of Stablecoins; and 7 Auction Mechanism for Issuing Stablecoins.
Alternating Direction Method of Multipliers (ADMM)
Class of algorithms to solve distributed convex optimisation problems by breaking them into smaller pieces, and distributing between multiple parties[BPC11]. Itself a variant of the augmented Lagrangian methods that use partial updates for the dual variable, it requires exchanges of information between neighbors for every iteration until converging to the result.
In this paper, multiple optimisation problems expressed in Model Predictive Control will be decomposed with ADMM techniques in order to decentralise their computation between multiple parties: 5 Decentralised Prediction of Currency Prices through Deep Learning; 6 Decentralised Stabilisation of Stablecoins; and 8 Decentralised Implementation of Auction Mechanism.
Also called “reverse game theory”, is a field of game theory and economics in which a “game designer” chooses the game structure where players act rationally and engineers incentives or economic mechanisms, toward desired objectives pursuing a predetermined game’s outcome.
Secure Multi-Party Computation
Protocols for secure multi-party computation (MPC) enable multiple parties to jointly compute a function over inputs without disclosing said inputs (i.e., secure distributed computation). MPC protocols usually aim to at least satisfy the conditions of inputs privacy (i.e., the only information that can be inferred about private inputs is whatever can be inferred from the output of the function alone) and correctness (adversarial parties should not be able to force honest parties to output an incorrect result). Multiple security models are available: semi-honest, where corrupted parties are passive adversaries that do not deviate from the protocol; covert, where adversaries may deviate arbitrarily from the protocol specification in an attempt to cheat, but do not wish to be “caught” doing so ; and malicious security, where corrupted parties may arbitrarily deviate from the protocol.
We utilise the framework SPDZ[DPSZ11], a multi-party protocol with malicious security.
Zero-knowledge proofs are proofs that prove that a certain statement is true and nothing else, without revealing the prover’s secret for this statement. Additionally, zero-knowledge proofs of knowledge also prove that the prover indeed knows the secret.
In this paper, zero-knowledge proofs are used to prove that a local computation was executed correctly.
4 Economic Models
We formalise a basic model of a cryptocurrency111DISCLAIMER: the simplified models in the present paper are only for illustrative purposes. Complex and parameterised models are needed for real-world settings. issuing variable block rewards and periodically auctioning a variable amount of unissued coins from its uncapped and dynamically adjusted supply: all these three variables are constantly adjusted by a dynamical system using Stochastic Model Predictive Control in order to maintain price stability (i.e., controlled variables).
Let denote the time slots used by the blockchain. Let denote the maximum supply of a cryptocurrency, the supply that is visible on-chain, the initially issued supply by an initial offering event (i.e., an initial auction) and is the amount of cryptocurrency yet to be issued. Then, we have:
Periodically, miners are being rewarded for successfully processing blocks with a variable amount of block rewards, :
Auctions are carried out to issue coins from the pool of , each auction releasing a variable amount of auctioned coins, , with denoting the amount of coins demanded by participant , :
Let denote the market price of a coin at time in a currency (i.e., the number of cryptocurrency coins that one unit of currency -EUR, JPY, USD- will buy at time ) and we adopt a geometric Brownian motion model:
where is a Weiner process. Let be the controlled variables. Thus, in order to maintain price stability, these controlled variables will expand when the price is increasing and contract when the price is lowering:
4.1 Economic Model for an Algorithmic Stablecoin
Consider the stochastic linear state space system in the form
where are state space matrices,
is the state vector,is the input vector, is the output vector, is the vector of controlled variables, is the noise vector of the process, and is the vector of measurement noise. Let be the length of the prediction and receding horizon control and define the vectors
Define the following exchange rate function measuring the cumulative exchange rate between the price of a currency (e.g., EUR, JPY, USD) and a stablecoin in the stochastic state space system 4.17 in the following time steps,
Let be the spot price of the stablecoin cryptocurrency denominated in a currency (e.g., EUR, JPY, USD). Then, the cumulative exchange rate at time is
Following a criterion of social welfare maximisation, users and holders of the stablecoin prefer to minimise the volatility of the exchange rate, with the following equation describing the minimisation problem
determines the trade-off between the expected exchange rate and the exchange rate variance.
4.2 Economic Model for a Collaterised Stablecoin
We extend the basic model of a cryptocurrency (4), with a reserve backing every issued coin with units of the reserve asset: for example, for a 1 to 1 peg against a currency (e.g., EUR, JPY, USD), and for an overcollaterised stablecoin backed with other cryptocurrencies. Then, we have:
In order to maintain price stability, could also be a controlled variable that will increase when the price is lowering and contract when the price is increasing:
4.3 Economic Model for a Central-Banked Currency
The framework and results of this paper could also be applied to the monetary policy conducted by central banks, just by representing their models in the framework of Model Predictive Control in a way similar to the previous 4.1 Economic Model for an Algorithmic Stablecoin.
The Taylor rule[Tay93] is an approximation of the responsiveness of the nominal short-term interest rate as applied by the central bank to changes in inflation and output , according to the following formula
where a standard model describes the evolution of the economy
describing the dynamic relationship between the manipulated input and the two controlled outputs and . At equilibrium, we obtain , , and . Equations 4.28 and 4.29 can be rewritten in the terms of deviation variables from the equilibrium point, as
The cost function of the central bank is of the standard optimal control form
where is the discount factor, is the expected value of at time using all information available at time and model 4.30; is the input value at time decided on at time ; and the function is usually defined as
with and the values of and determine the trade-off between the output gap and inflation.
The decentralised implementation of the previous Model Predictive Control 4.36 using the ADMM decomposition technique is left as an exercise to the central banker.
4.3.1 Closed-Loop Stability
The characteristic equation for the matrix is:
is an eigenvalue of matrix. The closed-loop system is stable when both eigenvalues of are inside the unit disk (Jury[Jur74] and Routh[Rou77]-Hurtwiz[Hur95] stability criteria), if and only if,
4.3.2 On Negative Interests
In case the central bank wants to implement negative interest rates, said (LABEL:zero-lower-bound
) must be removed. A possible implementation of negative interests for a cryptocurrency starts by considering coinage epochs and then defining a depreciation rate for every coinage epoch as time elapses. In the basic model of a cryptocurrency (4), we could add the following equation:
where is the amount of minted coins at time and is the depreciation of coins minted at time evaluated at time , for example,
for a 1% depreciation rate for every coinage epoch since the first epoch.
5 Decentralised Prediction of Currency Prices through Deep Learning
As noted in previous publications about predicting markets using Stochastic Model Predictive Control techniques[PB17]
, this approach is only justifiable only for consistent prediction of the direction of price changes (i.e., sign changes): thus, it’s a requisite to use artificial intelligence techniques to predict price movements in order to maintain price stability. Of course, price data can be shifted by one sampling interval to the past, thereby making the economic models independent of any predictive power: however, the correct formulation is to use any potential good estimate of step-ahead prices as this is the core of Stochastic Model Predictive Control. Therefore, the exchange rate function4.21 of the models is formulated with one step-ahead prices ().
A neural network haslayers, each defined by a linear operator
and a neural non-linear activation function. A layer computes and outputs the non-linear function:
on input activations . By nesting the layers, composite functions are obtained, for example,
where the collection of weight matrices is . Training a neural network for deep learning is the task of finding the that matches the output activations to targets , given inputs
: it’s equivalent to the following minimisation problem, given loss function,
And this is equivalent to solving the following problem:
where a new variable stores the output of layer , , and the output of the link function is represented as a vector of activations . By following the penalty method, a ridge penalty function is added to obtain the following unconstrained problem
where and are constants controlling the weight of each constraint, and is a Lagrange multiplier term. The advantage of the previous formulation resides in that each sub-step has a simple closed-form solution with only one variable, thus these sub-problems can be solved globally.
The update steps of each variable in the minimisation problem 5.7 are considered as follows:
To obtain , each layer minimises : the solution of this least square problem is
where is the pseudo-inverse of .
To obtain , another least-squares problem must be solved. The solution is
The update for requires minimising
Finally, the update of the Lagrange multiplier is given by
All the previous steps are listed in the next Algorithm 1:
do for do end for until converged;
6 Decentralised Stabilisation of Stablecoins
Following the 4.1 Economic Model for an Algorithmic Stablecoin and its minimisation problem (4.22), the expectation of the exchange rate and the variance of the exchange rate are traded off in a mean-variance Optimal Control Problem with the following objective function
with determining the trade-off between the expected exchange rate and the exchange rate variance. Estimates of prices for the expected exchange rate, , and the variance, , are introduced as follows
where is sampled from the distribution and is the set of scenarios: when the number of scenarios is large, then
The open-loop input trajectory is defined as the trajectory, , that minimises (6.4), with being some input constraint set. For the stochastic linear system (4.17), can be expressed as the solution to the following Optimal Control Problem,
6.1 ADMM Decomposition
The Optimal Control Problem (6.5) is re-written as
The previous Optimal Control Problem (6.10) is then transformed into ADMM form,
with the optimisation variables defined as
6.2 Decentralised Iterated Computation
where the augmented Lagrangian with penalty parameter is defined as
and is a scaled dual variable.