1 Introduction
The Holy Grail of a stablecoin[Her18], an asset with all the benefits of decentralisation but none of the volatility, remains the most elusive singlehorned 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 mediumofpayment a value on average only 1.44% higher than to the nonanonymous mediumofpayment.
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 shortterm 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 malicioussecurity.
Contributions
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 malicioussecurity.

dynamical models of economic systems: currency prediction with deep learning, and stabilisation and emission of stablecoins.

decomposition of Model Predictive Controllers with consensusADMM 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.
2 Related
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.
3 Background
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 multiparty computation, and zeroknowledge proofs. A highlevel and conceptual rendering of the interrelationship between these techniques can be found in Figure 1.
Blockchains
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 ProofofWork calculations over the most recently appended data by miners. Permissioned blockchains have identity systems to limit participation and do not rely on ProofsofWork. Blockchainbased smart contracts are computer programs executed by the nodes and implementing selfenforced contracts. They are usually executed by all or many nodes (
onchain smart contracts), thus their code must be designed to minimise execution costs. Lately, offchain 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 CentralBanked 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.
Mechanism Design
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 MultiParty Computation
Protocols for secure multiparty 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: semihonest, 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 multiparty protocol with malicious security.
ZeroKnowledge Proofs
Zeroknowledge proofs are proofs that prove that a certain statement is true and nothing else, without revealing the prover’s secret for this statement. Additionally, zeroknowledge proofs of knowledge also prove that the prover indeed knows the secret.
In this paper, zeroknowledge proofs are used to prove that a local computation was executed correctly.
4 Economic Models
We formalise a basic model of a cryptocurrency^{1}^{1}1DISCLAIMER: the simplified models in the present paper are only for illustrative purposes. Complex and parameterised models are needed for realworld 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 onchain, 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:
(4.1)  
(4.2)  
(4.3) 
Periodically, miners are being rewarded for successfully processing blocks with a variable amount of block rewards, :
(4.4)  
(4.5)  
(4.6) 
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 , :
(4.7)  
(4.8)  
(4.9)  
(4.10)  
(4.11) 
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:
(4.12)  
(4.13) 
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:
(4.14)  
(4.15)  
(4.16) 
4.1 Economic Model for an Algorithmic Stablecoin
Consider the stochastic linear state space system in the form
(4.17)  
(4.18)  
(4.19) 
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 vectorsDefine 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,
(4.20) 
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
(4.21) 
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
(4.22) 
with
determines the tradeoff 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:
(4.23)  
(4.24) 
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:
(4.25)  
(4.26) 
4.3 Economic Model for a CentralBanked 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 shortterm interest rate as applied by the central bank to changes in inflation and output , according to the following formula
(4.27) 
where a standard model describes the evolution of the economy
(4.28)  
(4.29) 
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
(4.30) 
where
(4.31)  
(4.32)  
(4.33) 
The cost function of the central bank is of the standard optimal control form
(4.34) 
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
(4.35) 
with and . The previous equations 4.34 and 4.35 can be reformulated as an objective for Model Predictive Control as
(4.36) 
where
(4.37)  
(4.38)  
(4.39)  
(4.40)  
(4.41)  
(4.42) 
with and the values of and determine the tradeoff 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 ClosedLoop Stability
Theorem 1.
Proof.
The characteristic equation for the matrix is:
(4.48) 
where
is an eigenvalue of matrix
. The closedloop 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,(4.49)  
(4.50)  
(4.51) 
∎
Similar stability results can be derived for the Model Predictive Controllers of the 4.1 Economic Model for an Algorithmic Stablecoin and the 4.2 Economic Model for a Collaterised Stablecoin.
4.3.2 On Negative Interests
The Model Predictive Controller for the Taylor rule (4.36)(4.42) includes a constraint for the zero lower bound on the interest rate, (LABEL:zerolowerbound):
In case the central bank wants to implement negative interest rates, said (LABEL:zerolowerbound
) 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:(4.52)  
(4.53)  
(4.54) 
where is the amount of minted coins at time and is the depreciation of coins minted at time evaluated at time , for example,
(4.55)  
(4.56) 
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 stepahead prices as this is the core of Stochastic Model Predictive Control. Therefore, the exchange rate function
4.21 of the models is formulated with one stepahead prices ().A neural network has
layers, each defined by a linear operatorand a neural nonlinear activation function
. A layer computes and outputs the nonlinear function:(5.1) 
on input activations . By nesting the layers, composite functions are obtained, for example,
(5.2) 
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
,(5.3) 
And this is equivalent to solving the following problem:
(5.4)  
subject to  (5.5)  
(5.6) 
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
(5.7) 
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 substep has a simple closedform solution with only one variable, thus these subproblems 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
(5.8) where is the pseudoinverse of .

To obtain , another leastsquares problem must be solved. The solution is
(5.9) 
The update for requires minimising

Finally, the update of the Lagrange multiplier is given by
(5.10)
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 meanvariance Optimal Control Problem with the following objective function
(6.1) 
with determining the tradeoff 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
(6.2)  
(6.3) 
where is sampled from the distribution and is the set of scenarios: when the number of scenarios is large, then
(6.4) 
The openloop 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,
(6.5)  
subject to  (6.6)  
(6.7)  
(6.8)  
(6.9) 
where  
The previous Optimal Control Problem (6.5) is a convex optimisation problem when is a convex set and is a convex function: an ADMMbased decomposition algorithm for (6.5) is presented below.
6.1 ADMM Decomposition
The Optimal Control Problem (6.5) is rewritten as
(6.10)  
subject to  (6.11)  
(6.12)  
(6.13)  
(6.14)  
(6.15) 
where
The previous Optimal Control Problem (6.10) is then transformed into ADMM form,
(6.16)  
subject to  (6.17) 
with the optimisation variables defined as
(6.18)  
(6.19) 
where
(6.20)  
(6.21) 
(6.22)  
(6.24)  
(6.25) 
6.2 Decentralised Iterated Computation
The Lagrangian of (6.16) and (6.17) is
(6.26) 
where is a vector of Lagrangian multipliers for (6.17). In ADMM, points satisfying the optimality conditions for (6.16) and (6.17) are obtained via the recursions with iteration number
(6.27)  
(6.28)  
(6.29) 
where the augmented Lagrangian with penalty parameter is defined as
and is a scaled dual variable.
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