With the increasing adoption of the Proof of Stake (PoS) blockchain, it is timely to study the economy created by such blockchain. In this chapter, we will survey recent progress on the trading and wealth evolution in a cryptocurrency where the new coins are issued according to the PoS protocol. We first consider the wealth evolution in the PoS protocol assuming no trading, and focus on the problem of decentralisation. Next we consider each miner's trading incentive and strategy through the lens of optimal control, where the miner needs to trade off PoS mining and trading. Finally, we study the collective behavior of the miners in a PoS trading environment by a mean field model. We use both stochastic and analytic tools in our study. A list of open problems are also presented.
The Metaverse has emerged as the next generation of the Internet. It aims to provide an immersive, persistent virtual space where people can live, learn, work and interact with each other. However, the existing technology is inadequate to guarantee high visual quality and ultra-low latency service for the Metaverse players. Mobile Edge Computing (MEC) is a paradigm where proximal edge servers are utilized to perform computation-intensive and latency-sensitive tasks like image processing and video analysis. In MEC, the large amount of data is processed by edge servers closest to where it is captured, thus significantly reducing the latency and providing almost real-time performance. In this paper, we integrate fundamental elements (5G and 6G wireless communications, Blockchain, digital twin and artificial intelligence) into the MEC framework to facilitate the Metaverse. We also elaborate on the research problems and applications in the MEC-enabled Metaverse. Finally, we provide a case study to establish a thorough knowledge of the user utility maximization problem in a real-world scenario and gain some insights about trends in potential research directions.
With rapid development of blockchain technology as well as integration of various application areas, performance evaluation, performance optimization, and dynamic decision in blockchain systems are playing an increasingly important role in developing new blockchain technology. This paper provides a recent systematic overview of this class of research, and especially, developing mathematical modeling and basic theory of blockchain systems. Important examples include (a) performance evaluation: Markov processes, queuing theory, Markov reward processes, random walks, fluid and diffusion approximations, and martingale theory; (b) performance optimization: Linear programming, nonlinear programming, integer programming, and multi-objective programming; (c) optimal control and dynamic decision: Markov decision processes, and stochastic optimal control; and (d) artificial intelligence: Machine learning, deep reinforcement learning, and federated learning. So far, a little research has focused on these research lines. We believe that the basic theory with mathematical methods, algorithms and simulations of blockchain systems discussed in this paper will strongly support future development and continuous innovation of blockchain technology.
The control of future power grids is migrating from a centralized to a distributed/decentralized scheme to enable a massive penetration of distributed energy resources and bring extreme enhancements of autonomous operations in terms of grid resilience, security, and reliability. Most effort has been on the design of distributed/decentralized controllers; however, the guarantees of the proper execution of the controls are also essential but relatively less emphasized. A common assumption is that local controllers would fully follow the designated controller dynamics based on the data received from communication channels. Such an assumption could be risky because proper execution of the controller dynamics is then built on trust in secure communication and computation. On the other hand, it is impractical for a verifier to repeat all the computations involved in the controls to verify the computational integrity. In this work, we leverage a type of cryptography technology, known as zero-knowledge scalable transparent arguments of knowledge to verify the computational integrity of control algorithms, such that verifiers can check the computational integrity with much less computational burden. The method presented here converts the challenge of data integrity into a subset of computational integrity. In this proof-of-concept paper, our focus will be on projected linear dynamics that are commonly seen in distributed/decentralized power system controllers. In particular, we have derived polynomial conditions in the context of zk-STARKs for the projected linear dynamics.
The goal of cryptocurrencies is decentralization. In principle, all currencies have equal status. Unlike traditional stock markets, there is no default currency of denomination (fiat), thus the trading pairs can be set freely. However, it is impractical to set up a trading market between every two currencies. In order to control management costs and ensure sufficient liquidity, we must give priority to covering those large-volume trading pairs and ensure that all coins are reachable. We note that this is an optimization problem. Its particularity lies in: 1) the trading volume between most (>99.5%) possible trading pairs cannot be directly observed. 2) It satisfies the connectivity constraint, that is, all currencies are guaranteed to be tradable. To solve this problem, we use a two-stage process: 1) Fill in missing values based on a regularized, truncated eigenvalue decomposition, where the regularization term is used to control what extent missing values should be limited to zero. 2) Search for the optimal trading pairs, based on a branch and bound process, with heuristic search and pruning strategies. The experimental results show that: 1) If the number of denominated coins is not limited, we will get a more decentralized trading pair settings, which advocates the establishment of trading pairs directly between large currency pairs. 2) There is a certain room for optimization in all exchanges. The setting of inappropriate trading pairs is mainly caused by subjectively setting small coins to quote, or failing to track emerging big coins in time. 3) Too few trading pairs will lead to low coverage; too many trading pairs will need to be adjusted with markets frequently. Exchanges should consider striking an appropriate balance between them.
A trading system is said to be {robust} if it generates a robust return regardless of market direction. To this end, a consistently positive expected trading gain is often used as a robustness metric for a trading system. In this paper, we propose a new class of trading policies called the {double linear policy} in an asset trading scenario when the transaction costs are involved. Unlike many existing papers, we first show that the desired robust positive expected gain may disappear when transaction costs are involved. Then we quantify under what conditions the desired positivity can still be preserved. In addition, we conduct heavy Monte-Carlo simulations for an underlying asset whose prices are governed by a geometric Brownian motion with jumps to validate our theory. A more realistic backtesting example involving historical data for cryptocurrency Bitcoin-USD is also studied.
Recent financial bubbles such as the emergence of cryptocurrencies and "meme stocks" have gained increasing attention from both retail and institutional investors. In this paper, we propose a game-theoretic model on optimal liquidation in the presence of an asset bubble. Our setup allows the influx of players to fuel the price of the asset. Moreover, traders will enter the market at possibly different times and take advantage of the uptrend at the risk of an inevitable crash. In particular, we consider two types of crashes: an endogenous burst which results from excessive selling, and an exogenous burst which cannot be anticipated and is independent from the actions of the traders. The popularity of asset bubbles suggests a large-population setting, which naturally leads to a mean field game (MFG) formulation. We introduce a class of MFGs with varying entry times. In particular, an equilibrium will depend on the entry-weighted average of conditional optimal strategies. To incorporate the exogenous burst time, we adopt the method of progressive enlargement of filtrations. We prove existence of MFG equilibria using the weak formulation in a generalized setup, and we show that the equilibrium strategy can be decomposed into before-and-after-burst segments, each part containing only the market information. We also perform numerical simulations of the solution, which allow us to provide some intriguing results on the relationship between the bubble burst and equilibrium strategies.
Theo Diamandis, Alex Evans, Tarun Chitra, Guillermo Angeris
Public blockchains implement a fee mechanism to allocate scarce computational resources across competing transactions. Most existing fee market designs utilize a joint, fungible unit of account (e.g., gas in Ethereum) to price otherwise non-fungible resources such as bandwidth, computation, and storage, by hardcoding their relative prices. Fixing the relative price of each resource in this way inhibits granular price discovery, limiting scalability and opening up the possibility of denial-of-service attacks. As a result, many prominent networks such as Ethereum and Solana have proposed multi-dimensional fee markets. In this paper, we provide a principled way to design fee markets that efficiently price multiple non-fungible resources. Starting from a loss function specified by the network designer, we show how to compute dynamic prices that align the network's incentives (to minimize the loss) with those of the users and miners (to maximize their welfare), even as demand for these resources changes. Our pricing mechanism follows from a natural decomposition of the network designer's problem into two parts that are related to each other via the resource prices. These results can be used to efficiently set fees in order to improve network performance.
Matheus V. X. Ferreira, Ye Lin Sally Hahn, S. Matthew Weinberg, Catherine Yu
Cryptographic Self-Selection is a subroutine used to select a leader for modern proof-of-stake consensus protocols, such as Algorand. In cryptographic self-selection, each round $r$ has a seed $Q_r$. In round $r$, each account owner is asked to digitally sign $Q_r$, hash their digital signature to produce a credential, and then broadcast this credential to the entire network. A publicly-known function scores each credential in a manner so that the distribution of the lowest scoring credential is identical to the distribution of stake owned by each account. The user who broadcasts the lowest-scoring credential is the leader for round $r$, and their credential becomes the seed $Q_{r+1}$. Such protocols leave open the possibility of a selfish-mining style attack: a user who owns multiple accounts that each produce low-scoring credentials in round $r$ can selectively choose which ones to broadcast in order to influence the seed for round $r+1$. Indeed, the user can pre-compute their credentials for round $r+1$ for each potential seed, and broadcast only the credential (among those with a low enough score to be the leader) that produces the most favorable seed. We consider an adversary who wishes to maximize the expected fraction of rounds in which an account they own is the leader. We show such an adversary always benefits from deviating from the intended protocol, regardless of the fraction of the stake controlled. We characterize the optimal strategy; first by proving the existence of optimal positive recurrent strategies whenever the adversary owns last than $38\%$ of the stake. Then, we provide a Markov Decision Process formulation to compute the optimal strategy.
Organizations are often unable to align the interests of all stakeholders with the financial success of the organization (e.g. due to regulation). However, continuous organizations (COs) introduce a paradigm shift. COs offer immediate liquidity, are permission-less and can align incentives. CO shares are issued continuously in the form of tokens via a smart contract on a blockchain. Token prices are designed to increase as more tokens are minted. When the share supply is low, near-zero prices make it advantageous to buy and hold tokens until interest in the CO increases, enabling a profitable sale. This attribute of COs, known as investment efficiency, is desirable. Yet, it can yield allocative inefficiency via the "holdout problem," i.e. latecomers may find a CO more valuable than early tokenholders, but be unable to attain the same token holdings due to inflated prices. With the aim of increasing overall equity, we introduce a voting mechanism into COs. We show this balances allocative and investment efficiency, and may dissuade speculative trading behaviors, thereby decreasing investment risk.
We investigate the instantaneous and limiting behavior of an n-node blockchain which is under continuous monitoring of the IT department of a company but faces non-stop cyber attacks from a single hacker. The blockchain is functional as far as no data stored on it has been changed, deleted, or locked. Once the IT department detects the attack from the hacker, it will immediately re-set the blockchain, rendering all previous efforts of the hacker in vain. The hacker will not stop until the blockchain is dysfunctional. For arbitrary distributions of the hacking times and detecting times, we derive the limiting functional probability, instantaneous functional probability, and mean functional time of the blockchain. We also show that all these quantities are increasing functions of the number of nodes, substantiating the intuition that the more nodes a blockchain has, the harder it is for a hacker to succeed in a cyber attack.
Abstract We develop a continuous‐time control approach to optimal trading in a Proof‐of‐Stake (PoS) blockchain, formulated as a consumption‐investment problem that aims to strike the optimal balance between a participant's (or agent's) utility from holding/trading stakes and utility from consumption. We present solutions via dynamic programming and the Hamilton–Jacobi–Bellman (HJB) equations. When the utility functions are linear or convex, we derive close‐form solutions and show that the bang‐bang strategy is optimal (i.e., always buy or sell at full capacity). Furthermore, we bring out the explicit connection between the rate of return in trading/holding stakes and the participant's risk‐adjusted valuation of the stakes. In particular, we show when a participant is risk‐neutral or risk‐seeking, corresponding to the risk‐adjusted valuation being a martingale or a sub‐martingale, the optimal strategy must be to either buy all the time, sell all the time, or first buy then sell, and with both buying and selling executed at full capacity. We also propose a risk‐control version of the consumption‐investment problem; and for a special case, the “stake‐parity” problem, we show a mean‐reverting strategy is optimal.
The Ethereum ecosystem was maintained by a distributed global network of computers that required massive amounts of computational power. Previous work on estimating the energy use and emissions of the Ethereum network has relied on top-down economic analysis and rough estimates of hardware efficiency and emissions factors. In this work we provide a bottom-up analysis that works from hashrate to an energy usage estimate, and from mining locations to an emissions factor estimate, and combines these for an overall emissions estimate. We analyze the entire history of PoW Ethereum, from creation to the merge.
In this paper, we provide a novel dynamic decision method of blockchain selfish mining by applying the sensitivity-based optimization theory. Our aim is to find the optimal dynamic blockchain-pegged policy of the dishonest mining pool. To study the selfish mining attacks, two mining pools is designed by means of different competitive criterions, where the honest mining pool follows a two-block leading competitive criterion, while the dishonest mining pool follows a modification of two-block leading competitive criterion through using a blockchain-pegged policy. To find the optimal blockchain-pegged policy, we set up a policy-based continuous-time Markov process and analyze some key factors. Based on this, we discuss monotonicity and optimality of the long-run average profit with respect to the blockchain-pegged reward and prove the structure of the optimal blockchain-pegged policy. We hope the methodology and results derived in this paper can shed light on the dynamic decision research on the selfish mining attacks of blockchain selfish mining.
Energy storage provides an effective way of shifting temporal energy demands and supplies, which enables significant cost reduction under time-of-use energy pricing plans. Despite its promising benefits, the cost of present energy storage remains expensive, presenting a major obstacle to practical deployment. A more viable solution to improve the cost-effectiveness is by sharing energy storage, such as community sharing, cloud energy storage and peer-to-peer sharing. However, revealing private energy demand data to an external energy storage operator may compromise user privacy, and is susceptible to data misuses and breaches. In this paper, we explore a novel approach to support energy storage sharing with privacy protection, based on privacy-preserving blockchain and secure multi-party computation. We present an integrated solution to enable privacy-preserving energy storage sharing, such that energy storage service scheduling and cost-sharing can be attained without the knowledge of individual users' demands. It also supports auditing and verification by the grid operator via blockchain. Furthermore, our privacy-preserving solution can safeguard against a majority of dishonest users, who may collude in cheating, without requiring a trusted third-party. We implemented our solution as a smart contract on real-world Ethereum blockchain platform, and provided empirical evaluation in this paper 1 .
In this paper, we study a pricing problem of the multiple reset put option, which allows the holder to reset several times a current strike price to obtain an at-the-money European put option. We formulate the pricing problem as a multiple optimal stopping problem, then reduce it to a sequence of single optimal stopping problems and study the associated free-boundary problems. We solve this sequence of problems by induction in the number of remaining reset rights and exploit probabilistic arguments such as local time-space calculus on curves. As a result, we characterize each optimal reset boundary as the unique solution to a nonlinear integral equation and derive the reset premium representations for the option prices. We propose that the multiple reset options can be used as cryptocurrency derivatives and an attractive alternative to standard European options due to the extreme volatility of underlying cryptocurrencies.
A decentralized blockchain is a distributed ledger that is often used as a platform for exchanging goods and services. This ledger is maintained by a network of nodes that obeys a set of rules, called a consensus protocol, which helps to resolve inconsistencies among local copies of a blockchain. In this paper, we build a mathematical framework for the consensus protocol designer, specifying (a) the measurement of a resource which nodes strategically invest in and compete for to win the right to build new blocks in the blockchain; and (b) a payoff function for such efforts. Thus, the equilibrium of an associated stochastic differential game can be implemented by selecting nodes in proportion to this specified resource and penalizing dishonest nodes by its loss. This associated, induced game can be further analyzed using mean field games. The problem can be broken down into two coupled PDEs, where an individual node's optimal control path is solved using a Hamilton-Jacobi-Bellman equation, and where the evolution of states distribution is characterized by a Fokker-Planck equation. We develop numerical methods to compute the mean field equilibrium for both steady states at the infinite time horizon and evolutionary dynamics. As an example, we show how the mean field equilibrium can be applied to the Bitcoin blockchain mechanism design. We demonstrate that a blockchain can be viewed as a mechanism that operates in a decentralized setup and propagates properties of the mean field equilibrium over time, such as the underlying security of the blockchain.
The rise of Ethereum and other blockchains that support smart contracts has led to the creation of decentralized exchanges (DEXs), such as Uniswap, Balancer, Curve, mStable, and SushiSwap, which enable agents to trade cryptocurrencies without trusting a centralized authority. While traditional exchanges use order books to match and execute trades, DEXs are typically organized as constant function market makers (CFMMs). CFMMs accept and reject proposed trades based on the evaluation of a function that depends on the proposed trade and the current reserves of the DEX. For trades that involve only two assets, CFMMs are easy to understand, via two functions that give the quantity of one asset that must be tendered to receive a given quantity of the other, and vice versa. When more than two assets are being exchanged, it is harder to understand the landscape of possible trades. We observe that various problems of choosing a multi-asset trade can be formulated as convex optimization problems, and can therefore be reliably and efficiently solved.
Internet of quantum blockchains (IoB) will be the future Internet. In this paper, we make two new contributions to IoB: developing a block based quantum channel networking technology to handle its security modeling in face of the quantum supremacy and establishing IoB based FinTech platform model with dynamic pricing for stable digital currency. The interaction between our new contributions is also addressed. In doing so, we establish a generalized IoB security model by quantum channel networking in terms of both time and space quantum entanglements with quantum key distribution (QKD). Our IoB can interact with general structured things (e.g., supply chain systems) having online trading and payment capability via stable digital currency and can handle vector-valued data streams requiring synchronized services. Thus, within our designed QKD, a generalized random number generator for private and public keys is proposed by a mixed zero-sum and non-zero-sum resource-competition pricing policy. The effectiveness of this policy is justified by diffusion modeling with approximation theory and numerical implementations.
Matheus V. X. Ferreira, Daniel J. Moroz, David C. Parkes, Mitchell Stern
In recent years, prominent blockchain systems such as Bitcoin and Ethereum have experienced explosive growth in transaction volume, leading to frequent surges in demand for limited block space and causing transaction fees to fluctuate by orders of magnitude. Existing systems sell space using first-price auctions; however, users find it difficult to estimate how much they need to bid in order to get their transactions accepted onto the chain. If they bid too low, their transactions can have long confirmation times. If they bid too high, they pay larger fees than necessary. In light of these issues, new transaction fee mechanisms have been proposed, most notably EIP-1559, aiming to provide better usability. EIP-1559 is a history-dependent mechanism that relies on block utilization to adjust a base fee. We propose an alternative design -- a {\em dynamic posted-price mechanism} -- which uses not only block utilization but also observable bids from past blocks to compute a posted price for subsequent blocks. We show its potential to reduce price volatility by providing examples for which the prices of EIP-1559 are unstable while the prices of the proposed mechanism are stable. More generally, whenever the demand for the blockchain stabilizes, we ask if our mechanism is able to converge to a stable state. Our main result provides sufficient conditions in a probabilistic setting for which the proposed mechanism is approximately welfare optimal and the prices are stable. Our main technical contribution towards establishing stability is an iterative algorithm that, given oracle access to a Lipschitz continuous and strictly concave function $f$, converges to a fixed point of $f$.
Pietro Ferraro, Lianna Zhao, Christopher King, Robert Shorten
This paper describes the use of Distributed Ledger Technologies as a mean to enforce social contracts and to orchestrate the behaviour of agents in a smart city environment. Specifically, we present a scheme to price personalised risk in sharing economy applications. We provide proofs for the convergence of the proposed stochastic system and we validate our approach through the use of extensive Monte Carlo simulations.
Constant function market makers (CFMMs) such as Uniswap, Balancer, Curve, and mStable, among many others, make up some of the largest decentralized exchanges on Ethereum and other blockchains. Because all transactions are public in current implementations, a natural next question is if there exist similar decentralized exchanges which are privacy-preserving; i.e., if a transaction's quantities are hidden from the public view, then an adversary cannot correctly reconstruct the traded quantities from other public information. In this note, we show that privacy is impossible with the usual implementations of CFMMs under most reasonable models of an adversary and provide some mitigating strategies.
Dynamic Portfolio optimization is the process of distribution and rebalancing of a fund into different financial assets such as stocks, cryptocurrencies, etc, in consecutive trading periods to maximize accumulated profits or minimize risks over a time horizon. This field saw huge developments in recent years, because of the increased computational power and increased research in sequential decision making through control theory. Recently Reinforcement Learning(RL) has been an important tool in the development of sequential and dynamic portfolio optimization theory. In this paper, we design a Deep Reinforcement Learning(DRL) framework as an autonomous portfolio optimization agent consisting of a Latent Feature State Space(LFSS) Module for filtering and feature extraction of financial data which is used as a state space for deep RL model. We develop an extensive RL agent with high efficiency and performance advantages over several benchmarks and model-free RL agents used in prior work. The noisy and non-stationary behaviour of daily asset prices in the financial market is addressed through Kalman Filter. Autoencoders, ZoomSVD, and restricted Boltzmann machines were the models used and compared in the module to extract relevant time series features as state space. We simulate weekly data, with practical constraints and transaction costs, on a portfolio of S&P 500 stocks. We introduce a new benchmark based on technical indicator Kd-Index and Mean-Variance Model as compared to equal weighted portfolio used in most of the prior work. The study confirms that the proposed RL portfolio agent with state space function in the form of LFSS module gives robust results with an attractive performance profile over baseline RL agents and given benchmarks.
With the booming of smart grid, The ubiquitously deployed smart meters constitutes an energy internet of things. This paper develops a novel blockchain-based transactive energy management system for IoT-aided smart homes. We consider a holistic set of options for smart homes to participate in transactive energy. Smart homes can interact with the grid to perform vertical transactions, e.g., feeding in extra solar energy to the grid and providing demand response service to alleviate the grid load. Smart homes can also interact with peer users to perform horizontal transactions, e.g., peer-to-peer energy trading. However, conventional transactive energy management method suffers from the drawbacks of low efficiency, privacy leakage, and single-point failure. To address these challenges, we develop a privacy-preserving distributed algorithm that enables users to optimally manage their energy usages in parallel via the smart contract on the blockchain. Further, we design an efficient blockchain system tailored for IoT devices and develop the smart contract to support the holistic transactive energy management system. Finally, we evaluate the feasibility and performance of the blockchain-based transactive energy management system through extensive simulations and experiments. The results show that the blockchain-based transactive energy management system is feasible on practical IoT devices and reduces the overall cost by 25%.