Trading under the Proof-of-Stake Protocol
– a Continuous-Time Control Approach

As a digital exchange vehicle, blockchain technology has been successfully deployed in many applications including cryptocurrency [18], healthcare [9], supply chain [8], electoral voting [27], and non-fungible tokens [26]. A blockchain is a growing chain of accounting records, called blocks, which are jointly maintained by participants of the system using cryptography. Consider for instance Bitcoin – a peer to peer decentralized payment system. In contrast to traditional payment processing networks, Bitcoin provides a permissionless environment in which everyone is free to participate. At the core of Bitcoin is the consensus protocol known as Proof of Work (PoW), in which “miners” compete with each other by solving a hashing puzzle so as to validate an ever-growing log of transactions (the “longest chain”) to update a distributed ledger; and the miner who solves the puzzle first receives a reward (a number of coins). Thus, while the competition is open to all participants, the chance of winning is proportional to a miner’s computing power.

Despite its popularity, the PoW protocol has some obvious drawbacks. Competition among miners has led to exploding levels of energy consumption in Bitcoin mining, [17, 20]. [1, 3, 7] pointed out that PoW mining will lead to centralization, violating the core tenet of decentralization. To solve the problem of energy efficiency, [14, 28] introduced another consensus protocol – Proof of Stake (PoS), which is a bidding mechanism to select a miner to validate the new block. Participants who choose to join the bidding process are required to commit certain stakes (coins they own), and the winning probability is proportional to the stakes committed. Hence, a participant in a PoS blockchain is a “bidder”, and only the winning bidder becomes the miner who does the validation. As yet the PoS protocol has not been as popular as PoW. However, it is catching up quickly, and blockchain developers have strong incentives to switch from a PoW to a PoS ecosystem. A prominent case in this direction is Ethereum 2.0, where two parallel chains – Mainnet (PoW) and Beacon Chain (PoS) are expected soon to merge into one unified PoS blockchain [10].

There has been an active stream of recent studies on PoS in the research literature; and here we briefly mention several that relate closely to our study. In [22] it is shown that the PoS protocol is “without waste” from an economic standpoint. Issues of stability and decentralization of the PoS protocol are examined in [21, 24]. Specifically, it is shown in [21] that for large owners of initial wealth in a PoS system their shares of the total wealth will remain stable in the long run (i.e., proportions to the total wealth will remain constant), and hence the rich-get-richer phenomenon will not happen. [24] further extends this to medium and small participants, and reveals a phase transition in share stability among those different types of participants. In [21, 25], various aspects of the consumption-investment problem in PoS are examined, and certain conditions are identified under which a participant may have no incentive to trade with others. This leads to the complementing question, given a participant does prefer to trade, what is the optimal trading strategy?

Motivated by the above question, the objective of our study here is to develop a continuous-time control approach to optimal trading in a PoS blockchain. While the control (or game) approach has been proposed in previous studies [4, 5, 15], they are all for the PoW protocol. To the best of our knowledge, ours is the first control model developed for optimal trading under the PoS protocol.

Here is an overview of our main results. We first formulate the consumption-investment problem, which aims to strike a balance between a participant’s utility from holding/trading stakes and utility from consumption. It takes the form of a deterministic control problem with the real-time trading strategy being the control variable. We start with a detailed analysis on a special case that we call the “stake-hoarding” problem (Proposition 3.1), where we bring out the possible scenario of monopoly. We then solve the general consumption-investment problem via dynamic programming and the Hamilton-Jacobi-Bellman (HJB) equations (Theorem 3.4).

When the utility functions are linear or convex, more explicit solutions can be obtained, and we show that the bang-bang control is optimal, i.e., always buy or sell at full capacity (Propositions 4.1 and 4.3). Along with the optimal trading strategy, we are also able to 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 other words, the participant’s risk sensitivity is explicitly accounted for in the trading strategy. In particular, 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 either buy all the time, sell all the time, or first buy then sell (with both buying and selling executed at full capacity).

Finally, we propose a risk control version of the consumption-investment problem, by adding a penalty term to control the level of stake holding so as to reduce the level of concentration risk (Theorem 5.1). A special case is a “stake-parity” problem, where the participant’s holding is controlled at a level that tries to track the system-wide average. We show that the “mean-reverting” strategy is the optimal solution to the stake-parity problem (Proposition 5.2).

The rest of the paper is organized as follows. Section 2 details the formulation of the consumption-investment problem under the PoS protocol. Section 3 presents the optimal solution to the problem, and Section 4 focuses on the special case of linear and convex utility functions. Section 5 presents extensions to risk-control objectives. Concluding remarks are summarized in Section 6.

This section introduces the problem of trading under the PoS protocol in continuous time, and formulate a control model to solve the problem. First, collected below are some conventions that will be used throughout this paper.

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  $\mathbb{R}$ denotes the set of real numbers, and $\mathbb{R}_{+}$ denotes the set of nonnegative real numbers.

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  For $x,y\in\mathbb{R}$, $x\wedge y$ denotes the smaller number of $x$ and $y$; $x\vee y$ denotes the larger number of $x$ and $y$.

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  The symbol $x=o(y)$ means $\frac{x}{y}$ decays towards zero as $y\to\infty$.

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  For a random variable $X$, $\mathbb{E}(X)$ denotes the expectation of $X$.

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  Let $\Omega$ be a subset of $\mathbb{R}$. A function $f\in\mathcal{C}^{k}(\Omega)$ if it is $k$-time continuously differentiable in $\Omega$.

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  For $f\in\mathcal{C}^{1}([0,T])$, $f^{\prime}(t)$ denotes the derivative of $f$. For $f\in\mathcal{C}^{1}([0,T]\times\Omega)$, $\partial_{t}f$ (resp. $\partial_{x}f$) denotes the partial derivative of $f$ with respect to $t$ (resp. $x$).

Time is continuous, indexed by $t\in[0,T]$, for a fixed $T>0$ representing the length of a finite horizon. Let $\{N(t),\,0\leq t\leq T\}$ (with $N(0):=N$) denote the process of the total volume of stakes, which are issued over time by the PoS protocol, and can either be deterministic or stochastic. For ease of presentation, we consider a deterministic process $N(t)$, which is increasing in time and sufficiently smooth, with the derivative $N^{\prime}(t)$ representing the instantaneous rate of “reward” — additional stakes (or “coins”) injected into the system specified (exogenously) by the PoS protocol. For instance, we will consider below, as a special case, the process $N(t)$ of a polynomial form:

Then, $N^{\prime}_{\alpha}(t)=\alpha(N^{\frac{1}{\alpha}}+t)^{\alpha-1}$, and $N^{\prime\prime}_{\alpha}(t)=\alpha(\alpha-1)(N^{\frac{1}{\alpha}}+t)^{\alpha-2}$, so the parametric family (2.1) covers different rewarding schemes according to the values of $\alpha$.

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  For $0<\alpha<1$, we have $N^{\prime\prime}_{\alpha}(t)<0$ so the process $N_{\alpha}(t)$ corresponds to a decreasing reward (e.g. Bitcoin);

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  For $\alpha=1$, the process $N_{1}(t)=N+t$ gives a rate one constant reward (e.g. Blackcoin);

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  For $\alpha>1$, we get $N^{\prime\prime}_{\alpha}(t)>0$ and hence, the process $N_{\alpha}(t)$ amounts to an increasing reward (e.g. EOS).

Let $K\geq 2$ denote the total number of participants in the system, who are indexed by $k\in[K]:=\{1,\ldots,K\}$. For each participant $k$, let $\{X_{k}(t),\,0\leq t\leq T\}$ (with $X_{k}(0)=x_{k}$) denote the process of the number of stakes that participant $k$ holds, with $X_{k}(t)\geq 0$ and $\sum_{k=1}^{K}X_{k}(t)=N(t)$ for all $t\in[0,T]$. In the (discrete-time) PoS protocol, in each round of the bidding process, individual participants commit stakes so as to be selected to validate the block and receive a reward; and the winning probability is $X_{k}(t)/N(t)$ for participant $k$, i.e., proportional to the number of stakes committed. (For instance, each round in Ethereum takes about $10$ seconds, corresponding to the block-generation time [6].) For our continuous-time PoS model here, in which the time required for each round of voting is “infinitesimal,” imagine there are $M$ rounds of bidding during any given time interval $[t,t+\Delta t]$. In each round participant $k$ gets either some stake(s) or nothing; so the average total number of stakes $k$ will get over the $M$ rounds is (by law of large numbers when $M$ is large),

Hence, replacing $\Delta t$ by the infinitesimal $dt$, we know participant $k$ will receive (on average) $\frac{X_{k}(t)}{N(t)}N^{\prime}(t)dt$ stakes, where $\frac{X_{k}(t)}{N(t)}$ is $k$’s winning probability, and $N^{\prime}(t)dt$ is the reward issued by the blockchain in $[t,t+dt]$.

Participants are allowed to trade (buy or sell) their stakes. Participant $k$ will buy $\nu_{k}(t)dt$ stakes in $[t,t+dt]$ if $\nu_{k}(t)>0$, and sell $-\nu_{k}(t)dt$ stakes if $\nu_{k}(t)<0$. This leads to the following dynamics of participant $k$’s stakes under trading:

where $\tau_{k}:=\inf\{t>0:X_{k}(t)=0\}$ is the first time at which the process $X_{k}(t)$ reaches zero. It is reasonable to stop the trading process if a participant runs out of stakes, or gets all available stakes:

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  If $\mathcal{T}_{k}=\tau_{k}$, then participant $k$ liquidates all his stakes by time $\tau_{k}$, and $X_{k}(\mathcal{T}_{k})=0$;

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  If $\mathcal{T}_{k}=\max_{j\neq k}\tau_{j}$, then participant $k$ gets all issued stakes by time $\max_{j\neq k}\tau_{j}$, and hence $X_{k}(\mathcal{T}_{k})=N(\mathcal{T}_{k})$.

We set $X_{k}(t)=X_{k}(\mathcal{T}_{k})$ for $t>\mathcal{T}_{k}$.

The problem is for each participant $k$ to decide how to trade stakes with others under the PoS protocol. Let $\{P(t),\,0\leq t\leq T\}$ be the price process of each (unit of) stake, which is a stochastic process assumed to be independent of the dynamics in (2.2). (This assumption has appeared in recent studies (e.g., [21]), and is somehow a reflection of the reality that the crypto price tends to be affected by market shocks such as macroeconomics, geopolitics, breaking news, etc much more than by trading activities.) Here, the price $P(t)$ of each stake is measured in terms of an underlying risk-free asset (referred to as “cash” for simplicity); and let $b_{k}(t)$ denote the (units of) risk-free asset that participant $k$ holds at time $t$, and let $r>0$ denote the risk-free (interest) rate. Also note that all $K$ participants are allowed to trade stakes (with cash) only internally among themselves, whereas each participants can only exchange cash with an external source (say, a bank).

The decision for each participant $k$ at $t$ is hence a tuple $(\nu_{k}(t),b_{k}(t))$. Let $\{c_{k}(t),\,0\leq t\leq T\}$ be the process of consumption, or cash flow of participant $k$, which follows the dynamics below:

with

Set $b_{k}(t)=b_{k}(\mathcal{T}_{k})$ and $\nu_{k}(t)=0$ for $t>\mathcal{T}_{k}$.

In (C1), if $db_{k}(t)<0$, the participant sells the risk-free asset to get cash either for buying stakes, or for consumption; if $db_{k}(t)>0$, the participant adds more risk-free asset. Thus, (C1) is a self-financing condition in which $rb_{k}(t)dt-db_{k}(t)$ is the net change (in value of the risk-free asset held) used to finance new stakes $P(t)\nu_{k}(t)dt$ and consumption $dc(t)$. The requirements in (C2) are all in the spirit of disallowing shorting on either the risk free asset $b_{k}(t)$ or the stakes $X_{k}(t)$. In some PoS blockchains, there is a minimum requirement for bidding (e.g. 32 ETHs for Ethereum). In this case, we can impose a lower bound on the process $X_{k}(t)$, to prevent it from falling below this threshold. The analysis will be similar. We also require that the trading strategy be bounded: there is $\overline{\nu}_{k}>0$ such that

The objective of participant $k$ is:

where $\beta_{k}>0$ is a discount factor, a parameter measuring the risk sensitivity of participant $k$; $\ell_{k}(\cdot)$ and $h_{k}(\cdot)$ are two utility functions representing, respectively, the running profit and the terminal profit.

While generally following Merton’s consumption-investment framework, our formulation as presented above takes into account some distinct features of PoS blockchains and cryptocurrencies. One notable point is, the utilities $\ell$ and $h$ in the objective are expressed as functions of the number of stakes $X_{k}(t)$, as opposed to their total value $P(t)X_{k}(t)$. To the extent that $P(t)$ is treated as exogenous (as explained above), this difference may seem to be trivial. Yet, it is a reflection of the more substantial fact that crypto-participants tend to mentally decouple the utility of holding stakes from their monetary value at any given time. For instance, holding $1$ ETH may be equivalent to $\$5,000$ for one person, and $\$500$ for another, and neither will be influenced by the ETH market price at the time, which could be say, about $\$1,500$.

Throughout below, the following conditions will be assumed:

Here we study the consumption-investment problem for participant $k$ in (2.3). To lighten notation, omit the subscript $k$, and write out the problem in full as follows, where (C0) is a repeat of the state dynamics in (2.2):

where $\mathcal{T}:=\inf\{t>0:X(t)=0\mbox{ or }N(t)\}\wedge T$.

Note that the expectation in the objective function is with respect to $P(t)$, which is involved in $dc_{k}(t)$ via (C1). Denote

Substituting the constraint (C1) into the objective function, and taking into account

along with (3.2), we have

where $b(0)=0$ is used in the last equality. Hence,

Next, suppose $\beta\geq r$, a condition that will be assumed below (and readily justified as the risk premium associated with the valuation of any stake over the risk-free asset). Then, from the $J_{1}(b)$ expression in (3.3), and taking into account $b(t)\geq 0$ as constrained in (C2), we have $\sup_{b}J_{1}(b)=0$ with the optimality binding at $b_{*}(t)=0$ for all $t$. Consequently, the problem in (3.1) is reduced to

where (C2’) is (C2) without the constraints on $b(\cdot)$.

In summary, the key fact here is that the objective $U(x)$ is separable in the control variables $(\nu(t),b(t))$; hence the problem in (3.1) is decomposed into two optimal control problems, one on the risk-free asset $b(t)$, and the other on the trading of stakes $\nu(t)$, as specified in (3.3) and (3.4). Moreover, under the condition $\beta\geq r$, the consumption-investment problem is reduced to the one in (3.5), where the objective function $J_{2}(\nu)$ – refer to (3.3) – takes the form of a tradeoff between the utility from holding stakes ($\ell(X(t))$ and $h(X(\mathcal{T}))$) and the dis-utility of reducing consumption ($-\widetilde{P}_{\beta}(t)\nu(t)$). Thus, the optimal trading strategy needs to strike a balance between these two opposing terms.

Before we present the optimal solution to the consumption-investment problem in (3.5), we make a digression to first study a simple degenerate case of $\widetilde{P}_{\beta}(t)\equiv 0$. This special case removes the tradeoff mentioned above, so the solution becomes a one-sided strategy of always accumulating (or “hoarding”) the stakes at full capacity ($\overline{\nu}$). Yet, as the analysis below will show, there are still some interesting (and subtle) issues involved. More importantly, this special case provides a very accessible path to finding the optimal solution via dynamic programming and the HJB equation.