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Market Equilibria and Risk Diversification in Blockchain Mining Economies

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Mathematical Research for Blockchain Economy (MARBLE 2022)

Abstract

The success of blockchain-based applications, most notably cryptocurrencies, has brought the allocation of mining resources at the epicenter of academic and entrepreneurial attention. Critical for the stability of these markets is the question of how miners should adjust their allocations over time in response to changes in their environment and in other miners’ strategies. In this paper, we present a proportional response (PR) protocol that makes these adjustments for any risk profile of a miner. The protocol has low informational requirements and is particularly suitable for such distributed settings. When the environment is static, we formally show that the PR protocol attains stability by converging to the market equilibrium. For dynamic environments, we carry out an empirical study with actual data from four popular cryptocurrencies. We find that running the PR protocol with higher risk diversification is beneficial both to the market by curbing volatile re-allocations (and, thus, increasing market stability), and to individual miners by improving their profits after accounting for factor mobility (switching) costs.

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Notes

  1. 1.

    Here, mining refers to any form of expense or investment of scarce resources such as computational power in PoW or native tokens in PoS.

  2. 2.

    Since PoW mining resembles an oligopoly, conventional oligopolistic competition suggests that cost asymmetries cannot be very large among active miners [2, 55]. Similarly, in PoS, all miners (or validators) experience the same opportunity cost when committing their stake.

  3. 3.

    This period is before the implementation of the EIP-1559 update in the Etheurem transaction fee market [41, 56, 62, 66].

  4. 4.

    In particular, fluctuations in mining resources, e.g., hashrate in PoW, cause volatility in cryptocurrency prices [51, 52] which in turn may destabilize blockchain-based applications.

  5. 5.

    Different ways to calculate the factor mobility cost led to the same conclusion. Here, we follow one that is standard in portfolio optimization, see e.g., [59].

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Correspondence to Stefanos Leonardos .

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A Technical Materials: Proof of Theorem 1

A Technical Materials: Proof of Theorem 1

Before we proceed with our empirical results in Sect. 4, we provide the technical details of the proof of Theorem 1. Our proof of Theorem 1 consists of two steps. The first involves the formulation of a convex program in the spending (budget) domain that captures the market equilibrium (ME) of the underlying blockchain mining economy with quasi-CES utilities. The second involves the derivation of a general Mirror Descent (MD) protocol which converges to the optimal solution of the convex program from the previous step. The proof concludes by showing that the (PR-QCES) protocol is an instantiation of this MD protocol.

Fig. 7
figure 7

Convex optimization framework in Theorem 1. The Eisenberg-Gale (EG) [39], its dual (D) and transformed dual (TD) and the Shmyrev-type convex program (SH-QCES) which captures the market equilibrium spending [68]

Part I: Convex Program Framework. This part utilizes the convex optimization framework in the study of Fisher markets with linear or quasi-linear utilities, see e.g., [6, 23, 25]. As with quasi-linear utilities, the main challenge in this case is to guess a convex program that correctly captures the market equilibria also for general quasi-CES utilities [31, 35]. The convex optimization framework that we use to capture the ME spending in quasi-CES Fisher markets is summarized in Fig. 7. Our starting point is a Shmyrev-type convex program proposed by [23] which captures the ME spending in the case of quasi-linear utilities. Our first task is to appropriately modify it so that it captures the ME spending of a quasi-CES Fisher market. The resulting convex program is

(SH-QCES)

where \(F(\textbf{b},\textbf{w})\) is the following function:

$$\begin{aligned} F(\textbf{b},\textbf{w}) :={}&-\sum \nolimits _{i=1}^n \frac{1}{{\rho _i}}\sum \nolimits _{k=1}^m b_{ij}\ln [(v_k)^{{\rho _i}} (b_{ij})^{{\rho _i}-1}]+\sum \nolimits _{k=1}^m p_k\ln {p_k}\\ {}&+\sum \nolimits _{i=1}^n \left[ w_i + \frac{{\rho _i}-1}{{\rho _i}}\cdot (K_i - w_i) \ln (K_i - w_i) \right] . \end{aligned}$$

Here, we used the notation \(\textbf{w}=(w_i)_{i\in N}\) and for more clarity, we also wrote \(\textbf{p}=(p_k)_{k\in M}\) to denote the exogenously given aggregate spending vector (cf. \(\tilde{\textbf{b}}\) in the formulation of Theorem 1). While the first and second constraints in (SH-QCES) fully determine the values of \(p_k\) and \(w_i\) in terms of \(b_{ij}\), it will be more instructive to retain them as separate variables in this part. Our main task is to show that the solutions of (SH-QCES) are solutions to our initial problem, i.e., that they correspond to the ME spending of \(\varGamma \).

Lemma 1

The unique minimum point of (SH-QCES) corresponds to the unique market equilibrium spending of \(\varGamma \).

Proof

We verify that the optimality condition of the convex program (SH-QCES) is the same as the market equilibrium condition. The claim is immediate for \({\rho _i}=1\), so we restrict attention to \({\rho _i}<1\). To determine the optimality condition of (SH-QCES), we take the partial derivatives of F with respect to \(b_{ij}\) and \(w_i\)

$$\begin{aligned} \frac{\partial }{\partial b_{ij}}F(\textbf{b},\textbf{w})&= \frac{1}{{\rho _i}}\left( 1 - {\rho _i}\ln v_k \right) + \frac{1-{\rho _i}}{{\rho _i}}\cdot \ln b_{ij}+ \ln p_k\\\frac{\partial }{\partial w_i} F(\textbf{b},\textbf{w})&= \frac{1}{{\rho _i}} \left[ 1 - ({\rho _i}-1) \ln (K_i - w_i) \right] . \end{aligned}$$

Since \((1-{\rho _i})/{\rho _i}> 0\), \(\lim _{b_{ij}\searrow 0} \frac{1-{\rho _i}}{{\rho _i}}\cdot \ln b_{ij}= -\infty \). Hence, at each minimum point, \(b_{ij}\) must be strictly positive. In turn, since \(b_{ij}\) is in the relative interior of the domain at each minimum point, and we have the constraint \(\sum _{k=1}^m b_{ij}\le K_i\), it must hold that all \(\frac{\partial }{\partial b_{ij}}F(\textbf{b},\textbf{w})\) are identical for all \(k\in M\). Equivalently, \(\frac{(v_k)^{\rho _i}(b_{ij})^{{\rho _i}-1}}{(p_k)^{\rho _i}}\) are identical for all k. Thus, depending on whether \(w_i>0\) or \(w_i=0\), we have the following two cases. If \(K_i> w_i > 0\), then \(\frac{\partial }{\partial b_{ij}}F(\textbf{b},\textbf{w}) = \frac{\partial }{\partial w_i} F(\textbf{b},\textbf{w})\), which implies \(\frac{(v_k)^{\rho _i}(b_{ij})^{{\rho _i}-1}}{(p_k)^{\rho _i}} = (K_i - w_i)^{{\rho _i}-1}\) for all \(k\in M\). If \(w_i = 0\), then \(\frac{\partial }{\partial b_{ij}}F(\textbf{b},\textbf{w}) \le \frac{\partial }{\partial w_i} F(\textbf{b},\textbf{w})\), which implies \(\frac{(v_k)^{\rho _i}(b_{ij})^{{\rho _i}-1}}{(p_k)^{\rho _i}} \ge (K_i - w_i)^{{\rho _i}-1}\) for all k.

To determine the ME condition, we need to find the rate of change in a miner’s utility w.r.t. changes in spending on mining cryptocurrency k. Due to the cost-homogeneity assumption, in (3), \(v_{ij}= v_k / p_k\). Since aggregate expenditures \(\textbf{p}\) are considered as independent signals in the market, the rate is

$$ \frac{\partial }{\partial b_{ik}} u_i(\textbf{b}\mid \textbf{p}) = \left( \sum \nolimits _{k=1}^m \frac{(v_k)^{\rho _i}(b_{ik})^{\rho _i}}{(p_k)^{\rho _i}} \right) ^{1/{\rho _i}-1} \cdot \frac{(v_k)^{\rho _i}(b_{ik})^{{\rho _i}-1}}{(p_k)^{\rho _i}} - 1.$$

Since \({\rho _i}-1 < 0\) and, hence, \(\lim _{b_{ik}\searrow 0} (b_{ik})^{{\rho _i}-1} = +\infty \), at the market equilibrium, \(b_{ik}\) must be strictly positive. Thus, at the ME, each \(b_{ij}\) is in the relative interior of the domain. Together with the constraint \(\sum _k b_{ij}\le K_i\), this implies that \(\frac{\partial }{\partial b_{ik}} u_i(\textbf{b}\mid \textbf{p})\) are identical for all \(k\in M\), which in turn implies that \(\frac{(v_k)^{\rho _i}(b_{ij})^{{\rho _i}-1}}{(p_k)^{\rho _i}}\) must be identical for all \(k\in M\). We denote this (common) value by \(z_i\). Then

$$\begin{aligned} \frac{\partial }{\partial b_{ik}} u_i(\textbf{b};\textbf{p})&= \left( \sum \nolimits _{k=1}^m z_i b_{ik}\right) ^{1/{\rho _i}-1} \cdot z_i - 1\\ {}&= (z_i)^{1/{\rho _i}} \left( \sum \nolimits _{k=1}^m b_{ij}\right) ^{1/{\rho _i}-1} - 1 \\&= (z_i)^{1/{\rho _i}} (K_i -w_i)^{1/{\rho _i}-1} - 1. \end{aligned}$$

Again, there are two cases. If \(K_i> w_i > 0\), i.e., if \(w_i\) is in the relative interior of its domain, then the above derivative must be zero, i.e., \(z_i = (K_i - w_i)^{{\rho _i}-1}\) for all i. If \(w_i = 0\), then the above derivative at ME is positive or zero, i.e., \(z_i \ge (K_i - w_i)^{{\rho _i}-1}\) for all i.    \(\square \)

Part II: From Mirror Descent to Proportional Response. As mentioned above, a useful observation in (SH-QCES) is that the first and second constraints determine the values of \(p_k,w_i\) in terms of the \(b_{ij}\)’s. Thus, we may rewrite F as a function of \(\textbf{b}\) only. Then, the convex program (SH-QCES) has only the variables \(\textbf{b}\), and the only constraints on \(\textbf{b}\) are \(b_{ij}\ge 0\) and \(\sum _{k=1}^m b_{ij}\le K_i\). Using this formulation, we can conveniently compute a ME spending by using standard optimization methods like Mirror Descent (MD). Our task in this part, will be to show that the objective function, \(F(\textbf{b})\), of (SH-QCES) is 1-Bregman convex which implies convergence of the MD protocol and hence, of the (PR-QCES) protocol. To begin, we introduce some minimal additional notation and recap a general result about MD [6, 17] below.

Let C be a compact and convex set. The Bregman divergence, \(d_h\), generated by a convex regularizer function h is defined as

$$\begin{aligned} d_h(\textbf{b},\textbf{a}) := h(\textbf{b}) - \left[ h(\textbf{a}) +\langle \nabla h(\textbf{a}),\textbf{b}-\textbf{a}\rangle \right] . \end{aligned}$$
(7)

for any \(\textbf{b}\in C, \textbf{a}\in \textsf{rint}(C)\) where \(\textsf{rint}(C)\) is the relative interior of C. Due to convexity of the function h, \(d_h(\textbf{b},\textbf{a})\) is convex in \(\textbf{b}\), and its value is always non-negative. The Kullback-Leibler divergence (KL-divergence) between \(\textbf{b}\) and \(\textbf{a}\) is \(\textrm{KL}(\textbf{b}\Vert \textbf{a}) := \sum _k b_k \cdot \ln \frac{b_k}{a_k} - \sum _k b_k + \sum _k a_k\), which is same as the Bregman divergence \(d_h\) with regularizer \(h\left( \textbf{b}\right) := \sum _k(b_k \cdot \ln b_k - b_k)\). A function f is L-Bregman convex w.r.t. Bregman divergence \(d_h\) if for any \(\textbf{b}\in C\) and \(\textbf{a}\in \textsf{rint}(C)\),

$$f(\textbf{a}) + \langle \nabla f(\textbf{a}),\textbf{b}-\textbf{a}\rangle \le f(\textbf{b}) \le f(\textbf{a}) + \langle \nabla f(\textbf{a}),\textbf{b}-\textbf{a}\rangle + L \cdot d_h(\textbf{b},\textbf{a}).$$

For the problem of minimizing a convex function \(f(\textbf{b})\) subject to \(\textbf{b}\in C\), the Mirror Descent (MD) method w.r.t. Bregman divergence \(d_h\) is given by the update rule in Algorithm 2. In the MD update rule, \(1/\xi > 0\) is the step-size, which may vary with t (and typically diminishes with t). However, in the current application of distributed dynamics, time-varying step-size and thus, update rule is undesirable or even impracticable since this will require from the agents/firms to keep track with a global clock.

figure b

Theorem 2

([6]) Suppose that f is an L-Bregman convex function w.r.t. \(d_h\) and let \(\textbf{b}(T)\) be the point reached after T applications of the MD update rule in Algorithm 2 with parameter \(\xi = 1/L\). Then

$$\begin{aligned} \textstyle f(\textbf{b}(T)) - f(\textbf{b}^*) ~\le ~ L \cdot d_h(\textbf{b}^*, \textbf{b}(0))/T. \end{aligned}$$

Using the above, we are now ready to show that the objective function of the (SH-QCES) is a 1-Bregman convex function w.r.t. the KL-divergence. This is the statement of Lemma 2. Its proof closely mirrors an analogous statement in [23] and is, thus, omitted.

Lemma 2

The objective function F of (SH-QCES) is a 1-Bregman convex function w.r.t. the divergence \(\sum _{i=1}^n \frac{1}{{\rho _i}}\cdot \textrm{KL}(x'_i || x_i)\).

We now turn to the derivation of the (PR-QCES) protocol from the MD algorithm for a suitable choice of \(\xi \). For the convex program (SH-QCES), the MD rule (Algorithm 2) is

$$\begin{aligned}&(\textbf{b}(t+1),\textbf{w}(t+1)) = {{\,\mathrm{arg\,min}\,}}_{(\textbf{b},\textbf{w})\in C} \left\{ \sum \nolimits _{i=1}^n \frac{1}{{\rho _i}}\cdot \textrm{KL}(\textbf{b}_i \Vert \textbf{b}_i(t))+ \right. \\&\left. +\sum \nolimits _{i=1}^n \sum \nolimits _{k=1}^m \frac{(b_{ij}- b_{ij}(t))}{{\rho _i}}\cdot \left( 1 - \ln \frac{(v_k)^{\rho _i}b_{ij}(t)^{{\rho _i}-1}}{p_k(t)^{\rho _i}}\right) \right. \\ {}&\left. +\sum \nolimits _{i=1}^n \frac{1}{{\rho _i}} \left[ 1 - ({\rho _i}-1)\cdot \ln (K_i - w_i(t)) \right] \cdot (w_i - w_i(t)) \right\} . \end{aligned}$$

Since \(\sum _{k=1}^m b_{ij}+ w_i\) is constant in the domain C, we may ignore any term that does not depend on \(\textbf{b}\) and \(\textbf{w}\), and any positive constant in the objective function and simplify the above update rule to

$$\begin{aligned}&(\textbf{b}(t+1),\textbf{w}(t+1)) = {{\,\mathrm{arg\,min}\,}}_{(\textbf{b},\textbf{w})\in C} \bigg \{ \sum \nolimits _{i=1}^n (1-{\rho _i}) \ln (K_i - w_i(t)) \cdot w_i \\ {}&\left. -\sum \nolimits _{i=1}^n \sum \nolimits _{k=1}^m \left( \ln \frac{(v_k)^{\rho _i}b_{ij}(t)^{{\rho _i}-1}}{p_k(t)^{\rho _i}}\cdot b_{ij}- b_{ij}\ln \frac{b_{ij}}{b_{ij}(t)}+ b_{ij}\right) \right\} \\&\phantom {(\textbf{b}(t+1),\textbf{w}(t+1))} \triangleq {{\,\mathrm{arg\,min}\,}}_{(\textbf{b},\textbf{w})\in C}\overline{F}(\textbf{b},\textbf{w}). \end{aligned}$$

Concerning the partial derivatives of \(\overline{F}\), we have

$$\begin{aligned} \frac{\partial }{\partial b_{ij}} \overline{F}(\textbf{b},\textbf{w})&= \ln b_{ij}- \ln \frac{(v_k)^{\rho _i}b_{ij}(t)^{\rho _i}}{p_k(t)^{\rho _i}} \\ \frac{\partial }{\partial w_i} \overline{F}(\textbf{b},\textbf{w})&=(1-{\rho _i})\ln (K_i - w_i(t)). \end{aligned}$$

As before, for each fixed i, the values of \(\ln b_{ij}- \ln \frac{(v_k)^{\rho _i}b_{ij}(t)^{\rho _i}}{p_k(t)^{\rho _i}}\) for all \(k\in M\) are identical. In other words, there exists \(c_i > 0\) such that \(b_{ij}= c_i \cdot (v_k)^{\rho _i}b_{ij}(t)^{\rho _i}/p_k(t)^{\rho _i}\). As before, there are two cases which depend on

$$\begin{aligned} S_i \triangleq \sum \nolimits _{k\in M} (v_k)^{\rho _i}b_{ij}(t)^{\rho _i}/p_k(t)^{\rho _i}= \sum \nolimits _{k\in M} (v_{ik} b_{ik})^{\rho _i}. \end{aligned}$$

If \(S_i \ge K_i \cdot (K_i - w_i(t))^{{\rho _i}-1}\), then we set \(b_{ij}(t+1) = K_i \cdot (v_{ik} b_{ik}) / S_i\), for each \(k\in M\), and \(w_i(t+1) = 0\). At this point, we have

$$\begin{aligned} \frac{\partial }{\partial b_{ij}} \overline{F}(\textbf{b},\textbf{w}) = \ln \frac{K_i}{S_i} \le \frac{\partial }{\partial w_i} \overline{F}(\textbf{b},\textbf{w}), \end{aligned}$$

so the optimality condition is satisfied. If \(S_i < K_i \cdot (K_i - w_i(t))^{{\rho _i}-1}\), then, we set \(b_{ij}(t+1) = (K_i - w_i(t))^{1-{\rho _i}} \cdot (v_{ik} b_{ik})^{\rho _i}\), for each k, and \(w_i(t+1) = K_i - \sum _{k=1}^m b_{ij}(t+1) > 0\). At this point, we have \(\frac{\partial }{\partial b_{ij}} \overline{F}(\textbf{b},\textbf{w}) = \frac{\partial }{\partial w_i} \overline{F}(\textbf{b},\textbf{w})\), so the optimality condition is satisfied.

Theorem 2 now guarantees that the updates of (PR-QCES) converge to an optimal solution of (SH-QCES). This shows that the (PR-QCES) dynamics converge to the ME of a Fisher market with quasi-CES utilities for any \(0<\rho _i\le 1\) and concludes the proof of Theorem 1. Note that the previous proof cannot be extended in a straightforward way to values of \(\rho _i<0\), since in that case, a direct calculation shows that F is neither convex nor concave function which implies that the above argument does not apply.

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Cheung, Y.K., Leonardos, S., Sridhar, S., Piliouras, G. (2023). Market Equilibria and Risk Diversification in Blockchain Mining Economies. In: Pardalos, P., Kotsireas, I., Guo, Y., Knottenbelt, W. (eds) Mathematical Research for Blockchain Economy. MARBLE 2022. Lecture Notes in Operations Research. Springer, Cham. https://doi.org/10.1007/978-3-031-18679-0_2

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