1 Introduction

With the advancement of the blockchain technologies (a.k.a., distributed ledger technology), distributed applications (DApps) are burgeoning. Starting from the bitcoin, many altcoins have been proposed to achieve different goals. At the time of this writing, there are almost 1,600 cryptocurrencies with market capitalization totaling approximate $456 billion^[1], and among them Bitcoin, Ethereum and Ripple are the top three market-caped crypto-currencies.

Miners in these peer-to-peer networks play the important role of maintaining the integrity of the underlying blockchains, incentivized to earn digital currencies and transaction fees. Mining involves executing a distributed consensus protocol on how to achieve agreement of the underlying ledger when there is no central authority in presence. Among them, proof-of-work (PoW)^[1] and proof-of-stake (PoS)^[2] are the widely adopted consensus protocols by existing crypto-currencies.

For example, in the PoW framework, during a given average time period (e.g., every 10 minutes for Bitcoin), miners participate in a winner-take-all competition to extend the next block on the longest block chain by solving some cryptographic hashing proof-of-work, a mathematical puzzle. As a concrete case, in Bitcoin network, the puzzle goes as follows^[3]: Given a difficulty d > 0, a challenge c and a nonce x (usually bit-strings), a function

is called a Proof-of-Work (PoW) function if it has the following two properties: (i) Fd(c, x) is fast to compute, given d, c, and x; and (ii) for fixed parameters d and c, finding x such that Fd(c; x) = TRUE is computationally difficult but feasible. The difficulty d is used to adjust the time to find such an x.

With miners equipped with certain computing power (a.k.a., hashing power in Bitcoin network) and a large number of different cryptocurrencies to mine, they are facing the challenge on how to allocate their computing power to compete in mining each cryptocurrency to maximize their expected payoffs. Due to the competitive nature of the mining protocol, all miners together form a non-cooperative allocation game. This work aims to answer the following questions associated with the aforementioned game:

1. Does Nash Equilibrium (NE) exist?

2. Is NE unique?

3. Can the NE be computed efficiently?

We offer affirmative answers to all three questions for our game. We show that the NE allocation is unique and follows a proportional rule (Theorem 1) where each miner allocates his total computing power to a given cryptocurrency proportional to the percentage of the award among all currencies, while his expected revenue is proportional to the percentage of the hashing power possessed and the total award.

The equilibrium analysis of the allocation game is of both theoretical and practical relevance. On the theoretical side, we set up a succinct backbone model which admits a closed-form solution via non-trivial technical analysis. On the practical side, we provide insights which can help mining pool managers or individual miners in making the most important operational/ tactical decisions, namely how to allocate the hashing power when facing under reward and peer competition.

To filter out the most salient factors that are of managerial relevance, we made some simplifications in the modelling, such as the the assumption that the cost to purchase certain hashing power is independent from the price of the currencies. However, this type of deviations from the realism on one hand may be a good approximation to reality and on the other hand is to be expected in an early attempt to apprehend an otherwise complex problem. Also, this work focuses on static games, and leave the discussion of dynamic games to future research.

2 Relevant Literature

Our computing power allocation game is relevant to several areas, including game theory, portfolio management, and market equilibrium models.

Several blockchain games (mainly non-cooperative in nature) are proposed in the recent literature to address and improve upon the limitations of existing distributed consensus mechanism in various crypt-currencies^{[4, 5, 6, 7, 8]}, while some other games (mainly non-cooperative in nature) focus on the application layer without invoking any protocol technicality, such as the mining pool games^{[7, 9, 10, 11, 12]}. Our computing power allocation game is non-cooperative and focuses on the application layer; namely the allocation of mining resource. Furthermore, these games all deal with a single currency, which is a major difference from the game investigated in this work. Dimitri^[13] studied a special case of our game where there is only one currency available for mining. [14] is a survey on this line of research.

Our computing power allocation game is similar to the extensively-studied general blotto game in the game theory literature^{[15, 16, 17, 18]}, but the two models have completely different utility functions to suit different applications in mind.

The resource allocation nature is also relevant to the large literature on portfolio management^[19], and the market equilibrium model, in particular the Fisher market^{[20, 21]}. However, the portfolio management literature usually assume that the supply of assets is independent from the allocation decision. And the Fisher market models focus on finding market-clearing prices and the allocation rule at market equilibrium.

The readers are referred to the survey^[22] for research perspectives and challenges for Bitcoin and cryptocurrencies.

We describe the the hashing power allocation game in Section 3. We consider games without and with a risk-free asset in Sections 4 and 5, respectively.

3 The Hashing Power Allocation Game

There are n miners N = {1, 2, · · · , n} with computing powers h = (h₁, h₂, · · · , h_n)^T∈R+n(the cost to possess such a computing power, expressed in fiat currency such as US dollar) and there are m cryptocurrencies M = {1, 2, · · · ,m} available for mining. When there is a risk-free asset with return r, we introduce a dummy labeled as currency 0.

Miner i ∈ N allocates x_ij ≥ 0 of his computing power to mine cryptocurrency j ∈ M. Let x_i0 be the hashing power allocated to the dummy risk-free asset whose return is known as r for any amount allocated. Evidently, we have

For each cryptocurrency j ∈ M, the n miners play a winner-take-all game and the winner is rewarded with uncertain reward vector R = (R₁,R₂, · · · ,R_m)^T (expressed in fiat currency such as US dollar) with mean vector E[R] = μ^T = (μ₁, μ₂, · · · , μ_m)^T. Miner i ∈ N wins cryptocurrency j ∈ M with probability proportional to its allocated computing power

and his profit for mining cryptocurrency j ∈ M is given by

Moreover the profit obtained with the risk-free asset is given by

Therefore miner i’s total profit is given by

where x=x1x2⋯xnT=xijn×m∈R+n×mand

The mean profit of miner i is given by

4 Game without Risk-free Asset

We recall the result without a risk-free asset. When miners allocate all their funds to mining, the NE for the miners game can be obtained by solving the following n optimization problems based on (3): For any given i ∈ N,

Theorem 1 (see [23]) Assume that ∑1mμℓ≥∑1nhℓ.The hashing power game with risk-neutral miners (4) admits the following unique NE

along with each miner i’s expected profit

This result was obtained in our conference proceedings^[23]. However the proof therein contains some errors. In this journal version we corrected all errors and include the full proof of Theorem 1 in the appendix.

5 Game with a Risk-free Asset

The main contribution of this work is to include a risk-free asset. When miners have the option to allocate their funds to both mining and the risk-free asset, the NE for the miners game can be obtained by solving the following n optimization problems based on (3): For any given i ∈ N,

Let ¯h_i = h_i − x_i0. This problem is equivalent to the following parametric problem:

Using Theorem 1, the last problem is reduced to the following optimization problem

Theorem 2 Assume that h₁≤ h₂≤ · · · ≤ h_n. The solution to problem (7) is as follows.

1.  If ∑1n−1hℓ⋅∑1muj∑1nhℓ2≥1+r,then ¯h_i = h_i for i = 1, 2, · · · , n. This case includes all thoseminers who spend their entire cash.

2.  If n−1n2⋅∑1muj<(1+r)h1, then h¯i=h^=n−1(1+r)n2∑1muj,i=1,2,⋯,n.This caseincludes all those miners who invest the same amount on the cryptocurrencies and invest the remaining amounts into the bond, earning an interest of r, respectively.

3. Otherwise, there exists an index 1 ≤ k ≤ n − 1 such that

Find h_k ≤ ˆh < h_k+1 such that

then

In this case, miners are divided into two classes: The first class {1, 2, · · · , k} contain those miners who spend their entire cash; and the second class {k+1, k+2, · · · , n} contains all those miners who invest the same amount on the cryptocurrencies and invest the remaining amounts into the bond, earning an interest of r, respectively.

The hashing power game with a risk-free asset (5) admits the following unique NE

along with each miner i’s expected payoff

Proof Let fih¯i=h¯i∑1mμℓ∑1nhℓ−(1+r)h¯i, then fi′h¯i=∑1nh¯ℓ−h¯i∑1nhℓ2⋅∑1mμℓ−(1+r).If ∑1n−1hℓ⋅∑1muj∑1nhℓ2≥1+r , then fi′h¯i≥0 at h¯i=hi for i=1,2,⋯,n,so the solution to (7) is _i = h_i for i = 1, 2, · · · , n. Otherwise, some or all ¯h_i =ˆh at the solution (7) so that fi′h¯i≥0for i = 1, 2, · · · , n. The conclusion is obtained.

We make the following observations based on Theorem 2:

1. At the NE, the return keeps the same for all miners, that ∑1mμℓ∑1nhℓ−(1+r).

2. At the NE, the marginal profit fi′h¯i=∑1nh¯ℓ−h¯i∑1nh¯ℓ2⋅∑1mμℓ−(1+r)is larger for miner i with smaller h¯i.

3. Miner i’s expected profit h¯i∑1nh¯ℓ⋅∑1mμℓ−(1+r)h¯i+rhiincludes two parts. Part rh_i is the risk-free return for h_i. Part h¯i∑1nh¯ℓ⋅∑1mμℓ−(1+r)h¯iis the excess return.

4. The total social welfare is increasing. The rate of increasing is larger than r.

6 Conclusion

Many future research problems are worth pursuing, such as the risk-averse miners and dynamic versions of the game.