The Model

Consider a network with \(N\) identical miners, indexed \(i\in\{1,2,\dots,N\}\), competing to be the first to solve a computational puzzle. Each miner \(i\) chooses a computing technology \(x_{i}\in\mathbb{R}_{+}\) at a cost of \(c(x_{i})\) at time \(t=0\) to compete to be the first to solve a cryptographic puzzle. The network determines a difficulty level \(K\in\mathbb{R}_{++}\), representing the threshold number of computations required to solve the puzzle. The network sets its target puzzle solution time as \(\delta^{*}\in\mathbb{R}_{+}\). The difficulty of the puzzle adjusts dynamically to ensure that \(\delta^{*}\) is realized on average. All miners face the same cost function \(c:\mathbb{R}_{+}\to\mathbb{R}_{+}\), where \(c(x_{i})\) is an analytic strictly increasing convex monotone function. A miner \(i\) chooses some technology \(x_{i}>0\) only if they can meet the minimum effort cost required to complete \(K\) computations, otherwise they do not compete and choose \(x_{i}=0\).
\begin{equation} \par X_{i}(t)\sim\text{Poisson}(x_{i})\,,\par \nonumber \\ \end{equation}
The technology for solving computational puzzles is represented by a stochastic Poisson process
where \(x_{i}\) gives the expected number of computations miner \(i\) will complete in a time interval \(t\) given their choice of technology. We assume that the Poisson processes over miners are independent and operate in parallel.
\begin{equation} \par t_{i}\sim\text{Gamma}(K,x_{i})\,.\par \nonumber \\ \end{equation}
The Poisson process \(X_{i}(t)\) gives a random time \(t_{i}\in\mathbb{R}_{+}\) at which \(K\) computations are completed by miner \(i\) that is independent between the miners. The distribution of the random variable \(t_{i}\) is
\begin{equation} \par \gamma_{K,x_{i}}(t_{i})=\frac{t_{i}^{K-1}}{\Gamma(K)}x_{i}^{K}e^{-x_{i}t_{i}}\,,\par \nonumber \\ \end{equation}
We denote the probability density function of this random variable by
where \(\Gamma(\cdot)\) is the Gamma function. The probability that miner \(i\) completes \(K\) computations within time \(t_{i}\) increases with their technology \(x_{i}\).
\begin{equation} \par P-c(x_{i})\,.\par \nonumber \\ \end{equation}
The network determines a prize \(P\in\mathbb{R}_{+}\) won by the first miner to complete the threshold of \(K\) computations. If \(t=(t_{1},t_{2},\dots,t_{N})\) is a realization of the success times of miners and \(t_{i}<t_{j}\) for all \(j\neq i\), then miner \(i\) receives a payoff
\begin{equation} \par W_{i}=\{t\in\mathbb{R}^{N}_{+}:t_{i}<t_{j},~{}\text{for all}~{}j\neq i\}\,.\par \nonumber \\ \end{equation}
All other miners \(j\neq i\) receive a payoff of \(-c(x_{j})\). Let \(W_{i}\subseteq\mathbb{R}^{N}_{+}\) be the set of time realizations for which miner \(i\) is the first to solve the puzzle
The probability of miner \(i\) winning by realizing a time profile \(t\in W_{i}\) given the strategy profile \((x_{i},x_{-i})\) is given by the cumulative density function of the minimum order statistic of the gamma random variable \(t_{i}\), across all \(N\) miners
\begin{equation} \pi(W_{i};K,x_{i},x_{-i}):=\mathbb{P}(t\in W_{i})=\prod_{-i}\left[1-\int_{0}^{t_{i}}\gamma_{K,x_{-}i}(t_{i})~{}dt_{i}\right]\,.\nonumber \\ \end{equation}
We note \(\pi(W_{i};K,x_{i},x_{-i})\) is an infinitely differentiable function.
The expected payoff of a strategy profile \(x=(x_{1},x_{2},\dots,x_{N})\) for miner \(i\), is therefore
\begin{equation} \label{u}U_{i}(x_{i})=P\pi(W_{i};K,x_{i},x_{-i})-c(x_{i})\,=\mathbb{E}(P)-c(x_{i})\,.\\ \end{equation}
Each miner \(i\) chooses the technology \(x_{i}\) that maximizes their expected payoff.
.\label{nasheq}
A Nash equilibrium is an action profile \(x^{*}=(x^{*}_{1},x^{*}_{2},\dots,x^{*}_{N})\) representing the technology for each miner \(i=1,2,\dots,N\) given a fixed difficulty level \(K\) of the computational puzzle if
  1. \(U_{i}(x^{*}_{i},x^{*}_{-i})\geq U_{i}(x_{i},x^{*}_{-i})\) for all miners \(i=1,2,\dots,N\), where \(x^{*}_{i}\neq x_{i}\), and
  2. The expected time required to solve the computational puzzle given \((K,x^{*})\) is \(\delta_{K}\in\mathbb{R}_{+}\), where
\begin{equation} \label{et}\delta_{K}:=\mathbb{E}_{K}(t)=\mathbb{E}_{K}(\min\{t_{1},t_{2},\dots,t_{N}\})\,.\\ \end{equation} .
A symmetric equilibrium is a pair \((K^{*},x^{*})\in\mathbb{R}_{+}^{2}\,,K\geq 1\) such that
  1. \((K^{*},x^{*})\) is a Nash equilibrium for all \(i\) by Definition \ref{nasheq}, and
  2. The expected time of completion is \(\delta^{*}\), where \(\delta^{*}\) is the target solution time set by the network.