Welfare analysis

Monopolist miner

We first consider the monopoly case, where the monopolist can be thought of as the central bank. When \(N=1\), the monopolist miner receives the prize, \(P\), with probability \(1\). Therefore, monopolist’s payoff is
\begin{equation} U_{m}(x_{m})=P-c(x_{m})\,.\nonumber \\ \end{equation} \begin{equation} \delta_{1}=\mathbb{E}_{1}(t)=\frac{1}{x_{m}}\,.\nonumber \\ \end{equation}
To maximize its payoff, the monopolist chooses the minimum amount of technology required to reach a solution. Since the monopolist’s choice of technology influences the dynamic difficulty \(K\) of the puzzle, it chooses the minimum technology required to reach a solution when \(K=1\). If the target time is \(\delta\), then
\begin{equation} x_{m}^{*}=\frac{1}{\delta_{1}}\,.\nonumber \\ \end{equation}
Hence, the monopolist’s optimal technology is
Let \(s_{i}\in[0,1]\) denote the share of currency held by miner \(i\), with \(\sum_{i}s_{i}=1\). If the monopolist is unregulated, then it can determine how the bitcoins are distributed in the economy. However, if \(s_{m}=1\), then the currency becomes worthless in the following period, since no others in the network will trade with it. As the monopolist’s share of the currency increases, the greater the currency depreciates in the following period.
\begin{equation} \max_{s_{m}}s_{m}e^{\prime}(s_{m})\,,\nonumber \\ \end{equation}
Thus, the unregulated monopolist seeks to maximize the value of its share of the currency in the following period
\begin{equation} e^{\prime}(s_{m})+s_{m}\frac{\partial e^{\prime}(s_{m})}{\partial s_{m}}=0\,.\nonumber \\ \end{equation}
where \(e^{\prime}\) represents the exchange rate in the following period. The first order condition which \(s_{m}^{*}\) must satisfy is
The monopolist always maximises with some \(s_{m}\in(0,1)\) provided \(e(s)>0\) for all \(s\in[0,1)\).
\begin{equation} \max_{s_{m}}\,(s_{m}+h^{\prime})e^{\prime}(s_{m})+he\,,\nonumber \\ \end{equation}
Now we suppose that the monopolist is regulated. The problem then becomes
where \(h\) is the future expected share of currency held by the monopolist arising from transaction processing fees.
\begin{equation} e^{\prime}(s)+(s+h)\frac{\partial e^{\prime}(s_{m})}{\partial s_{m}}+he\,.\nonumber \\ \end{equation} \begin{equation} h^{*}=-\frac{e(0)}{\frac{\partial e}{\partial s}}\,>0\,,\nonumber \\ \end{equation}
The first order condition is If the monopolist takes no share of the currency in a given period, it must be the case that
which is positive since \(e(s)\geq 0\) and \(\frac{\partial e}{\partial s}<0\) for all \(s\in[0,1)\). Thus, if the transaction processing fees offered are equal to \(h^{*}\), then the monopolist maximizes by choosing \(s=0\). Since the monopolist is guaranteed a payoff of \(P-c(x_{m})\), it must be the case that this is equal to \(h^{*}\). The cost of the monopolist’s investment in technology at equilibrium is therefore \(c(x_{m})=P-h^{*}\) where \(h^{*}>0\).
If the monopoly is contestable, then the monopolist’s profits from transaction processing fees will be driven down to \(h=0\) by competition. The monopolist’s problem returns to the unregulated case, with \(s_{m}^{*}>0\).

Duopolist miners

Adding another miner allows each only imperfect control over the difficulty of computations. As a miner increases its technology, that difficulty rises. This both makes it harder for them to achieve the required number of computations but also increases their relative probability of winning. This externality is what causes an increased technological choice.
Thus, the first order condition for a miner is
\begin{equation} \par P\frac{\partial\pi}{\partial x_{i}}-\frac{\partial c}{\partial x_{i}}=P\frac{\partial\pi}{\partial\mathbb{E}(t_{i})}\left(\frac{\partial\mathbb{E}(t_{i})}{\partial K}\frac{dK}{dx_{i}}+\frac{\partial\mathbb{E}(t_{i})}{\partial x_{i}}\right)-\frac{\partial c}{\partial x_{i}}\,.\par \nonumber \\ \end{equation}
Miner’s choices impact on the difficulty of computations. For this reason, they will actually have a reduced incentive to invest in technologies for faster computation. While one expects that they will invest more in aggregate than a monopolist would, they will likely earn profits in equilibrium. Expanding \(N\) thus reduces the impact of a miners choice on computational difficulty in equilibrium but also on their own probability of winning.
Choice of technology of miners will increase until equilibrium is reached, where the expected payoff for both miners is zero. Therefore, the aggregate cost of equilibrium investment under duopoly is \(2c(x^{*})=P\).
Given the same prize \(P\), we find \(2c(x^{*})>c(x_{m})\), such that from a cost perspective, mining is most efficient in the monopolist’s case.
In order to seize the system, one of the duopolists must increase their mining investment to increase their probability of winning to secure the prize. From equilibrium where both miners earn zero expected profits, an increase in technological investment results in an expected net loss. However, increasing technology also decreases the expected time required to solve the puzzle below \(\delta^{*}\), in turn increasing the difficulty of the puzzle \(K\). Given the symmetry of the miners, the other duopolist will know that this is not sustainable. Thus, the mining system itself prevents a miner from seizing the system due to its dynamic protocol, as well as the random process underlying the puzzles.
However, in equilibrium for \(N\) miners it will also be the case that the aggregate cost of investment is equal to the prize, since there are zero expected profits. As \(N\) increases at equilibrium, there will more miners participating in the race but each with less technology, meaning that at a given difficulty level \(K\), the puzzle takes longer to complete until the Bitcoin protocol adjusts \(K\) such that \(\delta^{*}\) is realized. As such, we have seen the formation of mining pools in the network, whereby individual miners are able to combine their technology to achieve faster computations, and thus, a higher probability of winning.