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Bitcoin, blockchain, selfish mining, proof-of-work
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\title{On profitability of selfish mining}

\subjclass[2010]{68M01, 60G40, 91A60.}
\keywords{Bitcoin, blockchain, selfish mining, proof-of-work.}


\author[C. Grunspan]{Cyril Grunspan}
\address{Cyril Grunspan\newline{}\indent L\'eonard de Vinci P\^ole Univ, Finance Lab, Labex R\'efi\newline{}\indent Paris, France, }
\email{cyril.grunspan@devinci.fr}
\author[R. P\'{e}rez-Marco]{Ricardo P\'{e}rez-Marco}
\address{Ricardo P\'{e}rez-Marco\newline{}\indent CNRS, IMJ-PRG, Labex R\'efi \newline{}\indent Paris, France}
\email{ricardo.perez.marco@gmail.com}
\address{\tiny Author's Bitcoin Beer Address (ABBA)\footnote{\tiny Send some bitcoins to support our research at the pub.}:\newline{}\indent 1KrqVxqQFyUY9WuWcR5EHGVvhCS841LPLn} 

\address{\includegraphics[scale=0.5]{natoshi}}


\begin{document}

\begin{abstract}
  We review the so called selfish mining strategy in the Bitcoin network and properly evaluate the cost of the attack and its profitability. 
  The expected duration of the attack has been ignored in the literature but is critical. We prove that such strategy 
  can only be profitable after a difficulty adjustment. Therefore, it is an attack on the 
  difficulty adjustment algorithm. We propose an improvement of Bitcoin protocol making it inmune to a selfish mining attack.
\end{abstract}

{\maketitle}

\section{Introduction}

The stability of Bitcoin protocol \cite{N08} relies on rules aligned with self-interest of participants in the network. One 
of its basic rules is that miners make public blocks as soon as they are validated. 
``Selfish Mining'' is a deviant mining strategy described in {\cite{ES14}} where a major mining 
operator withholds mined blocks and releases them with a well timed strategy in order to invalidate the maximum number of blocks 
mined by the rest of the network. The authors of {\cite{ES14}} make the unfounded claim that the selfish mining strategy breaks 
the Bitcoin protocol {\cite{ES13}} which has not been observed in practice.

\medskip

Other researchers have proposed other 
selfish mining strategies which are supposed to be ``optimal'' {\cite{SSZ16}}. The selfish mining strategy is 
presented in courses and textbooks on Bitcoin such  as  {\cite{BFGMN16}} (in chapter 5 under 
the name ``Temporary Block-Withholding Attack'') or {\cite{W17}}.

\medskip

All these articles do not make a proper analysis of the costs of the attack, and, more critically, do 
ignore time considerations. More precisely, 
the Markov model used in these papers is flawed by inception since it does not incorporate an analysis of 
the time duration of the attack. The main goal of this article 
is to carry out a proper analysis of profitability that is lacking in the literature. It turns out that without difficulty adjustments 
the strategy is unsound. 

\bigskip

\section{Selfish mining strategy.} \label{sec_SM}

We describe the selfish mining strategy presented in  {\cite{ES14}}. The selfish miner attack starts by validating 
and not broadcasting a block, then continuing mining secretly on top of this block. Then he proceeds as follows:


\begin{enumerate}
  \item \label{case1} If the advance of the selfish miner is only equal to $1$ block and the
  honest miners discover a block then the selfish miner broadcasts immediately
  the block he has mined secretly. A competition then follows. The selfish miner 
  is assumed to be sufficiently well connected with the rest of the network
  so that a fraction $0\leq\gamma \leq 1$ of the honest network accepts his block
  proposal and starts mining on top of it.
  
  \medskip
  
  
  \item \label{compet}If the advance of the selfish miner is $2$ blocks and
  the honest miners discover a block, then the selfish miner broadcasts
  immediately the two blocks he had mined secretly. Then, the whole network
  switches to his fork.
  
  \medskip
  
  \item \label{case3} If the advance of the selfish miner is greater than $2$, then the
  selfish miner releases a block as soon as the honest miners discovers one.
  
  \medskip
  
  \item In other cases, the selfish miner keeps on mining secretly on top of his
  fork.
\end{enumerate}

Note that if the advance of the miner is greater than $2$, then at some
point the advance of the selfish miner will be equal to $2$ (because we assume that his
hashrate is less than $50\%$, otherwise other more efficient attacks are possible) and 
then, according to the second point, the whole
network ends up adopting the fork proposed by the selfish miner.
Therefore, the blocks made public by the selfish miner when his advance is
greater than $2$ always end up being accepted by the network.

\


In  {\cite{ES14}} it is assumed that the fraction $\gamma$ stays always constant.
This is not accurate since $\gamma$ depends on the timing of the discovery of a new block mined 
by the network, and therefore cannot be constant.
But it is necessary to assume it constant for the sake of the Markov chain model presented in {\cite{ES14}} 
and these authors made such assumption.  
The analysis of the necessary capital to 
reach a stable regime is not done. But, more importantly, the cost of deviating from 
the Bitcoin protocol are not properly accounted. This is fundamental in order to compute the profitability of such a rogue 
strategy. The time analysis is also critical to estimate the profitability and is ignored.

\

\section{Profitability of selfish vs. honest mining.}

\subsection{Profitability in the literature.}
Most of the articles on selfish mining strategies fail to consider, explicitely or implicitely, the cost of mining.
In particular this is flatly ignored in  {\cite{ES14}}. In other articles like {\cite{CKNW16}} we can read surprising 
statements as:

\medskip

\begin{quote}
  \textit{\textbf{Costs.} There is a configurable parameter $c_m$ for every miner
  $m$ that denotes the cost (i.e. in electricity) for miner $m$ to mine. For
  our simulations, we always set $c_m = 0$ because we aren't looking at this
  aspect of mining.}
\end{quote}

\medskip

Obviously if mining blocks was without any energy cost, anyone could try to
reverse the history of bitcoin transactions and try double spend attacks, and 
there will be no economic incentive to respect the Bitcoin protocol. The cost of energy 
is what makes expensive to falsify the blockchain, and this is at the base of Bitcoin security.
  
\medskip

Other authors, like in \cite{But13}, present Selfish Mining in these terms:

\textit{``This [Selfish Mining] does reduce the attacker's revenue in the short term,
  but it reduces everyone else's revenue even more, so neutral nodes now have
  the incentive to join the attacker's coalition to increase their own
  revenue. Eventually, the attacker's coalition would expand to above 50\% in
  size, potentially giving the attacker a high degree of control over the
  network.''}


\medskip

In  other words, according to this interpretation, a miner who looses money would be attracted to join another miner who also looses money. And
this only because the attacker miner persuades the honest miner that he is loosing less money (!).  Aside from the 
fact that the honest miners will be unable to know for sure the difference of profitability, this naive 
scenario is very implausible. 

From this point of view, selfish mining is justified only as a preparation of a 51\%
attack to take control of the network. A more likely outcome, is that the honest pools noticing someone selfish mining, will employ the obvious 
defensive strategy and start 
to selfish mining themselves making the network stall. This outcome will be highly non profitable for everyone. A form of Nash equilibrium 
is at play that avoids the described scenario.

\medskip

Aside from these creative interpretations, what is really laking in the literature is a proper accounting and understanding 
of the cost of selfish mining attack, with a proper time analysis.


\medskip

\subsection{Cost of mining.}
The key idea in order to evaluate properly the profitability of selfish mining is to compare it to the profitability 
of honest mining. We assume that miners are in the mining business because the operating cost (equipment, energy power, salaries, etc)
are compensated by the block reward of newly generated bitcoins plus transaction 
fees\footnote{There may be other non-economic incentives, as transforming non-internationally 
circulating currency into bitcoins, that we cannot consider.}. 
Therefore the cost of mining per unit time is independent of the strategy if the hashrate is operate at full capacity. 
So we have $C_S=C_H$ where $C_S$, resp. $C_H$, is the cost of mining 
per unit time for the selfish, resp. honest, mining strategy. We denote $C_0=C_S=C_H$. 
Note that this cost is not only independent of the strategy but also
of the block mined being accepted or not by the network.

\medskip

\subsection{Profit and Loss.} Our goal is to evaluate the profit and loss (PnL) and compare the PnL of selfish mining 
and honest mining. Profit and loss
is the revenue $R$ minus the costs $C$,
$$
PnL=R-C \ .
$$
We take a conservative, but sufficient, setup where block 
rewards are reduced to the bounty $b>0$ of newly generated bitcoins (thus we don't consider transactions fees, and maybe double spend attempts 
that may increase the profitability). Unless otherwise stated, we assume that we are not near a halving, so that the reward $b$ stays constant. 

What is important, for any business, is the $PnL$ per unit of time, and not the $PnL$ per block solved or accepted by the network. It is worth 
noting that PnL per block or per unit of time are not equivalent since the strategy employed does delay the speed of validation of blocks in 
the network. 


\subsection{Profit and Loss per unit of time.} A non-stop attack strategy, as selfish mining, consists in a consecutive sequence of ``attack cycles''. 

In the 
case of selfish mining it starts when both selfish miners and honest miners are working on top of the same public blockchain. The selfish miners
aim to getn an advance of their secretly mined blockchain. If, at the beginning, the honest miners find the new block, then a new cycle starts. If
the selfish miners succeed in building an advantage, then the cycles lasts until the honest miners catch-up and force the selfish miners to releases
all of their secretly mined blocks. Then a new cycle starts. 

\medskip

For this type of strategies of games with repetition,  the asymptotic $PnL$ per unit of time, $PnLt_\infty$, can be evaluated. This is the content of the following Theorem. 

\bigskip

\begin{theorem}[Profitability of an attack]
 Let $R$, $C$ and $T$ be the random variables corresponding respectively to the revenue, cost and duration of a cycle for a repetition strategy. We assume 
 these random variables to be integrable. Then in the long run the $PnL$ per unit 
 of time of the repeated strategy is
 $$
 PnLt_\infty =\frac{\EE[R]-\EE[C]}{\EE[T]} \ .
 $$
 We call $PnLt_\infty$ the profitability of the strategy.
\end{theorem}


\begin{proof}
Let $R_i$, $C_i$ and $T_i$, be the corresponding values for the $i$-cycle. The $(R_i)$ (resp. $(C_i)$, $(T_i)$) are i.i.d. random variables.
The $PnL_n$ after $n$-cycles is given by
$$
PnL_n = \frac{\sum_{i=1}^n R_i -\sum_{i=1}^n C_i}{\sum_{i=1}^n T_i} = \frac{\frac{1}{n}\sum_{i=1}^n R_i -\frac{1}{n}\sum_{i=1}^n C_i}
{\frac{1}{n}\sum_{i=1}^n T_i}\ .
$$
By the strong law of large numbers \cite{R12} we have that almost surely
$$
\lim_{n\to +\infty } PnL_n = \frac{\EE[R]-\EE[C]}{\EE[T]} \ .
$$
\end{proof}

We consider integrable games with random variables $R$, $C$, and $T$ are all integrable.

\begin{definition}[Integrable games]
A game or strategy is integrable when $R$, $C$, and $T$ are integrable.
An integrable game or strategy has constant cost if the cost per unit time is a constant $C_0$, then
$$
\EE[C] =C_0 \EE[T] \ .
$$
\end{definition}

We will see below that the selfish mining strategy is integrable and has constant cost. We have:

\begin{corollary}
 For an integrable constant cost strategy we have
$$
PnL_\infty = \frac{\EE[R]}{\EE[T]}-C_0 \ .
$$
\end{corollary}

Therefore, in order to compare two integrable constant cost strategies we only need to compare $\frac{\EE[R]}{\EE[T]}$.

\begin{definition}
The \textit{revenue ratio} of a strategy $\xi$ is defined as
$$
P(\xi) =\frac{\EE[R]}{\EE[T]} \ .
$$
The \textit{revenue ratio} is the benchmark for profitability of integrable constant cost strategies, more precisely, 
strategy $\xi$ is more profitable than strategy $\xi'$ if and only if
$$
P(\xi')\leq P(\xi) \ .
$$
\end{definition}

\medskip

In the article {\cite{ES14}} the authors only consider a ``relative revenue''. This is a non-standard 
notion in accountability that they define as the ratio
\begin{eqnarray*}
 \frac{\mathbb{E} [R_S]}{\mathbb{E} [R_S]+\mathbb{E} [R_N]}
\end{eqnarray*}
where $R_S$ and $R_N$ are the revenues of the selfish miners and the rest of the network. 
The effect of the selfish mining strategy 
is to reduce both $\mathbb{E} [R_S]$ 
and $\mathbb{E} [R_N]$, and to increase this relative revenue on certain conditions on $q$ and $\gamma$. 
Obviously, increasing the relative revenue 
at the cost of reducing its own revenue is not a sound strategy. Also discussing profitability via a Markov model without 
time duration consideration of the strategy is unsound. The time dynamics is absent in the Markov model, and cannot take 
account situations where the expected time of some attack cycles may take much longer which will impact the $PnLt$.


\section{Mining strategies.}

Let us fix some notations. The reward per block is $b>0$. We have two sets of miners (for example, honest miners and attacking miners for an attack). 
The progression of blocks are described by two independent Poisson processes $N$ and $N'$ respectively, 
with parameter $\alpha$, resp. $\alpha'$.
Interblock valdation times are denoted by $(T_i)$ for the honest miners and $(T'_i)$ for the selfish miner. We denote 
for $n\geq 1$
\begin{align*}
 S_n &= T_1+T_2+\ldots +T_n  \ ,\\
 S'_n &= T'_1+T'_2+\ldots +T'_n \ .\\
\end{align*}
We recall (see \cite{GPM17}) that the random variables $(T_i)$ (resp. $(T'_i)$) follow an exponential density with parameter $\alpha$ (resp. $\alpha'$) 
and the random variable $S_n$ (resp. $S'_n$) follows a gamma density distribution with parameter $(n,\alpha)$ (resp. $(n,\alpha')$). We denote 
\begin{align*}
 \tau_0 &= \frac{1}{\alpha + \alpha'} \ ,\\
 p &= \frac{\alpha}{\alpha + \alpha'} \ ,\\
 q &= \frac{\alpha'}{\alpha + \alpha'} \ ,
\end{align*}
We have $p+q=1$ and if $\alpha'<\alpha$ then $0<q<1/2<p<1$ (and $\tau_0 =10 \text{ min}$ for 
the Bitcoin network in normal conditions). The quantities $p$ and $q$ 
represent relative hashrates, and also probabilities of finding the next block by each group 
of miners as the following elementary computations show \cite{R12}:


\begin{lemma}
 We have
 \begin{align*}
  \PP [T_1 < T'_1] &= p \ , \\
  \PP [T'_1 < T_1] &= q \ , \\
  \PP [T'_1 <T_1<S'_2] &= pq \ ,\\
  \EE[T_1\wedge T'_1] &= \tau_0 \ .
 \end{align*}
\end{lemma}

\subsection{Honest strategy.}

The cycle for the honest strategy last until a block is found. So the stopping time for the honest strategy is
$$
\tau_H = T_1'\wedge T_1 \ .
$$
We have $\EE[\tau_H] =\tau_0$. 

The reward in a cycle is $0$ or $b$ depending who is the miner, thus 
$$
\EE[R(\tau_H)] =p\cdot 0+q\cdot b =qb\ ,
$$
and we have,

\begin{theorem} \label{honest}
We have that $\tau_H$ and $ R (\tau_H)$ are integrable and 
\begin{align*}
\EE [R (\tau_H)] &= qb  \ ,\\
\EE [\tau_H]  &= \tau_0 \ .
\end{align*}
Therefore,
$$
P(H)=\frac{\EE [R (\tau_H)]}{\EE [\tau_H]} = \frac{qb}{\tau_0} \ .
$$
\end{theorem}

\subsection{Selfish mining strategy}

We consider now the selfish mining strategy as described in Section \ref{sec_SM}. We assume that the hashrate of the attackers is less than that 
of the honest miners (i.e. $\alpha'<\alpha$).
We denote by $\tau_{SM, \gamma}$ the duration time of an attack cycle.

\begin{lemma}
We have
\begin{align*}
  \tau_{SM, \gamma} & = \inf  \{ t\geq T_1 ; N(t) = N'(t) -1 + 2  \cdot \boldsymbol{1}_{T_1 < T'_1} +   2  \cdot \boldsymbol{1}_{T'_1 < T_1<S_2<S'_2} \} \ ,
\end{align*}
and the stopping time $\tau_{SM, \gamma}$ is finite  almost surely.
\end{lemma}

\begin{proof}
Note that if $T_1 < T'_1$, then $ \tau_{SM, \gamma} = T_1$. If $T'_1 < T_1<S_2<S'_2$ then $ \tau_{SM, \gamma} = S_2$. 
If  $T'_1 < T_1<S'_2<S_2$ then $ \tau_{SM, \gamma} = S'_2$. Otherwise we have $S'_2 \leq T_1$ and $N(T_1) = 1 \leq N'(T_1) -1$, and in that case 
 $\tau_{SM, \gamma} = \inf  \{ t\geq T_1 ; N(t) = N'(t) -1 \}$ exists and is finite almost surely since $\alpha' < \alpha$.  
\end{proof}

\begin{theorem}\label{profitsmg}
 We have that $\tau_{SM, \gamma}$ and $R (\tau_{SM, \gamma})$ are integrable, and
 \begin{align*}
\EE [R (\tau_{SM, \gamma})] &=  \frac{(1+pq)(p - q) + pq}{p - q}\, qb -(1-\gamma) p^2q  \, b \ ,\\
\EE [\tau_{SM,\gamma}]  &= \frac{(1+pq) (p - q) + pq}{p - q} \, \tau_0  \ .
\end{align*}
Therefore,
$$
P(SM,\gamma)=\frac{\EE [R (\tau_{SM, \gamma})]}{\EE [\tau_{SM, \gamma}]} =
  \frac{qb}{\tau_0} -  (1 - \gamma) \frac{p^2 q (p - q)  b}{\left((1+pq)(p - q) + pq \right) \tau_0}\ .
$$
\end{theorem}


\begin{corollary}
  For $\gamma <1$, we have that 
  $$
  P(SM,\gamma)=\frac{\EE [R (\tau_{SM, \gamma})]}{\EE [\tau_{SM, \gamma}]} < \frac{qb}{\tau_0} = P(H) \ ,
  $$
  so, the Selfish Mining strategy with $\gamma <1$ is strictly less profitable than the honest strategy. 
\end{corollary}

\begin{proof}[Proof of the Theorem]
For any $t_0\in \RR$, the stopping time $\tau_{SM, \gamma} \wedge t_0$ is bounded. Moreover, the compensated Poisson process 
$N(t) - \alpha t$ (resp. $N'(t) -\alpha' t$) is a well known martingale. So, using Doob's theorem \cite{R12}, we have:
\begin{align*}
&\alpha' \EE [\tau_{SM, \gamma} \wedge t_0]  =  \EE [N' (\tau_{SM, \gamma} \wedge t_0)]\\
  &=   \EE [N' (\tau_{SM, \gamma} \wedge t_0) | \tau_{SM, \gamma} < t_0] \cdot \PP [\tau_{SM, \gamma} < t_0] + 
  \EE [N' (\tau_{SM, \gamma} \wedge t_0) | \tau_{SM, \gamma} > t_0] \cdot \PP [\tau_{SM, \gamma} > t_0] \\
   &=  \EE [N' (\tau_{SM, \gamma}) | \tau_{SM, \gamma} < t_0] \cdot \PP [\tau_{SM, \gamma} < t_0]
  +\EE [N' (t_0) | \tau_{SM, \gamma} > t_0] \cdot \PP [\tau_{SM, \gamma} > t_0]\\
   &=   \EE [N (\tau_{SM, \gamma}) + 1 - 2 (\boldsymbol{1}_{T_1 < T'_1}
  +\boldsymbol{1}_{S'_1 < S_1 < S_2 < S'_2}) | \tau_{SM, \gamma} < t_0] \cdot \PP [\tau_{SM, \gamma} < t_0] \\
  & \ \ \ + \EE [N' (t_0)] \cdot \PP [\tau_{SM, \gamma} > t_0] \\
  &=  \EE [N (\tau_{SM, \gamma}) | \tau_{SM, \gamma} < t_0] \cdot \PP [\tau_{SM, \gamma} < t_0]
  +\PP [\tau_{SM, \gamma} < t_0] \\
  & \ \ \  +  \EE [N' (t_0)] \cdot \PP [\tau_{SM, \gamma} > t_0]  - 2 \cdot \EE [\boldsymbol{1}_{T_1 < T'_1}
  +\boldsymbol{1}_{S'_1 < S_1 < S_2 < S'_2} | \tau_{SM, \gamma} < t_0] \cdot \PP [\tau_{SM, \gamma} <
  t_0]\\
  & =  \EE [N (\tau_{SM, \gamma} \wedge t_0)] -\EE [N (\tau_{SM, \gamma} \wedge t_0) |
  \tau_{SM, \gamma} > t_0] \cdot \PP [\tau_{SM, \gamma} > t_0] +\PP [\tau_{SM, \gamma} < t_0] \\
  & \ \ \ -  2\cdot \EE [\boldsymbol{1}_{T_1 < T'_1} +\boldsymbol{1}_{S'_1 < S_1 < S_2
  < S'_2} | \tau_{SM, \gamma} < t_0] \cdot \PP [\tau_{SM, \gamma} < t_0] + \alpha' t_0 \cdot
  \PP [\tau_{SM, \gamma} > t_0]\\
  & =  \alpha \EE [\tau_{SM, \gamma} \wedge t_0] -\EE [N (t_0)] \cdot
  \PP [\tau_{SM, \gamma} > t_0] +\PP [\tau_{SM, \gamma} < t_0] \\ 
  & \ \ \  -  2 \cdot \EE [(\boldsymbol{1}_{T_1 < T'_1} +\boldsymbol{1}_{S'_1 < S_1 <
  S_2 < S'_2}) \cdot \boldsymbol{1}_{\tau_{SM, \gamma} < t_0}] + \alpha' t_0 \cdot \PP
  [\tau_{SM, \gamma} > t_0]
\end{align*}
and,
\begin{align*}
  (\alpha' - \alpha) \EE [\tau_{SM, \gamma} \boldsymbol{1}_{\tau_{SM, \gamma} > t_0}] & =  
  (\alpha' - \alpha) \EE [\tau_{SM, \gamma} \wedge t_0] - (\alpha' - \alpha) t_0
  \cdot \PP [\tau_{SM, \gamma} > t_0]\\
  &=\PP [\tau_{SM, \gamma} < t_0] - 2\cdot\EE [(\boldsymbol{1}_{T_1 < T'_1}
  +\boldsymbol{1}_{S'_1 < S_1 < S_2 < S'_2}) \cdot \boldsymbol{1}_{\tau_{SM, \gamma} < t_0}]
\end{align*}
The monotone convergence theorem implies that $\EE [\tau_{SM, \gamma}]$ if finite
and
\begin{align*}
  (\alpha' - \alpha) \EE [\tau_{SM, \gamma}] & =  1 - 2\cdot \EE
  [(\boldsymbol{1}_{T_1 < T'_1} +\boldsymbol{1}_{S'_1 < S_1 < S_2 < S'_2})]\\
  & =  1 - 2 (p + p^2 q)
\end{align*}
This gives
$$
\EE [\tau_{SM,\gamma}]  = \frac{(1+pq) (p - q) + pq}{p - q} \, \tau_0   \ .
$$
Also we have
\begin{align*}
  \EE [N' (\tau_{SM, \gamma}) \cdot \boldsymbol{1}_{\tau_{SM, \gamma} < t_0}] & = \EE [N'
  (\tau_{SM, \gamma}) | \tau_{SM, \gamma} < t_0] \cdot \PP [\tau_{SM, \gamma} < t_0]\\
  & = \EE [N' (\tau_{SM, \gamma} \wedge t_0)] -\EE [N' (\tau_{SM, \gamma} \wedge t_0) |
  \tau_{SM, \gamma} > t_0] \cdot \PP [\tau_{SM, \gamma} > t_0]\\
  & = \alpha' \EE [\tau_{SM, \gamma} \wedge t_0] -\EE [N' (t_0)] \cdot
  \PP [\tau_{SM, \gamma} > t_0]\\
  & = \alpha' \EE [\tau_{SM, \gamma} \wedge t_0] - \alpha' t_0 \cdot \PP
  [\tau_{SM, \gamma} > t_0]\\
  & = \alpha' \EE [\tau_{SM, \gamma} \cdot \boldsymbol{1}_{\tau_{SM, \gamma} < t_0}]
\end{align*}
So, by the monotone convergence theorem again, we get $\EE [N' (\tau_{SM, \gamma})]
= \alpha' \EE [\tau_{SM, \gamma}]$. In the same way, $\EE [N (\tau_{SM, \gamma})] = \alpha
\EE [\tau_{SM, \gamma}]$.


 Finally, we observe that at the end of an attack cycle, the selfish miner has no orphan block 
 unless $T'_1 < T_1 < S_2 < S'_2$ and the second block of the honest miners is found by a fraction $1- \gamma$ of all 
 honest miners. This event occurs with a probability $p^2 q (1-\gamma)$ and, in this case the selfish, miner has exactly one orphan block. 
Therefore, we have 
\begin{align*}
  \EE [R(\tau_{SM, \gamma})] &= \EE \left[ N'\left( \tau_{SM, \gamma} \right)\right] b- p^2 q (1-\gamma) b\\
  &= \alpha' \EE [\tau_{SM, \gamma}]b  - p^2 q (1-\gamma)b
\end{align*}
and we get 
$$
\EE [R (\tau_{SM, \gamma})] =  \frac{(1+pq)(p - q) + pq}{p - q}\, qb -(1-\gamma) p^2q  \, b  \ .
$$

\end{proof}
\begin{remark*}
The introduction of $t_0$ in the above proof is a technical point so that we can use Doob's stopping time theorem.  
\end{remark*}

\subsection{Poisson games.}
Theorem \ref{profitsmg} is a variation of the following result. 
\begin{theorem}[Poisson games]
Let $N_1$ and $N_2$ be two independent Poisson processes with parameters $\alpha_1$ and $\alpha_2$ with $\alpha_1 < \alpha_2$ and $N_1(0) = N_2(0) = 0$. 
Then, the stopping time 
$$
\tau = \inf \{ t>0 ; N_1(t) = N_2(t) + 1\}
$$ 
is finite a.s. and integrable. Moreover, we have $\EE [\tau] = \frac{1}{\alpha_1 - \alpha_2}$,
$\EE[N_1(\tau)] =  \frac{\alpha_1}{\alpha_1 - \alpha_2}$, $\EE[N_2(\tau)] =  \frac{\alpha_2}{\alpha_1 - \alpha_2}$. 
\end{theorem}



\subsection{Apparent hashrate}
As discussed before, the good notion for accounting is the revenue of the miner by unit of time. However, we can also compute the revenue of the miner by unit of block. 
We compute the proportion $q'$ of blocks mined by the selfish miner in the official blockchain. This represents the apparent hashrate of the selfish miner.

\begin{proposition}\label{qprime}
We have $q' = \frac{((1 + pq) (p - q) + pq) q - (1 - \gamma) p^2 q (p - q)}{p^2 q + p - q}$
\end{proposition}

\begin{proof}

In all cases we observe that after one cycle of attack, the number of official blocks mined is always $\frac{N(\tau_{SM, \gamma}) + N'(\tau_{SM, \gamma}) + 1}{2}$. 
So, by the proof of Theorem \ref{profitsmg}, we get

\begin{align*}
  \EE \left [\frac{N(\tau_{SM, \gamma}) + N'(\tau_{SM, \gamma}) + 1}{2}\right ]  = 1+\frac{p^2 q}{p-q} \ .
\end{align*}



Hence, to get $n$ validated blocks in the official blockchain, the selfish miner needs to repeat his attack $X_n$ times with 

\begin{align}
  \EE [X_n]  =  \frac{n}{1+\frac{p^2 q}{p-q}} \ . \label{exn}
\end{align}

Therefore the average number of blocks mined by the selfish miner in a sequence of $n$ blocks is (we take $n\leq 2016$ to avoid a difficulty adjustment) 

\begin{align*}
  q' & = \frac{\EE [X_n]}{n} \cdot \frac{\EE [R(\tau_{SM, \gamma})]}{b}\\
&= \frac{((1 + pq) (p - q) + pq) q - (1 - \gamma) p^2 q (p - q)}{p^2 q + p - q}
\end{align*}

\end{proof}

We can rearrange this expression to get formula (8) for $R_{\text{pool}}$ from \cite{ES14},
$$
q'=\frac{q (1  - q)^2  (4 q + \gamma (1 - 2 q)) - q^3}{1 - q (1 + q (2 - q))} \ .
$$ 



\section{Selfish mining and difficulty adjustment}
We examine now the impact of a difficulty's adjustment on the selfish mining strategy. 
After $n_0=2016$ blocks have been validated, the protocol modifies the speed of mining $\alpha$ and 
$\alpha'$ by a factor $\delta = \frac{\tilde{S}_{n_0}}{n_0 t_0}$ where $ \tilde{S}_{n_0}$ is the time needed
by the network to validate the sequence of $n_0$ blocks and $t_0=10 \text{ min}$.

\begin{proposition}\label{expd}
In presence of a single selfish miner with relative hashrate $q$, after the validation by the network of $n_0$ blocks, 
the parameter $\delta$ updating the difficulty's adjustment satisfies 
$$
\EE [\delta] = \frac{p - q + pq (p - q) + pq}{p^2 q + p - q} \ .
$$
\end{proposition}

\begin{proof}
Before the difficulty adjustment we have $\frac{1}{\alpha + \alpha'} = t_0$. So,  by (\ref{exn}) and Theorem \ref{profitsmg} with  $n_0 = 2016$, we have, 

\begin{eqnarray*}
 \EE[\tilde{S}_{n_0}] & = &  \EE[X_{n_0}] \cdot \EE[\tau_{SM, \gamma}]\\
&=& \frac{p - q + pq (p - q) + pq}{p^2 q + p - q}\cdot n_0 t_0
\end{eqnarray*}

\end{proof}

We prove now that after a difficulty adjustment, the apparent hashrate and the profitability ratio coincide.

\begin{theorem}\label{profitpost}
After a difficulty adjustment, the profitability ratio of the selfish miner is $P(SM, \gamma) = \frac{q' b}{t_0}$
\end{theorem}

\begin{proof}
Before a difficulty adjustment, the speeds of validation $\alpha$ and $\alpha'$ were $\alpha = \frac{p}{t_0}$ and $\alpha' = \frac{q}{t_0}$. 
After a difficulty adjustment these quantities are both multiplied by a factor $\delta$. So, the profitability ratio $P(SM, \gamma)$ is also multiplied by $\delta$. 
Therefore, by Theorem \ref{profitsmg} and Proposition \ref{expd}, we get
$$P(SM, \gamma) = \left( qb - (1 - \gamma) \frac{p^2 q (p - q) b}{(1 + pq) (p - q) + pq} \right) \cdot \frac{p - q + pq (p - q) + pq}{p^2 q + p - q} \cdot \frac{b}{t_0}$$
After arranging this expression, we get $P(SM, \gamma) = \frac{q' b}{t_0}$.
\end{proof}

\begin{corollary}
After a period of difficulty adjustment, the selfish mining strategy becomes more profitable than staying honest  
if $\gamma > \frac{p - 2 q}{p - q} = \frac{1 - 3 q}{1 - 2 q}$.
\end{corollary}

\begin{proof}
The selfish mining strategy outperforms the strategy of "staying honest forever" if $P(SM, \gamma) >  \frac{q b}{t_0}$. 
The condition is equivalent to $q'>q$. Hence we get the result by Proposition \ref{qprime}.
\end{proof}
This condition is also equivalent to the condition $q>\frac{1 - \gamma}{3 - 2 \gamma}$ from \cite{ES14}. 
Note, and this point is crucial, that after a first period of difficulty's adjustment, the difficulty remains constant on average. 
So, the selfish mining strategy becomes more profitable than the honest strategy if the hashrates of the miners stay the same.

\section{A proposition to prevent selfish mining}

\subsection{The origin of the problem}
Basically, the atttack exploits the difficulty adjustment law. The protocol underestimates the real 
hashing power in the network since
only the blocks that are in the (official) blockchain are taken into account.
The number of orphan blocks grows in the presence of a selfish miner and a significant amount of honest hashrate is lost. 
The average time used by the network to validate blocks increases. 
After 2016 blocks, a difficulty adjustment is done automatically
ignoring the production of orphan blocks. Despite the fact that the
total hashing power of the network remains the same, the new difficulty
is lower than it should be, and the block validation time decreases. 
So the revenue per unit of time of the selfish miner improves and makes the attack profitable.

\subsection{A new difficulty adjustment formula}

To mitigate this attack, the idea is to incorporate the count 
of orphan blocks in the difficulty adjustment formula. This can be implemented with 
miners indicating the presence of ``uncles' in the blocks they mine by including
their header and peers relaying this data. Nodes would not need to broadcast whole orphan blocks but only their
headers. It is possible to incentive miners to include proofs of existence of uncles 
in their blocks by including a rule that, in case of competition between two blocks 
with the same height, nodes should always broadcast {\bf the block with the most proof-of-work} i.e., 
the block which includes the most proofs of existence of ''uncles". According to {\cite{PZ17}}, 
this rule would also be profitable to honest miners in case of blocks competition with selfish miners.
At the end of a period of $n_0 = 2016$ blocks validated by the network, the
new formula of difficulty adjustment would be
\begin{eqnarray}
  D_{\text{new}} & = & D_{\text{old}} \cdot \frac{(n_0 + n') \tau_0}{S_{n_0}} 
  \label{f}
\end{eqnarray}
where $n'$ is the total number of orphan blocks mined during this period of
time and $S_{n_0}$ is the time used by the network to validate the $n_0$
blocks (and evaluated with the formula $S_{n_0} = T_{n_0} - T_1$ where $T_i$
denotes the timestamp in the header of block $i$).

\subsection{Analysis of the formula}Let  $\omega$ be the average
number of orphan blocks observed during a period of $\tau_0 = 600$ sec. So, on
average, every $\tau_0$, there are $\omega$ orphan blocks and $(1 - \omega)$
non-orphan blocks. Only the last ones will add to the official blockchain. The time used by the network to grow the blockchain by
$n_0 = 2016$ blocks is then
$S_{n_0} = \frac{n_0 \tau_0}{1 - \omega}$. During this interval, we observe
$n' = \frac{n_0 \omega}{1 - \omega}$ orphan blocks. Thus we have $\frac{(n_0 + n')
\tau_0}{S_{n_0}} = 1$ and Formula (\ref{f}) cannot lead to a fall of difficulty.

\section{Conclusion}
Selfish mining is a trick that slows down the network and reduces the mining difficulty. The attack diminishes the profitability of honest miners
and the one of selfish miners before a difficulty adjustment. Selfish mining only becomes profitable after lowering the difficulty. 
Another way to achieve that would be to withdraw 
from the network and start mining another cryptocurrency with the same
hashing function. The existence of other cryptocurrenies with the 
same validation algorithm that allows to switch mining without cost is a vector of attack in itself.
When the attacker withdraws and comes back the mining difficulty will increase again after 2016 blocks 
unless the miner executes a selfish mining strategy. Then after a first difficulty's adjustment, the difficulty mining will stay constant on average. 
% This implies that the selfish mining strategy outperforms the honest strategy on the long term if  $q>\frac{1 - \gamma}{3 - 2 \gamma}$. In particular, 
% if $q>33\%$, it is always the case.

\medskip

Selfish mining is an attack on the Bitcoin protocol, but the arguments present in the literature do not properly justify the attack. They lack of a correct 
evaluation of the cost of the attack and a proper analysis of profit and loss per unit of time. To compare the profitability of different mining strategies 
one needs to compute the average length of their cycles and their profitability ratio, that is a new notion introduced in this article. 

\medskip

The attack exploits a flaw in the difficulty adjustment formula. The parameter used to update the mining difficulty is supposed 
to measure the actual hashing power of the network. In the presence of a selfish miner, this is no longer true. 

\medskip

We have proposed a formula that corrects this anomaly by taking into account the production of orphan blocks.
We propose to reinforce the protocol which states that the official blockchain is the chain which contains the 
most proof-of-work by requiring peers to give priority to those containing the most proof-of-work (with ''uncles").

The proposed formula, if adopted, would not eliminate the possibility of selfish mining but it would make it non-profitable compared to honest mining 
even after a difficulty adjustment. So this will keep the individual incentives properly aligned in the protocol rules, as intended in the original 
inception of Bitcoin \cite{N08}.

%\newpage


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