# Mining Pool Selection under Block WithHolding Attack +
## 5 Numerical Results
### Results and Discussions
#### 5.2.2 One-Side Attack Case  
Next, we consider the one-side attack case, where pool 1 attacks pool 2 with attack size \(a_{12}=0.015\). Other parameters are set to the same as those in Figure 3. Under this setting, we obtain four NEs: \((0,1)\), \((0.55,0.45)\), \((0.77,0.23)\) and \((1,0)\) by calculating (27). Figure 3(a) shows the phase portrait of the replicator dynamics. We can see from Figure 3(a) that the population states converge to \((0.55,0.45)\) from the initial state with \(0\leq x_{1}<0.77\) and converge to \((1,0)\) from \(0.77<x_{1}\leq 1\). This means that only \((1,0)\) and \((0.55,0.45)\) are stable, i.e., they are ESSs. We consider three initial population states for simulations, i.e., \((x_{1},x_{2})=(0.20,0.80)\), \((x_{1},x_{2})=(0.75,0.25)\) and \((x_{1},x_{2})=(0.85,0.15)\). Figure 3(b) shows the simulation results of the evolution of population states over time. We can see from Figure 3(b) that the population state converges to different points from different initial states, indicating the existence of two stable population states, which are \((x_{1},x_{2})=(1,0)\) and \((x_{1},x_{2})=(0.55,0.45)\). The simulation results are consistent with the theoretical ones, verifying the correctness of our analysis. Comparing the stable population states with those in the no-attack case, we can see that the population fraction of pool 1 increases from 0.4 (no-attack case) to 0.55 or to 1.0 (one-side attack). This indicates that pools can attracts miners to join by launching the BWH attack. We proceed to investigate the impacts of attack size on the stable population states, for which we fix the initial population state at \((0.5,0.5)\). Figure 4(a),b show the simulation results for small attack size (i.e., in the region \([0,0.1]\)) and large attack size (i.e., in the region \([0.9,1.0]\)), respectively. From Figure 4(a), we can see that as the attack size increases, the population of the attacking pool (i.e., pool 1) increases, while that of the attacked pool (i.e., pool 2) decreases. The results show that using more hash power for BWH attack attracts  
Figure 4: One-side attack case with \(a_{12}=0.015\). Figure 3: No-attack case. more miners to join, when the attack size is small. However, this is not the case when the attack size is sufficiently large, as shown in Figure 4(b). We can see from Figure 4(b) that, for the case of the large attack size, the population of the attacking pool decreases as the attack size increases, while that of the attacked pool increases. The reason is that the larger the attack size is, the less hash power is used for mining, leading to decreased mining revenue, which dominates the trend of the total revenue of the pool. This indicates that devoting too much hash power for BWH attack may discourage miners to join the attacking pool. Similar observations have also been reported in [25]. We now examine the importance of the attacking revenue (i.e., revenue from the attacked pool) in the pool selection, for which we show in Figure 6 how the attack size of pool 1 affects the stable population states for both the case with attacking revenue (this paper) and without attacking revenue ([25]). We fix the initial population state as \((0.5,0.5)\) and also consider two regions, i.e., small attack size (i.e., 0.0 to 0.1) and large attack size (i.e., 0.9 to 1.0). We can see from Figure 5(a) that, for the case without considering the attacking revenue, miners prefer to join the attacked pool (i.e., pool 2) as the attack size increases. This is unreasonable, since intuitively joining the attacked pool will earn less revenues than joining the attacking pool, due to the advantage brought by BWH attack. We can see from Figure 5(a) that increasing the attack size to a sufficiently large (or even to 1) has no impact on the mining selection of miners in the case without considering the attacking revenue.