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title: '**Part 1: Response to Common Questions from Both Reviewers**'

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We sincerely appreciate reviewers' insightful suggestions and constructive comments. These comments are extremely valuable and helpful for improving our paper. For the concerns, we respond as follows.

# **Part 1: Response to Common Questions from Both Reviewers**

1.1 **It is infeasible to expose the hidden blocks (or block headers) of the selfish pool.** Although the spy in the selfish pool can observe the hash values of all hidden blocks, the corresponding nonce value (i.e., the puzzle solution) is only known to the (selfish) pool manager. Thus, we cannot convince honest miners that there are several valid blocks by just exposing the block hash values without the corresponding nonce value.

1.2 **Response to the concern about our model and strategy.** We would like to emphasize that our paper explores the idea of spy in the strategic mining setting for the first time. Potentially there is a possibility to get another strategy by utilizing the hash information, but simply mining after the private block sometimes works worse than our insightful mining. Here we give a simple counter-example. Suppose that the strategic player follows the private branch, releases the block when successful, and the selfish pool mines after it. If both have the mining power of 49%, the strategic player will obtain about half the revenue. However, Section 5.1 illustrates that it can obtain almost all revenue with insightful mining. The intuition behind this is that, if mining on the opposite side with honest miners, it is possible to invalidate the selfish pool's hidden block (recall that selfish pool is the major rival rather than honest miners).

1.3 **The intuition why Theorem 3.1 is correct.** Recall that the insightful mining does two things: (1) If selfish pool finds the first block, it behaves honestly but tie-breaks against the selfish pool. (2) If the insightful pool finds the first block, it mines selfishly. We explain the intuition of Theorem 3.1 from two aspects. First, when the selfish pool takes the lead (first case), the insightful pool cooperates with honest miners as a whole. But in the second case, when the insightful pool is taking the lead, the selfish pool still competes with honest miners (inducing honest miners to waste their mining power on a stale branch), which causes their internal friction. The second intuition is that, when breaking ties, the insightful pool can clearly know which is selfish pool and play against it, while the choice of the selfish pool is uniform. These reasons help insightful mining to get more revenue than selfish mining.

# **Part 2: Response to Reviewer #567A**
2.1 We have responded to some of your concerns in Part 1.

2.2 It seems that there are some **misunderstandings of the game setting**. As we have emphasized at the beginning of Section 4, insightful mining is a well-defined strategy and can be adopted directly. If there is no selfish pool in the system, insightful mining will look the same as selfish mining. For the equilibrium with multiple insightful pools, only one insightful pool is hiding blocks at any given time, as others will mine honestly when detecting its selfish behavior.

2.3 Either choosing not to extend hidden blocks or turning off mining equipment is practical. However, they are not within our strategy space. In addition, like previous work, we don't consider the mining cost.

2.4 The meaning of "prompt" is that by generating blocks on opposing branch, the selfish pool will be prompted/encouraged to reveal its hidden blocks one by one. In this process, the selfish pool doesn't know the insightful pool exists. "Avoid being detected" is to avoid being realized that insighful pool is behaving strategically. 

# **Part 3: Response to Reviewer #567B**
3.1 We have responded to some of your concerns in Part 1.

3.2 Thank you for pointing out the connection between our work and [Eyal and Sirer 2014] with the help of $\gamma$. Regarding your second concern about the model, it is quite an interesting idea to think of insightful mining with the framework of [Eyal and Sirer 2014]. In fact, we did carefully read this seminal paper and are also familiar with its results and techniques.

However, we have to say that **our work cannot be directly analyzed by using existing results of [Eyal and Sirer 2014]**. For the first case where the selfish pool finds the first block, after formalizing the corresponding $\gamma$ value, we can only know the revenue of the selfish pool and of the others. But for the latter, because insightful pool and honest miners don't behave exactly the same, we cannot calculate their respective revenue. For the second case where insightful pool finds the first block, due to the competition of honest miners and selfish pool, we cannot know their effective hashing power, neither individually nor as a whole. Thus the revenue of each player cannot be calculated either. In conclusion, it is technically unfeasible to analyze our work with the results of [Eyal and Sirer 2014] and to further obtain Theorem 3.1 and Theorem 4.2. We believe all techniques developed in our paper are necessary.

3.3 Finally, we would like to explain why **it doesn't suffice to look at Theorem 4.1(2) and (3) with the logic of [Eyal and Sirer 2014]**. At a high level, the reason is each player's decision can affect others' utility, so it is a game (but not optimization in [Eyal and Sirer 2014]). Let's look at miner 2's relative revenue w.r.t. miner 1's decision. When miner 1 behaves honestly, miner 2's relative revenue is exactly proportional to its hashing power (say $m_2$). But when miner 1 adopts IM ($m_1\geq 1/3$), the relative revenue of miner 2 is lower than $m_2$ (see Line6 of Page14 for the specific revenue function). As a result, the threshold of miner 2 to adopt IM is also lower than (the original) $1/3$, and the exact bound is $g(m_1)$. Although there are some calculations, the roadmap is classic equilibrium analysis.

3.4 Again, we sincerely appreciate your valuable comments!