Post-block Reward Mining Strategy

Bitcoins rules call for fixed coinbase block rewards to disappear over time and be replaced by transaction fees as the sole source of revenue for miners. In this question well explore the implications of this transition on miner behavior.

Well assume that the block size is increased so that miners can always include all available transactions in a block (e.g. assume blocks can have infinite size).

Well also assume that new transactions are broadcast at a constant rate, all including the same transaction fee. This means that the value of any block is directly proportional to the time since the last block was found. Well say that the block reward is R(t) = rt where t is the time in minutes since the last block, and r is a constant.

Well assume the marginal cost of computing each hash is c and current target value for proof of work is a (i.e. the SHA-256 hash puzzle is solved if H() < a). Well also assume that miners can turn their equipment on and off instantaneously and pay nothing when its off.

Given these assumptions, miners clearly will not want to mine immediately after a block is found, as the value starts at 0, because the transaction pool is empty.

a. How long will miners wait after a block is found before turning on their hardware?

Hint: Consider the cost of each hash compared to the expected revenue it generates.

b. Assume that the target level a₀ is set to ensure the average block time would be 10 minutes if all miners were mining constantly (without stopping). After 2016 blocks using the above strategy, what will the new target a₁ be (in terms of c, r, and a₀)? Assume the total amount of mining hardware is not changing.

Hint: After the amount of time found above, the expected time to the next block is 10 minutes

c. Another possible mining strategy is the following: immediately after a block is found, it is not profitable to attempt to find a new block (as the transaction pool is empty) but it is profitable to try to re-mine the previous block, mining fewer transactions so that mining on top of this new block will be more attractive than mining on top of the original block. Suppose you are the only miner executing this strategy. Suppose the previous block was found after t0 time (so that the block reward for the previous is rt₀), and t₁ is the time since the previous block was found. For what value of t₁ should you switch from trying to steal the previous block, to trying to mine a fresh block?