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The Gap Game @ CESC

The Gap Game

Itay Tsabary, Technion

Headline result: miners in PoW are incentivized to shut off their hardware occasionally.

Proof of Work

  • Find a value for which a protocol-defined condition holds
    • In Bitcoin, hash(block) < K
  • Memory-less process
  • PoW is very expensive

Mining incentives

  • First incentive is the minting of new currency (i.e., inflation)
  • Second incentive is transaction fees
    • The sum of transaction fees within mempool = your mining incentive
    • You choose the optimal subset of these transactions
      • Because mining is a memory-less process, this is fine.
  • Miners are willing to mine when their expected reward is above their

The Gap Game

  • If transaction fees are most of the mining incentive, what happens?
    • Note: this is what happens in Bitcoin as block reward approaches 0, and as contention for block space increases
  • The longer you wait, the more transaction fees accumulate
    • But if transaction fees at time T are below your operating cost as a miner, it's not rational to mine
    • Therefore, a rational miner will wait to turn on their miner until the best possible block in the mempool is above their mining cost
    • These gaps are called Mining Gaps

Mining Gap

  • These gaps in mining were first formalized by Carlsten, et al in 2016.
  • Carlsten also uses a simplified model with unbounded block size

Gap Game Setup

  • Create a game theoretic model of cryptocurrencies
    • Incentives + expenses
    • Players, strategies, etc.

What's a miner's expected profit?

  • Composition of three steps:
    • Block time probability distribution function
    • Conditional expected profit (assuming a block time)
    • Expected profit using law of total expectation
  • Create a set of random miners
    • Sample a miner and have them randomly change strategy at each tick
    • Wait until they reach equilibrium


  • Bunch of simulated rollouts
  • Long story short: rational miners turn off their machines
    • The fewer miners in the system, the larger the mining gap should be
    • The more dominant the fees, the larger the gap
  • Estimate:
    • Assuming all parameters stay the same as today, but minting reward halves twice
    • In 10 years, we'll have an average 20% mining gap


  • When PoW block rewards approach 0, mining security goes down
  • Miners will be incentivized to turn off their machines (or point them at different chains) until enough profitable transactions accumulate
  • This is bad!
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