  In blockchains such as Bitcoin and Ethereum, users compete in a transaction
fee auction to get their transactions confirmed in the next block. A line of
recent works set forth the desiderata for a "dream" transaction fee mechanism
(TFM), and explored whether such a mechanism existed. A dream TFM should
satisfy 1) user incentive compatibility (UIC), i.e., truthful bidding should be
a user's dominant strategy; 2) miner incentive compatibility (MIC), i.e., the
miner's dominant strategy is to faithfully implement the prescribed mechanism;
and 3) miner-user side contract proofness (SCP), i.e., no coalition of the
miner and one or more user(s) can increase their joint utility by deviating
from the honest behavior. The weakest form of SCP is called 1-SCP, where we
only aim to provide resilience against the collusion of the miner and a single
user. Sadly, despite the various attempts, to the best of knowledge, no
existing mechanism can satisfy all three properties in all situations.
  Since the TFM departs from classical mechanism design in modeling and
assumptions, to date, our understanding of the design space is relatively
little. In this paper, we further unravel the mathematical structure of
transaction fee mechanism design by proving the following results:
  - Can we have a dream TFM?
  - Rethinking the incentive compatibility notions.
  - Do the new design elements make a difference?
