  Participation in permissionless blockchains results in competition over
system resources, which needs to be controlled with fees. Ethereum's current
fee mechanism is implemented via a first-price auction that results in
unpredictable fees as well as other inefficiencies. EIP-1559 is a recent,
improved proposal that introduces a number of innovative features such as a
dynamically adaptive base fee that is burned, instead of being paid to the
miners. Despite intense interest in understanding its properties, several basic
questions such as whether and under what conditions does this protocol
self-stabilize have remained elusive thus far.
  We perform a thorough analysis of the resulting fee market dynamic mechanism
via a combination of tools from game theory and dynamical systems. We start by
providing bounds on the step-size of the base fee update rule that suffice for
global convergence to equilibrium via Lyapunov arguments. In the negative
direction, we show that for larger step-sizes instability and even formally
chaotic behavior are possible under a wide range of settings. We complement
these qualitative results with quantitative bounds on the resulting range of
base fees. We conclude our analysis with a thorough experimental case study
that corroborates our theoretical findings.
