  \emph{Automated Market Makers} (\emph{AMMs}) are a cornerstone of
decentralized finance (DeFi) blockchain-based platforms.
  They are smart contracts, enabling the direct exchange of virtual tokens by
maintaining \emph{liquidity pools}.
  Traders exchange tokens with the contract, paying a fee; liquidity comes from
\emph{liquidity providers}, paid by those fees.
  But despite growing demand, the performance of AMMs is limited.
  State-of-the-art blockchain platforms allow for parallel execution of
transactions.
  However, we show that AMMs do not enjoy these gains, since their operations
are not commutative so transactions using them must be serialized.
  We present \emph{SAMM}, an AMM comprising multiple independent \emph{shards}.
  All shards are smart contracts operating in the same chain, but they allow
for parallel execution as each is independent.
  The challenge is that trading in a standard AMM is cheaper if its liquidity
pool is larger.
  Therefore, we show that simply using multiple smaller AMMs results in traders
splitting each trade among all AMMs, which worsens performance.
  SAMM addresses this issue with a novel design of the trading fees.
  Traders are incentivized to use only a single smallest shard.
  We show that all Subgame-Perfect Nash Equilibria (SPNE) fit the desired
behavior: Liquidity providers balance the liquidity among all pools, so the
system converges to the state where trades are evenly distributed.
  Evaluation in the Sui blockchain shows that SAMM's throughput is over
fivefold that of traditional AMMs, approaching the system's limit.
  SAMM is a directly deployable open-source smart contract, allowing trading at
scale for individuals and DeFi applications.
