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The central challenge in addressing the security-performance trade-off in blockchain systems has moved from consensus protocols to enabling parallel execution. Various smart contracts (stateful blockchain programs) already benefit from parallelization. But we show that non-parallelizable operations within smart contracts remain a bottleneck. A natural approach is to apply sharding, that is, partitioning a single logical service into multiple identical parallelizable shards. However, we observe that this introduces game-theoretic issues in various applications: Participants may be incentivized to overload the system. We focus on Automated Market Makers (AMMs), which enable token exchange: Traders swap tokens with the AMM for a fee, while liquidity providers pre-fund the AMM and receive these fees. We show experimentally that their demand will soon reach the single-thread performance limit. We present SAMM, a sharded AMM system where multiple independent AMM shards operate on the same blockchain. The key challenge is ensuring SAMM’s security against overloading and destabilization attacks. SAMM addresses this with careful mechanism design; we provide both game-theoretic analysis and simulation-based validation. We implement SAMM on Sui and Solana testnets, achieving 5x and 16x throughput improvements, respectively, with potential for more as blockchain parallelism advances. SAMM is deployable and has already attracted industry interest.