Should payments for environmental services be used to implement zero-deforestation supply chain policies? The case of soy in the Brazilian Cerrado

•Payments for environmental services (PES) have a mixed record on achieving their social and environmental goals.•Soy industry actors are considering using PES to end soy-driven deforestation the Brazilian Cerrado.•We find that a PES would have a lower likely effectiveness and cost-effectiveness than an MEM, but higher legitimacy among Brazilian soy actors.•Both approaches have equity drawbacks – MEM disadvantages soy farmers with forested land and PES unfairly privileges these soy farmers.•Mixing an MEM with a PES for only the most vulnerable actors would be a better solution to end deforestation in the Cerrado.