We formally introduce and solve the synthesis problem for LTL goals in the case of multiple, even contradicting, assumptions about the environment. Our solution concept is based on ``best-effort strategies'' which are agent plans that, for each of the environment specifications individually, achieve the agent goal against a maximal set of environments satisfying that specification. By means of a novel automata theoretic characterization we demonstrate that this best-effort synthesis for multiple environments is 2ExpTime-complete, i.e., no harder than plain LTL synthesis. We study an important case in which the environment specifications are increasingly indeterminate, and show that as in the case of a single environment, best-effort strategies always exist for this setting.
Moreover, we show that in this setting the set of solutions are exactly the strategies formed as follows: amongst the best-effort agent strategies for ɸ under the environment specification E1, find those that do a best-effort for ɸ under (the more indeterminate) environment specification E2, and amongst those find those that do a best-effort for ɸ under the environment specification E3, etc.