  Within the framework of quantum refereed steering games, quantum steerability
can be certified without any assumption on the underlying state nor the
measurements involved. Such a scheme is termed the
measurement-device-independent (MDI) scenario. Here we introduce a measure of
steerability in an MDI scenario, i.e., the result merely depends on the
observed statistics and the quantum inputs. We prove that such a measure
satisfies the convex steering monotone. Moreover, it is robust against not only
measurement biases but also losses. We also experimentally estimate the amount
of the measure with an entangled photon source. As two by-products, our
experimental results provide lower bounds on an entanglement measure of the
underlying state and an incompatible measure of the involved measurement. Our
research paves a way for exploring one-side device-independent quantum
information processing within an MDI framework.
