In this paper, two multiple description coding schemes are developed, based on prediction-induced randomly offset quantizers and unequal-deadzone-induced near-uniformly offset quantizers, respectively. In both schemes, each description encodes one source subset with a small quantization stepsize, and other subsets are predictively coded with a large quantization stepsize. In the first method, due to predictive coding, the quantization bins that a coefficient belongs to in different descriptions are randomly overlapped. The optimal reconstruction is obtained by finding the intersection of all received bins. In the second method, joint dequantization is also used, but near-uniform offsets are created among different low-rate quantizers by quantizing the predictions and by employing unequal deadzones. By generalizing the recently developed random quantization theory, the closed-form expression of the expected distortion is obtained for the first method, and a lower bound is obtained for the second method. The schemes are then applied to lapped transform-based multiple description image coding. The closed-form expressions enable the optimization of the lapped transform. An iterative algorithm is also developed to facilitate the optimization. Theoretical analyzes and image coding results show that both schemes achieve better performance than other methods in this category.