In this paper, a multiple-relaxation-time color-gradient lattice Boltzmann model is used to simulate the deformation and breakup of a confined droplet in a simple shear flow with power-law rheology. This model is demonstrated to be accurate in simulating power-law fluids with a broad range of power-law indices. Through a preliminary study, we find that the non-Newtonian rheology of the matrix fluid has a more significant effect on the droplet deformation than that of the droplet. Compared to the Newtonian case, the non-Newtonian rheology is found to strengthen the droplet deformation in the shear-thickening matrix fluid but weaken the deformation in the shear-thinning matrix fluid. The extent of droplet deformation increases with increasing power-law index, and the rate of increase is more rapid near the smallest and largest indices considered than for intermediate values. We then systematically investigate the influence of the capillary number, geometrical confinement, and viscosity ratio on the deformation of a Newtonian droplet in power-law matrix fluids. The non-Newtonian effect on droplet deformation increases with increasing the capillary number or the wall confinement. In all the matrix fluids considered, the viscosity ratio noticeably affects the droplet deformation only when the capillary number is not less than 0.15, and the maximum deformation occurs at the viscosity ratio of unity for a constant capillary number. Finally, the critical capillary number, above which the droplet breakup occurs, is investigated for various confinement ratios in three different power-law matrix fluids. As the confinement ratio increases, the critical capillary number exhibits an overall increasing trend in the shear-thinning matrix fluid and an overall decreasing trend in the shear-thickening matrix fluid, both distinct from that in Newtonian case where the critical capillary number first decreases and then increases. The mode of droplet breakup is found to depend on the confinement ratio and the power-law index of the matrix fluid, and a high confinement ratio or a low power-law index favors the ternary breakup. The present study can provide useful suggestions and guidance for precise control of droplet behavior in microfluidic applications where non-Newtonian rheology is often encountered. (C) 2017 The Society of Rheology.