This study addresses the problem of adaptive polarimetric detector (APD) and optimal polarimetric design for the distributed multiple-input-multiple-output radar in compound-Gaussian clutter with inverse-gamma distributed texture component. We derive the APD by maximizing a posteriori estimation and performing a generalized likelihood ratio test. The false alarm probability for the detector is analyzed to validate the corresponding constant false alarm rate property. Furthermore, based on the concepts of game theory, we formulate an optimal polarimetric design as a two players zero-sum game, which further improves the performance of the proposed detector. Simulation results show that the proposed detector outperforms its counterparts, and the optimal polarimetric design algorithm can efficiently enhance the detection performance.