In　this　paper,　we　propose　a　method　for　finding　a　Nash　equilibrium　of　two-person　games　with　alternating　offers.　The　proposed　method　is　referred　to　as　the　inexact　proximal　alternating　direction　method.　In　this　method,　the　idea　of　alternating　direction　method　simulates　alternating　offers　in　the　game,　while　the　inexact　solutions　of　subproblems　can　be　matched　to　the　assumptions　of　incomplete　information　and　bounded　individual　rationality　in　practice.　The　convergence　of　the　proposed　method　is　proved　under　some　suitable　conditions.　Numerical　tests　show　that　the　proposed　method　is　competitive　to　the　state-of-the-art　algorithms.