Taiwan has been bothered by the issues of homosexuality for so many decades. LGBT (Lesbian, Gay, Bisexual and Transgender) community has being tried to legalize same-sex marriage. Same-Sex Marriage, one of the Diverse Family Formation Issues promoted by Taiwan Alliance to Promote Civil Partnership Rights, an amendment to the Civil Code was approved by a legislative committee on Dec.26, 2016. The bill now is being sent for cross-party negotiations and the Civil Code was announced unconstitutionally by the Grand Justices Council for limiting same-sax marriage. In this paper, Multiple Streams Approach (MSA) was applied to analyze the predecision processes and the arguments of Same-Sex Marriage issue for understanding how it moved from problem identification to public and policy issue. Through five characters of the MSA, problem stream, politics stream, policy stream, policy window and policy entrepreneurs, we find that although the above three streams are coupling at the right time, the policy entrepreneurs, especially the Judicial Department that Kingdon ignored should be the key sector in this issue. And we also find that the period of policy window may not disappear quickly when this window is not closed completely.