Opinion　dynamics　on　social　networks　have　received　considerable　attentions　in　recent　years.　Nevertheless,　just　a　few　works　have　theoretically　analyzed　the　condition　in　which　a　certain　opinion　can　spread　in　the　whole　structured　population.　In　this　article,　we　propose　an　evolutionary　game　approach　for　a　binary　opinion　model　to　explore　the　conditions　for　an　opinion＇s　spreading.　Inspired　by　real-life　observations,　we　assume　that　an　agent＇s　choice　to　select　an　opinion　is　not　random　but　is　based　on　a　score　rooted　from　both　public　knowledge　and　the　interactions　with　neighbors.　By　means　of　coalescing　random　walks,　we　obtain　a　condition　in　which　opinion　A　can　be　favored　to　spread　on　social　networks　in　the　weak　selection　limit.　We　find　that　the　successfully　spreading　condition　of　opinion　A　is　closely　related　to　the　basic　scores　of　binary　opinions,　the　feedback　scores　on　opinion　interactions,　and　the　structural　parameters　including　the　edge　weights,　the　weighted　degrees　of　vertices,　and　the　average　degree　of　the　network.　In　particular,　when　individuals　adjust　their　opinions　based　solely　on　the　public　information,　the　vitality　of　opinion　A　depends　exclusively　on　the　difference　of　basic　scores　of　A　and　B.　When　there　are　no　negative　(positive)　feedback　interactions　between　connected　individuals,　we　find　that　the　success　of　opinion　A　depends　on　the　ratio　of　the　obtained　positive　(negative)　feedback　scores　of　competing　opinions.　To　complete　our　study,　we　perform　computer　simulations　on　fully　connected,　small-world,　and　scale-free　networks,　respectively,　which　support　and　confirm　our　theoretical　findings.