We　study　evolutionary　games　on　scale-free　networks　with　tunable　degree　distribution.　The　heterogeneity　of　networks　is　controlled　by　the　exponent　of　power-law　degree　distribution.　It　is　found　that　there　exists　an　optimal　value　of　the　exponent,　for　which　the　level　of　cooperation　becomes　highest.　This　phenomenon　indicates　that,　although　degree　heterogeneity　plays　an　important　role　in　maintaining　cooperation,　too　strong　heterogeneity　may　counterintuitively　encourage　defection.　We　observe　that,　for　strong　heterogeneity,　large-degree　nodes　are　no　longer　certainly　occupied　by　cooperators.　Our　results　yield　insights　into　the　effect　of　degree　heterogeneity　on　cooperation.　Copyright　(C)　EPLA,　2012