The　game　coloring　number　of　the　square　of　a　graph　G,　denoted　by　gcol(　G2),　was　first　studied　by　Esperet　and　Zhu.　The　(a,b)-game　coloring　number,　denoted　by　(a,b)-gcol(G),　is　defined　like　the　game　coloring　number,　except　that　on　each　turn　Alice　makes　a　moves　and　Bob　makes　b　moves.　For　a　graph　G,　the　maximum　average　degree　of　G　is　defined　as　Mad(G)=max2|E(H)||V(H)|:H　is　a　subgraph　of　G.　Let　k　be　an　integer.　In　this　paper,　by　introducing　a　new　parameter　rG,　which　is　defined　through　orientations　and　orderings　of　the　vertices　of　G,　we　show　that　if　a<Mad(G)2≤k,　then　(a,1)-gcol(　G2)≤kΔ(G)+⌊(1+1a)　rG⌋+　rG+2.　This　implies　that　if　G　is　a　partial　k-tree　and　a<k,　then　(a,1)-gcol(　G2)≤kΔ(G)+(1+1a)(　k2+3k+22)+2;　if　G　is　planar,　then　there　exists　a　constant　C　such　that　gcol(　G2)　≤5Δ(G)+C.　These　improve　previous　corresponding　known　results.　For　a<k<Mad(G)　and　Δ(G)<2k-2,　we　prove　that　(a,1)-gcol(　G2)≤(3k-2)Δ(G)-　k2+4k+2.　©　2012　Elsevier　B.V.　All　rights　reserved.