In　this　article,　the　general　value　iteration　(GVI)　algorithm　for　discrete-time　zero-sum　games　is　investigated.　The　theoretical　analysis　focuses　on　stability　properties　of　the　systems　and　also　the　admissibility　properties　of　the　iterative　policy　pair.　A　new　criterion　is　established　to　determine　the　admissibility　of　the　current　policy　pair.　Besides,　based　on　the　admissibility　criterion,　the　improved　GVI　algorithm　toward　zero-sum　games　is　developed　to　guarantee　that　all　iterative　policy　pairs　are　admissible　if　the　current　policy　pair　satisfies　the　criterion.　On　the　basis　of　the　attraction　domain,　we　demonstrate　that　the　state　trajectory　will　stay　in　the　region　using　the　fixed　or　the　evolving　policy　pair　if　the　initial　state　belongs　to　the　domain.　It　is　emphasized　that　the　evolving　policy　pair　can　stabilize　the　controlled　system.　These　theoretical　results　are　applied　to　linear　and　nonlinear　systems　via　offline　and　online　critic　control　design.