In　this　paper,　an　adaptive　critic　design　with　performance　guarantee　is　established　based　on　the　discounted　value　iteration　algorithm　to　settle　with　the　optimal　regulation　problem　for　discrete-time　zero-sum　games.　Value　iteration　is　implemented　to　obtain　the　approximate　optimal　solutions　to　the　Hamilton-Jacobi-Isaacs　equation　for　nonlinear　systems　and　the　game　algebraic　Riccati　equation　for　linear　systems.　Then,　we　focus　on　system　stability　affected　by　the　introduction　of　the　discount　factor　and　the　admissibility　of　the　policy　pairs　in　the　value　iteration　process.　The　appropriate　selection　range　of　the　discount　factor　and　the　criteria　for　ensuring　system　stability　are　established　to　assist　in　obtaining　the　stabilized　optimal　policy　pair,　which　not　only　makes　the　cost　function　converge　to　the　optimal　value,　but　also　guarantees　the　asymptotic　stability　of　the　closed-loop　system.　Finally,　practical　examples　for　the　power　system　and　the　ball-beam　system　are　conducted　to　demonstrate　the　effectiveness　of　the　presented　method.