In　this　paper,　a　discounted　value　iteration　algorithm　is　developed　to　settle　with　the　optimal　regulation　problem　for　linear　discrete-time　zero-sum　games.　The　Hamilton-Jacobi-Isaacs　equation　and　the　game　algebraic　Riccati　equation　are　provided　with　the　discount　factor.　In　addition,　because　of　the　implementation　of　the　discount　factor,　the　stability　of　the　system　might　be　affected.　Then,　we　establish　the　stability　condition　to　seek　out　the　stabilized　optimal　control　strategies,　which　not　merely　make　the　cost　function　converge　to　the　optimal　value,　but　also　ensure　the　system　achieves　asymptotic　stability.　Furthermore,　the　corresponding　stability　analysis　is　discussed.　Finally,　a　practical　example　of　the　one-tank　system　is　conducted　to　illustrate　the　performance　of　the　presented　method.　©　2024　IEEE.