Physical　layer　security　has　emerged　as　a　promising　technique　to　safeguard　the　information　security　in　wireless　networks.　In　this　paper,　we　investigate　the　physical　layer　security　issue　for　a　wireless　network　where　there　coexist　multiple　users　with　security　concerns.　Specifically,　we　tackle　the　problem　from　a　distributed　perspective　and　formulate　the　secure　transmissions　at　different　users　as　a　non-cooperative　game.　Consider　the　practical　situation　that　the　legitimate　transmitter　may　not　always　have　the　perfect　information　regarding　the　channel　state　information　of　the　eavesdropper,　we　adopt　the　robust　secrecy　rate　to　combat　the　potential　worst　cases.　Accordingly,　the　robust　Nash　equilibrium　is　employed　as　the　solution　to　the　resource　competition　game　among　the　users.　Further,　we　analyze　properties　of　the　equilibrium　and　derive　the　optimal　transmission　strategy　for　each　individual　user　to　maximize　its　own　robust　secrecy　rate,　following　which　the　distributed　algorithm　is　proposed　for　the　network-wide　competition　to　reach　the　equilibrium.　Finally,　simulation　results　are　provided　to　corroborate　our　theoretical　findings.