It looks to me as if Dudley Smith gave up at the very end of L.A. Confidential by throwing away his handgun. Why didn't he fight with Exley, who was weak and might not have pulled the trigger before him?
Exley has the drop on him with a shotgun.
Exley has a shotgun aimed at him while he still has his pistol pointed down at White. Unless Exley freezes, Smith has no chance at that range--a 12 gauge load of buckshot would shred his body with eight pellets, each 1/3 of an inch in diameter with very little chance of missing. His pistol could more easily miss, or fail to kill Exley. He doesn't want to take this gamble on Exley freezing because...
He thinks he can get Exley to go along with a phony story by promising to advance his career.
The dialogue is pretty clear:
You gonna shoot me? Arrest me? Good lad; always the politician. Let me do the talking. When I'm done, they'll make you chief of detectives.
Smith thinks that the uncompromising Bud White is dead and tells Exley that if he goes along with what Smith intends to tell police (i.e., that Smith and Exley were working together to kill all the bad guys) then he will get Exley a huge promotion. Throughout the film, Exley is seen by other characters as a careerist who looks out for his own advancement first and foremost. Smith is banking on this reputation that Exley will do the wrong thing and back up his story to get the big promotion. He fails to see that Exley is actually much more interested in justice than people give him credit for.