The story of Death Note is a supernatural crime thriller, a battle of wits between the ambitious high school student Light Yagami and the world's best detective, L. Light has more than his wits on his side, though -- he has the cursed Death Note, a shinigami notebook that can kill people when their names are written inside. In the story, Light aimed to become the god of a new world order with the notebook's power.
Light clashed with L and eventually triumphed over him, and assumed L's place to disguise himself further from his remaining enemies. In the end, though, the rivals Near and Mello cornered Light/Kira and defeated him. But, in theory, Light could have easily defeated Near, and perhaps it never would have even reached that point. Light made some mistakes that someone like him never should have made.
How Light Yagami Lost
Light Yagami suffered a number of mishaps and defeats in the story of Death Note, and arguably, his very first defeat is what made all the rest possible. Early in the story, Light fell for L's trick on TV and killed a man named Lind L. Tailor, who claimed to be L himself. This was nothing but bait to smoke out Kira, and it worked. Once Light killed Lind L. Tailor, the real L confirmed Kira's existence, location, and ability to take lives remotely.
The battle of wits began at that point, and Light soon made another mishap, when he made it obvious that he was hacking into the NPA's computer mainframe. Light did that to pit L and the police against each other, but his plan completely backfired, and his actions made it clear to L that Kira had a personal connection to an NPA member. Thus, Light did the bulk of L's work for him, especially since L and the police called a ceasefire.
Light recovered from these mistakes and defeated L in the end when he set up Kyosuke Higuchi as a third Kira and used a complex plan to regain the notebook and force Rem the shinigami to kill both L and Watari. Then, Light/L/Kira went up against Mello and Near, and though Mello died, Near survived. He and Light were waging a battle of wits based on the use of proxy Kiras and decoy notebooks.
Light's plan could have worked, and nearly did, but his attorney ally Teru Mikami got reckless and retrieved the real notebook to kill Kiyomi Takada. This allowed Near to swap that notebook for a fake, and trick Mikami into using it. In the end, Light lost to Near because he chose a follower who was a bit too loyal and proactive.
How Light Yagami Could Have Won
Light Yagami could have won most effectively if he didn't fall for the Lind L. Tailor trick. If so, L would have gained no leads on Kira's location or even Kira's existence, and Light could continue to kill criminals by means L couldn't even imagine. Furthermore, Light could have refrained from hacking into the NPA mainframe, and prevented L from getting a lead there, either. Light never really had a need for pitting the police against L and doing so only benefitted L, especially when L found a handful of police officers who were actually loyal to him.
If all that didn't happen, then Light could have defeated Near and Mello by having a slightly more restrained successor as Kira. Teru Mikami was fanatical about Kira, to the point that he often worried Light with his ruthless methods that actually contradicted Light's ideals at times. Light's downfall was really Mikami's fault since he acted against Light's orders when he retrieved the real notebook to kill Kiyomi Takada without being ordered to. Light was already prepared to kill Kiyomi himself -- Teru didn't have to make a single move. If that scenario had played out, Light's actual plan to kill Near would have worked. In fact, Near himself admitted that if it weren't for Mikami's reckless move, Light would have won.
Why Light Yagami Should Have Won
Light Yagami's downfall was his hubris and defiant attitude, almost like a tragic Greek theater hero. He had the intelligence, notebook, and goals necessary to create his new world and forever elude capture. However, he arrogantly killed the fake L out of spite, even though he had no need to kill Lind L. Tailor -- or the real L -- at all. Light even stated that without the notebook, L had no proof, and Kira would never be caught. Getting cocky and killing Lind L. Tailor out of sheer defiance cost Light everything.
Light Yagami could have and should have, won this battle of wits if his hubris wasn't so extreme, especially when that attitude can betray him. At one point, L stated that he understood Kira's arrogant and bratty side, because he, too, was a sore loser who would never quit. If that common ground was eliminated from the story, and Light and L fought only with their wits and detective skills, L would have stood no chance. Death Note could have been a very different crime thriller where the antihero Light Yagami won after all, and his world would become a reality for all time. The story of Death Note owes that to Light Yagami.