First of all, please don't be lame and just say "this isn't a battle." It's a battle between one version of Lex Luthor and the others.
Now I just finished Red Son, and I am struck by how amazingly intelligent, tempered, and competent this version of Lex Luthor was. If you haven't read Superman: Red Son, , please do not read the spoilers that follow. It's one of the greatest Superman stories ever told, and that makes it one of the greatest comic books ever told. If there is even a microscopic chance that you will read it in the future, do not open these spoilers.
He actually won in the end, straight up, no help, against all odds. He beat Superman, beat him in almost every way possible.
Alright, first of all, he regularly played chess matches against a dozen chess masters simultaneously, keeping track of each individual game and effortlessly checkmating everybody, just to keep his mind distracted while he also did other things like study theoretical physics, which he also did while planning his schemes. When he was older, he was doing the same thing but now playing over 90 different chess games at once. He could apparently talk people into suicide within a few minutes. He somehow created a design for a box within a box within an infinite amount of boxes, each layer a micron thick, so that it would be impossible to break open, and the Green Lanterns followed this design to trap Superman.
Most importantly, he created a 40-year long masterplan to ultimately break and expose Superman morally while simultaneously making himself (Lex Luthor) look heroic, and then after Superman's will was shattered and everybody hated him, he was forced into a situation where he had to sacrifice his own life. This plan took 40 years to unfold perfectly and involved innumerable variables to ensure all went perfectly which Lex was said to have plotted the calculations to the tenth decimal point. The thing he failed on was Superman actually being killed when he sacrificed himself. Apparently he dropped a decimal when calculating Superman's mass, and so the six black holes created over his head didn't kill him.
But Lex Luthor was manipulating the American Government for decades, he won the presidency, and within six months he brought America from a literal apocalypse to it's most prosperous state in centuries. All of these things were very minor details to his plan. Again that was just six months, the whole plan took 40 years.
Lex Luthor was not only facing the Soviet Union, not only facing the goddamn Batman, not only facing Braniac, and not only facing a Superman who is godlike and invulnerable enough to fly MFTL+++ speeds and tank six black holes opening on top of him, not only was he facing ALL of that, but he was also facing one of the most intellectual, calculated, and patient versions of Superman that I can think of. This version of Superman was a communist, and he ran the Soviet Union so efficiently, that by his 63ed year he had humanely taken over the whole planet and created one system of government and lifestyle where crime didn't exist, accidents didn't happen, technology rose, and nobody ever complained. Not even in private. (Because everybody was healthy, happy and satisfied). And when I say he did this "humanely", I mean he did this without ever invading or forcing any nations to become Communist. He said that if he wanted to force the world with his fists, he could have been Global Leader overnight. But every single one of his achievements and victories was accomplished by "winning the argument." Since he was Superman he could think at superspeed, sense and keep track of seemingly everything on Earth at once, and most impressively, he himself also outsmarted and beat the likes of Batman and Brainiac along with his usual entire rogues gallery throughout the course of his life.
Bottom line, Lex was going against one of the best versions of Superman. That, along with the thoroughness in which Lex wanted to defeat Superman (just killing him wasn't enough. He wanted to defame Superman's legacy and publicly break his will, while giving himself the legacy of Superman, THEN force Superman into killing himself) are why it took him 40 years to arrange every event perfectly. Along the way he got Batman killed, Braniac killed, Wonder Woman forced into retirement, and Green Lantern becoming Superman's enemy. And he succeeded in it all.
Now here is the best part. This plan took 40 years to come to fruition. Some would say that is a long time. But after getting rid of Superman, Lex led mankind into a new renascence, first becoming the elected Global President, then creating an entirely new system of Government, which did away with politicians and created an era where everybody was an artist or a scientist, leading to a technological revolution, all disease and prebirth defects were eradicated, the average lifespan was over 800 years, and Lex himself lived over 4,000 years. This means that the mere 40 years he spent on Superman, which sounds like such a legendary, all-consuming plot, was actually a very small and insignificant section of his life, the equivalent of one Summer back when you were a kid. In Lex's lifetime, he ushered humanity into the most advanced race in the entire universe, and his lineage continued running the planet for another billion years, until finally, a descendant was born whose intelligence surpassed Lex's. That means Lex was the most intelligent in the universe, and stayed that way until a billion years after his death. Despite all this, on his deathbed, Lex still claimed that his greatest single accomplishment was "defeating the alien." I suspect Lex never really had a true rival in his lifetime, "defeating the alien" is just the most significant single thing he's done, but looking at how long Lex lived and how much more he accomplished, Superman really didn't keep up whatsoever.
Bottom line, is there a version of Luthor who is smarter than this version?