As David Greenberg points out, it is a myth that Nixon acquiesced in JFK's election in 1960:
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"More to the point, while Nixon publicly pooh-poohed a challenge, his allies aggressively pursued one. Much of this history has, incredibly, been forgotten as biographers such as Stephen E. Ambrose have propounded Nixon's line. But a glance at the 1960 newspapers shows that GOP leaders tried to undo the results. They knew it was a longshot, but the effort continued right up until the electoral college certified Kennedy's win on Dec. 19.
GOP leaders had reason to conduct even a doomed campaign. Even if they ultimately lost, they reasoned, they could still taint Kennedy's victory. They could claim he had no mandate, galvanize their rank and file and build a winning issue for upcoming elections.
So on Nov. 11, three days after the election, Thruston B. Morton, a Kentucky senator and the Republican Party's national chairman, launched bids for recounts or investigations in not just Illinois and Texas but also Delaware, Michigan, Minnesota, Missouri, New Jersey, New Mexico, Nevada, Pennsylvania and South Carolina. A few days later, Robert H. Finch and Leonard W. Hall, two Nixon intimates, sent agents to conduct what they called "field checks" in eight of those 11 battlegrounds.
In New Jersey, local Republicans obtained court orders for recounts; Texans brought suit in federal court. Illinois witnessed the most vigorous crusade. Nixon aide Peter Flanigan encouraged the creation of a Chicago-area Nixon Recount Committee. As late as Nov. 23, Republican National Committee general counsel H. Meade Alcorn Jr. was still predicting Nixon would take Illinois. On Dec. 2, Morton dramatically flew into Chicago while a Cook County recount was underway.
All the while, leading Republicans, as well as Nixon himself, claimed that the vice president had nothing to do with the efforts. But the involvement of such close aides as Finch and Hall makes such an assertion implausible. More likely, Nixon was shrewd enough not to muddy himself directly in the grimy business because, as he later wrote, getting involved would mean "charges of 'sore loser' would follow me through history and remove any possibility of a further political career."
In the end, these efforts came to naught. In some cases (New Jersey, Illinois), the GOP obtained recounts but they didn't change the final vote. In other cases (Texas), judges rejected the GOP's bids for recounts. As it turned out (as several academic studies have concluded), some fraud existed, but not enough to alter the outcome.
Still, because of the ruckus that the Republicans kicked up, they succeeded in tainting Kennedy's triumph. Nixon proved canny enough to wage his battles while escaping opprobrium. Nixon may have been a loser, but he was nobody's fool."
http://articles.latimes.com/2000/nov/10/local/me-49741
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Inicdentally, even if Nixon had been able to reverse the results in Illinois, JFK would still have won: Illinois' 27 electoral votes would not be enough to deprive him of an Electoral College majority.
http://psephos.adam-carr.net/countries/u/usa/pres/1960.txt To be sure, if Nixon could reverse the results in Illinois *and Texas as well* he would win, but this was a most unlikely prospect; Texas Republicans knew that "they can't outcount Lyndon Johnson"...
https://books.google.com/books?id=XJ3fg3Dw80AC&pg=PA97
BTW, in my view, one Republican could justly complain that the 1960 race was probably stolen from him in Cook County. However, his name was not Richard Nixon but Ben Adamowski.
https://groups.google.com/d/msg/soc.history.what-if/Vcz8NQPl9CU/mlFfVzTyFlMJ https://groups.google.com/d/msg/soc.history.what-if/O8dBNFLSn9E/eHo29bstohgJ