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Problems of multicandidate races in U.S. presidential elections motivated the modern invention and advocacy of approval voting; but it has not previously been recognized that the first four presidential elections (1788–1800) were conducted using a variant of approval voting. That experiment ended disastrously in 1800 with the infamous Electoral College tie between Jefferson and Burr. The tie, this paper shows, resulted less from miscalculation than from a strategic tension built into approval voting, which forces two leaders appealing to the same voters to play a game of Chicken. Because the Burr Dilemma poses a significant difficulty for approval voting, this paper urges that researchers give more attention to “instant runoff” reform options, especially the alternative vote and the Coombs rule.