The Constitution has only one statement regarding the establishment and governing of elections (and one Article establishing the Electoral College).
Article I, Section, 4, Clause 1:
Library of Congress
The Times, Places and Manner of holding Elections for Senators and Representatives, shall be prescribed in each State by the Legislature thereof; but the Congress may at any time by Law make or alter such Regulations, except as to the Places of chusing Senators.
Based on this law, local elections were held and managed state by state in the earliest days of the republic—and to this day. The federal government does not manage any elections. After the revolution, keeping this power with the states was a big prerequisite for many states before they would ratify the constitution. Between 1800 and 1807, New Jersey voters cast secret ballots in their home townships. They went to a local tavern, and either brought their own ballot or used one provided at the tavern. They wrote down the names of their preferred candidates and placed their slips of paper in a locked wooden ballot box with a slit in the top.

An election official took down the names of each voter as they came in. When voting was finished, the town clerk compared the number of votes cast against the written list. If there were too many voting slips, they would only record the number of votes based on their list. The rest were thrown away.
Methods were similar across the 13 original states. In New Jersey between 1776 and 1807, the voter also had to swear an oath in front of the election official that he was 21 years old, lived in the township, and owned property worth at least 50 pounds. Although the young United States had turned to dollars in the 1790’s, they still used pounds to evaluate property.
“50 pounds of property” was the equivalent to $133.33 in today’s money.
Presidential elections took several days, since each state had its own designated days and methods. Moreover, there was not a federal election commission to relay the results to the public, or the world. After local elections were completed, the Electoral College process was triggered. It would be weeks before anyone knew who the President was. In the vacuum, news organizations stepped up to collect results and report them.
Furthermore, when the telegraph came about, many worried that news of one state’s results might influence the vote in other states. Hence, Congress established the first Tuesday, following the first Monday in November, as Election Day across all states.

In 1848, The Associated Press announced Zachary Taylor’s election as president. And news organizations have been doing it ever since, with the advantage that they have no particular stake in the outcome.
But why hasn’t the federal government ever changed this process so that they would announce the winner? Because after the Civil War, Southern states, in particular, resisted any effort to give the federal government any more power.

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