Neither candidate running for U.S. Senate in Michigan seems to want voters to know whether they are the Republican or the Democrat.

For Senator Gary Peters, the motorcycle guy and one of only two Democrats running for re-election in a state that Mr. Trump won in 2016, the path to victory becomes much easier if he can persuade enough of Mr. Trump’s supporters to vote for him.

For John James, the self-described nonpartisan Republican, success will come from the opposite situation: If enough people voting for former Vice President Joseph R. Biden Jr. cross over to support a Republican for the Senate.

The race — one of a handful that could tip the balance in the Senate — is among the most contested and expensive in the country. Polls show a tighter race than Democrats had anticipated, and both sides are fighting for the few remaining independent, undecided voters. In an election in which the parties have focused on getting their base to turn out, Michigan stands out as a place where winning the middle could make the difference.