Secret ballots protect people from extortion and coercion. [causal]
Inability to assume an enforceable promise, like inability to perform the activity demanded, may protect one from an extortionate threat. The mandatory secret ballot is a nuisance to the voter who would like to sell his vote, but protection to the one who would fear coercion.
XREF: Connects to democratic theory and discussions of voting integrity, as in the Federalist Papers and contemporary debates about voter intimidation.
Thomas C. Schelling, The Strategy of Conflict, loc. 722