  A fundamental principle of individual rational choice is Sen's $\gamma$
axiom, also known as expansion consistency, stating that any alternative chosen
from each of two menus must be chosen from the union of the menus. Expansion
consistency can also be formulated in the setting of social choice. In voting
theory, it states that any candidate chosen from two fields of candidates must
be chosen from the combined field of candidates. An important special case of
the axiom is binary expansion consistency, which states that any candidate
chosen from an initial field of candidates and chosen in a head-to-head match
with a new candidate must also be chosen when the new candidate is added to the
field, thereby ruling out spoiler effects. In this paper, we study the tension
between this weakening of expansion consistency and weakenings of resoluteness,
an axiom demanding the choice of a single candidate in any election. As is well
known, resoluteness is inconsistent with basic fairness conditions on social
choice, namely anonymity and neutrality. Here we prove that even significant
weakenings of resoluteness, which are consistent with anonymity and neutrality,
are inconsistent with binary expansion consistency. The proofs make use of SAT
solving, with the correctness of a SAT encoding formally verified in the Lean
Theorem Prover, as well as a strategy for generalizing impossibility theorems
obtained for special types of voting methods (namely majoritarian and pairwise
voting methods) to impossibility theorems for arbitrary voting methods. This
proof strategy may be of independent interest for its potential applicability
to other impossibility theorems in social choice.
