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Appendix
Here we provide proofs for the propositions and facts stated in the text. In some cases we
actually prove more general results than given there; for clarity we include the text of all propositions
here. We begin with technical definitions of those terms defined verbally in the text and required
for the proofs: order preservation, party neutrality, and symmetry.
As stated in the text, we focus our analytic results on the special case of two outcomes’ arising
from an incumbent’s actions: success due to them, or failure.
Since there are two parties, these
create four possible combinations of outcomes: 1) success with a D incumbent, 2) failure with a D
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