Figure 1. Illustration of Ladha’s Study 0q0q 120 b 150 a a m Note: c 1 is the cut point on the first roll call, regardless if legislators adopt a sincere or sophisticated voting strategy. C 2 is the cutpoint on the second roll call if legislators adopt a sincere voting strategy. C 2 ’ is the cutpoint on the second roll call if they adopt a sophisticated voting strategy. Figure 3. Cutpoints Under Sincere and Sophisticated Voting 0q0q 150 a a b 100 m 60 c 2 ’ = (q+b)/2 135 c 1 = (b+a 1 )/2 185 c 2 = (b+a 2 )/2 Figure 2. Voting Tree for the Alternatives in Figure 1 a1a1 b (a 1 ) a2a2 a1a1 a2a2 b (b) (a 1 ) qq (b) qb (q) qa2a2 a1a1 a2a2

Figure 4. An Opportunity For Sophisticated Voting q median ba Note: The median legislator proposes a bill, b, equal to his own ideal point. The status quo, q, is to the left of the bill, which is to the left of the amendment, a. The legislators have symmetric preferences and a is farther from the median’s ideal point than q. Thus a is a “potential killer amendment”. That is, if it defeats b in the first round of voting, then it would lose to q in the next round.

Figure 5. Standard Veto Model (When Legislators Vote Sophisticatedly) 5.35 status quo 5.45 equilibrium bill 5.50 Median ideal pt. of Cong President’s ideal pt.

Figure 6. Veto Model (When Legislators CanNOT Vote Sophisticatedly) 5.35 status quo 5.45 bill 5.50 median 5.40 filibuster pivot 5.46 amendment Legislators whose outcome preferences favor the amendment (since it kills the bill). Legislators whose outcome preferences oppose the amendment (since it kills the bill). Legislators whose position-taking preferences oppose the amendment. Legislators whose position-taking preferences favor the amendment. Contingent-a legislatorsDominant-b legislators Contingent-b legislators Note: The figure illustrates legislators’ position-taking and outcome preferences when the amendment ($5.46) is paired against the bill ($5.45). Note that there are several contingent-a legislators and several contingent-b legislators. Hence, the Dual Conflict assumption is satisfied. Accordingly, Proposition 1 implies that the only possible equilibrium is where all legislators vote sincerely, which implies that the amendment defeats the bill.